diff --git a/Core_DOM/CITATION b/Core_DOM/CITATION new file mode 100644 index 0000000..5246cbd --- /dev/null +++ b/Core_DOM/CITATION @@ -0,0 +1,37 @@ +An overview of the formalization is given in: + + Achim D. Brucker and Michael Herzberg. A Formal Semantics of the Core DOM + in Isabelle/HOL. In The 2018 Web Conference Companion (WWW). Pages 741-749, + ACM Press, 2018. doi:10.1145/3184558.3185980 + +A BibTeX entry for LaTeX users is +@InProceedings{ brucker.ea:core-dom:2018, + abstract = {At its core, the Document Object Model (DOM) defines a tree-like + data structure for representing documents in general and HTML + documents in particular. It forms the heart of any rendering engine + of modern web browsers. Formalizing the key concepts of the DOM is + a pre-requisite for the formal reasoning over client-side JavaScript + programs as well as for the analysis of security concepts in modern + web browsers. In this paper, we present a formalization of the core DOM, + with focus on the node-tree and the operations defined on node-trees, + in Isabelle/HOL. We use the formalization to verify the functional + correctness of the most important functions defined in the DOM standard. + Moreover, our formalization is (1) extensible, i.e., can be extended without + the need of re-proving already proven properties and (2) executable, i.e., + we can generate executable code from our specification.}, + address = {New York, NY, USA}, + author = {Achim D. Brucker and Michael Herzberg}, + booktitle= {The 2018 Web Conference Companion (WWW)}, + conf_date= {April 23-27, 2018}, + doi = {10.1145/3184558.3185980}, + editor = {Pierre{-}Antoine Champin and Fabien L. Gandon and Mounia Lalmas and Panagiotis G. Ipeirotis}, + isbn = {978-1-4503-5640-4/18/04}, + keywords = {Document Object Model, DOM, Formal Semantics, Isabelle/HOL}, + location = {Lyon, France}, + pages = {741--749}, + pdf = {https://www.brucker.ch/bibliography/download/2018/brucker.ea-core-dom-2018.pdf}, + publisher= {ACM Press}, + title = {A Formal Semantics of the Core {DOM} in {Isabelle/HOL}}, + url = {https://www.brucker.ch/bibliography/abstract/brucker.ea-core-dom-2018}, + year = {2018}, +} diff --git a/Core_DOM/Core_DOM.thy b/Core_DOM/Core_DOM.thy new file mode 100644 index 0000000..6b48724 --- /dev/null +++ b/Core_DOM/Core_DOM.thy @@ -0,0 +1,39 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\The Core DOM\ +text\This theory is the main entry point of our formalization of the core DOM.\ + +theory Core_DOM +imports + "Core_DOM_Heap_WF" +begin + + +end diff --git a/Core_DOM/Core_DOM_Basic_Datatypes.thy b/Core_DOM/Core_DOM_Basic_Datatypes.thy new file mode 100644 index 0000000..fa409d9 --- /dev/null +++ b/Core_DOM/Core_DOM_Basic_Datatypes.thy @@ -0,0 +1,66 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + *******************************************************************************\***) + +section\Basic Data Types\ +text\ + \label{sec:Core_DOM_Basic_Datatypes} + This theory formalizes the primitive data types used by the DOM standard~\cite{dom-specification}. +\ +theory Core_DOM_Basic_Datatypes + imports + Main +begin + +type_synonym USVString = string +text\ + In the official standard, the type @{type "USVString"} corresponds to the set of all possible + sequences of Unicode scalar values. As we are not interested in analyzing the specifics of Unicode + strings, we just model @{type "USVString"} using the standard type @{type "string"} of Isabelle/HOL. +\ + +type_synonym DOMString = string +text\ + In the official standard, the type @{type "DOMString"} corresponds to the set of all possible + sequences of code units, commonly interpreted as UTF-16 encoded strings. Again, as we are not + interested in analyzing the specifics of Unicode strings, we just model @{type "DOMString"} using + the standard type @{type "string"} of Isabelle/HOL. +\ + +type_synonym doctype = DOMString + +paragraph\Examples\ +definition html :: doctype + where "html = ''''" + +hide_const id + +text \This dummy locale is used to create scoped definitions by using global interpretations + and defines.\ +locale l_dummy +end diff --git a/Core_DOM/Core_DOM_Functions.thy b/Core_DOM/Core_DOM_Functions.thy new file mode 100644 index 0000000..851d76c --- /dev/null +++ b/Core_DOM/Core_DOM_Functions.thy @@ -0,0 +1,3515 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Querying and Modifying the DOM\ +text\In this theory, we are formalizing the functions for querying and modifying +the DOM.\ + +theory Core_DOM_Functions +imports + "monads/DocumentMonad" +begin + +text \If we do not declare show\_variants, then all abbreviations that contain + constants that are overloaded by using adhoc\_overloading get immediately unfolded.\ +declare [[show_variants]] + +subsection \Various Functions\ + +lemma insort_split: "x \ set (insort y xs) \ (x = y \ x \ set xs)" + apply(induct xs) + by(auto) + +lemma concat_map_distinct: + "distinct (concat (map f xs)) \ y \ set (concat (map f xs)) \ \!x \ set xs. y \ set (f x)" + apply(induct xs) + by(auto) + +lemma concat_map_all_distinct: "distinct (concat (map f xs)) \ x \ set xs \ distinct (f x)" + apply(induct xs) + by(auto) + +lemma distinct_concat_map_I: + assumes "distinct xs" + and "\x. x \ set xs \ distinct (f x)" +and "\x y. x \ set xs \ y \ set xs \ x \ y \ (set (f x)) \ (set (f y)) = {}" +shows "distinct (concat ((map f xs)))" + using assms + apply(induct xs) + by(auto) + +lemma distinct_concat_map_E: + assumes "distinct (concat ((map f xs)))" + shows "\x y. x \ set xs \ y \ set xs \ x \ y \ (set (f x)) \ (set (f y)) = {}" + and "\x. x \ set xs \ distinct (f x)" + using assms + apply(induct xs) + by(auto) + +lemma bind_is_OK_E3 [elim]: + assumes "h \ ok (f \ g)" and "pure f h" + obtains x where "h \ f \\<^sub>r x" and "h \ ok (g x)" + using assms + by(auto simp add: bind_def returns_result_def returns_heap_def is_OK_def execute_def pure_def + split: sum.splits) + + +subsection \Basic Functions\ + +subsubsection \get\_child\_nodes\ + +locale l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +begin + +definition get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) element_ptr \ unit \ (_, (_) node_ptr list) dom_prog" + where + "get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr _ = get_M element_ptr RElement.child_nodes" + +definition get_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) character_data_ptr \ unit \ (_, (_) node_ptr list) dom_prog" + where + "get_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r _ _ = return []" + +definition get_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) document_ptr \ unit \ (_, (_) node_ptr list) dom_prog" + where + "get_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr _ = do { + doc_elem \ get_M document_ptr document_element; + (case doc_elem of + Some element_ptr \ return [cast element_ptr] + | None \ return []) + }" + +definition a_get_child_nodes_tups :: "(((_) object_ptr \ bool) \ ((_) object_ptr \ unit + \ (_, (_) node_ptr list) dom_prog)) list" + where + "a_get_child_nodes_tups = [ + (is_element_ptr, get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \ the \ cast), + (is_character_data_ptr, get_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r \ the \ cast), + (is_document_ptr, get_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \ the \ cast) + ]" + +definition a_get_child_nodes :: "(_) object_ptr \ (_, (_) node_ptr list) dom_prog" + where + "a_get_child_nodes ptr = invoke a_get_child_nodes_tups ptr ()" + +definition a_get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + where + "a_get_child_nodes_locs ptr \ + (if is_element_ptr_kind ptr then {preserved (get_M (the (cast ptr)) RElement.child_nodes)} else {}) \ + (if is_document_ptr_kind ptr then {preserved (get_M (the (cast ptr)) RDocument.document_element)} else {}) \ + {preserved (get_M ptr RObject.nothing)}" + +definition first_child :: "(_) object_ptr \ (_, (_) node_ptr option) dom_prog" + where + "first_child ptr = do { + children \ a_get_child_nodes ptr; + return (case children of [] \ None | child#_ \ Some child)}" +end + +locale l_get_child_nodes_defs = + fixes get_child_nodes :: "(_) object_ptr \ (_, (_) node_ptr list) dom_prog" + fixes get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + +locale l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_type_wf type_wf + + l_known_ptr known_ptr + + l_get_child_nodes_defs get_child_nodes get_child_nodes_locs + + l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + for type_wf :: "(_) heap \ bool" + and known_ptr :: "(_) object_ptr \ bool" + and get_child_nodes :: "(_) object_ptr \ (_, (_) node_ptr list) dom_prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + + assumes known_ptr_impl: "known_ptr = DocumentClass.known_ptr" + assumes type_wf_impl: "type_wf = DocumentClass.type_wf" + assumes get_child_nodes_impl: "get_child_nodes = a_get_child_nodes" + assumes get_child_nodes_locs_impl: "get_child_nodes_locs = a_get_child_nodes_locs" +begin +lemmas get_child_nodes_def = get_child_nodes_impl[unfolded a_get_child_nodes_def] +lemmas get_child_nodes_locs_def = get_child_nodes_locs_impl[unfolded a_get_child_nodes_locs_def] + +lemma get_child_nodes_split: + "P (invoke (a_get_child_nodes_tups @ xs) ptr ()) = + ((known_ptr ptr \ P (get_child_nodes ptr)) + \ (\(known_ptr ptr) \ P (invoke xs ptr ())))" + by(auto simp add: known_ptr_impl get_child_nodes_impl a_get_child_nodes_def a_get_child_nodes_tups_def + known_ptr_defs CharacterDataClass.known_ptr_defs ElementClass.known_ptr_defs + NodeClass.known_ptr_defs + split: invoke_splits) + +lemma get_child_nodes_split_asm: + "P (invoke (a_get_child_nodes_tups @ xs) ptr ()) = + (\((known_ptr ptr \ \P (get_child_nodes ptr)) + \ (\(known_ptr ptr) \ \P (invoke xs ptr ()))))" + by(auto simp add: known_ptr_impl get_child_nodes_impl a_get_child_nodes_def + a_get_child_nodes_tups_def known_ptr_defs CharacterDataClass.known_ptr_defs + ElementClass.known_ptr_defs NodeClass.known_ptr_defs + split: invoke_splits) + +lemmas get_child_nodes_splits = get_child_nodes_split get_child_nodes_split_asm + +lemma get_child_nodes_ok [simp]: + assumes "known_ptr ptr" + assumes "type_wf h" + assumes "ptr |\| object_ptr_kinds h" + shows "h \ ok (get_child_nodes ptr)" + using assms(1) assms(2) assms(3) + apply(auto simp add: known_ptr_impl type_wf_impl get_child_nodes_def a_get_child_nodes_tups_def)[1] + apply(split invoke_splits, rule conjI)+ + apply((rule impI)+, drule(1) known_ptr_not_document_ptr, drule(1) known_ptr_not_character_data_ptr, + drule(1) known_ptr_not_element_ptr) + apply(auto simp add: NodeClass.known_ptr_defs)[1] + apply(auto simp add: get_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def dest: get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok + split: list.splits option.splits intro!: bind_is_OK_I2)[1] + apply(auto simp add: get_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def)[1] + apply (auto simp add: get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def CharacterDataClass.type_wf_defs + DocumentClass.type_wf_defs intro!: bind_is_OK_I2 split: option.splits)[1] + using get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok \type_wf h\[unfolded type_wf_impl] by blast + +lemma get_child_nodes_ptr_in_heap [simp]: + assumes "h \ get_child_nodes ptr \\<^sub>r children" + shows "ptr |\| object_ptr_kinds h" + using assms + by(auto simp add: get_child_nodes_impl a_get_child_nodes_def invoke_ptr_in_heap + dest: is_OK_returns_result_I) + +lemma get_child_nodes_pure [simp]: + "pure (get_child_nodes ptr) h" + apply (auto simp add: get_child_nodes_impl a_get_child_nodes_def a_get_child_nodes_tups_def)[1] + apply(split invoke_splits, rule conjI)+ + by(auto simp add: get_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def get_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def intro!: bind_pure_I split: option.splits) + +lemma get_child_nodes_reads: "reads (get_child_nodes_locs ptr) (get_child_nodes ptr) h h'" + apply(simp add: get_child_nodes_locs_impl get_child_nodes_impl a_get_child_nodes_def + a_get_child_nodes_tups_def a_get_child_nodes_locs_def) + apply(split invoke_splits, rule conjI)+ + apply(auto)[1] + apply(auto simp add: get_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def intro: reads_subset[OF reads_singleton] + reads_subset[OF check_in_heap_reads] + intro!: reads_bind_pure reads_subset[OF return_reads] split: option.splits)[1] (* slow: ca 1min *) + apply(auto simp add: get_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def intro: reads_subset[OF check_in_heap_reads] + intro!: reads_bind_pure reads_subset[OF return_reads] )[1] + apply(auto simp add: get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def intro: reads_subset[OF reads_singleton] + reads_subset[OF check_in_heap_reads] intro!: reads_bind_pure reads_subset[OF return_reads] + split: option.splits) + done +end + +locale l_get_child_nodes = l_type_wf + l_known_ptr + l_get_child_nodes_defs + + assumes get_child_nodes_reads: "reads (get_child_nodes_locs ptr) (get_child_nodes ptr) h h'" + assumes get_child_nodes_ok: "type_wf h \ known_ptr ptr \ ptr |\| object_ptr_kinds h + \ h \ ok (get_child_nodes ptr)" + assumes get_child_nodes_ptr_in_heap: "h \ ok (get_child_nodes ptr) \ ptr |\| object_ptr_kinds h" + assumes get_child_nodes_pure [simp]: "pure (get_child_nodes ptr) h" + +global_interpretation l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs defines + get_child_nodes = l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes and + get_child_nodes_locs = l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes_locs + . + +interpretation + i_get_child_nodes?: l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs + by(auto simp add: l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def get_child_nodes_def get_child_nodes_locs_def) +declare l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma get_child_nodes_is_l_get_child_nodes [instances]: + "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs" + apply(unfold_locales) + using get_child_nodes_reads get_child_nodes_ok get_child_nodes_ptr_in_heap get_child_nodes_pure + by blast+ + + +paragraph \new\_element\ + +locale l_new_element_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs + for type_wf :: "(_) heap \ bool" + and known_ptr :: "(_) object_ptr \ bool" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" +begin +lemma get_child_nodes_new_element: + "ptr' \ cast new_element_ptr \ h \ new_element \\<^sub>r new_element_ptr \ h \ new_element \\<^sub>h h' + \ r \ get_child_nodes_locs ptr' \ r h h'" + by (auto simp add: get_child_nodes_locs_def new_element_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t new_element_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + new_element_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t split: prod.splits if_splits option.splits + elim!: bind_returns_result_E bind_returns_heap_E intro: is_element_ptr_kind_obtains) + +lemma new_element_no_child_nodes: + "h \ new_element \\<^sub>r new_element_ptr \ h \ new_element \\<^sub>h h' + \ h' \ get_child_nodes (cast new_element_ptr) \\<^sub>r []" + apply(auto simp add: get_child_nodes_def a_get_child_nodes_tups_def + split: prod.splits elim!: bind_returns_result_E bind_returns_heap_E)[1] + apply(split invoke_splits, rule conjI)+ + apply(auto intro: new_element_is_element_ptr)[1] + by(auto simp add: new_element_ptr_in_heap get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def check_in_heap_def + new_element_child_nodes intro!: bind_pure_returns_result_I + intro: new_element_is_element_ptr elim!: new_element_ptr_in_heap) +end + +locale l_new_element_get_child_nodes = l_new_element + l_get_child_nodes + + assumes get_child_nodes_new_element: + "ptr' \ cast new_element_ptr \ h \ new_element \\<^sub>r new_element_ptr + \ h \ new_element \\<^sub>h h' \ r \ get_child_nodes_locs ptr' \ r h h'" + assumes new_element_no_child_nodes: + "h \ new_element \\<^sub>r new_element_ptr \ h \ new_element \\<^sub>h h' + \ h' \ get_child_nodes (cast new_element_ptr) \\<^sub>r []" + +interpretation i_new_element_get_child_nodes?: + l_new_element_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs + by(unfold_locales) +declare l_new_element_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma new_element_get_child_nodes_is_l_new_element_get_child_nodes [instances]: + "l_new_element_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs" + using new_element_is_l_new_element get_child_nodes_is_l_get_child_nodes + apply(auto simp add: l_new_element_get_child_nodes_def l_new_element_get_child_nodes_axioms_def)[1] + using get_child_nodes_new_element new_element_no_child_nodes + by fast+ + + +paragraph \new\_character\_data\ + +locale l_new_character_data_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs + for type_wf :: "(_) heap \ bool" + and known_ptr :: "(_) object_ptr \ bool" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" +begin +lemma get_child_nodes_new_character_data: + "ptr' \ cast new_character_data_ptr \ h \ new_character_data \\<^sub>r new_character_data_ptr + \ h \ new_character_data \\<^sub>h h' \ r \ get_child_nodes_locs ptr' \ r h h'" + by (auto simp add: get_child_nodes_locs_def new_character_data_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + new_character_data_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t new_character_data_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t + split: prod.splits if_splits option.splits + elim!: bind_returns_result_E bind_returns_heap_E + intro: is_character_data_ptr_kind_obtains) + +lemma new_character_data_no_child_nodes: + "h \ new_character_data \\<^sub>r new_character_data_ptr \ h \ new_character_data \\<^sub>h h' + \ h' \ get_child_nodes (cast new_character_data_ptr) \\<^sub>r []" + apply(auto simp add: get_child_nodes_def a_get_child_nodes_tups_def + split: prod.splits elim!: bind_returns_result_E bind_returns_heap_E)[1] + apply(split invoke_splits, rule conjI)+ + apply(auto intro: new_character_data_is_character_data_ptr)[1] + by(auto simp add: new_character_data_ptr_in_heap get_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + check_in_heap_def new_character_data_child_nodes + intro!: bind_pure_returns_result_I + intro: new_character_data_is_character_data_ptr elim!: new_character_data_ptr_in_heap) +end + +locale l_new_character_data_get_child_nodes = l_new_character_data + l_get_child_nodes + + assumes get_child_nodes_new_character_data: + "ptr' \ cast new_character_data_ptr \ h \ new_character_data \\<^sub>r new_character_data_ptr + \ h \ new_character_data \\<^sub>h h' \ r \ get_child_nodes_locs ptr' \ r h h'" + assumes new_character_data_no_child_nodes: + "h \ new_character_data \\<^sub>r new_character_data_ptr \ h \ new_character_data \\<^sub>h h' + \ h' \ get_child_nodes (cast new_character_data_ptr) \\<^sub>r []" + +interpretation i_new_character_data_get_child_nodes?: + l_new_character_data_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs + by(unfold_locales) +declare l_new_character_data_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma new_character_data_get_child_nodes_is_l_new_character_data_get_child_nodes [instances]: + "l_new_character_data_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs" + using new_character_data_is_l_new_character_data get_child_nodes_is_l_get_child_nodes + apply(simp add: l_new_character_data_get_child_nodes_def l_new_character_data_get_child_nodes_axioms_def) + using get_child_nodes_new_character_data new_character_data_no_child_nodes + by fast + + + +paragraph \new\_document\ + +locale l_new_document_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs + for type_wf :: "(_) heap \ bool" + and known_ptr :: "(_) object_ptr \ bool" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" +begin +lemma get_child_nodes_new_document: + "ptr' \ cast new_document_ptr \ h \ new_document \\<^sub>r new_document_ptr + \ h \ new_document \\<^sub>h h' \ r \ get_child_nodes_locs ptr' \ r h h'" + by (auto simp add: get_child_nodes_locs_def new_document_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t new_document_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + new_document_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t split: prod.splits if_splits option.splits + elim!: bind_returns_result_E bind_returns_heap_E + intro: is_document_ptr_kind_obtains) + +lemma new_document_no_child_nodes: + "h \ new_document \\<^sub>r new_document_ptr \ h \ new_document \\<^sub>h h' + \ h' \ get_child_nodes (cast new_document_ptr) \\<^sub>r []" + apply(auto simp add: get_child_nodes_def a_get_child_nodes_tups_def + split: prod.splits + elim!: bind_returns_result_E bind_returns_heap_E)[1] + apply(split invoke_splits, rule conjI)+ + apply(auto intro: new_document_is_document_ptr)[1] + by(auto simp add: new_document_ptr_in_heap get_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def check_in_heap_def + new_document_document_element + intro!: bind_pure_returns_result_I + intro: new_document_is_document_ptr elim!: new_document_ptr_in_heap split: option.splits) +end + +locale l_new_document_get_child_nodes = l_new_document + l_get_child_nodes + + assumes get_child_nodes_new_document: + "ptr' \ cast new_document_ptr \ h \ new_document \\<^sub>r new_document_ptr + \ h \ new_document \\<^sub>h h' \ r \ get_child_nodes_locs ptr' \ r h h'" + assumes new_document_no_child_nodes: + "h \ new_document \\<^sub>r new_document_ptr \ h \ new_document \\<^sub>h h' + \ h' \ get_child_nodes (cast new_document_ptr) \\<^sub>r []" + +interpretation i_new_document_get_child_nodes?: + l_new_document_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs + by(unfold_locales) +declare l_new_document_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma new_document_get_child_nodes_is_l_new_document_get_child_nodes [instances]: + "l_new_document_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs" + using new_document_is_l_new_document get_child_nodes_is_l_get_child_nodes + apply(simp add: l_new_document_get_child_nodes_def l_new_document_get_child_nodes_axioms_def) + using get_child_nodes_new_document new_document_no_child_nodes + by fast + +subsubsection \set\_child\_nodes\ + +locale l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +begin +definition set_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: + "(_) element_ptr \ (_) node_ptr list \ (_, unit) dom_prog" + where + "set_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr children = put_M element_ptr RElement.child_nodes_update children" + +definition set_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: + "(_) character_data_ptr \ (_) node_ptr list \ (_, unit) dom_prog" + where + "set_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r _ _ = error HierarchyRequestError" + +definition set_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) document_ptr \ (_) node_ptr list \ (_, unit) dom_prog" + where + "set_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr children = do { + (case children of + [] \ put_M document_ptr document_element_update None + | child # [] \ (case cast child of + Some element_ptr \ put_M document_ptr document_element_update (Some element_ptr) + | None \ error HierarchyRequestError) + | _ \ error HierarchyRequestError) + }" + +definition a_set_child_nodes_tups :: + "(((_) object_ptr \ bool) \ ((_) object_ptr \ (_) node_ptr list \ (_, unit) dom_prog)) list" + where + "a_set_child_nodes_tups \ [ + (is_element_ptr, set_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \ the \ cast), + (is_character_data_ptr, set_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r \ the \ cast), + (is_document_ptr, set_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \ the \ cast) + ]" + +definition a_set_child_nodes :: "(_) object_ptr \ (_) node_ptr list \ (_, unit) dom_prog" + where + "a_set_child_nodes ptr children = invoke a_set_child_nodes_tups ptr (children)" +lemmas set_child_nodes_defs = a_set_child_nodes_def + +definition a_set_child_nodes_locs :: "(_) object_ptr \ (_, unit) dom_prog set" + where + "a_set_child_nodes_locs ptr \ + (if is_element_ptr_kind ptr + then all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast ptr)) RElement.child_nodes_update) else {}) \ + (if is_document_ptr_kind ptr + then all_args (put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast ptr)) document_element_update) else {})" +end + +locale l_set_child_nodes_defs = + fixes set_child_nodes :: "(_) object_ptr \ (_) node_ptr list \ (_, unit) dom_prog" + fixes set_child_nodes_locs :: "(_) object_ptr \ (_, unit) dom_prog set" + +locale l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_type_wf type_wf + + l_known_ptr known_ptr + + l_set_child_nodes_defs set_child_nodes set_child_nodes_locs + + l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + for type_wf :: "(_) heap \ bool" + and known_ptr :: "(_) object_ptr \ bool" + and set_child_nodes :: "(_) object_ptr \ (_) node_ptr list \ (_, unit) dom_prog" + and set_child_nodes_locs :: "(_) object_ptr \ (_, unit) dom_prog set" + + assumes known_ptr_impl: "known_ptr = DocumentClass.known_ptr" + assumes type_wf_impl: "type_wf = DocumentClass.type_wf" + assumes set_child_nodes_impl: "set_child_nodes = a_set_child_nodes" + assumes set_child_nodes_locs_impl: "set_child_nodes_locs = a_set_child_nodes_locs" +begin +lemmas set_child_nodes_def = set_child_nodes_impl[unfolded a_set_child_nodes_def] +lemmas set_child_nodes_locs_def = set_child_nodes_locs_impl[unfolded a_set_child_nodes_locs_def] + +lemma set_child_nodes_split: + "P (invoke (a_set_child_nodes_tups @ xs) ptr (children)) = + ((known_ptr ptr \ P (set_child_nodes ptr children)) + \ (\(known_ptr ptr) \ P (invoke xs ptr (children))))" + by(auto simp add: known_ptr_impl set_child_nodes_impl a_set_child_nodes_def + a_set_child_nodes_tups_def known_ptr_defs CharacterDataClass.known_ptr_defs + ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: invoke_splits) + +lemma set_child_nodes_split_asm: + "P (invoke (a_set_child_nodes_tups @ xs) ptr (children)) = + (\((known_ptr ptr \ \P (set_child_nodes ptr children)) + \ (\(known_ptr ptr) \ \P (invoke xs ptr (children)))))" + by(auto simp add: known_ptr_impl set_child_nodes_impl a_set_child_nodes_def + a_set_child_nodes_tups_def known_ptr_defs CharacterDataClass.known_ptr_defs + ElementClass.known_ptr_defs NodeClass.known_ptr_defs split: invoke_splits)[1] +lemmas set_child_nodes_splits = set_child_nodes_split set_child_nodes_split_asm + +lemma set_child_nodes_writes: "writes (set_child_nodes_locs ptr) (set_child_nodes ptr children) h h'" + apply(simp add: set_child_nodes_locs_impl set_child_nodes_impl a_set_child_nodes_def + a_set_child_nodes_tups_def a_set_child_nodes_locs_def) + apply(split invoke_splits, rule conjI)+ + apply(auto)[1] + apply(auto simp add: set_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def intro!: writes_bind_pure + intro: writes_union_right_I split: list.splits)[1] + apply(auto intro: writes_union_right_I split: option.splits)[1] + apply(auto intro: writes_union_right_I split: option.splits)[1] + apply(auto intro: writes_union_right_I split: option.splits)[1] + apply(auto intro: writes_union_right_I split: option.splits)[1] + apply(auto intro: writes_union_right_I split: option.splits)[1] + apply(auto intro: writes_union_right_I split: option.splits)[1] (*slow: ca. 1min *) + apply(auto simp add: set_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def intro!: writes_bind_pure)[1] + apply(auto simp add: set_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def intro: writes_union_left_I + intro!: writes_bind_pure split: list.splits option.splits)[1] + done + +lemma set_child_nodes_pointers_preserved: + assumes "w \ set_child_nodes_locs object_ptr" + assumes "h \ w \\<^sub>h h'" + shows "object_ptr_kinds h = object_ptr_kinds h'" + using assms(1) object_ptr_kinds_preserved[OF writes_singleton2 assms(2)] + by(auto simp add: set_child_nodes_locs_impl all_args_def a_set_child_nodes_locs_def + split: if_splits) + +lemma set_child_nodes_typess_preserved: + assumes "w \ set_child_nodes_locs object_ptr" + assumes "h \ w \\<^sub>h h'" + shows "type_wf h = type_wf h'" + using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)] + by(auto simp add: set_child_nodes_locs_impl type_wf_impl all_args_def a_set_child_nodes_locs_def + split: if_splits) +end + +locale l_set_child_nodes = l_type_wf + l_set_child_nodes_defs + + assumes set_child_nodes_writes: + "writes (set_child_nodes_locs ptr) (set_child_nodes ptr children) h h'" + assumes set_child_nodes_pointers_preserved: + "w \ set_child_nodes_locs object_ptr \ h \ w \\<^sub>h h' \ object_ptr_kinds h = object_ptr_kinds h'" + assumes set_child_nodes_types_preserved: + "w \ set_child_nodes_locs object_ptr \ h \ w \\<^sub>h h' \ type_wf h = type_wf h'" + +global_interpretation l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs defines + set_child_nodes = l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_child_nodes and + set_child_nodes_locs = l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_child_nodes_locs . + +interpretation + i_set_child_nodes?: l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs + apply(unfold_locales) + by (auto simp add: set_child_nodes_def set_child_nodes_locs_def) +declare l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + + +lemma set_child_nodes_is_l_set_child_nodes [instances]: + "l_set_child_nodes type_wf set_child_nodes set_child_nodes_locs" + apply(unfold_locales) + using set_child_nodes_pointers_preserved set_child_nodes_typess_preserved set_child_nodes_writes + by blast+ + + +paragraph \get\_child\_nodes\ + +locale l_set_child_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin + +lemma set_child_nodes_get_child_nodes: + assumes "known_ptr ptr" + assumes "type_wf h" + assumes "h \ set_child_nodes ptr children \\<^sub>h h'" + shows "h' \ get_child_nodes ptr \\<^sub>r children" +proof - + have "h \ check_in_heap ptr \\<^sub>r ()" + using assms set_child_nodes_impl[unfolded a_set_child_nodes_def] invoke_ptr_in_heap + by (metis (full_types) check_in_heap_ptr_in_heap is_OK_returns_heap_I is_OK_returns_result_E + old.unit.exhaust) + then have ptr_in_h: "ptr |\| object_ptr_kinds h" + by (simp add: check_in_heap_ptr_in_heap is_OK_returns_result_I) + + have "type_wf h'" + apply(unfold type_wf_impl) + apply(rule subst[where P=id, OF type_wf_preserved[OF set_child_nodes_writes assms(3), + unfolded all_args_def], simplified]) + by(auto simp add: all_args_def assms(2)[unfolded type_wf_impl] + set_child_nodes_locs_impl[unfolded a_set_child_nodes_locs_def] + split: if_splits) + have "h' \ check_in_heap ptr \\<^sub>r ()" + using check_in_heap_reads set_child_nodes_writes assms(3) \h \ check_in_heap ptr \\<^sub>r ()\ + apply(rule reads_writes_separate_forwards) + by(auto simp add: all_args_def set_child_nodes_locs_impl[unfolded a_set_child_nodes_locs_def]) + then have "ptr |\| object_ptr_kinds h'" + using check_in_heap_ptr_in_heap by blast + with assms ptr_in_h \type_wf h'\ show ?thesis + apply(auto simp add: get_child_nodes_impl set_child_nodes_impl type_wf_impl known_ptr_impl + a_get_child_nodes_def a_get_child_nodes_tups_def a_set_child_nodes_def + a_set_child_nodes_tups_def + del: bind_pure_returns_result_I2 + intro!: bind_pure_returns_result_I2)[1] + apply(split invoke_splits, rule conjI) + apply(split invoke_splits, rule conjI) + apply(split invoke_splits, rule conjI) + apply(auto simp add: NodeClass.known_ptr_defs + dest!: known_ptr_not_document_ptr known_ptr_not_character_data_ptr + known_ptr_not_element_ptr)[1] + apply(auto simp add: NodeClass.known_ptr_defs + dest!: known_ptr_not_document_ptr known_ptr_not_character_data_ptr + known_ptr_not_element_ptr)[1] + apply(auto simp add: get_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def set_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok + split: list.splits option.splits + intro!: bind_pure_returns_result_I2 + dest: get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok; auto dest: returns_result_eq + dest!: document_put_get[where getter = document_element])[1] (* slow, ca 1min *) + apply(auto simp add: get_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def set_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def)[1] + by(auto simp add: get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def set_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def dest: element_put_get) +qed + +lemma set_child_nodes_get_child_nodes_different_pointers: + assumes "ptr \ ptr'" + assumes "w \ set_child_nodes_locs ptr" + assumes "h \ w \\<^sub>h h'" + assumes "r \ get_child_nodes_locs ptr'" + shows "r h h'" + using assms + apply(auto simp add: get_child_nodes_locs_impl set_child_nodes_locs_impl all_args_def + a_set_child_nodes_locs_def a_get_child_nodes_locs_def + split: if_splits option.splits )[1] + apply(rule is_document_ptr_kind_obtains) + apply(simp) + apply(rule is_document_ptr_kind_obtains) + apply(auto)[1] + apply(auto)[1] + apply(rule is_element_ptr_kind_obtains) + apply(auto)[1] + apply(auto)[1] + apply(rule is_element_ptr_kind_obtains) + apply(auto) + done +end + +locale l_set_child_nodes_get_child_nodes = l_get_child_nodes + l_set_child_nodes + + assumes set_child_nodes_get_child_nodes: + "type_wf h \ known_ptr ptr + \ h \ set_child_nodes ptr children \\<^sub>h h' \ h' \ get_child_nodes ptr \\<^sub>r children" + assumes set_child_nodes_get_child_nodes_different_pointers: + "ptr \ ptr' \ w \ set_child_nodes_locs ptr \ h \ w \\<^sub>h h' + \ r \ get_child_nodes_locs ptr' \ r h h'" + +interpretation + i_set_child_nodes_get_child_nodes?: l_set_child_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf + known_ptr get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs + by unfold_locales +declare l_set_child_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_child_nodes_get_child_nodes_is_l_set_child_nodes_get_child_nodes [instances]: + "l_set_child_nodes_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs + set_child_nodes set_child_nodes_locs" + using get_child_nodes_is_l_get_child_nodes set_child_nodes_is_l_set_child_nodes + apply(auto simp add: l_set_child_nodes_get_child_nodes_def l_set_child_nodes_get_child_nodes_axioms_def)[1] + using set_child_nodes_get_child_nodes apply blast + using set_child_nodes_get_child_nodes_different_pointers apply metis + done + + +subsubsection \get\_attribute\ + +locale l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +begin +definition a_get_attribute :: "(_) element_ptr \ attr_key \ (_, attr_value option) dom_prog" + where + "a_get_attribute ptr k = do {m \ get_M ptr attrs; return (fmlookup m k)}" +lemmas get_attribute_defs = a_get_attribute_def + +definition a_get_attribute_locs :: "(_) element_ptr \ ((_) heap \ (_) heap \ bool) set" + where + "a_get_attribute_locs element_ptr = {preserved (get_M element_ptr attrs)}" +end + +locale l_get_attribute_defs = + fixes get_attribute :: "(_) element_ptr \ attr_key \ (_, attr_value option) dom_prog" + fixes get_attribute_locs :: "(_) element_ptr \ ((_) heap \ (_) heap \ bool) set" + +locale l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_type_wf type_wf + + l_get_attribute_defs get_attribute get_attribute_locs + + l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + for type_wf :: "(_) heap \ bool" + and get_attribute :: "(_) element_ptr \ attr_key \ (_, attr_value option) dom_prog" + and get_attribute_locs :: "(_) element_ptr \ ((_) heap \ (_) heap \ bool) set" + + assumes type_wf_impl: "type_wf = DocumentClass.type_wf" + assumes get_attribute_impl: "get_attribute = a_get_attribute" + assumes get_attribute_locs_impl: "get_attribute_locs = a_get_attribute_locs" +begin +lemma get_attribute_pure [simp]: "pure (get_attribute ptr k) h" + by (auto simp add: bind_pure_I get_attribute_impl[unfolded a_get_attribute_def]) + +lemma get_attribute_ok: + "type_wf h \ element_ptr |\| element_ptr_kinds h \ h \ ok (get_attribute element_ptr k)" + apply(unfold type_wf_impl) + unfolding get_attribute_impl[unfolded a_get_attribute_def] using get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok + by (metis bind_is_OK_pure_I return_ok ElementMonad.get_M_pure) + +lemma get_attribute_ptr_in_heap: + "h \ ok (get_attribute element_ptr k) \ element_ptr |\| element_ptr_kinds h" + unfolding get_attribute_impl[unfolded a_get_attribute_def] + by (meson DocumentMonad.get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap bind_is_OK_E is_OK_returns_result_I) + +lemma get_attribute_reads: + "reads (get_attribute_locs element_ptr) (get_attribute element_ptr k) h h'" + by(auto simp add: get_attribute_impl[unfolded a_get_attribute_def] + get_attribute_locs_impl[unfolded a_get_attribute_locs_def] + reads_insert_writes_set_right + intro!: reads_bind_pure) +end + +locale l_get_attribute = l_type_wf + l_get_attribute_defs + +assumes get_attribute_reads: + "reads (get_attribute_locs element_ptr) (get_attribute element_ptr k) h h'" +assumes get_attribute_ok: + "type_wf h \ element_ptr |\| element_ptr_kinds h \ h \ ok (get_attribute element_ptr k)" +assumes get_attribute_ptr_in_heap: + "h \ ok (get_attribute element_ptr k) \ element_ptr |\| element_ptr_kinds h" +assumes get_attribute_pure [simp]: "pure (get_attribute element_ptr k) h" + +global_interpretation l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs defines + get_attribute = l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_attribute and + get_attribute_locs = l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_attribute_locs . + +interpretation + i_get_attribute?: l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_attribute get_attribute_locs + apply(unfold_locales) + by (auto simp add: get_attribute_def get_attribute_locs_def) +declare l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma get_attribute_is_l_get_attribute [instances]: + "l_get_attribute type_wf get_attribute get_attribute_locs" + apply(unfold_locales) + using get_attribute_reads get_attribute_ok get_attribute_ptr_in_heap get_attribute_pure + by blast+ + + +subsubsection \set\_attribute\ + +locale l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +begin + +definition + a_set_attribute :: "(_) element_ptr \ attr_key \ attr_value option \ (_, unit) dom_prog" + where + "a_set_attribute ptr k v = do { + m \ get_M ptr attrs; + put_M ptr attrs_update (if v = None then fmdrop k m else fmupd k (the v) m) + }" + +definition a_set_attribute_locs :: "(_) element_ptr \ (_, unit) dom_prog set" + where + "a_set_attribute_locs element_ptr \ all_args (put_M element_ptr attrs_update)" +end + +locale l_set_attribute_defs = + fixes set_attribute :: "(_) element_ptr \ attr_key \ attr_value option \ (_, unit) dom_prog" + fixes set_attribute_locs :: "(_) element_ptr \ (_, unit) dom_prog set" + +locale l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_type_wf type_wf + + l_set_attribute_defs set_attribute set_attribute_locs + + l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + for type_wf :: "(_) heap \ bool" + and set_attribute :: "(_) element_ptr \ attr_key \ attr_value option \ (_, unit) dom_prog" + and set_attribute_locs :: "(_) element_ptr \ (_, unit) dom_prog set" + + assumes type_wf_impl: "type_wf = DocumentClass.type_wf" + assumes set_attribute_impl: "set_attribute = a_set_attribute" + assumes set_attribute_locs_impl: "set_attribute_locs = a_set_attribute_locs" +begin +lemmas set_attribute_def = set_attribute_impl[folded a_set_attribute_def] +lemmas set_attribute_locs_def = set_attribute_locs_impl[unfolded a_set_attribute_locs_def] + +lemma set_attribute_ok: "type_wf h \ element_ptr |\| element_ptr_kinds h \ h \ ok (set_attribute element_ptr k v)" + apply(unfold type_wf_impl) + unfolding set_attribute_impl[unfolded a_set_attribute_def] using get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok + by(metis (no_types, lifting) DocumentClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ElementMonad.get_M_pure bind_is_OK_E + bind_is_OK_pure_I is_OK_returns_result_I) + +lemma set_attribute_writes: + "writes (set_attribute_locs element_ptr) (set_attribute element_ptr k v) h h'" + by(auto simp add: set_attribute_impl[unfolded a_set_attribute_def] + set_attribute_locs_impl[unfolded a_set_attribute_locs_def] + intro: writes_bind_pure) +end + +locale l_set_attribute = l_type_wf + l_set_attribute_defs + + assumes set_attribute_writes: + "writes (set_attribute_locs element_ptr) (set_attribute element_ptr k v) h h'" + assumes set_attribute_ok: + "type_wf h \ element_ptr |\| element_ptr_kinds h \ h \ ok (set_attribute element_ptr k v)" + +global_interpretation l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs defines + set_attribute = l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_attribute and + set_attribute_locs = l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_attribute_locs . +interpretation + i_set_attribute?: l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_attribute set_attribute_locs + apply(unfold_locales) + by (auto simp add: set_attribute_def set_attribute_locs_def) +declare l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_attribute_is_l_set_attribute [instances]: + "l_set_attribute type_wf set_attribute set_attribute_locs" + apply(unfold_locales) + using set_attribute_ok set_attribute_writes + by blast+ + + +paragraph \get\_attribute\ + +locale l_set_attribute_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin + +lemma set_attribute_get_attribute: + "h \ set_attribute ptr k v \\<^sub>h h' \ h' \ get_attribute ptr k \\<^sub>r v" + by(auto simp add: set_attribute_impl[unfolded a_set_attribute_def] + get_attribute_impl[unfolded a_get_attribute_def] + elim!: bind_returns_heap_E2 + intro!: bind_pure_returns_result_I + elim: element_put_get) +end + +locale l_set_attribute_get_attribute = l_get_attribute + l_set_attribute + + assumes set_attribute_get_attribute: + "h \ set_attribute ptr k v \\<^sub>h h' \ h' \ get_attribute ptr k \\<^sub>r v" + +interpretation + i_set_attribute_get_attribute?: l_set_attribute_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf + get_attribute get_attribute_locs set_attribute set_attribute_locs + by(unfold_locales) +declare l_set_attribute_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_attribute_get_attribute_is_l_set_attribute_get_attribute [instances]: + "l_set_attribute_get_attribute type_wf get_attribute get_attribute_locs set_attribute set_attribute_locs" + using get_attribute_is_l_get_attribute set_attribute_is_l_set_attribute + apply(simp add: l_set_attribute_get_attribute_def l_set_attribute_get_attribute_axioms_def) + using set_attribute_get_attribute + by blast + +paragraph \get\_child\_nodes\ + +locale l_set_attribute_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin +lemma set_attribute_get_child_nodes: + "\w \ set_attribute_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_child_nodes_locs ptr'. r h h'))" + by(auto simp add: set_attribute_locs_def get_child_nodes_locs_def all_args_def + intro: element_put_get_preserved[where setter=attrs_update]) +end + +locale l_set_attribute_get_child_nodes = + l_set_attribute + + l_get_child_nodes + + assumes set_attribute_get_child_nodes: + "\w \ set_attribute_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_child_nodes_locs ptr'. r h h'))" + +interpretation + i_set_attribute_get_child_nodes?: l_set_attribute_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf + set_attribute set_attribute_locs known_ptr get_child_nodes get_child_nodes_locs + by unfold_locales +declare l_set_attribute_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_attribute_get_child_nodes_is_l_set_attribute_get_child_nodes [instances]: + "l_set_attribute_get_child_nodes type_wf set_attribute set_attribute_locs known_ptr + get_child_nodes get_child_nodes_locs" + using set_attribute_is_l_set_attribute get_child_nodes_is_l_get_child_nodes + apply(simp add: l_set_attribute_get_child_nodes_def l_set_attribute_get_child_nodes_axioms_def) + using set_attribute_get_child_nodes + by blast + + +subsubsection \get\_disconnected\_nodes\ + +locale l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +begin +definition a_get_disconnected_nodes :: "(_) document_ptr + \ (_, (_) node_ptr list) dom_prog" + where + "a_get_disconnected_nodes document_ptr = get_M document_ptr disconnected_nodes" +lemmas get_disconnected_nodes_defs = a_get_disconnected_nodes_def + +definition a_get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + where + "a_get_disconnected_nodes_locs document_ptr = {preserved (get_M document_ptr disconnected_nodes)}" +end + +locale l_get_disconnected_nodes_defs = + fixes get_disconnected_nodes :: "(_) document_ptr \ (_, (_) node_ptr list) dom_prog" + fixes get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + +locale l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_type_wf type_wf + + l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs + + l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + for type_wf :: "(_) heap \ bool" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + + assumes type_wf_impl: "type_wf = DocumentClass.type_wf" + assumes get_disconnected_nodes_impl: "get_disconnected_nodes = a_get_disconnected_nodes" + assumes get_disconnected_nodes_locs_impl: "get_disconnected_nodes_locs = a_get_disconnected_nodes_locs" +begin +lemmas + get_disconnected_nodes_def = get_disconnected_nodes_impl[unfolded a_get_disconnected_nodes_def] +lemmas + get_disconnected_nodes_locs_def = get_disconnected_nodes_locs_impl[unfolded a_get_disconnected_nodes_locs_def] + +lemma get_disconnected_nodes_ok: + "type_wf h \ document_ptr |\| document_ptr_kinds h \ h \ ok (get_disconnected_nodes document_ptr)" + apply(unfold type_wf_impl) + unfolding get_disconnected_nodes_impl[unfolded a_get_disconnected_nodes_def] using get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok + by fast + +lemma get_disconnected_nodes_ptr_in_heap: + "h \ ok (get_disconnected_nodes document_ptr) \ document_ptr |\| document_ptr_kinds h" + unfolding get_disconnected_nodes_impl[unfolded a_get_disconnected_nodes_def] + by (simp add: DocumentMonad.get_M_ptr_in_heap) + +lemma get_disconnected_nodes_pure [simp]: "pure (get_disconnected_nodes document_ptr) h" + unfolding get_disconnected_nodes_impl[unfolded a_get_disconnected_nodes_def] by simp + +lemma get_disconnected_nodes_reads: + "reads (get_disconnected_nodes_locs document_ptr) (get_disconnected_nodes document_ptr) h h'" + by(simp add: get_disconnected_nodes_impl[unfolded a_get_disconnected_nodes_def] + get_disconnected_nodes_locs_impl[unfolded a_get_disconnected_nodes_locs_def] + reads_bind_pure reads_insert_writes_set_right) +end + +locale l_get_disconnected_nodes = l_type_wf + l_get_disconnected_nodes_defs + + assumes get_disconnected_nodes_reads: + "reads (get_disconnected_nodes_locs document_ptr) (get_disconnected_nodes document_ptr) h h'" + assumes get_disconnected_nodes_ok: + "type_wf h \ document_ptr |\| document_ptr_kinds h \ h \ ok (get_disconnected_nodes document_ptr)" + assumes get_disconnected_nodes_ptr_in_heap: + "h \ ok (get_disconnected_nodes document_ptr) \ document_ptr |\| document_ptr_kinds h" + assumes get_disconnected_nodes_pure [simp]: + "pure (get_disconnected_nodes document_ptr) h" + +global_interpretation l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs defines + get_disconnected_nodes = l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_nodes and + get_disconnected_nodes_locs = l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_nodes_locs . +interpretation + i_get_disconnected_nodes?: l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes + get_disconnected_nodes_locs + apply(unfold_locales) + by (auto simp add: get_disconnected_nodes_def get_disconnected_nodes_locs_def) +declare l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma get_disconnected_nodes_is_l_get_disconnected_nodes [instances]: + "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs" + apply(simp add: l_get_disconnected_nodes_def) + using get_disconnected_nodes_reads get_disconnected_nodes_ok get_disconnected_nodes_ptr_in_heap + get_disconnected_nodes_pure + by blast+ + + +paragraph \set\_child\_nodes\ + +locale l_set_child_nodes_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + CD: l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin +lemma set_child_nodes_get_disconnected_nodes: + "\w \ a_set_child_nodes_locs ptr. (h \ w \\<^sub>h h' \ (\r \ a_get_disconnected_nodes_locs ptr'. r h h'))" + by(auto simp add: a_set_child_nodes_locs_def a_get_disconnected_nodes_locs_def all_args_def) +end + +locale l_set_child_nodes_get_disconnected_nodes = l_set_child_nodes + l_get_disconnected_nodes + + assumes set_child_nodes_get_disconnected_nodes: + "\w \ set_child_nodes_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_disconnected_nodes_locs ptr'. r h h'))" + +interpretation + i_set_child_nodes_get_disconnected_nodes?: l_set_child_nodes_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf + known_ptr set_child_nodes set_child_nodes_locs + get_disconnected_nodes get_disconnected_nodes_locs + by(unfold_locales) +declare l_set_child_nodes_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_child_nodes_get_disconnected_nodes_is_l_set_child_nodes_get_disconnected_nodes [instances]: + "l_set_child_nodes_get_disconnected_nodes type_wf set_child_nodes set_child_nodes_locs + get_disconnected_nodes get_disconnected_nodes_locs" + using set_child_nodes_is_l_set_child_nodes get_disconnected_nodes_is_l_get_disconnected_nodes + apply(simp add: l_set_child_nodes_get_disconnected_nodes_def + l_set_child_nodes_get_disconnected_nodes_axioms_def) + using set_child_nodes_get_disconnected_nodes + by fast + + +paragraph \set\_attribute\ + +locale l_set_attribute_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin +lemma set_attribute_get_disconnected_nodes: + "\w \ a_set_attribute_locs ptr. (h \ w \\<^sub>h h' \ (\r \ a_get_disconnected_nodes_locs ptr'. r h h'))" + by(auto simp add: a_set_attribute_locs_def a_get_disconnected_nodes_locs_def all_args_def) +end + +locale l_set_attribute_get_disconnected_nodes = l_set_attribute + l_get_disconnected_nodes + + assumes set_attribute_get_disconnected_nodes: + "\w \ set_attribute_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_disconnected_nodes_locs ptr'. r h h'))" + +interpretation + i_set_attribute_get_disconnected_nodes?: l_set_attribute_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf + set_attribute set_attribute_locs get_disconnected_nodes get_disconnected_nodes_locs + by(unfold_locales) +declare l_set_attribute_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_attribute_get_disconnected_nodes_is_l_set_attribute_get_disconnected_nodes [instances]: + "l_set_attribute_get_disconnected_nodes type_wf set_attribute set_attribute_locs + get_disconnected_nodes get_disconnected_nodes_locs" + using set_attribute_is_l_set_attribute get_disconnected_nodes_is_l_get_disconnected_nodes + apply(simp add: l_set_attribute_get_disconnected_nodes_def + l_set_attribute_get_disconnected_nodes_axioms_def) + using set_attribute_get_disconnected_nodes + by fast + + +paragraph \new\_element\ + +locale l_new_element_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs + for type_wf :: "(_) heap \ bool" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" +begin +lemma get_disconnected_nodes_new_element: + "h \ new_element \\<^sub>r new_element_ptr \ h \ new_element \\<^sub>h h' + \ r \ get_disconnected_nodes_locs ptr' \ r h h'" + by(auto simp add: get_disconnected_nodes_locs_def new_element_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t) +end + +locale l_new_element_get_disconnected_nodes = l_get_disconnected_nodes_defs + + assumes get_disconnected_nodes_new_element: + "h \ new_element \\<^sub>r new_element_ptr \ h \ new_element \\<^sub>h h' + \ r \ get_disconnected_nodes_locs ptr' \ r h h'" + +interpretation i_new_element_get_disconnected_nodes?: + l_new_element_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes + get_disconnected_nodes_locs + by unfold_locales +declare l_new_element_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma new_element_get_disconnected_nodes_is_l_new_element_get_disconnected_nodes [instances]: + "l_new_element_get_disconnected_nodes get_disconnected_nodes_locs" + by (simp add: get_disconnected_nodes_new_element l_new_element_get_disconnected_nodes_def) + + +paragraph \new\_character\_data\ + +locale l_new_character_data_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs + for type_wf :: "(_) heap \ bool" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" +begin +lemma get_disconnected_nodes_new_character_data: + "h \ new_character_data \\<^sub>r new_character_data_ptr \ h \ new_character_data \\<^sub>h h' + \ r \ get_disconnected_nodes_locs ptr' \ r h h'" + by(auto simp add: get_disconnected_nodes_locs_def new_character_data_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t) +end + +locale l_new_character_data_get_disconnected_nodes = l_get_disconnected_nodes_defs + + assumes get_disconnected_nodes_new_character_data: + "h \ new_character_data \\<^sub>r new_character_data_ptr \ h \ new_character_data \\<^sub>h h' + \ r \ get_disconnected_nodes_locs ptr' \ r h h'" + +interpretation i_new_character_data_get_disconnected_nodes?: + l_new_character_data_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes + get_disconnected_nodes_locs + by unfold_locales +declare l_new_character_data_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma new_character_data_get_disconnected_nodes_is_l_new_character_data_get_disconnected_nodes [instances]: + "l_new_character_data_get_disconnected_nodes get_disconnected_nodes_locs" + by (simp add: get_disconnected_nodes_new_character_data l_new_character_data_get_disconnected_nodes_def) + + +paragraph \new\_document\ + +locale l_new_document_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs + for type_wf :: "(_) heap \ bool" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" +begin +lemma get_disconnected_nodes_new_document_different_pointers: + "new_document_ptr \ ptr' \ h \ new_document \\<^sub>r new_document_ptr \ h \ new_document \\<^sub>h h' + \ r \ get_disconnected_nodes_locs ptr' \ r h h'" + by(auto simp add: get_disconnected_nodes_locs_def new_document_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t) + +lemma new_document_no_disconnected_nodes: + "h \ new_document \\<^sub>r new_document_ptr \ h \ new_document \\<^sub>h h' + \ h' \ get_disconnected_nodes new_document_ptr \\<^sub>r []" + by(simp add: get_disconnected_nodes_def new_document_disconnected_nodes) + +end + +interpretation i_new_document_get_disconnected_nodes?: + l_new_document_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs + by unfold_locales +declare l_new_document_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +locale l_new_document_get_disconnected_nodes = l_get_disconnected_nodes_defs + + assumes get_disconnected_nodes_new_document_different_pointers: + "new_document_ptr \ ptr' \ h \ new_document \\<^sub>r new_document_ptr \ h \ new_document \\<^sub>h h' + \ r \ get_disconnected_nodes_locs ptr' \ r h h'" + assumes new_document_no_disconnected_nodes: + "h \ new_document \\<^sub>r new_document_ptr \ h \ new_document \\<^sub>h h' + \ h' \ get_disconnected_nodes new_document_ptr \\<^sub>r []" + +lemma new_document_get_disconnected_nodes_is_l_new_document_get_disconnected_nodes [instances]: + "l_new_document_get_disconnected_nodes get_disconnected_nodes get_disconnected_nodes_locs" + apply (auto simp add: l_new_document_get_disconnected_nodes_def)[1] + using get_disconnected_nodes_new_document_different_pointers apply fast + using new_document_no_disconnected_nodes apply blast + done + + + +subsubsection \set\_disconnected\_nodes\ + +locale l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +begin + +definition a_set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ (_, unit) dom_prog" + where + "a_set_disconnected_nodes document_ptr disc_nodes = put_M document_ptr disconnected_nodes_update disc_nodes" +lemmas set_disconnected_nodes_defs = a_set_disconnected_nodes_def + +definition a_set_disconnected_nodes_locs :: "(_) document_ptr \ (_, unit) dom_prog set" + where + "a_set_disconnected_nodes_locs document_ptr \ all_args (put_M document_ptr disconnected_nodes_update)" +end + +locale l_set_disconnected_nodes_defs = + fixes set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ (_, unit) dom_prog" + fixes set_disconnected_nodes_locs :: "(_) document_ptr \ (_, unit) dom_prog set" + +locale l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_type_wf type_wf + + l_set_disconnected_nodes_defs set_disconnected_nodes set_disconnected_nodes_locs + + l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + for type_wf :: "(_) heap \ bool" + and set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ (_, unit) dom_prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ (_, unit) dom_prog set" + + assumes type_wf_impl: "type_wf = DocumentClass.type_wf" + assumes set_disconnected_nodes_impl: "set_disconnected_nodes = a_set_disconnected_nodes" + assumes set_disconnected_nodes_locs_impl: "set_disconnected_nodes_locs = a_set_disconnected_nodes_locs" +begin +lemmas set_disconnected_nodes_def = set_disconnected_nodes_impl[unfolded a_set_disconnected_nodes_def] +lemmas set_disconnected_nodes_locs_def = set_disconnected_nodes_locs_impl[unfolded a_set_disconnected_nodes_locs_def] +lemma set_disconnected_nodes_ok: + "type_wf h \ document_ptr |\| document_ptr_kinds h \ h \ ok (set_disconnected_nodes document_ptr node_ptrs)" + by (simp add: type_wf_impl put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok set_disconnected_nodes_impl[unfolded a_set_disconnected_nodes_def]) + +lemma set_disconnected_nodes_ptr_in_heap: + "h \ ok (set_disconnected_nodes document_ptr disc_nodes) \ document_ptr |\| document_ptr_kinds h" + by (simp add: set_disconnected_nodes_impl[unfolded a_set_disconnected_nodes_def] + DocumentMonad.put_M_ptr_in_heap) + +lemma set_disconnected_nodes_writes: + "writes (set_disconnected_nodes_locs document_ptr) (set_disconnected_nodes document_ptr disc_nodes) h h'" + by(auto simp add: set_disconnected_nodes_impl[unfolded a_set_disconnected_nodes_def] + set_disconnected_nodes_locs_impl[unfolded a_set_disconnected_nodes_locs_def] + intro: writes_bind_pure) + +lemma set_disconnected_nodes_pointers_preserved: + assumes "w \ set_disconnected_nodes_locs object_ptr" + assumes "h \ w \\<^sub>h h'" + shows "object_ptr_kinds h = object_ptr_kinds h'" + using assms(1) object_ptr_kinds_preserved[OF writes_singleton2 assms(2)] + by(auto simp add: all_args_def set_disconnected_nodes_locs_impl[unfolded + a_set_disconnected_nodes_locs_def] + split: if_splits) + +lemma set_disconnected_nodes_typess_preserved: + assumes "w \ set_disconnected_nodes_locs object_ptr" + assumes "h \ w \\<^sub>h h'" + shows "type_wf h = type_wf h'" + using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)] + apply(unfold type_wf_impl) + by(auto simp add: all_args_def + set_disconnected_nodes_locs_impl[unfolded a_set_disconnected_nodes_locs_def] + split: if_splits) +end + +locale l_set_disconnected_nodes = l_type_wf + l_set_disconnected_nodes_defs + + assumes set_disconnected_nodes_writes: + "writes (set_disconnected_nodes_locs document_ptr) (set_disconnected_nodes document_ptr disc_nodes) h h'" + assumes set_disconnected_nodes_ok: + "type_wf h \ document_ptr |\| document_ptr_kinds h \ h \ ok (set_disconnected_nodes document_ptr disc_noded)" + assumes set_disconnected_nodes_ptr_in_heap: + "h \ ok (set_disconnected_nodes document_ptr disc_noded) \ document_ptr |\| document_ptr_kinds h" + assumes set_disconnected_nodes_pointers_preserved: + "w \ set_disconnected_nodes_locs document_ptr \ h \ w \\<^sub>h h' \ object_ptr_kinds h = object_ptr_kinds h'" + assumes set_disconnected_nodes_types_preserved: + "w \ set_disconnected_nodes_locs document_ptr \ h \ w \\<^sub>h h' \ type_wf h = type_wf h'" + +global_interpretation l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs defines + set_disconnected_nodes = l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_disconnected_nodes and + set_disconnected_nodes_locs = l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_disconnected_nodes_locs . +interpretation + i_set_disconnected_nodes?: l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes + set_disconnected_nodes_locs + apply unfold_locales + by (auto simp add: set_disconnected_nodes_def set_disconnected_nodes_locs_def) +declare l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_disconnected_nodes_is_l_set_disconnected_nodes [instances]: + "l_set_disconnected_nodes type_wf set_disconnected_nodes set_disconnected_nodes_locs" + apply(simp add: l_set_disconnected_nodes_def) + using set_disconnected_nodes_ok set_disconnected_nodes_writes set_disconnected_nodes_pointers_preserved + set_disconnected_nodes_ptr_in_heap set_disconnected_nodes_typess_preserved + by blast+ + + +paragraph \get\_disconnected\_nodes\ + +locale l_set_disconnected_nodes_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin +lemma set_disconnected_nodes_get_disconnected_nodes: + assumes "h \ a_set_disconnected_nodes document_ptr disc_nodes \\<^sub>h h'" + shows "h' \ a_get_disconnected_nodes document_ptr \\<^sub>r disc_nodes" + using assms + by(auto simp add: a_get_disconnected_nodes_def a_set_disconnected_nodes_def) + +lemma set_disconnected_nodes_get_disconnected_nodes_different_pointers: + assumes "ptr \ ptr'" + assumes "w \ a_set_disconnected_nodes_locs ptr" + assumes "h \ w \\<^sub>h h'" + assumes "r \ a_get_disconnected_nodes_locs ptr'" + shows "r h h'" + using assms + by(auto simp add: all_args_def a_set_disconnected_nodes_locs_def a_get_disconnected_nodes_locs_def + split: if_splits option.splits ) +end + +locale l_set_disconnected_nodes_get_disconnected_nodes = l_get_disconnected_nodes + + l_set_disconnected_nodes + + assumes set_disconnected_nodes_get_disconnected_nodes: + "h \ set_disconnected_nodes document_ptr disc_nodes \\<^sub>h h' + \ h' \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes" + assumes set_disconnected_nodes_get_disconnected_nodes_different_pointers: + "ptr \ ptr' \ w \ set_disconnected_nodes_locs ptr \ h \ w \\<^sub>h h' + \ r \ get_disconnected_nodes_locs ptr' \ r h h'" + +interpretation i_set_disconnected_nodes_get_disconnected_nodes?: + l_set_disconnected_nodes_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes + get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs + by unfold_locales +declare l_set_disconnected_nodes_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_disconnected_nodes_get_disconnected_nodes_is_l_set_disconnected_nodes_get_disconnected_nodes [instances]: + "l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs + set_disconnected_nodes set_disconnected_nodes_locs" + using set_disconnected_nodes_is_l_set_disconnected_nodes get_disconnected_nodes_is_l_get_disconnected_nodes + apply(simp add: l_set_disconnected_nodes_get_disconnected_nodes_def + l_set_disconnected_nodes_get_disconnected_nodes_axioms_def) + using set_disconnected_nodes_get_disconnected_nodes + set_disconnected_nodes_get_disconnected_nodes_different_pointers + by fast+ + + +paragraph \get\_child\_nodes\ + +locale l_set_disconnected_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin +lemma set_disconnected_nodes_get_child_nodes: + "\w \ set_disconnected_nodes_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_child_nodes_locs ptr'. r h h'))" + by(auto simp add: set_disconnected_nodes_locs_impl[unfolded a_set_disconnected_nodes_locs_def] + get_child_nodes_locs_impl[unfolded a_get_child_nodes_locs_def] all_args_def) +end + +locale l_set_disconnected_nodes_get_child_nodes = l_set_disconnected_nodes_defs + l_get_child_nodes_defs + + assumes set_disconnected_nodes_get_child_nodes [simp]: + "\w \ set_disconnected_nodes_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_child_nodes_locs ptr'. r h h'))" + +interpretation + i_set_disconnected_nodes_get_child_nodes?: l_set_disconnected_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + type_wf + set_disconnected_nodes set_disconnected_nodes_locs + known_ptr get_child_nodes get_child_nodes_locs + by unfold_locales +declare l_set_disconnected_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_disconnected_nodes_get_child_nodes_is_l_set_disconnected_nodes_get_child_nodes [instances]: + "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs" + using set_disconnected_nodes_is_l_set_disconnected_nodes get_child_nodes_is_l_get_child_nodes + apply(simp add: l_set_disconnected_nodes_get_child_nodes_def) + using set_disconnected_nodes_get_child_nodes + by fast + + +subsubsection \get\_tag\_name\ + +locale l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +begin +definition a_get_tag_name :: "(_) element_ptr \ (_, tag_type) dom_prog" + where + "a_get_tag_name element_ptr = get_M element_ptr tag_type" + +definition a_get_tag_name_locs :: "(_) element_ptr \ ((_) heap \ (_) heap \ bool) set" + where + "a_get_tag_name_locs element_ptr \ {preserved (get_M element_ptr tag_type)}" +end + +locale l_get_tag_name_defs = + fixes get_tag_name :: "(_) element_ptr \ (_, tag_type) dom_prog" + fixes get_tag_name_locs :: "(_) element_ptr \ ((_) heap \ (_) heap \ bool) set" + +locale l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_type_wf type_wf + + l_get_tag_name_defs get_tag_name get_tag_name_locs + + l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + for type_wf :: "(_) heap \ bool" + and get_tag_name :: "(_) element_ptr \ (_, tag_type) dom_prog" + and get_tag_name_locs :: "(_) element_ptr \ ((_) heap \ (_) heap \ bool) set" + + assumes type_wf_impl: "type_wf = DocumentClass.type_wf" + assumes get_tag_name_impl: "get_tag_name = a_get_tag_name" + assumes get_tag_name_locs_impl: "get_tag_name_locs = a_get_tag_name_locs" +begin +lemmas get_tag_name_def = get_tag_name_impl[unfolded a_get_tag_name_def] +lemmas get_tag_name_locs_def = get_tag_name_locs_impl[unfolded a_get_tag_name_locs_def] + + + +lemma get_tag_name_ok: + "type_wf h \ element_ptr |\| element_ptr_kinds h \ h \ ok (get_tag_name element_ptr)" + apply(unfold type_wf_impl get_tag_name_impl[unfolded a_get_tag_name_def]) + using get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok + by blast + +lemma get_tag_name_pure [simp]: "pure (get_tag_name element_ptr) h" + unfolding get_tag_name_impl[unfolded a_get_tag_name_def] + by simp + +lemma get_tag_name_ptr_in_heap [simp]: + assumes "h \ get_tag_name element_ptr \\<^sub>r children" + shows "element_ptr |\| element_ptr_kinds h" + using assms + by(auto simp add: get_tag_name_impl[unfolded a_get_tag_name_def] get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap + dest: is_OK_returns_result_I) + +lemma get_tag_name_reads: "reads (get_tag_name_locs element_ptr) (get_tag_name element_ptr) h h'" + by(simp add: get_tag_name_impl[unfolded a_get_tag_name_def] + get_tag_name_locs_impl[unfolded a_get_tag_name_locs_def] reads_bind_pure + reads_insert_writes_set_right) +end + +locale l_get_tag_name = l_type_wf + l_get_tag_name_defs + + assumes get_tag_name_reads: + "reads (get_tag_name_locs element_ptr) (get_tag_name element_ptr) h h'" + assumes get_tag_name_ok: + "type_wf h \ element_ptr |\| element_ptr_kinds h \ h \ ok (get_tag_name element_ptr)" + assumes get_tag_name_ptr_in_heap: + "h \ ok (get_tag_name element_ptr) \ element_ptr |\| element_ptr_kinds h" + assumes get_tag_name_pure [simp]: + "pure (get_tag_name element_ptr) h" + + +global_interpretation l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs defines + get_tag_name = l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_tag_name and + get_tag_name_locs = l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_tag_name_locs . + +interpretation + i_get_tag_name?: l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs + apply(unfold_locales) + by (auto simp add: get_tag_name_def get_tag_name_locs_def) +declare l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma get_tag_name_is_l_get_tag_name [instances]: + "l_get_tag_name type_wf get_tag_name get_tag_name_locs" + apply(unfold_locales) + using get_tag_name_reads get_tag_name_ok get_tag_name_ptr_in_heap get_tag_name_pure + by blast+ + + +paragraph \set\_disconnected\_nodes\ + +locale l_set_disconnected_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin +lemma set_disconnected_nodes_get_tag_name: + "\w \ a_set_disconnected_nodes_locs ptr. (h \ w \\<^sub>h h' \ (\r \ a_get_tag_name_locs ptr'. r h h'))" + by(auto simp add: a_set_disconnected_nodes_locs_def a_get_tag_name_locs_def all_args_def) +end + +locale l_set_disconnected_nodes_get_tag_name = l_set_disconnected_nodes + l_get_tag_name + + assumes set_disconnected_nodes_get_tag_name: + "\w \ set_disconnected_nodes_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_tag_name_locs ptr'. r h h'))" + +interpretation + i_set_disconnected_nodes_get_tag_name?: l_set_disconnected_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf + set_disconnected_nodes set_disconnected_nodes_locs + get_tag_name get_tag_name_locs + by unfold_locales +declare l_set_disconnected_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_disconnected_nodes_get_tag_name_is_l_set_disconnected_nodes_get_tag_name [instances]: + "l_set_disconnected_nodes_get_tag_name type_wf set_disconnected_nodes set_disconnected_nodes_locs + get_tag_name get_tag_name_locs" + using set_disconnected_nodes_is_l_set_disconnected_nodes get_tag_name_is_l_get_tag_name + apply(simp add: l_set_disconnected_nodes_get_tag_name_def l_set_disconnected_nodes_get_tag_name_axioms_def) + using set_disconnected_nodes_get_tag_name + by fast + + +paragraph \set\_child\_nodes\ + +locale l_set_child_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin +lemma set_child_nodes_get_tag_name: + "\w \ set_child_nodes_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_tag_name_locs ptr'. r h h'))" + by(auto simp add: set_child_nodes_locs_def get_tag_name_locs_def all_args_def + intro: element_put_get_preserved[where getter=tag_type and setter=child_nodes_update]) +end + +locale l_set_child_nodes_get_tag_name = l_set_child_nodes + l_get_tag_name + + assumes set_child_nodes_get_tag_name: + "\w \ set_child_nodes_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_tag_name_locs ptr'. r h h'))" + +interpretation + i_set_child_nodes_get_tag_name?: l_set_child_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr + set_child_nodes set_child_nodes_locs get_tag_name get_tag_name_locs + by unfold_locales +declare l_set_child_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_child_nodes_get_tag_name_is_l_set_child_nodes_get_tag_name [instances]: + "l_set_child_nodes_get_tag_name type_wf set_child_nodes set_child_nodes_locs get_tag_name get_tag_name_locs" + using set_child_nodes_is_l_set_child_nodes get_tag_name_is_l_get_tag_name + apply(simp add: l_set_child_nodes_get_tag_name_def l_set_child_nodes_get_tag_name_axioms_def) + using set_child_nodes_get_tag_name + by fast + + +subsubsection \set\_tag\_type\ + +locale l_set_tag_type\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +begin + +definition a_set_tag_type :: "(_) element_ptr \ tag_type \ (_, unit) dom_prog" + where + "a_set_tag_type ptr tag = do { + m \ get_M ptr attrs; + put_M ptr tag_type_update tag + }" +lemmas set_tag_type_defs = a_set_tag_type_def + +definition a_set_tag_type_locs :: "(_) element_ptr \ (_, unit) dom_prog set" + where + "a_set_tag_type_locs element_ptr \ all_args (put_M element_ptr tag_type_update)" +end + +locale l_set_tag_type_defs = + fixes set_tag_type :: "(_) element_ptr \ tag_type \ (_, unit) dom_prog" + fixes set_tag_type_locs :: "(_) element_ptr \ (_, unit) dom_prog set" + +locale l_set_tag_type\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_type_wf type_wf + + l_set_tag_type_defs set_tag_type set_tag_type_locs + + l_set_tag_type\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + for type_wf :: "(_) heap \ bool" + and set_tag_type :: "(_) element_ptr \ char list \ (_, unit) dom_prog" + and set_tag_type_locs :: "(_) element_ptr \ (_, unit) dom_prog set" + + assumes type_wf_impl: "type_wf = DocumentClass.type_wf" + assumes set_tag_type_impl: "set_tag_type = a_set_tag_type" + assumes set_tag_type_locs_impl: "set_tag_type_locs = a_set_tag_type_locs" +begin + +lemma set_tag_type_ok: + "type_wf h \ element_ptr |\| element_ptr_kinds h \ h \ ok (set_tag_type element_ptr tag)" + apply(unfold type_wf_impl) + unfolding set_tag_type_impl[unfolded a_set_tag_type_def] using get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok + by (metis (no_types, lifting) DocumentClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ElementMonad.get_M_pure bind_is_OK_E + bind_is_OK_pure_I is_OK_returns_result_I) + +lemma set_tag_type_writes: + "writes (set_tag_type_locs element_ptr) (set_tag_type element_ptr tag) h h'" + by(auto simp add: set_tag_type_impl[unfolded a_set_tag_type_def] + set_tag_type_locs_impl[unfolded a_set_tag_type_locs_def] intro: writes_bind_pure) + +lemma set_tag_type_pointers_preserved: + assumes "w \ set_tag_type_locs element_ptr" + assumes "h \ w \\<^sub>h h'" + shows "object_ptr_kinds h = object_ptr_kinds h'" + using assms(1) object_ptr_kinds_preserved[OF writes_singleton2 assms(2)] + by(auto simp add: all_args_def set_tag_type_locs_impl[unfolded a_set_tag_type_locs_def] + split: if_splits) + +lemma set_tag_type_typess_preserved: + assumes "w \ set_tag_type_locs element_ptr" + assumes "h \ w \\<^sub>h h'" + shows "type_wf h = type_wf h'" + apply(unfold type_wf_impl) + using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)] + by(auto simp add: all_args_def set_tag_type_locs_impl[unfolded a_set_tag_type_locs_def] + split: if_splits) +end + +locale l_set_tag_type = l_type_wf + l_set_tag_type_defs + + assumes set_tag_type_writes: + "writes (set_tag_type_locs element_ptr) (set_tag_type element_ptr tag) h h'" + assumes set_tag_type_ok: + "type_wf h \ element_ptr |\| element_ptr_kinds h \ h \ ok (set_tag_type element_ptr tag)" + assumes set_tag_type_pointers_preserved: + "w \ set_tag_type_locs element_ptr \ h \ w \\<^sub>h h' \ object_ptr_kinds h = object_ptr_kinds h'" + assumes set_tag_type_types_preserved: + "w \ set_tag_type_locs element_ptr \ h \ w \\<^sub>h h' \ type_wf h = type_wf h'" + + +global_interpretation l_set_tag_type\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs defines + set_tag_type = l_set_tag_type\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_tag_type and + set_tag_type_locs = l_set_tag_type\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_tag_type_locs . +interpretation + i_set_tag_type?: l_set_tag_type\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_type set_tag_type_locs + apply(unfold_locales) + by (auto simp add: set_tag_type_def set_tag_type_locs_def) +declare l_set_tag_type\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_tag_type_is_l_set_tag_type [instances]: + "l_set_tag_type type_wf set_tag_type set_tag_type_locs" + apply(simp add: l_set_tag_type_def) + using set_tag_type_ok set_tag_type_writes set_tag_type_pointers_preserved + set_tag_type_typess_preserved + by blast + +paragraph \get\_child\_nodes\ + +locale l_set_tag_type_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_tag_type\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin +lemma set_tag_type_get_child_nodes: + "\w \ set_tag_type_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_child_nodes_locs ptr'. r h h'))" + by(auto simp add: set_tag_type_locs_impl[unfolded a_set_tag_type_locs_def] + get_child_nodes_locs_impl[unfolded a_get_child_nodes_locs_def] all_args_def + intro: element_put_get_preserved[where setter=tag_type_update and getter=child_nodes]) +end + +locale l_set_tag_type_get_child_nodes = l_set_tag_type + l_get_child_nodes + + assumes set_tag_type_get_child_nodes: + "\w \ set_tag_type_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_child_nodes_locs ptr'. r h h'))" + +interpretation + i_set_tag_type_get_child_nodes?: l_set_tag_type_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf + set_tag_type set_tag_type_locs known_ptr + get_child_nodes get_child_nodes_locs + by unfold_locales +declare l_set_tag_type_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_tag_type_get_child_nodes_is_l_set_tag_type_get_child_nodes [instances]: + "l_set_tag_type_get_child_nodes type_wf set_tag_type set_tag_type_locs known_ptr get_child_nodes + get_child_nodes_locs" + using set_tag_type_is_l_set_tag_type get_child_nodes_is_l_get_child_nodes + apply(simp add: l_set_tag_type_get_child_nodes_def l_set_tag_type_get_child_nodes_axioms_def) + using set_tag_type_get_child_nodes + by fast + + +paragraph \get\_disconnected\_nodes\ + +locale l_set_tag_type_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_tag_type\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin +lemma set_tag_type_get_disconnected_nodes: + "\w \ set_tag_type_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_disconnected_nodes_locs ptr'. r h h'))" + by(auto simp add: set_tag_type_locs_impl[unfolded a_set_tag_type_locs_def] + get_disconnected_nodes_locs_impl[unfolded a_get_disconnected_nodes_locs_def] + all_args_def) +end + +locale l_set_tag_type_get_disconnected_nodes = l_set_tag_type + l_get_disconnected_nodes + + assumes set_tag_type_get_disconnected_nodes: + "\w \ set_tag_type_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_disconnected_nodes_locs ptr'. r h h'))" + +interpretation + i_set_tag_type_get_disconnected_nodes?: l_set_tag_type_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf + set_tag_type set_tag_type_locs get_disconnected_nodes + get_disconnected_nodes_locs + by unfold_locales +declare l_set_tag_type_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_tag_type_get_disconnected_nodes_is_l_set_tag_type_get_disconnected_nodes [instances]: + "l_set_tag_type_get_disconnected_nodes type_wf set_tag_type set_tag_type_locs get_disconnected_nodes + get_disconnected_nodes_locs" + using set_tag_type_is_l_set_tag_type get_disconnected_nodes_is_l_get_disconnected_nodes + apply(simp add: l_set_tag_type_get_disconnected_nodes_def + l_set_tag_type_get_disconnected_nodes_axioms_def) + using set_tag_type_get_disconnected_nodes + by fast + + +subsubsection \set\_val\ + +locale l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +begin + +definition a_set_val :: "(_) character_data_ptr \ DOMString \ (_, unit) dom_prog" + where + "a_set_val ptr v = do { + m \ get_M ptr val; + put_M ptr val_update v + }" +lemmas set_val_defs = a_set_val_def + +definition a_set_val_locs :: "(_) character_data_ptr \ (_, unit) dom_prog set" + where + "a_set_val_locs character_data_ptr \ all_args (put_M character_data_ptr val_update)" +end + +locale l_set_val_defs = + fixes set_val :: "(_) character_data_ptr \ DOMString \ (_, unit) dom_prog" + fixes set_val_locs :: "(_) character_data_ptr \ (_, unit) dom_prog set" + +locale l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_type_wf type_wf + + l_set_val_defs set_val set_val_locs + + l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + for type_wf :: "(_) heap \ bool" + and set_val :: "(_) character_data_ptr \ char list \ (_, unit) dom_prog" + and set_val_locs :: "(_) character_data_ptr \ (_, unit) dom_prog set" + + assumes type_wf_impl: "type_wf = DocumentClass.type_wf" + assumes set_val_impl: "set_val = a_set_val" + assumes set_val_locs_impl: "set_val_locs = a_set_val_locs" +begin + +lemma set_val_ok: + "type_wf h \ character_data_ptr |\| character_data_ptr_kinds h \ h \ ok (set_val character_data_ptr tag)" + apply(unfold type_wf_impl) + unfolding set_val_impl[unfolded a_set_val_def] using get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ok put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ok + by (metis (no_types, lifting) DocumentClass.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a CharacterDataMonad.get_M_pure + bind_is_OK_E bind_is_OK_pure_I is_OK_returns_result_I) + +lemma set_val_writes: "writes (set_val_locs character_data_ptr) (set_val character_data_ptr tag) h h'" + by(auto simp add: set_val_impl[unfolded a_set_val_def] set_val_locs_impl[unfolded a_set_val_locs_def] + intro: writes_bind_pure) + +lemma set_val_pointers_preserved: + assumes "w \ set_val_locs character_data_ptr" + assumes "h \ w \\<^sub>h h'" + shows "object_ptr_kinds h = object_ptr_kinds h'" + using assms(1) object_ptr_kinds_preserved[OF writes_singleton2 assms(2)] + by(auto simp add: all_args_def set_val_locs_impl[unfolded a_set_val_locs_def] split: if_splits) + +lemma set_val_typess_preserved: + assumes "w \ set_val_locs character_data_ptr" + assumes "h \ w \\<^sub>h h'" + shows "type_wf h = type_wf h'" + apply(unfold type_wf_impl) + using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)] + by(auto simp add: all_args_def set_val_locs_impl[unfolded a_set_val_locs_def] split: if_splits) +end + +locale l_set_val = l_type_wf + l_set_val_defs + + assumes set_val_writes: + "writes (set_val_locs character_data_ptr) (set_val character_data_ptr tag) h h'" + assumes set_val_ok: + "type_wf h \ character_data_ptr |\| character_data_ptr_kinds h \ h \ ok (set_val character_data_ptr tag)" + assumes set_val_pointers_preserved: + "w \ set_val_locs character_data_ptr \ h \ w \\<^sub>h h' \ object_ptr_kinds h = object_ptr_kinds h'" + assumes set_val_types_preserved: + "w \ set_val_locs character_data_ptr \ h \ w \\<^sub>h h' \ type_wf h = type_wf h'" + + +global_interpretation l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs defines + set_val = l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_val and + set_val_locs = l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_val_locs . +interpretation + i_set_val?: l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_val set_val_locs + apply(unfold_locales) + by (auto simp add: set_val_def set_val_locs_def) +declare l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_val_is_l_set_val [instances]: "l_set_val type_wf set_val set_val_locs" + apply(simp add: l_set_val_def) + using set_val_ok set_val_writes set_val_pointers_preserved set_val_typess_preserved + by blast + +paragraph \get\_child\_nodes\ + +locale l_set_val_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin +lemma set_val_get_child_nodes: + "\w \ set_val_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_child_nodes_locs ptr'. r h h'))" + by(auto simp add: set_val_locs_impl[unfolded a_set_val_locs_def] + get_child_nodes_locs_impl[unfolded a_get_child_nodes_locs_def] all_args_def) +end + +locale l_set_val_get_child_nodes = l_set_val + l_get_child_nodes + + assumes set_val_get_child_nodes: + "\w \ set_val_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_child_nodes_locs ptr'. r h h'))" + +interpretation + i_set_val_get_child_nodes?: l_set_val_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_val set_val_locs known_ptr + get_child_nodes get_child_nodes_locs + by unfold_locales +declare l_set_val_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_val_get_child_nodes_is_l_set_val_get_child_nodes [instances]: + "l_set_val_get_child_nodes type_wf set_val set_val_locs known_ptr get_child_nodes get_child_nodes_locs" + using set_val_is_l_set_val get_child_nodes_is_l_get_child_nodes + apply(simp add: l_set_val_get_child_nodes_def l_set_val_get_child_nodes_axioms_def) + using set_val_get_child_nodes + by fast + + +paragraph \get\_disconnected\_nodes\ + +locale l_set_val_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin +lemma set_val_get_disconnected_nodes: + "\w \ set_val_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_disconnected_nodes_locs ptr'. r h h'))" + by(auto simp add: set_val_locs_impl[unfolded a_set_val_locs_def] + get_disconnected_nodes_locs_impl[unfolded a_get_disconnected_nodes_locs_def] + all_args_def) +end + +locale l_set_val_get_disconnected_nodes = l_set_val + l_get_disconnected_nodes + + assumes set_val_get_disconnected_nodes: + "\w \ set_val_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_disconnected_nodes_locs ptr'. r h h'))" + +interpretation + i_set_val_get_disconnected_nodes?: l_set_val_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_val + set_val_locs get_disconnected_nodes get_disconnected_nodes_locs + by unfold_locales +declare l_set_val_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_val_get_disconnected_nodes_is_l_set_val_get_disconnected_nodes [instances]: + "l_set_val_get_disconnected_nodes type_wf set_val set_val_locs get_disconnected_nodes get_disconnected_nodes_locs" + using set_val_is_l_set_val get_disconnected_nodes_is_l_get_disconnected_nodes + apply(simp add: l_set_val_get_disconnected_nodes_def l_set_val_get_disconnected_nodes_axioms_def) + using set_val_get_disconnected_nodes + by fast + + + +subsubsection \get\_parent\ + +locale l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_get_child_nodes_defs get_child_nodes get_child_nodes_locs + for get_child_nodes :: "(_::linorder) object_ptr \ (_, (_) node_ptr list) dom_prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" +begin +definition a_get_parent :: "(_) node_ptr \ (_, (_::linorder) object_ptr option) dom_prog" + where + "a_get_parent node_ptr = do { + check_in_heap (cast node_ptr); + parent_ptrs \ object_ptr_kinds_M \ filter_M (\ptr. do { + children \ get_child_nodes ptr; + return (node_ptr \ set children) + }); + (if parent_ptrs = [] + then return None + else return (Some (hd parent_ptrs))) + }" + +definition + "a_get_parent_locs \ (\ptr. get_child_nodes_locs ptr \ {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing)})" +end + +locale l_get_parent_defs = + fixes get_parent :: "(_) node_ptr \ (_, (_::linorder) object_ptr option) dom_prog" + fixes get_parent_locs :: "((_) heap \ (_) heap \ bool) set" + +locale l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs + + l_known_ptrs known_ptr known_ptrs + + l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_child_nodes get_child_nodes_locs + + l_get_parent_defs get_parent get_parent_locs + for known_ptr :: "(_::linorder) object_ptr \ bool" + and type_wf :: "(_) heap \ bool" + and get_child_nodes (* :: "(_) object_ptr \ (_, (_) node_ptr list) dom_prog" *) + and get_child_nodes_locs (* :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" *) + and known_ptrs :: "(_) heap \ bool" + and get_parent :: "(_) node_ptr \ ((_) heap, exception, (_) object_ptr option) prog" + and get_parent_locs (* :: "((_) heap \ (_) heap \ bool) set" *) + + assumes get_parent_impl: "get_parent = a_get_parent" + assumes get_parent_locs_impl: "get_parent_locs = a_get_parent_locs" +begin +lemmas get_parent_def = get_parent_impl[unfolded a_get_parent_def] +lemmas get_parent_locs_def = get_parent_locs_impl[unfolded a_get_parent_locs_def] + +lemma get_parent_pure [simp]: "pure (get_parent ptr) h" + using get_child_nodes_pure + by(auto simp add: get_parent_def intro!: bind_pure_I filter_M_pure_I) + +lemma get_parent_ok [simp]: + assumes "type_wf h" + assumes "known_ptrs h" + assumes "ptr |\| node_ptr_kinds h" + shows "h \ ok (get_parent ptr)" + using assms get_child_nodes_ok get_child_nodes_pure + by(auto simp add: get_parent_impl[unfolded a_get_parent_def] known_ptrs_known_ptr + intro!: bind_is_OK_pure_I filter_M_pure_I filter_M_is_OK_I bind_pure_I) + +lemma get_parent_ptr_in_heap [simp]: "h \ ok (get_parent node_ptr) \ node_ptr |\| node_ptr_kinds h" + using get_parent_def is_OK_returns_result_I check_in_heap_ptr_in_heap + by (metis (no_types, lifting) bind_returns_heap_E get_parent_pure node_ptr_kinds_commutes pure_pure) + +lemma get_parent_parent_in_heap: + assumes "h \ get_parent child_node \\<^sub>r Some parent" + shows "parent |\| object_ptr_kinds h" + using assms get_child_nodes_pure + by(auto simp add: get_parent_def elim!: bind_returns_result_E2 + dest!: filter_M_not_more_elements[where x=parent] + intro!: filter_M_pure_I bind_pure_I + split: if_splits) + +lemma get_parent_child_dual: + assumes "h \ get_parent child \\<^sub>r Some ptr" + obtains children where "h \ get_child_nodes ptr \\<^sub>r children" and "child \ set children" + using assms get_child_nodes_pure + by(auto simp add: get_parent_def bind_pure_I + dest!: filter_M_holds_for_result + elim!: bind_returns_result_E2 + intro!: filter_M_pure_I + split: if_splits) + +lemma get_parent_reads: "reads get_parent_locs (get_parent node_ptr) h h'" + using get_child_nodes_reads[unfolded reads_def] + by(auto simp add: get_parent_def get_parent_locs_def + intro!: reads_bind_pure reads_subset[OF check_in_heap_reads] + reads_subset[OF get_child_nodes_reads] reads_subset[OF return_reads] + reads_subset[OF object_ptr_kinds_M_reads] filter_M_reads filter_M_pure_I bind_pure_I) + +lemma get_parent_reads_pointers: "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing) \ get_parent_locs" + by(auto simp add: get_parent_locs_def) +end + +locale l_get_parent = l_type_wf + l_known_ptrs + l_get_parent_defs + l_get_child_nodes + + assumes get_parent_reads: + "reads get_parent_locs (get_parent node_ptr) h h'" + assumes get_parent_ok: + "type_wf h \ known_ptrs h \ node_ptr |\| node_ptr_kinds h \ h \ ok (get_parent node_ptr)" + assumes get_parent_ptr_in_heap: + "h \ ok (get_parent node_ptr) \ node_ptr |\| node_ptr_kinds h" + assumes get_parent_pure [simp]: + "pure (get_parent node_ptr) h" + assumes get_parent_parent_in_heap: + "h \ get_parent child_node \\<^sub>r Some parent \ parent |\| object_ptr_kinds h" + assumes get_parent_child_dual: + "h \ get_parent child \\<^sub>r Some ptr \ (\children. h \ get_child_nodes ptr \\<^sub>r children + \ child \ set children \ thesis) \ thesis" + assumes get_parent_reads_pointers: + "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing) \ get_parent_locs" + +global_interpretation l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_child_nodes get_child_nodes_locs defines + get_parent = "l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_parent get_child_nodes" and + get_parent_locs = "l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_parent_locs get_child_nodes_locs" . + +interpretation + i_get_parent?: l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs + get_parent get_parent_locs + using instances + apply(simp add: l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def) + apply(simp add: get_parent_def get_parent_locs_def) + done +declare l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma get_parent_is_l_get_parent [instances]: + "l_get_parent type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs" + using instances + apply(auto simp add: l_get_parent_def l_get_parent_axioms_def)[1] + using get_parent_reads get_parent_ok get_parent_ptr_in_heap get_parent_pure + get_parent_parent_in_heap get_parent_child_dual + using get_parent_reads_pointers + by blast+ + + +paragraph \set\_disconnected\_nodes\ + +locale l_set_disconnected_nodes_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_disconnected_nodes_get_child_nodes + set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs + + l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + type_wf set_disconnected_nodes set_disconnected_nodes_locs + + l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs + for known_ptr :: "(_::linorder) object_ptr \ bool" + and type_wf :: "(_) heap \ bool" + and set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap, exception, unit) prog set" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and known_ptrs :: "(_) heap \ bool" + and get_parent :: "(_) node_ptr \ ((_) heap, exception, (_) object_ptr option) prog" + and get_parent_locs :: "((_) heap \ (_) heap \ bool) set" +begin +lemma set_disconnected_nodes_get_parent [simp]: + "\w \ set_disconnected_nodes_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_parent_locs. r h h'))" + by(auto simp add: get_parent_locs_def set_disconnected_nodes_locs_def all_args_def) +end + +locale l_set_disconnected_nodes_get_parent = l_set_disconnected_nodes_defs + l_get_parent_defs + + assumes set_disconnected_nodes_get_parent [simp]: + "\w \ set_disconnected_nodes_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_parent_locs. r h h'))" + +interpretation i_set_disconnected_nodes_get_parent?: + l_set_disconnected_nodes_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf set_disconnected_nodes + set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs + using instances + by (simp add: l_set_disconnected_nodes_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) +declare l_set_disconnected_nodes_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_disconnected_nodes_get_parent_is_l_set_disconnected_nodes_get_parent [instances]: + "l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs get_parent_locs" + by(simp add: l_set_disconnected_nodes_get_parent_def) + + + +subsubsection \get\_root\_node\ + +locale l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_get_parent_defs get_parent get_parent_locs + for get_parent :: "(_) node_ptr \ ((_) heap, exception, (_::linorder) object_ptr option) prog" + and get_parent_locs :: "((_) heap \ (_) heap \ bool) set" +begin +partial_function (dom_prog) + a_get_ancestors :: "(_::linorder) object_ptr \ (_, (_) object_ptr list) dom_prog" + where + "a_get_ancestors ptr = do { + check_in_heap ptr; + ancestors \ (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of + Some node_ptr \ do { + parent_ptr_opt \ get_parent node_ptr; + (case parent_ptr_opt of + Some parent_ptr \ a_get_ancestors parent_ptr + | None \ return []) + } + | None \ return []); + return (ptr # ancestors) + }" + +definition "a_get_ancestors_locs = get_parent_locs" + +definition a_get_root_node :: "(_) object_ptr \ (_, (_) object_ptr) dom_prog" + where + "a_get_root_node ptr = do { + ancestors \ a_get_ancestors ptr; + return (last ancestors) + }" +definition "a_get_root_node_locs = a_get_ancestors_locs" +end + +locale l_get_ancestors_defs = + fixes get_ancestors :: "(_::linorder) object_ptr \ (_, (_) object_ptr list) dom_prog" + fixes get_ancestors_locs :: "((_) heap \ (_) heap \ bool) set" + +locale l_get_root_node_defs = + fixes get_root_node :: "(_) object_ptr \ (_, (_) object_ptr) dom_prog" + fixes get_root_node_locs :: "((_) heap \ (_) heap \ bool) set" + +locale l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_parent + + l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + + l_get_ancestors_defs + + l_get_root_node_defs + + assumes get_ancestors_impl: "get_ancestors = a_get_ancestors" + assumes get_ancestors_locs_impl: "get_ancestors_locs = a_get_ancestors_locs" + assumes get_root_node_impl: "get_root_node = a_get_root_node" + assumes get_root_node_locs_impl: "get_root_node_locs = a_get_root_node_locs" +begin +lemmas get_ancestors_def = a_get_ancestors.simps[folded get_ancestors_impl] +lemmas get_ancestors_locs_def = a_get_ancestors_locs_def[folded get_ancestors_locs_impl] +lemmas get_root_node_def = a_get_root_node_def[folded get_root_node_impl get_ancestors_impl] +lemmas get_root_node_locs_def = a_get_root_node_locs_def[folded get_root_node_locs_impl + get_ancestors_locs_impl] + +lemma get_ancestors_pure [simp]: + "pure (get_ancestors ptr) h" +proof - + have "\ptr h h' x. h \ get_ancestors ptr \\<^sub>r x \ h \ get_ancestors ptr \\<^sub>h h' \ h = h'" + proof (induct rule: a_get_ancestors.fixp_induct[folded get_ancestors_impl]) + case 1 + then show ?case + by(rule admissible_dom_prog) + next + case 2 + then show ?case + by simp + next + case (3 f) + then show ?case + using get_parent_pure + apply(auto simp add: pure_returns_heap_eq pure_def + split: option.splits + elim!: bind_returns_heap_E bind_returns_result_E + dest!: pure_returns_heap_eq[rotated, OF check_in_heap_pure])[1] + apply (meson option.simps(3) returns_result_eq) + by (metis get_parent_pure pure_returns_heap_eq) + qed + then show ?thesis + by (meson pure_eq_iff) +qed + + +lemma get_root_node_pure [simp]: "pure (get_root_node ptr) h" + by(auto simp add: get_root_node_def bind_pure_I) + + +lemma get_ancestors_ptr_in_heap: + assumes "h \ ok (get_ancestors ptr)" + shows "ptr |\| object_ptr_kinds h" + using assms + by(auto simp add: get_ancestors_def check_in_heap_ptr_in_heap + elim!: bind_is_OK_E dest: is_OK_returns_result_I) + +lemma get_ancestors_ptr: + assumes "h \ get_ancestors ptr \\<^sub>r ancestors" + shows "ptr \ set ancestors" + using assms + apply(simp add: get_ancestors_def) + by(auto elim!: bind_returns_result_E2 split: option.splits intro!: bind_pure_I) + +lemma get_ancestors_not_node: + assumes "h \ get_ancestors ptr \\<^sub>r ancestors" + assumes "\is_node_ptr_kind ptr" + shows "ancestors = [ptr]" + using assms + apply(simp add: get_ancestors_def) + by(auto elim!: bind_returns_result_E2 split: option.splits) + +lemma get_root_node_no_parent: + "h \ get_parent node_ptr \\<^sub>r None \ h \ get_root_node (cast node_ptr) \\<^sub>r cast node_ptr" + apply(auto simp add: check_in_heap_def get_root_node_def get_ancestors_def + intro!: bind_pure_returns_result_I )[1] + using get_parent_ptr_in_heap by blast + +end + +locale l_get_ancestors = l_get_ancestors_defs + + assumes get_ancestors_pure [simp]: "pure (get_ancestors node_ptr) h" + assumes get_ancestors_ptr_in_heap: "h \ ok (get_ancestors ptr) \ ptr |\| object_ptr_kinds h" + assumes get_ancestors_ptr: "h \ get_ancestors ptr \\<^sub>r ancestors \ ptr \ set ancestors" + +locale l_get_root_node = l_get_root_node_defs + l_get_parent_defs + + assumes get_root_node_pure[simp]: + "pure (get_root_node ptr) h" + assumes get_root_node_no_parent: + "h \ get_parent node_ptr \\<^sub>r None \ h \ get_root_node (cast node_ptr) \\<^sub>r cast node_ptr" + +global_interpretation l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_parent get_parent_locs + defines get_root_node = "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node get_parent" + and get_root_node_locs = "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node_locs get_parent_locs" + and get_ancestors = "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors get_parent" + and get_ancestors_locs = "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_locs get_parent_locs" + . +declare a_get_ancestors.simps [code] + +interpretation + i_get_root_node?: l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs + get_child_nodes get_child_nodes_locs get_ancestors get_ancestors_locs + get_root_node get_root_node_locs + using instances + apply(simp add: l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def) + by(simp add: get_root_node_def get_root_node_locs_def get_ancestors_def get_ancestors_locs_def) +declare l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma get_ancestors_is_l_get_ancestors [instances]: "l_get_ancestors get_ancestors" + unfolding l_get_ancestors_def + using get_ancestors_pure get_ancestors_ptr get_ancestors_ptr_in_heap + by blast + +lemma get_root_node_is_l_get_root_node [instances]: "l_get_root_node get_root_node get_parent" + apply(simp add: l_get_root_node_def) + using get_root_node_no_parent + by fast + + +paragraph \set\_disconnected\_nodes\ + +locale l_set_disconnected_nodes_get_ancestors\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_disconnected_nodes_get_parent + set_disconnected_nodes set_disconnected_nodes_locs get_parent get_parent_locs + + l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs + get_ancestors get_ancestors_locs get_root_node get_root_node_locs + + l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + type_wf set_disconnected_nodes set_disconnected_nodes_locs + for known_ptr :: "(_::linorder) object_ptr \ bool" + and set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap, exception, unit) prog set" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and get_parent :: "(_) node_ptr \ ((_) heap, exception, (_) object_ptr option) prog" + and get_parent_locs :: "((_) heap \ (_) heap \ bool) set" + and type_wf :: "(_) heap \ bool" + and known_ptrs :: "(_) heap \ bool" + and get_ancestors :: "(_) object_ptr \ ((_) heap, exception, (_) object_ptr list) prog" + and get_ancestors_locs :: "((_) heap \ (_) heap \ bool) set" + and get_root_node :: "(_) object_ptr \ ((_) heap, exception, (_) object_ptr) prog" + and get_root_node_locs :: "((_) heap \ (_) heap \ bool) set" +begin +lemma set_disconnected_nodes_get_ancestors: + "\w \ set_disconnected_nodes_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_ancestors_locs. r h h'))" + by(auto simp add: get_parent_locs_def set_disconnected_nodes_locs_def get_ancestors_locs_def + all_args_def) +end + +locale l_set_disconnected_nodes_get_ancestors = l_set_disconnected_nodes_defs + l_get_ancestors_defs + + assumes set_disconnected_nodes_get_ancestors: + "\w \ set_disconnected_nodes_locs ptr. (h \ w \\<^sub>h h' \ (\r \ get_ancestors_locs. r h h'))" + +interpretation + i_set_disconnected_nodes_get_ancestors?: l_set_disconnected_nodes_get_ancestors\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr + set_disconnected_nodes set_disconnected_nodes_locs + get_child_nodes get_child_nodes_locs get_parent + get_parent_locs type_wf known_ptrs get_ancestors + get_ancestors_locs get_root_node get_root_node_locs + using instances + by (simp add: l_set_disconnected_nodes_get_ancestors\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) +declare l_set_disconnected_nodes_get_ancestors\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + + +lemma set_disconnected_nodes_get_ancestors_is_l_set_disconnected_nodes_get_ancestors [instances]: + "l_set_disconnected_nodes_get_ancestors set_disconnected_nodes_locs get_ancestors_locs" + using instances + apply(simp add: l_set_disconnected_nodes_get_ancestors_def) + using set_disconnected_nodes_get_ancestors + by fast + + + +subsubsection \get\_owner\_document\ + +locale l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs + + l_get_root_node_defs get_root_node get_root_node_locs + for get_root_node :: "(_::linorder) object_ptr \ ((_) heap, exception, (_) object_ptr) prog" + and get_root_node_locs :: "((_) heap \ (_) heap \ bool) set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" +begin + +definition a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) node_ptr \ unit \ (_, (_) document_ptr) dom_prog" + where + "a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr _ = do { + root \ get_root_node (cast node_ptr); + (case cast root of + Some document_ptr \ return document_ptr + | None \ do { + ptrs \ document_ptr_kinds_M; + candidates \ filter_M (\document_ptr. do { + disconnected_nodes \ get_disconnected_nodes document_ptr; + return (root \ cast ` set disconnected_nodes) + }) ptrs; + return (hd candidates) + }) + }" + +definition + a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) document_ptr \ unit \ (_, (_) document_ptr) dom_prog" + where + "a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr _ = do { + document_ptrs \ document_ptr_kinds_M; + (if document_ptr \ set document_ptrs then return document_ptr else error SegmentationFault)}" + +definition + a_get_owner_document_tups :: "(((_) object_ptr \ bool) \ ((_) object_ptr \ unit + \ (_, (_) document_ptr) dom_prog)) list" + where + "a_get_owner_document_tups = [ + (is_element_ptr, a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r \ the \ cast), + (is_character_data_ptr, a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r \ the \ cast), + (is_document_ptr, a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \ the \ cast) + ]" + +definition a_get_owner_document :: "(_) object_ptr \ (_, (_) document_ptr) dom_prog" + where + "a_get_owner_document ptr = invoke a_get_owner_document_tups ptr ()" +end + +locale l_get_owner_document_defs = + fixes get_owner_document :: "(_::linorder) object_ptr \ (_, (_) document_ptr) dom_prog" + +locale l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_known_ptr known_ptr + + l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs + + l_get_root_node get_root_node get_root_node_locs + + l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_root_node get_root_node_locs get_disconnected_nodes + get_disconnected_nodes_locs + + l_get_owner_document_defs get_owner_document + for known_ptr :: "(_::linorder) object_ptr \ bool" + and type_wf :: "(_) heap \ bool" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and get_root_node :: "(_) object_ptr \ ((_) heap, exception, (_) object_ptr) prog" + and get_root_node_locs :: "((_) heap \ (_) heap \ bool) set" + and get_owner_document :: "(_) object_ptr \ ((_) heap, exception, (_) document_ptr) prog" + + assumes known_ptr_impl: "known_ptr = a_known_ptr" + assumes get_owner_document_impl: "get_owner_document = a_get_owner_document" +begin +lemmas known_ptr_def = known_ptr_impl[unfolded a_known_ptr_def] +lemmas get_owner_document_def = a_get_owner_document_def[folded get_owner_document_impl] + +lemma get_owner_document_split: + "P (invoke (a_get_owner_document_tups @ xs) ptr ()) = + ((known_ptr ptr \ P (get_owner_document ptr)) + \ (\(known_ptr ptr) \ P (invoke xs ptr ())))" + by(auto simp add: get_owner_document_def a_get_owner_document_tups_def known_ptr_def + CharacterDataClass.known_ptr_defs ElementClass.known_ptr_defs + NodeClass.known_ptr_defs + split: invoke_splits option.splits) + +lemma get_owner_document_split_asm: + "P (invoke (a_get_owner_document_tups @ xs) ptr ()) = + (\((known_ptr ptr \ \P (get_owner_document ptr)) + \ (\(known_ptr ptr) \ \P (invoke xs ptr ()))))" + by(auto simp add: get_owner_document_def a_get_owner_document_tups_def known_ptr_def + CharacterDataClass.known_ptr_defs ElementClass.known_ptr_defs + NodeClass.known_ptr_defs + split: invoke_splits) +lemmas get_owner_document_splits = get_owner_document_split get_owner_document_split_asm + +lemma get_owner_document_pure [simp]: + "pure (get_owner_document ptr) h" +proof - + have "\node_ptr. pure (a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr ()) h" + by(auto simp add: a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + intro!: bind_pure_I filter_M_pure_I + split: option.splits) + moreover have "\document_ptr. pure (a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr ()) h" + by(auto simp add: a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def bind_pure_I) + ultimately show ?thesis + by(auto simp add: get_owner_document_def a_get_owner_document_tups_def + intro!: bind_pure_I + split: invoke_splits) +qed + +lemma get_owner_document_ptr_in_heap: + assumes "h \ ok (get_owner_document ptr)" + shows "ptr |\| object_ptr_kinds h" + using assms + by(auto simp add: get_owner_document_def invoke_ptr_in_heap dest: is_OK_returns_heap_I) +end + +locale l_get_owner_document = l_get_owner_document_defs + + assumes get_owner_document_ptr_in_heap: + "h \ ok (get_owner_document ptr) \ ptr |\| object_ptr_kinds h" + assumes get_owner_document_pure [simp]: + "pure (get_owner_document ptr) h" + +global_interpretation l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_root_node get_root_node_locs + get_disconnected_nodes get_disconnected_nodes_locs + defines get_owner_document_tups = + "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document_tups get_root_node get_disconnected_nodes" + and get_owner_document = + "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document get_root_node get_disconnected_nodes" + and get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r = + "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r get_root_node get_disconnected_nodes" + . +interpretation + i_get_owner_document?: l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent get_parent_locs known_ptr type_wf + get_disconnected_nodes get_disconnected_nodes_locs get_root_node + get_root_node_locs get_owner_document + using instances + apply(auto simp add: l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def)[1] + by(auto simp add: get_owner_document_tups_def get_owner_document_def get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def)[1] +declare l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma get_owner_document_is_l_get_owner_document [instances]: + "l_get_owner_document get_owner_document" + using get_owner_document_ptr_in_heap + by(auto simp add: l_get_owner_document_def) + +subsubsection \remove\_child\ + +locale l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_get_child_nodes_defs get_child_nodes get_child_nodes_locs + + l_set_child_nodes_defs set_child_nodes set_child_nodes_locs + + l_get_parent_defs get_parent get_parent_locs + + l_get_owner_document_defs get_owner_document + + l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs + + l_set_disconnected_nodes_defs set_disconnected_nodes set_disconnected_nodes_locs + for get_child_nodes :: "(_::linorder) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and set_child_nodes :: "(_) object_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_child_nodes_locs :: "(_) object_ptr \ ((_) heap, exception, unit) prog set" + and get_parent :: "(_) node_ptr \ ((_) heap, exception, (_) object_ptr option) prog" + and get_parent_locs :: "((_) heap \ (_) heap \ bool) set" + and get_owner_document :: "(_) object_ptr \ ((_) heap, exception, (_) document_ptr) prog" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap, exception, unit) prog set" +begin +definition a_remove_child :: "(_) object_ptr \ (_) node_ptr \ (_, unit) dom_prog" + where + "a_remove_child ptr child = do { + children \ get_child_nodes ptr; + if child \ set children then + error NotFoundError + else do { + owner_document \ get_owner_document (cast child); + disc_nodes \ get_disconnected_nodes owner_document; + set_disconnected_nodes owner_document (child # disc_nodes); + set_child_nodes ptr (remove1 child children) + } + }" + +definition a_remove_child_locs :: "(_) object_ptr \ (_) document_ptr \ (_, unit) dom_prog set" + where + "a_remove_child_locs ptr owner_document = set_child_nodes_locs ptr + \ set_disconnected_nodes_locs owner_document" + +definition a_remove :: "(_) node_ptr \ (_, unit) dom_prog" + where + "a_remove node_ptr = do { + parent_opt \ get_parent node_ptr; + (case parent_opt of + Some parent \ do { + a_remove_child parent node_ptr; + return () + } + | None \ return ()) + }" +end + +locale l_remove_child_defs = + fixes remove_child :: "(_::linorder) object_ptr \ (_) node_ptr \ (_, unit) dom_prog" + fixes remove_child_locs :: "(_) object_ptr \ (_) document_ptr \ (_, unit) dom_prog set" + +locale l_remove_defs = + fixes remove :: "(_) node_ptr \ (_, unit) dom_prog" + +locale l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + + l_remove_child_defs + + l_remove_defs + + l_get_parent + + l_get_owner_document + + l_set_child_nodes_get_child_nodes + + l_set_child_nodes_get_disconnected_nodes + + l_set_disconnected_nodes_get_disconnected_nodes + + l_set_disconnected_nodes_get_child_nodes + + assumes remove_child_impl: "remove_child = a_remove_child" + assumes remove_child_locs_impl: "remove_child_locs = a_remove_child_locs" + assumes remove_impl: "remove = a_remove" +begin +lemmas remove_child_def = a_remove_child_def[folded remove_child_impl] +lemmas remove_child_locs_def = a_remove_child_locs_def[folded remove_child_locs_impl] +lemmas remove_def = a_remove_def[folded remove_child_impl remove_impl] + +lemma remove_child_ptr_in_heap: + assumes "h \ ok (remove_child ptr child)" + shows "ptr |\| object_ptr_kinds h" +proof - + obtain children where children: "h \ get_child_nodes ptr \\<^sub>r children" + using assms + by(auto simp add: remove_child_def) + moreover have "children \ []" + using assms calculation + by(auto simp add: remove_child_def elim!: bind_is_OK_E2) + ultimately show ?thesis + using assms(1) get_child_nodes_ptr_in_heap by blast +qed + +lemma remove_child_in_disconnected_nodes: + (* assumes "known_ptrs h" *) + assumes "h \ remove_child ptr child \\<^sub>h h'" + assumes "h \ get_owner_document (cast child) \\<^sub>r owner_document" + assumes "h' \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes" + shows "child \ set disc_nodes" +proof - + obtain prev_disc_nodes h2 children where + disc_nodes: "h \ get_disconnected_nodes owner_document \\<^sub>r prev_disc_nodes" and + h2: "h \ set_disconnected_nodes owner_document (child # prev_disc_nodes) \\<^sub>h h2" and + h': "h2 \ set_child_nodes ptr (remove1 child children) \\<^sub>h h'" + using assms(1) + apply(auto simp add: remove_child_def + elim!: bind_returns_heap_E + dest!: returns_result_eq[OF assms(2)] pure_returns_heap_eq[rotated, OF get_owner_document_pure] + pure_returns_heap_eq[rotated, OF get_child_nodes_pure] + split: if_splits)[1] + by (metis get_disconnected_nodes_pure pure_returns_heap_eq) + have "h2 \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes" + apply(rule reads_writes_separate_backwards[OF get_disconnected_nodes_reads + set_child_nodes_writes h' assms(3)]) + by (simp add: set_child_nodes_get_disconnected_nodes) + then show ?thesis + by (metis (no_types, lifting) h2 set_disconnected_nodes_get_disconnected_nodes + list.set_intros(1) select_result_I2) +qed + +lemma remove_child_writes [simp]: + "writes (remove_child_locs ptr |h \ get_owner_document (cast child)|\<^sub>r) (remove_child ptr child) h h'" + apply(auto simp add: remove_child_def intro!: writes_bind_pure[OF get_child_nodes_pure] + writes_bind_pure[OF get_owner_document_pure] + writes_bind_pure[OF get_disconnected_nodes_pure])[1] + by(auto simp add: remove_child_locs_def set_disconnected_nodes_writes writes_union_right_I + set_child_nodes_writes writes_union_left_I + intro!: writes_bind) + +lemma remove_writes: + "writes (remove_child_locs (the |h \ get_parent child|\<^sub>r) |h \ get_owner_document (cast child)|\<^sub>r) (remove child) h h'" + by(auto simp add: remove_def intro!: writes_bind_pure split: option.splits) + +lemma remove_child_children_subset: + assumes "h \ remove_child parent child \\<^sub>h h'" + and "h \ get_child_nodes ptr \\<^sub>r children" + and "h' \ get_child_nodes ptr \\<^sub>r children'" + and known_ptrs: "known_ptrs h" + and type_wf: "type_wf h" + shows "set children' \ set children" +proof - + obtain ptr_children owner_document h2 disc_nodes where + owner_document: "h \ get_owner_document (cast child) \\<^sub>r owner_document" and + ptr_children: "h \ get_child_nodes parent \\<^sub>r ptr_children" and + disc_nodes: "h \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes" and + h2: "h \ set_disconnected_nodes owner_document (child # disc_nodes) \\<^sub>h h2" and + h': "h2 \ set_child_nodes parent (remove1 child ptr_children) \\<^sub>h h'" + using assms(1) + by(auto simp add: remove_child_def + elim!: bind_returns_heap_E + dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure] + pure_returns_heap_eq[rotated, OF get_disconnected_nodes_pure] + pure_returns_heap_eq[rotated, OF get_child_nodes_pure] + split: if_splits) + have "parent |\| object_ptr_kinds h" + using get_child_nodes_ptr_in_heap ptr_children by blast + have "object_ptr_kinds h = object_ptr_kinds h2" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h = object_ptr_kinds h'", + OF set_disconnected_nodes_writes h2]) + using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved + by (auto simp add: reflp_def transp_def) + have "type_wf h2" + using type_wf writes_small_big[where P="\h h'. type_wf h \ type_wf h'", + OF set_disconnected_nodes_writes h2] + using set_disconnected_nodes_types_preserved + by(auto simp add: reflp_def transp_def) + have "h2 \ get_child_nodes ptr \\<^sub>r children" + using get_child_nodes_reads set_disconnected_nodes_writes h2 assms(2) + apply(rule reads_writes_separate_forwards) + by (simp add: set_disconnected_nodes_get_child_nodes) + moreover have "h2 \ get_child_nodes parent \\<^sub>r ptr_children" + using get_child_nodes_reads set_disconnected_nodes_writes h2 ptr_children + apply(rule reads_writes_separate_forwards) + by (simp add: set_disconnected_nodes_get_child_nodes) + moreover have + "ptr \ parent \ h2 \ get_child_nodes ptr \\<^sub>r children = h' \ get_child_nodes ptr \\<^sub>r children" + using get_child_nodes_reads set_child_nodes_writes h' + apply(rule reads_writes_preserved) + by (metis set_child_nodes_get_child_nodes_different_pointers) + moreover have "h' \ get_child_nodes parent \\<^sub>r remove1 child ptr_children" + using h' set_child_nodes_get_child_nodes known_ptrs type_wf known_ptrs_known_ptr + \parent |\| object_ptr_kinds h\ \object_ptr_kinds h = object_ptr_kinds h2\ \type_wf h2\ + by fast + moreover have "set ( remove1 child ptr_children) \ set ptr_children" + by (simp add: set_remove1_subset) + ultimately show ?thesis + by (metis assms(3) order_refl returns_result_eq) +qed + + +lemma remove_child_pointers_preserved: + assumes "w \ remove_child_locs ptr owner_document" + assumes "h \ w \\<^sub>h h'" + shows "object_ptr_kinds h = object_ptr_kinds h'" + using assms + using set_child_nodes_pointers_preserved + using set_disconnected_nodes_pointers_preserved + unfolding remove_child_locs_def + by auto + +lemma remove_child_types_preserved: + assumes "w \ remove_child_locs ptr owner_document" + assumes "h \ w \\<^sub>h h'" + shows "type_wf h = type_wf h'" + using assms + using set_child_nodes_types_preserved + using set_disconnected_nodes_types_preserved + unfolding remove_child_locs_def + by auto +end + +locale l_remove_child = l_type_wf + l_known_ptrs + l_remove_child_defs + l_get_owner_document_defs + + l_get_child_nodes_defs + l_get_disconnected_nodes_defs + + assumes remove_child_writes: + "writes (remove_child_locs object_ptr |h \ get_owner_document (cast child)|\<^sub>r) (remove_child object_ptr child) h h'" + assumes remove_child_pointers_preserved: + "w \ remove_child_locs ptr owner_document \ h \ w \\<^sub>h h' \ object_ptr_kinds h = object_ptr_kinds h'" + assumes remove_child_types_preserved: + "w \ remove_child_locs ptr owner_document \ h \ w \\<^sub>h h' \ type_wf h = type_wf h'" + assumes remove_child_in_disconnected_nodes: + "known_ptrs h \ h \ remove_child ptr child \\<^sub>h h' + \ h \ get_owner_document (cast child) \\<^sub>r owner_document + \ h' \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes + \ child \ set disc_nodes" + assumes remove_child_ptr_in_heap: "h \ ok (remove_child ptr child) \ ptr |\| object_ptr_kinds h" + assumes remove_child_children_subset: + "known_ptrs h \ type_wf h \ h \ remove_child parent child \\<^sub>h h' + \ h \ get_child_nodes ptr \\<^sub>r children + \ h' \ get_child_nodes ptr \\<^sub>r children' + \ set children' \ set children" + +locale l_remove + + +global_interpretation l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_child_nodes get_child_nodes_locs set_child_nodes + set_child_nodes_locs get_parent get_parent_locs + get_owner_document get_disconnected_nodes + get_disconnected_nodes_locs set_disconnected_nodes + set_disconnected_nodes_locs + defines remove = + "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove get_child_nodes set_child_nodes get_parent get_owner_document + get_disconnected_nodes set_disconnected_nodes" + and remove_child = + "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child get_child_nodes set_child_nodes get_owner_document + get_disconnected_nodes set_disconnected_nodes" + and remove_child_locs = + "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child_locs set_child_nodes_locs set_disconnected_nodes_locs" + . +interpretation + i_remove_child?: l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes + set_child_nodes_locs get_parent get_parent_locs get_owner_document + get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes + set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf + known_ptr known_ptrs + using instances + apply(simp add: l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def) + by(simp add: remove_child_def remove_child_locs_def remove_def) +declare l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma remove_child_is_l_remove_child [instances]: + "l_remove_child type_wf known_ptr known_ptrs remove_child remove_child_locs get_owner_document + get_child_nodes get_disconnected_nodes" + using instances + apply(auto simp add: l_remove_child_def l_remove_child_axioms_def)[1] (*slow, ca 1min *) + using remove_child_pointers_preserved apply(blast) + using remove_child_pointers_preserved apply(blast) + using remove_child_types_preserved apply(blast) + using remove_child_types_preserved apply(blast) + using remove_child_in_disconnected_nodes apply(blast) + using remove_child_ptr_in_heap apply(blast) + using remove_child_children_subset apply(blast) + done + + + +subsubsection \adopt\_node\ + +locale l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_get_owner_document_defs get_owner_document + + l_get_parent_defs get_parent get_parent_locs + + l_remove_child_defs remove_child remove_child_locs + + l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs + + l_set_disconnected_nodes_defs set_disconnected_nodes set_disconnected_nodes_locs + for get_owner_document :: "(_::linorder) object_ptr \ ((_) heap, exception, (_) document_ptr) prog" + and get_parent :: "(_) node_ptr \ ((_) heap, exception, (_) object_ptr option) prog" + and get_parent_locs :: "((_) heap \ (_) heap \ bool) set" + and remove_child :: "(_) object_ptr \ (_) node_ptr \ ((_) heap, exception, unit) prog" + and remove_child_locs :: "(_) object_ptr \ (_) document_ptr \ ((_) heap, exception, unit) prog set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap, exception, unit) prog set" +begin +definition a_adopt_node :: "(_) document_ptr \ (_) node_ptr \ (_, unit) dom_prog" + where + "a_adopt_node document node = do { + old_document \ get_owner_document (cast node); + parent_opt \ get_parent node; + (case parent_opt of + Some parent \ do { + remove_child parent node + } | None \ do { + return () + }); + (if document \ old_document then do { + old_disc_nodes \ get_disconnected_nodes old_document; + set_disconnected_nodes old_document (remove1 node old_disc_nodes); + disc_nodes \ get_disconnected_nodes document; + set_disconnected_nodes document (node # disc_nodes) + } else do { + return () + }) + }" + +definition + a_adopt_node_locs :: "(_) object_ptr option \ (_) document_ptr \ (_) document_ptr \ (_, unit) dom_prog set" + where + "a_adopt_node_locs parent owner_document document_ptr = + ((if parent = None + then {} + else remove_child_locs (the parent) owner_document) \ set_disconnected_nodes_locs document_ptr + \ set_disconnected_nodes_locs owner_document)" +end + +locale l_adopt_node_defs = + fixes + adopt_node :: "(_) document_ptr \ (_) node_ptr \ (_, unit) dom_prog" + fixes + adopt_node_locs :: "(_) object_ptr option \ (_) document_ptr \ (_) document_ptr \ (_, unit) dom_prog set" + +global_interpretation l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_owner_document get_parent get_parent_locs remove_child + remove_child_locs get_disconnected_nodes + get_disconnected_nodes_locs set_disconnected_nodes + set_disconnected_nodes_locs + defines adopt_node = "l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node get_owner_document get_parent remove_child + get_disconnected_nodes set_disconnected_nodes" + and adopt_node_locs = "l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node_locs + remove_child_locs set_disconnected_nodes_locs" + . + +locale l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + get_owner_document get_parent get_parent_locs remove_child remove_child_locs get_disconnected_nodes + get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs + + l_adopt_node_defs + adopt_node adopt_node_locs + + l_get_owner_document + get_owner_document + + l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs + + l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent + get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs + set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf + known_ptr known_ptrs + + l_set_disconnected_nodes_get_disconnected_nodes + type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes + set_disconnected_nodes_locs + for get_owner_document :: "(_::linorder) object_ptr \ ((_) heap, exception, (_) document_ptr) prog" + and get_parent :: "(_) node_ptr \ ((_) heap, exception, (_) object_ptr option) prog" + and get_parent_locs :: "((_) heap \ (_) heap \ bool) set" + and remove_child :: "(_) object_ptr \ (_) node_ptr \ ((_) heap, exception, unit) prog" + and remove_child_locs :: "(_) object_ptr \ (_) document_ptr \ ((_) heap, exception, unit) prog set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap, exception, unit) prog set" + and adopt_node :: "(_) document_ptr \ (_) node_ptr \ ((_) heap, exception, unit) prog" + and adopt_node_locs :: "(_) object_ptr option \ (_) document_ptr \ (_) document_ptr + \ ((_) heap, exception, unit) prog set" + and known_ptr :: "(_) object_ptr \ bool" + and type_wf :: "(_) heap \ bool" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and known_ptrs :: "(_) heap \ bool" + and set_child_nodes :: "(_) object_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_child_nodes_locs :: "(_) object_ptr \ ((_) heap, exception, unit) prog set" + and remove :: "(_) node_ptr \ ((_) heap, exception, unit) prog" + + assumes adopt_node_impl: "adopt_node = a_adopt_node" + assumes adopt_node_locs_impl: "adopt_node_locs = a_adopt_node_locs" +begin +lemmas adopt_node_def = a_adopt_node_def[folded adopt_node_impl] +lemmas adopt_node_locs_def = a_adopt_node_locs_def[folded adopt_node_locs_impl] + +lemma adopt_node_writes: + shows "writes (adopt_node_locs |h \ get_parent node|\<^sub>r |h + \ get_owner_document (cast node)|\<^sub>r document_ptr) (adopt_node document_ptr node) h h'" + apply(auto simp add: adopt_node_def adopt_node_locs_def + intro!: writes_bind_pure[OF get_owner_document_pure] writes_bind_pure[OF get_parent_pure] + writes_bind_pure[OF get_disconnected_nodes_pure] + split: option.splits)[1] + apply(auto intro!: writes_bind)[1] + apply (simp add: set_disconnected_nodes_writes writes_union_right_I) + apply (simp add: set_disconnected_nodes_writes writes_union_left_I writes_union_right_I) + apply(auto intro!: writes_bind)[1] + apply (metis (no_types, lifting) remove_child_writes select_result_I2 writes_union_left_I) + apply (simp add: set_disconnected_nodes_writes writes_union_right_I) + by(auto intro: writes_subset[OF set_disconnected_nodes_writes] writes_subset[OF remove_child_writes]) + +lemma adopt_node_children_subset: + assumes "h \ adopt_node owner_document node \\<^sub>h h'" + and "h \ get_child_nodes ptr \\<^sub>r children" + and "h' \ get_child_nodes ptr \\<^sub>r children'" + and known_ptrs: "known_ptrs h" + and type_wf: "type_wf h" + shows "set children' \ set children" +proof - + obtain old_document parent_opt h2 where + old_document: "h \ get_owner_document (cast node) \\<^sub>r old_document" and + parent_opt: "h \ get_parent node \\<^sub>r parent_opt" and + h2: "h \ (case parent_opt of Some parent \ do { remove_child parent node } | None \ do { return ()}) \\<^sub>h h2" + and + h': "h2 \ (if owner_document \ old_document then do { + old_disc_nodes \ get_disconnected_nodes old_document; + set_disconnected_nodes old_document (remove1 node old_disc_nodes); + disc_nodes \ get_disconnected_nodes owner_document; + set_disconnected_nodes owner_document (node # disc_nodes) + } else do { return () }) \\<^sub>h h'" + using assms(1) + by(auto simp add: adopt_node_def + elim!: bind_returns_heap_E + dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure] + pure_returns_heap_eq[rotated, OF get_parent_pure]) + + have "h2 \ get_child_nodes ptr \\<^sub>r children'" + proof (cases "owner_document \ old_document") + case True + then obtain h3 old_disc_nodes disc_nodes where + old_disc_nodes: "h2 \ get_disconnected_nodes old_document \\<^sub>r old_disc_nodes" and + h3: "h2 \ set_disconnected_nodes old_document (remove1 node old_disc_nodes) \\<^sub>h h3" and + old_disc_nodes: "h3 \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes" and + h': "h3 \ set_disconnected_nodes owner_document (node # disc_nodes) \\<^sub>h h'" + using h' + by(auto elim!: bind_returns_heap_E + bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] ) + have "h3 \ get_child_nodes ptr \\<^sub>r children'" + using get_child_nodes_reads set_disconnected_nodes_writes h' assms(3) + apply(rule reads_writes_separate_backwards) + by (simp add: set_disconnected_nodes_get_child_nodes) + show ?thesis + using get_child_nodes_reads set_disconnected_nodes_writes h3 \h3 \ get_child_nodes ptr \\<^sub>r children'\ + apply(rule reads_writes_separate_backwards) + by (simp add: set_disconnected_nodes_get_child_nodes) + next + case False + then show ?thesis + using h' assms(3) by(auto) + qed + + show ?thesis + proof (insert h2, induct parent_opt) + case None + then show ?case + using assms + by(auto dest!: returns_result_eq[OF \h2 \ get_child_nodes ptr \\<^sub>r children'\]) + next + case (Some option) + then show ?case + using assms(2) \h2 \ get_child_nodes ptr \\<^sub>r children'\ remove_child_children_subset known_ptrs type_wf + by simp + qed +qed + +lemma adopt_node_child_in_heap: + assumes "h \ ok (adopt_node document_ptr child)" + shows "child |\| node_ptr_kinds h" + using assms + apply(auto simp add: adopt_node_def elim!: bind_is_OK_E)[1] + using get_owner_document_pure get_parent_ptr_in_heap pure_returns_heap_eq + by fast + +lemma adopt_node_pointers_preserved: + assumes "w \ adopt_node_locs parent owner_document document_ptr" + assumes "h \ w \\<^sub>h h'" + shows "object_ptr_kinds h = object_ptr_kinds h'" + using assms + using set_disconnected_nodes_pointers_preserved + using remove_child_pointers_preserved + unfolding adopt_node_locs_def + by (auto split: if_splits) + +lemma adopt_node_types_preserved: + assumes "w \ adopt_node_locs parent owner_document document_ptr" + assumes "h \ w \\<^sub>h h'" + shows "type_wf h = type_wf h'" + using assms + using remove_child_types_preserved + using set_disconnected_nodes_types_preserved + unfolding adopt_node_locs_def + by (auto split: if_splits) +end + +locale l_adopt_node = l_type_wf + l_known_ptrs + l_get_parent_defs + l_adopt_node_defs + l_get_child_nodes_defs + l_get_owner_document_defs + + assumes adopt_node_writes: + "writes (adopt_node_locs |h \ get_parent node|\<^sub>r + |h \ get_owner_document (cast node)|\<^sub>r document_ptr) (adopt_node document_ptr node) h h'" + assumes adopt_node_pointers_preserved: + "w \ adopt_node_locs parent owner_document document_ptr + \ h \ w \\<^sub>h h' \ object_ptr_kinds h = object_ptr_kinds h'" + assumes adopt_node_types_preserved: + "w \ adopt_node_locs parent owner_document document_ptr + \ h \ w \\<^sub>h h' \ type_wf h = type_wf h'" + assumes adopt_node_child_in_heap: + "h \ ok (adopt_node document_ptr child) \ child |\| node_ptr_kinds h" + assumes adopt_node_children_subset: + "h \ adopt_node owner_document node \\<^sub>h h' \ h \ get_child_nodes ptr \\<^sub>r children + \ h' \ get_child_nodes ptr \\<^sub>r children' + \ known_ptrs h \ type_wf h \ set children' \ set children" + +interpretation + i_adopt_node?: l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent get_parent_locs remove_child + remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs + set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs + known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes + set_child_nodes_locs remove + apply(unfold_locales) + by(auto simp add: adopt_node_def adopt_node_locs_def) +declare l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + + +lemma adopt_node_is_l_adopt_node [instances]: + "l_adopt_node type_wf known_ptr known_ptrs get_parent adopt_node adopt_node_locs get_child_nodes + get_owner_document" + using instances + by (simp add: l_adopt_node_axioms_def adopt_node_child_in_heap adopt_node_children_subset + adopt_node_pointers_preserved adopt_node_types_preserved adopt_node_writes + l_adopt_node_def) + + + +subsubsection \insert\_before\ + +locale l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_get_parent_defs get_parent get_parent_locs + + l_get_child_nodes_defs get_child_nodes get_child_nodes_locs + + l_set_child_nodes_defs set_child_nodes set_child_nodes_locs + + l_get_ancestors_defs get_ancestors get_ancestors_locs + + l_adopt_node_defs adopt_node adopt_node_locs + + l_set_disconnected_nodes_defs set_disconnected_nodes set_disconnected_nodes_locs + + l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs + + l_get_owner_document_defs get_owner_document + for get_parent :: "(_) node_ptr \ ((_) heap, exception, (_::linorder) object_ptr option) prog" + and get_parent_locs :: "((_) heap \ (_) heap \ bool) set" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and set_child_nodes :: "(_) object_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_child_nodes_locs :: "(_) object_ptr \ ((_) heap, exception, unit) prog set" + and get_ancestors :: "(_) object_ptr \ ((_) heap, exception, (_) object_ptr list) prog" + and get_ancestors_locs :: "((_) heap \ (_) heap \ bool) set" + and adopt_node :: "(_) document_ptr \ (_) node_ptr \ ((_) heap, exception, unit) prog" + and adopt_node_locs :: "(_) object_ptr option \ (_) document_ptr \ (_) document_ptr + \ ((_) heap, exception, unit) prog set" + and set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap, exception, unit) prog set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and get_owner_document :: "(_) object_ptr \ ((_) heap, exception, (_) document_ptr) prog" +begin + +definition a_next_sibling :: "(_) node_ptr \ (_, (_) node_ptr option) dom_prog" + where + "a_next_sibling node_ptr = do { + parent_opt \ get_parent node_ptr; + (case parent_opt of + Some parent \ do { + children \ get_child_nodes parent; + (case (dropWhile (\ptr. ptr = node_ptr) (dropWhile (\ptr. ptr \ node_ptr) children)) of + x#_ \ return (Some x) + | [] \ return None)} + | None \ return None) + }" + +fun insert_before_list :: "'xyz \ 'xyz option \ 'xyz list \ 'xyz list" + where + "insert_before_list v (Some reference) (x#xs) = (if reference = x + then v#x#xs else x # insert_before_list v (Some reference) xs)" + | "insert_before_list v (Some _) [] = [v]" + | "insert_before_list v None xs = xs @ [v]" + +definition a_insert_node :: "(_) object_ptr \ (_) node_ptr \ (_) node_ptr option + \ (_, unit) dom_prog" + where + "a_insert_node ptr new_child reference_child_opt = do { + children \ get_child_nodes ptr; + set_child_nodes ptr (insert_before_list new_child reference_child_opt children) + }" + +definition a_ensure_pre_insertion_validity :: "(_) node_ptr \ (_) object_ptr + \ (_) node_ptr option \ (_, unit) dom_prog" + where + "a_ensure_pre_insertion_validity node parent child_opt = do { + (if is_character_data_ptr_kind parent + then error HierarchyRequestError else return ()); + ancestors \ get_ancestors parent; + (if cast node \ set ancestors then error HierarchyRequestError else return ()); + (case child_opt of + Some child \ do { + child_parent \ get_parent child; + (if child_parent \ Some parent then error NotFoundError else return ())} + | None \ return ()); + children \ get_child_nodes parent; + (if children \ [] \ is_document_ptr parent + then error HierarchyRequestError else return ()); + (if is_character_data_ptr node \ is_document_ptr parent + then error HierarchyRequestError else return ()) + }" + +definition a_insert_before :: "(_) object_ptr \ (_) node_ptr + \ (_) node_ptr option \ (_, unit) dom_prog" + where + "a_insert_before ptr node child = do { + a_ensure_pre_insertion_validity node ptr child; + reference_child \ (if Some node = child + then a_next_sibling node + else return child); + owner_document \ get_owner_document ptr; + adopt_node owner_document node; + disc_nodes \ get_disconnected_nodes owner_document; + set_disconnected_nodes owner_document (remove1 node disc_nodes); + a_insert_node ptr node reference_child + }" + +definition a_insert_before_locs :: "(_) object_ptr \ (_) object_ptr option \ (_) document_ptr + \ (_) document_ptr \ (_, unit) dom_prog set" + where + "a_insert_before_locs ptr old_parent child_owner_document ptr_owner_document = + adopt_node_locs old_parent child_owner_document ptr_owner_document \ + set_child_nodes_locs ptr \ + set_disconnected_nodes_locs ptr_owner_document" +end + +locale l_insert_before_defs = + fixes insert_before :: "(_) object_ptr \ (_) node_ptr \ (_) node_ptr option \ (_, unit) dom_prog" + fixes insert_before_locs :: "(_) object_ptr \ (_) object_ptr option \ (_) document_ptr + \ (_) document_ptr \ (_, unit) dom_prog set" + +locale l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_insert_before_defs +begin +definition "a_append_child ptr child = insert_before ptr child None" +end + +locale l_append_child_defs = + fixes append_child :: "(_) object_ptr \ (_) node_ptr \ (_, unit) dom_prog" + +locale l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + get_parent get_parent_locs get_child_nodes get_child_nodes_locs set_child_nodes + set_child_nodes_locs get_ancestors get_ancestors_locs adopt_node adopt_node_locs + set_disconnected_nodes set_disconnected_nodes_locs get_disconnected_nodes + get_disconnected_nodes_locs get_owner_document + + l_insert_before_defs + insert_before insert_before_locs + + l_append_child_defs + append_child + + l_set_child_nodes_get_child_nodes + type_wf known_ptr get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs + + l_get_ancestors + get_ancestors get_ancestors_locs + + l_adopt_node + type_wf known_ptr known_ptrs get_parent get_parent_locs adopt_node adopt_node_locs + get_child_nodes get_child_nodes_locs get_owner_document + + l_set_disconnected_nodes + type_wf set_disconnected_nodes set_disconnected_nodes_locs + + l_get_disconnected_nodes + type_wf get_disconnected_nodes get_disconnected_nodes_locs + + l_get_owner_document + get_owner_document + + l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs + + l_set_disconnected_nodes_get_child_nodes + set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs + for get_parent :: "(_) node_ptr \ ((_) heap, exception, (_::linorder) object_ptr option) prog" + and get_parent_locs :: "((_) heap \ (_) heap \ bool) set" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and set_child_nodes :: "(_) object_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_child_nodes_locs :: "(_) object_ptr \ ((_) heap, exception, unit) prog set" + and get_ancestors :: "(_) object_ptr \ ((_) heap, exception, (_) object_ptr list) prog" + and get_ancestors_locs :: "((_) heap \ (_) heap \ bool) set" + and adopt_node :: "(_) document_ptr \ (_) node_ptr \ ((_) heap, exception, unit) prog" + and adopt_node_locs :: "(_) object_ptr option \ (_) document_ptr \ (_) document_ptr + \ ((_) heap, exception, unit) prog set" + and set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap, exception, unit) prog set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and get_owner_document :: "(_) object_ptr \ ((_) heap, exception, (_) document_ptr) prog" + and insert_before :: "(_) object_ptr \ (_) node_ptr \ (_) node_ptr option \ ((_) heap, exception, unit) prog" + and insert_before_locs :: "(_) object_ptr \ (_) object_ptr option \ (_) document_ptr + \ (_) document_ptr \ (_, unit) dom_prog set" + and append_child :: "(_) object_ptr \ (_) node_ptr \ ((_) heap, exception, unit) prog" + and type_wf :: "(_) heap \ bool" + and known_ptr :: "(_) object_ptr \ bool" + and known_ptrs :: "(_) heap \ bool" + + assumes insert_before_impl: "insert_before = a_insert_before" + assumes insert_before_locs_impl: "insert_before_locs = a_insert_before_locs" +begin +lemmas insert_before_def = a_insert_before_def[folded insert_before_impl] +lemmas insert_before_locs_def = a_insert_before_locs_def[folded insert_before_locs_impl] + +lemma next_sibling_pure [simp]: + "pure (a_next_sibling new_child) h" + by(auto simp add: a_next_sibling_def get_parent_pure intro!: bind_pure_I split: option.splits list.splits) + +lemma insert_before_list_in_set: "x \ set (insert_before_list v ref xs) \ x = v \ x \ set xs" + apply(induct v ref xs rule: insert_before_list.induct) + by(auto) + +lemma insert_before_list_distinct: "x \ set xs \ distinct xs \ distinct (insert_before_list x ref xs)" + by (induct x ref xs rule: insert_before_list.induct) + (auto simp add: insert_before_list_in_set) + +lemma insert_before_list_subset: "set xs \ set (insert_before_list x ref xs)" + apply(induct x ref xs rule: insert_before_list.induct) + by(auto) + +lemma insert_before_list_node_in_set: "x \ set (insert_before_list x ref xs)" + apply(induct x ref xs rule: insert_before_list.induct) + by(auto) + +lemma insert_node_writes: + "writes (set_child_nodes_locs ptr) (a_insert_node ptr new_child reference_child_opt) h h'" + by(auto simp add: a_insert_node_def set_child_nodes_writes + intro!: writes_bind_pure[OF get_child_nodes_pure]) + +lemma ensure_pre_insertion_validity_pure [simp]: + "pure (a_ensure_pre_insertion_validity node ptr child) h" + by(auto simp add: a_ensure_pre_insertion_validity_def + intro!: bind_pure_I + split: option.splits) + +lemma insert_before_reference_child_not_in_children: + assumes "h \ get_parent child \\<^sub>r Some parent" + and "ptr \ parent" + and "\is_character_data_ptr_kind ptr" + and "h \ get_ancestors ptr \\<^sub>r ancestors" + and "cast node \ set ancestors" + shows "h \ insert_before ptr node (Some child) \\<^sub>e NotFoundError" +proof - + have "h \ a_ensure_pre_insertion_validity node ptr (Some child) \\<^sub>e NotFoundError" + using assms unfolding insert_before_def a_ensure_pre_insertion_validity_def + by auto (simp | rule bind_returns_error_I2)+ + then show ?thesis + unfolding insert_before_def by auto +qed + +lemma insert_before_child_in_heap: + assumes "h \ ok (insert_before ptr node reference_child)" + shows "node |\| node_ptr_kinds h" + using assms + apply(auto simp add: insert_before_def elim!: bind_is_OK_E)[1] + by (metis (mono_tags, lifting) ensure_pre_insertion_validity_pure is_OK_returns_heap_I + l_get_owner_document.get_owner_document_pure local.adopt_node_child_in_heap + local.l_get_owner_document_axioms next_sibling_pure pure_returns_heap_eq return_pure) + +lemma insert_node_children_remain_distinct: + assumes insert_node: "h \ a_insert_node ptr new_child reference_child_opt \\<^sub>h h2" + and "h \ get_child_nodes ptr \\<^sub>r children" + and "new_child \ set children" + and "\ptr children. h \ get_child_nodes ptr \\<^sub>r children \ distinct children" + and known_ptr: "known_ptr ptr" + and type_wf: "type_wf h" + shows "\ptr children. h2 \ get_child_nodes ptr \\<^sub>r children \ distinct children" +proof - + fix ptr' children' + assume a1: "h2 \ get_child_nodes ptr' \\<^sub>r children'" + then show "distinct children'" + proof (cases "ptr = ptr'") + case True + have "h2 \ get_child_nodes ptr \\<^sub>r (insert_before_list new_child reference_child_opt children)" + using assms(1) assms(2) apply(auto simp add: a_insert_node_def elim!: bind_returns_heap_E)[1] + using returns_result_eq set_child_nodes_get_child_nodes known_ptr type_wf + using pure_returns_heap_eq by fastforce + then show ?thesis + using True a1 assms(2) assms(3) assms(4) insert_before_list_distinct returns_result_eq + by fastforce + next + case False + have "h \ get_child_nodes ptr' \\<^sub>r children'" + using get_child_nodes_reads insert_node_writes insert_node a1 + apply(rule reads_writes_separate_backwards) + by (meson False set_child_nodes_get_child_nodes_different_pointers) + then show ?thesis + using assms(4) by blast + qed +qed + +lemma insert_before_writes: + "writes (insert_before_locs ptr |h \ get_parent child|\<^sub>r + |h \ get_owner_document (cast child)|\<^sub>r |h \ get_owner_document ptr|\<^sub>r) (insert_before ptr child ref) h h'" + apply(auto simp add: insert_before_def insert_before_locs_def a_insert_node_def + intro!: writes_bind)[1] + apply (metis (no_types, hide_lams) ensure_pre_insertion_validity_pure local.adopt_node_writes + local.get_owner_document_pure next_sibling_pure pure_returns_heap_eq + select_result_I2 sup_commute writes_union_right_I) + apply (metis (no_types, hide_lams) ensure_pre_insertion_validity_pure next_sibling_pure + pure_returns_heap_eq select_result_I2 set_disconnected_nodes_writes + writes_union_right_I) + apply (simp add: set_child_nodes_writes writes_union_left_I writes_union_right_I) + apply (metis (no_types, hide_lams) adopt_node_writes ensure_pre_insertion_validity_pure + get_owner_document_pure pure_returns_heap_eq select_result_I2 writes_union_left_I) + apply (metis (no_types, hide_lams) ensure_pre_insertion_validity_pure pure_returns_heap_eq + select_result_I2 set_disconnected_nodes_writes writes_union_right_I) + by (simp add: set_child_nodes_writes writes_union_left_I writes_union_right_I) +end + + +locale l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_append_child_defs + + l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + + assumes append_child_impl: "append_child = a_append_child" +begin + +lemmas append_child_def = a_append_child_def[folded append_child_impl] +end + +locale l_insert_before = l_insert_before_defs + +locale l_append_child = l_append_child_defs + +global_interpretation l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_parent get_parent_locs get_child_nodes + get_child_nodes_locs set_child_nodes set_child_nodes_locs get_ancestors get_ancestors_locs + adopt_node adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs + get_disconnected_nodes get_disconnected_nodes_locs get_owner_document + defines + next_sibling = "l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_next_sibling get_parent get_child_nodes" and + insert_node = "l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_node get_child_nodes set_child_nodes" and + ensure_pre_insertion_validity = "l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_ensure_pre_insertion_validity + get_parent get_child_nodes get_ancestors" and + insert_before = "l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_before get_parent get_child_nodes + set_child_nodes get_ancestors adopt_node set_disconnected_nodes + get_disconnected_nodes get_owner_document" and + insert_before_locs = "l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_before_locs set_child_nodes_locs + adopt_node_locs set_disconnected_nodes_locs" + . + +global_interpretation l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs insert_before + defines append_child = "l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_append_child insert_before" + . + +interpretation + i_insert_before?: l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent get_parent_locs get_child_nodes + get_child_nodes_locs set_child_nodes set_child_nodes_locs get_ancestors get_ancestors_locs + adopt_node adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs get_disconnected_nodes + get_disconnected_nodes_locs get_owner_document insert_before insert_before_locs append_child + type_wf known_ptr known_ptrs + apply(simp add: l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def instances) + by (simp add: insert_before_def insert_before_locs_def) +declare l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +interpretation i_append_child?: l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M append_child insert_before insert_before_locs + apply(simp add: l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances append_child_def) + done +declare l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + + + + +subsubsection \create\_element\ + +locale l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs + + l_set_disconnected_nodes_defs set_disconnected_nodes set_disconnected_nodes_locs + + l_set_tag_type_defs set_tag_type set_tag_type_locs + for get_disconnected_nodes :: + "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: + "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and set_disconnected_nodes :: + "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: + "(_) document_ptr \ ((_) heap, exception, unit) prog set" + and set_tag_type :: + "(_) element_ptr \ char list \ ((_) heap, exception, unit) prog" + and set_tag_type_locs :: + "(_) element_ptr \ ((_) heap, exception, unit) prog set" +begin +definition a_create_element :: "(_) document_ptr \ tag_type \ (_, (_) element_ptr) dom_prog" + where + "a_create_element document_ptr tag = do { + new_element_ptr \ new_element; + set_tag_type new_element_ptr tag; + disc_nodes \ get_disconnected_nodes document_ptr; + set_disconnected_nodes document_ptr (cast new_element_ptr # disc_nodes); + return new_element_ptr + }" +end + +locale l_create_element_defs = + fixes create_element :: "(_) document_ptr \ tag_type \ (_, (_) element_ptr) dom_prog" + +global_interpretation l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_disconnected_nodes get_disconnected_nodes_locs + set_disconnected_nodes set_disconnected_nodes_locs + set_tag_type set_tag_type_locs + defines + create_element = "l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_element get_disconnected_nodes + set_disconnected_nodes set_tag_type" + . + +locale l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + + l_create_element_defs + + assumes create_element_impl: "create_element = a_create_element" +begin +lemmas create_element_def = a_create_element_def[folded create_element_impl] +end + +locale l_create_element = l_create_element_defs + +interpretation + i_create_element?: l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs + set_disconnected_nodes set_disconnected_nodes_locs set_tag_type + set_tag_type_locs create_element + by unfold_locales (simp add: create_element_def) +declare l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + + +subsubsection \create\_character\_data\ + +locale l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_set_val_defs set_val set_val_locs + + l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs + + l_set_disconnected_nodes_defs set_disconnected_nodes set_disconnected_nodes_locs + for set_val :: "(_) character_data_ptr \ char list \ ((_) heap, exception, unit) prog" + and set_val_locs :: "(_) character_data_ptr \ ((_) heap, exception, unit) prog set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap, exception, unit) prog set" +begin +definition a_create_character_data :: "(_) document_ptr \ string \ (_, (_) character_data_ptr) dom_prog" + where + "a_create_character_data document_ptr text = do { + new_character_data_ptr \ new_character_data; + set_val new_character_data_ptr text; + disc_nodes \ get_disconnected_nodes document_ptr; + set_disconnected_nodes document_ptr (cast new_character_data_ptr # disc_nodes); + return new_character_data_ptr + }" +end + +locale l_create_character_data_defs = + fixes create_character_data :: "(_) document_ptr \ string \ (_, (_) character_data_ptr) dom_prog" + +global_interpretation l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs set_val set_val_locs get_disconnected_nodes + get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs + defines create_character_data = "l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_character_data + set_val get_disconnected_nodes set_disconnected_nodes" + . + +locale l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + + l_create_character_data_defs + + assumes create_character_data_impl: "create_character_data = a_create_character_data" +begin +lemmas create_character_data_def = a_create_character_data_def[folded create_character_data_impl] +end + +locale l_create_character_data = l_create_character_data_defs + +interpretation + i_create_character_data?: l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_val set_val_locs get_disconnected_nodes + get_disconnected_nodes_locs set_disconnected_nodes + set_disconnected_nodes_locs create_character_data + by unfold_locales (simp add: create_character_data_def) +declare l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances] + + + +subsubsection \create\_character\_data\ + +locale l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +begin +definition a_create_document :: "(_, (_) document_ptr) dom_prog" + where + "a_create_document = new_document" +end + +locale l_create_document_defs = + fixes create_document :: "(_, (_) document_ptr) dom_prog" + +global_interpretation l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + defines create_document = "l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_document" + . + +locale l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs + + l_create_document_defs + + assumes create_document_impl: "create_document = a_create_document" +begin +lemmas + create_document_def = create_document_impl[unfolded create_document_def, unfolded a_create_document_def] +end + +locale l_create_document = l_create_document_defs + +interpretation + i_create_document?: l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_document + by(simp add: l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) +declare l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances] + + +subsubsection \tree\_order\ + +locale l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_get_child_nodes_defs get_child_nodes get_child_nodes_locs + for get_child_nodes :: "(_::linorder) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" +begin +partial_function (dom_prog) a_to_tree_order :: "(_) object_ptr \ (_, (_) object_ptr list) dom_prog" + where + "a_to_tree_order ptr = (do { + children \ get_child_nodes ptr; + treeorders \ map_M a_to_tree_order (map (cast) children); + return (ptr # concat treeorders) + })" +end + +locale l_to_tree_order_defs = + fixes to_tree_order :: "(_) object_ptr \ (_, (_) object_ptr list) dom_prog" + +global_interpretation l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_child_nodes get_child_nodes_locs defines + to_tree_order = "l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_to_tree_order get_child_nodes" . +declare a_to_tree_order.simps [code] + +locale l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs + + l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_child_nodes get_child_nodes_locs + + l_to_tree_order_defs to_tree_order + for known_ptr :: "(_::linorder) object_ptr \ bool" + and type_wf :: "(_) heap \ bool" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and to_tree_order :: "(_) object_ptr \ ((_) heap, exception, (_) object_ptr list) prog" + + assumes to_tree_order_impl: "to_tree_order = a_to_tree_order" +begin +lemmas to_tree_order_def = a_to_tree_order.simps[folded to_tree_order_impl] + +lemma to_tree_order_pure [simp]: "pure (to_tree_order ptr) h" +proof - + have "\ptr h h' x. h \ to_tree_order ptr \\<^sub>r x \ h \ to_tree_order ptr \\<^sub>h h' \ h = h'" + proof (induct rule: a_to_tree_order.fixp_induct[folded to_tree_order_impl]) + case 1 + then show ?case + by (rule admissible_dom_prog) + next + case 2 + then show ?case + by simp + next + case (3 f) + then have "\x h. pure (f x) h" + by (metis is_OK_returns_heap_E is_OK_returns_result_E pure_def) + then have "\xs h. pure (map_M f xs) h" + by(rule map_M_pure_I) + then show ?case + by(auto elim!: bind_returns_heap_E2) + qed + then show ?thesis + unfolding pure_def + by (metis is_OK_returns_heap_E is_OK_returns_result_E) +qed +end + +locale l_to_tree_order = + fixes to_tree_order :: "(_) object_ptr \ (_, (_) object_ptr list) dom_prog" + assumes to_tree_order_pure [simp]: "pure (to_tree_order ptr) h" + +interpretation + i_to_tree_order?: l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs + to_tree_order + apply(unfold_locales) + by (simp add: to_tree_order_def) +declare l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma to_tree_order_is_l_to_tree_order [instances]: "l_to_tree_order to_tree_order" + using to_tree_order_pure l_to_tree_order_def by blast + + + +subsubsection \first\_in\_tree\_order\ + +locale l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_to_tree_order_defs to_tree_order + for to_tree_order :: "(_) object_ptr \ ((_) heap, exception, (_) object_ptr list) prog" +begin +definition a_first_in_tree_order :: "(_) object_ptr \ ((_) object_ptr + \ (_, 'result option) dom_prog) \ (_, 'result option) dom_prog" + where + "a_first_in_tree_order ptr f = (do { + tree_order \ to_tree_order ptr; + results \ map_filter_M f tree_order; + (case results of + [] \ return None + | x#_\ return (Some x)) + })" +end + +locale l_first_in_tree_order_defs = + fixes first_in_tree_order :: "(_) object_ptr \ ((_) object_ptr \ (_, 'result option) dom_prog) + \ (_, 'result option) dom_prog" + +global_interpretation l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs to_tree_order defines + first_in_tree_order = "l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_first_in_tree_order to_tree_order" . + +locale l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs to_tree_order + + l_first_in_tree_order_defs first_in_tree_order + for to_tree_order :: "(_) object_ptr \ ((_) heap, exception, (_) object_ptr list) prog" + and first_in_tree_order :: "(_) object_ptr \ ((_) object_ptr \ ((_) heap, exception, 'result option) prog) + \ ((_) heap, exception, 'result option) prog" + +assumes first_in_tree_order_impl: "first_in_tree_order = a_first_in_tree_order" +begin +lemmas first_in_tree_order_def = first_in_tree_order_impl[unfolded a_first_in_tree_order_def] +end + +locale l_first_in_tree_order + +interpretation i_first_in_tree_order?: + l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order + by unfold_locales (simp add: first_in_tree_order_def) +declare l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + + + +subsubsection \get\_element\_by\ + +locale l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_first_in_tree_order_defs first_in_tree_order + + l_to_tree_order_defs to_tree_order + + l_get_attribute_defs get_attribute get_attribute_locs + for to_tree_order :: "(_::linorder) object_ptr \ ((_) heap, exception, (_) object_ptr list) prog" + and first_in_tree_order :: "(_) object_ptr \ ((_) object_ptr + \ ((_) heap, exception, (_) element_ptr option) prog) + \ ((_) heap, exception, (_) element_ptr option) prog" + and get_attribute :: "(_) element_ptr \ char list \ ((_) heap, exception, char list option) prog" + and get_attribute_locs :: "(_) element_ptr \ ((_) heap \ (_) heap \ bool) set" +begin +definition a_get_element_by_id :: "(_) object_ptr \ attr_value \ (_, (_) element_ptr option) dom_prog" + where + "a_get_element_by_id ptr iden = first_in_tree_order ptr (\ptr. (case cast ptr of + Some element_ptr \ do { + id_opt \ get_attribute element_ptr ''id''; + (if id_opt = Some iden then return (Some element_ptr) else return None) + } + | _ \ return None + ))" + +definition a_get_elements_by_class_name :: "(_) object_ptr \ attr_value \ (_, (_) element_ptr list) dom_prog" + where + "a_get_elements_by_class_name ptr class_name = to_tree_order ptr \ + map_filter_M (\ptr. (case cast ptr of + Some element_ptr \ do { + class_name_opt \ get_attribute element_ptr ''class''; + (if class_name_opt = Some class_name then return (Some element_ptr) else return None) + } + | _ \ return None))" + +definition a_get_elements_by_tag_name :: "(_) object_ptr \ attr_value \ (_, (_) element_ptr list) dom_prog" + where + "a_get_elements_by_tag_name ptr tag_name = to_tree_order ptr \ + map_filter_M (\ptr. (case cast ptr of + Some element_ptr \ do { + this_tag_name \ get_M element_ptr tag_type; + (if this_tag_name = tag_name then return (Some element_ptr) else return None) + } + | _ \ return None))" +end + +locale l_get_element_by_defs = + fixes get_element_by_id :: "(_) object_ptr \ attr_value \ (_, (_) element_ptr option) dom_prog" + fixes get_elements_by_class_name :: "(_) object_ptr \ attr_value \ (_, (_) element_ptr list) dom_prog" + fixes get_elements_by_tag_name :: "(_) object_ptr \ attr_value \ (_, (_) element_ptr list) dom_prog" + +global_interpretation +l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs to_tree_order first_in_tree_order get_attribute get_attribute_locs +defines + get_element_by_id = "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_element_by_id first_in_tree_order get_attribute" +and + get_elements_by_class_name = "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_elements_by_class_name to_tree_order get_attribute" +and + get_elements_by_tag_name = "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_elements_by_tag_name to_tree_order" . + +locale l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs to_tree_order first_in_tree_order get_attribute get_attribute_locs + + l_get_element_by_defs get_element_by_id get_elements_by_class_name get_elements_by_tag_name + + l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order + + l_to_tree_order to_tree_order + + l_get_attribute type_wf get_attribute get_attribute_locs + for to_tree_order :: "(_::linorder) object_ptr \ ((_) heap, exception, (_) object_ptr list) prog" + and first_in_tree_order :: "(_) object_ptr \ ((_) object_ptr \ ((_) heap, exception, (_) element_ptr option) prog) + \ ((_) heap, exception, (_) element_ptr option) prog" + and get_attribute :: "(_) element_ptr \ char list \ ((_) heap, exception, char list option) prog" + and get_attribute_locs :: "(_) element_ptr \ ((_) heap \ (_) heap \ bool) set" + and get_element_by_id :: "(_) object_ptr \ char list \ ((_) heap, exception, (_) element_ptr option) prog" + and get_elements_by_class_name :: "(_) object_ptr \ char list \ ((_) heap, exception, (_) element_ptr list) prog" + and get_elements_by_tag_name :: "(_) object_ptr \ char list \ ((_) heap, exception, (_) element_ptr list) prog" + and type_wf :: "(_) heap \ bool" + + assumes get_element_by_id_impl: "get_element_by_id = a_get_element_by_id" + assumes get_elements_by_class_name_impl: "get_elements_by_class_name = a_get_elements_by_class_name" + assumes get_elements_by_tag_name_impl: "get_elements_by_tag_name = a_get_elements_by_tag_name" +begin +lemmas + get_element_by_id_def = get_element_by_id_impl[unfolded a_get_element_by_id_def] +lemmas + get_elements_by_class_name_def = get_elements_by_class_name_impl[unfolded a_get_elements_by_class_name_def] +lemmas + get_elements_by_tag_name_def = get_elements_by_tag_name_impl[unfolded a_get_elements_by_tag_name_def] + +lemma get_element_by_id_result_in_tree_order: + assumes "h \ get_element_by_id ptr iden \\<^sub>r Some element_ptr" + assumes "h \ to_tree_order ptr \\<^sub>r to" + shows "cast element_ptr \ set to" + using assms + by(auto simp add: get_element_by_id_def first_in_tree_order_def + elim!: map_filter_M_pure_E[where y=element_ptr] bind_returns_result_E2 + dest!: bind_returns_result_E3[rotated, OF assms(2), rotated] + intro!: map_filter_M_pure map_M_pure_I bind_pure_I + split: option.splits list.splits if_splits) + +lemma get_elements_by_tag_name_pure [simp]: "pure (get_elements_by_tag_name ptr tag_name) h" + by(auto simp add: get_elements_by_tag_name_def + intro!: bind_pure_I map_filter_M_pure + split: option.splits) +end + +locale l_get_element_by = l_get_element_by_defs + l_to_tree_order_defs + + assumes get_element_by_id_result_in_tree_order: + "h \ get_element_by_id ptr iden \\<^sub>r Some element_ptr \ h \ to_tree_order ptr \\<^sub>r to + \ cast element_ptr \ set to" + assumes get_elements_by_tag_name_pure [simp]: "pure (get_elements_by_tag_name ptr tag_name) h" + +interpretation + i_get_element_by?: l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order get_attribute + get_attribute_locs get_element_by_id get_elements_by_class_name + get_elements_by_tag_name type_wf + using instances + apply(simp add: l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def) + by(simp add: get_element_by_id_def get_elements_by_class_name_def get_elements_by_tag_name_def) +declare l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma get_element_by_is_l_get_element_by [instances]: + "l_get_element_by get_element_by_id get_elements_by_tag_name to_tree_order" + apply(unfold_locales) + using get_element_by_id_result_in_tree_order get_elements_by_tag_name_pure + by fast+ +end diff --git a/Core_DOM/Core_DOM_Heap_WF.thy b/Core_DOM/Core_DOM_Heap_WF.thy new file mode 100644 index 0000000..c9a9f93 --- /dev/null +++ b/Core_DOM/Core_DOM_Heap_WF.thy @@ -0,0 +1,6209 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Wellformedness\ +text\In this theory, we discuss the wellformedness of the DOM. First, we define +wellformedness and, second, we show for all functions for querying and modifying the +DOM to what extend they preserve wellformendess.\ + +theory Core_DOM_Heap_WF +imports + "Core_DOM_Functions" +begin + +locale l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs = + l_get_child_nodes_defs get_child_nodes get_child_nodes_locs + + l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs + for get_child_nodes :: "(_::linorder) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" +begin +definition a_owner_document_valid :: "(_) heap \ bool" + where + "a_owner_document_valid h = (\node_ptr. node_ptr |\| node_ptr_kinds h \ + ((\document_ptr. document_ptr |\| document_ptr_kinds h + \ node_ptr \ set |h \ get_disconnected_nodes document_ptr|\<^sub>r) + \ (\parent_ptr. parent_ptr |\| object_ptr_kinds h + \ node_ptr \ set |h \ get_child_nodes parent_ptr|\<^sub>r)))" + + +definition a_parent_child_rel :: "(_) heap \ ((_) object_ptr \ (_) object_ptr) set" + where + "a_parent_child_rel h = {(parent, child). parent |\| object_ptr_kinds h + \ child \ cast ` set |h \ get_child_nodes parent|\<^sub>r}" + +definition a_acyclic_heap :: "(_) heap \ bool" + where + "a_acyclic_heap h = acyclic (a_parent_child_rel h)" + + +definition a_all_ptrs_in_heap :: "(_) heap \ bool" + where + "a_all_ptrs_in_heap h = ((\ptr children. (h \ get_child_nodes ptr \\<^sub>r children) + \ fset_of_list children |\| node_ptr_kinds h) + \ (\document_ptr disc_node_ptrs. (h \ get_disconnected_nodes document_ptr \\<^sub>r disc_node_ptrs) + \ fset_of_list disc_node_ptrs |\| node_ptr_kinds h))" + + +definition a_distinct_lists :: "(_) heap \ bool" + where + "a_distinct_lists h = distinct (concat ( + (map (\ptr. |h \ get_child_nodes ptr|\<^sub>r) |h \ object_ptr_kinds_M|\<^sub>r) + @ (map (\document_ptr. |h \ get_disconnected_nodes document_ptr|\<^sub>r) |h \ document_ptr_kinds_M|\<^sub>r) + ))" + +definition a_heap_is_wellformed :: "(_) heap \ bool" + where + "a_heap_is_wellformed h \ + a_acyclic_heap h \ a_all_ptrs_in_heap h \ a_distinct_lists h \ a_owner_document_valid h" +end + +locale l_heap_is_wellformed_defs = + fixes heap_is_wellformed :: "(_) heap \ bool" + fixes parent_child_rel :: "(_) heap \ ((_) object_ptr \ (_) object_ptr) set" + +global_interpretation l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_child_nodes get_child_nodes_locs + get_disconnected_nodes get_disconnected_nodes_locs +defines heap_is_wellformed = "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_heap_is_wellformed get_child_nodes + get_disconnected_nodes" + and parent_child_rel = "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_parent_child_rel get_child_nodes" + . + +locale l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs + + l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_child_nodes get_child_nodes_locs get_disconnected_nodes + get_disconnected_nodes_locs + + l_heap_is_wellformed_defs heap_is_wellformed parent_child_rel + + l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs + for known_ptr :: "(_::linorder) object_ptr \ bool" + and type_wf :: "(_) heap \ bool" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and heap_is_wellformed :: "(_) heap \ bool" + and parent_child_rel :: "(_) heap \ ((_) object_ptr \ (_) object_ptr) set" + + assumes heap_is_wellformed_impl: "heap_is_wellformed = a_heap_is_wellformed" + assumes parent_child_rel_impl: "parent_child_rel = a_parent_child_rel" +begin +lemmas heap_is_wellformed_def = heap_is_wellformed_impl[unfolded a_heap_is_wellformed_def] +lemmas parent_child_rel_def = parent_child_rel_impl[unfolded a_parent_child_rel_def] +lemmas acyclic_heap_def = a_acyclic_heap_def[folded parent_child_rel_impl] + +lemma parent_child_rel_node_ptr: + "(parent, child) \ parent_child_rel h \ is_node_ptr_kind child" + by(auto simp add: parent_child_rel_def) + +lemma parent_child_rel_child_nodes: + assumes "known_ptr parent" + and "h \ get_child_nodes parent \\<^sub>r children" + and "child \ set children" + shows "(parent, cast child) \ parent_child_rel h" + using assms + apply(auto simp add: parent_child_rel_def is_OK_returns_result_I )[1] + using get_child_nodes_ptr_in_heap by blast + +lemma parent_child_rel_child_nodes2: + assumes "known_ptr parent" + and "h \ get_child_nodes parent \\<^sub>r children" + and "child \ set children" + and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child = child_obj" + shows "(parent, child_obj) \ parent_child_rel h" + using assms parent_child_rel_child_nodes by blast + + +lemma parent_child_rel_finite: "finite (parent_child_rel h)" +proof - + have "parent_child_rel h = (\ptr \ set |h \ object_ptr_kinds_M|\<^sub>r. + (\child \ set |h \ get_child_nodes ptr|\<^sub>r. {(ptr, cast child)}))" + by(auto simp add: parent_child_rel_def) + moreover have "finite (\ptr \ set |h \ object_ptr_kinds_M|\<^sub>r. + (\child \ set |h \ get_child_nodes ptr|\<^sub>r. {(ptr, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)}))" + by simp + ultimately show ?thesis + by simp +qed + +lemma distinct_lists_no_parent: + assumes "a_distinct_lists h" + assumes "h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes" + assumes "node_ptr \ set disc_nodes" + shows "\(\parent_ptr. parent_ptr |\| object_ptr_kinds h + \ node_ptr \ set |h \ get_child_nodes parent_ptr|\<^sub>r)" + using assms + apply(auto simp add: a_distinct_lists_def)[1] +proof - + fix parent_ptr :: "(_) object_ptr" + assume a1: "parent_ptr |\| object_ptr_kinds h" + assume a2: "(\x\fset (object_ptr_kinds h). + set |h \ get_child_nodes x|\<^sub>r) \ (\x\fset (document_ptr_kinds h). + set |h \ get_disconnected_nodes x|\<^sub>r) = {}" + assume a3: "h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes" + assume a4: "node_ptr \ set disc_nodes" + assume a5: "node_ptr \ set |h \ get_child_nodes parent_ptr|\<^sub>r" + have f6: "parent_ptr \ fset (object_ptr_kinds h)" + using a1 by auto + have f7: "document_ptr \ fset (document_ptr_kinds h)" + using a3 by (meson fmember.rep_eq get_disconnected_nodes_ptr_in_heap is_OK_returns_result_I) + have "|h \ get_disconnected_nodes document_ptr|\<^sub>r = disc_nodes" + using a3 by simp + then show False + using f7 f6 a5 a4 a2 by blast +qed + + +lemma distinct_lists_disconnected_nodes: + assumes "a_distinct_lists h" + and "h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes" + shows "distinct disc_nodes" +proof - + have h1: "distinct (concat (map (\document_ptr. |h \ get_disconnected_nodes document_ptr|\<^sub>r) + |h \ document_ptr_kinds_M|\<^sub>r))" + using assms(1) + by(simp add: a_distinct_lists_def) + then show ?thesis + using concat_map_all_distinct[OF h1] assms(2) is_OK_returns_result_I get_disconnected_nodes_ok + by (metis (no_types, lifting) DocumentMonad.ptr_kinds_ptr_kinds_M + l_get_disconnected_nodes.get_disconnected_nodes_ptr_in_heap + l_get_disconnected_nodes_axioms select_result_I2) +qed + +lemma distinct_lists_children: + assumes "a_distinct_lists h" + and "known_ptr ptr" + and "h \ get_child_nodes ptr \\<^sub>r children" + shows "distinct children" +proof (cases "children = []", simp) + assume "children \ []" + have h1: "distinct (concat ((map (\ptr. |h \ get_child_nodes ptr|\<^sub>r) |h \ object_ptr_kinds_M|\<^sub>r)))" + using assms(1) + by(simp add: a_distinct_lists_def) + show ?thesis + using concat_map_all_distinct[OF h1] assms(2) assms(3) + by (metis (no_types, lifting) ObjectMonad.ptr_kinds_ptr_kinds_M get_child_nodes_ptr_in_heap + is_OK_returns_result_I select_result_I2) +qed + +lemma heap_is_wellformed_children_in_heap: + assumes "heap_is_wellformed h" + assumes "h \ get_child_nodes ptr \\<^sub>r children" + assumes "child \ set children" + shows "child |\| node_ptr_kinds h" + using assms + apply(auto simp add: heap_is_wellformed_def a_all_ptrs_in_heap_def)[1] + by (meson fset_of_list_elem fset_rev_mp) + +lemma heap_is_wellformed_one_parent: + assumes "heap_is_wellformed h" + assumes "h \ get_child_nodes ptr \\<^sub>r children" + assumes "h \ get_child_nodes ptr' \\<^sub>r children'" + assumes "set children \ set children' \ {}" + shows "ptr = ptr'" + using assms +proof (auto simp add: heap_is_wellformed_def a_distinct_lists_def)[1] + fix x :: "(_) node_ptr" + assume a1: "ptr \ ptr'" + assume a2: "h \ get_child_nodes ptr \\<^sub>r children" + assume a3: "h \ get_child_nodes ptr' \\<^sub>r children'" + assume a4: "distinct (concat (map (\ptr. |h \ get_child_nodes ptr|\<^sub>r) + (sorted_list_of_set (fset (object_ptr_kinds h)))))" + have f5: "|h \ get_child_nodes ptr|\<^sub>r = children" + using a2 by simp + have "|h \ get_child_nodes ptr'|\<^sub>r = children'" + using a3 by (meson select_result_I2) + then have "ptr \ set (sorted_list_of_set (fset (object_ptr_kinds h))) + \ ptr' \ set (sorted_list_of_set (fset (object_ptr_kinds h))) + \ set children \ set children' = {}" + using f5 a4 a1 by (meson distinct_concat_map_E(1)) + then show False + using a3 a2 by (metis (no_types) assms(4) finite_fset fmember.rep_eq is_OK_returns_result_I + local.get_child_nodes_ptr_in_heap set_sorted_list_of_set) +qed + +lemma parent_child_rel_child: + "h \ get_child_nodes ptr \\<^sub>r children \ child \ set children \ (ptr, cast child) \ parent_child_rel h" + by (simp add: is_OK_returns_result_I get_child_nodes_ptr_in_heap parent_child_rel_def) + +lemma parent_child_rel_acyclic: "heap_is_wellformed h \ acyclic (parent_child_rel h)" + by (simp add: acyclic_heap_def local.heap_is_wellformed_def) + +lemma heap_is_wellformed_disconnected_nodes_distinct: + "heap_is_wellformed h \ h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes \ distinct disc_nodes" + using distinct_lists_disconnected_nodes local.heap_is_wellformed_def by blast + +lemma parent_child_rel_parent_in_heap: + "(parent, child_ptr) \ parent_child_rel h \ parent |\| object_ptr_kinds h" + using local.parent_child_rel_def by blast + +lemma parent_child_rel_child_in_heap: + "heap_is_wellformed h \ type_wf h \ known_ptr parent + \ (parent, child_ptr) \ parent_child_rel h \ child_ptr |\| object_ptr_kinds h" + apply(auto simp add: heap_is_wellformed_def parent_child_rel_def a_all_ptrs_in_heap_def)[1] + using get_child_nodes_ok + by (meson fin_mono fset_of_list_elem returns_result_select_result) + +lemma heap_is_wellformed_disc_nodes_in_heap: + "heap_is_wellformed h \ h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes + \ node \ set disc_nodes \ node |\| node_ptr_kinds h" + by (meson fset_mp fset_of_list_elem local.a_all_ptrs_in_heap_def local.heap_is_wellformed_def) + +lemma heap_is_wellformed_one_disc_parent: + "heap_is_wellformed h \ h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes + \ h \ get_disconnected_nodes document_ptr' \\<^sub>r disc_nodes' + \ set disc_nodes \ set disc_nodes' \ {} \ document_ptr = document_ptr'" + using DocumentMonad.ptr_kinds_ptr_kinds_M concat_append distinct_append distinct_concat_map_E(1) + is_OK_returns_result_I local.a_distinct_lists_def local.get_disconnected_nodes_ptr_in_heap + local.heap_is_wellformed_def select_result_I2 +proof - + assume a1: "heap_is_wellformed h" + assume a2: "h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes" + assume a3: "h \ get_disconnected_nodes document_ptr' \\<^sub>r disc_nodes'" + assume a4: "set disc_nodes \ set disc_nodes' \ {}" + have f5: "|h \ get_disconnected_nodes document_ptr|\<^sub>r = disc_nodes" + using a2 by (meson select_result_I2) + have f6: "|h \ get_disconnected_nodes document_ptr'|\<^sub>r = disc_nodes'" + using a3 by (meson select_result_I2) + have "\nss nssa. \ distinct (concat (nss @ nssa)) \ distinct (concat nssa::(_) node_ptr list)" + by (metis (no_types) concat_append distinct_append) + then have "distinct (concat (map (\d. |h \ get_disconnected_nodes d|\<^sub>r) |h \ document_ptr_kinds_M|\<^sub>r))" + using a1 local.a_distinct_lists_def local.heap_is_wellformed_def by blast + then show ?thesis + using f6 f5 a4 a3 a2 by (meson DocumentMonad.ptr_kinds_ptr_kinds_M distinct_concat_map_E(1) + is_OK_returns_result_I local.get_disconnected_nodes_ptr_in_heap) +qed + +lemma heap_is_wellformed_children_disc_nodes_different: + "heap_is_wellformed h \ h \ get_child_nodes ptr \\<^sub>r children + \ h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes + \ set children \ set disc_nodes = {}" + by (metis (no_types, hide_lams) disjoint_iff_not_equal distinct_lists_no_parent + is_OK_returns_result_I local.get_child_nodes_ptr_in_heap + local.heap_is_wellformed_def select_result_I2) + +lemma heap_is_wellformed_children_disc_nodes: + "heap_is_wellformed h \ node_ptr |\| node_ptr_kinds h + \ \(\parent \ fset (object_ptr_kinds h). node_ptr \ set |h \ get_child_nodes parent|\<^sub>r) + \ (\document_ptr \ fset (document_ptr_kinds h). node_ptr \ set |h \ get_disconnected_nodes document_ptr|\<^sub>r)" + apply(auto simp add: heap_is_wellformed_def a_distinct_lists_def a_owner_document_valid_def)[1] + by (meson fmember.rep_eq) +lemma heap_is_wellformed_children_distinct: + "heap_is_wellformed h \ h \ get_child_nodes ptr \\<^sub>r children \ distinct children" + by (metis (no_types, lifting) ObjectMonad.ptr_kinds_ptr_kinds_M concat_append distinct_append + distinct_concat_map_E(2) is_OK_returns_result_I local.a_distinct_lists_def + local.get_child_nodes_ptr_in_heap local.heap_is_wellformed_def + select_result_I2) +end + +locale l_heap_is_wellformed = l_type_wf + l_known_ptr + l_heap_is_wellformed_defs + + l_get_child_nodes_defs + l_get_disconnected_nodes_defs + +assumes heap_is_wellformed_children_in_heap: + "heap_is_wellformed h \ h \ get_child_nodes ptr \\<^sub>r children \ child \ set children + \ child |\| node_ptr_kinds h" +assumes heap_is_wellformed_disc_nodes_in_heap: + "heap_is_wellformed h \ h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes + \ node \ set disc_nodes \ node |\| node_ptr_kinds h" +assumes heap_is_wellformed_one_parent: + "heap_is_wellformed h \ h \ get_child_nodes ptr \\<^sub>r children + \ h \ get_child_nodes ptr' \\<^sub>r children' + \ set children \ set children' \ {} \ ptr = ptr'" +assumes heap_is_wellformed_one_disc_parent: + "heap_is_wellformed h \ h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes + \ h \ get_disconnected_nodes document_ptr' \\<^sub>r disc_nodes' + \ set disc_nodes \ set disc_nodes' \ {} \ document_ptr = document_ptr'" +assumes heap_is_wellformed_children_disc_nodes_different: + "heap_is_wellformed h \ h \ get_child_nodes ptr \\<^sub>r children + \ h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes + \ set children \ set disc_nodes = {}" +assumes heap_is_wellformed_disconnected_nodes_distinct: + "heap_is_wellformed h \ h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes + \ distinct disc_nodes" +assumes heap_is_wellformed_children_distinct: + "heap_is_wellformed h \ h \ get_child_nodes ptr \\<^sub>r children \ distinct children" +assumes heap_is_wellformed_children_disc_nodes: + "heap_is_wellformed h \ node_ptr |\| node_ptr_kinds h + \ \(\parent \ fset (object_ptr_kinds h). node_ptr \ set |h \ get_child_nodes parent|\<^sub>r) + \ (\document_ptr \ fset (document_ptr_kinds h). node_ptr \ set |h \ get_disconnected_nodes document_ptr|\<^sub>r)" +assumes parent_child_rel_child: + "h \ get_child_nodes ptr \\<^sub>r children + \ child \ set children \ (ptr, cast child) \ parent_child_rel h" +assumes parent_child_rel_finite: + "heap_is_wellformed h \ finite (parent_child_rel h)" +assumes parent_child_rel_acyclic: + "heap_is_wellformed h \ acyclic (parent_child_rel h)" +assumes parent_child_rel_node_ptr: + "(parent, child_ptr) \ parent_child_rel h \ is_node_ptr_kind child_ptr" +assumes parent_child_rel_parent_in_heap: + "(parent, child_ptr) \ parent_child_rel h \ parent |\| object_ptr_kinds h" +assumes parent_child_rel_child_in_heap: + "heap_is_wellformed h \ type_wf h \ known_ptr parent + \ (parent, child_ptr) \ parent_child_rel h \ child_ptr |\| object_ptr_kinds h" + +interpretation i_heap_is_wellformed?: l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes + get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs + heap_is_wellformed parent_child_rel + apply(unfold_locales) + by(auto simp add: heap_is_wellformed_def parent_child_rel_def) +declare l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + + +lemma heap_is_wellformed_is_l_heap_is_wellformed [instances]: + "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes + get_disconnected_nodes" + apply(auto simp add: l_heap_is_wellformed_def)[1] + using heap_is_wellformed_children_in_heap + apply blast + using heap_is_wellformed_disc_nodes_in_heap + apply blast + using heap_is_wellformed_one_parent + apply blast + using heap_is_wellformed_one_disc_parent + apply blast + using heap_is_wellformed_children_disc_nodes_different + apply blast + using heap_is_wellformed_disconnected_nodes_distinct + apply blast + using heap_is_wellformed_children_distinct + apply blast + using heap_is_wellformed_children_disc_nodes + apply blast + using parent_child_rel_child + apply (blast) + using parent_child_rel_child + apply(blast) + using parent_child_rel_finite + apply blast + using parent_child_rel_acyclic + apply blast + using parent_child_rel_node_ptr + apply blast + using parent_child_rel_parent_in_heap + apply blast + using parent_child_rel_child_in_heap + apply blast + done + +subsection \get\_parent\ + +locale l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs + + l_heap_is_wellformed + type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs + get_disconnected_nodes get_disconnected_nodes_locs + for known_ptr :: "(_::linorder) object_ptr \ bool" + and type_wf :: "(_) heap \ bool" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and known_ptrs :: "(_) heap \ bool" + and get_parent :: "(_) node_ptr \ ((_) heap, exception, (_) object_ptr option) prog" + and get_parent_locs :: "((_) heap \ (_) heap \ bool) set" + and heap_is_wellformed :: "(_) heap \ bool" + and parent_child_rel :: "(_) heap \ ((_) object_ptr \ (_) object_ptr) set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" +begin +lemma child_parent_dual: + assumes heap_is_wellformed: "heap_is_wellformed h" + assumes "h \ get_child_nodes ptr \\<^sub>r children" + assumes "child \ set children" + assumes "known_ptrs h" + assumes type_wf: "type_wf h" + shows "h \ get_parent child \\<^sub>r Some ptr" +proof - + obtain ptrs where ptrs: "h \ object_ptr_kinds_M \\<^sub>r ptrs" + by(simp add: object_ptr_kinds_M_defs) + then have h1: "ptr \ set ptrs" + using get_child_nodes_ok assms(2) is_OK_returns_result_I + by (metis (no_types, hide_lams) ObjectMonad.ptr_kinds_ptr_kinds_M + \\thesis. (\ptrs. h \ object_ptr_kinds_M \\<^sub>r ptrs \ thesis) \ thesis\ + get_child_nodes_ptr_in_heap returns_result_eq select_result_I2) + + let ?P = "(\ptr. get_child_nodes ptr \ (\children. return (child \ set children)))" + let ?filter = "filter_M ?P ptrs" + + have "h \ ok ?filter" + using ptrs type_wf + using get_child_nodes_ok + apply(auto intro!: filter_M_is_OK_I bind_is_OK_pure_I get_child_nodes_ok simp add: bind_pure_I)[1] + using assms(4) local.known_ptrs_known_ptr by blast + then obtain parent_ptrs where parent_ptrs: "h \ ?filter \\<^sub>r parent_ptrs" + by auto + + have h5: "\!x. x \ set ptrs \ h \ Heap_Error_Monad.bind (get_child_nodes x) + (\children. return (child \ set children)) \\<^sub>r True" + apply(auto intro!: bind_pure_returns_result_I)[1] + using heap_is_wellformed_one_parent + proof - + have "h \ (return (child \ set children)::((_) heap, exception, bool) prog) \\<^sub>r True" + by (simp add: assms(3)) + then show + "\z. z \ set ptrs \ h \ Heap_Error_Monad.bind (get_child_nodes z) + (\ns. return (child \ set ns)) \\<^sub>r True" + by (metis (no_types) assms(2) bind_pure_returns_result_I2 h1 is_OK_returns_result_I + local.get_child_nodes_pure select_result_I2) + next + fix x y + assume 0: "x \ set ptrs" + and 1: "h \ Heap_Error_Monad.bind (get_child_nodes x) + (\children. return (child \ set children)) \\<^sub>r True" + and 2: "y \ set ptrs" + and 3: "h \ Heap_Error_Monad.bind (get_child_nodes y) + (\children. return (child \ set children)) \\<^sub>r True" + and 4: "(\h ptr children ptr' children'. heap_is_wellformed h + \ h \ get_child_nodes ptr \\<^sub>r children \ h \ get_child_nodes ptr' \\<^sub>r children' + \ set children \ set children' \ {} \ ptr = ptr')" + then show "x = y" + by (metis (no_types, lifting) bind_returns_result_E disjoint_iff_not_equal heap_is_wellformed + return_returns_result) + qed + + have "child |\| node_ptr_kinds h" + using heap_is_wellformed_children_in_heap heap_is_wellformed assms(2) assms(3) + by fast + moreover have "parent_ptrs = [ptr]" + apply(rule filter_M_ex1[OF parent_ptrs h1 h5]) + using ptrs assms(2) assms(3) + by(auto simp add: object_ptr_kinds_M_defs bind_pure_I intro!: bind_pure_returns_result_I) + ultimately show ?thesis + using ptrs parent_ptrs + by(auto simp add: bind_pure_I get_parent_def + elim!: bind_returns_result_E2 + intro!: bind_pure_returns_result_I filter_M_pure_I) (*slow, ca 1min *) +qed + +lemma parent_child_rel_parent: + assumes "heap_is_wellformed h" + and "h \ get_parent child_node \\<^sub>r Some parent" + shows "(parent, cast child_node) \ parent_child_rel h" + using assms parent_child_rel_child get_parent_child_dual by auto + +lemma heap_wellformed_induct [consumes 1, case_names step]: + assumes "heap_is_wellformed h" + and step: "\parent. (\children child. h \ get_child_nodes parent \\<^sub>r children + \ child \ set children \ P (cast child)) \ P parent" + shows "P ptr" +proof - + fix ptr + have "wf ((parent_child_rel h)\)" + by (simp add: assms(1) finite_acyclic_wf_converse parent_child_rel_acyclic parent_child_rel_finite) + then show "?thesis" + proof (induct rule: wf_induct_rule) + case (less parent) + then show ?case + using assms child_parent_dual parent_child_rel_parent + by (meson converse_iff parent_child_rel_child) + qed +qed + +lemma heap_wellformed_induct2 [consumes 3, case_names not_in_heap empty_children step]: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + and not_in_heap: "\parent. parent |\| object_ptr_kinds h \ P parent" + and empty_children: "\parent. h \ get_child_nodes parent \\<^sub>r [] \ P parent" + and step: "\parent children child. h \ get_child_nodes parent \\<^sub>r children + \ child \ set children \ P (cast child) \ P parent" + shows "P ptr" +proof(insert assms(1), induct rule: heap_wellformed_induct) + case (step parent) + then show ?case + proof(cases "parent |\| object_ptr_kinds h") + case True + then obtain children where children: "h \ get_child_nodes parent \\<^sub>r children" + using get_child_nodes_ok assms(2) assms(3) + by (meson is_OK_returns_result_E local.known_ptrs_known_ptr) + then show ?thesis + proof (cases "children = []") + case True + then show ?thesis + using children empty_children + by simp + next + case False + then show ?thesis + using assms(6) children last_in_set step.hyps by blast + qed + next + case False + then show ?thesis + by (simp add: not_in_heap) + qed +qed + +lemma heap_wellformed_induct_rev [consumes 1, case_names step]: + assumes "heap_is_wellformed h" + and step: "\child. (\parent child_node. cast child_node = child + \ h \ get_parent child_node \\<^sub>r Some parent \ P parent) \ P child" + shows "P ptr" +proof - + fix ptr + have "wf ((parent_child_rel h))" + by (simp add: assms(1) local.parent_child_rel_acyclic local.parent_child_rel_finite + wf_iff_acyclic_if_finite) + + then show "?thesis" + proof (induct rule: wf_induct_rule) + case (less child) + show ?case + using assms get_parent_child_dual + by (metis less.hyps parent_child_rel_parent) + qed +qed +end + +interpretation i_get_parent_wf?: l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes + get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed + parent_child_rel get_disconnected_nodes + using instances + by(simp add: l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) +declare l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + + +locale l_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs + heap_is_wellformed parent_child_rel get_disconnected_nodes get_disconnected_nodes_locs + + l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes + get_disconnected_nodes_locs heap_is_wellformed parent_child_rel + for known_ptr :: "(_::linorder) object_ptr \ bool" + and type_wf :: "(_) heap \ bool" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and known_ptrs :: "(_) heap \ bool" + and get_parent :: "(_) node_ptr \ ((_) heap, exception, (_) object_ptr option) prog" + and get_parent_locs :: "((_) heap \ (_) heap \ bool) set" + and heap_is_wellformed :: "(_) heap \ bool" + and parent_child_rel :: "(_) heap \ ((_) object_ptr \ (_) object_ptr) set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" +begin +lemma preserves_wellformedness_writes_needed: + assumes heap_is_wellformed: "heap_is_wellformed h" + and "h \ f \\<^sub>h h'" + and "writes SW f h h'" + and preserved_get_child_nodes: + "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \object_ptr. \r \ get_child_nodes_locs object_ptr. r h h'" + and preserved_get_disconnected_nodes: + "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \document_ptr. \r \ get_disconnected_nodes_locs document_ptr. r h h'" + and preserved_object_pointers: + "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" +shows "heap_is_wellformed h'" +proof - + have object_ptr_kinds_eq3: "object_ptr_kinds h = object_ptr_kinds h'" + using assms(2) assms(3) object_ptr_kinds_preserved preserved_object_pointers by blast + then have object_ptr_kinds_eq: + "\ptrs. h \ object_ptr_kinds_M \\<^sub>r ptrs = h' \ object_ptr_kinds_M \\<^sub>r ptrs" + unfolding object_ptr_kinds_M_defs by simp + then have object_ptr_kinds_eq2: "|h \ object_ptr_kinds_M|\<^sub>r = |h' \ object_ptr_kinds_M|\<^sub>r" + using select_result_eq by force + then have node_ptr_kinds_eq2: "|h \ node_ptr_kinds_M|\<^sub>r = |h' \ node_ptr_kinds_M|\<^sub>r" + using node_ptr_kinds_M_eq by auto + then have node_ptr_kinds_eq3: "node_ptr_kinds h = node_ptr_kinds h'" + by auto + have document_ptr_kinds_eq2: "|h \ document_ptr_kinds_M|\<^sub>r = |h' \ document_ptr_kinds_M|\<^sub>r" + using object_ptr_kinds_eq2 document_ptr_kinds_M_eq by auto + then have document_ptr_kinds_eq3: "document_ptr_kinds h = document_ptr_kinds h'" + by auto + have children_eq: + "\ptr children. h \ get_child_nodes ptr \\<^sub>r children = h' \ get_child_nodes ptr \\<^sub>r children" + apply(rule reads_writes_preserved[OF get_child_nodes_reads assms(3) assms(2)]) + using preserved_get_child_nodes by fast + then have children_eq2: "\ptr. |h \ get_child_nodes ptr|\<^sub>r = |h' \ get_child_nodes ptr|\<^sub>r" + using select_result_eq by force + have disconnected_nodes_eq: + "\document_ptr disconnected_nodes. + h \ get_disconnected_nodes document_ptr \\<^sub>r disconnected_nodes + = h' \ get_disconnected_nodes document_ptr \\<^sub>r disconnected_nodes" + apply(rule reads_writes_preserved[OF get_disconnected_nodes_reads assms(3) assms(2)]) + using preserved_get_disconnected_nodes by fast + then have disconnected_nodes_eq2: + "\document_ptr. |h \ get_disconnected_nodes document_ptr|\<^sub>r + = |h' \ get_disconnected_nodes document_ptr|\<^sub>r" + using select_result_eq by force + have get_parent_eq: "\ptr parent. h \ get_parent ptr \\<^sub>r parent = h' \ get_parent ptr \\<^sub>r parent" + apply(rule reads_writes_preserved[OF get_parent_reads assms(3) assms(2)]) + using preserved_get_child_nodes preserved_object_pointers unfolding get_parent_locs_def by fast + have "a_acyclic_heap h" + using heap_is_wellformed by (simp add: heap_is_wellformed_def) + have "parent_child_rel h' \ parent_child_rel h" + proof + fix x + assume "x \ parent_child_rel h'" + then show "x \ parent_child_rel h" + by(simp add: parent_child_rel_def children_eq2 object_ptr_kinds_eq3) + qed + then have "a_acyclic_heap h'" + using \a_acyclic_heap h\ acyclic_heap_def acyclic_subset by blast + + moreover have "a_all_ptrs_in_heap h" + using heap_is_wellformed by (simp add: heap_is_wellformed_def) + then have "a_all_ptrs_in_heap h'" + by(auto simp add: a_all_ptrs_in_heap_def node_ptr_kinds_def node_ptr_kinds_eq2 + object_ptr_kinds_eq3 children_eq disconnected_nodes_eq) + + moreover have h0: "a_distinct_lists h" + using heap_is_wellformed by (simp add: heap_is_wellformed_def) + have h1: "map (\ptr. |h \ get_child_nodes ptr|\<^sub>r) (sorted_list_of_set (fset (object_ptr_kinds h))) + = map (\ptr. |h' \ get_child_nodes ptr|\<^sub>r) (sorted_list_of_set (fset (object_ptr_kinds h')))" + by (simp add: children_eq2 object_ptr_kinds_eq3) + have h2: "map (\document_ptr. |h \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h))) + = map (\document_ptr. |h' \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h')))" + using disconnected_nodes_eq document_ptr_kinds_eq2 select_result_eq by force + have "a_distinct_lists h'" + using h0 + by(simp add: a_distinct_lists_def h1 h2) + + moreover have "a_owner_document_valid h" + using heap_is_wellformed by (simp add: heap_is_wellformed_def) + then have "a_owner_document_valid h'" + by(auto simp add: a_owner_document_valid_def children_eq2 disconnected_nodes_eq2 + object_ptr_kinds_eq3 node_ptr_kinds_eq3 document_ptr_kinds_eq3) + ultimately show ?thesis + by (simp add: heap_is_wellformed_def) +qed +end + +interpretation i_get_parent_wf2?: l_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes + get_child_nodes_locs known_ptrs get_parent get_parent_locs + heap_is_wellformed parent_child_rel get_disconnected_nodes + get_disconnected_nodes_locs + using l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms + by (simp add: l_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) + +declare l_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] +locale l_get_parent_wf = l_type_wf + l_known_ptrs + l_heap_is_wellformed_defs + + l_get_child_nodes_defs + l_get_parent_defs + + assumes child_parent_dual: + "heap_is_wellformed h + \ type_wf h + \ known_ptrs h + \ h \ get_child_nodes ptr \\<^sub>r children + \ child \ set children + \ h \ get_parent child \\<^sub>r Some ptr" + assumes heap_wellformed_induct [consumes 1, case_names step]: + "heap_is_wellformed h + \ (\parent. (\children child. h \ get_child_nodes parent \\<^sub>r children + \ child \ set children \ P (cast child)) \ P parent) + \ P ptr" + assumes heap_wellformed_induct_rev [consumes 1, case_names step]: + "heap_is_wellformed h + \ (\child. (\parent child_node. cast child_node = child + \ h \ get_parent child_node \\<^sub>r Some parent \ P parent) \ P child) + \ P ptr" + assumes parent_child_rel_parent: "heap_is_wellformed h + \ h \ get_parent child_node \\<^sub>r Some parent + \ (parent, cast child_node) \ parent_child_rel h" + +lemma get_parent_wf_is_l_get_parent_wf [instances]: + "l_get_parent_wf type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel + get_child_nodes get_parent" + using known_ptrs_is_l_known_ptrs + apply(auto simp add: l_get_parent_wf_def l_get_parent_wf_axioms_def)[1] + using child_parent_dual heap_wellformed_induct heap_wellformed_induct_rev parent_child_rel_parent + by metis+ + + + +subsection \get\_disconnected\_nodes\ + + + +subsection \set\_disconnected\_nodes\ + + +subsubsection \get\_disconnected\_nodes\ + +locale l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_set_disconnected_nodes_get_disconnected_nodes + type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes + set_disconnected_nodes_locs + + l_heap_is_wellformed + type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs + get_disconnected_nodes get_disconnected_nodes_locs + for known_ptr :: "(_) object_ptr \ bool" + and type_wf :: "(_) heap \ bool" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap, exception, unit) prog set" + and heap_is_wellformed :: "(_) heap \ bool" + and parent_child_rel :: "(_) heap \ ((_) object_ptr \ (_) object_ptr) set" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" +begin + +lemma remove_from_disconnected_nodes_removes: + assumes "heap_is_wellformed h" + assumes "h \ get_disconnected_nodes ptr \\<^sub>r disc_nodes" + assumes "h \ set_disconnected_nodes ptr (remove1 node_ptr disc_nodes) \\<^sub>h h'" + assumes "h' \ get_disconnected_nodes ptr \\<^sub>r disc_nodes'" + shows "node_ptr \ set disc_nodes'" + using assms + by (metis distinct_remove1_removeAll heap_is_wellformed_disconnected_nodes_distinct + set_disconnected_nodes_get_disconnected_nodes member_remove remove_code(1) + returns_result_eq) +end + +locale l_set_disconnected_nodes_get_disconnected_nodes_wf = l_heap_is_wellformed + + l_set_disconnected_nodes_get_disconnected_nodes + + assumes remove_from_disconnected_nodes_removes: + "heap_is_wellformed h \ h \ get_disconnected_nodes ptr \\<^sub>r disc_nodes + \ h \ set_disconnected_nodes ptr (remove1 node_ptr disc_nodes) \\<^sub>h h' + \ h' \ get_disconnected_nodes ptr \\<^sub>r disc_nodes' + \ node_ptr \ set disc_nodes'" + +interpretation i_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M?: + l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_disconnected_nodes + get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs heap_is_wellformed + parent_child_rel get_child_nodes + using instances + by (simp add: l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) +declare l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma set_disconnected_nodes_get_disconnected_nodes_wf_is_l_set_disconnected_nodes_get_disconnected_nodes_wf [instances]: + "l_set_disconnected_nodes_get_disconnected_nodes_wf type_wf known_ptr heap_is_wellformed parent_child_rel + get_child_nodes get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes + set_disconnected_nodes_locs" + apply(auto simp add: l_set_disconnected_nodes_get_disconnected_nodes_wf_def + l_set_disconnected_nodes_get_disconnected_nodes_wf_axioms_def instances)[1] + using remove_from_disconnected_nodes_removes apply fast + done + + +subsection \get\_root\_node\ + +locale l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_heap_is_wellformed + type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs + get_disconnected_nodes get_disconnected_nodes_locs + + l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs + + l_get_parent_wf + type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel get_child_nodes + get_child_nodes_locs get_parent get_parent_locs + + l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs + get_ancestors get_ancestors_locs get_root_node get_root_node_locs + for known_ptr :: "(_::linorder) object_ptr \ bool" + and type_wf :: "(_) heap \ bool" + and known_ptrs :: "(_) heap \ bool" + and heap_is_wellformed :: "(_) heap \ bool" + and parent_child_rel :: "(_) heap \ ((_) object_ptr \ (_) object_ptr) set" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and get_parent :: "(_) node_ptr \ ((_) heap, exception, (_) object_ptr option) prog" + and get_parent_locs :: "((_) heap \ (_) heap \ bool) set" + and get_ancestors :: "(_) object_ptr \ ((_) heap, exception, (_) object_ptr list) prog" + and get_ancestors_locs :: "((_) heap \ (_) heap \ bool) set" + and get_root_node :: "(_) object_ptr \ ((_) heap, exception, (_) object_ptr) prog" + and get_root_node_locs :: "((_) heap \ (_) heap \ bool) set" + +begin +lemma get_ancestors_reads: + assumes "heap_is_wellformed h" + shows "reads get_ancestors_locs (get_ancestors node_ptr) h h'" +proof (insert assms(1), induct rule: heap_wellformed_induct_rev) + case (step child) + then show ?case + using [[simproc del: Product_Type.unit_eq]] get_parent_reads[unfolded reads_def] + apply(simp (no_asm) add: get_ancestors_def) + by(auto simp add: get_ancestors_locs_def reads_subset[OF return_reads] get_parent_reads_pointers + intro!: reads_bind_pure reads_subset[OF check_in_heap_reads] + reads_subset[OF get_parent_reads] reads_subset[OF get_child_nodes_reads] + split: option.splits) +qed + +lemma get_ancestors_ok: + assumes "heap_is_wellformed h" + and "ptr |\| object_ptr_kinds h" + and "known_ptrs h" + and type_wf: "type_wf h" + shows "h \ ok (get_ancestors ptr)" +proof (insert assms(1) assms(2), induct rule: heap_wellformed_induct_rev) + case (step child) + then show ?case + using assms(3) assms(4) + apply(simp (no_asm) add: get_ancestors_def) + apply(simp add: assms(1) get_parent_parent_in_heap) + by(auto intro!: bind_is_OK_pure_I bind_pure_I get_parent_ok split: option.splits) +qed + +lemma get_root_node_ptr_in_heap: + assumes "h \ ok (get_root_node ptr)" + shows "ptr |\| object_ptr_kinds h" + using assms + unfolding get_root_node_def + using get_ancestors_ptr_in_heap + by auto + + +lemma get_root_node_ok: + assumes "heap_is_wellformed h" "known_ptrs h" "type_wf h" + and "ptr |\| object_ptr_kinds h" + shows "h \ ok (get_root_node ptr)" + unfolding get_root_node_def + using assms get_ancestors_ok + by auto + + +lemma get_ancestors_parent: + assumes "heap_is_wellformed h" + and "h \ get_parent child \\<^sub>r Some parent" + shows "h \ get_ancestors (cast child) \\<^sub>r (cast child) # parent # ancestors + \ h \ get_ancestors parent \\<^sub>r parent # ancestors" +proof + assume a1: "h \ get_ancestors (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child # parent # ancestors" + then have "h \ Heap_Error_Monad.bind (check_in_heap (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)) + (\_. Heap_Error_Monad.bind (get_parent child) + (\x. Heap_Error_Monad.bind (case x of None \ return [] | Some x \ get_ancestors x) + (\ancestors. return (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child # ancestors)))) + \\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child # parent # ancestors" + by(simp add: get_ancestors_def) + then show "h \ get_ancestors parent \\<^sub>r parent # ancestors" + using assms(2) apply(auto elim!: bind_returns_result_E2 split: option.splits)[1] + using returns_result_eq by fastforce +next + assume "h \ get_ancestors parent \\<^sub>r parent # ancestors" + then show "h \ get_ancestors (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child # parent # ancestors" + using assms(2) + apply(simp (no_asm) add: get_ancestors_def) + apply(auto intro!: bind_pure_returns_result_I split: option.splits)[1] + by (metis (full_types) assms(2) check_in_heap_ptr_in_heap is_OK_returns_result_I + local.get_parent_ptr_in_heap node_ptr_kinds_commutes old.unit.exhaust + select_result_I) +qed + + +lemma get_ancestors_never_empty: + assumes "heap_is_wellformed h" + and "h \ get_ancestors child \\<^sub>r ancestors" + shows "ancestors \ []" +proof(insert assms(2), induct arbitrary: ancestors rule: heap_wellformed_induct_rev[OF assms(1)]) + case (1 child) + then show ?case + proof (induct "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r child") + case None + then show ?case + apply(simp add: get_ancestors_def) + by(auto elim!: bind_returns_result_E2 split: option.splits) + next + case (Some child_node) + then obtain parent_opt where parent_opt: "h \ get_parent child_node \\<^sub>r parent_opt" + apply(simp add: get_ancestors_def) + by(auto elim!: bind_returns_result_E2 split: option.splits) + with Some show ?case + proof(induct parent_opt) + case None + then show ?case + apply(simp add: get_ancestors_def) + by(auto elim!: bind_returns_result_E2 split: option.splits) + next + case (Some option) + then show ?case + apply(simp add: get_ancestors_def) + by(auto elim!: bind_returns_result_E2 split: option.splits) + qed + qed +qed + + + +lemma get_ancestors_subset: + assumes "heap_is_wellformed h" + and "h \ get_ancestors ptr \\<^sub>r ancestors" + and "ancestor \ set ancestors" + and "h \ get_ancestors ancestor \\<^sub>r ancestor_ancestors" +and type_wf: "type_wf h" +and known_ptrs: "known_ptrs h" + shows "set ancestor_ancestors \ set ancestors" +proof (insert assms(1) assms(2) assms(3), induct ptr arbitrary: ancestors + rule: heap_wellformed_induct_rev) + case (step child) + have "child |\| object_ptr_kinds h" + using get_ancestors_ptr_in_heap step(2) by auto + (* then have "h \ check_in_heap child \\<^sub>r ()" + using returns_result_select_result by force *) + show ?case + proof (induct "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r child") + case None + then have "ancestors = [child]" + using step(2) step(3) + by(auto simp add: get_ancestors_def elim!: bind_returns_result_E2) + show ?case + using step(2) step(3) + apply(auto simp add: \ancestors = [child]\)[1] + using assms(4) returns_result_eq by fastforce + next + case (Some child_node) + note s1 = Some + obtain parent_opt where parent_opt: "h \ get_parent child_node \\<^sub>r parent_opt" + using \child |\| object_ptr_kinds h\ assms(1) Some[symmetric] get_parent_ok[OF type_wf known_ptrs] + by (metis (no_types, lifting) is_OK_returns_result_E known_ptrs get_parent_ok + l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms node_ptr_casts_commute node_ptr_kinds_commutes) + then show ?case + proof (induct parent_opt) + case None + then have "ancestors = [child]" + using step(2) step(3) s1 + apply(simp add: get_ancestors_def) + by(auto elim!: bind_returns_result_E2 split: option.splits dest: returns_result_eq) + show ?case + using step(2) step(3) + apply(auto simp add: \ancestors = [child]\)[1] + using assms(4) returns_result_eq by fastforce + next + case (Some parent) + have "h \ Heap_Error_Monad.bind (check_in_heap child) + (\_. Heap_Error_Monad.bind + (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r child of None \ return [] + | Some node_ptr \ Heap_Error_Monad.bind (get_parent node_ptr) + (\parent_ptr_opt. case parent_ptr_opt of None \ return [] + | Some x \ get_ancestors x)) + (\ancestors. return (child # ancestors))) + \\<^sub>r ancestors" + using step(2) + by(simp add: get_ancestors_def) + moreover obtain tl_ancestors where tl_ancestors: "ancestors = child # tl_ancestors" + using calculation + by(auto elim!: bind_returns_result_E2 split: option.splits) + ultimately have "h \ get_ancestors parent \\<^sub>r tl_ancestors" + using s1 Some + by(auto elim!: bind_returns_result_E2 split: option.splits dest: returns_result_eq) + show ?case + using step(1)[OF s1[symmetric, simplified] Some \h \ get_ancestors parent \\<^sub>r tl_ancestors\] + step(3) + apply(auto simp add: tl_ancestors)[1] + by (metis assms(4) insert_iff list.simps(15) local.step(2) returns_result_eq tl_ancestors) + qed + qed +qed + +lemma get_ancestors_also_parent: + assumes "heap_is_wellformed h" + and "h \ get_ancestors some_ptr \\<^sub>r ancestors" + and "cast child \ set ancestors" + and "h \ get_parent child \\<^sub>r Some parent" + and type_wf: "type_wf h" + and known_ptrs: "known_ptrs h" + shows "parent \ set ancestors" +proof - + obtain child_ancestors where child_ancestors: "h \ get_ancestors (cast child) \\<^sub>r child_ancestors" + by (meson assms(1) assms(4) get_ancestors_ok is_OK_returns_result_I known_ptrs + local.get_parent_ptr_in_heap node_ptr_kinds_commutes returns_result_select_result + type_wf) + then have "parent \ set child_ancestors" + apply(simp add: get_ancestors_def) + by(auto elim!: bind_returns_result_E2 split: option.splits dest!: returns_result_eq[OF assms(4)] + get_ancestors_ptr) + then show ?thesis + using assms child_ancestors get_ancestors_subset by blast +qed + +lemma get_ancestors_obtains_children: + assumes "heap_is_wellformed h" + and "ancestor \ ptr" + and "ancestor \ set ancestors" + and "h \ get_ancestors ptr \\<^sub>r ancestors" + and type_wf: "type_wf h" + and known_ptrs: "known_ptrs h" + obtains children ancestor_child where "h \ get_child_nodes ancestor \\<^sub>r children" + and "ancestor_child \ set children" and "cast ancestor_child \ set ancestors" +proof - + assume 0: "(\children ancestor_child. + h \ get_child_nodes ancestor \\<^sub>r children \ + ancestor_child \ set children \ cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ancestor_child \ set ancestors + \ thesis)" + have "\child. h \ get_parent child \\<^sub>r Some ancestor \ cast child \ set ancestors" + proof (insert assms(1) assms(2) assms(3) assms(4), induct ptr arbitrary: ancestors + rule: heap_wellformed_induct_rev) + case (step child) + have "child |\| object_ptr_kinds h" + using get_ancestors_ptr_in_heap step(4) by auto + show ?case + proof (induct "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r child") + case None + then have "ancestors = [child]" + using step(3) step(4) + by(auto simp add: get_ancestors_def elim!: bind_returns_result_E2) + show ?case + using step(2) step(3) step(4) + by(auto simp add: \ancestors = [child]\) + next + case (Some child_node) + note s1 = Some + obtain parent_opt where parent_opt: "h \ get_parent child_node \\<^sub>r parent_opt" + using \child |\| object_ptr_kinds h\ assms(1) Some[symmetric] + using get_parent_ok known_ptrs type_wf + by (metis (no_types, lifting) is_OK_returns_result_E node_ptr_casts_commute + node_ptr_kinds_commutes) + then show ?case + proof (induct parent_opt) + case None + then have "ancestors = [child]" + using step(2) step(3) step(4) s1 + apply(simp add: get_ancestors_def) + by(auto elim!: bind_returns_result_E2 split: option.splits dest: returns_result_eq) + show ?case + using step(2) step(3) step(4) + by(auto simp add: \ancestors = [child]\) + next + case (Some parent) + have "h \ Heap_Error_Monad.bind (check_in_heap child) + (\_. Heap_Error_Monad.bind + (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r child of None \ return [] + | Some node_ptr \ Heap_Error_Monad.bind (get_parent node_ptr) + (\parent_ptr_opt. case parent_ptr_opt of None \ return [] + | Some x \ get_ancestors x)) + (\ancestors. return (child # ancestors))) + \\<^sub>r ancestors" + using step(4) + by(simp add: get_ancestors_def) + moreover obtain tl_ancestors where tl_ancestors: "ancestors = child # tl_ancestors" + using calculation + by(auto elim!: bind_returns_result_E2 split: option.splits) + ultimately have "h \ get_ancestors parent \\<^sub>r tl_ancestors" + using s1 Some + by(auto elim!: bind_returns_result_E2 split: option.splits dest: returns_result_eq) + (* have "ancestor \ parent" *) + have "ancestor \ set tl_ancestors" + using tl_ancestors step(2) step(3) by auto + show ?case + proof (cases "ancestor \ parent") + case True + show ?thesis + using step(1)[OF s1[symmetric, simplified] Some True + \ancestor \ set tl_ancestors\ \h \ get_ancestors parent \\<^sub>r tl_ancestors\] + using tl_ancestors by auto + next + case False + have "child \ set ancestors" + using step(4) get_ancestors_ptr by simp + then show ?thesis + using Some False s1[symmetric] by(auto) + qed + qed + qed + qed + then obtain child where child: "h \ get_parent child \\<^sub>r Some ancestor" + and in_ancestors: "cast child \ set ancestors" + by auto + then obtain children where + children: "h \ get_child_nodes ancestor \\<^sub>r children" and + child_in_children: "child \ set children" + using get_parent_child_dual by blast + show thesis + using 0[OF children child_in_children] child assms(3) in_ancestors by blast +qed + +lemma get_ancestors_parent_child_rel: + assumes "heap_is_wellformed h" + and "h \ get_ancestors child \\<^sub>r ancestors" + and known_ptrs: "known_ptrs h" + and type_wf: "type_wf h" +shows "(ptr, child) \ (parent_child_rel h)\<^sup>* \ ptr \ set ancestors" +proof (safe) + assume 3: "(ptr, child) \ (parent_child_rel h)\<^sup>*" + show "ptr \ set ancestors" + proof (insert 3, induct ptr rule: heap_wellformed_induct[OF assms(1)]) + case (1 ptr) + then show ?case + proof (cases "ptr = child") + case True + then show ?thesis + by (metis (no_types, lifting) assms(2) bind_returns_result_E get_ancestors_def + in_set_member member_rec(1) return_returns_result) + next + case False + obtain ptr_child where + ptr_child: "(ptr, ptr_child) \ (parent_child_rel h) \ (ptr_child, child) \ (parent_child_rel h)\<^sup>*" + using converse_rtranclE[OF 1(2)] \ptr \ child\ + by metis + then obtain ptr_child_node + where ptr_child_ptr_child_node: "ptr_child = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr_child_node" + using ptr_child node_ptr_casts_commute3 parent_child_rel_node_ptr + by (metis ) + then obtain children where + children: "h \ get_child_nodes ptr \\<^sub>r children" and + ptr_child_node: "ptr_child_node \ set children" + proof - + assume a1: "\children. \h \ get_child_nodes ptr \\<^sub>r children; ptr_child_node \ set children\ + \ thesis" + + have "ptr |\| object_ptr_kinds h" + using local.parent_child_rel_parent_in_heap ptr_child by blast + moreover have "ptr_child_node \ set |h \ get_child_nodes ptr|\<^sub>r" + by (metis calculation known_ptrs local.get_child_nodes_ok local.known_ptrs_known_ptr + local.parent_child_rel_child ptr_child ptr_child_ptr_child_node + returns_result_select_result type_wf) + ultimately show ?thesis + using a1 get_child_nodes_ok type_wf known_ptrs + by (meson local.known_ptrs_known_ptr returns_result_select_result) + qed + moreover have "(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr_child_node, child) \ (parent_child_rel h)\<^sup>*" + using ptr_child ptr_child_ptr_child_node by auto + ultimately have "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr_child_node \ set ancestors" + using 1 by auto + moreover have "h \ get_parent ptr_child_node \\<^sub>r Some ptr" + using assms(1) children ptr_child_node child_parent_dual + using known_ptrs type_wf by blast + ultimately show ?thesis + using get_ancestors_also_parent assms type_wf by blast + qed + qed + next + assume 3: "ptr \ set ancestors" + show "(ptr, child) \ (parent_child_rel h)\<^sup>*" + proof (insert 3, induct ptr rule: heap_wellformed_induct[OF assms(1)]) + case (1 ptr) + then show ?case + proof (cases "ptr = child") + case True + then show ?thesis + by simp + next + case False + then obtain children ptr_child_node where + children: "h \ get_child_nodes ptr \\<^sub>r children" and + ptr_child_node: "ptr_child_node \ set children" and + ptr_child_node_in_ancestors: "cast ptr_child_node \ set ancestors" + using 1(2) assms(2) get_ancestors_obtains_children assms(1) + using known_ptrs type_wf by blast + then have "(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr_child_node, child) \ (parent_child_rel h)\<^sup>*" + using 1(1) by blast + + moreover have "(ptr, cast ptr_child_node) \ parent_child_rel h" + using children ptr_child_node assms(1) parent_child_rel_child_nodes2 + using child_parent_dual known_ptrs parent_child_rel_parent type_wf + by blast + + ultimately show ?thesis + by auto + qed + qed +qed + +lemma get_root_node_parent_child_rel: + assumes "heap_is_wellformed h" + and "h \ get_root_node child \\<^sub>r root" + and known_ptrs: "known_ptrs h" + and type_wf: "type_wf h" + shows "(root, child) \ (parent_child_rel h)\<^sup>*" + using assms get_ancestors_parent_child_rel + apply(auto simp add: get_root_node_def elim!: bind_returns_result_E2)[1] + using get_ancestors_never_empty last_in_set by blast + + +lemma get_ancestors_eq: + assumes "heap_is_wellformed h" + and "heap_is_wellformed h'" + and "\object_ptr w. object_ptr \ ptr \ w \ get_child_nodes_locs object_ptr \ w h h'" + and pointers_preserved: "\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + and known_ptrs: "known_ptrs h" + and known_ptrs': "known_ptrs h'" + and "h \ get_ancestors ptr \\<^sub>r ancestors" + and type_wf: "type_wf h" + and type_wf': "type_wf h'" + shows "h' \ get_ancestors ptr \\<^sub>r ancestors" +proof - + have object_ptr_kinds_eq3: "object_ptr_kinds h = object_ptr_kinds h'" + using pointers_preserved object_ptr_kinds_preserved_small by blast + then have object_ptr_kinds_M_eq: + "\ptrs. h \ object_ptr_kinds_M \\<^sub>r ptrs = h' \ object_ptr_kinds_M \\<^sub>r ptrs" + by(simp add: object_ptr_kinds_M_defs) + then have object_ptr_kinds_eq: "|h \ object_ptr_kinds_M|\<^sub>r = |h' \ object_ptr_kinds_M|\<^sub>r" + by(simp) + have "h' \ ok (get_ancestors ptr)" + using get_ancestors_ok get_ancestors_ptr_in_heap object_ptr_kinds_eq3 assms(1) known_ptrs + known_ptrs' assms(2) assms(7) type_wf' + by blast + then obtain ancestors' where ancestors': "h' \ get_ancestors ptr \\<^sub>r ancestors'" + by auto + + obtain root where root: "h \ get_root_node ptr \\<^sub>r root" + proof - + assume 0: "(\root. h \ get_root_node ptr \\<^sub>r root \ thesis)" + show thesis + apply(rule 0) + using assms(7) + by(auto simp add: get_root_node_def elim!: bind_returns_result_E2 split: option.splits) + qed + + have children_eq: + "\p children. p \ ptr \ h \ get_child_nodes p \\<^sub>r children = h' \ get_child_nodes p \\<^sub>r children" + using get_child_nodes_reads assms(3) + apply(simp add: reads_def reflp_def transp_def preserved_def) + by blast + + have "acyclic (parent_child_rel h)" + using assms(1) local.parent_child_rel_acyclic by auto + have "acyclic (parent_child_rel h')" + using assms(2) local.parent_child_rel_acyclic by blast + have 2: "\c parent_opt. cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c \ set ancestors \ set ancestors' + \ h \ get_parent c \\<^sub>r parent_opt = h' \ get_parent c \\<^sub>r parent_opt" + proof - + fix c parent_opt + assume 1: " cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c \ set ancestors \ set ancestors'" + + obtain ptrs where ptrs: "h \ object_ptr_kinds_M \\<^sub>r ptrs" + by simp + + let ?P = "(\ptr. Heap_Error_Monad.bind (get_child_nodes ptr) (\children. return (c \ set children)))" + have children_eq_True: "\p. p \ set ptrs \ h \ ?P p \\<^sub>r True \ h' \ ?P p \\<^sub>r True" + proof - + fix p + assume "p \ set ptrs" + then show "h \ ?P p \\<^sub>r True \ h' \ ?P p \\<^sub>r True" + proof (cases "p = ptr") + case True + have "(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c, ptr) \ (parent_child_rel h)\<^sup>*" + using get_ancestors_parent_child_rel 1 assms by blast + then have "(ptr, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c) \ (parent_child_rel h)" + proof (cases "cast c = ptr") + case True + then show ?thesis + using \acyclic (parent_child_rel h)\ by(auto simp add: acyclic_def) + next + case False + then have "(ptr, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c) \ (parent_child_rel h)\<^sup>*" + using \acyclic (parent_child_rel h)\ False rtrancl_eq_or_trancl rtrancl_trancl_trancl + \(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c, ptr) \ (parent_child_rel h)\<^sup>*\ + by (metis acyclic_def) + then show ?thesis + using r_into_rtrancl by auto + qed + obtain children where children: "h \ get_child_nodes ptr \\<^sub>r children" + using type_wf + by (metis \h' \ ok get_ancestors ptr\ assms(1) get_ancestors_ptr_in_heap get_child_nodes_ok + heap_is_wellformed_def is_OK_returns_result_E known_ptrs local.known_ptrs_known_ptr + object_ptr_kinds_eq3) + then have "c \ set children" + using \(ptr, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c) \ (parent_child_rel h)\ assms(1) + using parent_child_rel_child_nodes2 + using child_parent_dual known_ptrs parent_child_rel_parent + type_wf by blast + with children have "h \ ?P p \\<^sub>r False" + by(auto simp add: True) + + moreover have "(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c, ptr) \ (parent_child_rel h')\<^sup>*" + using get_ancestors_parent_child_rel assms(2) ancestors' 1 known_ptrs' type_wf + type_wf' by blast + then have "(ptr, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c) \ (parent_child_rel h')" + proof (cases "cast c = ptr") + case True + then show ?thesis + using \acyclic (parent_child_rel h')\ by(auto simp add: acyclic_def) + next + case False + then have "(ptr, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c) \ (parent_child_rel h')\<^sup>*" + using \acyclic (parent_child_rel h')\ False rtrancl_eq_or_trancl rtrancl_trancl_trancl + \(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c, ptr) \ (parent_child_rel h')\<^sup>*\ + by (metis acyclic_def) + then show ?thesis + using r_into_rtrancl by auto + qed + then have "(ptr, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c) \ (parent_child_rel h')" + using r_into_rtrancl by auto + obtain children' where children': "h' \ get_child_nodes ptr \\<^sub>r children'" + using type_wf type_wf' + by (meson \h' \ ok (get_ancestors ptr)\ assms(2) get_ancestors_ptr_in_heap + get_child_nodes_ok is_OK_returns_result_E known_ptrs' + local.known_ptrs_known_ptr) + then have "c \ set children'" + using \(ptr, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c) \ (parent_child_rel h')\ assms(2) type_wf type_wf' + using parent_child_rel_child_nodes2 child_parent_dual known_ptrs' parent_child_rel_parent + by auto + with children' have "h' \ ?P p \\<^sub>r False" + by(auto simp add: True) + + ultimately show ?thesis + by (metis returns_result_eq) + next + case False + then show ?thesis + using children_eq ptrs + by (metis (no_types, lifting) bind_pure_returns_result_I bind_returns_result_E + get_child_nodes_pure return_returns_result) + qed + qed + have "\pa. pa \ set ptrs \ h \ ok (get_child_nodes pa + \ (\children. return (c \ set children))) = h' \ ok ( get_child_nodes pa + \ (\children. return (c \ set children)))" + using assms(1) assms(2) object_ptr_kinds_eq ptrs type_wf type_wf' + by (metis (no_types, lifting) ObjectMonad.ptr_kinds_ptr_kinds_M bind_is_OK_pure_I + get_child_nodes_ok get_child_nodes_pure known_ptrs' + local.known_ptrs_known_ptr return_ok select_result_I2) + have children_eq_False: + "\pa. pa \ set ptrs \ h \ get_child_nodes pa + \ (\children. return (c \ set children)) \\<^sub>r False = h' \ get_child_nodes pa + \ (\children. return (c \ set children)) \\<^sub>r False" + proof + fix pa + assume "pa \ set ptrs" + and "h \ get_child_nodes pa \ (\children. return (c \ set children)) \\<^sub>r False" + have "h \ ok (get_child_nodes pa \ (\children. return (c \ set children))) + \ h' \ ok ( get_child_nodes pa \ (\children. return (c \ set children)))" + using \pa \ set ptrs\ \\pa. pa \ set ptrs \ h \ ok (get_child_nodes pa + \ (\children. return (c \ set children))) = h' \ ok ( get_child_nodes pa + \ (\children. return (c \ set children)))\ + by auto + moreover have "h \ get_child_nodes pa \ (\children. return (c \ set children)) \\<^sub>r False + \ h' \ get_child_nodes pa \ (\children. return (c \ set children)) \\<^sub>r False" + by (metis (mono_tags, lifting) \\pa. pa \ set ptrs + \ h \ get_child_nodes pa + \ (\children. return (c \ set children)) \\<^sub>r True = h' \ get_child_nodes pa + \ (\children. return (c \ set children)) \\<^sub>r True\ \pa \ set ptrs\ + calculation is_OK_returns_result_I returns_result_eq returns_result_select_result) + ultimately show "h' \ get_child_nodes pa \ (\children. return (c \ set children)) \\<^sub>r False" + using \h \ get_child_nodes pa \ (\children. return (c \ set children)) \\<^sub>r False\ + by auto + next + fix pa + assume "pa \ set ptrs" + and "h' \ get_child_nodes pa \ (\children. return (c \ set children)) \\<^sub>r False" + have "h' \ ok (get_child_nodes pa \ (\children. return (c \ set children))) + \ h \ ok ( get_child_nodes pa \ (\children. return (c \ set children)))" + using \pa \ set ptrs\ \\pa. pa \ set ptrs + \ h \ ok (get_child_nodes pa + \ (\children. return (c \ set children))) = h' \ ok ( get_child_nodes pa + \ (\children. return (c \ set children)))\ + by auto + moreover have "h' \ get_child_nodes pa \ (\children. return (c \ set children)) \\<^sub>r False + \ h \ get_child_nodes pa \ (\children. return (c \ set children)) \\<^sub>r False" + by (metis (mono_tags, lifting) + \\pa. pa \ set ptrs \ h \ get_child_nodes pa + \ (\children. return (c \ set children)) \\<^sub>r True = h' \ get_child_nodes pa + \ (\children. return (c \ set children)) \\<^sub>r True\ \pa \ set ptrs\ + calculation is_OK_returns_result_I returns_result_eq returns_result_select_result) + ultimately show "h \ get_child_nodes pa \ (\children. return (c \ set children)) \\<^sub>r False" + using \h' \ get_child_nodes pa \ (\children. return (c \ set children)) \\<^sub>r False\ by blast + qed + + have filter_eq: "\xs. h \ filter_M ?P ptrs \\<^sub>r xs = h' \ filter_M ?P ptrs \\<^sub>r xs" + proof (rule filter_M_eq) + show + "\xs x. pure (Heap_Error_Monad.bind (get_child_nodes x) (\children. return (c \ set children))) h" + by(auto intro!: bind_pure_I) + next + show + "\xs x. pure (Heap_Error_Monad.bind (get_child_nodes x) (\children. return (c \ set children))) h'" + by(auto intro!: bind_pure_I) + next + fix xs b x + assume 0: "x \ set ptrs" + then show "h \ Heap_Error_Monad.bind (get_child_nodes x) (\children. return (c \ set children)) \\<^sub>r b + = h' \ Heap_Error_Monad.bind (get_child_nodes x) (\children. return (c \ set children)) \\<^sub>r b" + apply(induct b) + using children_eq_True apply blast + using children_eq_False apply blast + done + qed + + show "h \ get_parent c \\<^sub>r parent_opt = h' \ get_parent c \\<^sub>r parent_opt" + apply(simp add: get_parent_def) + apply(rule bind_cong_2) + apply(simp) + apply(simp) + apply(simp add: check_in_heap_def node_ptr_kinds_def object_ptr_kinds_eq3) + apply(rule bind_cong_2) + apply(auto simp add: object_ptr_kinds_M_eq object_ptr_kinds_eq3)[1] + apply(auto simp add: object_ptr_kinds_M_eq object_ptr_kinds_eq3)[1] + apply(auto simp add: object_ptr_kinds_M_eq object_ptr_kinds_eq3)[1] + apply(rule bind_cong_2) + apply(auto intro!: filter_M_pure_I bind_pure_I)[1] + apply(auto intro!: filter_M_pure_I bind_pure_I)[1] + apply(auto simp add: filter_eq (* dest!: returns_result_eq[OF ptrs] *)) + using filter_eq ptrs apply auto[1] + using filter_eq ptrs by auto + qed + + have "ancestors = ancestors'" + proof(insert assms(1) assms(7) ancestors' 2, induct ptr arbitrary: ancestors ancestors' + rule: heap_wellformed_induct_rev) + case (step child) + show ?case + using step(2) step(3) step(4) + apply(simp add: get_ancestors_def) + apply(auto intro!: elim!: bind_returns_result_E2 split: option.splits)[1] + using returns_result_eq apply fastforce + apply (meson option.simps(3) returns_result_eq) + by (metis IntD1 IntD2 option.inject returns_result_eq step.hyps) + qed + then show ?thesis + using assms(5) ancestors' + by simp +qed + +lemma get_ancestors_remains_not_in_ancestors: + assumes "heap_is_wellformed h" + and "heap_is_wellformed h'" + and "h \ get_ancestors ptr \\<^sub>r ancestors" + and "h' \ get_ancestors ptr \\<^sub>r ancestors'" + and "\p children children'. h \ get_child_nodes p \\<^sub>r children + \ h' \ get_child_nodes p \\<^sub>r children' \ set children' \ set children" + and "node \ set ancestors" + and object_ptr_kinds_eq3: "object_ptr_kinds h = object_ptr_kinds h'" + and known_ptrs: "known_ptrs h" + and type_wf: "type_wf h" + and type_wf': "type_wf h'" + shows "node \ set ancestors'" +proof - + have object_ptr_kinds_M_eq: + "\ptrs. h \ object_ptr_kinds_M \\<^sub>r ptrs = h' \ object_ptr_kinds_M \\<^sub>r ptrs" + using object_ptr_kinds_eq3 + by(simp add: object_ptr_kinds_M_defs) + then have object_ptr_kinds_eq: "|h \ object_ptr_kinds_M|\<^sub>r = |h' \ object_ptr_kinds_M|\<^sub>r" + by(simp) + + show ?thesis + proof (insert assms(1) assms(3) assms(4) assms(6), induct ptr arbitrary: ancestors ancestors' + rule: heap_wellformed_induct_rev) + case (step child) + have 1: "\p parent. h' \ get_parent p \\<^sub>r Some parent \ h \ get_parent p \\<^sub>r Some parent" + proof - + fix p parent + assume "h' \ get_parent p \\<^sub>r Some parent" + then obtain children' where + children': "h' \ get_child_nodes parent \\<^sub>r children'" and + p_in_children': "p \ set children'" + using get_parent_child_dual by blast + obtain children where children: "h \ get_child_nodes parent \\<^sub>r children" + using get_child_nodes_ok assms(1) get_child_nodes_ptr_in_heap object_ptr_kinds_eq children' + known_ptrs + using type_wf type_wf' + by (metis \h' \ get_parent p \\<^sub>r Some parent\ get_parent_parent_in_heap is_OK_returns_result_E + local.known_ptrs_known_ptr object_ptr_kinds_eq3) + have "p \ set children" + using assms(5) children children' p_in_children' + by blast + then show "h \ get_parent p \\<^sub>r Some parent" + using child_parent_dual assms(1) children known_ptrs type_wf by blast + qed + have "node \ child" + using assms(1) get_ancestors_parent_child_rel step.prems(1) step.prems(3) known_ptrs + using type_wf type_wf' + by blast + then show ?case + using step(2) step(3) + apply(simp add: get_ancestors_def) + using step(4) + apply(auto elim!: bind_returns_result_E2 split: option.splits)[1] + using 1 + apply (meson option.distinct(1) returns_result_eq) + by (metis "1" option.inject returns_result_eq step.hyps) + qed +qed + +lemma get_ancestors_ptrs_in_heap: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_ancestors ptr \\<^sub>r ancestors" + assumes "ptr' \ set ancestors" + shows "ptr' |\| object_ptr_kinds h" +proof (insert assms(4) assms(5), induct ancestors arbitrary: ptr) + case Nil + then show ?case + by(auto) +next + case (Cons a ancestors) + then obtain x where x: "h \ get_ancestors x \\<^sub>r a # ancestors" + by(auto simp add: get_ancestors_def[of a] elim!: bind_returns_result_E2 split: option.splits) + then have "x = a" + by(auto simp add: get_ancestors_def[of x] elim!: bind_returns_result_E2 split: option.splits) + then show ?case + using Cons.hyps Cons.prems(2) get_ancestors_ptr_in_heap x + by (metis assms(1) assms(2) assms(3) get_ancestors_obtains_children get_child_nodes_ptr_in_heap + is_OK_returns_result_I) +qed + + +lemma get_ancestors_prefix: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_ancestors ptr \\<^sub>r ancestors" + assumes "ptr' \ set ancestors" + assumes "h \ get_ancestors ptr' \\<^sub>r ancestors'" + shows "\pre. ancestors = pre @ ancestors'" +proof (insert assms(1) assms(5) assms(6), induct ptr' arbitrary: ancestors' + rule: heap_wellformed_induct) + case (step parent) + then show ?case + proof (cases "parent \ ptr" ) + case True + + then obtain children ancestor_child where "h \ get_child_nodes parent \\<^sub>r children" + and "ancestor_child \ set children" and "cast ancestor_child \ set ancestors" + using assms(1) assms(2) assms(3) assms(4) get_ancestors_obtains_children step.prems(1) by blast + then have "h \ get_parent ancestor_child \\<^sub>r Some parent" + using assms(1) assms(2) assms(3) child_parent_dual by blast + then have "h \ get_ancestors (cast ancestor_child) \\<^sub>r cast ancestor_child # ancestors'" + apply(simp add: get_ancestors_def) + using \h \ get_ancestors parent \\<^sub>r ancestors'\ get_parent_ptr_in_heap + by(auto simp add: check_in_heap_def is_OK_returns_result_I intro!: bind_pure_returns_result_I) + then show ?thesis + using step(1) \h \ get_child_nodes parent \\<^sub>r children\ \ancestor_child \ set children\ + \cast ancestor_child \ set ancestors\ \h \ get_ancestors (cast ancestor_child) \\<^sub>r cast ancestor_child # ancestors'\ + by fastforce + next + case False + then show ?thesis + by (metis append_Nil assms(4) returns_result_eq step.prems(2)) + qed +qed + + +lemma get_ancestors_same_root_node: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_ancestors ptr \\<^sub>r ancestors" + assumes "ptr' \ set ancestors" + assumes "ptr'' \ set ancestors" + shows "h \ get_root_node ptr' \\<^sub>r root_ptr \ h \ get_root_node ptr'' \\<^sub>r root_ptr" +proof - + have "ptr' |\| object_ptr_kinds h" + by (metis assms(1) assms(2) assms(3) assms(4) assms(5) get_ancestors_obtains_children + get_ancestors_ptr_in_heap get_child_nodes_ptr_in_heap is_OK_returns_result_I) + then obtain ancestors' where ancestors': "h \ get_ancestors ptr' \\<^sub>r ancestors'" + by (meson assms(1) assms(2) assms(3) get_ancestors_ok is_OK_returns_result_E) + then have "\pre. ancestors = pre @ ancestors'" + using get_ancestors_prefix assms by blast + moreover have "ptr'' |\| object_ptr_kinds h" + by (metis assms(1) assms(2) assms(3) assms(4) assms(6) get_ancestors_obtains_children + get_ancestors_ptr_in_heap get_child_nodes_ptr_in_heap is_OK_returns_result_I) + then obtain ancestors'' where ancestors'': "h \ get_ancestors ptr'' \\<^sub>r ancestors''" + by (meson assms(1) assms(2) assms(3) get_ancestors_ok is_OK_returns_result_E) + then have "\pre. ancestors = pre @ ancestors''" + using get_ancestors_prefix assms by blast + ultimately show ?thesis + using ancestors' ancestors'' + apply(auto simp add: get_root_node_def elim!: bind_returns_result_E2 + intro!: bind_pure_returns_result_I)[1] + apply (metis (no_types, lifting) assms(1) get_ancestors_never_empty last_appendR + returns_result_eq) + by (metis assms(1) get_ancestors_never_empty last_appendR returns_result_eq) +qed + +lemma get_root_node_parent_same: + assumes "h \ get_parent child \\<^sub>r Some ptr" + shows "h \ get_root_node (cast child) \\<^sub>r root \ h \ get_root_node ptr \\<^sub>r root" +proof + assume 1: " h \ get_root_node (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \\<^sub>r root" + show "h \ get_root_node ptr \\<^sub>r root" + using 1[unfolded get_root_node_def] assms + apply(simp add: get_ancestors_def) + apply(auto simp add: get_root_node_def dest: returns_result_eq elim!: bind_returns_result_E2 + intro!: bind_pure_returns_result_I split: option.splits)[1] + using returns_result_eq apply fastforce + using get_ancestors_ptr by fastforce +next + assume 1: " h \ get_root_node ptr \\<^sub>r root" + show "h \ get_root_node (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \\<^sub>r root" + apply(simp add: get_root_node_def) + using assms 1 + apply(simp add: get_ancestors_def) + apply(auto simp add: get_root_node_def dest: returns_result_eq elim!: bind_returns_result_E2 + intro!: bind_pure_returns_result_I split: option.splits)[1] + apply (simp add: check_in_heap_def is_OK_returns_result_I) + using get_ancestors_ptr get_parent_ptr_in_heap + apply (simp add: is_OK_returns_result_I) + by (meson list.distinct(1) list.set_cases local.get_ancestors_ptr) +qed + +lemma get_root_node_same_no_parent: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_root_node ptr \\<^sub>r cast child" + shows "h \ get_parent child \\<^sub>r None" +proof (insert assms(1) assms(4), induct ptr rule: heap_wellformed_induct_rev) + case (step c) + then show ?case + proof (cases "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r c") + case None + then have "c = cast child" + using step(2) + by(auto simp add: get_root_node_def get_ancestors_def[of c] elim!: bind_returns_result_E2) + then show ?thesis + using None by auto + next + case (Some child_node) + note s = this + then obtain parent_opt where parent_opt: "h \ get_parent child_node \\<^sub>r parent_opt" + by (metis (no_types, lifting) assms(2) assms(3) get_root_node_ptr_in_heap + is_OK_returns_result_I local.get_parent_ok node_ptr_casts_commute + node_ptr_kinds_commutes returns_result_select_result step.prems) + then show ?thesis + proof(induct parent_opt) + case None + then show ?case + using Some get_root_node_no_parent returns_result_eq step.prems by fastforce + next + case (Some parent) + then show ?case + using step s + apply(auto simp add: get_root_node_def get_ancestors_def[of c] + elim!: bind_returns_result_E2 split: option.splits list.splits)[1] + using get_root_node_parent_same step.hyps step.prems by auto + qed + qed +qed + +lemma get_root_node_not_node_same: + assumes "ptr |\| object_ptr_kinds h" + assumes "\is_node_ptr_kind ptr" + shows "h \ get_root_node ptr \\<^sub>r ptr" + using assms + apply(simp add: get_root_node_def get_ancestors_def) + by(auto simp add: get_root_node_def dest: returns_result_eq elim!: bind_returns_result_E2 + intro!: bind_pure_returns_result_I split: option.splits) + + +lemma get_root_node_root_in_heap: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_root_node ptr \\<^sub>r root" + shows "root |\| object_ptr_kinds h" + using assms + apply(auto simp add: get_root_node_def elim!: bind_returns_result_E2)[1] + by (simp add: get_ancestors_never_empty get_ancestors_ptrs_in_heap) + + +lemma get_root_node_same_no_parent_parent_child_rel: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_root_node ptr' \\<^sub>r ptr'" + shows "\(\p. (p, ptr') \ (parent_child_rel h))" + by (metis (no_types, lifting) assms(1) assms(2) assms(3) assms(4) get_root_node_same_no_parent + l_heap_is_wellformed.parent_child_rel_child local.child_parent_dual local.get_child_nodes_ok + local.known_ptrs_known_ptr local.l_heap_is_wellformed_axioms local.parent_child_rel_node_ptr + local.parent_child_rel_parent_in_heap node_ptr_casts_commute3 option.simps(3) returns_result_eq + returns_result_select_result) + +end + + +locale l_get_ancestors_wf = l_heap_is_wellformed_defs + l_known_ptrs + l_type_wf + l_get_ancestors_defs + + l_get_child_nodes_defs + l_get_parent_defs + + assumes get_ancestors_never_empty: + "heap_is_wellformed h \ h \ get_ancestors child \\<^sub>r ancestors \ ancestors \ []" + assumes get_ancestors_ok: + "heap_is_wellformed h \ ptr |\| object_ptr_kinds h \ known_ptrs h \ type_wf h + \ h \ ok (get_ancestors ptr)" + assumes get_ancestors_reads: + "heap_is_wellformed h \ reads get_ancestors_locs (get_ancestors node_ptr) h h'" + assumes get_ancestors_ptrs_in_heap: + "heap_is_wellformed h \ type_wf h \ known_ptrs h + \ h \ get_ancestors ptr \\<^sub>r ancestors \ ptr' \ set ancestors + \ ptr' |\| object_ptr_kinds h" + assumes get_ancestors_remains_not_in_ancestors: + "heap_is_wellformed h \ heap_is_wellformed h' \ h \ get_ancestors ptr \\<^sub>r ancestors + \ h' \ get_ancestors ptr \\<^sub>r ancestors' + \ (\p children children'. h \ get_child_nodes p \\<^sub>r children + \ h' \ get_child_nodes p \\<^sub>r children' + \ set children' \ set children) + \ node \ set ancestors + \ object_ptr_kinds h = object_ptr_kinds h' \ known_ptrs h + \ type_wf h \ type_wf h' \ node \ set ancestors'" + assumes get_ancestors_also_parent: + "heap_is_wellformed h \ h \ get_ancestors some_ptr \\<^sub>r ancestors + \ cast child_node \ set ancestors + \ h \ get_parent child_node \\<^sub>r Some parent \ type_wf h + \ known_ptrs h \ parent \ set ancestors" + assumes get_ancestors_obtains_children: + "heap_is_wellformed h \ ancestor \ ptr \ ancestor \ set ancestors + \ h \ get_ancestors ptr \\<^sub>r ancestors \ type_wf h \ known_ptrs h + \ (\children ancestor_child . h \ get_child_nodes ancestor \\<^sub>r children + \ ancestor_child \ set children + \ cast ancestor_child \ set ancestors + \ thesis) + \ thesis" + assumes get_ancestors_parent_child_rel: + "heap_is_wellformed h \ h \ get_ancestors child \\<^sub>r ancestors \ known_ptrs h \ type_wf h + \ (ptr, child) \ (parent_child_rel h)\<^sup>* \ ptr \ set ancestors" + +locale l_get_root_node_wf = l_heap_is_wellformed_defs + l_get_root_node_defs + l_type_wf + + l_known_ptrs + l_get_ancestors_defs + l_get_parent_defs + + assumes get_root_node_ok: + "heap_is_wellformed h \ known_ptrs h \ type_wf h \ ptr |\| object_ptr_kinds h + \ h \ ok (get_root_node ptr)" + assumes get_root_node_ptr_in_heap: + "h \ ok (get_root_node ptr) \ ptr |\| object_ptr_kinds h" + assumes get_root_node_root_in_heap: + "heap_is_wellformed h \ type_wf h \ known_ptrs h + \ h \ get_root_node ptr \\<^sub>r root \ root |\| object_ptr_kinds h" + assumes get_ancestors_same_root_node: + "heap_is_wellformed h \ type_wf h \ known_ptrs h + \ h \ get_ancestors ptr \\<^sub>r ancestors \ ptr' \ set ancestors + \ ptr'' \ set ancestors + \ h \ get_root_node ptr' \\<^sub>r root_ptr \ h \ get_root_node ptr'' \\<^sub>r root_ptr" + assumes get_root_node_same_no_parent: + "heap_is_wellformed h \ type_wf h \ known_ptrs h + \ h \ get_root_node ptr \\<^sub>r cast child \ h \ get_parent child \\<^sub>r None" + assumes get_root_node_not_node_same: + "ptr |\| object_ptr_kinds h \ \is_node_ptr_kind ptr + \ h \ get_root_node ptr \\<^sub>r ptr" + assumes get_root_node_parent_same: + "h \ get_parent child \\<^sub>r Some ptr + \ h \ get_root_node (cast child) \\<^sub>r root \ h \ get_root_node ptr \\<^sub>r root" + +interpretation i_get_root_node_wf?: + l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel + get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs + get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs + using instances + by(simp add: l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) +declare l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +lemma get_ancestors_wf_is_l_get_ancestors_wf [instances]: + "l_get_ancestors_wf heap_is_wellformed parent_child_rel known_ptr known_ptrs type_wf get_ancestors + get_ancestors_locs get_child_nodes get_parent" + using known_ptrs_is_l_known_ptrs + apply(auto simp add: l_get_ancestors_wf_def l_get_ancestors_wf_axioms_def)[1] + using get_ancestors_never_empty apply blast + using get_ancestors_ok apply blast + using get_ancestors_reads apply blast + using get_ancestors_ptrs_in_heap apply blast + using get_ancestors_remains_not_in_ancestors apply blast + using get_ancestors_also_parent apply blast + using get_ancestors_obtains_children apply blast + using get_ancestors_parent_child_rel apply blast + using get_ancestors_parent_child_rel apply blast + done + +lemma get_root_node_wf_is_l_get_root_node_wf [instances]: + "l_get_root_node_wf heap_is_wellformed get_root_node type_wf known_ptr known_ptrs + get_ancestors get_parent" + using known_ptrs_is_l_known_ptrs + apply(auto simp add: l_get_root_node_wf_def l_get_root_node_wf_axioms_def)[1] + using get_root_node_ok apply blast + using get_root_node_ptr_in_heap apply blast + using get_root_node_root_in_heap apply blast + using get_ancestors_same_root_node apply(blast, blast) + using get_root_node_same_no_parent apply blast + using get_root_node_not_node_same apply blast + using get_root_node_parent_same apply (blast, blast) + done + + +subsection \to\_tree\_order\ +(* lemma to_tree_order_reads: + assumes "a_heap_is_wellformed h" + shows "reads (all_ptrs (getter_preserved_set_ext \ {get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_preserved document_element} + \ {get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_preserved Element.child_nodes})) (to_tree_order ptr) h h'" + oops *) + +locale l_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_parent + + l_get_parent_wf + + l_heap_is_wellformed + (* l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M *) +begin + +lemma to_tree_order_ptr_in_heap: + assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h" + assumes "h \ ok (to_tree_order ptr)" + shows "ptr |\| object_ptr_kinds h" +proof(insert assms(1) assms(4), induct rule: heap_wellformed_induct) + case (step parent) + then show ?case + apply(auto simp add: to_tree_order_def[of parent] map_M_pure_I elim!: bind_is_OK_E3)[1] + using get_child_nodes_ptr_in_heap by blast +qed + +lemma to_tree_order_either_ptr_or_in_children: + assumes "h \ to_tree_order ptr \\<^sub>r nodes" + and "node \ set nodes" + and "h \ get_child_nodes ptr \\<^sub>r children" + and "node \ ptr" + obtains child child_to where "child \ set children" + and "h \ to_tree_order (cast child) \\<^sub>r child_to" and "node \ set child_to" +proof - + obtain treeorders where treeorders: "h \ map_M to_tree_order (map cast children) \\<^sub>r treeorders" + using assms + apply(auto simp add: to_tree_order_def elim!: bind_returns_result_E)[1] + using pure_returns_heap_eq returns_result_eq by fastforce + then have "node \ set (concat treeorders)" + using assms[simplified to_tree_order_def] + by(auto elim!: bind_returns_result_E4 dest: pure_returns_heap_eq) + then obtain treeorder where "treeorder \ set treeorders" + and node_in_treeorder: "node \ set treeorder" + by auto + then obtain child where "h \ to_tree_order (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \\<^sub>r treeorder" + and "child \ set children" + using assms[simplified to_tree_order_def] treeorders + by(auto elim!: map_M_pure_E2) + then show ?thesis + using node_in_treeorder returns_result_eq that by auto +qed + + +lemma to_tree_order_ptrs_in_heap: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ to_tree_order ptr \\<^sub>r to" + assumes "ptr' \ set to" + shows "ptr' |\| object_ptr_kinds h" +proof(insert assms(1) assms(4) assms(5), induct ptr arbitrary: to rule: heap_wellformed_induct) + case (step parent) + have "parent |\| object_ptr_kinds h" + using assms(1) assms(2) assms(3) step.prems(1) to_tree_order_ptr_in_heap by blast + then obtain children where children: "h \ get_child_nodes parent \\<^sub>r children" + by (meson assms(2) assms(3) get_child_nodes_ok is_OK_returns_result_E local.known_ptrs_known_ptr) + then show ?case + proof (cases "children = []") + case True + then have "to = [parent]" + using step(2) children + apply(auto simp add: to_tree_order_def[of parent] map_M_pure_I elim!: bind_returns_result_E2)[1] + by (metis list.distinct(1) list.map_disc_iff list.set_cases map_M_pure_E2 returns_result_eq) + then show ?thesis + using \parent |\| object_ptr_kinds h\ step.prems(2) by auto + next + case False + note f = this + then show ?thesis + using children step to_tree_order_either_ptr_or_in_children + proof (cases "ptr' = parent") + case True + then show ?thesis + using \parent |\| object_ptr_kinds h\ by blast + next + case False + then show ?thesis + using children step.hyps to_tree_order_either_ptr_or_in_children + by (metis step.prems(1) step.prems(2)) + qed + qed +qed + +lemma to_tree_order_ok: + assumes wellformed: "heap_is_wellformed h" + and "ptr |\| object_ptr_kinds h" + and "known_ptrs h" + and type_wf: "type_wf h" + shows "h \ ok (to_tree_order ptr)" +proof(insert assms(1) assms(2), induct rule: heap_wellformed_induct) + case (step parent) + then show ?case + using assms(3) type_wf + apply(simp add: to_tree_order_def) + apply(auto simp add: heap_is_wellformed_def intro!: map_M_ok_I bind_is_OK_pure_I map_M_pure_I)[1] + using get_child_nodes_ok known_ptrs_known_ptr apply blast + by (simp add: local.heap_is_wellformed_children_in_heap local.to_tree_order_def wellformed) +qed + +lemma to_tree_order_child_subset: + assumes "heap_is_wellformed h" + and "h \ to_tree_order ptr \\<^sub>r nodes" + and "h \ get_child_nodes ptr \\<^sub>r children" + and "node \ set children" + and "h \ to_tree_order (cast node) \\<^sub>r nodes'" + shows "set nodes' \ set nodes" +proof + fix x + assume a1: "x \ set nodes'" + moreover obtain treeorders + where treeorders: "h \ map_M to_tree_order (map cast children) \\<^sub>r treeorders" + using assms(2) assms(3) + apply(auto simp add: to_tree_order_def elim!: bind_returns_result_E)[1] + using pure_returns_heap_eq returns_result_eq by fastforce + then have "nodes' \ set treeorders" + using assms(4) assms(5) + by(auto elim!: map_M_pure_E dest: returns_result_eq) + moreover have "set (concat treeorders) \ set nodes" + using treeorders assms(2) assms(3) + by(auto simp add: to_tree_order_def elim!: bind_returns_result_E4 dest: pure_returns_heap_eq) + ultimately show "x \ set nodes" + by auto +qed + +lemma to_tree_order_ptr_in_result: + assumes "h \ to_tree_order ptr \\<^sub>r nodes" + shows "ptr \ set nodes" + using assms + apply(simp add: to_tree_order_def) + by(auto elim!: bind_returns_result_E2 intro!: map_M_pure_I bind_pure_I) + +lemma to_tree_order_subset: + assumes "heap_is_wellformed h" + and "h \ to_tree_order ptr \\<^sub>r nodes" + and "node \ set nodes" + and "h \ to_tree_order node \\<^sub>r nodes'" + and "known_ptrs h" + and type_wf: "type_wf h" + shows "set nodes' \ set nodes" +proof - + have "\nodes. h \ to_tree_order ptr \\<^sub>r nodes \ (\node. node \ set nodes + \ (\nodes'. h \ to_tree_order node \\<^sub>r nodes' \ set nodes' \ set nodes))" + proof(insert assms(1), induct ptr rule: heap_wellformed_induct) + case (step parent) + then show ?case + proof safe + fix nodes node nodes' x + assume 1: "(\children child. + h \ get_child_nodes parent \\<^sub>r children \ + child \ set children \ \nodes. h \ to_tree_order (cast child) \\<^sub>r nodes + \ (\node. node \ set nodes \ (\nodes'. h \ to_tree_order node \\<^sub>r nodes' + \ set nodes' \ set nodes)))" + and 2: "h \ to_tree_order parent \\<^sub>r nodes" + and 3: "node \ set nodes" + and "h \ to_tree_order node \\<^sub>r nodes'" + and "x \ set nodes'" + have h1: "(\children child nodes node nodes'. + h \ get_child_nodes parent \\<^sub>r children \ + child \ set children \ h \ to_tree_order (cast child) \\<^sub>r nodes + \ (node \ set nodes \ (h \ to_tree_order node \\<^sub>r nodes' \ set nodes' \ set nodes)))" + using 1 + by blast + obtain children where children: "h \ get_child_nodes parent \\<^sub>r children" + using 2 + by(auto simp add: to_tree_order_def elim!: bind_returns_result_E) + then have "set nodes' \ set nodes" + proof (cases "children = []") + case True + then show ?thesis + by (metis "2" "3" \h \ to_tree_order node \\<^sub>r nodes'\ children empty_iff list.set(1) + subsetI to_tree_order_either_ptr_or_in_children) + next + case False + then show ?thesis + proof (cases "node = parent") + case True + then show ?thesis + using "2" \h \ to_tree_order node \\<^sub>r nodes'\ returns_result_eq by fastforce + next + case False + then obtain child nodes_of_child where + "child \ set children" and + "h \ to_tree_order (cast child) \\<^sub>r nodes_of_child" and + "node \ set nodes_of_child" + using 2[simplified to_tree_order_def] 3 + to_tree_order_either_ptr_or_in_children[where node=node and ptr=parent] children + apply(auto elim!: bind_returns_result_E2 intro: map_M_pure_I)[1] + using is_OK_returns_result_E 2 a_all_ptrs_in_heap_def assms(1) heap_is_wellformed_def + using "3" by blast + then have "set nodes' \ set nodes_of_child" + using h1 + using \h \ to_tree_order node \\<^sub>r nodes'\ children by blast + moreover have "set nodes_of_child \ set nodes" + using "2" \child \ set children\ \h \ to_tree_order (cast child) \\<^sub>r nodes_of_child\ + assms children to_tree_order_child_subset by auto + ultimately show ?thesis + by blast + qed + qed + then show "x \ set nodes" + using \x \ set nodes'\ by blast + qed + qed + then show ?thesis + using assms by blast +qed + +lemma to_tree_order_parent: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ to_tree_order ptr \\<^sub>r nodes" + assumes "h \ get_parent child \\<^sub>r Some parent" + assumes "parent \ set nodes" + shows "cast child \ set nodes" +proof - + obtain nodes' where nodes': "h \ to_tree_order parent \\<^sub>r nodes'" + using assms to_tree_order_ok get_parent_parent_in_heap + by (meson get_parent_parent_in_heap is_OK_returns_result_E) + + then have "set nodes' \ set nodes" + using to_tree_order_subset assms + by blast + moreover obtain children where + children: "h \ get_child_nodes parent \\<^sub>r children" and + child: "child \ set children" + using assms get_parent_child_dual by blast + then obtain child_to where child_to: "h \ to_tree_order (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \\<^sub>r child_to" + by (meson assms(1) assms(2) assms(3) assms(5) is_OK_returns_result_E is_OK_returns_result_I + get_parent_ptr_in_heap node_ptr_kinds_commutes to_tree_order_ok) + then have "cast child \ set child_to" + apply(simp add: to_tree_order_def) + by(auto elim!: bind_returns_result_E2 map_M_pure_E + dest!: bind_returns_result_E3[rotated, OF children, rotated] intro!: map_M_pure_I) + + have "cast child \ set nodes'" + using nodes' child + apply(simp add: to_tree_order_def) + apply(auto elim!: bind_returns_result_E2 map_M_pure_E + dest!: bind_returns_result_E3[rotated, OF children, rotated] intro!: map_M_pure_I)[1] + using child_to \cast child \ set child_to\ returns_result_eq by fastforce + ultimately show ?thesis + by auto +qed + +lemma to_tree_order_child: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ to_tree_order ptr \\<^sub>r nodes" + assumes "h \ get_child_nodes parent \\<^sub>r children" + assumes "cast child \ ptr" + assumes "child \ set children" + assumes "cast child \ set nodes" +shows "parent \ set nodes" +proof(insert assms(1) assms(4) assms(6) assms(8), induct ptr arbitrary: nodes + rule: heap_wellformed_induct) + case (step p) + have "p |\| object_ptr_kinds h" + using \h \ to_tree_order p \\<^sub>r nodes\ to_tree_order_ptr_in_heap + using assms(1) assms(2) assms(3) by blast + then obtain children where children: "h \ get_child_nodes p \\<^sub>r children" + by (meson assms(2) assms(3) get_child_nodes_ok is_OK_returns_result_E local.known_ptrs_known_ptr) + then show ?case + proof (cases "children = []") + case True + then show ?thesis + using step(2) step(3) step(4) children + by(auto simp add: to_tree_order_def[of p] map_M_pure_I elim!: bind_returns_result_E2 + dest!: bind_returns_result_E3[rotated, OF children, rotated]) + next + case False + then obtain c child_to where + child: "c \ set children" and + child_to: "h \ to_tree_order (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c) \\<^sub>r child_to" and + "cast child \ set child_to" + using step(2) children + apply(auto simp add: to_tree_order_def[of p] map_M_pure_I elim!: bind_returns_result_E2 + dest!: bind_returns_result_E3[rotated, OF children, rotated])[1] + by (metis (full_types) assms(1) assms(2) assms(3) get_parent_ptr_in_heap + is_OK_returns_result_I l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.child_parent_dual + l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms node_ptr_kinds_commutes + returns_result_select_result step.prems(1) step.prems(2) step.prems(3) + to_tree_order_either_ptr_or_in_children to_tree_order_ok) + then have "set child_to \ set nodes" + using assms(1) child children step.prems(1) to_tree_order_child_subset by auto + + show ?thesis + proof (cases "c = child") + case True + then have "parent = p" + using step(3) children child assms(5) assms(7) + by (meson assms(1) assms(2) assms(3) child_parent_dual option.inject returns_result_eq) + + then show ?thesis + using step.prems(1) to_tree_order_ptr_in_result by blast + next + case False + then show ?thesis + using step(1)[OF children child child_to] step(3) step(4) + using \set child_to \ set nodes\ + using \cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \ set child_to\ by auto + qed + qed +qed + +lemma to_tree_order_node_ptrs: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ to_tree_order ptr \\<^sub>r nodes" + assumes "ptr' \ ptr" + assumes "ptr' \ set nodes" + shows "is_node_ptr_kind ptr'" +proof(insert assms(1) assms(4) assms(5) assms(6), induct ptr arbitrary: nodes + rule: heap_wellformed_induct) + case (step p) + have "p |\| object_ptr_kinds h" + using \h \ to_tree_order p \\<^sub>r nodes\ to_tree_order_ptr_in_heap + using assms(1) assms(2) assms(3) by blast + then obtain children where children: "h \ get_child_nodes p \\<^sub>r children" + by (meson assms(2) assms(3) get_child_nodes_ok is_OK_returns_result_E local.known_ptrs_known_ptr) + then show ?case + proof (cases "children = []") + case True + then show ?thesis + using step(2) step(3) step(4) children + by(auto simp add: to_tree_order_def[of p] map_M_pure_I elim!: bind_returns_result_E2 + dest!: bind_returns_result_E3[rotated, OF children, rotated])[1] + next + case False + then obtain c child_to where + child: "c \ set children" and + child_to: "h \ to_tree_order (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c) \\<^sub>r child_to" and + "ptr' \ set child_to" + using step(2) children + apply(auto simp add: to_tree_order_def[of p] map_M_pure_I elim!: bind_returns_result_E2 + dest!: bind_returns_result_E3[rotated, OF children, rotated])[1] + using step.prems(1) step.prems(2) step.prems(3) to_tree_order_either_ptr_or_in_children by blast + then have "set child_to \ set nodes" + using assms(1) child children step.prems(1) to_tree_order_child_subset by auto + + show ?thesis + proof (cases "cast c = ptr") + case True + then show ?thesis + using step \ptr' \ set child_to\ assms(5) child child_to children by blast + next + case False + then show ?thesis + using \ptr' \ set child_to\ child child_to children is_node_ptr_kind_cast step.hyps by blast + qed + qed +qed + +lemma to_tree_order_child2: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ to_tree_order ptr \\<^sub>r nodes" + assumes "cast child \ ptr" + assumes "cast child \ set nodes" + obtains parent where "h \ get_parent child \\<^sub>r Some parent" and "parent \ set nodes" +proof - + assume 1: "(\parent. h \ get_parent child \\<^sub>r Some parent \ parent \ set nodes \ thesis)" + show thesis + proof(insert assms(1) assms(4) assms(5) assms(6) 1, induct ptr arbitrary: nodes + rule: heap_wellformed_induct) + case (step p) + have "p |\| object_ptr_kinds h" + using \h \ to_tree_order p \\<^sub>r nodes\ to_tree_order_ptr_in_heap + using assms(1) assms(2) assms(3) by blast + then obtain children where children: "h \ get_child_nodes p \\<^sub>r children" + by (meson assms(2) assms(3) get_child_nodes_ok is_OK_returns_result_E local.known_ptrs_known_ptr) + then show ?case + proof (cases "children = []") + case True + then show ?thesis + using step(2) step(3) step(4) children + by(auto simp add: to_tree_order_def[of p] map_M_pure_I elim!: bind_returns_result_E2 + dest!: bind_returns_result_E3[rotated, OF children, rotated]) + next + case False + then obtain c child_to where + child: "c \ set children" and + child_to: "h \ to_tree_order (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r c) \\<^sub>r child_to" and + "cast child \ set child_to" + using step(2) children + apply(auto simp add: to_tree_order_def[of p] map_M_pure_I elim!: bind_returns_result_E2 + dest!: bind_returns_result_E3[rotated, OF children, rotated])[1] + using step.prems(1) step.prems(2) step.prems(3) to_tree_order_either_ptr_or_in_children + by blast + then have "set child_to \ set nodes" + using assms(1) child children step.prems(1) to_tree_order_child_subset by auto + + have "cast child |\| object_ptr_kinds h" + using assms(1) assms(2) assms(3) assms(4) assms(6) to_tree_order_ptrs_in_heap by blast + then obtain parent_opt where parent_opt: "h \ get_parent child \\<^sub>r parent_opt" + by (meson assms(2) assms(3) is_OK_returns_result_E get_parent_ok node_ptr_kinds_commutes) + then show ?thesis + proof (induct parent_opt) + case None + then show ?case + by (metis \cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \ set child_to\ assms(1) assms(2) assms(3) + cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject child child_parent_dual child_to children + option.distinct(1) returns_result_eq step.hyps) + next + case (Some option) + then show ?case + by (meson assms(1) assms(2) assms(3) get_parent_child_dual step.prems(1) step.prems(2) + step.prems(3) step.prems(4) to_tree_order_child) + qed + qed + qed +qed + +lemma to_tree_order_parent_child_rel: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ to_tree_order ptr \\<^sub>r to" + shows "(ptr, child) \ (parent_child_rel h)\<^sup>* \ child \ set to" +proof + assume 3: "(ptr, child) \ (parent_child_rel h)\<^sup>*" + show "child \ set to" + proof (insert 3, induct child rule: heap_wellformed_induct_rev[OF assms(1)]) + case (1 child) + then show ?case + proof (cases "ptr = child") + case True + then show ?thesis + using assms(4) + apply(simp add: to_tree_order_def) + by(auto simp add: map_M_pure_I elim!: bind_returns_result_E2) + next + case False + obtain child_parent where + "(ptr, child_parent) \ (parent_child_rel h)\<^sup>*" and + "(child_parent, child) \ (parent_child_rel h)" + using \ptr \ child\ + by (metis "1.prems" rtranclE) + obtain child_node where child_node: "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node = child" + using \(child_parent, child) \ parent_child_rel h\ node_ptr_casts_commute3 + parent_child_rel_node_ptr + by blast + then have "h \ get_parent child_node \\<^sub>r Some child_parent" + using \(child_parent, child) \ (parent_child_rel h)\ + by (meson assms(1) assms(2) assms(3) is_OK_returns_result_E l_get_parent_wf.child_parent_dual + l_heap_is_wellformed.parent_child_rel_child local.get_child_nodes_ok + local.known_ptrs_known_ptr local.l_get_parent_wf_axioms + local.l_heap_is_wellformed_axioms local.parent_child_rel_parent_in_heap) + then show ?thesis + using 1(1) child_node \(ptr, child_parent) \ (parent_child_rel h)\<^sup>*\ + using assms(1) assms(2) assms(3) assms(4) to_tree_order_parent by blast + qed + qed +next + assume "child \ set to" + then show "(ptr, child) \ (parent_child_rel h)\<^sup>*" + proof (induct child rule: heap_wellformed_induct_rev[OF assms(1)]) + case (1 child) + then show ?case + proof (cases "ptr = child") + case True + then show ?thesis + by simp + next + case False + then have "\parent. (parent, child) \ (parent_child_rel h)" + using 1(2) assms(4) to_tree_order_child2[OF assms(1) assms(2) assms(3) assms(4)] + to_tree_order_node_ptrs + by (metis assms(1) assms(2) assms(3) node_ptr_casts_commute3 parent_child_rel_parent) + then obtain child_node where child_node: "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node = child" + using node_ptr_casts_commute3 parent_child_rel_node_ptr by blast + then obtain child_parent where child_parent: "h \ get_parent child_node \\<^sub>r Some child_parent" + using \\parent. (parent, child) \ (parent_child_rel h)\ + by (metis "1.prems" False assms(1) assms(2) assms(3) assms(4) to_tree_order_child2) + then have "(child_parent, child) \ (parent_child_rel h)" + using assms(1) child_node parent_child_rel_parent by blast + moreover have "child_parent \ set to" + by (metis "1.prems" False assms(1) assms(2) assms(3) assms(4) child_node child_parent + get_parent_child_dual to_tree_order_child) + then have "(ptr, child_parent) \ (parent_child_rel h)\<^sup>*" + using 1 child_node child_parent by blast + ultimately show ?thesis + by auto + qed + qed +qed +end + +interpretation i_to_tree_order_wf?: l_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes + get_child_nodes_locs to_tree_order known_ptrs get_parent + get_parent_locs heap_is_wellformed parent_child_rel + get_disconnected_nodes get_disconnected_nodes_locs + using instances + apply(simp add: l_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) + done +declare l_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances] + +locale l_to_tree_order_wf = l_heap_is_wellformed_defs + l_type_wf + l_known_ptrs + + l_to_tree_order_defs + + l_get_parent_defs + l_get_child_nodes_defs + + assumes to_tree_order_ok: + "heap_is_wellformed h \ ptr |\| object_ptr_kinds h \ known_ptrs h \ type_wf h + \ h \ ok (to_tree_order ptr)" + assumes to_tree_order_ptrs_in_heap: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ h \ to_tree_order ptr \\<^sub>r to + \ ptr' \ set to \ ptr' |\| object_ptr_kinds h" + assumes to_tree_order_parent_child_rel: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ h \ to_tree_order ptr \\<^sub>r to + \ (ptr, child_ptr) \ (parent_child_rel h)\<^sup>* \ child_ptr \ set to" + assumes to_tree_order_child2: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ h \ to_tree_order ptr \\<^sub>r nodes + \ cast child \ ptr \ cast child \ set nodes + \ (\parent. h \ get_parent child \\<^sub>r Some parent + \ parent \ set nodes \ thesis) + \ thesis" + assumes to_tree_order_node_ptrs: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ h \ to_tree_order ptr \\<^sub>r nodes + \ ptr' \ ptr \ ptr' \ set nodes \ is_node_ptr_kind ptr'" + assumes to_tree_order_child: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ h \ to_tree_order ptr \\<^sub>r nodes + \ h \ get_child_nodes parent \\<^sub>r children \ cast child \ ptr + \ child \ set children \ cast child \ set nodes + \ parent \ set nodes" + assumes to_tree_order_ptr_in_result: + "h \ to_tree_order ptr \\<^sub>r nodes \ ptr \ set nodes" + assumes to_tree_order_parent: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ h \ to_tree_order ptr \\<^sub>r nodes + \ h \ get_parent child \\<^sub>r Some parent \ parent \ set nodes + \ cast child \ set nodes" + assumes to_tree_order_subset: + "heap_is_wellformed h \ h \ to_tree_order ptr \\<^sub>r nodes \ node \ set nodes + \ h \ to_tree_order node \\<^sub>r nodes' \ known_ptrs h + \ type_wf h \ set nodes' \ set nodes" + +lemma to_tree_order_wf_is_l_to_tree_order_wf [instances]: + "l_to_tree_order_wf heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs + to_tree_order get_parent get_child_nodes" + using instances + apply(auto simp add: l_to_tree_order_wf_def l_to_tree_order_wf_axioms_def)[1] + using to_tree_order_ok + apply blast + using to_tree_order_ptrs_in_heap + apply blast + using to_tree_order_parent_child_rel + apply(blast, blast) + using to_tree_order_child2 + apply blast + using to_tree_order_node_ptrs + apply blast + using to_tree_order_child + apply blast + using to_tree_order_ptr_in_result + apply blast + using to_tree_order_parent + apply blast + using to_tree_order_subset + apply blast + done + + +subsubsection \get\_root\_node\ + +locale l_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_ancestors + + l_get_ancestors_wf + + l_get_root_node + + l_get_root_node_wf + + l_to_tree_order_wf + + l_get_parent + + l_get_parent_wf +begin +lemma to_tree_order_get_root_node: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ to_tree_order ptr \\<^sub>r to" + assumes "ptr' \ set to" + assumes "h \ get_root_node ptr' \\<^sub>r root_ptr" + assumes "ptr'' \ set to" + shows "h \ get_root_node ptr'' \\<^sub>r root_ptr" +proof - + obtain ancestors' where ancestors': "h \ get_ancestors ptr' \\<^sub>r ancestors'" + by (meson assms(1) assms(2) assms(3) assms(4) assms(5) get_ancestors_ok is_OK_returns_result_E + to_tree_order_ptrs_in_heap ) + moreover have "ptr \ set ancestors'" + using \h \ get_ancestors ptr' \\<^sub>r ancestors'\ + using assms(1) assms(2) assms(3) assms(4) assms(5) get_ancestors_parent_child_rel + to_tree_order_parent_child_rel by blast + + ultimately have "h \ get_root_node ptr \\<^sub>r root_ptr" + using \h \ get_root_node ptr' \\<^sub>r root_ptr\ + using assms(1) assms(2) assms(3) get_ancestors_ptr get_ancestors_same_root_node by blast + + obtain ancestors'' where ancestors'': "h \ get_ancestors ptr'' \\<^sub>r ancestors''" + by (meson assms(1) assms(2) assms(3) assms(4) assms(7) get_ancestors_ok is_OK_returns_result_E + to_tree_order_ptrs_in_heap) + moreover have "ptr \ set ancestors''" + using \h \ get_ancestors ptr'' \\<^sub>r ancestors''\ + using assms(1) assms(2) assms(3) assms(4) assms(7) get_ancestors_parent_child_rel + to_tree_order_parent_child_rel by blast + ultimately show ?thesis + using \h \ get_root_node ptr \\<^sub>r root_ptr\ assms(1) assms(2) assms(3) get_ancestors_ptr + get_ancestors_same_root_node by blast +qed + +lemma to_tree_order_same_root: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_root_node ptr \\<^sub>r root_ptr" + assumes "h \ to_tree_order root_ptr \\<^sub>r to" + assumes "ptr' \ set to" + shows "h \ get_root_node ptr' \\<^sub>r root_ptr" +proof (insert assms(1)(* assms(4) assms(5) *) assms(6), induct ptr' rule: heap_wellformed_induct_rev) + case (step child) + then show ?case + proof (cases "h \ get_root_node child \\<^sub>r child") + case True + then have "child = root_ptr" + using assms(1) assms(2) assms(3) assms(5) step.prems + by (metis (no_types, lifting) get_root_node_same_no_parent node_ptr_casts_commute3 + option.simps(3) returns_result_eq to_tree_order_child2 to_tree_order_node_ptrs) + then show ?thesis + using True by blast + next + case False + then obtain child_node parent where "cast child_node = child" + and "h \ get_parent child_node \\<^sub>r Some parent" + by (metis assms(1) assms(2) assms(3) assms(4) assms(5) local.get_root_node_no_parent + local.get_root_node_not_node_same local.get_root_node_same_no_parent + local.to_tree_order_child2 local.to_tree_order_ptrs_in_heap node_ptr_casts_commute3 + step.prems) + then show ?thesis + proof (cases "child = root_ptr") + case True + then have "h \ get_root_node root_ptr \\<^sub>r root_ptr" + using assms(4) + using \cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node = child\ assms(1) assms(2) assms(3) + get_root_node_no_parent get_root_node_same_no_parent + by blast + then show ?thesis + using step assms(4) + using True by blast + next + case False + then have "parent \ set to" + using assms(5) step(2) to_tree_order_child \h \ get_parent child_node \\<^sub>r Some parent\ + \cast child_node = child\ + by (metis False assms(1) assms(2) assms(3) get_parent_child_dual) + then show ?thesis + using \cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node = child\ \h \ get_parent child_node \\<^sub>r Some parent\ + get_root_node_parent_same + using step.hyps by blast + qed + + qed +qed +end + +interpretation i_to_tree_order_wf_get_root_node_wf?: l_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + get_ancestors get_ancestors_locs heap_is_wellformed parent_child_rel known_ptr + known_ptrs type_wf get_child_nodes get_child_nodes_locs get_parent get_parent_locs + get_root_node get_root_node_locs to_tree_order + using instances + by(simp add: l_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) + +locale l_to_tree_order_wf_get_root_node_wf = l_type_wf + l_known_ptrs + l_to_tree_order_defs + + l_get_root_node_defs + l_heap_is_wellformed_defs + + assumes to_tree_order_get_root_node: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ h \ to_tree_order ptr \\<^sub>r to + \ ptr' \ set to \ h \ get_root_node ptr' \\<^sub>r root_ptr + \ ptr'' \ set to \ h \ get_root_node ptr'' \\<^sub>r root_ptr" + assumes to_tree_order_same_root: + "heap_is_wellformed h \ type_wf h \ known_ptrs h + \ h \ get_root_node ptr \\<^sub>r root_ptr + \ h \ to_tree_order root_ptr \\<^sub>r to \ ptr' \ set to + \ h \ get_root_node ptr' \\<^sub>r root_ptr" + +lemma to_tree_order_wf_get_root_node_wf_is_l_to_tree_order_wf_get_root_node_wf [instances]: + "l_to_tree_order_wf_get_root_node_wf type_wf known_ptr known_ptrs to_tree_order + get_root_node heap_is_wellformed" + using instances + apply(auto simp add: l_to_tree_order_wf_get_root_node_wf_def + l_to_tree_order_wf_get_root_node_wf_axioms_def)[1] + using to_tree_order_get_root_node apply blast + using to_tree_order_same_root apply blast + done + + +subsection \get\_owner\_document\ + +locale l_get_owner_document_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_known_ptrs + + l_heap_is_wellformed + + l_get_root_node + + l_get_ancestors + + l_get_ancestors_wf + + l_get_parent + + l_get_parent_wf + + l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin + +lemma get_owner_document_disconnected_nodes: + assumes "heap_is_wellformed h" + assumes "h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes" + assumes "node_ptr \ set disc_nodes" + assumes known_ptrs: "known_ptrs h" + assumes type_wf: "type_wf h" + shows "h \ get_owner_document (cast node_ptr) \\<^sub>r document_ptr" +proof - + have 2: "node_ptr |\| node_ptr_kinds h" + using assms heap_is_wellformed_disc_nodes_in_heap + by blast + have 3: "document_ptr |\| document_ptr_kinds h" + using assms(2) get_disconnected_nodes_ptr_in_heap by blast + have 0: + "\!document_ptr\set |h \ document_ptr_kinds_M|\<^sub>r. node_ptr \ set |h \ get_disconnected_nodes document_ptr|\<^sub>r" + by (metis (no_types, lifting) "3" DocumentMonad.ptr_kinds_ptr_kinds_M assms(1) assms(2) assms(3) + disjoint_iff_not_equal l_heap_is_wellformed.heap_is_wellformed_one_disc_parent + local.get_disconnected_nodes_ok local.l_heap_is_wellformed_axioms + returns_result_select_result select_result_I2 type_wf) + + have "h \ get_parent node_ptr \\<^sub>r None" + using heap_is_wellformed_children_disc_nodes_different child_parent_dual assms + using "2" disjoint_iff_not_equal local.get_parent_child_dual local.get_parent_ok + returns_result_select_result split_option_ex + by (metis (no_types, lifting)) + + then have 4: "h \ get_root_node (cast node_ptr) \\<^sub>r cast node_ptr" + using 2 get_root_node_no_parent + by blast + obtain document_ptrs where document_ptrs: "h \ document_ptr_kinds_M \\<^sub>r document_ptrs" + by simp + + then + have "h \ ok (filter_M (\document_ptr. do { + disconnected_nodes \ get_disconnected_nodes document_ptr; + return (((cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) \ cast ` set disconnected_nodes) + }) document_ptrs)" + using assms(1) get_disconnected_nodes_ok type_wf unfolding heap_is_wellformed_def + by(auto intro!: bind_is_OK_I2 filter_M_is_OK_I bind_pure_I) + then obtain candidates where + candidates: "h \ filter_M (\document_ptr. do { + disconnected_nodes \ get_disconnected_nodes document_ptr; + return (((cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) \ cast ` set disconnected_nodes) + }) document_ptrs \\<^sub>r candidates" + by auto + + + have eq: "\document_ptr. document_ptr |\| document_ptr_kinds h + \ node_ptr \ set |h \ get_disconnected_nodes document_ptr|\<^sub>r \ |h \ do { + disconnected_nodes \ get_disconnected_nodes document_ptr; + return (((cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) \ cast ` set disconnected_nodes) + }|\<^sub>r" + apply(auto dest!: get_disconnected_nodes_ok[OF type_wf] + intro!: select_result_I[where P=id, simplified] elim!: bind_returns_result_E2)[1] + apply(drule select_result_E[where P=id, simplified]) + by(auto elim!: bind_returns_result_E2) + + have filter: "filter (\document_ptr. |h \ do { + disconnected_nodes \ get_disconnected_nodes document_ptr; + return (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr \ cast ` set disconnected_nodes) + }|\<^sub>r) document_ptrs = [document_ptr]" + apply(rule filter_ex1) + using 0 document_ptrs apply(simp)[1] + using eq + using local.get_disconnected_nodes_ok apply auto[1] + using assms(2) assms(3) + apply(auto intro!: intro!: select_result_I[where P=id, simplified] + elim!: bind_returns_result_E2)[1] + using returns_result_eq apply fastforce + using document_ptrs 3 apply(simp) + using document_ptrs + by simp + have "h \ filter_M (\document_ptr. do { + disconnected_nodes \ get_disconnected_nodes document_ptr; + return (((cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) \ cast ` set disconnected_nodes) + }) document_ptrs \\<^sub>r [document_ptr]" + apply(rule filter_M_filter2) + using get_disconnected_nodes_ok document_ptrs 3 assms(1) type_wf filter + unfolding heap_is_wellformed_def + by(auto intro: bind_pure_I bind_is_OK_I2) + + with 4 document_ptrs have "h \ a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr () \\<^sub>r document_ptr" + by(auto simp add: a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + intro!: bind_pure_returns_result_I filter_M_pure_I bind_pure_I + split: option.splits)[1] + moreover have "known_ptr (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)" + using "4" assms(1) known_ptrs type_wf known_ptrs_known_ptr "2" node_ptr_kinds_commutes by blast + ultimately show ?thesis + using 2 + apply(auto simp add: known_ptr_impl get_owner_document_def a_get_owner_document_tups_def)[1] + apply(split invoke_splits, (rule conjI | rule impI)+)+ + apply(drule(1) known_ptr_not_document_ptr[folded known_ptr_impl]) + apply(drule(1) known_ptr_not_character_data_ptr) + apply(drule(1) known_ptr_not_element_ptr) + apply(simp add: NodeClass.known_ptr_defs) + by(auto split: option.splits intro!: bind_pure_returns_result_I) +qed + +lemma in_disconnected_nodes_no_parent: + assumes "heap_is_wellformed h" + and "h \ get_parent node_ptr \\<^sub>r None" + and "h \ get_owner_document (cast node_ptr) \\<^sub>r owner_document" + and "h \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes" + and known_ptrs: "known_ptrs h" + and type_wf: "type_wf h" + shows "node_ptr \ set disc_nodes" +proof - + have 2: "cast node_ptr |\| object_ptr_kinds h" + using assms(3) get_owner_document_ptr_in_heap by fast + then have 3: "h \ get_root_node (cast node_ptr) \\<^sub>r cast node_ptr" + using assms(2) local.get_root_node_no_parent by blast + + have "\(\parent_ptr. parent_ptr |\| object_ptr_kinds h \ node_ptr \ set |h \ get_child_nodes parent_ptr|\<^sub>r)" + apply(auto)[1] + using assms(2) child_parent_dual[OF assms(1)] type_wf + assms(1) assms(5) get_child_nodes_ok known_ptrs_known_ptr option.simps(3) + returns_result_eq returns_result_select_result + by (metis (no_types, hide_lams)) + moreover have "node_ptr |\| node_ptr_kinds h" + using assms(2) get_parent_ptr_in_heap by blast + thm heap_is_wellformed_children_disc_nodes_different + ultimately + have 0: "\document_ptr\set |h \ document_ptr_kinds_M|\<^sub>r. node_ptr \ set |h \ get_disconnected_nodes document_ptr|\<^sub>r" + by (metis DocumentMonad.ptr_kinds_ptr_kinds_M assms(1) finite_set_in heap_is_wellformed_children_disc_nodes) + then obtain document_ptr where + document_ptr: "document_ptr\set |h \ document_ptr_kinds_M|\<^sub>r" and + node_ptr_in_disc_nodes: "node_ptr \ set |h \ get_disconnected_nodes document_ptr|\<^sub>r" + by auto + then show ?thesis + using get_owner_document_disconnected_nodes known_ptrs type_wf assms + using DocumentMonad.ptr_kinds_ptr_kinds_M assms(1) assms(3) assms(4) get_disconnected_nodes_ok + returns_result_select_result select_result_I2 + by (metis (no_types, hide_lams) ) +qed +end + +locale l_get_owner_document_wf = l_heap_is_wellformed_defs + l_type_wf + l_known_ptrs + + l_get_disconnected_nodes_defs + l_get_owner_document_defs + + l_get_parent_defs + + assumes get_owner_document_disconnected_nodes: + "heap_is_wellformed h \ + known_ptrs h \ + type_wf h \ + h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes \ + node_ptr \ set disc_nodes \ + h \ get_owner_document (cast node_ptr) \\<^sub>r document_ptr" + assumes in_disconnected_nodes_no_parent: + "heap_is_wellformed h \ + h \ get_parent node_ptr \\<^sub>r None\ + h \ get_owner_document (cast node_ptr) \\<^sub>r owner_document \ + h \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes \ + known_ptrs h \ + type_wf h\ + node_ptr \ set disc_nodes" + +interpretation i_get_owner_document_wf?: + l_get_owner_document_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptrs type_wf heap_is_wellformed parent_child_rel + get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs + get_root_node get_root_node_locs get_parent get_parent_locs get_ancestors get_ancestors_locs + get_owner_document + using known_ptrs_is_l_known_ptrs + using heap_is_wellformed_is_l_heap_is_wellformed + using get_root_node_is_l_get_root_node + using get_ancestors_is_l_get_ancestors + using get_ancestors_wf_is_l_get_ancestors_wf + using get_parent_is_l_get_parent + using get_ancestors_wf_is_l_get_ancestors_wf + using get_parent_wf_is_l_get_parent_wf + using l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms + by(simp add: l_get_owner_document_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) + + + +lemma get_owner_document_wf_is_l_get_owner_document_wf [instances]: + "l_get_owner_document_wf heap_is_wellformed type_wf known_ptr known_ptrs get_disconnected_nodes + get_owner_document get_parent" + using known_ptrs_is_l_known_ptrs + apply(simp add: l_get_owner_document_wf_def l_get_owner_document_wf_axioms_def) + using get_owner_document_disconnected_nodes in_disconnected_nodes_no_parent + by fast + + +subsection \Preserving heap-wellformedness\ + + +subsection \set\_attribute\ + +locale l_set_attribute_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_set_attribute_get_disconnected_nodes + + l_set_attribute_get_child_nodes +begin +lemma set_attribute_preserves_wellformedness: + assumes "heap_is_wellformed h" + and "h \ set_attribute element_ptr k v \\<^sub>h h'" + shows "heap_is_wellformed h'" + thm preserves_wellformedness_writes_needed + apply(rule preserves_wellformedness_writes_needed[OF assms set_attribute_writes]) + using set_attribute_get_child_nodes + apply(fast) + using set_attribute_get_disconnected_nodes apply(fast) + by(auto simp add: all_args_def set_attribute_locs_def) +end + + +subsection \remove\_child\ + +locale l_remove_child_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_heap_is_wellformed + + l_set_disconnected_nodes_get_child_nodes +begin +lemma remove_child_removes_parent: + assumes wellformed: "heap_is_wellformed h" + and remove_child: "h \ remove_child ptr child \\<^sub>h h2" + and known_ptrs: "known_ptrs h" + and type_wf: "type_wf h" + shows "h2 \ get_parent child \\<^sub>r None" +proof - + obtain children where children: "h \ get_child_nodes ptr \\<^sub>r children" + using remove_child remove_child_def by auto + then have "child \ set children" + using remove_child remove_child_def + by(auto elim!: bind_returns_heap_E dest: returns_result_eq split: if_splits) + then have h1: "\other_ptr other_children. other_ptr \ ptr + \ h \ get_child_nodes other_ptr \\<^sub>r other_children \ child \ set other_children" + using assms(1) known_ptrs type_wf child_parent_dual + by (meson child_parent_dual children option.inject returns_result_eq) + + have known_ptr: "known_ptr ptr" + using known_ptrs + by (meson is_OK_returns_heap_I l_known_ptrs.known_ptrs_known_ptr l_known_ptrs_axioms + remove_child remove_child_ptr_in_heap) + + obtain owner_document disc_nodes h' where + owner_document: "h \ get_owner_document (cast child) \\<^sub>r owner_document" and + disc_nodes: "h \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes" and + h': "h \ set_disconnected_nodes owner_document (child # disc_nodes) \\<^sub>h h'" and + h2: "h' \ set_child_nodes ptr (remove1 child children) \\<^sub>h h2" + using assms children unfolding remove_child_def + apply(auto split: if_splits elim!: bind_returns_heap_E)[1] + by (metis (full_types) get_child_nodes_pure get_disconnected_nodes_pure + get_owner_document_pure pure_returns_heap_eq returns_result_eq) + + have "object_ptr_kinds h = object_ptr_kinds h2" + using remove_child_writes remove_child unfolding remove_child_locs_def + apply(rule writes_small_big) + using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved + by(auto simp add: reflp_def transp_def) + then have "|h \ object_ptr_kinds_M|\<^sub>r = |h2 \ object_ptr_kinds_M|\<^sub>r" + unfolding object_ptr_kinds_M_defs by simp + + have "type_wf h'" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", + OF set_disconnected_nodes_writes h'] + using set_disconnected_nodes_types_preserved type_wf + by(auto simp add: reflp_def transp_def) + have "type_wf h2" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", + OF remove_child_writes remove_child] unfolding remove_child_locs_def + using set_disconnected_nodes_types_preserved set_child_nodes_types_preserved type_wf + apply(auto simp add: reflp_def transp_def)[1] + by blast + then obtain children' where children': "h2 \ get_child_nodes ptr \\<^sub>r children'" + using h2 set_child_nodes_get_child_nodes known_ptr + by (metis \object_ptr_kinds h = object_ptr_kinds h2\ children get_child_nodes_ok + get_child_nodes_ptr_in_heap is_OK_returns_result_E is_OK_returns_result_I) + + have "child \ set children'" + by (metis (mono_tags, lifting) \type_wf h'\ children children' distinct_remove1_removeAll h2 + known_ptr local.heap_is_wellformed_children_distinct + local.set_child_nodes_get_child_nodes member_remove remove_code(1) select_result_I2 + wellformed) + + + moreover have "\other_ptr other_children. other_ptr \ ptr + \ h' \ get_child_nodes other_ptr \\<^sub>r other_children \ child \ set other_children" + proof - + fix other_ptr other_children + assume a1: "other_ptr \ ptr" and a3: "h' \ get_child_nodes other_ptr \\<^sub>r other_children" + have "h \ get_child_nodes other_ptr \\<^sub>r other_children" + using get_child_nodes_reads set_disconnected_nodes_writes h' a3 + apply(rule reads_writes_separate_backwards) + using set_disconnected_nodes_get_child_nodes by fast + show "child \ set other_children" + using \h \ get_child_nodes other_ptr \\<^sub>r other_children\ a1 h1 by blast + qed + then have "\other_ptr other_children. other_ptr \ ptr + \ h2 \ get_child_nodes other_ptr \\<^sub>r other_children \ child \ set other_children" + proof - + fix other_ptr other_children + assume a1: "other_ptr \ ptr" and a3: "h2 \ get_child_nodes other_ptr \\<^sub>r other_children" + have "h' \ get_child_nodes other_ptr \\<^sub>r other_children" + using get_child_nodes_reads set_child_nodes_writes h2 a3 + apply(rule reads_writes_separate_backwards) + using set_disconnected_nodes_get_child_nodes a1 set_child_nodes_get_child_nodes_different_pointers + by metis + then show "child \ set other_children" + using \\other_ptr other_children. \other_ptr \ ptr; h' \ get_child_nodes other_ptr \\<^sub>r other_children\ + \ child \ set other_children\ a1 by blast + qed + ultimately have ha: "\other_ptr other_children. h2 \ get_child_nodes other_ptr \\<^sub>r other_children + \ child \ set other_children" + by (metis (full_types) children' returns_result_eq) + moreover obtain ptrs where ptrs: "h2 \ object_ptr_kinds_M \\<^sub>r ptrs" + by (simp add: object_ptr_kinds_M_defs) + moreover have "\ptr. ptr \ set ptrs \ h2 \ ok (get_child_nodes ptr)" + using \type_wf h2\ ptrs get_child_nodes_ok known_ptr + using \object_ptr_kinds h = object_ptr_kinds h2\ known_ptrs local.known_ptrs_known_ptr by auto + ultimately show "h2 \ get_parent child \\<^sub>r None" + apply(auto simp add: get_parent_def intro!: bind_pure_returns_result_I filter_M_pure_I bind_pure_I)[1] + proof - + have "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child |\| object_ptr_kinds h" + using get_owner_document_ptr_in_heap owner_document by blast + then show "h2 \ check_in_heap (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \\<^sub>r ()" + by (simp add: \object_ptr_kinds h = object_ptr_kinds h2\ check_in_heap_def) + next + show "(\other_ptr other_children. h2 \ get_child_nodes other_ptr \\<^sub>r other_children + \ child \ set other_children) \ + ptrs = sorted_list_of_set (fset (object_ptr_kinds h2)) \ + (\ptr. ptr |\| object_ptr_kinds h2 \ h2 \ ok get_child_nodes ptr) \ + h2 \ filter_M (\ptr. Heap_Error_Monad.bind (get_child_nodes ptr) + (\children. return (child \ set children))) (sorted_list_of_set (fset (object_ptr_kinds h2))) \\<^sub>r []" + by(auto intro!: filter_M_empty_I bind_pure_I) + qed +qed +end + +locale l_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_remove_child_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin + +lemma remove_child_parent_child_rel_subset: + assumes "heap_is_wellformed h" + and "h \ remove_child ptr child \\<^sub>h h'" + and "known_ptrs h" + and type_wf: "type_wf h" + shows "parent_child_rel h' \ parent_child_rel h" +proof (standard, safe) + + obtain owner_document children_h h2 disconnected_nodes_h where + owner_document: "h \ get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \\<^sub>r owner_document" and + children_h: "h \ get_child_nodes ptr \\<^sub>r children_h" and + child_in_children_h: "child \ set children_h" and + disconnected_nodes_h: "h \ get_disconnected_nodes owner_document \\<^sub>r disconnected_nodes_h" and + h2: "h \ set_disconnected_nodes owner_document (child # disconnected_nodes_h) \\<^sub>h h2" and + h': "h2 \ set_child_nodes ptr (remove1 child children_h) \\<^sub>h h'" + using assms(2) + apply(auto simp add: remove_child_def elim!: bind_returns_heap_E + dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure] + pure_returns_heap_eq[rotated, OF get_child_nodes_pure] + split: if_splits)[1] + using pure_returns_heap_eq by fastforce + have object_ptr_kinds_eq3: "object_ptr_kinds h = object_ptr_kinds h'" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h = object_ptr_kinds h'", + OF remove_child_writes assms(2)]) + unfolding remove_child_locs_def + using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have object_ptr_kinds_eq: "\ptrs. h \ object_ptr_kinds_M \\<^sub>r ptrs = h' \ object_ptr_kinds_M \\<^sub>r ptrs" + unfolding object_ptr_kinds_M_defs by simp + then have object_ptr_kinds_eq2: "|h \ object_ptr_kinds_M|\<^sub>r = |h' \ object_ptr_kinds_M|\<^sub>r" + using select_result_eq by force + then have node_ptr_kinds_eq2: "|h \ node_ptr_kinds_M|\<^sub>r = |h' \ node_ptr_kinds_M|\<^sub>r" + using node_ptr_kinds_M_eq by auto + then have node_ptr_kinds_eq3: "node_ptr_kinds h = node_ptr_kinds h'" + using node_ptr_kinds_M_eq by auto + have document_ptr_kinds_eq2: "|h \ document_ptr_kinds_M|\<^sub>r = |h' \ document_ptr_kinds_M|\<^sub>r" + using object_ptr_kinds_eq2 document_ptr_kinds_M_eq by auto + then have document_ptr_kinds_eq3: "document_ptr_kinds h = document_ptr_kinds h'" + using document_ptr_kinds_M_eq by auto + have children_eq: + "\ptr' children. ptr \ ptr' \ h \ get_child_nodes ptr' \\<^sub>r children = h' \ get_child_nodes ptr' \\<^sub>r children" + apply(rule reads_writes_preserved[OF get_child_nodes_reads remove_child_writes assms(2)]) + unfolding remove_child_locs_def + using set_disconnected_nodes_get_child_nodes set_child_nodes_get_child_nodes_different_pointers + by fast + then have children_eq2: + "\ptr' children. ptr \ ptr' \ |h \ get_child_nodes ptr'|\<^sub>r = |h' \ get_child_nodes ptr'|\<^sub>r" + using select_result_eq by force + have disconnected_nodes_eq: + "\document_ptr disconnected_nodes. document_ptr \ owner_document + \ h \ get_disconnected_nodes document_ptr \\<^sub>r disconnected_nodes + = h' \ get_disconnected_nodes document_ptr \\<^sub>r disconnected_nodes" + apply(rule reads_writes_preserved[OF get_disconnected_nodes_reads remove_child_writes assms(2)]) + unfolding remove_child_locs_def + using set_child_nodes_get_disconnected_nodes set_disconnected_nodes_get_disconnected_nodes_different_pointers + by (metis (no_types, lifting) Un_iff owner_document select_result_I2) + then have disconnected_nodes_eq2: + "\document_ptr. document_ptr \ owner_document + \ |h \ get_disconnected_nodes document_ptr|\<^sub>r = |h' \ get_disconnected_nodes document_ptr|\<^sub>r" + using select_result_eq by force + + have "h2 \ get_child_nodes ptr \\<^sub>r children_h" + apply(rule reads_writes_separate_forwards[OF get_child_nodes_reads set_disconnected_nodes_writes h2 children_h] ) + by (simp add: set_disconnected_nodes_get_child_nodes) + + have "known_ptr ptr" + using assms(3) + using children_h get_child_nodes_ptr_in_heap local.known_ptrs_known_ptr by blast + have "type_wf h2" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_disconnected_nodes_writes h2] + using set_disconnected_nodes_types_preserved type_wf + by(auto simp add: reflp_def transp_def) + then have "type_wf h'" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_child_nodes_writes h'] + using set_child_nodes_types_preserved + by(auto simp add: reflp_def transp_def) + + have children_h': "h' \ get_child_nodes ptr \\<^sub>r remove1 child children_h" + using assms(2) owner_document h2 disconnected_nodes_h children_h + apply(auto simp add: remove_child_def split: if_splits)[1] + apply(drule bind_returns_heap_E3) + apply(auto split: if_splits)[1] + apply(simp) + apply(auto split: if_splits)[1] + apply(drule bind_returns_heap_E3) + apply(auto)[1] + apply(simp) + apply(drule bind_returns_heap_E3) + apply(auto)[1] + apply(simp) + apply(drule bind_returns_heap_E4) + apply(auto)[1] + apply(simp) + using \type_wf h2\ set_child_nodes_get_child_nodes \known_ptr ptr\ h' + by blast + + fix parent child + assume a1: "(parent, child) \ parent_child_rel h'" + then show "(parent, child) \ parent_child_rel h" + proof (cases "parent = ptr") + case True + then show ?thesis + using a1 remove_child_removes_parent[OF assms(1) assms(2)] children_h children_h' + get_child_nodes_ptr_in_heap + apply(auto simp add: parent_child_rel_def object_ptr_kinds_eq )[1] + by (metis notin_set_remove1) + next + case False + then show ?thesis + using a1 + by(auto simp add: parent_child_rel_def object_ptr_kinds_eq3 children_eq2) + qed +qed + + +lemma remove_child_heap_is_wellformed_preserved: + assumes "heap_is_wellformed h" + and "h \ remove_child ptr child \\<^sub>h h'" + and "known_ptrs h" + and type_wf: "type_wf h" + shows "type_wf h'" and "known_ptrs h'" and "heap_is_wellformed h'" +proof - + obtain owner_document children_h h2 disconnected_nodes_h where + owner_document: "h \ get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \\<^sub>r owner_document" and + children_h: "h \ get_child_nodes ptr \\<^sub>r children_h" and + child_in_children_h: "child \ set children_h" and + disconnected_nodes_h: "h \ get_disconnected_nodes owner_document \\<^sub>r disconnected_nodes_h" and + h2: "h \ set_disconnected_nodes owner_document (child # disconnected_nodes_h) \\<^sub>h h2" and + h': "h2 \ set_child_nodes ptr (remove1 child children_h) \\<^sub>h h'" + using assms(2) + apply(auto simp add: remove_child_def elim!: bind_returns_heap_E + dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure] + pure_returns_heap_eq[rotated, OF get_child_nodes_pure] split: if_splits)[1] + using pure_returns_heap_eq by fastforce + + have object_ptr_kinds_eq3: "object_ptr_kinds h = object_ptr_kinds h'" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h = object_ptr_kinds h'", + OF remove_child_writes assms(2)]) + unfolding remove_child_locs_def + using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have object_ptr_kinds_eq: "\ptrs. h \ object_ptr_kinds_M \\<^sub>r ptrs = h' \ object_ptr_kinds_M \\<^sub>r ptrs" + unfolding object_ptr_kinds_M_defs by simp + then have object_ptr_kinds_eq2: "|h \ object_ptr_kinds_M|\<^sub>r = |h' \ object_ptr_kinds_M|\<^sub>r" + using select_result_eq by force + then have node_ptr_kinds_eq2: "|h \ node_ptr_kinds_M|\<^sub>r = |h' \ node_ptr_kinds_M|\<^sub>r" + using node_ptr_kinds_M_eq by auto + then have node_ptr_kinds_eq3: "node_ptr_kinds h = node_ptr_kinds h'" + using node_ptr_kinds_M_eq by auto + have document_ptr_kinds_eq2: "|h \ document_ptr_kinds_M|\<^sub>r = |h' \ document_ptr_kinds_M|\<^sub>r" + using object_ptr_kinds_eq2 document_ptr_kinds_M_eq by auto + then have document_ptr_kinds_eq3: "document_ptr_kinds h = document_ptr_kinds h'" + using document_ptr_kinds_M_eq by auto + have children_eq: + "\ptr' children. ptr \ ptr' \ h \ get_child_nodes ptr' \\<^sub>r children = h' \ get_child_nodes ptr' \\<^sub>r children" + apply(rule reads_writes_preserved[OF get_child_nodes_reads remove_child_writes assms(2)]) + unfolding remove_child_locs_def + using set_disconnected_nodes_get_child_nodes set_child_nodes_get_child_nodes_different_pointers + by fast + then have children_eq2: + "\ptr' children. ptr \ ptr' \ |h \ get_child_nodes ptr'|\<^sub>r = |h' \ get_child_nodes ptr'|\<^sub>r" + using select_result_eq by force + have disconnected_nodes_eq: "\document_ptr disconnected_nodes. document_ptr \ owner_document + \ h \ get_disconnected_nodes document_ptr \\<^sub>r disconnected_nodes + = h' \ get_disconnected_nodes document_ptr \\<^sub>r disconnected_nodes" + apply(rule reads_writes_preserved[OF get_disconnected_nodes_reads remove_child_writes assms(2)]) + unfolding remove_child_locs_def + using set_child_nodes_get_disconnected_nodes set_disconnected_nodes_get_disconnected_nodes_different_pointers + by (metis (no_types, lifting) Un_iff owner_document select_result_I2) + then have disconnected_nodes_eq2: + "\document_ptr. document_ptr \ owner_document + \ |h \ get_disconnected_nodes document_ptr|\<^sub>r = |h' \ get_disconnected_nodes document_ptr|\<^sub>r" + using select_result_eq by force + + have "h2 \ get_child_nodes ptr \\<^sub>r children_h" + apply(rule reads_writes_separate_forwards[OF get_child_nodes_reads set_disconnected_nodes_writes h2 children_h] ) + by (simp add: set_disconnected_nodes_get_child_nodes) + + show "known_ptrs h'" + using object_ptr_kinds_eq3 known_ptrs_preserved \known_ptrs h\ by blast + + have "known_ptr ptr" + using assms(3) + using children_h get_child_nodes_ptr_in_heap local.known_ptrs_known_ptr by blast +have "type_wf h2" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_disconnected_nodes_writes h2] + using set_disconnected_nodes_types_preserved type_wf + by(auto simp add: reflp_def transp_def) + then show "type_wf h'" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_child_nodes_writes h'] + using set_child_nodes_types_preserved + by(auto simp add: reflp_def transp_def) + + have children_h': "h' \ get_child_nodes ptr \\<^sub>r remove1 child children_h" + using assms(2) owner_document h2 disconnected_nodes_h children_h + apply(auto simp add: remove_child_def split: if_splits)[1] + apply(drule bind_returns_heap_E3) + apply(auto split: if_splits)[1] + apply(simp) + apply(auto split: if_splits)[1] + apply(drule bind_returns_heap_E3) + apply(auto)[1] + apply(simp) + apply(drule bind_returns_heap_E3) + apply(auto)[1] + apply(simp) + apply(drule bind_returns_heap_E4) + apply(auto)[1] + apply simp + using \type_wf h2\ set_child_nodes_get_child_nodes \known_ptr ptr\ h' + by blast + + have disconnected_nodes_h2: "h2 \ get_disconnected_nodes owner_document \\<^sub>r child # disconnected_nodes_h" + using owner_document assms(2) h2 disconnected_nodes_h + apply (auto simp add: remove_child_def split: if_splits)[1] + apply(drule bind_returns_heap_E2) + apply(auto split: if_splits)[1] + apply(simp) + by(auto simp add: local.set_disconnected_nodes_get_disconnected_nodes split: if_splits) + then have disconnected_nodes_h': "h' \ get_disconnected_nodes owner_document \\<^sub>r child # disconnected_nodes_h" + apply(rule reads_writes_separate_forwards[OF get_disconnected_nodes_reads set_child_nodes_writes h']) + by (simp add: set_child_nodes_get_disconnected_nodes) + + moreover have "a_acyclic_heap h" + using assms(1) by (simp add: heap_is_wellformed_def) + have "parent_child_rel h' \ parent_child_rel h" + proof (standard, safe) + fix parent child + assume a1: "(parent, child) \ parent_child_rel h'" + then show "(parent, child) \ parent_child_rel h" + proof (cases "parent = ptr") + case True + then show ?thesis + using a1 remove_child_removes_parent[OF assms(1) assms(2)] children_h children_h' + get_child_nodes_ptr_in_heap + apply(auto simp add: parent_child_rel_def object_ptr_kinds_eq )[1] + by (metis imageI notin_set_remove1) + next + case False + then show ?thesis + using a1 + by(auto simp add: parent_child_rel_def object_ptr_kinds_eq3 children_eq2) + qed + qed + then have "a_acyclic_heap h'" + using \a_acyclic_heap h\ acyclic_heap_def acyclic_subset by blast + + moreover have "a_all_ptrs_in_heap h" + using assms(1) by (simp add: heap_is_wellformed_def) + then have "a_all_ptrs_in_heap h'" + apply(auto simp add: a_all_ptrs_in_heap_def node_ptr_kinds_eq3 disconnected_nodes_eq)[1] + apply (metis (no_types, lifting) type_wf assms(3) children_eq2 children_h children_h' + fset_of_list_subset fsubsetD get_child_nodes_ok get_child_nodes_ptr_in_heap + is_OK_returns_result_E is_OK_returns_result_I local.known_ptrs_known_ptr + object_ptr_kinds_eq3 select_result_I2 set_remove1_subset) + by (metis (no_types, lifting) + \\thesis. (\owner_document children_h h2 disconnected_nodes_h. + \h \ get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \\<^sub>r owner_document; + h \ get_child_nodes ptr \\<^sub>r children_h; child \ set children_h; + h \ get_disconnected_nodes owner_document \\<^sub>r disconnected_nodes_h; + h \ set_disconnected_nodes owner_document (child # disconnected_nodes_h) \\<^sub>h h2; + h2 \ set_child_nodes ptr (remove1 child children_h) \\<^sub>h h'\ \ thesis) \ thesis\ + disconnected_nodes_h disconnected_nodes_eq disconnected_nodes_h' fset_mp fset_of_list_elem + returns_result_eq set_ConsD) + + moreover have "a_owner_document_valid h" + using assms(1) by (simp add: heap_is_wellformed_def) + then have "a_owner_document_valid h'" + apply(auto simp add: a_owner_document_valid_def object_ptr_kinds_eq3 document_ptr_kinds_eq3 + node_ptr_kinds_eq3)[1] + proof - + fix node_ptr + assume 0: "\node_ptr. node_ptr |\| node_ptr_kinds h' + \ (\document_ptr. document_ptr |\| document_ptr_kinds h' + \ node_ptr \ set |h \ get_disconnected_nodes document_ptr|\<^sub>r) + \ (\parent_ptr. parent_ptr |\| object_ptr_kinds h' + \ node_ptr \ set |h \ get_child_nodes parent_ptr|\<^sub>r)" + and 1: "node_ptr |\| node_ptr_kinds h'" + and 2: "\parent_ptr. parent_ptr |\| object_ptr_kinds h' + \ node_ptr \ set |h' \ get_child_nodes parent_ptr|\<^sub>r" + then show "\document_ptr. document_ptr |\| document_ptr_kinds h' + \ node_ptr \ set |h' \ get_disconnected_nodes document_ptr|\<^sub>r" + proof (cases "node_ptr = child") + case True + show ?thesis + apply(rule exI[where x=owner_document]) + using children_eq2 disconnected_nodes_eq2 children_h children_h' disconnected_nodes_h' True + by (metis (no_types, lifting) get_disconnected_nodes_ptr_in_heap is_OK_returns_result_I + list.set_intros(1) select_result_I2) + next + case False + then show ?thesis + using 0 1 2 children_eq2 children_h children_h' disconnected_nodes_eq2 disconnected_nodes_h + disconnected_nodes_h' + apply(auto simp add: children_eq2 disconnected_nodes_eq2 dest!: select_result_I2)[1] + by (metis children_eq2 disconnected_nodes_eq2 in_set_remove1 list.set_intros(2)) + qed + qed + + moreover + { + have h0: "a_distinct_lists h" + using assms(1) by (simp add: heap_is_wellformed_def) + moreover have ha1: "(\x\set |h \ object_ptr_kinds_M|\<^sub>r. set |h \ get_child_nodes x|\<^sub>r) + \ (\x\set |h \ document_ptr_kinds_M|\<^sub>r. set |h \ get_disconnected_nodes x|\<^sub>r) = {}" + using \a_distinct_lists h\ + unfolding a_distinct_lists_def + by(auto) + have ha2: "ptr |\| object_ptr_kinds h" + using children_h get_child_nodes_ptr_in_heap by blast + have ha3: "child \ set |h \ get_child_nodes ptr|\<^sub>r" + using child_in_children_h children_h + by(simp) + have child_not_in: "\document_ptr. document_ptr |\| document_ptr_kinds h + \ child \ set |h \ get_disconnected_nodes document_ptr|\<^sub>r" + using ha1 ha2 ha3 + apply(simp) + using IntI by fastforce + moreover have "distinct |h \ object_ptr_kinds_M|\<^sub>r" + apply(rule select_result_I) + by(auto simp add: object_ptr_kinds_M_defs) + moreover have "distinct |h \ document_ptr_kinds_M|\<^sub>r" + apply(rule select_result_I) + by(auto simp add: document_ptr_kinds_M_defs) + ultimately have "a_distinct_lists h'" + proof(simp (no_asm) add: a_distinct_lists_def, safe) + assume 1: "a_distinct_lists h" + and 3: "distinct |h \ object_ptr_kinds_M|\<^sub>r" + + assume 1: "a_distinct_lists h" + and 3: "distinct |h \ object_ptr_kinds_M|\<^sub>r" + have 4: "distinct (concat ((map (\ptr. |h \ get_child_nodes ptr|\<^sub>r) |h \ object_ptr_kinds_M|\<^sub>r)))" + using 1 by(auto simp add: a_distinct_lists_def) + show "distinct (concat (map (\ptr. |h' \ get_child_nodes ptr|\<^sub>r) + (sorted_list_of_set (fset (object_ptr_kinds h')))))" + proof(rule distinct_concat_map_I[OF 3[unfolded object_ptr_kinds_eq2], simplified]) + fix x + assume 5: "x |\| object_ptr_kinds h'" + then have 6: "distinct |h \ get_child_nodes x|\<^sub>r" + using 4 distinct_concat_map_E object_ptr_kinds_eq2 by fastforce + obtain children where children: "h \ get_child_nodes x \\<^sub>r children" + and distinct_children: "distinct children" + by (metis "5" "6" type_wf assms(3) get_child_nodes_ok local.known_ptrs_known_ptr + object_ptr_kinds_eq3 select_result_I) + obtain children' where children': "h' \ get_child_nodes x \\<^sub>r children'" + using children children_eq children_h' by fastforce + then have "distinct children'" + proof (cases "ptr = x") + case True + then show ?thesis + using children distinct_children children_h children_h' + by (metis children' distinct_remove1 returns_result_eq) + next + case False + then show ?thesis + using children distinct_children children_eq[OF False] + using children' distinct_lists_children h0 + using select_result_I2 by fastforce + qed + + then show "distinct |h' \ get_child_nodes x|\<^sub>r" + using children' by(auto simp add: ) + next + fix x y + assume 5: "x |\| object_ptr_kinds h'" and 6: "y |\| object_ptr_kinds h'" and 7: "x \ y" + obtain children_x where children_x: "h \ get_child_nodes x \\<^sub>r children_x" + by (metis "5" type_wf assms(3) get_child_nodes_ok is_OK_returns_result_E + local.known_ptrs_known_ptr object_ptr_kinds_eq3) + obtain children_y where children_y: "h \ get_child_nodes y \\<^sub>r children_y" + by (metis "6" type_wf assms(3) get_child_nodes_ok is_OK_returns_result_E + local.known_ptrs_known_ptr object_ptr_kinds_eq3) + obtain children_x' where children_x': "h' \ get_child_nodes x \\<^sub>r children_x'" + using children_eq children_h' children_x by fastforce + obtain children_y' where children_y': "h' \ get_child_nodes y \\<^sub>r children_y'" + using children_eq children_h' children_y by fastforce + have "distinct (concat (map (\ptr. |h \ get_child_nodes ptr|\<^sub>r) |h \ object_ptr_kinds_M|\<^sub>r))" + using h0 by(auto simp add: a_distinct_lists_def) + then have 8: "set children_x \ set children_y = {}" + using "7" assms(1) children_x children_y local.heap_is_wellformed_one_parent by blast + have "set children_x' \ set children_y' = {}" + proof (cases "ptr = x") + case True + then have "ptr \ y" + by(simp add: 7) + have "children_x' = remove1 child children_x" + using children_h children_h' children_x children_x' True returns_result_eq by fastforce + moreover have "children_y' = children_y" + using children_y children_y' children_eq[OF \ptr \ y\] by auto + ultimately show ?thesis + using 8 set_remove1_subset by fastforce + next + case False + then show ?thesis + proof (cases "ptr = y") + case True + have "children_y' = remove1 child children_y" + using children_h children_h' children_y children_y' True returns_result_eq by fastforce + moreover have "children_x' = children_x" + using children_x children_x' children_eq[OF \ptr \ x\] by auto + ultimately show ?thesis + using 8 set_remove1_subset by fastforce + next + case False + have "children_x' = children_x" + using children_x children_x' children_eq[OF \ptr \ x\] by auto + moreover have "children_y' = children_y" + using children_y children_y' children_eq[OF \ptr \ y\] by auto + ultimately show ?thesis + using 8 by simp + qed + qed + then show "set |h' \ get_child_nodes x|\<^sub>r \ set |h' \ get_child_nodes y|\<^sub>r = {}" + using children_x' children_y' + by (metis (no_types, lifting) select_result_I2) + qed + next + assume 2: "distinct |h \ document_ptr_kinds_M|\<^sub>r" + then have 4: "distinct (sorted_list_of_set (fset (document_ptr_kinds h')))" + by simp + have 3: "distinct (concat (map (\document_ptr. |h \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h')))))" + using h0 + by(simp add: a_distinct_lists_def document_ptr_kinds_eq3) + + show "distinct (concat (map (\document_ptr. |h' \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h')))))" + proof(rule distinct_concat_map_I[OF 4[unfolded document_ptr_kinds_eq3]]) + fix x + assume 4: "x \ set (sorted_list_of_set (fset (document_ptr_kinds h')))" + have 5: "distinct |h \ get_disconnected_nodes x|\<^sub>r" + using distinct_lists_disconnected_nodes[OF h0] 4 get_disconnected_nodes_ok + by (simp add: type_wf document_ptr_kinds_eq3 select_result_I) + show "distinct |h' \ get_disconnected_nodes x|\<^sub>r" + proof (cases "x = owner_document") + case True + have "child \ set |h \ get_disconnected_nodes x|\<^sub>r" + using child_not_in document_ptr_kinds_eq2 "4" by fastforce + moreover have "|h' \ get_disconnected_nodes x|\<^sub>r = child # |h \ get_disconnected_nodes x|\<^sub>r" + using disconnected_nodes_h' disconnected_nodes_h unfolding True + by(simp) + ultimately show ?thesis + using 5 unfolding True + by simp + next + case False + show ?thesis + using "5" False disconnected_nodes_eq2 by auto + qed + next + fix x y + assume 4: "x \ set (sorted_list_of_set (fset (document_ptr_kinds h')))" + and 5: "y \ set (sorted_list_of_set (fset (document_ptr_kinds h')))" and "x \ y" + obtain disc_nodes_x where disc_nodes_x: "h \ get_disconnected_nodes x \\<^sub>r disc_nodes_x" + using 4 get_disconnected_nodes_ok[OF \type_wf h\, of x] document_ptr_kinds_eq2 + by auto + obtain disc_nodes_y where disc_nodes_y: "h \ get_disconnected_nodes y \\<^sub>r disc_nodes_y" + using 5 get_disconnected_nodes_ok[OF \type_wf h\, of y] document_ptr_kinds_eq2 + by auto + obtain disc_nodes_x' where disc_nodes_x': "h' \ get_disconnected_nodes x \\<^sub>r disc_nodes_x'" + using 4 get_disconnected_nodes_ok[OF \type_wf h'\, of x] document_ptr_kinds_eq2 + by auto + obtain disc_nodes_y' where disc_nodes_y': "h' \ get_disconnected_nodes y \\<^sub>r disc_nodes_y'" + using 5 get_disconnected_nodes_ok[OF \type_wf h'\, of y] document_ptr_kinds_eq2 + by auto + have "distinct + (concat (map (\document_ptr. |h \ get_disconnected_nodes document_ptr|\<^sub>r) |h \ document_ptr_kinds_M|\<^sub>r))" + using h0 by (simp add: a_distinct_lists_def) + then have 6: "set disc_nodes_x \ set disc_nodes_y = {}" + using \x \ y\ assms(1) disc_nodes_x disc_nodes_y local.heap_is_wellformed_one_disc_parent + by blast + + have "set disc_nodes_x' \ set disc_nodes_y' = {}" + proof (cases "x = owner_document") + case True + then have "y \ owner_document" + using \x \ y\ by simp + then have "disc_nodes_y' = disc_nodes_y" + using disconnected_nodes_eq[OF \y \ owner_document\] disc_nodes_y disc_nodes_y' + by auto + have "disc_nodes_x' = child # disc_nodes_x" + using disconnected_nodes_h' disc_nodes_x disc_nodes_x' True disconnected_nodes_h returns_result_eq + by fastforce + have "child \ set disc_nodes_y" + using child_not_in disc_nodes_y 5 + using document_ptr_kinds_eq2 by fastforce + then show ?thesis + apply(unfold \disc_nodes_x' = child # disc_nodes_x\ \disc_nodes_y' = disc_nodes_y\) + using 6 by auto + next + case False + then show ?thesis + proof (cases "y = owner_document") + case True + then have "disc_nodes_x' = disc_nodes_x" + using disconnected_nodes_eq[OF \x \ owner_document\] disc_nodes_x disc_nodes_x' by auto + have "disc_nodes_y' = child # disc_nodes_y" + using disconnected_nodes_h' disc_nodes_y disc_nodes_y' True disconnected_nodes_h returns_result_eq + by fastforce + have "child \ set disc_nodes_x" + using child_not_in disc_nodes_x 4 + using document_ptr_kinds_eq2 by fastforce + then show ?thesis + apply(unfold \disc_nodes_y' = child # disc_nodes_y\ \disc_nodes_x' = disc_nodes_x\) + using 6 by auto + next + case False + have "disc_nodes_x' = disc_nodes_x" + using disconnected_nodes_eq[OF \x \ owner_document\] disc_nodes_x disc_nodes_x' by auto + have "disc_nodes_y' = disc_nodes_y" + using disconnected_nodes_eq[OF \y \ owner_document\] disc_nodes_y disc_nodes_y' by auto + then show ?thesis + apply(unfold \disc_nodes_y' = disc_nodes_y\ \disc_nodes_x' = disc_nodes_x\) + using 6 by auto + qed + qed + then show "set |h' \ get_disconnected_nodes x|\<^sub>r \ set |h' \ get_disconnected_nodes y|\<^sub>r = {}" + using disc_nodes_x' disc_nodes_y' by auto + qed + next +fix x xa xb +assume 1: "xa \ fset (object_ptr_kinds h')" + and 2: "x \ set |h' \ get_child_nodes xa|\<^sub>r" + and 3: "xb \ fset (document_ptr_kinds h')" + and 4: "x \ set |h' \ get_disconnected_nodes xb|\<^sub>r" + obtain disc_nodes where disc_nodes: "h \ get_disconnected_nodes xb \\<^sub>r disc_nodes" + using 3 get_disconnected_nodes_ok[OF \type_wf h\, of xb] document_ptr_kinds_eq2 by auto + obtain disc_nodes' where disc_nodes': "h' \ get_disconnected_nodes xb \\<^sub>r disc_nodes'" + using 3 get_disconnected_nodes_ok[OF \type_wf h'\, of xb] document_ptr_kinds_eq2 by auto + + obtain children where children: "h \ get_child_nodes xa \\<^sub>r children" + by (metis "1" type_wf assms(3) finite_set_in get_child_nodes_ok is_OK_returns_result_E + local.known_ptrs_known_ptr object_ptr_kinds_eq3) + obtain children' where children': "h' \ get_child_nodes xa \\<^sub>r children'" + using children children_eq children_h' by fastforce + have "\x. x \ set |h \ get_child_nodes xa|\<^sub>r \ x \ set |h \ get_disconnected_nodes xb|\<^sub>r \ False" + using 1 3 + apply(fold \ object_ptr_kinds h = object_ptr_kinds h'\) + apply(fold \ document_ptr_kinds h = document_ptr_kinds h'\) + using children disc_nodes h0 apply(auto simp add: a_distinct_lists_def)[1] + by (metis (no_types, lifting) h0 local.distinct_lists_no_parent select_result_I2) + then have 5: "\x. x \ set children \ x \ set disc_nodes \ False" + using children disc_nodes by fastforce + have 6: "|h' \ get_child_nodes xa|\<^sub>r = children'" + using children' by (simp add: ) + have 7: "|h' \ get_disconnected_nodes xb|\<^sub>r = disc_nodes'" + using disc_nodes' by (simp add: ) + have "False" + proof (cases "xa = ptr") + case True + have "distinct children_h" + using children_h distinct_lists_children h0 \known_ptr ptr\ by blast + have "|h' \ get_child_nodes ptr|\<^sub>r = remove1 child children_h" + using children_h' + by(simp add: ) + have "children = children_h" + using True children children_h by auto + show ?thesis + using disc_nodes' children' 5 2 4 children_h \distinct children_h\ disconnected_nodes_h' + apply(auto simp add: 6 7 + \xa = ptr\ \|h' \ get_child_nodes ptr|\<^sub>r = remove1 child children_h\ \children = children_h\)[1] + by (metis (no_types, lifting) disc_nodes disconnected_nodes_eq2 disconnected_nodes_h + select_result_I2 set_ConsD) + next + case False + have "children' = children" + using children' children children_eq[OF False[symmetric]] + by auto + then show ?thesis + proof (cases "xb = owner_document") + case True + then show ?thesis + using disc_nodes disconnected_nodes_h disconnected_nodes_h' + using "2" "4" "5" "6" "7" False \children' = children\ assms(1) child_in_children_h + child_parent_dual children children_h disc_nodes' get_child_nodes_ptr_in_heap + list.set_cases list.simps(3) option.simps(1) returns_result_eq set_ConsD + by (metis (no_types, hide_lams) assms(3) type_wf) + next + case False + then show ?thesis + using "2" "4" "5" "6" "7" \children' = children\ disc_nodes disc_nodes' + disconnected_nodes_eq returns_result_eq + by metis + qed + qed + then show "x \ {}" + by simp + qed + } + + ultimately show "heap_is_wellformed h'" + using heap_is_wellformed_def by blast +qed + +lemma remove_child_removes_child: + assumes wellformed: "heap_is_wellformed h" + and remove_child: "h \ remove_child ptr' child \\<^sub>h h'" + and children: "h' \ get_child_nodes ptr \\<^sub>r children" +and known_ptrs: "known_ptrs h" +and type_wf: "type_wf h" +shows "child \ set children" +proof - + obtain owner_document children_h h2 disconnected_nodes_h where + owner_document: "h \ get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \\<^sub>r owner_document" and + children_h: "h \ get_child_nodes ptr' \\<^sub>r children_h" and + child_in_children_h: "child \ set children_h" and + disconnected_nodes_h: "h \ get_disconnected_nodes owner_document \\<^sub>r disconnected_nodes_h" and + h2: "h \ set_disconnected_nodes owner_document (child # disconnected_nodes_h) \\<^sub>h h2" and + h': "h2 \ set_child_nodes ptr' (remove1 child children_h) \\<^sub>h h'" + using assms(2) + apply(auto simp add: remove_child_def elim!: bind_returns_heap_E + dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure] + pure_returns_heap_eq[rotated, OF get_child_nodes_pure] + split: if_splits)[1] + using pure_returns_heap_eq + by fastforce + have "object_ptr_kinds h = object_ptr_kinds h'" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h = object_ptr_kinds h'", + OF remove_child_writes remove_child]) + unfolding remove_child_locs_def + using set_child_nodes_pointers_preserved set_disconnected_nodes_pointers_preserved + by (auto simp add: reflp_def transp_def) + moreover have "type_wf h'" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF remove_child_writes assms(2)] + using set_child_nodes_types_preserved set_disconnected_nodes_types_preserved type_wf + unfolding remove_child_locs_def + apply(auto simp add: reflp_def transp_def) + by blast + ultimately show ?thesis + using remove_child_removes_parent remove_child_heap_is_wellformed_preserved child_parent_dual + by (meson children known_ptrs local.known_ptrs_preserved option.distinct(1) remove_child + returns_result_eq type_wf wellformed) +qed + +lemma remove_child_removes_first_child: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_child_nodes ptr \\<^sub>r node_ptr # children" + assumes "h \ remove_child ptr node_ptr \\<^sub>h h'" + shows "h' \ get_child_nodes ptr \\<^sub>r children" +proof - + obtain h2 disc_nodes owner_document where + "h \ get_owner_document (cast node_ptr) \\<^sub>r owner_document" and + "h \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes" and + h2: "h \ set_disconnected_nodes owner_document (node_ptr # disc_nodes) \\<^sub>h h2" and + "h2 \ set_child_nodes ptr children \\<^sub>h h'" + using assms(5) + apply(auto simp add: remove_child_def + dest!: bind_returns_heap_E3[rotated, OF assms(4) get_child_nodes_pure, rotated])[1] + by(auto elim!: bind_returns_heap_E + bind_returns_heap_E2[rotated,OF get_owner_document_pure, rotated] + bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated]) + have "known_ptr ptr" + by (meson assms(3) assms(4) is_OK_returns_result_I get_child_nodes_ptr_in_heap known_ptrs_known_ptr) + moreover have "h2 \ get_child_nodes ptr \\<^sub>r node_ptr # children" + apply(rule reads_writes_separate_forwards[OF get_child_nodes_reads set_disconnected_nodes_writes h2 assms(4)]) + using set_disconnected_nodes_get_child_nodes set_child_nodes_get_child_nodes_different_pointers + by fast + moreover have "type_wf h2" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_disconnected_nodes_writes h2] + using \type_wf h\ set_disconnected_nodes_types_preserved + by(auto simp add: reflp_def transp_def) + ultimately show ?thesis + using set_child_nodes_get_child_nodes\h2 \ set_child_nodes ptr children \\<^sub>h h'\ + by fast +qed + +lemma remove_removes_child: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_child_nodes ptr \\<^sub>r node_ptr # children" + assumes "h \ remove node_ptr \\<^sub>h h'" + shows "h' \ get_child_nodes ptr \\<^sub>r children" +proof - + have "h \ get_parent node_ptr \\<^sub>r Some ptr" + using child_parent_dual assms by fastforce + show ?thesis + using assms remove_child_removes_first_child + by(auto simp add: remove_def + dest!: bind_returns_heap_E3[rotated, OF \h \ get_parent node_ptr \\<^sub>r Some ptr\, rotated] + bind_returns_heap_E3[rotated, OF assms(4) get_child_nodes_pure, rotated]) +qed + +lemma remove_for_all_empty_children: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_child_nodes ptr \\<^sub>r children" + assumes "h \ forall_M remove children \\<^sub>h h'" + shows "h' \ get_child_nodes ptr \\<^sub>r []" + using assms +proof(induct children arbitrary: h h') + case Nil + then show ?case + by simp +next + case (Cons a children) + have "h \ get_parent a \\<^sub>r Some ptr" + using child_parent_dual Cons by fastforce + with Cons show ?case + proof(auto elim!: bind_returns_heap_E)[1] + fix h2 + assume 0: "(\h h'. heap_is_wellformed h \ type_wf h \ known_ptrs h + \ h \ get_child_nodes ptr \\<^sub>r children + \ h \ forall_M remove children \\<^sub>h h' \ h' \ get_child_nodes ptr \\<^sub>r [])" + and 1: "heap_is_wellformed h" + and 2: "type_wf h" + and 3: "known_ptrs h" + and 4: "h \ get_child_nodes ptr \\<^sub>r a # children" + and 5: "h \ get_parent a \\<^sub>r Some ptr" + and 7: "h \ remove a \\<^sub>h h2" + and 8: "h2 \ forall_M remove children \\<^sub>h h'" + then have "h2 \ get_child_nodes ptr \\<^sub>r children" + using remove_removes_child by blast + + moreover have "heap_is_wellformed h2" + using 7 1 2 3 remove_child_heap_is_wellformed_preserved(3) + by(auto simp add: remove_def + elim!: bind_returns_heap_E + bind_returns_heap_E2[rotated, OF get_parent_pure, rotated] + split: option.splits) + moreover have "type_wf h2" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF remove_writes 7] + using \type_wf h\ remove_child_types_preserved + by(auto simp add: a_remove_child_locs_def reflp_def transp_def) + moreover have "object_ptr_kinds h = object_ptr_kinds h2" + using 7 + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h = object_ptr_kinds h'", + OF remove_writes]) + using remove_child_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have "known_ptrs h2" + using 3 known_ptrs_preserved by blast + + ultimately show "h' \ get_child_nodes ptr \\<^sub>r []" + using 0 8 by fast + qed +qed +end + +locale l_remove_child_wf2 = l_type_wf + l_known_ptrs + l_remove_child_defs + l_heap_is_wellformed_defs + + l_get_child_nodes_defs + l_remove_defs + + assumes remove_child_preserves_type_wf: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ h \ remove_child ptr child \\<^sub>h h' + \ type_wf h'" + assumes remove_child_preserves_known_ptrs: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ h \ remove_child ptr child \\<^sub>h h' + \ known_ptrs h'" + assumes remove_child_heap_is_wellformed_preserved: + "type_wf h \ known_ptrs h \ heap_is_wellformed h \ h \ remove_child ptr child \\<^sub>h h' + \ heap_is_wellformed h'" + assumes remove_child_removes_child: + "heap_is_wellformed h \ h \ remove_child ptr' child \\<^sub>h h' \ h' \ get_child_nodes ptr \\<^sub>r children + \ known_ptrs h \ type_wf h + \ child \ set children" + assumes remove_child_removes_first_child: + "heap_is_wellformed h \ type_wf h \ known_ptrs h + \ h \ get_child_nodes ptr \\<^sub>r node_ptr # children + \ h \ remove_child ptr node_ptr \\<^sub>h h' + \ h' \ get_child_nodes ptr \\<^sub>r children" + assumes remove_removes_child: + "heap_is_wellformed h \ type_wf h \ known_ptrs h + \ h \ get_child_nodes ptr \\<^sub>r node_ptr # children + \ h \ remove node_ptr \\<^sub>h h' \ h' \ get_child_nodes ptr \\<^sub>r children" + assumes remove_for_all_empty_children: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ h \ get_child_nodes ptr \\<^sub>r children + \ h \ forall_M remove children \\<^sub>h h' \ h' \ get_child_nodes ptr \\<^sub>r []" + +interpretation i_remove_child_wf2?: l_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs + set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document + get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes + set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs + heap_is_wellformed parent_child_rel + by unfold_locales + +lemma remove_child_wf2_is_l_remove_child_wf2 [instances]: + "l_remove_child_wf2 type_wf known_ptr known_ptrs remove_child heap_is_wellformed get_child_nodes remove" + apply(auto simp add: l_remove_child_wf2_def l_remove_child_wf2_axioms_def instances)[1] + using remove_child_heap_is_wellformed_preserved apply(fast, fast, fast) + using remove_child_removes_child apply fast + using remove_child_removes_first_child apply fast + using remove_removes_child apply fast + using remove_for_all_empty_children apply fast + done + + + +subsection \adopt\_node\ + +locale l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_parent_wf + + l_remove_child_wf2 + + l_heap_is_wellformed +begin +lemma adopt_node_removes_first_child: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ adopt_node owner_document node \\<^sub>h h'" + assumes "h \ get_child_nodes ptr' \\<^sub>r node # children" + shows "h' \ get_child_nodes ptr' \\<^sub>r children" +proof - + obtain old_document parent_opt h2 where + old_document: "h \ get_owner_document (cast node) \\<^sub>r old_document" and + parent_opt: "h \ get_parent node \\<^sub>r parent_opt" and + h2: "h \ (case parent_opt of Some parent \ do { remove_child parent node } + | None \ do { return ()}) \\<^sub>h h2" and + h': "h2 \ (if owner_document \ old_document then do { + old_disc_nodes \ get_disconnected_nodes old_document; + set_disconnected_nodes old_document (remove1 node old_disc_nodes); + disc_nodes \ get_disconnected_nodes owner_document; + set_disconnected_nodes owner_document (node # disc_nodes) + } else do { return () }) \\<^sub>h h'" + using assms(4) + by(auto simp add: adopt_node_def elim!: bind_returns_heap_E + dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure] + pure_returns_heap_eq[rotated, OF get_parent_pure]) + have "h2 \ get_child_nodes ptr' \\<^sub>r children" + using h2 remove_child_removes_first_child assms(1) assms(2) assms(3) assms(5) + by (metis list.set_intros(1) local.child_parent_dual option.simps(5) parent_opt returns_result_eq) + then + show ?thesis + using h' + by(auto elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] + dest!: reads_writes_separate_forwards[OF get_child_nodes_reads set_disconnected_nodes_writes] + split: if_splits) +qed +end + +locale l_adopt_node_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_root_node + + l_get_owner_document_wf + + l_remove_child_wf2 + + l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin + +lemma adopt_node_removes_child: + assumes wellformed: "heap_is_wellformed h" + and adopt_node: "h \ adopt_node owner_document node_ptr \\<^sub>h h2" + and children: "h2 \ get_child_nodes ptr \\<^sub>r children" + and known_ptrs: "known_ptrs h" + and type_wf: "type_wf h" + shows "node_ptr \ set children" +proof - + obtain old_document parent_opt h' where + old_document: "h \ get_owner_document (cast node_ptr) \\<^sub>r old_document" and + parent_opt: "h \ get_parent node_ptr \\<^sub>r parent_opt" and + h': "h \ (case parent_opt of Some parent \ remove_child parent node_ptr | None \ return () ) \\<^sub>h h'" + using adopt_node get_parent_pure + by(auto simp add: adopt_node_def + elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_owner_document_pure, rotated] + bind_returns_heap_E2[rotated, OF get_parent_pure, rotated] + bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] + split: if_splits) + + then have "h' \ get_child_nodes ptr \\<^sub>r children" + using adopt_node + apply(auto simp add: adopt_node_def + dest!: bind_returns_heap_E3[rotated, OF old_document, rotated] + bind_returns_heap_E3[rotated, OF parent_opt, rotated] + elim!: bind_returns_heap_E4[rotated, OF h', rotated])[1] + apply(auto split: if_splits + elim!: bind_returns_heap_E + bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated])[1] + apply (simp add: set_disconnected_nodes_get_child_nodes children + reads_writes_preserved[OF get_child_nodes_reads set_disconnected_nodes_writes]) + using children by blast + show ?thesis + proof(insert parent_opt h', induct parent_opt) + case None + then show ?case + using child_parent_dual wellformed known_ptrs type_wf + \h' \ get_child_nodes ptr \\<^sub>r children\ returns_result_eq + by fastforce + next + case (Some option) + then show ?case + using remove_child_removes_child \h' \ get_child_nodes ptr \\<^sub>r children\ known_ptrs type_wf + wellformed + by auto + qed +qed + +lemma adopt_node_preserves_wellformedness: + assumes "heap_is_wellformed h" + and "h \ adopt_node document_ptr child \\<^sub>h h'" + and known_ptrs: "known_ptrs h" + and type_wf: "type_wf h" + shows "heap_is_wellformed h'" +proof - + obtain old_document parent_opt h2 where + old_document: "h \ get_owner_document (cast child) \\<^sub>r old_document" + and + parent_opt: "h \ get_parent child \\<^sub>r parent_opt" + and + h2: "h \ (case parent_opt of Some parent \ remove_child parent child | None \ return ()) \\<^sub>h h2" + and + h': "h2 \ (if document_ptr \ old_document then do { + old_disc_nodes \ get_disconnected_nodes old_document; + set_disconnected_nodes old_document (remove1 child old_disc_nodes); + disc_nodes \ get_disconnected_nodes document_ptr; + set_disconnected_nodes document_ptr (child # disc_nodes) + } else do { + return () + }) \\<^sub>h h'" + using assms(2) + by(auto simp add: adopt_node_def elim!: bind_returns_heap_E + dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure] + pure_returns_heap_eq[rotated, OF get_parent_pure]) + + have object_ptr_kinds_h_eq3: "object_ptr_kinds h = object_ptr_kinds h2" + using h2 apply(simp split: option.splits) + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h = object_ptr_kinds h'", + OF remove_child_writes]) + using remove_child_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have object_ptr_kinds_M_eq_h: + "\ptrs. h \ object_ptr_kinds_M \\<^sub>r ptrs = h2 \ object_ptr_kinds_M \\<^sub>r ptrs" + unfolding object_ptr_kinds_M_defs by simp + then have object_ptr_kinds_eq_h: "|h \ object_ptr_kinds_M|\<^sub>r = |h2 \ object_ptr_kinds_M|\<^sub>r" + by simp + then have node_ptr_kinds_eq_h: "|h \ node_ptr_kinds_M|\<^sub>r = |h2 \ node_ptr_kinds_M|\<^sub>r" + using node_ptr_kinds_M_eq by blast + + have wellformed_h2: "heap_is_wellformed h2" + using h2 remove_child_heap_is_wellformed_preserved known_ptrs type_wf + by (metis (no_types, lifting) assms(1) option.case_eq_if pure_returns_heap_eq return_pure) + then show ?thesis + proof(cases "document_ptr = old_document") + case True + then show ?thesis + using h' wellformed_h2 by auto + next + case False + then obtain h3 old_disc_nodes disc_nodes_document_ptr_h3 where + docs_neq: "document_ptr \ old_document" and + old_disc_nodes: "h2 \ get_disconnected_nodes old_document \\<^sub>r old_disc_nodes" and + h3: "h2 \ set_disconnected_nodes old_document (remove1 child old_disc_nodes) \\<^sub>h h3" and + disc_nodes_document_ptr_h3: + "h3 \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes_document_ptr_h3" and + h': "h3 \ set_disconnected_nodes document_ptr (child # disc_nodes_document_ptr_h3) \\<^sub>h h'" + using h' + by(auto elim!: bind_returns_heap_E + bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] ) + + have object_ptr_kinds_h2_eq3: "object_ptr_kinds h2 = object_ptr_kinds h3" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h = object_ptr_kinds h'", + OF set_disconnected_nodes_writes h3]) + using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have object_ptr_kinds_M_eq_h2: + "\ptrs. h2 \ object_ptr_kinds_M \\<^sub>r ptrs = h3 \ object_ptr_kinds_M \\<^sub>r ptrs" + by(simp add: object_ptr_kinds_M_defs) + then have object_ptr_kinds_eq_h2: "|h2 \ object_ptr_kinds_M|\<^sub>r = |h3 \ object_ptr_kinds_M|\<^sub>r" + by(simp) + then have node_ptr_kinds_eq_h2: "|h2 \ node_ptr_kinds_M|\<^sub>r = |h3 \ node_ptr_kinds_M|\<^sub>r" + using node_ptr_kinds_M_eq by blast + then have node_ptr_kinds_eq3_h2: "node_ptr_kinds h2 = node_ptr_kinds h3" + by auto + have document_ptr_kinds_eq2_h2: "|h2 \ document_ptr_kinds_M|\<^sub>r = |h3 \ document_ptr_kinds_M|\<^sub>r" + using object_ptr_kinds_eq_h2 document_ptr_kinds_M_eq by auto + then have document_ptr_kinds_eq3_h2: "document_ptr_kinds h2 = document_ptr_kinds h3" + using object_ptr_kinds_eq_h2 document_ptr_kinds_M_eq by auto + have children_eq_h2: + "\ptr children. h2 \ get_child_nodes ptr \\<^sub>r children = h3 \ get_child_nodes ptr \\<^sub>r children" + using get_child_nodes_reads set_disconnected_nodes_writes h3 + apply(rule reads_writes_preserved) + by (simp add: set_disconnected_nodes_get_child_nodes) + then have children_eq2_h2: "\ptr. |h2 \ get_child_nodes ptr|\<^sub>r = |h3 \ get_child_nodes ptr|\<^sub>r" + using select_result_eq by force + + have object_ptr_kinds_h3_eq3: "object_ptr_kinds h3 = object_ptr_kinds h'" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h = object_ptr_kinds h'", + OF set_disconnected_nodes_writes h']) + using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have object_ptr_kinds_M_eq_h3: + "\ptrs. h3 \ object_ptr_kinds_M \\<^sub>r ptrs = h' \ object_ptr_kinds_M \\<^sub>r ptrs" + by(simp add: object_ptr_kinds_M_defs) + then have object_ptr_kinds_eq_h3: "|h3 \ object_ptr_kinds_M|\<^sub>r = |h' \ object_ptr_kinds_M|\<^sub>r" + by(simp) + then have node_ptr_kinds_eq_h3: "|h3 \ node_ptr_kinds_M|\<^sub>r = |h' \ node_ptr_kinds_M|\<^sub>r" + using node_ptr_kinds_M_eq by blast + then have node_ptr_kinds_eq3_h3: "node_ptr_kinds h3 = node_ptr_kinds h'" + by auto + have document_ptr_kinds_eq2_h3: "|h3 \ document_ptr_kinds_M|\<^sub>r = |h' \ document_ptr_kinds_M|\<^sub>r" + using object_ptr_kinds_eq_h3 document_ptr_kinds_M_eq by auto + then have document_ptr_kinds_eq3_h3: "document_ptr_kinds h3 = document_ptr_kinds h'" + using object_ptr_kinds_eq_h3 document_ptr_kinds_M_eq by auto + have children_eq_h3: + "\ptr children. h3 \ get_child_nodes ptr \\<^sub>r children = h' \ get_child_nodes ptr \\<^sub>r children" + using get_child_nodes_reads set_disconnected_nodes_writes h' + apply(rule reads_writes_preserved) + by (simp add: set_disconnected_nodes_get_child_nodes) + then have children_eq2_h3: "\ptr. |h3 \ get_child_nodes ptr|\<^sub>r = |h' \ get_child_nodes ptr|\<^sub>r" + using select_result_eq by force + + have disconnected_nodes_eq_h2: + "\doc_ptr disc_nodes. old_document \ doc_ptr + \ h2 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes = h3 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes" + using get_disconnected_nodes_reads set_disconnected_nodes_writes h3 + apply(rule reads_writes_preserved) + by (simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers) + then have disconnected_nodes_eq2_h2: + "\doc_ptr. old_document \ doc_ptr + \ |h2 \ get_disconnected_nodes doc_ptr|\<^sub>r = |h3 \ get_disconnected_nodes doc_ptr|\<^sub>r" + using select_result_eq by force + obtain disc_nodes_old_document_h2 where disc_nodes_old_document_h2: + "h2 \ get_disconnected_nodes old_document \\<^sub>r disc_nodes_old_document_h2" + using old_disc_nodes by blast + then have disc_nodes_old_document_h3: + "h3 \ get_disconnected_nodes old_document \\<^sub>r remove1 child disc_nodes_old_document_h2" + using h3 old_disc_nodes returns_result_eq set_disconnected_nodes_get_disconnected_nodes + by fastforce + have "distinct disc_nodes_old_document_h2" + using disc_nodes_old_document_h2 local.heap_is_wellformed_disconnected_nodes_distinct wellformed_h2 + by blast + + + have "type_wf h2" + proof (insert h2, induct parent_opt) + case None + then show ?case + using type_wf by simp + next + case (Some option) + then show ?case + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF remove_child_writes] + type_wf remove_child_types_preserved + by (simp add: reflp_def transp_def) + qed + then have "type_wf h3" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_disconnected_nodes_writes h3] + using set_disconnected_nodes_types_preserved + by(auto simp add: reflp_def transp_def) + then have "type_wf h'" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_disconnected_nodes_writes h'] + using set_disconnected_nodes_types_preserved + by(auto simp add: reflp_def transp_def) + + have disconnected_nodes_eq_h3: + "\doc_ptr disc_nodes. document_ptr \ doc_ptr + \ h3 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes = h' \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes" + using get_disconnected_nodes_reads set_disconnected_nodes_writes h' + apply(rule reads_writes_preserved) + by (simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers) + then have disconnected_nodes_eq2_h3: + "\doc_ptr. document_ptr \ doc_ptr + \ |h3 \ get_disconnected_nodes doc_ptr|\<^sub>r = |h' \ get_disconnected_nodes doc_ptr|\<^sub>r" + using select_result_eq by force + have disc_nodes_document_ptr_h2: + "h2 \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes_document_ptr_h3" + using disconnected_nodes_eq_h2 docs_neq disc_nodes_document_ptr_h3 by auto + have disc_nodes_document_ptr_h': " + h' \ get_disconnected_nodes document_ptr \\<^sub>r child # disc_nodes_document_ptr_h3" + using h' disc_nodes_document_ptr_h3 + using set_disconnected_nodes_get_disconnected_nodes by blast + + have document_ptr_in_heap: "document_ptr |\| document_ptr_kinds h2" + using disc_nodes_document_ptr_h3 document_ptr_kinds_eq2_h2 get_disconnected_nodes_ok assms(1) + unfolding heap_is_wellformed_def + using disc_nodes_document_ptr_h2 get_disconnected_nodes_ptr_in_heap by blast + have old_document_in_heap: "old_document |\| document_ptr_kinds h2" + using disc_nodes_old_document_h3 document_ptr_kinds_eq2_h2 get_disconnected_nodes_ok assms(1) + unfolding heap_is_wellformed_def + using get_disconnected_nodes_ptr_in_heap old_disc_nodes by blast + + have "child \ set disc_nodes_old_document_h2" + proof (insert parent_opt h2, induct parent_opt) + case None + then have "h = h2" + by(auto) + moreover have "a_owner_document_valid h" + using assms(1) heap_is_wellformed_def by(simp add: heap_is_wellformed_def) + ultimately show ?case + using old_document disc_nodes_old_document_h2 None(1) child_parent_dual[OF assms(1)] + in_disconnected_nodes_no_parent assms(1) known_ptrs type_wf by blast + next + case (Some option) + then show ?case + apply(simp split: option.splits) + using assms(1) disc_nodes_old_document_h2 old_document remove_child_in_disconnected_nodes known_ptrs + by blast + qed + have "child \ set (remove1 child disc_nodes_old_document_h2)" + using disc_nodes_old_document_h3 h3 known_ptrs wellformed_h2 \distinct disc_nodes_old_document_h2\ + by auto + have "child \ set disc_nodes_document_ptr_h3" + proof - + have "a_distinct_lists h2" + using heap_is_wellformed_def wellformed_h2 by blast + then have 0: "distinct (concat (map (\document_ptr. |h2 \ get_disconnected_nodes document_ptr|\<^sub>r) + |h2 \ document_ptr_kinds_M|\<^sub>r))" + by(simp add: a_distinct_lists_def) + show ?thesis + using distinct_concat_map_E(1)[OF 0] \child \ set disc_nodes_old_document_h2\ + disc_nodes_old_document_h2 disc_nodes_document_ptr_h2 + by (meson \type_wf h2\ docs_neq known_ptrs local.get_owner_document_disconnected_nodes + local.known_ptrs_preserved object_ptr_kinds_h_eq3 returns_result_eq wellformed_h2) + qed + + have child_in_heap: "child |\| node_ptr_kinds h" + using get_owner_document_ptr_in_heap[OF is_OK_returns_result_I[OF old_document]] + node_ptr_kinds_commutes by blast + have "a_acyclic_heap h2" + using wellformed_h2 by (simp add: heap_is_wellformed_def) + have "parent_child_rel h' \ parent_child_rel h2" + proof + fix x + assume "x \ parent_child_rel h'" + then show "x \ parent_child_rel h2" + using object_ptr_kinds_h2_eq3 object_ptr_kinds_h3_eq3 children_eq2_h2 children_eq2_h3 + mem_Collect_eq object_ptr_kinds_M_eq_h3 select_result_eq split_cong + unfolding parent_child_rel_def + by(simp) + qed + then have "a_acyclic_heap h'" + using \a_acyclic_heap h2\ acyclic_heap_def acyclic_subset by blast + + moreover have "a_all_ptrs_in_heap h2" + using wellformed_h2 by (simp add: heap_is_wellformed_def) + then have "a_all_ptrs_in_heap h3" + apply(auto simp add: a_all_ptrs_in_heap_def node_ptr_kinds_eq3_h2 children_eq_h2)[1] + by (metis (mono_tags, lifting) disc_nodes_old_document_h2 disc_nodes_old_document_h3 + disconnected_nodes_eq_h2 fset_of_list_elem fset_rev_mp returns_result_eq + set_remove1_subset subsetCE) + then have "a_all_ptrs_in_heap h'" + apply(auto simp add: a_all_ptrs_in_heap_def node_ptr_kinds_eq3_h3 children_eq_h3)[1] + by (metis (no_types) NodeMonad.ptr_kinds_ptr_kinds_M child_in_heap disc_nodes_document_ptr_h' + disc_nodes_document_ptr_h3 disconnected_nodes_eq_h3 fset_mp fset_of_list_elem + node_ptr_kinds_eq_h node_ptr_kinds_eq_h2 node_ptr_kinds_eq_h3 select_result_I2 + set_ConsD) + + moreover have "a_owner_document_valid h2" + using wellformed_h2 by (simp add: heap_is_wellformed_def) + then have "a_owner_document_valid h'" + apply(simp add: a_owner_document_valid_def node_ptr_kinds_eq_h2 node_ptr_kinds_eq3_h3 + object_ptr_kinds_eq_h2 object_ptr_kinds_eq_h3 document_ptr_kinds_eq2_h2 + document_ptr_kinds_eq2_h3 children_eq2_h2 children_eq2_h3 ) + by (metis (no_types) disc_nodes_document_ptr_h' disc_nodes_document_ptr_h2 + disc_nodes_old_document_h2 disc_nodes_old_document_h3 + disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 document_ptr_in_heap + document_ptr_kinds_eq3_h2 document_ptr_kinds_eq3_h3 in_set_remove1 + list.set_intros(1) list.set_intros(2) node_ptr_kinds_eq3_h2 + node_ptr_kinds_eq3_h3 object_ptr_kinds_h2_eq3 object_ptr_kinds_h3_eq3 + select_result_I2) + + have a_distinct_lists_h2: "a_distinct_lists h2" + using wellformed_h2 by (simp add: heap_is_wellformed_def) + then have "a_distinct_lists h'" + apply(auto simp add: a_distinct_lists_def object_ptr_kinds_eq_h3 object_ptr_kinds_eq_h2 + children_eq2_h2 children_eq2_h3)[1] + proof - + assume 1: "distinct (concat (map (\ptr. |h' \ get_child_nodes ptr|\<^sub>r) + (sorted_list_of_set (fset (object_ptr_kinds h')))))" + and 2: "distinct (concat (map (\document_ptr. |h2 \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h2)))))" + and 3: "(\x\fset (object_ptr_kinds h'). set |h' \ get_child_nodes x|\<^sub>r) + \ (\x\fset (document_ptr_kinds h2). set |h2 \ get_disconnected_nodes x|\<^sub>r) = {}" + show "distinct (concat (map (\document_ptr. |h' \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h')))))" + proof(rule distinct_concat_map_I) + show "distinct (sorted_list_of_set (fset (document_ptr_kinds h')))" + by(auto simp add: document_ptr_kinds_M_def ) + next + fix x + assume a1: "x \ set (sorted_list_of_set (fset (document_ptr_kinds h')))" + have 4: "distinct |h2 \ get_disconnected_nodes x|\<^sub>r" + using a_distinct_lists_h2 "2" a1 concat_map_all_distinct document_ptr_kinds_eq2_h2 + document_ptr_kinds_eq2_h3 + by fastforce + then show "distinct |h' \ get_disconnected_nodes x|\<^sub>r" + proof (cases "old_document \ x") + case True + then show ?thesis + proof (cases "document_ptr \ x") + case True + then show ?thesis + using disconnected_nodes_eq2_h2[OF \old_document \ x\] + disconnected_nodes_eq2_h3[OF \document_ptr \ x\] 4 + by(auto) + next + case False + then show ?thesis + using disc_nodes_document_ptr_h3 disc_nodes_document_ptr_h' 4 + \child \ set disc_nodes_document_ptr_h3\ + by(auto simp add: disconnected_nodes_eq2_h2[OF \old_document \ x\] ) + qed + next + case False + then show ?thesis + by (metis (no_types, hide_lams) \distinct disc_nodes_old_document_h2\ + disc_nodes_old_document_h3 disconnected_nodes_eq2_h3 + distinct_remove1 docs_neq select_result_I2) + qed + next + fix x y + assume a0: "x \ set (sorted_list_of_set (fset (document_ptr_kinds h')))" + and a1: "y \ set (sorted_list_of_set (fset (document_ptr_kinds h')))" + and a2: "x \ y" + + moreover have 5: "set |h2 \ get_disconnected_nodes x|\<^sub>r \ set |h2 \ get_disconnected_nodes y|\<^sub>r = {}" + using 2 calculation + by (auto simp add: document_ptr_kinds_eq3_h2 document_ptr_kinds_eq3_h3 dest: distinct_concat_map_E(1)) + ultimately show "set |h' \ get_disconnected_nodes x|\<^sub>r \ set |h' \ get_disconnected_nodes y|\<^sub>r = {}" + proof(cases "old_document = x") + case True + have "old_document \ y" + using \x \ y\ \old_document = x\ by simp + have "document_ptr \ x" + using docs_neq \old_document = x\ by auto + show ?thesis + proof(cases "document_ptr = y") + case True + then show ?thesis + using 5 True select_result_I2[OF disc_nodes_document_ptr_h'] + select_result_I2[OF disc_nodes_document_ptr_h2] + select_result_I2[OF disc_nodes_old_document_h2] + select_result_I2[OF disc_nodes_old_document_h3] \old_document = x\ + by (metis (no_types, lifting) \child \ set (remove1 child disc_nodes_old_document_h2)\ + \document_ptr \ x\ disconnected_nodes_eq2_h3 disjoint_iff_not_equal + notin_set_remove1 set_ConsD) + next + case False + then show ?thesis + using 5 select_result_I2[OF disc_nodes_document_ptr_h'] + select_result_I2[OF disc_nodes_document_ptr_h2] + select_result_I2[OF disc_nodes_old_document_h2] + select_result_I2[OF disc_nodes_old_document_h3] + disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 \old_document = x\ + docs_neq \old_document \ y\ + by (metis (no_types, lifting) disjoint_iff_not_equal notin_set_remove1) + qed + next + case False + then show ?thesis + proof(cases "old_document = y") + case True + then show ?thesis + proof(cases "document_ptr = x") + case True + show ?thesis + using 5 select_result_I2[OF disc_nodes_document_ptr_h'] + select_result_I2[OF disc_nodes_document_ptr_h2] + select_result_I2[OF disc_nodes_old_document_h2] + select_result_I2[OF disc_nodes_old_document_h3] + \old_document \ x\ \old_document = y\ \document_ptr = x\ + apply(simp) + by (metis (no_types, lifting) \child \ set (remove1 child disc_nodes_old_document_h2)\ + disconnected_nodes_eq2_h3 disjoint_iff_not_equal notin_set_remove1) + next + case False + then show ?thesis + using 5 select_result_I2[OF disc_nodes_document_ptr_h'] + select_result_I2[OF disc_nodes_document_ptr_h2] + select_result_I2[OF disc_nodes_old_document_h2] + select_result_I2[OF disc_nodes_old_document_h3] + \old_document \ x\ \old_document = y\ \document_ptr \ x\ + by (metis (no_types, lifting) disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 + disjoint_iff_not_equal docs_neq notin_set_remove1) + qed + next + case False + have "set |h2 \ get_disconnected_nodes y|\<^sub>r \ set disc_nodes_old_document_h2 = {}" + by (metis DocumentMonad.ptr_kinds_M_ok DocumentMonad.ptr_kinds_M_ptr_kinds False + \type_wf h2\ a1 disc_nodes_old_document_h2 document_ptr_kinds_M_def + document_ptr_kinds_eq2_h2 document_ptr_kinds_eq2_h3 + l_ptr_kinds_M.ptr_kinds_ptr_kinds_M local.get_disconnected_nodes_ok + local.heap_is_wellformed_one_disc_parent returns_result_select_result + wellformed_h2) + then show ?thesis + proof(cases "document_ptr = x") + case True + then have "document_ptr \ y" + using \x \ y\ by auto + have "set |h2 \ get_disconnected_nodes y|\<^sub>r \ set disc_nodes_old_document_h2 = {}" + using \set |h2 \ get_disconnected_nodes y|\<^sub>r \ set disc_nodes_old_document_h2 = {}\ + by blast + then show ?thesis + using 5 select_result_I2[OF disc_nodes_document_ptr_h'] + select_result_I2[OF disc_nodes_document_ptr_h2] + select_result_I2[OF disc_nodes_old_document_h2] + select_result_I2[OF disc_nodes_old_document_h3] + \old_document \ x\ \old_document \ y\ \document_ptr = x\ \document_ptr \ y\ + \child \ set disc_nodes_old_document_h2\ disconnected_nodes_eq2_h2 + disconnected_nodes_eq2_h3 + \set |h2 \ get_disconnected_nodes y|\<^sub>r \ set disc_nodes_old_document_h2 = {}\ + by(auto) + next + case False + then show ?thesis + proof(cases "document_ptr = y") + case True + have f1: "set |h2 \ get_disconnected_nodes x|\<^sub>r \ set disc_nodes_document_ptr_h3 = {}" + using 2 a1 document_ptr_in_heap document_ptr_kinds_eq2_h2 document_ptr_kinds_eq2_h3 + \document_ptr \ x\ select_result_I2[OF disc_nodes_document_ptr_h3, symmetric] + disconnected_nodes_eq2_h2[OF docs_neq[symmetric], symmetric] + by (simp add: "5" True) + moreover have f1: + "set |h2 \ get_disconnected_nodes x|\<^sub>r \ set |h2 \ get_disconnected_nodes old_document|\<^sub>r = {}" + using 2 a1 old_document_in_heap document_ptr_kinds_eq2_h2 document_ptr_kinds_eq2_h3 + \old_document \ x\ + by (metis (no_types, lifting) a0 distinct_concat_map_E(1) document_ptr_kinds_eq3_h2 + document_ptr_kinds_eq3_h3 finite_fset fmember.rep_eq set_sorted_list_of_set) + ultimately show ?thesis + using 5 select_result_I2[OF disc_nodes_document_ptr_h'] + select_result_I2[OF disc_nodes_old_document_h2] \old_document \ x\ + \document_ptr \ x\ \document_ptr = y\ + \child \ set disc_nodes_old_document_h2\ disconnected_nodes_eq2_h2 + disconnected_nodes_eq2_h3 + by auto + next + case False + then show ?thesis + using 5 + select_result_I2[OF disc_nodes_old_document_h2] \old_document \ x\ + \document_ptr \ x\ \document_ptr \ y\ + \child \ set disc_nodes_old_document_h2\ + disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 + by (metis \set |h2 \ get_disconnected_nodes y|\<^sub>r \ set disc_nodes_old_document_h2 = {}\ + empty_iff inf.idem) + qed + qed + qed + qed + qed + next + fix x xa xb + assume 0: "distinct (concat (map (\ptr. |h' \ get_child_nodes ptr|\<^sub>r) + (sorted_list_of_set (fset (object_ptr_kinds h')))))" + and 1: "distinct (concat (map (\document_ptr. |h2 \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h2)))))" + and 2: "(\x\fset (object_ptr_kinds h'). set |h' \ get_child_nodes x|\<^sub>r) + \ (\x\fset (document_ptr_kinds h2). set |h2 \ get_disconnected_nodes x|\<^sub>r) = {}" + and 3: "xa |\| object_ptr_kinds h'" + and 4: "x \ set |h' \ get_child_nodes xa|\<^sub>r" + and 5: "xb |\| document_ptr_kinds h'" + and 6: "x \ set |h' \ get_disconnected_nodes xb|\<^sub>r" + then show False + using \child \ set disc_nodes_old_document_h2\ disc_nodes_document_ptr_h' + disc_nodes_document_ptr_h2 disc_nodes_old_document_h2 disc_nodes_old_document_h3 + disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 document_ptr_kinds_eq2_h2 + document_ptr_kinds_eq2_h3 old_document_in_heap + apply(auto)[1] + apply(cases "xb = old_document") + proof - + assume a1: "xb = old_document" + assume a2: "h2 \ get_disconnected_nodes old_document \\<^sub>r disc_nodes_old_document_h2" + assume a3: "h3 \ get_disconnected_nodes old_document \\<^sub>r remove1 child disc_nodes_old_document_h2" + assume a4: "x \ set |h' \ get_child_nodes xa|\<^sub>r" + assume "document_ptr_kinds h2 = document_ptr_kinds h'" + assume a5: "(\x\fset (object_ptr_kinds h'). set |h' \ get_child_nodes x|\<^sub>r) + \ (\x\fset (document_ptr_kinds h'). set |h2 \ get_disconnected_nodes x|\<^sub>r) = {}" + have f6: "old_document |\| document_ptr_kinds h'" + using a1 \xb |\| document_ptr_kinds h'\ by blast + have f7: "|h2 \ get_disconnected_nodes old_document|\<^sub>r = disc_nodes_old_document_h2" + using a2 by simp + have "x \ set disc_nodes_old_document_h2" + using f6 a3 a1 by (metis (no_types) \type_wf h'\ \x \ set |h' \ get_disconnected_nodes xb|\<^sub>r\ + disconnected_nodes_eq_h3 docs_neq get_disconnected_nodes_ok returns_result_eq + returns_result_select_result set_remove1_subset subsetCE) + then have "set |h' \ get_child_nodes xa|\<^sub>r \ set |h2 \ get_disconnected_nodes xb|\<^sub>r = {}" + using f7 f6 a5 a4 \xa |\| object_ptr_kinds h'\ + by fastforce + then show ?thesis + using \x \ set disc_nodes_old_document_h2\ a1 a4 f7 by blast + next + assume a1: "xb \ old_document" + assume a2: "h2 \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes_document_ptr_h3" + assume a3: "h2 \ get_disconnected_nodes old_document \\<^sub>r disc_nodes_old_document_h2" + assume a4: "xa |\| object_ptr_kinds h'" + assume a5: "h' \ get_disconnected_nodes document_ptr \\<^sub>r child # disc_nodes_document_ptr_h3" + assume a6: "old_document |\| document_ptr_kinds h'" + assume a7: "x \ set |h' \ get_disconnected_nodes xb|\<^sub>r" + assume a8: "x \ set |h' \ get_child_nodes xa|\<^sub>r" + assume a9: "document_ptr_kinds h2 = document_ptr_kinds h'" + assume a10: "\doc_ptr. old_document \ doc_ptr + \ |h2 \ get_disconnected_nodes doc_ptr|\<^sub>r = |h3 \ get_disconnected_nodes doc_ptr|\<^sub>r" + assume a11: "\doc_ptr. document_ptr \ doc_ptr + \ |h3 \ get_disconnected_nodes doc_ptr|\<^sub>r = |h' \ get_disconnected_nodes doc_ptr|\<^sub>r" + assume a12: "(\x\fset (object_ptr_kinds h'). set |h' \ get_child_nodes x|\<^sub>r) + \ (\x\fset (document_ptr_kinds h'). set |h2 \ get_disconnected_nodes x|\<^sub>r) = {}" + have f13: "\d. d \ set |h' \ document_ptr_kinds_M|\<^sub>r \ h2 \ ok get_disconnected_nodes d" + using a9 \type_wf h2\ get_disconnected_nodes_ok + by simp + then have f14: "|h2 \ get_disconnected_nodes old_document|\<^sub>r = disc_nodes_old_document_h2" + using a6 a3 by simp + have "x \ set |h2 \ get_disconnected_nodes xb|\<^sub>r" + using a12 a8 a4 \xb |\| document_ptr_kinds h'\ + by (meson UN_I disjoint_iff_not_equal fmember.rep_eq) + then have "x = child" + using f13 a11 a10 a7 a5 a2 a1 + by (metis (no_types, lifting) select_result_I2 set_ConsD) + then have "child \ set disc_nodes_old_document_h2" + using f14 a12 a8 a6 a4 + by (metis \type_wf h'\ adopt_node_removes_child assms(1) assms(2) type_wf + get_child_nodes_ok known_ptrs local.known_ptrs_known_ptr object_ptr_kinds_h2_eq3 + object_ptr_kinds_h3_eq3 object_ptr_kinds_h_eq3 returns_result_select_result) + then show ?thesis + using \child \ set disc_nodes_old_document_h2\ by fastforce + qed + qed + ultimately show ?thesis + using \type_wf h'\ \a_owner_document_valid h'\ heap_is_wellformed_def by blast + qed +qed + +lemma adopt_node_node_in_disconnected_nodes: + assumes wellformed: "heap_is_wellformed h" + and adopt_node: "h \ adopt_node owner_document node_ptr \\<^sub>h h'" + and "h' \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes" + and known_ptrs: "known_ptrs h" + and type_wf: "type_wf h" + shows "node_ptr \ set disc_nodes" +proof - + obtain old_document parent_opt h2 where + old_document: "h \ get_owner_document (cast node_ptr) \\<^sub>r old_document" and + parent_opt: "h \ get_parent node_ptr \\<^sub>r parent_opt" and + h2: "h \ (case parent_opt of Some parent \ remove_child parent node_ptr | None \ return ()) \\<^sub>h h2" + and + h': "h2 \ (if owner_document \ old_document then do { + old_disc_nodes \ get_disconnected_nodes old_document; + set_disconnected_nodes old_document (remove1 node_ptr old_disc_nodes); + disc_nodes \ get_disconnected_nodes owner_document; + set_disconnected_nodes owner_document (node_ptr # disc_nodes) + } else do { + return () + }) \\<^sub>h h'" + using assms(2) + by(auto simp add: adopt_node_def elim!: bind_returns_heap_E + dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure] + pure_returns_heap_eq[rotated, OF get_parent_pure]) + + show ?thesis + proof (cases "owner_document = old_document") + case True + then show ?thesis + proof (insert parent_opt h2, induct parent_opt) + case None + then have "h = h'" + using h2 h' by(auto) + then show ?case + using in_disconnected_nodes_no_parent assms None old_document by blast + next + case (Some parent) + then show ?case + using remove_child_in_disconnected_nodes known_ptrs True h' assms(3) old_document by auto + qed + next + case False + then show ?thesis + using assms(3) h' list.set_intros(1) select_result_I2 set_disconnected_nodes_get_disconnected_nodes + apply(auto elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated])[1] + proof - + fix x and h'a and xb + assume a1: "h' \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes" + assume a2: "\h document_ptr disc_nodes h'. h \ set_disconnected_nodes document_ptr disc_nodes \\<^sub>h h' + \ h' \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes" + assume "h'a \ set_disconnected_nodes owner_document (node_ptr # xb) \\<^sub>h h'" + then have "node_ptr # xb = disc_nodes" + using a2 a1 by (meson returns_result_eq) + then show ?thesis + by (meson list.set_intros(1)) + qed + qed +qed +end + +interpretation i_adopt_node_wf?: l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent get_parent_locs + remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs + set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs known_ptr + type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes set_child_nodes_locs + remove heap_is_wellformed parent_child_rel + by(simp add: l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances) +declare l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + +interpretation i_adopt_node_wf2?: l_adopt_node_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent get_parent_locs + remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs + set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs known_ptr + type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes set_child_nodes_locs + remove heap_is_wellformed parent_child_rel get_root_node get_root_node_locs + by(simp add: l_adopt_node_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances) +declare l_adopt_node_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances] + + +locale l_adopt_node_wf = l_heap_is_wellformed + l_known_ptrs + l_type_wf + l_adopt_node_defs + + l_get_child_nodes_defs + l_get_disconnected_nodes_defs + + assumes adopt_node_preserves_wellformedness: + "heap_is_wellformed h \ h \ adopt_node document_ptr child \\<^sub>h h' \ known_ptrs h + \ type_wf h \ heap_is_wellformed h'" + assumes adopt_node_removes_child: + "heap_is_wellformed h \ h \ adopt_node owner_document node_ptr \\<^sub>h h2 + \ h2 \ get_child_nodes ptr \\<^sub>r children \ known_ptrs h + \ type_wf h \ node_ptr \ set children" + assumes adopt_node_node_in_disconnected_nodes: + "heap_is_wellformed h \ h \ adopt_node owner_document node_ptr \\<^sub>h h' + \ h' \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes + \ known_ptrs h \ type_wf h \ node_ptr \ set disc_nodes" + assumes adopt_node_removes_first_child: "heap_is_wellformed h \ type_wf h \ known_ptrs h + \ h \ adopt_node owner_document node \\<^sub>h h' + \ h \ get_child_nodes ptr' \\<^sub>r node # children + \ h' \ get_child_nodes ptr' \\<^sub>r children" + +lemma adopt_node_wf_is_l_adopt_node_wf [instances]: + "l_adopt_node_wf type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes + get_disconnected_nodes known_ptrs adopt_node" + using heap_is_wellformed_is_l_heap_is_wellformed known_ptrs_is_l_known_ptrs + apply(auto simp add: l_adopt_node_wf_def l_adopt_node_wf_axioms_def)[1] + using adopt_node_preserves_wellformedness apply blast + using adopt_node_removes_child apply blast + using adopt_node_node_in_disconnected_nodes apply blast + using adopt_node_removes_first_child apply blast + done + + +subsection \insert\_before\ + +locale l_insert_before_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_adopt_node_wf + + l_set_disconnected_nodes_get_child_nodes + + l_heap_is_wellformed +begin +lemma insert_before_removes_child: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "ptr \ ptr'" + assumes "h \ insert_before ptr node child \\<^sub>h h'" + assumes "h \ get_child_nodes ptr' \\<^sub>r node # children" + shows "h' \ get_child_nodes ptr' \\<^sub>r children" +proof - + obtain owner_document h2 h3 disc_nodes reference_child where + "h \ (if Some node = child then a_next_sibling node else return child) \\<^sub>r reference_child" and + "h \ get_owner_document ptr \\<^sub>r owner_document" and + h2: "h \ adopt_node owner_document node \\<^sub>h h2" and + "h2 \ get_disconnected_nodes owner_document \\<^sub>r disc_nodes" and + h3: "h2 \ set_disconnected_nodes owner_document (remove1 node disc_nodes) \\<^sub>h h3" and + h': "h3 \ a_insert_node ptr node reference_child \\<^sub>h h'" + using assms(5) + by(auto simp add: insert_before_def a_ensure_pre_insertion_validity_def + elim!: bind_returns_heap_E bind_returns_result_E + bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated] + bind_returns_heap_E2[rotated, OF get_parent_pure, rotated] + bind_returns_heap_E2[rotated, OF get_ancestors_pure, rotated] + bind_returns_heap_E2[rotated, OF get_owner_document_pure, rotated] + bind_returns_heap_E2[rotated, OF next_sibling_pure, rotated] + bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] + split: if_splits option.splits) + + have "h2 \ get_child_nodes ptr' \\<^sub>r children" + using h2 adopt_node_removes_first_child assms(1) assms(2) assms(3) assms(6) + by simp + then have "h3 \ get_child_nodes ptr' \\<^sub>r children" + using h3 + by(auto simp add: set_disconnected_nodes_get_child_nodes + dest!: reads_writes_separate_forwards[OF get_child_nodes_reads set_disconnected_nodes_writes]) + then show ?thesis + using h' assms(4) + apply(auto simp add: a_insert_node_def + elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated])[1] + by(auto simp add: set_child_nodes_get_child_nodes_different_pointers + elim!: reads_writes_separate_forwards[OF get_child_nodes_reads set_child_nodes_writes]) +qed +end + +locale l_insert_before_wf = l_heap_is_wellformed_defs + l_type_wf + l_known_ptrs + + l_insert_before_defs + l_get_child_nodes_defs + +assumes insert_before_removes_child: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ ptr \ ptr' + \ h \ insert_before ptr node child \\<^sub>h h' + \ h \ get_child_nodes ptr' \\<^sub>r node # children + \ h' \ get_child_nodes ptr' \\<^sub>r children" + +interpretation i_insert_before_wf?: l_insert_before_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent get_parent_locs + get_child_nodes get_child_nodes_locs set_child_nodes + set_child_nodes_locs get_ancestors get_ancestors_locs + adopt_node adopt_node_locs set_disconnected_nodes + set_disconnected_nodes_locs get_disconnected_nodes + get_disconnected_nodes_locs get_owner_document insert_before + insert_before_locs append_child type_wf known_ptr known_ptrs + heap_is_wellformed parent_child_rel + by(simp add: l_insert_before_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances) +declare l_insert_before_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances] + +lemma insert_before_wf_is_l_insert_before_wf [instances]: + "l_insert_before_wf heap_is_wellformed type_wf known_ptr known_ptrs insert_before get_child_nodes" + apply(auto simp add: l_insert_before_wf_def l_insert_before_wf_axioms_def instances)[1] + using insert_before_removes_child apply fast + done + +locale l_insert_before_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_insert_before_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_set_child_nodes_get_disconnected_nodes + + l_remove_child + + l_get_root_node_wf + + l_set_disconnected_nodes_get_disconnected_nodes_wf + + l_set_disconnected_nodes_get_ancestors + + l_get_ancestors_wf + + l_get_owner_document + + l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +begin +lemma insert_before_heap_is_wellformed_preserved: + assumes wellformed: "heap_is_wellformed h" + and insert_before: "h \ insert_before ptr node child \\<^sub>h h'" + and known_ptrs: "known_ptrs h" + and type_wf: "type_wf h" + shows "heap_is_wellformed h'" and "type_wf h'" and "known_ptrs h'" +proof - + obtain ancestors reference_child owner_document h2 h3 disconnected_nodes_h2 where + ancestors: "h \ get_ancestors ptr \\<^sub>r ancestors" and + node_not_in_ancestors: "cast node \ set ancestors" and + reference_child: + "h \ (if Some node = child then a_next_sibling node else return child) \\<^sub>r reference_child" and + owner_document: "h \ get_owner_document ptr \\<^sub>r owner_document" and + h2: "h \ adopt_node owner_document node \\<^sub>h h2" and + disconnected_nodes_h2: "h2 \ get_disconnected_nodes owner_document \\<^sub>r disconnected_nodes_h2" and + h3: "h2 \ set_disconnected_nodes owner_document (remove1 node disconnected_nodes_h2) \\<^sub>h h3" and + h': "h3 \ a_insert_node ptr node reference_child \\<^sub>h h'" + using assms(2) + by(auto simp add: insert_before_def a_ensure_pre_insertion_validity_def + elim!: bind_returns_heap_E bind_returns_result_E + bind_returns_heap_E2[rotated, OF get_parent_pure, rotated] + bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated] + bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] + bind_returns_heap_E2[rotated, OF get_ancestors_pure, rotated] + bind_returns_heap_E2[rotated, OF next_sibling_pure, rotated] + bind_returns_heap_E2[rotated, OF get_owner_document_pure, rotated] + split: if_splits option.splits) + + have "known_ptr ptr" + by (meson get_owner_document_ptr_in_heap is_OK_returns_result_I known_ptrs + l_known_ptrs.known_ptrs_known_ptr l_known_ptrs_axioms owner_document) + + have "type_wf h2" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF adopt_node_writes h2] + using type_wf adopt_node_types_preserved + by(auto simp add: a_remove_child_locs_def reflp_def transp_def) + then have "type_wf h3" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_disconnected_nodes_writes h3] + using set_disconnected_nodes_types_preserved + by(auto simp add: reflp_def transp_def) + then show "type_wf h'" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF insert_node_writes h'] + using set_child_nodes_types_preserved + by(auto simp add: reflp_def transp_def) + + have object_ptr_kinds_M_eq3_h: "object_ptr_kinds h = object_ptr_kinds h2" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h = object_ptr_kinds h'", + OF adopt_node_writes h2]) + using adopt_node_pointers_preserved + apply blast + by (auto simp add: reflp_def transp_def) + then have object_ptr_kinds_M_eq_h: "\ptrs. h \ object_ptr_kinds_M \\<^sub>r ptrs = h2 \ object_ptr_kinds_M \\<^sub>r ptrs" + by(simp add: object_ptr_kinds_M_defs ) + then have object_ptr_kinds_M_eq2_h: "|h \ object_ptr_kinds_M|\<^sub>r = |h2 \ object_ptr_kinds_M|\<^sub>r" + by simp + then have node_ptr_kinds_eq2_h: "|h \ node_ptr_kinds_M|\<^sub>r = |h2 \ node_ptr_kinds_M|\<^sub>r" + using node_ptr_kinds_M_eq by blast + + have "known_ptrs h2" + using known_ptrs object_ptr_kinds_M_eq3_h known_ptrs_preserved by blast + + have wellformed_h2: "heap_is_wellformed h2" + using adopt_node_preserves_wellformedness[OF wellformed h2] known_ptrs type_wf . + + have object_ptr_kinds_M_eq3_h2: "object_ptr_kinds h2 = object_ptr_kinds h3" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h = object_ptr_kinds h'", + OF set_disconnected_nodes_writes h3]) + unfolding a_remove_child_locs_def + using set_disconnected_nodes_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have object_ptr_kinds_M_eq_h2: "\ptrs. h2 \ object_ptr_kinds_M \\<^sub>r ptrs = h3 \ object_ptr_kinds_M \\<^sub>r ptrs" + by(simp add: object_ptr_kinds_M_defs) + then have object_ptr_kinds_M_eq2_h2: "|h2 \ object_ptr_kinds_M|\<^sub>r = |h3 \ object_ptr_kinds_M|\<^sub>r" + by simp + then have node_ptr_kinds_eq2_h2: "|h2 \ node_ptr_kinds_M|\<^sub>r = |h3 \ node_ptr_kinds_M|\<^sub>r" + using node_ptr_kinds_M_eq by blast + have document_ptr_kinds_eq2_h2: "|h2 \ document_ptr_kinds_M|\<^sub>r = |h3 \ document_ptr_kinds_M|\<^sub>r" + using object_ptr_kinds_M_eq2_h2 document_ptr_kinds_M_eq by auto + + have "known_ptrs h3" + using object_ptr_kinds_M_eq3_h2 known_ptrs_preserved \known_ptrs h2\ by blast + + have object_ptr_kinds_M_eq3_h': "object_ptr_kinds h3 = object_ptr_kinds h'" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h = object_ptr_kinds h'", + OF insert_node_writes h']) + unfolding a_remove_child_locs_def + using set_child_nodes_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have object_ptr_kinds_M_eq_h3: + "\ptrs. h3 \ object_ptr_kinds_M \\<^sub>r ptrs = h' \ object_ptr_kinds_M \\<^sub>r ptrs" + by(simp add: object_ptr_kinds_M_defs) + then have object_ptr_kinds_M_eq2_h3: + "|h3 \ object_ptr_kinds_M|\<^sub>r = |h' \ object_ptr_kinds_M|\<^sub>r" + by simp + then have node_ptr_kinds_eq2_h3: "|h3 \ node_ptr_kinds_M|\<^sub>r = |h' \ node_ptr_kinds_M|\<^sub>r" + using node_ptr_kinds_M_eq by blast + have document_ptr_kinds_eq2_h3: "|h3 \ document_ptr_kinds_M|\<^sub>r = |h' \ document_ptr_kinds_M|\<^sub>r" + using object_ptr_kinds_M_eq2_h3 document_ptr_kinds_M_eq by auto + + show "known_ptrs h'" + using object_ptr_kinds_M_eq3_h' known_ptrs_preserved \known_ptrs h3\ by blast + + have disconnected_nodes_eq_h2: + "\doc_ptr disc_nodes. owner_document \ doc_ptr + \ h2 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes = h3 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes" + using get_disconnected_nodes_reads set_disconnected_nodes_writes h3 + apply(rule reads_writes_preserved) + by (auto simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers) + then have disconnected_nodes_eq2_h2: + "\doc_ptr. doc_ptr \ owner_document + \ |h2 \ get_disconnected_nodes doc_ptr|\<^sub>r = |h3 \ get_disconnected_nodes doc_ptr|\<^sub>r" + using select_result_eq by force + have disconnected_nodes_h3: + "h3 \ get_disconnected_nodes owner_document \\<^sub>r remove1 node disconnected_nodes_h2" + using h3 set_disconnected_nodes_get_disconnected_nodes + by blast + + have disconnected_nodes_eq_h3: + "\doc_ptr disc_nodes. h3 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes + = h' \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes" + using get_disconnected_nodes_reads insert_node_writes h' + apply(rule reads_writes_preserved) + using set_child_nodes_get_disconnected_nodes by fast + then have disconnected_nodes_eq2_h3: + "\doc_ptr. |h3 \ get_disconnected_nodes doc_ptr|\<^sub>r = |h' \ get_disconnected_nodes doc_ptr|\<^sub>r" + using select_result_eq by force + + have children_eq_h2: + "\ptr' children. h2 \ get_child_nodes ptr' \\<^sub>r children = h3 \ get_child_nodes ptr' \\<^sub>r children" + using get_child_nodes_reads set_disconnected_nodes_writes h3 + apply(rule reads_writes_preserved) + by (auto simp add: set_disconnected_nodes_get_child_nodes) + then have children_eq2_h2: + "\ptr'. |h2 \ get_child_nodes ptr'|\<^sub>r = |h3 \ get_child_nodes ptr'|\<^sub>r" + using select_result_eq by force + + have children_eq_h3: + "\ptr' children. ptr \ ptr' + \ h3 \ get_child_nodes ptr' \\<^sub>r children = h' \ get_child_nodes ptr' \\<^sub>r children" + using get_child_nodes_reads insert_node_writes h' + apply(rule reads_writes_preserved) + by (auto simp add: set_child_nodes_get_child_nodes_different_pointers) + then have children_eq2_h3: + "\ptr'. ptr \ ptr' \ |h3 \ get_child_nodes ptr'|\<^sub>r = |h' \ get_child_nodes ptr'|\<^sub>r" + using select_result_eq by force + obtain children_h3 where children_h3: "h3 \ get_child_nodes ptr \\<^sub>r children_h3" + using h' a_insert_node_def by auto + have children_h': "h' \ get_child_nodes ptr \\<^sub>r insert_before_list node reference_child children_h3" + using h' \type_wf h3\ \known_ptr ptr\ + by(auto simp add: a_insert_node_def elim!: bind_returns_heap_E2 + dest!: set_child_nodes_get_child_nodes returns_result_eq[OF children_h3]) + + have ptr_in_heap: "ptr |\| object_ptr_kinds h3" + using children_h3 get_child_nodes_ptr_in_heap by blast + have node_in_heap: "node |\| node_ptr_kinds h" + using h2 adopt_node_child_in_heap by fast + have child_not_in_any_children: + "\p children. h2 \ get_child_nodes p \\<^sub>r children \ node \ set children" + using wellformed h2 adopt_node_removes_child \type_wf h\ \known_ptrs h\ by auto + have "node \ set disconnected_nodes_h2" + using disconnected_nodes_h2 h2 adopt_node_node_in_disconnected_nodes assms(1) + \type_wf h\ \known_ptrs h\ by blast + have node_not_in_disconnected_nodes: + "\d. d |\| document_ptr_kinds h3 \ node \ set |h3 \ get_disconnected_nodes d|\<^sub>r" + proof - + fix d + assume "d |\| document_ptr_kinds h3" + show "node \ set |h3 \ get_disconnected_nodes d|\<^sub>r" + proof (cases "d = owner_document") + case True + then show ?thesis + using disconnected_nodes_h2 wellformed_h2 h3 remove_from_disconnected_nodes_removes + wellformed_h2 \d |\| document_ptr_kinds h3\ disconnected_nodes_h3 + by fastforce + next + case False + then have + "set |h2 \ get_disconnected_nodes d|\<^sub>r \ set |h2 \ get_disconnected_nodes owner_document|\<^sub>r = {}" + using distinct_concat_map_E(1) wellformed_h2 + by (metis (no_types, lifting) \d |\| document_ptr_kinds h3\ \type_wf h2\ + disconnected_nodes_h2 document_ptr_kinds_M_def document_ptr_kinds_eq2_h2 + l_ptr_kinds_M.ptr_kinds_ptr_kinds_M local.get_disconnected_nodes_ok + local.heap_is_wellformed_one_disc_parent returns_result_select_result + select_result_I2) + then show ?thesis + using disconnected_nodes_eq2_h2[OF False] \node \ set disconnected_nodes_h2\ + disconnected_nodes_h2 by fastforce + qed + qed + + have "cast node \ ptr" + using ancestors node_not_in_ancestors get_ancestors_ptr + by fast + + obtain ancestors_h2 where ancestors_h2: "h2 \ get_ancestors ptr \\<^sub>r ancestors_h2" + using get_ancestors_ok object_ptr_kinds_M_eq2_h2 \known_ptrs h2\ \type_wf h2\ + by (metis is_OK_returns_result_E object_ptr_kinds_M_eq3_h2 ptr_in_heap wellformed_h2) + have ancestors_h3: "h3 \ get_ancestors ptr \\<^sub>r ancestors_h2" + using get_ancestors_reads set_disconnected_nodes_writes h3 + apply(rule reads_writes_separate_forwards) + using \heap_is_wellformed h2\ ancestors_h2 + by (auto simp add: set_disconnected_nodes_get_ancestors) + have node_not_in_ancestors_h2: "cast node \ set ancestors_h2" + apply(rule get_ancestors_remains_not_in_ancestors[OF assms(1) wellformed_h2 ancestors ancestors_h2]) + using adopt_node_children_subset using h2 \known_ptrs h\ \ type_wf h\ apply(blast) + using node_not_in_ancestors apply(blast) + using object_ptr_kinds_M_eq3_h apply(blast) + using \known_ptrs h\ apply(blast) + using \type_wf h\ apply(blast) + using \type_wf h2\ by blast + + moreover have "a_acyclic_heap h'" + proof - + have "acyclic (parent_child_rel h2)" + using wellformed_h2 by (simp add: heap_is_wellformed_def acyclic_heap_def) + then have "acyclic (parent_child_rel h3)" + by(auto simp add: parent_child_rel_def object_ptr_kinds_M_eq3_h2 children_eq2_h2) + moreover have "cast node \ {x. (x, ptr) \ (parent_child_rel h2)\<^sup>*}" + using get_ancestors_parent_child_rel node_not_in_ancestors_h2 \known_ptrs h2\ \type_wf h2\ + using ancestors_h2 wellformed_h2 by blast + then have "cast node \ {x. (x, ptr) \ (parent_child_rel h3)\<^sup>*}" + by(auto simp add: parent_child_rel_def object_ptr_kinds_M_eq3_h2 children_eq2_h2) + moreover have "parent_child_rel h' = insert (ptr, cast node) ((parent_child_rel h3))" + using children_h3 children_h' ptr_in_heap + apply(auto simp add: parent_child_rel_def object_ptr_kinds_M_eq3_h' children_eq2_h3 + insert_before_list_node_in_set)[1] + apply (metis (no_types, lifting) children_eq2_h3 insert_before_list_in_set select_result_I2) + by (metis (no_types, lifting) children_eq2_h3 imageI insert_before_list_in_set select_result_I2) + ultimately show ?thesis + by(auto simp add: acyclic_heap_def) + qed + + + moreover have "a_all_ptrs_in_heap h2" + using wellformed_h2 by (simp add: heap_is_wellformed_def) + have "a_all_ptrs_in_heap h'" + proof - + have "a_all_ptrs_in_heap h3" + using \a_all_ptrs_in_heap h2\ + apply(auto simp add: a_all_ptrs_in_heap_def object_ptr_kinds_M_eq2_h2 node_ptr_kinds_eq2_h2 + children_eq_h2)[1] + using disconnected_nodes_eq2_h2 disconnected_nodes_h2 disconnected_nodes_h3 + using node_ptr_kinds_eq2_h2 apply auto[1] + by (metis (no_types, lifting) NodeMonad.ptr_kinds_ptr_kinds_M disconnected_nodes_eq_h2 + disconnected_nodes_h2 disconnected_nodes_h3 fset_mp fset_of_list_subset + node_ptr_kinds_eq2_h2 select_result_I2 set_remove1_subset) + have "set children_h3 \ set |h' \ node_ptr_kinds_M|\<^sub>r" + using children_h3 \a_all_ptrs_in_heap h3\ + apply(auto simp add: a_all_ptrs_in_heap_def node_ptr_kinds_eq2_h3)[1] + by (metis (no_types, hide_lams) fset_mp fset_of_list_elem node_ptr_kinds_commutes object_ptr_kinds_M_eq3_h') + then have "set (insert_before_list node reference_child children_h3) \ set |h' \ node_ptr_kinds_M|\<^sub>r" + using node_in_heap + apply(auto simp add: node_ptr_kinds_eq2_h node_ptr_kinds_eq2_h2 node_ptr_kinds_eq2_h3)[1] + by (metis (no_types, hide_lams) contra_subsetD finite_set_in insert_before_list_in_set + node_ptr_kinds_commutes object_ptr_kinds_M_eq3_h object_ptr_kinds_M_eq3_h' + object_ptr_kinds_M_eq3_h2) + then show ?thesis + using \a_all_ptrs_in_heap h3\ + apply(auto simp add: object_ptr_kinds_M_eq3_h' a_all_ptrs_in_heap_def node_ptr_kinds_def + node_ptr_kinds_eq2_h3 disconnected_nodes_eq_h3)[1] + using children_eq_h3 children_h' + by (metis (no_types, hide_lams) NodeMonad.ptr_kinds_ptr_kinds_M + \set (insert_before_list node reference_child children_h3) \ set |h' \ node_ptr_kinds_M|\<^sub>r\ + fset.map_comp fset_mp fset_of_list_elem node_ptr_kinds_def node_ptr_kinds_eq2_h3 + returns_result_eq subsetCE) + qed + + + moreover have "a_distinct_lists h2" + using wellformed_h2 by (simp add: heap_is_wellformed_def) + then have "a_distinct_lists h3" + proof(auto simp add: a_distinct_lists_def object_ptr_kinds_M_eq2_h2 document_ptr_kinds_eq2_h2 + children_eq2_h2 intro!: distinct_concat_map_I)[1] + fix x + assume 1: "x |\| document_ptr_kinds h3" + and 2: "distinct (concat (map (\document_ptr. |h2 \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h3)))))" + show "distinct |h3 \ get_disconnected_nodes x|\<^sub>r" + using distinct_concat_map_E(2)[OF 2] select_result_I2[OF disconnected_nodes_h3] + disconnected_nodes_eq2_h2 select_result_I2[OF disconnected_nodes_h2] 1 + by (metis (full_types) distinct_remove1 finite_fset fmember.rep_eq set_sorted_list_of_set) + next + fix x y xa + assume 1: "distinct (concat (map (\document_ptr. |h2 \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h3)))))" + and 2: "x |\| document_ptr_kinds h3" + and 3: "y |\| document_ptr_kinds h3" + and 4: "x \ y" + and 5: "xa \ set |h3 \ get_disconnected_nodes x|\<^sub>r" + and 6: "xa \ set |h3 \ get_disconnected_nodes y|\<^sub>r" + show False + proof (cases "x = owner_document") + case True + then have "y \ owner_document" + using 4 by simp + show ?thesis + using distinct_concat_map_E(1)[OF 1] + using 2 3 4 5 6 select_result_I2[OF disconnected_nodes_h3] select_result_I2[OF disconnected_nodes_h2] + apply(auto simp add: True disconnected_nodes_eq2_h2[OF \y \ owner_document\])[1] + by (metis (no_types, hide_lams) disconnected_nodes_eq2_h2 disjoint_iff_not_equal notin_set_remove1) + next + case False + then show ?thesis + proof (cases "y = owner_document") + case True + then show ?thesis + using distinct_concat_map_E(1)[OF 1] + using 2 3 4 5 6 select_result_I2[OF disconnected_nodes_h3] select_result_I2[OF disconnected_nodes_h2] + apply(auto simp add: True disconnected_nodes_eq2_h2[OF \x \ owner_document\])[1] + by (metis (no_types, hide_lams) disconnected_nodes_eq2_h2 disjoint_iff_not_equal notin_set_remove1) + next + case False + then show ?thesis + using distinct_concat_map_E(1)[OF 1, simplified, OF 2 3 4] 5 6 + using disconnected_nodes_eq2_h2 disconnected_nodes_h2 disconnected_nodes_h3 + disjoint_iff_not_equal finite_fset fmember.rep_eq notin_set_remove1 select_result_I2 + set_sorted_list_of_set + by (metis (no_types, lifting)) + qed + qed + next + fix x xa xb + assume 1: "(\x\fset (object_ptr_kinds h3). set |h3 \ get_child_nodes x|\<^sub>r) + \ (\x\fset (document_ptr_kinds h3). set |h2 \ get_disconnected_nodes x|\<^sub>r) = {}" + and 2: "xa |\| object_ptr_kinds h3" + and 3: "x \ set |h3 \ get_child_nodes xa|\<^sub>r" + and 4: "xb |\| document_ptr_kinds h3" + and 5: "x \ set |h3 \ get_disconnected_nodes xb|\<^sub>r" + have 6: "set |h3 \ get_child_nodes xa|\<^sub>r \ set |h2 \ get_disconnected_nodes xb|\<^sub>r = {}" + using 1 2 4 + by (metis \type_wf h2\ children_eq2_h2 document_ptr_kinds_commutes known_ptrs + local.get_child_nodes_ok local.get_disconnected_nodes_ok + local.heap_is_wellformed_children_disc_nodes_different local.known_ptrs_known_ptr + object_ptr_kinds_M_eq3_h object_ptr_kinds_M_eq3_h2 returns_result_select_result + wellformed_h2) + show False + proof (cases "xb = owner_document") + case True + then show ?thesis + using select_result_I2[OF disconnected_nodes_h3,folded select_result_I2[OF disconnected_nodes_h2]] + by (metis (no_types, lifting) "3" "5" "6" disjoint_iff_not_equal notin_set_remove1) + next + case False + show ?thesis + using 2 3 4 5 6 unfolding disconnected_nodes_eq2_h2[OF False] by auto + qed + qed + then have "a_distinct_lists h'" + proof(auto simp add: a_distinct_lists_def document_ptr_kinds_eq2_h3 object_ptr_kinds_M_eq2_h3 + disconnected_nodes_eq2_h3 intro!: distinct_concat_map_I)[1] + fix x + assume 1: "distinct (concat (map (\ptr. |h3 \ get_child_nodes ptr|\<^sub>r) + (sorted_list_of_set (fset (object_ptr_kinds h')))))" and + 2: "x |\| object_ptr_kinds h'" + have 3: "\p. p |\| object_ptr_kinds h' \ distinct |h3 \ get_child_nodes p|\<^sub>r" + using 1 by (auto elim: distinct_concat_map_E) + show "distinct |h' \ get_child_nodes x|\<^sub>r" + proof(cases "ptr = x") + case True + show ?thesis + using 3[OF 2] children_h3 children_h' + by(auto simp add: True insert_before_list_distinct + dest: child_not_in_any_children[unfolded children_eq_h2]) + next + case False + show ?thesis + using children_eq2_h3[OF False] 3[OF 2] by auto + qed + next + fix x y xa + assume 1: "distinct (concat (map (\ptr. |h3 \ get_child_nodes ptr|\<^sub>r) + (sorted_list_of_set (fset (object_ptr_kinds h')))))" + and 2: "x |\| object_ptr_kinds h'" + and 3: "y |\| object_ptr_kinds h'" + and 4: "x \ y" + and 5: "xa \ set |h' \ get_child_nodes x|\<^sub>r" + and 6: "xa \ set |h' \ get_child_nodes y|\<^sub>r" + have 7:"set |h3 \ get_child_nodes x|\<^sub>r \ set |h3 \ get_child_nodes y|\<^sub>r = {}" + using distinct_concat_map_E(1)[OF 1] 2 3 4 by auto + show False + proof (cases "ptr = x") + case True + then have "ptr \ y" + using 4 by simp + then show ?thesis + using children_h3 children_h' child_not_in_any_children[unfolded children_eq_h2] 5 6 + apply(auto simp add: True children_eq2_h3[OF \ptr \ y\])[1] + by (metis (no_types, hide_lams) "3" "7" \type_wf h3\ children_eq2_h3 disjoint_iff_not_equal + get_child_nodes_ok insert_before_list_in_set known_ptrs local.known_ptrs_known_ptr + object_ptr_kinds_M_eq3_h object_ptr_kinds_M_eq3_h' + object_ptr_kinds_M_eq3_h2 returns_result_select_result select_result_I2) + next + case False + then show ?thesis + proof (cases "ptr = y") + case True + then show ?thesis + using children_h3 children_h' child_not_in_any_children[unfolded children_eq_h2] 5 6 + apply(auto simp add: True children_eq2_h3[OF \ptr \ x\])[1] + by (metis (no_types, hide_lams) "2" "4" "7" IntI \known_ptrs h3\ \type_wf h'\ + children_eq_h3 empty_iff insert_before_list_in_set local.get_child_nodes_ok + local.known_ptrs_known_ptr object_ptr_kinds_M_eq3_h' + returns_result_select_result select_result_I2) + next + case False + then show ?thesis + using children_eq2_h3[OF \ptr \ x\] children_eq2_h3[OF \ptr \ y\] 5 6 7 by auto + qed + qed + next + fix x xa xb + assume 1: " (\x\fset (object_ptr_kinds h'). set |h3 \ get_child_nodes x|\<^sub>r) + \ (\x\fset (document_ptr_kinds h'). set |h' \ get_disconnected_nodes x|\<^sub>r) = {} " + and 2: "xa |\| object_ptr_kinds h'" + and 3: "x \ set |h' \ get_child_nodes xa|\<^sub>r" + and 4: "xb |\| document_ptr_kinds h'" + and 5: "x \ set |h' \ get_disconnected_nodes xb|\<^sub>r" + have 6: "set |h3 \ get_child_nodes xa|\<^sub>r \ set |h' \ get_disconnected_nodes xb|\<^sub>r = {}" + using 1 2 3 4 5 + proof - + have "\h d. \ type_wf h \ d |\| document_ptr_kinds h \ h \ ok get_disconnected_nodes d" + using local.get_disconnected_nodes_ok by satx + then have "h' \ ok get_disconnected_nodes xb" + using "4" \type_wf h'\ by fastforce + then have f1: "h3 \ get_disconnected_nodes xb \\<^sub>r |h' \ get_disconnected_nodes xb|\<^sub>r" + by (simp add: disconnected_nodes_eq_h3) + have "xa |\| object_ptr_kinds h3" + using "2" object_ptr_kinds_M_eq3_h' by blast + then show ?thesis + using f1 \local.a_distinct_lists h3\ local.distinct_lists_no_parent by fastforce + qed + show False + proof (cases "ptr = xa") + case True + show ?thesis + using 6 node_not_in_disconnected_nodes 3 4 5 select_result_I2[OF children_h'] + select_result_I2[OF children_h3] True disconnected_nodes_eq2_h3 + by (metis (no_types, lifting) "2" DocumentMonad.ptr_kinds_ptr_kinds_M + \a_distinct_lists h3\ \type_wf h'\ disconnected_nodes_eq_h3 + distinct_lists_no_parent document_ptr_kinds_eq2_h3 get_disconnected_nodes_ok + insert_before_list_in_set object_ptr_kinds_M_eq3_h' returns_result_select_result) + + next + case False + then show ?thesis + using 1 2 3 4 5 children_eq2_h3[OF False] by fastforce + qed + qed + + moreover have "a_owner_document_valid h2" + using wellformed_h2 by (simp add: heap_is_wellformed_def) + then have "a_owner_document_valid h'" + apply(auto simp add: a_owner_document_valid_def object_ptr_kinds_M_eq2_h2 + object_ptr_kinds_M_eq2_h3 node_ptr_kinds_eq2_h2 node_ptr_kinds_eq2_h3 + document_ptr_kinds_eq2_h2 document_ptr_kinds_eq2_h3 children_eq2_h2)[1] + apply(auto simp add: document_ptr_kinds_eq2_h2[simplified] document_ptr_kinds_eq2_h3[simplified] + object_ptr_kinds_M_eq2_h2[simplified] object_ptr_kinds_M_eq2_h3[simplified] + node_ptr_kinds_eq2_h2[simplified] node_ptr_kinds_eq2_h3[simplified])[1] + apply(auto simp add: disconnected_nodes_eq2_h3[symmetric])[1] + proof - + fix node_ptr + assume 0: "\node_ptr. node_ptr |\| node_ptr_kinds h' + \ (\document_ptr. document_ptr |\| document_ptr_kinds h' + \ node_ptr \ set |h2 \ get_disconnected_nodes document_ptr|\<^sub>r) + \ (\parent_ptr. parent_ptr |\| object_ptr_kinds h' + \ node_ptr \ set |h3 \ get_child_nodes parent_ptr|\<^sub>r)" + and 1: "node_ptr |\| node_ptr_kinds h'" + and 2: "\parent_ptr. parent_ptr |\| object_ptr_kinds h' + \ node_ptr \ set |h' \ get_child_nodes parent_ptr|\<^sub>r" + then have "(\document_ptr. document_ptr |\| document_ptr_kinds h' + \ node_ptr \ set |h2 \ get_disconnected_nodes document_ptr|\<^sub>r)" + by (metis (no_types, lifting) children_eq2_h3 children_h' children_h3 + insert_before_list_in_set select_result_I2) + then show "\document_ptr. document_ptr |\| document_ptr_kinds h' + \ node_ptr \ set |h3 \ get_disconnected_nodes document_ptr|\<^sub>r" + by (metis (no_types, hide_lams) "2" children_h' disconnected_nodes_eq2_h2 + disconnected_nodes_h2 disconnected_nodes_h3 in_set_remove1 + insert_before_list_in_set object_ptr_kinds_M_eq3_h' + ptr_in_heap select_result_I2) + qed + + ultimately show "heap_is_wellformed h'" + by (simp add: heap_is_wellformed_def) +qed +end + +locale l_insert_before_wf2 = l_type_wf + l_known_ptrs + l_insert_before_defs + + l_heap_is_wellformed_defs + l_get_child_nodes_defs + l_remove_defs + + assumes insert_before_preserves_type_wf: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ h \ insert_before ptr child ref \\<^sub>h h' + \ type_wf h'" + assumes insert_before_preserves_known_ptrs: + "heap_is_wellformed h \ type_wf h \ known_ptrs h \ h \ insert_before ptr child ref \\<^sub>h h' + \ known_ptrs h'" + assumes insert_before_heap_is_wellformed_preserved: + "type_wf h \ known_ptrs h \ heap_is_wellformed h \ h \ insert_before ptr child ref \\<^sub>h h' + \ heap_is_wellformed h'" + +interpretation i_insert_before_wf2?: l_insert_before_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent get_parent_locs + get_child_nodes get_child_nodes_locs set_child_nodes + set_child_nodes_locs get_ancestors get_ancestors_locs + adopt_node adopt_node_locs set_disconnected_nodes + set_disconnected_nodes_locs get_disconnected_nodes + get_disconnected_nodes_locs get_owner_document insert_before + insert_before_locs append_child type_wf known_ptr known_ptrs + heap_is_wellformed parent_child_rel remove_child + remove_child_locs get_root_node get_root_node_locs + by(simp add: l_insert_before_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances) +declare l_insert_before_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances] + +lemma insert_before_wf2_is_l_insert_before_wf2 [instances]: + "l_insert_before_wf2 type_wf known_ptr known_ptrs insert_before heap_is_wellformed" + apply(auto simp add: l_insert_before_wf2_def l_insert_before_wf2_axioms_def instances)[1] + using insert_before_heap_is_wellformed_preserved apply(fast, fast, fast) + done + +locale l_append_child_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_insert_before_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_insert_before_wf2 + + l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + + l_get_child_nodes +begin + +lemma append_child_children: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_child_nodes ptr \\<^sub>r xs" + assumes "h \ append_child ptr node \\<^sub>h h'" + assumes "node \ set xs" + shows "h' \ get_child_nodes ptr \\<^sub>r xs @ [node]" +proof - + + obtain ancestors owner_document h2 h3 disconnected_nodes_h2 where + ancestors: "h \ get_ancestors ptr \\<^sub>r ancestors" and + node_not_in_ancestors: "cast node \ set ancestors" and + owner_document: "h \ get_owner_document ptr \\<^sub>r owner_document" and + h2: "h \ adopt_node owner_document node \\<^sub>h h2" and + disconnected_nodes_h2: "h2 \ get_disconnected_nodes owner_document \\<^sub>r disconnected_nodes_h2" and + h3: "h2 \ set_disconnected_nodes owner_document (remove1 node disconnected_nodes_h2) \\<^sub>h h3" and + h': "h3 \ a_insert_node ptr node None \\<^sub>h h'" + using assms(5) + by(auto simp add: append_child_def insert_before_def a_ensure_pre_insertion_validity_def + elim!: bind_returns_heap_E bind_returns_result_E + bind_returns_heap_E2[rotated, OF get_parent_pure, rotated] + bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated] + bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] + bind_returns_heap_E2[rotated, OF get_ancestors_pure, rotated] + bind_returns_heap_E2[rotated, OF next_sibling_pure, rotated] + bind_returns_heap_E2[rotated, OF get_owner_document_pure, rotated] + split: if_splits option.splits) + + have "\parent. |h \ get_parent node|\<^sub>r = Some parent \ parent \ ptr" + using assms(1) assms(4) assms(6) + by (metis (no_types, lifting) assms(2) assms(3) h2 is_OK_returns_heap_I is_OK_returns_result_E + local.adopt_node_child_in_heap local.get_parent_child_dual local.get_parent_ok + select_result_I2) + have "h2 \ get_child_nodes ptr \\<^sub>r xs" + using get_child_nodes_reads adopt_node_writes h2 assms(4) + apply(rule reads_writes_separate_forwards) + using \\parent. |h \ get_parent node|\<^sub>r = Some parent \ parent \ ptr\ + apply(auto simp add: adopt_node_locs_def remove_child_locs_def)[1] + by (meson local.set_child_nodes_get_child_nodes_different_pointers) + + have "h3 \ get_child_nodes ptr \\<^sub>r xs" + using get_child_nodes_reads set_disconnected_nodes_writes h3 \h2 \ get_child_nodes ptr \\<^sub>r xs\ + apply(rule reads_writes_separate_forwards) + by(auto) + + have "ptr |\| object_ptr_kinds h" + by (meson ancestors is_OK_returns_result_I local.get_ancestors_ptr_in_heap) + then + have "known_ptr ptr" + using assms(3) + using local.known_ptrs_known_ptr by blast + + have "type_wf h2" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF adopt_node_writes h2] + using adopt_node_types_preserved \type_wf h\ + by(auto simp add: adopt_node_locs_def remove_child_locs_def reflp_def transp_def split: if_splits) + then + have "type_wf h3" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_disconnected_nodes_writes h3] + using set_disconnected_nodes_types_preserved + by(auto simp add: reflp_def transp_def) + + show "h' \ get_child_nodes ptr \\<^sub>r xs@[node]" + using h' + apply(auto simp add: a_insert_node_def + dest!: bind_returns_heap_E3[rotated, OF \h3 \ get_child_nodes ptr \\<^sub>r xs\ + get_child_nodes_pure, rotated])[1] + using \type_wf h3\ set_child_nodes_get_child_nodes \known_ptr ptr\ + by metis +qed + +lemma append_child_for_all_on_children: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_child_nodes ptr \\<^sub>r xs" + assumes "h \ forall_M (append_child ptr) nodes \\<^sub>h h'" + assumes "set nodes \ set xs = {}" + assumes "distinct nodes" + shows "h' \ get_child_nodes ptr \\<^sub>r xs@nodes" + using assms + apply(induct nodes arbitrary: h xs) + apply(simp) +proof(auto elim!: bind_returns_heap_E)[1]fix a nodes h xs h'a + assume 0: "(\h xs. heap_is_wellformed h \ type_wf h \ known_ptrs h + \ h \ get_child_nodes ptr \\<^sub>r xs \ h \ forall_M (append_child ptr) nodes \\<^sub>h h' + \ set nodes \ set xs = {} \ h' \ get_child_nodes ptr \\<^sub>r xs @ nodes)" + and 1: "heap_is_wellformed h" + and 2: "type_wf h" + and 3: "known_ptrs h" + and 4: "h \ get_child_nodes ptr \\<^sub>r xs" + and 5: "h \ append_child ptr a \\<^sub>r ()" + and 6: "h \ append_child ptr a \\<^sub>h h'a" + and 7: "h'a \ forall_M (append_child ptr) nodes \\<^sub>h h'" + and 8: "a \ set xs" + and 9: "set nodes \ set xs = {}" + and 10: "a \ set nodes" + and 11: "distinct nodes" + then have "h'a \ get_child_nodes ptr \\<^sub>r xs @ [a]" + using append_child_children 6 + using "1" "2" "3" "4" "8" by blast + + moreover have "heap_is_wellformed h'a" and "type_wf h'a" and "known_ptrs h'a" + using insert_before_heap_is_wellformed_preserved insert_before_preserves_known_ptrs + insert_before_preserves_type_wf 1 2 3 6 append_child_def + by metis+ + moreover have "set nodes \ set (xs @ [a]) = {}" + using 9 10 + by auto + ultimately show "h' \ get_child_nodes ptr \\<^sub>r xs @ a # nodes" + using 0 7 + by fastforce +qed + + +lemma append_child_for_all_on_no_children: + assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" + assumes "h \ get_child_nodes ptr \\<^sub>r []" + assumes "h \ forall_M (append_child ptr) nodes \\<^sub>h h'" + assumes "distinct nodes" + shows "h' \ get_child_nodes ptr \\<^sub>r nodes" + using assms append_child_for_all_on_children + by force +end + +interpretation i_append_child_wf?: l_append_child_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent + get_parent_locs remove_child remove_child_locs + get_disconnected_nodes get_disconnected_nodes_locs + set_disconnected_nodes set_disconnected_nodes_locs + adopt_node adopt_node_locs known_ptr type_wf get_child_nodes + get_child_nodes_locs known_ptrs set_child_nodes + set_child_nodes_locs remove get_ancestors get_ancestors_locs + insert_before insert_before_locs append_child heap_is_wellformed + parent_child_rel + by(auto simp add: l_append_child_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances) + + + +subsection \create\_element\ + +locale l_create_element_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs + get_disconnected_nodes get_disconnected_nodes_locs + heap_is_wellformed parent_child_rel + + l_new_element_get_disconnected_nodes get_disconnected_nodes get_disconnected_nodes_locs + + l_set_tag_type_get_disconnected_nodes type_wf set_tag_type set_tag_type_locs + get_disconnected_nodes get_disconnected_nodes_locs + + l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes + set_disconnected_nodes_locs set_tag_type set_tag_type_locs create_element + + l_new_element_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs + + l_set_tag_type_get_child_nodes type_wf set_tag_type set_tag_type_locs known_ptr + get_child_nodes get_child_nodes_locs + + l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes set_disconnected_nodes_locs + get_child_nodes get_child_nodes_locs + + l_set_disconnected_nodes type_wf set_disconnected_nodes set_disconnected_nodes_locs + + l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes + get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs + + l_new_element type_wf + + l_known_ptrs known_ptr known_ptrs + for known_ptr :: "(_::linorder) object_ptr \ bool" + and known_ptrs :: "(_) heap \ bool" + and type_wf :: "(_) heap \ bool" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and heap_is_wellformed :: "(_) heap \ bool" + and parent_child_rel :: "(_) heap \ ((_) object_ptr \ (_) object_ptr) set" + and set_tag_type :: "(_) element_ptr \ char list \ ((_) heap, exception, unit) prog" + and set_tag_type_locs :: "(_) element_ptr \ ((_) heap, exception, unit) prog set" + and set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap, exception, unit) prog set" + and create_element :: "(_) document_ptr \ char list \ ((_) heap, exception, (_) element_ptr) prog" +begin +lemma create_element_preserves_wellformedness: + assumes "heap_is_wellformed h" + and "h \ create_element document_ptr tag \\<^sub>h h'" + and "type_wf h" + and "known_ptrs h" + shows "heap_is_wellformed h'" +proof - + obtain new_element_ptr h2 h3 disc_nodes_h3 where + new_element_ptr: "h \ new_element \\<^sub>r new_element_ptr" and + h2: "h \ new_element \\<^sub>h h2" and + h3: "h2 \ set_tag_type new_element_ptr tag \\<^sub>h h3" and + disc_nodes_h3: "h3 \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes_h3" and + h': "h3 \ set_disconnected_nodes document_ptr (cast new_element_ptr # disc_nodes_h3) \\<^sub>h h'" + using assms(2) + by(auto simp add: create_element_def + elim!: bind_returns_heap_E + bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] ) + + have "new_element_ptr \ set |h \ element_ptr_kinds_M|\<^sub>r" + using new_element_ptr ElementMonad.ptr_kinds_ptr_kinds_M h2 + using new_element_ptr_not_in_heap by blast + then have "cast new_element_ptr \ set |h \ node_ptr_kinds_M|\<^sub>r" + by simp + then have "cast new_element_ptr \ set |h \ object_ptr_kinds_M|\<^sub>r" + by simp + + have object_ptr_kinds_eq_h: "object_ptr_kinds h2 = object_ptr_kinds h |\| {|cast new_element_ptr|}" + using new_element_new_ptr h2 new_element_ptr by blast + then have node_ptr_kinds_eq_h: "node_ptr_kinds h2 = node_ptr_kinds h |\| {|cast new_element_ptr|}" + apply(simp add: node_ptr_kinds_def) + by force + then have element_ptr_kinds_eq_h: "element_ptr_kinds h2 = element_ptr_kinds h |\| {|new_element_ptr|}" + apply(simp add: element_ptr_kinds_def) + by force + have character_data_ptr_kinds_eq_h: "character_data_ptr_kinds h2 = character_data_ptr_kinds h" + using object_ptr_kinds_eq_h + by(auto simp add: node_ptr_kinds_def character_data_ptr_kinds_def) + have document_ptr_kinds_eq_h: "document_ptr_kinds h2 = document_ptr_kinds h" + using object_ptr_kinds_eq_h + by(auto simp add: document_ptr_kinds_def) + + have object_ptr_kinds_eq_h2: "object_ptr_kinds h3 = object_ptr_kinds h2" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h' = object_ptr_kinds h", OF set_tag_type_writes h3]) + using set_tag_type_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have document_ptr_kinds_eq_h2: "document_ptr_kinds h3 = document_ptr_kinds h2" + by (auto simp add: document_ptr_kinds_def) + have node_ptr_kinds_eq_h2: "node_ptr_kinds h3 = node_ptr_kinds h2" + using object_ptr_kinds_eq_h2 + by(auto simp add: node_ptr_kinds_def) + + have object_ptr_kinds_eq_h3: "object_ptr_kinds h' = object_ptr_kinds h3" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h' = object_ptr_kinds h", + OF set_disconnected_nodes_writes h']) + using set_disconnected_nodes_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have document_ptr_kinds_eq_h3: "document_ptr_kinds h' = document_ptr_kinds h3" + by (auto simp add: document_ptr_kinds_def) + have node_ptr_kinds_eq_h3: "node_ptr_kinds h' = node_ptr_kinds h3" + using object_ptr_kinds_eq_h3 + by(auto simp add: node_ptr_kinds_def) + + + have "document_ptr |\| document_ptr_kinds h" + using disc_nodes_h3 document_ptr_kinds_eq_h object_ptr_kinds_eq_h2 + get_disconnected_nodes_ptr_in_heap \type_wf h\ document_ptr_kinds_def + by (metis is_OK_returns_result_I) + + have children_eq_h: "\(ptr'::(_) object_ptr) children. ptr' \ cast new_element_ptr + \ h \ get_child_nodes ptr' \\<^sub>r children = h2 \ get_child_nodes ptr' \\<^sub>r children" + using get_child_nodes_reads h2 get_child_nodes_new_element[rotated, OF new_element_ptr h2] + apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1] + by blast+ + then have children_eq2_h: "\ptr'. ptr' \ cast new_element_ptr + \ |h \ get_child_nodes ptr'|\<^sub>r = |h2 \ get_child_nodes ptr'|\<^sub>r" + using select_result_eq by force + + have "h2 \ get_child_nodes (cast new_element_ptr) \\<^sub>r []" + using new_element_ptr h2 new_element_ptr_in_heap[OF h2 new_element_ptr] + new_element_is_element_ptr[OF new_element_ptr] new_element_no_child_nodes + by blast + have disconnected_nodes_eq_h: + "\doc_ptr disc_nodes. h \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes + = h2 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes" + using get_disconnected_nodes_reads h2 get_disconnected_nodes_new_element[OF new_element_ptr h2] + apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1] + by blast+ + then have disconnected_nodes_eq2_h: + "\doc_ptr. |h \ get_disconnected_nodes doc_ptr|\<^sub>r = |h2 \ get_disconnected_nodes doc_ptr|\<^sub>r" + using select_result_eq by force + + have children_eq_h2: + "\ptr' children. h2 \ get_child_nodes ptr' \\<^sub>r children = h3 \ get_child_nodes ptr' \\<^sub>r children" + using get_child_nodes_reads set_tag_type_writes h3 + apply(rule reads_writes_preserved) + by(auto simp add: set_tag_type_get_child_nodes) + then have children_eq2_h2: "\ptr'. |h2 \ get_child_nodes ptr'|\<^sub>r = |h3 \ get_child_nodes ptr'|\<^sub>r" + using select_result_eq by force + have disconnected_nodes_eq_h2: + "\doc_ptr disc_nodes. h2 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes + = h3 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes" + using get_disconnected_nodes_reads set_tag_type_writes h3 + apply(rule reads_writes_preserved) + by(auto simp add: set_tag_type_get_disconnected_nodes) + then have disconnected_nodes_eq2_h2: + "\doc_ptr. |h2 \ get_disconnected_nodes doc_ptr|\<^sub>r = |h3 \ get_disconnected_nodes doc_ptr|\<^sub>r" + using select_result_eq by force + + have "type_wf h2" + using \type_wf h\ new_element_types_preserved h2 by blast + then have "type_wf h3" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_tag_type_writes h3] + using set_tag_type_types_preserved + by(auto simp add: reflp_def transp_def) + then have "type_wf h'" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_disconnected_nodes_writes h'] + using set_disconnected_nodes_types_preserved + by(auto simp add: reflp_def transp_def) + + have children_eq_h3: + "\ptr' children. h3 \ get_child_nodes ptr' \\<^sub>r children = h' \ get_child_nodes ptr' \\<^sub>r children" + using get_child_nodes_reads set_disconnected_nodes_writes h' + apply(rule reads_writes_preserved) + by(auto simp add: set_disconnected_nodes_get_child_nodes) + then have children_eq2_h3: "\ptr'. |h3 \ get_child_nodes ptr'|\<^sub>r = |h' \ get_child_nodes ptr'|\<^sub>r" + using select_result_eq by force + have disconnected_nodes_eq_h3: + "\doc_ptr disc_nodes. document_ptr \ doc_ptr + \ h3 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes + = h' \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes" + using get_disconnected_nodes_reads set_disconnected_nodes_writes h' + apply(rule reads_writes_preserved) + by(auto simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers) + then have disconnected_nodes_eq2_h3: + "\doc_ptr. document_ptr \ doc_ptr + \ |h3 \ get_disconnected_nodes doc_ptr|\<^sub>r = |h' \ get_disconnected_nodes doc_ptr|\<^sub>r" + using select_result_eq by force + + have disc_nodes_document_ptr_h2: "h2 \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes_h3" + using disconnected_nodes_eq_h2 disc_nodes_h3 by auto + then have disc_nodes_document_ptr_h: "h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes_h3" + using disconnected_nodes_eq_h by auto + then have "cast new_element_ptr \ set disc_nodes_h3" + using \heap_is_wellformed h\ + using \cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \ set |h \ node_ptr_kinds_M|\<^sub>r\ + a_all_ptrs_in_heap_def heap_is_wellformed_def + by (meson NodeMonad.ptr_kinds_ptr_kinds_M fset_mp fset_of_list_elem ) + + have "acyclic (parent_child_rel h)" + using \heap_is_wellformed h\ + by (simp add: heap_is_wellformed_def acyclic_heap_def) + also have "parent_child_rel h = parent_child_rel h2" + proof(auto simp add: parent_child_rel_def)[1] + fix a x + assume 0: "a |\| object_ptr_kinds h" + and 1: "x \ set |h \ get_child_nodes a|\<^sub>r" + then show "a |\| object_ptr_kinds h2" + by (simp add: object_ptr_kinds_eq_h) + next + fix a x + assume 0: "a |\| object_ptr_kinds h" + and 1: "x \ set |h \ get_child_nodes a|\<^sub>r" + then show "x \ set |h2 \ get_child_nodes a|\<^sub>r" + by (metis ObjectMonad.ptr_kinds_ptr_kinds_M + \cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \ set |h \ object_ptr_kinds_M|\<^sub>r\ children_eq2_h) + next + fix a x + assume 0: "a |\| object_ptr_kinds h2" + and 1: "x \ set |h2 \ get_child_nodes a|\<^sub>r" + then show "a |\| object_ptr_kinds h" + using object_ptr_kinds_eq_h \h2 \ get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr) \\<^sub>r []\ + by(auto) + next + fix a x + assume 0: "a |\| object_ptr_kinds h2" + and 1: "x \ set |h2 \ get_child_nodes a|\<^sub>r" + then show "x \ set |h \ get_child_nodes a|\<^sub>r" + by (metis (no_types, lifting) + \h2 \ get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr) \\<^sub>r []\ + children_eq2_h empty_iff empty_set image_eqI select_result_I2) + qed + also have "\ = parent_child_rel h3" + by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h2 children_eq2_h2) + also have "\ = parent_child_rel h'" + by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h3 children_eq2_h3) + finally have "a_acyclic_heap h'" + by (simp add: acyclic_heap_def) + + have "a_all_ptrs_in_heap h" + using \heap_is_wellformed h\ by (simp add: heap_is_wellformed_def) + then have "a_all_ptrs_in_heap h2" + apply(auto simp add: a_all_ptrs_in_heap_def)[1] + using node_ptr_kinds_eq_h + \cast new_element_ptr \ set |h \ node_ptr_kinds_M|\<^sub>r\ + \h2 \ get_child_nodes (cast new_element_ptr) \\<^sub>r []\ + apply (metis (no_types, hide_lams) children_eq_h fempty_iff fset_mp fset_of_list_simps(1) + funionCI select_result_I2) + by (simp add: disconnected_nodes_eq_h fset_rev_mp node_ptr_kinds_eq_h) + then have "a_all_ptrs_in_heap h3" + by(auto simp add: a_all_ptrs_in_heap_def object_ptr_kinds_eq_h2 node_ptr_kinds_def + children_eq_h2 disconnected_nodes_eq_h2) + then have "a_all_ptrs_in_heap h'" + apply(auto simp add: a_all_ptrs_in_heap_def object_ptr_kinds_eq_h3 node_ptr_kinds_def children_eq_h3 )[1] + using disconnected_nodes_eq_h3 object_ptr_kinds_eq_h object_ptr_kinds_eq_h2 + by (metis (no_types, lifting) disc_nodes_h3 finsertCI fset.map_comp fset_mp fset_of_list_elem + funion_finsert_right h' local.set_disconnected_nodes_get_disconnected_nodes + node_ptr_kinds_def node_ptr_kinds_eq_h select_result_I2 set_ConsD) + + have "\p. p |\| object_ptr_kinds h \ cast new_element_ptr \ set |h \ get_child_nodes p|\<^sub>r" + using \heap_is_wellformed h\ \cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \ set |h \ node_ptr_kinds_M|\<^sub>r\ + heap_is_wellformed_children_in_heap + by (meson NodeMonad.ptr_kinds_ptr_kinds_M a_all_ptrs_in_heap_def assms(3) assms(4) fset_mp + fset_of_list_elem get_child_nodes_ok known_ptrs_known_ptr returns_result_select_result) + then have "\p. p |\| object_ptr_kinds h2 \ cast new_element_ptr \ set |h2 \ get_child_nodes p|\<^sub>r" + using children_eq2_h + apply(auto simp add: object_ptr_kinds_eq_h)[1] + using \h2 \ get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr) \\<^sub>r []\ apply auto[1] + by (metis ObjectMonad.ptr_kinds_ptr_kinds_M + \cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \ set |h \ object_ptr_kinds_M|\<^sub>r\) + then have "\p. p |\| object_ptr_kinds h3 \ cast new_element_ptr \ set |h3 \ get_child_nodes p|\<^sub>r" + using object_ptr_kinds_eq_h2 children_eq2_h2 by auto + then have new_element_ptr_not_in_any_children: + "\p. p |\| object_ptr_kinds h' \ cast new_element_ptr \ set |h' \ get_child_nodes p|\<^sub>r" + using object_ptr_kinds_eq_h3 children_eq2_h3 by auto + + have "a_distinct_lists h" + using \heap_is_wellformed h\ + by (simp add: heap_is_wellformed_def) + then have "a_distinct_lists h2" + + using \h2 \ get_child_nodes (cast new_element_ptr) \\<^sub>r []\ + apply(auto simp add: a_distinct_lists_def object_ptr_kinds_eq_h document_ptr_kinds_eq_h + disconnected_nodes_eq2_h intro!: distinct_concat_map_I)[1] + apply (metis distinct_sorted_list_of_set finite_fset sorted_list_of_set_insert) + apply(case_tac "x=cast new_element_ptr") + apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1] + apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1] + apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1] + apply (metis IntI assms(1) assms(3) assms(4) empty_iff local.get_child_nodes_ok + local.heap_is_wellformed_one_parent local.known_ptrs_known_ptr returns_result_select_result) + apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1] + by (metis \local.a_distinct_lists h\ \type_wf h2\ disconnected_nodes_eq_h document_ptr_kinds_eq_h + local.distinct_lists_no_parent local.get_disconnected_nodes_ok returns_result_select_result) + + then have "a_distinct_lists h3" + by(auto simp add: a_distinct_lists_def disconnected_nodes_eq2_h2 document_ptr_kinds_eq_h2 + children_eq2_h2 object_ptr_kinds_eq_h2) + then have "a_distinct_lists h'" + proof(auto simp add: a_distinct_lists_def disconnected_nodes_eq2_h3 children_eq2_h3 + object_ptr_kinds_eq_h3 document_ptr_kinds_eq_h3 + intro!: distinct_concat_map_I)[1] + fix x + assume "distinct (concat (map (\document_ptr. |h3 \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h3)))))" + and "x |\| document_ptr_kinds h3" + then show "distinct |h' \ get_disconnected_nodes x|\<^sub>r" + using document_ptr_kinds_eq_h3 disconnected_nodes_eq_h3 h' set_disconnected_nodes_get_disconnected_nodes + by (metis (no_types, lifting) \cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \ set disc_nodes_h3\ + \a_distinct_lists h3\ \type_wf h'\ disc_nodes_h3 distinct.simps(2) + distinct_lists_disconnected_nodes get_disconnected_nodes_ok returns_result_eq + returns_result_select_result) + next + fix x y xa + assume "distinct (concat (map (\document_ptr. |h3 \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h3)))))" + and "x |\| document_ptr_kinds h3" + and "y |\| document_ptr_kinds h3" + and "x \ y" + and "xa \ set |h' \ get_disconnected_nodes x|\<^sub>r" + and "xa \ set |h' \ get_disconnected_nodes y|\<^sub>r" + moreover have "set |h3 \ get_disconnected_nodes x|\<^sub>r \ set |h3 \ get_disconnected_nodes y|\<^sub>r = {}" + using calculation by(auto dest: distinct_concat_map_E(1)) + ultimately show "False" + apply(-) + apply(cases "x = document_ptr") + apply (metis (no_types) NodeMonad.ptr_kinds_ptr_kinds_M + \a_all_ptrs_in_heap h\ + \cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \ set |h \ node_ptr_kinds_M|\<^sub>r\ + a_all_ptrs_in_heap_def assms(3) disc_nodes_h3 disconnected_nodes_eq2_h + disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 disjoint_iff_not_equal + document_ptr_kinds_eq_h document_ptr_kinds_eq_h2 fset_mp fset_of_list_elem + get_disconnected_nodes_ok h' returns_result_select_result select_result_I2 + set_ConsD set_disconnected_nodes_get_disconnected_nodes) + apply(cases "y = document_ptr" ) + apply (metis (no_types) NodeMonad.ptr_kinds_ptr_kinds_M + \a_all_ptrs_in_heap h\ + \cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \ set |h \ node_ptr_kinds_M|\<^sub>r\ + a_all_ptrs_in_heap_def assms(3) disc_nodes_h3 disconnected_nodes_eq2_h + disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 disjoint_iff_not_equal + document_ptr_kinds_eq_h document_ptr_kinds_eq_h2 fset_mp fset_of_list_elem + get_disconnected_nodes_ok h' returns_result_select_result select_result_I2 + set_ConsD set_disconnected_nodes_get_disconnected_nodes) + using disconnected_nodes_eq2_h3 by auto + next + fix x xa xb + assume 2: "(\x\fset (object_ptr_kinds h3). set |h' \ get_child_nodes x|\<^sub>r) + \ (\x\fset (document_ptr_kinds h3). set |h3 \ get_disconnected_nodes x|\<^sub>r) = {}" + and 3: "xa |\| object_ptr_kinds h3" + and 4: "x \ set |h' \ get_child_nodes xa|\<^sub>r" + and 5: "xb |\| document_ptr_kinds h3" + and 6: "x \ set |h' \ get_disconnected_nodes xb|\<^sub>r" + show "False" + using disc_nodes_document_ptr_h disconnected_nodes_eq2_h3 + apply - + apply(cases "xb = document_ptr") + apply (metis (no_types, hide_lams) "3" "4" "6" + \\p. p |\| object_ptr_kinds h3 + \ cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \ set |h3 \ get_child_nodes p|\<^sub>r\ + \a_distinct_lists h3\ children_eq2_h3 disc_nodes_h3 distinct_lists_no_parent h' + select_result_I2 set_ConsD set_disconnected_nodes_get_disconnected_nodes) + by (metis "3" "4" "5" "6" \a_distinct_lists h3\ \type_wf h3\ children_eq2_h3 + distinct_lists_no_parent get_disconnected_nodes_ok returns_result_select_result) + qed + + have "a_owner_document_valid h" + using \heap_is_wellformed h\ by (simp add: heap_is_wellformed_def) + then have "a_owner_document_valid h'" + using disc_nodes_h3 \document_ptr |\| document_ptr_kinds h\ + apply(auto simp add: a_owner_document_valid_def)[1] + apply(auto simp add: object_ptr_kinds_eq_h object_ptr_kinds_eq_h3 )[1] + apply(auto simp add: object_ptr_kinds_eq_h2)[1] + apply(auto simp add: document_ptr_kinds_eq_h document_ptr_kinds_eq_h3 )[1] + apply(auto simp add: document_ptr_kinds_eq_h2)[1] + apply(auto simp add: node_ptr_kinds_eq_h node_ptr_kinds_eq_h3 )[1] + apply(auto simp add: node_ptr_kinds_eq_h2 node_ptr_kinds_eq_h )[1] + apply(auto simp add: children_eq2_h2[symmetric] children_eq2_h3[symmetric] + disconnected_nodes_eq2_h disconnected_nodes_eq2_h2 + disconnected_nodes_eq2_h3)[1] + apply (metis (no_types, lifting) document_ptr_kinds_eq_h h' list.set_intros(1) + local.set_disconnected_nodes_get_disconnected_nodes select_result_I2) + apply(simp add: object_ptr_kinds_eq_h) + proof - + fix node_ptr :: "(_) node_ptr" + assume a1: "\node_ptr. node_ptr |\| node_ptr_kinds h \ (\document_ptr. document_ptr |\| document_ptr_kinds h \ node_ptr \ set |h \ get_disconnected_nodes document_ptr|\<^sub>r) \ (\parent_ptr. parent_ptr |\| object_ptr_kinds h \ node_ptr \ set |h \ get_child_nodes parent_ptr|\<^sub>r)" + assume a2: "node_ptr |\| node_ptr_kinds h" + assume a3: "\parent_ptr. (parent_ptr = cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \ node_ptr \ set |h' \ get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr)|\<^sub>r) \ (parent_ptr |\| object_ptr_kinds h \ node_ptr \ set |h' \ get_child_nodes parent_ptr|\<^sub>r)" + assume a4: "document_ptr |\| document_ptr_kinds h" + assume a5: "h3 \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes_h3" + obtain dd :: "(_) node_ptr \ (_) document_ptr" where + "\x0. (\v1. v1 |\| document_ptr_kinds h \ x0 \ set |h \ get_disconnected_nodes v1|\<^sub>r) = (dd x0 |\| document_ptr_kinds h \ x0 \ set |h \ get_disconnected_nodes (dd x0)|\<^sub>r)" + by moura + then have f6: "dd node_ptr |\| document_ptr_kinds h \ node_ptr \ set |h \ get_disconnected_nodes (dd node_ptr)|\<^sub>r" + using a3 a2 a1 by (metis (no_types) \cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \ set |h \ object_ptr_kinds_M|\<^sub>r\ children_eq2_h children_eq2_h2 children_eq2_h3 l_ptr_kinds_M.ptr_kinds_ptr_kinds_M object_ptr_kinds_M_def) + moreover + { assume "|h \ get_disconnected_nodes (dd node_ptr)|\<^sub>r \ disc_nodes_h3" + then have "document_ptr \ dd node_ptr" + using a5 disconnected_nodes_eq2_h disconnected_nodes_eq2_h2 by force + then have "\d. d |\| document_ptr_kinds h2 \ node_ptr \ set |h' \ get_disconnected_nodes d|\<^sub>r" + using f6 disconnected_nodes_eq2_h disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 document_ptr_kinds_eq_h by auto } + ultimately show "\d. d |\| document_ptr_kinds h2 \ node_ptr \ set |h' \ get_disconnected_nodes d|\<^sub>r" + using a4 by (metis (no_types) document_ptr_kinds_eq_h h' insert_iff list.set(2) local.set_disconnected_nodes_get_disconnected_nodes select_result_I2) + qed + show "heap_is_wellformed h'" + using \a_acyclic_heap h'\ \a_all_ptrs_in_heap h'\ \a_distinct_lists h'\ \a_owner_document_valid h'\ + by(simp add: heap_is_wellformed_def) +qed +end + +interpretation i_create_element_wf?: l_create_element_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptrs type_wf + get_child_nodes get_child_nodes_locs get_disconnected_nodes + get_disconnected_nodes_locs heap_is_wellformed parent_child_rel + set_tag_type set_tag_type_locs + set_disconnected_nodes set_disconnected_nodes_locs create_element + using instances + by(auto simp add: l_create_element_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) +declare l_create_element_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances] + + +subsection \create\_character\_data\ + +locale l_create_character_data_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes + get_disconnected_nodes_locs heap_is_wellformed parent_child_rel + + l_new_character_data_get_disconnected_nodes + get_disconnected_nodes get_disconnected_nodes_locs + + l_set_val_get_disconnected_nodes + type_wf set_val set_val_locs get_disconnected_nodes get_disconnected_nodes_locs + + l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + set_val set_val_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes + set_disconnected_nodes_locs create_character_data + + l_new_character_data_get_child_nodes + type_wf known_ptr get_child_nodes get_child_nodes_locs + + l_set_val_get_child_nodes + type_wf set_val set_val_locs known_ptr get_child_nodes get_child_nodes_locs + + l_set_disconnected_nodes_get_child_nodes + set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs + + l_set_disconnected_nodes + type_wf set_disconnected_nodes set_disconnected_nodes_locs + + l_set_disconnected_nodes_get_disconnected_nodes + type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes + set_disconnected_nodes_locs + + l_new_character_data + type_wf + + l_known_ptrs + known_ptr known_ptrs + for known_ptr :: "(_::linorder) object_ptr \ bool" + and type_wf :: "(_) heap \ bool" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and heap_is_wellformed :: "(_) heap \ bool" + and parent_child_rel :: "(_) heap \ ((_) object_ptr \ (_) object_ptr) set" + and set_val :: "(_) character_data_ptr \ char list \ ((_) heap, exception, unit) prog" + and set_val_locs :: "(_) character_data_ptr \ ((_) heap, exception, unit) prog set" + and set_disconnected_nodes :: + "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap, exception, unit) prog set" + and create_character_data :: + "(_) document_ptr \ char list \ ((_) heap, exception, (_) character_data_ptr) prog" + and known_ptrs :: "(_) heap \ bool" +begin + +lemma create_character_data_preserves_wellformedness: + assumes "heap_is_wellformed h" + and "h \ create_character_data document_ptr text \\<^sub>h h'" + and "type_wf h" + and "known_ptrs h" + shows "heap_is_wellformed h'" +proof - + obtain new_character_data_ptr h2 h3 disc_nodes_h3 where + new_character_data_ptr: "h \ new_character_data \\<^sub>r new_character_data_ptr" and + h2: "h \ new_character_data \\<^sub>h h2" and + h3: "h2 \ set_val new_character_data_ptr text \\<^sub>h h3" and + disc_nodes_h3: "h3 \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes_h3" and + h': "h3 \ set_disconnected_nodes document_ptr (cast new_character_data_ptr # disc_nodes_h3) \\<^sub>h h'" + using assms(2) + by(auto simp add: create_character_data_def + elim!: bind_returns_heap_E + bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] ) + + + have "new_character_data_ptr \ set |h \ character_data_ptr_kinds_M|\<^sub>r" + using new_character_data_ptr CharacterDataMonad.ptr_kinds_ptr_kinds_M h2 + using new_character_data_ptr_not_in_heap by blast + then have "cast new_character_data_ptr \ set |h \ node_ptr_kinds_M|\<^sub>r" + by simp + then have "cast new_character_data_ptr \ set |h \ object_ptr_kinds_M|\<^sub>r" + by simp + + + + have object_ptr_kinds_eq_h: + "object_ptr_kinds h2 = object_ptr_kinds h |\| {|cast new_character_data_ptr|}" + using new_character_data_new_ptr h2 new_character_data_ptr by blast + then have node_ptr_kinds_eq_h: + "node_ptr_kinds h2 = node_ptr_kinds h |\| {|cast new_character_data_ptr|}" + apply(simp add: node_ptr_kinds_def) + by force + then have character_data_ptr_kinds_eq_h: + "character_data_ptr_kinds h2 = character_data_ptr_kinds h |\| {|new_character_data_ptr|}" + apply(simp add: character_data_ptr_kinds_def) + by force + have element_ptr_kinds_eq_h: "element_ptr_kinds h2 = element_ptr_kinds h" + using object_ptr_kinds_eq_h + by(auto simp add: node_ptr_kinds_def element_ptr_kinds_def) + have document_ptr_kinds_eq_h: "document_ptr_kinds h2 = document_ptr_kinds h" + using object_ptr_kinds_eq_h + by(auto simp add: document_ptr_kinds_def) + + have object_ptr_kinds_eq_h2: "object_ptr_kinds h3 = object_ptr_kinds h2" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h' = object_ptr_kinds h", + OF set_val_writes h3]) + using set_val_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have document_ptr_kinds_eq_h2: "document_ptr_kinds h3 = document_ptr_kinds h2" + by (auto simp add: document_ptr_kinds_def) + have node_ptr_kinds_eq_h2: "node_ptr_kinds h3 = node_ptr_kinds h2" + using object_ptr_kinds_eq_h2 + by(auto simp add: node_ptr_kinds_def) + + have object_ptr_kinds_eq_h3: "object_ptr_kinds h' = object_ptr_kinds h3" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h' = object_ptr_kinds h", + OF set_disconnected_nodes_writes h']) + using set_disconnected_nodes_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have document_ptr_kinds_eq_h3: "document_ptr_kinds h' = document_ptr_kinds h3" + by (auto simp add: document_ptr_kinds_def) + have node_ptr_kinds_eq_h3: "node_ptr_kinds h' = node_ptr_kinds h3" + using object_ptr_kinds_eq_h3 + by(auto simp add: node_ptr_kinds_def) + + + have "document_ptr |\| document_ptr_kinds h" + using disc_nodes_h3 document_ptr_kinds_eq_h object_ptr_kinds_eq_h2 + get_disconnected_nodes_ptr_in_heap \type_wf h\ document_ptr_kinds_def + by (metis is_OK_returns_result_I) + + have children_eq_h: "\(ptr'::(_) object_ptr) children. ptr' \ cast new_character_data_ptr + \ h \ get_child_nodes ptr' \\<^sub>r children = h2 \ get_child_nodes ptr' \\<^sub>r children" + using get_child_nodes_reads h2 get_child_nodes_new_character_data[rotated, OF new_character_data_ptr h2] + apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1] + by blast+ + then have children_eq2_h: + "\ptr'. ptr' \ cast new_character_data_ptr + \ |h \ get_child_nodes ptr'|\<^sub>r = |h2 \ get_child_nodes ptr'|\<^sub>r" + using select_result_eq by force + have object_ptr_kinds_eq_h: + "object_ptr_kinds h2 = object_ptr_kinds h |\| {|cast new_character_data_ptr|}" + using new_character_data_new_ptr h2 new_character_data_ptr by blast + then have node_ptr_kinds_eq_h: + "node_ptr_kinds h2 = node_ptr_kinds h |\| {|cast new_character_data_ptr|}" + apply(simp add: node_ptr_kinds_def) + by force + then have character_data_ptr_kinds_eq_h: + "character_data_ptr_kinds h2 = character_data_ptr_kinds h |\| {|new_character_data_ptr|}" + apply(simp add: character_data_ptr_kinds_def) + by force + have element_ptr_kinds_eq_h: "element_ptr_kinds h2 = element_ptr_kinds h" + using object_ptr_kinds_eq_h + by(auto simp add: node_ptr_kinds_def element_ptr_kinds_def) + have document_ptr_kinds_eq_h: "document_ptr_kinds h2 = document_ptr_kinds h" + using object_ptr_kinds_eq_h + by(auto simp add: document_ptr_kinds_def) + + have object_ptr_kinds_eq_h2: "object_ptr_kinds h3 = object_ptr_kinds h2" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h' = object_ptr_kinds h", + OF set_val_writes h3]) + using set_val_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have document_ptr_kinds_eq_h2: "document_ptr_kinds h3 = document_ptr_kinds h2" + by (auto simp add: document_ptr_kinds_def) + have node_ptr_kinds_eq_h2: "node_ptr_kinds h3 = node_ptr_kinds h2" + using object_ptr_kinds_eq_h2 + by(auto simp add: node_ptr_kinds_def) + + have object_ptr_kinds_eq_h3: "object_ptr_kinds h' = object_ptr_kinds h3" + apply(rule writes_small_big[where P="\h h'. object_ptr_kinds h' = object_ptr_kinds h", + OF set_disconnected_nodes_writes h']) + using set_disconnected_nodes_pointers_preserved + by (auto simp add: reflp_def transp_def) + then have document_ptr_kinds_eq_h3: "document_ptr_kinds h' = document_ptr_kinds h3" + by (auto simp add: document_ptr_kinds_def) + have node_ptr_kinds_eq_h3: "node_ptr_kinds h' = node_ptr_kinds h3" + using object_ptr_kinds_eq_h3 + by(auto simp add: node_ptr_kinds_def) + + + have "document_ptr |\| document_ptr_kinds h" + using disc_nodes_h3 document_ptr_kinds_eq_h object_ptr_kinds_eq_h2 + get_disconnected_nodes_ptr_in_heap \type_wf h\ document_ptr_kinds_def + by (metis is_OK_returns_result_I) + + have children_eq_h: "\(ptr'::(_) object_ptr) children. ptr' \ cast new_character_data_ptr + \ h \ get_child_nodes ptr' \\<^sub>r children = h2 \ get_child_nodes ptr' \\<^sub>r children" + using get_child_nodes_reads h2 get_child_nodes_new_character_data[rotated, OF new_character_data_ptr h2] + apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1] + by blast+ + then have children_eq2_h: "\ptr'. ptr' \ cast new_character_data_ptr + \ |h \ get_child_nodes ptr'|\<^sub>r = |h2 \ get_child_nodes ptr'|\<^sub>r" + using select_result_eq by force + + have "h2 \ get_child_nodes (cast new_character_data_ptr) \\<^sub>r []" + using new_character_data_ptr h2 new_character_data_ptr_in_heap[OF h2 new_character_data_ptr] + new_character_data_is_character_data_ptr[OF new_character_data_ptr] + new_character_data_no_child_nodes + by blast + have disconnected_nodes_eq_h: + "\doc_ptr disc_nodes. h \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes + = h2 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes" + using get_disconnected_nodes_reads h2 + get_disconnected_nodes_new_character_data[OF new_character_data_ptr h2] + apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1] + by blast+ + then have disconnected_nodes_eq2_h: + "\doc_ptr. |h \ get_disconnected_nodes doc_ptr|\<^sub>r = |h2 \ get_disconnected_nodes doc_ptr|\<^sub>r" + using select_result_eq by force + + have children_eq_h2: + "\ptr' children. h2 \ get_child_nodes ptr' \\<^sub>r children = h3 \ get_child_nodes ptr' \\<^sub>r children" + using get_child_nodes_reads set_val_writes h3 + apply(rule reads_writes_preserved) + by(auto simp add: set_val_get_child_nodes) + then have children_eq2_h2: + "\ptr'. |h2 \ get_child_nodes ptr'|\<^sub>r = |h3 \ get_child_nodes ptr'|\<^sub>r" + using select_result_eq by force + have disconnected_nodes_eq_h2: + "\doc_ptr disc_nodes. h2 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes + = h3 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes" + using get_disconnected_nodes_reads set_val_writes h3 + apply(rule reads_writes_preserved) + by(auto simp add: set_val_get_disconnected_nodes) + then have disconnected_nodes_eq2_h2: + "\doc_ptr. |h2 \ get_disconnected_nodes doc_ptr|\<^sub>r = |h3 \ get_disconnected_nodes doc_ptr|\<^sub>r" + using select_result_eq by force + + have "type_wf h2" + using \type_wf h\ new_character_data_types_preserved h2 by blast + then have "type_wf h3" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_val_writes h3] + using set_val_types_preserved + by(auto simp add: reflp_def transp_def) + then have "type_wf h'" + using writes_small_big[where P="\h h'. type_wf h \ type_wf h'", OF set_disconnected_nodes_writes h'] + using set_disconnected_nodes_types_preserved + by(auto simp add: reflp_def transp_def) + + have children_eq_h3: + "\ptr' children. h3 \ get_child_nodes ptr' \\<^sub>r children = h' \ get_child_nodes ptr' \\<^sub>r children" + using get_child_nodes_reads set_disconnected_nodes_writes h' + apply(rule reads_writes_preserved) + by(auto simp add: set_disconnected_nodes_get_child_nodes) + then have children_eq2_h3: + " \ptr'. |h3 \ get_child_nodes ptr'|\<^sub>r = |h' \ get_child_nodes ptr'|\<^sub>r" + using select_result_eq by force + have disconnected_nodes_eq_h3: "\doc_ptr disc_nodes. document_ptr \ doc_ptr + \ h3 \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes + = h' \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes" + using get_disconnected_nodes_reads set_disconnected_nodes_writes h' + apply(rule reads_writes_preserved) + by(auto simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers) + then have disconnected_nodes_eq2_h3: "\doc_ptr. document_ptr \ doc_ptr + \ |h3 \ get_disconnected_nodes doc_ptr|\<^sub>r = |h' \ get_disconnected_nodes doc_ptr|\<^sub>r" + using select_result_eq by force + + have disc_nodes_document_ptr_h2: "h2 \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes_h3" + using disconnected_nodes_eq_h2 disc_nodes_h3 by auto + then have disc_nodes_document_ptr_h: "h \ get_disconnected_nodes document_ptr \\<^sub>r disc_nodes_h3" + using disconnected_nodes_eq_h by auto + then have "cast new_character_data_ptr \ set disc_nodes_h3" + using \heap_is_wellformed h\ using \cast new_character_data_ptr \ set |h \ node_ptr_kinds_M|\<^sub>r\ + a_all_ptrs_in_heap_def heap_is_wellformed_def + by (meson NodeMonad.ptr_kinds_ptr_kinds_M fset_mp fset_of_list_elem ) + + have "acyclic (parent_child_rel h)" + using \heap_is_wellformed h\ + by (simp add: heap_is_wellformed_def acyclic_heap_def) + also have "parent_child_rel h = parent_child_rel h2" + proof(auto simp add: parent_child_rel_def)[1] + fix a x + assume 0: "a |\| object_ptr_kinds h" + and 1: "x \ set |h \ get_child_nodes a|\<^sub>r" + then show "a |\| object_ptr_kinds h2" + by (simp add: object_ptr_kinds_eq_h) + next + fix a x + assume 0: "a |\| object_ptr_kinds h" + and 1: "x \ set |h \ get_child_nodes a|\<^sub>r" + then show "x \ set |h2 \ get_child_nodes a|\<^sub>r" + by (metis ObjectMonad.ptr_kinds_ptr_kinds_M + \cast new_character_data_ptr \ set |h \ object_ptr_kinds_M|\<^sub>r\ children_eq2_h) + next + fix a x + assume 0: "a |\| object_ptr_kinds h2" + and 1: "x \ set |h2 \ get_child_nodes a|\<^sub>r" + then show "a |\| object_ptr_kinds h" + using object_ptr_kinds_eq_h \h2 \ get_child_nodes (cast new_character_data_ptr) \\<^sub>r []\ + by(auto) + next + fix a x + assume 0: "a |\| object_ptr_kinds h2" + and 1: "x \ set |h2 \ get_child_nodes a|\<^sub>r" + then show "x \ set |h \ get_child_nodes a|\<^sub>r" + by (metis (no_types, lifting) \h2 \ get_child_nodes (cast new_character_data_ptr) \\<^sub>r []\ + children_eq2_h empty_iff empty_set image_eqI select_result_I2) + qed + also have "\ = parent_child_rel h3" + by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h2 children_eq2_h2) + also have "\ = parent_child_rel h'" + by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h3 children_eq2_h3) + finally have "a_acyclic_heap h'" + by (simp add: acyclic_heap_def) + + have "a_all_ptrs_in_heap h" + using \heap_is_wellformed h\ by (simp add: heap_is_wellformed_def) + then have "a_all_ptrs_in_heap h2" + apply(auto simp add: a_all_ptrs_in_heap_def)[1] + using node_ptr_kinds_eq_h \cast new_character_data_ptr \ set |h \ node_ptr_kinds_M|\<^sub>r\ + \h2 \ get_child_nodes (cast new_character_data_ptr) \\<^sub>r []\ + apply (metis (no_types, hide_lams) children_eq_h fempty_iff fset_mp fset_of_list_simps(1) + funionCI select_result_I2) + by (simp add: disconnected_nodes_eq_h fset_rev_mp node_ptr_kinds_eq_h) + then have "a_all_ptrs_in_heap h3" + by(auto simp add: a_all_ptrs_in_heap_def object_ptr_kinds_eq_h2 node_ptr_kinds_def + children_eq_h2 disconnected_nodes_eq_h2) + then have "a_all_ptrs_in_heap h'" + apply(auto simp add: a_all_ptrs_in_heap_def object_ptr_kinds_eq_h3 node_ptr_kinds_def + children_eq_h3 )[1] + using disconnected_nodes_eq_h3 object_ptr_kinds_eq_h object_ptr_kinds_eq_h2 + by (metis (no_types, lifting) disc_nodes_h3 finsertCI fset.map_comp fset_mp fset_of_list_elem + funion_finsert_right h' local.set_disconnected_nodes_get_disconnected_nodes + node_ptr_kinds_def node_ptr_kinds_eq_h select_result_I2 set_ConsD) + + have "\p. p |\| object_ptr_kinds h \ cast new_character_data_ptr \ set |h \ get_child_nodes p|\<^sub>r" + using \heap_is_wellformed h\ \cast new_character_data_ptr \ set |h \ node_ptr_kinds_M|\<^sub>r\ + heap_is_wellformed_children_in_heap + by (meson NodeMonad.ptr_kinds_ptr_kinds_M a_all_ptrs_in_heap_def assms(3) assms(4) fset_mp + fset_of_list_elem get_child_nodes_ok known_ptrs_known_ptr returns_result_select_result) + then have "\p. p |\| object_ptr_kinds h2 \ cast new_character_data_ptr \ set |h2 \ get_child_nodes p|\<^sub>r" + using children_eq2_h + apply(auto simp add: object_ptr_kinds_eq_h)[1] + using \h2 \ get_child_nodes (cast new_character_data_ptr) \\<^sub>r []\ apply auto[1] + by (metis ObjectMonad.ptr_kinds_ptr_kinds_M \cast new_character_data_ptr \ set |h \ object_ptr_kinds_M|\<^sub>r\) + then have "\p. p |\| object_ptr_kinds h3 \ cast new_character_data_ptr \ set |h3 \ get_child_nodes p|\<^sub>r" + using object_ptr_kinds_eq_h2 children_eq2_h2 by auto + then have new_character_data_ptr_not_in_any_children: + "\p. p |\| object_ptr_kinds h' \ cast new_character_data_ptr \ set |h' \ get_child_nodes p|\<^sub>r" + using object_ptr_kinds_eq_h3 children_eq2_h3 by auto + + have "a_distinct_lists h" + using \heap_is_wellformed h\ + by (simp add: heap_is_wellformed_def) + then have "a_distinct_lists h2" + using \h2 \ get_child_nodes (cast new_character_data_ptr) \\<^sub>r []\ + apply(auto simp add: a_distinct_lists_def object_ptr_kinds_eq_h document_ptr_kinds_eq_h + disconnected_nodes_eq2_h intro!: distinct_concat_map_I)[1] + apply (metis distinct_sorted_list_of_set finite_fset sorted_list_of_set_insert) + apply(case_tac "x=cast new_character_data_ptr") + apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1] + apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1] + apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1] + apply (metis IntI assms(1) assms(3) assms(4) empty_iff local.get_child_nodes_ok + local.heap_is_wellformed_one_parent local.known_ptrs_known_ptr + returns_result_select_result) + apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1] + by (metis \local.a_distinct_lists h\ \type_wf h2\ disconnected_nodes_eq_h document_ptr_kinds_eq_h + local.distinct_lists_no_parent local.get_disconnected_nodes_ok returns_result_select_result) + then have "a_distinct_lists h3" + by(auto simp add: a_distinct_lists_def disconnected_nodes_eq2_h2 document_ptr_kinds_eq_h2 + children_eq2_h2 object_ptr_kinds_eq_h2)[1] + then have "a_distinct_lists h'" + proof(auto simp add: a_distinct_lists_def disconnected_nodes_eq2_h3 children_eq2_h3 + object_ptr_kinds_eq_h3 document_ptr_kinds_eq_h3 intro!: distinct_concat_map_I)[1] + fix x + assume "distinct (concat (map (\document_ptr. |h3 \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h3)))))" + and "x |\| document_ptr_kinds h3" + then show "distinct |h' \ get_disconnected_nodes x|\<^sub>r" + using document_ptr_kinds_eq_h3 disconnected_nodes_eq_h3 h' set_disconnected_nodes_get_disconnected_nodes + by (metis (no_types, lifting) \cast new_character_data_ptr \ set disc_nodes_h3\ + \a_distinct_lists h3\ \type_wf h'\ disc_nodes_h3 distinct.simps(2) + distinct_lists_disconnected_nodes get_disconnected_nodes_ok returns_result_eq + returns_result_select_result) + next + fix x y xa + assume "distinct (concat (map (\document_ptr. |h3 \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h3)))))" + and "x |\| document_ptr_kinds h3" + and "y |\| document_ptr_kinds h3" + and "x \ y" + and "xa \ set |h' \ get_disconnected_nodes x|\<^sub>r" + and "xa \ set |h' \ get_disconnected_nodes y|\<^sub>r" + moreover have "set |h3 \ get_disconnected_nodes x|\<^sub>r \ set |h3 \ get_disconnected_nodes y|\<^sub>r = {}" + using calculation by(auto dest: distinct_concat_map_E(1)) + ultimately show "False" + apply(-) + apply(cases "x = document_ptr") + apply (metis (no_types) NodeMonad.ptr_kinds_ptr_kinds_M \a_all_ptrs_in_heap h\ + \cast new_character_data_ptr \ set |h \ node_ptr_kinds_M|\<^sub>r\ + a_all_ptrs_in_heap_def assms(3) disc_nodes_h3 + disconnected_nodes_eq2_h disconnected_nodes_eq2_h2 + disconnected_nodes_eq2_h3 disjoint_iff_not_equal + document_ptr_kinds_eq_h document_ptr_kinds_eq_h2 fset_mp + fset_of_list_elem get_disconnected_nodes_ok h' + returns_result_select_result select_result_I2 set_ConsD + set_disconnected_nodes_get_disconnected_nodes) + apply(cases "y = document_ptr" ) + apply (metis (no_types) NodeMonad.ptr_kinds_ptr_kinds_M + \a_all_ptrs_in_heap h\ \cast new_character_data_ptr \ set |h \ node_ptr_kinds_M|\<^sub>r\ + a_all_ptrs_in_heap_def assms(3) disc_nodes_h3 disconnected_nodes_eq2_h + disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 disjoint_iff_not_equal + document_ptr_kinds_eq_h document_ptr_kinds_eq_h2 fset_mp fset_of_list_elem + get_disconnected_nodes_ok h' returns_result_select_result select_result_I2 set_ConsD + set_disconnected_nodes_get_disconnected_nodes) + using disconnected_nodes_eq2_h3 by auto + next + fix x xa xb + assume 2: "(\x\fset (object_ptr_kinds h3). set |h' \ get_child_nodes x|\<^sub>r) + \ (\x\fset (document_ptr_kinds h3). set |h3 \ get_disconnected_nodes x|\<^sub>r) = {}" + and 3: "xa |\| object_ptr_kinds h3" + and 4: "x \ set |h' \ get_child_nodes xa|\<^sub>r" + and 5: "xb |\| document_ptr_kinds h3" + and 6: "x \ set |h' \ get_disconnected_nodes xb|\<^sub>r" + show "False" + using disc_nodes_document_ptr_h disconnected_nodes_eq2_h3 + apply - + apply(cases "xb = document_ptr") + apply (metis (no_types, hide_lams) "3" "4" "6" + \\p. p |\| object_ptr_kinds h3 \ cast new_character_data_ptr \ set |h3 \ get_child_nodes p|\<^sub>r\ + \a_distinct_lists h3\ children_eq2_h3 disc_nodes_h3 distinct_lists_no_parent h' + select_result_I2 set_ConsD set_disconnected_nodes_get_disconnected_nodes) + by (metis "3" "4" "5" "6" \a_distinct_lists h3\ \type_wf h3\ children_eq2_h3 + distinct_lists_no_parent get_disconnected_nodes_ok returns_result_select_result) + qed + + have "a_owner_document_valid h" + using \heap_is_wellformed h\ by (simp add: heap_is_wellformed_def) + then have "a_owner_document_valid h'" + using disc_nodes_h3 \document_ptr |\| document_ptr_kinds h\ + apply(simp add: a_owner_document_valid_def) + apply(simp add: object_ptr_kinds_eq_h object_ptr_kinds_eq_h3 ) + apply(simp add: object_ptr_kinds_eq_h2) + apply(simp add: document_ptr_kinds_eq_h document_ptr_kinds_eq_h3 ) + apply(simp add: document_ptr_kinds_eq_h2) + apply(simp add: node_ptr_kinds_eq_h node_ptr_kinds_eq_h3 ) + apply(simp add: node_ptr_kinds_eq_h2 node_ptr_kinds_eq_h ) + apply(auto simp add: children_eq2_h2[symmetric] children_eq2_h3[symmetric] disconnected_nodes_eq2_h + disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3)[1] + apply (metis (no_types, lifting) document_ptr_kinds_eq_h h' list.set_intros(1) + local.set_disconnected_nodes_get_disconnected_nodes select_result_I2) + apply(simp add: object_ptr_kinds_eq_h) + by (metis (no_types, lifting) ObjectMonad.ptr_kinds_ptr_kinds_M + \cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr \ set |h \ object_ptr_kinds_M|\<^sub>r\ + children_eq2_h disconnected_nodes_eq2_h3 document_ptr_kinds_eq_h h' list.set_intros(2) + local.set_disconnected_nodes_get_disconnected_nodes select_result_I2) + + show "heap_is_wellformed h'" + using \a_acyclic_heap h'\ \a_all_ptrs_in_heap h'\ \a_distinct_lists h'\ \a_owner_document_valid h'\ + by(simp add: heap_is_wellformed_def) +qed +end + +interpretation i_create_character_data_wf?: l_create_character_data_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf + get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs + heap_is_wellformed parent_child_rel set_val set_val_locs set_disconnected_nodes + set_disconnected_nodes_locs create_character_data known_ptrs + using instances + by (auto simp add: l_create_character_data_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) + + +subsection \create\_document\ + +locale l_create_document_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = + l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes + get_disconnected_nodes_locs heap_is_wellformed parent_child_rel + + l_new_document_get_disconnected_nodes + get_disconnected_nodes get_disconnected_nodes_locs + + l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M + create_document + + l_new_document_get_child_nodes + type_wf known_ptr get_child_nodes get_child_nodes_locs + + l_new_document + type_wf + + l_known_ptrs + known_ptr known_ptrs + for known_ptr :: "(_::linorder) object_ptr \ bool" + and type_wf :: "(_) heap \ bool" + and get_child_nodes :: "(_) object_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_child_nodes_locs :: "(_) object_ptr \ ((_) heap \ (_) heap \ bool) set" + and get_disconnected_nodes :: "(_) document_ptr \ ((_) heap, exception, (_) node_ptr list) prog" + and get_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap \ (_) heap \ bool) set" + and heap_is_wellformed :: "(_) heap \ bool" + and parent_child_rel :: "(_) heap \ ((_) object_ptr \ (_) object_ptr) set" + and set_val :: "(_) character_data_ptr \ char list \ ((_) heap, exception, unit) prog" + and set_val_locs :: "(_) character_data_ptr \ ((_) heap, exception, unit) prog set" + and set_disconnected_nodes :: "(_) document_ptr \ (_) node_ptr list \ ((_) heap, exception, unit) prog" + and set_disconnected_nodes_locs :: "(_) document_ptr \ ((_) heap, exception, unit) prog set" + and create_document :: "((_) heap, exception, (_) document_ptr) prog" + and known_ptrs :: "(_) heap \ bool" +begin + +lemma create_document_preserves_wellformedness: + assumes "heap_is_wellformed h" + and "h \ create_document \\<^sub>h h'" + and "type_wf h" + and "known_ptrs h" + shows "heap_is_wellformed h'" +proof - + obtain new_document_ptr where + new_document_ptr: "h \ new_document \\<^sub>r new_document_ptr" and + h': "h \ new_document \\<^sub>h h'" + using assms(2) + apply(simp add: create_document_def) + using new_document_ok by blast + + have "new_document_ptr \ set |h \ document_ptr_kinds_M|\<^sub>r" + using new_document_ptr DocumentMonad.ptr_kinds_ptr_kinds_M + using new_document_ptr_not_in_heap h' by blast + then have "cast new_document_ptr \ set |h \ object_ptr_kinds_M|\<^sub>r" + by simp + + have "new_document_ptr |\| document_ptr_kinds h" + using new_document_ptr DocumentMonad.ptr_kinds_ptr_kinds_M + using new_document_ptr_not_in_heap h' by blast + then have "cast new_document_ptr |\| object_ptr_kinds h" + by simp + + have object_ptr_kinds_eq: "object_ptr_kinds h' = object_ptr_kinds h |\| {|cast new_document_ptr|}" + using new_document_new_ptr h' new_document_ptr by blast + then have node_ptr_kinds_eq: "node_ptr_kinds h' = node_ptr_kinds h" + apply(simp add: node_ptr_kinds_def) + by force + then have character_data_ptr_kinds_eq_h: "character_data_ptr_kinds h' = character_data_ptr_kinds h" + by(simp add: character_data_ptr_kinds_def) + have element_ptr_kinds_eq_h: "element_ptr_kinds h' = element_ptr_kinds h" + using object_ptr_kinds_eq + by(auto simp add: node_ptr_kinds_def element_ptr_kinds_def) + have document_ptr_kinds_eq_h: "document_ptr_kinds h' = document_ptr_kinds h |\| {|new_document_ptr|}" + using object_ptr_kinds_eq + apply(auto simp add: document_ptr_kinds_def)[1] + by (metis (no_types, lifting) document_ptr_kinds_commutes document_ptr_kinds_def finsertI1 fset.map_comp) + + + have children_eq: + "\(ptr'::(_) object_ptr) children. ptr' \ cast new_document_ptr + \ h \ get_child_nodes ptr' \\<^sub>r children = h' \ get_child_nodes ptr' \\<^sub>r children" + using get_child_nodes_reads h' get_child_nodes_new_document[rotated, OF new_document_ptr h'] + apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1] + by blast+ + then have children_eq2: "\ptr'. ptr' \ cast new_document_ptr + \ |h \ get_child_nodes ptr'|\<^sub>r = |h' \ get_child_nodes ptr'|\<^sub>r" + using select_result_eq by force + + + have "h' \ get_child_nodes (cast new_document_ptr) \\<^sub>r []" + using new_document_ptr h' new_document_ptr_in_heap[OF h' new_document_ptr] + new_document_is_document_ptr[OF new_document_ptr] new_document_no_child_nodes + by blast + have disconnected_nodes_eq_h: + "\doc_ptr disc_nodes. doc_ptr \ new_document_ptr + \ h \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes = h' \ get_disconnected_nodes doc_ptr \\<^sub>r disc_nodes" + using get_disconnected_nodes_reads h' get_disconnected_nodes_new_document_different_pointers new_document_ptr + apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1] + by (metis(full_types) \\thesis. (\new_document_ptr. + \h \ new_document \\<^sub>r new_document_ptr; h \ new_document \\<^sub>h h'\ \ thesis) \ thesis\ + local.get_disconnected_nodes_new_document_different_pointers new_document_ptr)+ + then have disconnected_nodes_eq2_h: "\doc_ptr. doc_ptr \ new_document_ptr + \ |h \ get_disconnected_nodes doc_ptr|\<^sub>r = |h' \ get_disconnected_nodes doc_ptr|\<^sub>r" + using select_result_eq by force + have "h' \ get_disconnected_nodes new_document_ptr \\<^sub>r []" + using h' local.new_document_no_disconnected_nodes new_document_ptr by blast + + have "type_wf h'" + using \type_wf h\ new_document_types_preserved h' by blast + + have "acyclic (parent_child_rel h)" + using \heap_is_wellformed h\ + by (simp add: heap_is_wellformed_def acyclic_heap_def) + also have "parent_child_rel h = parent_child_rel h'" + proof(auto simp add: parent_child_rel_def)[1] + fix a x + assume 0: "a |\| object_ptr_kinds h" + and 1: "x \ set |h \ get_child_nodes a|\<^sub>r" + then show "a |\| object_ptr_kinds h'" + by (simp add: object_ptr_kinds_eq) + next + fix a x + assume 0: "a |\| object_ptr_kinds h" + and 1: "x \ set |h \ get_child_nodes a|\<^sub>r" + then show "x \ set |h' \ get_child_nodes a|\<^sub>r" + by (metis ObjectMonad.ptr_kinds_ptr_kinds_M + \cast new_document_ptr \ set |h \ object_ptr_kinds_M|\<^sub>r\ children_eq2) + next + fix a x + assume 0: "a |\| object_ptr_kinds h'" + and 1: "x \ set |h' \ get_child_nodes a|\<^sub>r" + then show "a |\| object_ptr_kinds h" + using object_ptr_kinds_eq \h' \ get_child_nodes (cast new_document_ptr) \\<^sub>r []\ + by(auto) + next + fix a x + assume 0: "a |\| object_ptr_kinds h'" + and 1: "x \ set |h' \ get_child_nodes a|\<^sub>r" + then show "x \ set |h \ get_child_nodes a|\<^sub>r" + by (metis (no_types, lifting) \h' \ get_child_nodes (cast new_document_ptr) \\<^sub>r []\ + children_eq2 empty_iff empty_set image_eqI select_result_I2) + qed + finally have "a_acyclic_heap h'" + by (simp add: acyclic_heap_def) + + have "a_all_ptrs_in_heap h" + using \heap_is_wellformed h\ by (simp add: heap_is_wellformed_def) + then have "a_all_ptrs_in_heap h'" + apply(auto simp add: a_all_ptrs_in_heap_def)[1] + apply (metis ObjectMonad.ptr_kinds_ptr_kinds_M + \cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr \ set |h \ object_ptr_kinds_M|\<^sub>r\ + \parent_child_rel h = parent_child_rel h'\ assms(1) children_eq fset_of_list_elem + local.heap_is_wellformed_children_in_heap local.parent_child_rel_child + local.parent_child_rel_parent_in_heap node_ptr_kinds_eq) + by (metis (no_types, lifting) ObjectMonad.ptr_kinds_ptr_kinds_M + \cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr \ set |h \ object_ptr_kinds_M|\<^sub>r\ + \h' \ get_child_nodes (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr) \\<^sub>r []\ + \parent_child_rel h = parent_child_rel h'\ assms(1) disconnected_nodes_eq_h + fset_of_list_elem h' local.heap_is_wellformed_disc_nodes_in_heap + local.new_document_no_disconnected_nodes local.parent_child_rel_child + local.parent_child_rel_parent_in_heap new_document_ptr node_ptr_kinds_eq + select_result_I2) + + have "a_distinct_lists h" + using \heap_is_wellformed h\ + by (simp add: heap_is_wellformed_def) + then have "a_distinct_lists h'" + using \h' \ get_disconnected_nodes new_document_ptr \\<^sub>r []\ + \h' \ get_child_nodes (cast new_document_ptr) \\<^sub>r []\ + + apply(auto simp add: children_eq2[symmetric] a_distinct_lists_def insort_split object_ptr_kinds_eq + document_ptr_kinds_eq_h disconnected_nodes_eq2_h intro!: distinct_concat_map_I)[1] + apply (metis distinct_sorted_list_of_set finite_fset sorted_list_of_set_insert) + + apply(auto simp add: dest: distinct_concat_map_E)[1] + apply(auto simp add: dest: distinct_concat_map_E)[1] + using \new_document_ptr |\| document_ptr_kinds h\ + apply(auto simp add: distinct_insort dest: distinct_concat_map_E)[1] + using disconnected_nodes_eq_h + apply (metis assms(1) assms(3) disconnected_nodes_eq2_h local.get_disconnected_nodes_ok + local.heap_is_wellformed_disconnected_nodes_distinct + returns_result_select_result) + proof - + fix x :: "(_) document_ptr" and y :: "(_) document_ptr" and xa :: "(_) node_ptr" + assume a1: "x \ y" + assume a2: "x |\| document_ptr_kinds h" + assume a3: "x \ new_document_ptr" + assume a4: "y |\| document_ptr_kinds h" + assume a5: "y \ new_document_ptr" + assume a6: "distinct (concat (map (\document_ptr. |h \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h)))))" + assume a7: "xa \ set |h' \ get_disconnected_nodes x|\<^sub>r" + assume a8: "xa \ set |h' \ get_disconnected_nodes y|\<^sub>r" + have f9: "xa \ set |h \ get_disconnected_nodes x|\<^sub>r" + using a7 a3 disconnected_nodes_eq2_h by presburger + have f10: "xa \ set |h \ get_disconnected_nodes y|\<^sub>r" + using a8 a5 disconnected_nodes_eq2_h by presburger + have f11: "y \ set (sorted_list_of_set (fset (document_ptr_kinds h)))" + using a4 by simp + have "x \ set (sorted_list_of_set (fset (document_ptr_kinds h)))" + using a2 by simp + then show False + using f11 f10 f9 a6 a1 by (meson disjoint_iff_not_equal distinct_concat_map_E(1)) + next + fix x xa xb + assume 0: "h' \ get_disconnected_nodes new_document_ptr \\<^sub>r []" + and 1: "h' \ get_child_nodes (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr) \\<^sub>r []" + and 2: "distinct (concat (map (\ptr. |h \ get_child_nodes ptr|\<^sub>r) + (sorted_list_of_set (fset (object_ptr_kinds h)))))" + and 3: "distinct (concat (map (\document_ptr. |h \ get_disconnected_nodes document_ptr|\<^sub>r) + (sorted_list_of_set (fset (document_ptr_kinds h)))))" + and 4: "(\x\fset (object_ptr_kinds h). set |h \ get_child_nodes x|\<^sub>r) + \ (\x\fset (document_ptr_kinds h). set |h \ get_disconnected_nodes x|\<^sub>r) = {}" + and 5: "x \ set |h \ get_child_nodes xa|\<^sub>r" + and 6: "x \ set |h' \ get_disconnected_nodes xb|\<^sub>r" + and 7: "xa |\| object_ptr_kinds h" + and 8: "xa \ cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr" + and 9: "xb |\| document_ptr_kinds h" + and 10: "xb \ new_document_ptr" + then show "False" + + by (metis \local.a_distinct_lists h\ assms(3) disconnected_nodes_eq2_h + local.distinct_lists_no_parent local.get_disconnected_nodes_ok + returns_result_select_result) + qed + + have "a_owner_document_valid h" + using \heap_is_wellformed h\ by (simp add: heap_is_wellformed_def) + then have "a_owner_document_valid h'" + apply(auto simp add: a_owner_document_valid_def)[1] + by (metis \cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr |\| object_ptr_kinds h\ + \new_document_ptr |\| document_ptr_kinds h\ assms(3) assms(4) children_eq + children_eq2 disconnected_nodes_eq2_h disconnected_nodes_eq_h + is_OK_returns_result_E is_OK_returns_result_I local.get_child_nodes_ok + local.get_child_nodes_ptr_in_heap local.get_disconnected_nodes_ok + local.get_disconnected_nodes_ptr_in_heap local.known_ptrs_known_ptr node_ptr_kinds_eq) + + show "heap_is_wellformed h'" + using \a_acyclic_heap h'\ \a_all_ptrs_in_heap h'\ \a_distinct_lists h'\ \a_owner_document_valid h'\ + by(simp add: heap_is_wellformed_def) +qed +end + +interpretation i_create_document_wf?: l_create_document_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes + get_child_nodes_locs get_disconnected_nodes + get_disconnected_nodes_locs heap_is_wellformed parent_child_rel + set_val set_val_locs set_disconnected_nodes + set_disconnected_nodes_locs create_document known_ptrs + using instances + by (auto simp add: l_create_document_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def) +declare l_create_document_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances] + + +end diff --git a/Core_DOM/Core_DOM_Tests.thy b/Core_DOM/Core_DOM_Tests.thy new file mode 100644 index 0000000..3819600 --- /dev/null +++ b/Core_DOM/Core_DOM_Tests.thy @@ -0,0 +1,40 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Core DOM Test Cases\ +text\This theory aggregates the individual test cases for the core DOM.\ + +theory Core_DOM_Tests + imports + "tests/Document_adoptNode" + "tests/Document_getElementById" + "tests/Node_insertBefore" + "tests/Node_removeChild" +begin +end diff --git a/Core_DOM/ROOT b/Core_DOM/ROOT new file mode 100644 index 0000000..b1ad135 --- /dev/null +++ b/Core_DOM/ROOT @@ -0,0 +1,10 @@ +chapter AFP + +session "Core_DOM" (AFP) = "HOL-Library" + + options [timeout = 4800, document = pdf, document_variants="document:outline=/proof,/ML",document_output=output] + theories + Core_DOM + Core_DOM_Tests + document_files + "root.tex" + "root.bib" diff --git a/Core_DOM/classes/BaseClass.thy b/Core_DOM/classes/BaseClass.thy new file mode 100644 index 0000000..02b4fa6 --- /dev/null +++ b/Core_DOM/classes/BaseClass.thy @@ -0,0 +1,74 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\The Class Infrastructure\ +text\In this theory, we introduce the basic infrastructure for our encoding +of classes.\ +theory BaseClass + imports + "~~/src/HOL/Library/Finite_Map" + "../pointers/Ref" + "../Core_DOM_Basic_Datatypes" +begin + +named_theorems instances + +consts get :: 'a +consts put :: 'a +consts delete :: 'a + +text \Overall, the definition of the class types follows closely the one of the pointer + types. Instead of datatypes, we use records for our classes. This allows us to, first, + make use of record inheritance, which is, in addition to the type synonyms of + previous class types, the second place where the inheritance relationship of + our types manifest. Second, we get a convenient notation to define classes, in + addition to automatically generated getter and setter functions.\ + +text \Along with our class types, we also develop our heap type, which is a finite + map at its core. It is important to note that while the map stores a mapping + from @{term "object_ptr"} to @{term "Object"}, we restrict the type variables + of the record extension slot of @{term "Object"} in such a way that allows + down-casting, but requires a bit of taking-apart and re-assembling of our records + before they are stored in the heap.\ + +text \Throughout the theory files, we will use underscore case to reference pointer + types, and camel case for class types.\ + +text \Every class type contains at least one attribute; nothing. This is used for + two purposes: first, the record package does not allow records without any + attributes. Second, we will use the getter of nothing later to check whether a + class of the correct type could be retrieved, for which we will be able to use + our infrastructure regarding the behaviour of getters across different heaps.\ + + +locale l_type_wf = fixes type_wf :: "'heap \ bool" + +locale l_known_ptr = fixes known_ptr :: "'ptr \ bool" + +end diff --git a/Core_DOM/classes/CharacterDataClass.thy b/Core_DOM/classes/CharacterDataClass.thy new file mode 100644 index 0000000..7af6a67 --- /dev/null +++ b/Core_DOM/classes/CharacterDataClass.thy @@ -0,0 +1,347 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\CharacterData\ +text\In this theory, we introduce the types for the CharacterData class.\ +theory CharacterDataClass + imports + ElementClass +begin + +subsubsection\CharacterData\ + +text\The type @{type "DOMString"} is a type synonym for @{type "string"}, defined + \autoref{sec:Core_DOM_Basic_Datatypes}.\ + +record RCharacterData = RNode + + nothing :: unit + val :: DOMString +register_default_tvars "'CharacterData RCharacterData_ext" +type_synonym 'CharacterData CharacterData = "'CharacterData option RCharacterData_scheme" +register_default_tvars "'CharacterData CharacterData" +type_synonym ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Node, + 'Element, 'CharacterData) Node + = "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, + 'CharacterData option RCharacterData_ext + 'Node, 'Element) Node" +register_default_tvars "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Node, + 'Element, 'CharacterData) Node" +type_synonym ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Object, 'Node, + 'Element, 'CharacterData) Object + = "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Object, + 'CharacterData option RCharacterData_ext + 'Node, + 'Element) Object" +register_default_tvars "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Object, + 'Node, 'Element, 'CharacterData) Object" + +type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, + 'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData) heap + = "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, 'shadow_root_ptr, + 'Object, 'CharacterData option RCharacterData_ext + 'Node, 'Element) heap" +register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, + 'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData) heap" + + +definition character_data_ptr_kinds :: "(_) heap \ (_) character_data_ptr fset" + where + "character_data_ptr_kinds heap = the |`| (cast |`| (ffilter is_character_data_ptr_kind + (node_ptr_kinds heap)))" + +lemma character_data_ptr_kinds_simp [simp]: + "character_data_ptr_kinds (Heap (fmupd (cast character_data_ptr) character_data (the_heap h))) + = {|character_data_ptr|} |\| character_data_ptr_kinds h" + apply(auto simp add: character_data_ptr_kinds_def)[1] + by force + +definition character_data_ptrs :: "(_) heap \ _ character_data_ptr fset" + where + "character_data_ptrs heap = ffilter is_character_data_ptr (character_data_ptr_kinds heap)" + +abbreviation "character_data_ptr_exts heap \ character_data_ptr_kinds heap - character_data_ptrs heap" + +definition cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a :: "(_) Node \ (_) CharacterData option" + where + "cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a node = (case RNode.more node of + Inr (Inl character_data) \ Some (RNode.extend (RNode.truncate node) character_data) + | _ \ None)" +adhoc_overloading cast cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a + +abbreviation cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a :: "(_) Object \ (_) CharacterData option" + where + "cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a obj \ (case cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e obj of Some node \ cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a node + | None \ None)" +adhoc_overloading cast cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a + +definition cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e :: "(_) CharacterData \ (_) Node" + where + "cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e character_data = RNode.extend (RNode.truncate character_data) + (Inr (Inl (RNode.more character_data)))" +adhoc_overloading cast cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e + +abbreviation cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t :: "(_) CharacterData \ (_) Object" + where + "cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr \ cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t (cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr)" +adhoc_overloading cast cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + +consts is_character_data_kind :: 'a +definition is_character_data_kind\<^sub>N\<^sub>o\<^sub>d\<^sub>e :: "(_) Node \ bool" + where + "is_character_data_kind\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr \ cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr \ None" + +adhoc_overloading is_character_data_kind is_character_data_kind\<^sub>N\<^sub>o\<^sub>d\<^sub>e +lemmas is_character_data_kind_def = is_character_data_kind\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + +abbreviation is_character_data_kind\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t :: "(_) Object \ bool" + where + "is_character_data_kind\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr \ cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr \ None" +adhoc_overloading is_character_data_kind is_character_data_kind\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + +lemma character_data_ptr_kinds_commutes [simp]: + "cast character_data_ptr |\| node_ptr_kinds h + \ character_data_ptr |\| character_data_ptr_kinds h" + apply(auto simp add: character_data_ptr_kinds_def)[1] + by (metis character_data_ptr_casts_commute2 comp_eq_dest_lhs ffmember_filter fimage_eqI + is_character_data_ptr_kind_none + option.distinct(1) option.sel) + +definition get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a :: "(_) character_data_ptr \ (_) heap \ (_) CharacterData option" + where + "get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr h = Option.bind (get\<^sub>N\<^sub>o\<^sub>d\<^sub>e (cast character_data_ptr) h) cast" +adhoc_overloading get get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a + +locale l_type_wf_def\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a +begin +definition a_type_wf :: "(_) heap \ bool" + where + "a_type_wf h = (ElementClass.type_wf h + \ (\character_data_ptr. character_data_ptr |\| character_data_ptr_kinds h + \ get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr h \ None))" +end +global_interpretation l_type_wf_def\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a defines type_wf = a_type_wf . +lemmas type_wf_defs = a_type_wf_def + +locale l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a = l_type_wf type_wf for type_wf :: "((_) heap \ bool)" + + assumes type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a: "type_wf h \ CharacterDataClass.type_wf h" + +sublocale l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a \ l_type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + apply(unfold_locales) + using ElementClass.a_type_wf_def + by (meson CharacterDataClass.a_type_wf_def l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_axioms l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def) + +locale l_get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_lemmas = l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a +begin +sublocale l_get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas by unfold_locales + +lemma get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_type_wf: + assumes "type_wf h" + shows "character_data_ptr |\| character_data_ptr_kinds h + \ get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr h \ None" + using l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_axioms assms + apply(simp add: type_wf_defs get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def) + by (metis NodeClass.get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_type_wf bind_eq_None_conv character_data_ptr_kinds_commutes + l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def local.l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e_axioms option.distinct(1)) +end + +global_interpretation l_get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_lemmas type_wf + by unfold_locales + +definition put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a :: "(_) character_data_ptr \ (_) CharacterData \ (_) heap \ (_) heap" + where + "put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr character_data = put\<^sub>N\<^sub>o\<^sub>d\<^sub>e (cast character_data_ptr) + (cast character_data)" +adhoc_overloading put put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a + +lemma put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ptr_in_heap: + assumes "put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr character_data h = h'" + shows "character_data_ptr |\| character_data_ptr_kinds h'" + using assms put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_ptr_in_heap + unfolding put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def character_data_ptr_kinds_def + by (metis character_data_ptr_kinds_commutes character_data_ptr_kinds_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_ptr_in_heap) + +lemma put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_put_ptrs: + assumes "put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr character_data h = h'" + shows "object_ptr_kinds h' = object_ptr_kinds h |\| {|cast character_data_ptr|}" + using assms + by (simp add: put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_put_ptrs) + + +lemma cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inject [simp]: "cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e x = cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e y \ x = y" + apply(simp add: cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def RObject.extend_def RNode.extend_def) + by (metis (full_types) RNode.surjective old.unit.exhaust) + +lemma cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_none [simp]: + "cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a node = None \ \ (\character_data. cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e character_data = node)" + apply(auto simp add: cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def RObject.extend_def RNode.extend_def + split: sum.splits)[1] + by (metis (full_types) RNode.select_convs(2) RNode.surjective old.unit.exhaust) + +lemma cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_some [simp]: + "cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a node = Some character_data \ cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e character_data = node" + by(auto simp add: cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def RObject.extend_def RNode.extend_def + split: sum.splits) + +lemma cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_inv [simp]: + "cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a (cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e character_data) = Some character_data" + by simp + +lemma cast_element_not_character_data [simp]: + "(cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e element \ cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e character_data)" + "(cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e character_data \ cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e element)" + by(auto simp add: cast\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def RNode.extend_def) + +lemma get_CharacterData_simp1 [simp]: + "get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr (put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr character_data h) + = Some character_data" + by(auto simp add: get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def) +lemma get_CharacterData_simp2 [simp]: + "character_data_ptr \ character_data_ptr' \ get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr + (put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr' character_data h) = get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr h" + by(auto simp add: get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def) + +lemma get_CharacterData_simp3 [simp]: + "get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr (put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr f h) = get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr h" + by(auto simp add: get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def) +lemma get_CharacterData_simp4 [simp]: + "get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a element_ptr (put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t character_data_ptr f h) = get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a element_ptr h" + by(auto simp add: get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + +lemma new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a [simp]: + assumes "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')" + shows "get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr h = get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr h'" + using assms + by(auto simp add: new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def) + + + +abbreviation "create_character_data_obj val_arg + \ \ RObject.nothing = (), RNode.nothing = (), RCharacterData.nothing = (), val = val_arg, \ = None \" + +definition new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a :: "(_) heap \ ((_) character_data_ptr \ (_) heap)" + where + "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = + (let new_character_data_ptr = character_data_ptr.Ref (Suc (fMax (character_data_ptr.the_ref + |`| (character_data_ptrs h)))) in + (new_character_data_ptr, put new_character_data_ptr (create_character_data_obj '''') h))" + +lemma new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ptr_in_heap: + assumes "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = (new_character_data_ptr, h')" + shows "new_character_data_ptr |\| character_data_ptr_kinds h'" + using assms + unfolding new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def Let_def + using put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ptr_in_heap by blast + +lemma new_character_data_ptr_new: + "character_data_ptr.Ref (Suc (fMax (finsert 0 (character_data_ptr.the_ref |`| character_data_ptrs h)))) + |\| character_data_ptrs h" + by (metis Suc_n_not_le_n character_data_ptr.sel(1) fMax_ge fimage_finsert finsertI1 finsertI2 set_finsert) + +lemma new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ptr_not_in_heap: + assumes "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = (new_character_data_ptr, h')" + shows "new_character_data_ptr |\| character_data_ptr_kinds h" + using assms + unfolding new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def + by (metis Pair_inject character_data_ptrs_def fMax_finsert fempty_iff ffmember_filter fimage_is_fempty is_character_data_ptr_ref max_0L new_character_data_ptr_new) + +lemma new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_new_ptr: + assumes "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = (new_character_data_ptr, h')" + shows "object_ptr_kinds h' = object_ptr_kinds h |\| {|cast new_character_data_ptr|}" + using assms + by (metis Pair_inject new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_put_ptrs) + +lemma new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_is_character_data_ptr: + assumes "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = (new_character_data_ptr, h')" + shows "is_character_data_ptr new_character_data_ptr" + using assms + by(auto simp add: new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def Let_def) + +lemma new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t [simp]: + assumes "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = (new_character_data_ptr, h')" + assumes "ptr \ cast new_character_data_ptr" + shows "get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h = get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h'" + using assms + by(auto simp add: new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def Let_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def) + +lemma new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_get\<^sub>N\<^sub>o\<^sub>d\<^sub>e [simp]: + assumes "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = (new_character_data_ptr, h')" + assumes "ptr \ cast new_character_data_ptr" + shows "get\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr h = get\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr h'" + using assms + by(auto simp add: new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def Let_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def) + +lemma new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t [simp]: + assumes "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = (new_character_data_ptr, h')" + shows "get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h = get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h'" + using assms + by(auto simp add: new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def Let_def) + +lemma new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a [simp]: + assumes "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = (new_character_data_ptr, h')" + assumes "ptr \ new_character_data_ptr" + shows "get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr h = get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr h'" + using assms + by(auto simp add: new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def Let_def) + + +locale l_known_ptr\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a +begin +definition a_known_ptr :: "(_) object_ptr \ bool" + where + "a_known_ptr ptr = (known_ptr ptr \ is_character_data_ptr ptr)" + +lemma known_ptr_not_character_data_ptr: + "\is_character_data_ptr ptr \ a_known_ptr ptr \ known_ptr ptr" + by(simp add: a_known_ptr_def) +end +global_interpretation l_known_ptr\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a defines known_ptr = a_known_ptr . +lemmas known_ptr_defs = a_known_ptr_def + + +locale l_known_ptrs\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr \ bool" +begin +definition a_known_ptrs :: "(_) heap \ bool" + where + "a_known_ptrs h = (\ptr. ptr |\| object_ptr_kinds h \ known_ptr ptr)" + +lemma known_ptrs_known_ptr: "a_known_ptrs h \ ptr |\| object_ptr_kinds h \ known_ptr ptr" + by(simp add: a_known_ptrs_def) + +lemma known_ptrs_preserved: + "object_ptr_kinds h = object_ptr_kinds h' \ a_known_ptrs h = a_known_ptrs h'" + by(auto simp add: a_known_ptrs_def) +lemma known_ptrs_subset: + "object_ptr_kinds h' |\| object_ptr_kinds h \ a_known_ptrs h \ a_known_ptrs h'" + by(auto simp add: a_known_ptrs_def) +end +global_interpretation l_known_ptrs\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a known_ptr defines known_ptrs = a_known_ptrs . +lemmas known_ptrs_defs = a_known_ptrs_def + +lemma known_ptrs_is_l_known_ptrs: "l_known_ptrs known_ptr known_ptrs" + using known_ptrs_known_ptr known_ptrs_preserved l_known_ptrs_def known_ptrs_subset + by blast + +end diff --git a/Core_DOM/classes/DocumentClass.thy b/Core_DOM/classes/DocumentClass.thy new file mode 100644 index 0000000..594e014 --- /dev/null +++ b/Core_DOM/classes/DocumentClass.thy @@ -0,0 +1,336 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Document\ +text\In this theory, we introduce the types for the Document class.\ +theory DocumentClass + imports + CharacterDataClass +begin + +text\The type @{type "doctype"} is a type synonym for @{type "string"}, defined + in \autoref{sec:Core_DOM_Basic_Datatypes}.\ + +record ('node_ptr, 'element_ptr, 'character_data_ptr) RDocument = RObject + + nothing :: unit + doctype :: doctype + document_element :: "(_) element_ptr option" + disconnected_nodes :: "('node_ptr, 'element_ptr, 'character_data_ptr) node_ptr list" +type_synonym + ('node_ptr, 'element_ptr, 'character_data_ptr, 'Document) Document + = "('node_ptr, 'element_ptr, 'character_data_ptr, 'Document option) RDocument_scheme" +register_default_tvars + "('node_ptr, 'element_ptr, 'character_data_ptr, 'Document) Document" +type_synonym + ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Object, 'Node, + 'Element, 'CharacterData, 'Document) Object + = "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, + ('node_ptr, 'element_ptr, 'character_data_ptr, 'Document option) + RDocument_ext + 'Object, 'Node, 'Element, 'CharacterData) Object" +register_default_tvars "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, + 'Object, 'Node, 'Element, 'CharacterData, 'Document) Object" + +type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, + 'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData, 'Document) heap + = "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, + 'shadow_root_ptr, + ('node_ptr, 'element_ptr, 'character_data_ptr, 'Document option) RDocument_ext + 'Object, 'Node, + 'Element, 'CharacterData) heap" +register_default_tvars + "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, + 'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData, 'Document) heap" + + +definition document_ptr_kinds :: "(_) heap \ (_) document_ptr fset" + where + "document_ptr_kinds heap = the |`| (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |`| + (ffilter is_document_ptr_kind (object_ptr_kinds heap)))" + +definition document_ptrs :: "(_) heap \ (_) document_ptr fset" + where + "document_ptrs heap = ffilter is_document_ptr (document_ptr_kinds heap)" + +definition cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t :: "(_) Object \ (_) Document option" + where + "cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t obj = (case RObject.more obj of + Inr (Inl document) \ Some (RObject.extend (RObject.truncate obj) document) + | _ \ None)" +adhoc_overloading cast cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t + +definition cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t:: "(_) Document \ (_) Object" + where + "cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t document = (RObject.extend (RObject.truncate document) + (Inr (Inl (RObject.more document))))" +adhoc_overloading cast cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + +definition is_document_kind :: "(_) Object \ bool" + where + "is_document_kind ptr \ cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr \ None" + +lemma document_ptr_kinds_simp [simp]: + "document_ptr_kinds (Heap (fmupd (cast document_ptr) document (the_heap h))) + = {|document_ptr|} |\| document_ptr_kinds h" + apply(auto simp add: document_ptr_kinds_def)[1] + by force + +lemma document_ptr_kinds_commutes [simp]: + "cast document_ptr |\| object_ptr_kinds h \ document_ptr |\| document_ptr_kinds h" + apply(auto simp add: object_ptr_kinds_def document_ptr_kinds_def)[1] + by (metis (no_types, lifting) document_ptr_casts_commute2 document_ptr_document_ptr_cast + ffmember_filter fimage_eqI fset.map_comp option.sel) + +definition get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t :: "(_) document_ptr \ (_) heap \ (_) Document option" + where + "get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr h = Option.bind (get (cast document_ptr) h) cast" +adhoc_overloading get get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t + +locale l_type_wf_def\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t +begin +definition a_type_wf :: "(_) heap \ bool" + where + "a_type_wf h = (CharacterDataClass.type_wf h \ + (\document_ptr. document_ptr |\| document_ptr_kinds h \ get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr h \ None))" +end +global_interpretation l_type_wf_def\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t defines type_wf = a_type_wf . +lemmas type_wf_defs = a_type_wf_def + +locale l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t = l_type_wf type_wf for type_wf :: "((_) heap \ bool)" + + assumes type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: "type_wf h \ DocumentClass.type_wf h" + +sublocale l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t \ l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a + apply(unfold_locales) + by (metis (full_types) type_wf_defs l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_axioms l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + +locale l_get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas = l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t +begin +sublocale l_get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_lemmas by unfold_locales +lemma get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_type_wf: + assumes "type_wf h" + shows "document_ptr |\| document_ptr_kinds h \ get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr h \ None" + using l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_axioms assms + apply(simp add: type_wf_defs get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + by (metis bind_eq_None_conv document_ptr_kinds_commutes local.get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf + option.distinct(1)) +end + +global_interpretation l_get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf by unfold_locales + +definition put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t :: "(_) document_ptr \ (_) Document \ (_) heap \ (_) heap" + where + "put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr document = put (cast document_ptr) (cast document)" +adhoc_overloading put put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t + +lemma put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: + assumes "put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr document h = h'" + shows "document_ptr |\| document_ptr_kinds h'" + using assms + unfolding put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + by (metis document_ptr_kinds_commutes put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ptr_in_heap) + +lemma put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_put_ptrs: + assumes "put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr document h = h'" + shows "object_ptr_kinds h' = object_ptr_kinds h |\| {|cast document_ptr|}" + using assms + by (simp add: put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_put_ptrs) + + +lemma cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_inject [simp]: "cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x = cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t y \ x = y" + apply(simp add: cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def RObject.extend_def) + by (metis (full_types) RObject.surjective old.unit.exhaust) + +lemma cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_none [simp]: + "cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t obj = None \ \ (\document. cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t document = obj)" + apply(auto simp add: cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def RObject.extend_def + split: sum.splits)[1] + by (metis (full_types) RObject.select_convs(2) RObject.surjective old.unit.exhaust) + +lemma cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_some [simp]: + "cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t obj = Some document \ cast document = obj" + by(auto simp add: cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def RObject.extend_def + split: sum.splits) + +lemma cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_inv [simp]: "cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t (cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t document) = Some document" + by simp + +lemma cast_document_not_node [simp]: + "cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t document \ cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t node" + "cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t node \ cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t document" + by(auto simp add: cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def RObject.extend_def) + +lemma get_document_ptr_simp1 [simp]: + "get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr (put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr document h) = Some document" + by(auto simp add: get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) +lemma get_document_ptr_simp2 [simp]: + "document_ptr \ document_ptr' + \ get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr (put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr' document h) = get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr h" + by(auto simp add: get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + + +lemma get_document_ptr_simp3 [simp]: + "get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr (put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr f h) = get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr h" + by(auto simp add: get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) +lemma get_document_ptr_simp4 [simp]: "get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr (put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr f h) = get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr h" + by(auto simp add: get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def) +lemma get_document_ptr_simp5 [simp]: + "get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr (put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr f h) = get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr h" + by(auto simp add: get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) +lemma get_document_ptr_simp6 [simp]: "get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr (put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr f h) = get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr h" + by(auto simp add: get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def) + +lemma new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t [simp]: + assumes "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')" + shows "get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h = get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h'" + using assms + by(auto simp add: new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def) + +lemma new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t [simp]: + assumes "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = (new_character_data_ptr, h')" + shows "get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h = get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h'" + using assms + by(auto simp add: new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def Let_def) + + + +abbreviation + create_document_obj :: "char list \ (_) element_ptr option \ (_) node_ptr list \ (_) Document" + where + "create_document_obj doctype_arg document_element_arg disconnected_nodes_arg + \ \ RObject.nothing = (), RDocument.nothing = (), doctype = doctype_arg, + document_element = document_element_arg, + disconnected_nodes = disconnected_nodes_arg, \ = None \" + +definition new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t :: "(_)heap \ ((_) document_ptr \ (_) heap)" + where + "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = + (let new_document_ptr = document_ptr.Ref (Suc (fMax (document_ptr.the_ref |`| (document_ptrs h)))) + in + (new_document_ptr, put new_document_ptr (create_document_obj '''' None []) h))" + +lemma new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: + assumes "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')" + shows "new_document_ptr |\| document_ptr_kinds h'" + using assms + unfolding new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def + using put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap by blast + +lemma new_document_ptr_new: + "document_ptr.Ref (Suc (fMax (finsert 0 (document_ptr.the_ref |`| document_ptrs h)))) + |\| document_ptrs h" + by (metis Suc_n_not_le_n document_ptr.sel(1) fMax_ge fimage_finsert finsertI1 finsertI2 set_finsert) + +lemma new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_not_in_heap: + assumes "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')" + shows "new_document_ptr |\| document_ptr_kinds h" + using assms + unfolding new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + by (metis Pair_inject document_ptrs_def fMax_finsert fempty_iff ffmember_filter + fimage_is_fempty is_document_ptr_ref max_0L new_document_ptr_new) + +lemma new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_new_ptr: + assumes "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')" + shows "object_ptr_kinds h' = object_ptr_kinds h |\| {|cast new_document_ptr|}" + using assms + by (metis Pair_inject new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_put_ptrs) + +lemma new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_is_document_ptr: + assumes "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')" + shows "is_document_ptr new_document_ptr" + using assms + by(auto simp add: new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def) + +lemma new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t [simp]: + assumes "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')" + assumes "ptr \ cast new_document_ptr" + shows "get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h = get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h'" + using assms + by(auto simp add: new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + +lemma new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>N\<^sub>o\<^sub>d\<^sub>e [simp]: + assumes "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')" + shows "get\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr h = get\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr h'" + using assms + apply(simp add: new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + by(auto simp add: get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def) + +lemma new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t [simp]: + assumes "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')" + shows "get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h = get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h'" + using assms + by(auto simp add: new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def) + +lemma new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a [simp]: + assumes "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')" + shows "get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr h = get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr h'" + using assms + by(auto simp add: new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def) + +lemma new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t [simp]: + assumes "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')" + assumes "ptr \ new_document_ptr" + shows "get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h = get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h'" + using assms + by(auto simp add: new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def) + + +locale l_known_ptr\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t +begin +definition a_known_ptr :: "(_) object_ptr \ bool" + where + "a_known_ptr ptr = (known_ptr ptr \ is_document_ptr ptr)" + +lemma known_ptr_not_document_ptr: "\is_document_ptr ptr \ a_known_ptr ptr \ known_ptr ptr" + by(simp add: a_known_ptr_def) +end +global_interpretation l_known_ptr\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t defines known_ptr = a_known_ptr . +lemmas known_ptr_defs = a_known_ptr_def + + +locale l_known_ptrs\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr \ bool" +begin +definition a_known_ptrs :: "(_) heap \ bool" + where + "a_known_ptrs h = (\ptr. ptr |\| object_ptr_kinds h \ known_ptr ptr)" + +lemma known_ptrs_known_ptr: "a_known_ptrs h \ ptr |\| object_ptr_kinds h \ known_ptr ptr" + by(simp add: a_known_ptrs_def) + +lemma known_ptrs_preserved: + "object_ptr_kinds h = object_ptr_kinds h' \ a_known_ptrs h = a_known_ptrs h'" + by(auto simp add: a_known_ptrs_def) +lemma known_ptrs_subset: + "object_ptr_kinds h' |\| object_ptr_kinds h \ a_known_ptrs h \ a_known_ptrs h'" + by(auto simp add: a_known_ptrs_def) +end +global_interpretation l_known_ptrs\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t known_ptr defines known_ptrs = a_known_ptrs . +lemmas known_ptrs_defs = a_known_ptrs_def + +lemma known_ptrs_is_l_known_ptrs [instances]: "l_known_ptrs known_ptr known_ptrs" + using known_ptrs_known_ptr known_ptrs_preserved l_known_ptrs_def known_ptrs_subset by blast + +end diff --git a/Core_DOM/classes/ElementClass.thy b/Core_DOM/classes/ElementClass.thy new file mode 100644 index 0000000..02538f4 --- /dev/null +++ b/Core_DOM/classes/ElementClass.thy @@ -0,0 +1,311 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Element\ +text\In this theory, we introduce the types for the Element class.\ +theory ElementClass + imports + NodeClass + "../pointers/ShadowRootPointer" +begin +text\The type @{type "DOMString"} is a type synonym for @{type "string"}, define + in \autoref{sec:Core_DOM_Basic_Datatypes}.\ +type_synonym attr_key = DOMString +type_synonym attr_value = DOMString +type_synonym attrs = "(attr_key, attr_value) fmap" +type_synonym tag_type = DOMString +record ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr) RElement = RNode + + nothing :: unit + tag_type :: tag_type + child_nodes :: "('node_ptr, 'element_ptr, 'character_data_ptr) node_ptr list" + attrs :: attrs + shadow_root_opt :: "'shadow_root_ptr shadow_root_ptr option" +type_synonym + ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Element) Element + = "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Element option) RElement_scheme" +register_default_tvars + "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Element) Element" +type_synonym + ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Node, 'Element) Node + = "(('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Element option) RElement_ext + 'Node) Node" +register_default_tvars + "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Node, 'Element) Node" +type_synonym + ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Object, 'Node, 'Element) Object + = "('Object, ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Element option) RElement_ext + 'Node) Object" +register_default_tvars + "('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Object, 'Node, 'Element) Object" + +type_synonym + ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, 'shadow_root_ptr, 'Object, 'Node, 'Element) heap + = "('document_ptr document_ptr + 'shadow_root_ptr shadow_root_ptr + 'object_ptr, 'element_ptr element_ptr + 'character_data_ptr character_data_ptr + 'node_ptr, 'Object, + ('node_ptr, 'element_ptr, 'character_data_ptr, 'shadow_root_ptr, 'Element option) RElement_ext + 'Node) heap" +register_default_tvars + "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, 'shadow_root_ptr, 'Object, 'Node, 'Element) heap" + +definition element_ptr_kinds :: "(_) heap \ (_) element_ptr fset" + where + "element_ptr_kinds heap = the |`| (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |`| (ffilter is_element_ptr_kind (node_ptr_kinds heap)))" + +lemma element_ptr_kinds_simp [simp]: + "element_ptr_kinds (Heap (fmupd (cast element_ptr) element (the_heap h))) = {|element_ptr|} |\| element_ptr_kinds h" + apply(auto simp add: element_ptr_kinds_def)[1] + by force + +definition element_ptrs :: "(_) heap \ (_) element_ptr fset" + where + "element_ptrs heap = ffilter is_element_ptr (element_ptr_kinds heap)" + +definition cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t :: "(_) Node \ (_) Element option" + where + "cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t node = (case RNode.more node of Inl element \ Some (RNode.extend (RNode.truncate node) element) | _ \ None)" +adhoc_overloading cast cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + +abbreviation cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t :: "(_) Object \ (_) Element option" + where + "cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t obj \ (case cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e obj of Some node \ cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t node | None \ None)" +adhoc_overloading cast cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + +definition cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e :: "(_) Element \ (_) Node" + where + "cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e element = RNode.extend (RNode.truncate element) (Inl (RNode.more element))" +adhoc_overloading cast cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e + +abbreviation cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t :: "(_) Element \ (_) Object" + where + "cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr \ cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t (cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr)" +adhoc_overloading cast cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + +consts is_element_kind :: 'a +definition is_element_kind\<^sub>N\<^sub>o\<^sub>d\<^sub>e :: "(_) Node \ bool" + where + "is_element_kind\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr \ cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr \ None" + +adhoc_overloading is_element_kind is_element_kind\<^sub>N\<^sub>o\<^sub>d\<^sub>e +lemmas is_element_kind_def = is_element_kind\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + +abbreviation is_element_kind\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t :: "(_) Object \ bool" + where + "is_element_kind\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr \ cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr \ None" +adhoc_overloading is_element_kind is_element_kind\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + +lemma element_ptr_kinds_commutes [simp]: + "cast element_ptr |\| node_ptr_kinds h \ element_ptr |\| element_ptr_kinds h" + apply(auto simp add: node_ptr_kinds_def element_ptr_kinds_def)[1] + by (metis (no_types, lifting) element_ptr_casts_commute2 ffmember_filter fimage_eqI + fset.map_comp is_element_ptr_kind_none node_ptr_casts_commute3 + node_ptr_kinds_commutes node_ptr_kinds_def option.sel option.simps(3)) + +definition get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t :: "(_) element_ptr \ (_) heap \ (_) Element option" + where + "get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr h = Option.bind (get\<^sub>N\<^sub>o\<^sub>d\<^sub>e (cast element_ptr) h) cast" +adhoc_overloading get get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + +locale l_type_wf_def\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t +begin +definition a_type_wf :: "(_) heap \ bool" + where + "a_type_wf h = (NodeClass.type_wf h \ (\element_ptr. element_ptr |\| element_ptr_kinds h + \ get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr h \ None))" +end +global_interpretation l_type_wf_def\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t defines type_wf = a_type_wf . +lemmas type_wf_defs = a_type_wf_def + +locale l_type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t = l_type_wf type_wf for type_wf :: "((_) heap \ bool)" + + assumes type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: "type_wf h \ ElementClass.type_wf h" + +sublocale l_type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t \ l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e + apply(unfold_locales) + using NodeClass.a_type_wf_def + by (meson ElementClass.a_type_wf_def l_type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_axioms l_type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + +locale l_get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas = l_type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t +begin +sublocale l_get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas by unfold_locales + +lemma get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_type_wf: + assumes "type_wf h" + shows "element_ptr |\| element_ptr_kinds h \ get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr h \ None" + using l_type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_axioms assms + apply(simp add: type_wf_defs get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def l_type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + by (metis NodeClass.get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_type_wf bind_eq_None_conv element_ptr_kinds_commutes + option.distinct(1)) +end + +global_interpretation l_get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf + by unfold_locales + +definition put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t :: "(_) element_ptr \ (_) Element \ (_) heap \ (_) heap" + where + "put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr element = put\<^sub>N\<^sub>o\<^sub>d\<^sub>e (cast element_ptr) (cast element)" +adhoc_overloading put put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + +lemma put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: + assumes "put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr element h = h'" + shows "element_ptr |\| element_ptr_kinds h'" + using assms + unfolding put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def element_ptr_kinds_def + by (metis element_ptr_kinds_commutes element_ptr_kinds_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_ptr_in_heap) + +lemma put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_put_ptrs: + assumes "put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr element h = h'" + shows "object_ptr_kinds h' = object_ptr_kinds h |\| {|cast element_ptr|}" + using assms + by (simp add: put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_put_ptrs) + + + +lemma cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inject [simp]: + "cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e x = cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e y \ x = y" + apply(simp add: cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def RObject.extend_def RNode.extend_def) + by (metis (full_types) RNode.surjective old.unit.exhaust) + +lemma cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_none [simp]: + "cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t node = None \ \ (\element. cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e element = node)" + apply(auto simp add: cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def RObject.extend_def RNode.extend_def + split: sum.splits)[1] + by (metis (full_types) RNode.select_convs(2) RNode.surjective old.unit.exhaust) + +lemma cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_some [simp]: + "cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t node = Some element \ cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e element = node" + by(auto simp add: cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def RObject.extend_def RNode.extend_def + split: sum.splits) + +lemma cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_inv [simp]: "cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t (cast\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e element) = Some element" + by simp + +lemma get_elment_ptr_simp1 [simp]: + "get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr (put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr element h) = Some element" + by(auto simp add: get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) +lemma get_elment_ptr_simp2 [simp]: + "element_ptr \ element_ptr' + \ get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr (put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr' element h) = get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr h" + by(auto simp add: get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + + +abbreviation "create_element_obj tag_type_arg child_nodes_arg attrs_arg shadow_root_opt_arg + \ \ RObject.nothing = (), RNode.nothing = (), RElement.nothing = (), + tag_type = tag_type_arg, Element.child_nodes = child_nodes_arg, attrs = attrs_arg, + shadow_root_opt = shadow_root_opt_arg, \ = None \" + +definition new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t :: "(_) heap \ ((_) element_ptr \ (_) heap)" + where + "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = + (let new_element_ptr = element_ptr.Ref (Suc (fMax (finsert 0 (element_ptr.the_ref + |`| (element_ptrs h))))) + in + (new_element_ptr, put new_element_ptr (create_element_obj '''' [] fmempty None) h))" + +lemma new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: + assumes "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')" + shows "new_element_ptr |\| element_ptr_kinds h'" + using assms + unfolding new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def + using put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap by blast + +lemma new_element_ptr_new: + "element_ptr.Ref (Suc (fMax (finsert 0 (element_ptr.the_ref |`| element_ptrs h)))) |\| element_ptrs h" + by (metis Suc_n_not_le_n element_ptr.sel(1) fMax_ge fimage_finsert finsertI1 finsertI2 set_finsert) + +lemma new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_not_in_heap: + assumes "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')" + shows "new_element_ptr |\| element_ptr_kinds h" + using assms + unfolding new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + by (metis Pair_inject element_ptrs_def ffmember_filter new_element_ptr_new is_element_ptr_ref) + +lemma new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_new_ptr: + assumes "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')" + shows "object_ptr_kinds h' = object_ptr_kinds h |\| {|cast new_element_ptr|}" + using assms + by (metis Pair_inject new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_put_ptrs) + +lemma new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_is_element_ptr: + assumes "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')" + shows "is_element_ptr new_element_ptr" + using assms + by(auto simp add: new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def) + +lemma new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t [simp]: + assumes "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')" + assumes "ptr \ cast new_element_ptr" + shows "get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h = get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h'" + using assms + by(auto simp add: new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def) + +lemma new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>N\<^sub>o\<^sub>d\<^sub>e [simp]: + assumes "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')" + assumes "ptr \ cast new_element_ptr" + shows "get\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr h = get\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr h'" + using assms + by(auto simp add: new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + +lemma new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t [simp]: + assumes "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')" + assumes "ptr \ new_element_ptr" + shows "get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h = get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr h'" + using assms + by(auto simp add: new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def) + + +locale l_known_ptr\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t +begin +definition a_known_ptr :: "(_) object_ptr \ bool" + where + "a_known_ptr ptr = (known_ptr ptr \ is_element_ptr ptr)" + +lemma known_ptr_not_element_ptr: "\is_element_ptr ptr \ a_known_ptr ptr \ known_ptr ptr" + by(simp add: a_known_ptr_def) +end +global_interpretation l_known_ptr\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t defines known_ptr = a_known_ptr . +lemmas known_ptr_defs = a_known_ptr_def + + +locale l_known_ptrs\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr \ bool" +begin +definition a_known_ptrs :: "(_) heap \ bool" + where + "a_known_ptrs h = (\ptr. ptr |\| object_ptr_kinds h \ known_ptr ptr)" + +lemma known_ptrs_known_ptr: + "ptr |\| object_ptr_kinds h \ a_known_ptrs h \ known_ptr ptr" + by(simp add: a_known_ptrs_def) + +lemma known_ptrs_preserved: "object_ptr_kinds h = object_ptr_kinds h' \ a_known_ptrs h = a_known_ptrs h'" + by(auto simp add: a_known_ptrs_def) +lemma known_ptrs_subset: "object_ptr_kinds h' |\| object_ptr_kinds h \ a_known_ptrs h \ a_known_ptrs h'" + by(auto simp add: a_known_ptrs_def) +end +global_interpretation l_known_ptrs\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t known_ptr defines known_ptrs = a_known_ptrs . +lemmas known_ptrs_defs = a_known_ptrs_def + +lemma known_ptrs_is_l_known_ptrs: "l_known_ptrs known_ptr known_ptrs" + using known_ptrs_known_ptr known_ptrs_preserved l_known_ptrs_def known_ptrs_subset by blast + +end diff --git a/Core_DOM/classes/NodeClass.thy b/Core_DOM/classes/NodeClass.thy new file mode 100644 index 0000000..142c7fa --- /dev/null +++ b/Core_DOM/classes/NodeClass.thy @@ -0,0 +1,202 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + + +section\Node\ +text\In this theory, we introduce the types for the Node class.\ + +theory NodeClass + imports + ObjectClass + "../pointers/NodePointer" +begin + +subsubsection\Node\ + +record RNode = RObject + + nothing :: unit +register_default_tvars "'Node RNode_ext" +type_synonym 'Node Node = "'Node RNode_scheme" +register_default_tvars "'Node Node" +type_synonym ('Object, 'Node) Object = "('Node RNode_ext + 'Object) Object" +register_default_tvars "('Object, 'Node) Object" + +type_synonym ('object_ptr, 'node_ptr, 'Object, 'Node) heap + = "('node_ptr node_ptr + 'object_ptr, 'Node RNode_ext + 'Object) heap" +register_default_tvars + "('object_ptr, 'node_ptr, 'Object, 'Node) heap" + + +definition node_ptr_kinds :: "(_) heap \ (_) node_ptr fset" + where + "node_ptr_kinds heap = + (the |`| (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r |`| (ffilter is_node_ptr_kind (object_ptr_kinds heap))))" + +lemma node_ptr_kinds_simp [simp]: + "node_ptr_kinds (Heap (fmupd (cast node_ptr) node (the_heap h))) + = {|node_ptr|} |\| node_ptr_kinds h" + apply(auto simp add: node_ptr_kinds_def)[1] + by force + +definition cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e :: "(_) Object \ (_) Node option" + where + "cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e obj = (case RObject.more obj of Inl node + \ Some (RObject.extend (RObject.truncate obj) node) | _ \ None)" +adhoc_overloading cast cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e + +definition cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t:: "(_) Node \ (_) Object" + where + "cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t node = (RObject.extend (RObject.truncate node) (Inl (RObject.more node)))" +adhoc_overloading cast cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + +definition is_node_kind :: "(_) Object \ bool" + where + "is_node_kind ptr \ cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr \ None" + +definition get\<^sub>N\<^sub>o\<^sub>d\<^sub>e :: "(_) node_ptr \ (_) heap \ (_) Node option" + where + "get\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr h = Option.bind (get (cast node_ptr) h) cast" +adhoc_overloading get get\<^sub>N\<^sub>o\<^sub>d\<^sub>e + +locale l_type_wf_def\<^sub>N\<^sub>o\<^sub>d\<^sub>e +begin +definition a_type_wf :: "(_) heap \ bool" + where + "a_type_wf h = (ObjectClass.type_wf h + \ (\node_ptr. node_ptr |\| node_ptr_kinds h + \ get\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr h \ None))" +end +global_interpretation l_type_wf_def\<^sub>N\<^sub>o\<^sub>d\<^sub>e defines type_wf = a_type_wf . +lemmas type_wf_defs = a_type_wf_def + +locale l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e = l_type_wf type_wf for type_wf :: "((_) heap \ bool)" + + assumes type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e: "type_wf h \ NodeClass.type_wf h" + +sublocale l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e \ l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + apply(unfold_locales) + using ObjectClass.a_type_wf_def by auto + +locale l_get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas = l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e +begin +sublocale l_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas by unfold_locales +lemma get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_type_wf: + assumes "type_wf h" + shows "node_ptr |\| node_ptr_kinds h \ get\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr h \ None" + using l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e_axioms assms + apply(simp add: type_wf_defs get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def) + by (metis (mono_tags, lifting) bind_eq_None_conv ffmember_filter fimage_eqI + is_node_ptr_kind_cast get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf local.l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_axioms + node_ptr_casts_commute2 node_ptr_kinds_def option.sel option.simps(3)) +end + +global_interpretation l_get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas type_wf + by unfold_locales + +definition put\<^sub>N\<^sub>o\<^sub>d\<^sub>e :: "(_) node_ptr \ (_) Node \ (_) heap \ (_) heap" + where + "put\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr node = put (cast node_ptr) (cast node)" +adhoc_overloading put put\<^sub>N\<^sub>o\<^sub>d\<^sub>e + +lemma put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_ptr_in_heap: + assumes "put\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr node h = h'" + shows "node_ptr |\| node_ptr_kinds h'" + using assms + unfolding put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def node_ptr_kinds_def + by (metis ffmember_filter fimage_eqI is_node_ptr_kind_cast node_ptr_casts_commute2 + option.sel put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ptr_in_heap) + +lemma put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_put_ptrs: + assumes "put\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr node h = h'" + shows "object_ptr_kinds h' = object_ptr_kinds h |\| {|cast node_ptr|}" + using assms + by (simp add: put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_put_ptrs) + +lemma node_ptr_kinds_commutes [simp]: + "cast node_ptr |\| object_ptr_kinds h \ node_ptr |\| node_ptr_kinds h" + apply(auto simp add: node_ptr_kinds_def split: option.splits)[1] + by (metis (no_types, lifting) ffmember_filter fimage_eqI fset.map_comp + is_node_ptr_kind_none node_ptr_casts_commute2 + option.distinct(1) option.sel) + +lemma node_empty [simp]: + "\RObject.nothing = (), RNode.nothing = (), \ = RNode.more node\ = node" + by simp + +lemma cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_inject [simp]: "cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x = cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t y \ x = y" + apply(simp add: cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def RObject.extend_def) + by (metis (full_types) RObject.surjective old.unit.exhaust) + +lemma cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_none [simp]: + "cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e obj = None \ \ (\node. cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t node = obj)" + apply(auto simp add: cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def RObject.extend_def split: sum.splits)[1] + by (metis (full_types) RObject.select_convs(2) RObject.surjective old.unit.exhaust) + +lemma cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_some [simp]: "cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e obj = Some node \ cast node = obj" + by(auto simp add: cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def RObject.extend_def split: sum.splits) + +lemma cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inv [simp]: "cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e (cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t node) = Some node" + by simp + +locale l_known_ptr\<^sub>N\<^sub>o\<^sub>d\<^sub>e +begin +definition a_known_ptr :: "(_) object_ptr \ bool" + where + "a_known_ptr ptr = False" +end +global_interpretation l_known_ptr\<^sub>N\<^sub>o\<^sub>d\<^sub>e defines known_ptr = a_known_ptr . +lemmas known_ptr_defs = a_known_ptr_def + + +locale l_known_ptrs\<^sub>N\<^sub>o\<^sub>d\<^sub>e = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr \ bool" +begin +definition a_known_ptrs :: "(_) heap \ bool" + where + "a_known_ptrs h = (\ptr. ptr |\| object_ptr_kinds h \ known_ptr ptr)" + +lemma known_ptrs_known_ptr: "a_known_ptrs h \ ptr |\| object_ptr_kinds h \ known_ptr ptr" + by(simp add: a_known_ptrs_def) + +lemma known_ptrs_preserved: "object_ptr_kinds h = object_ptr_kinds h' \ a_known_ptrs h = a_known_ptrs h'" + by(auto simp add: a_known_ptrs_def) +lemma known_ptrs_subset: "object_ptr_kinds h' |\| object_ptr_kinds h \ a_known_ptrs h \ a_known_ptrs h'" + by(auto simp add: a_known_ptrs_def) +end +global_interpretation l_known_ptrs\<^sub>N\<^sub>o\<^sub>d\<^sub>e known_ptr defines known_ptrs = a_known_ptrs . +lemmas known_ptrs_defs = a_known_ptrs_def + +lemma known_ptrs_is_l_known_ptrs: "l_known_ptrs known_ptr known_ptrs" + using known_ptrs_known_ptr known_ptrs_preserved l_known_ptrs_def known_ptrs_subset by blast + +lemma get_node_ptr_simp1 [simp]: "get\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr (put\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr node h) = Some node" + by(auto simp add: get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def) +lemma get_node_ptr_simp2 [simp]: + "node_ptr \ node_ptr' \ get\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr (put\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr' node h) = get\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr h" + by(auto simp add: get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def) + +end diff --git a/Core_DOM/classes/ObjectClass.thy b/Core_DOM/classes/ObjectClass.thy new file mode 100644 index 0000000..310c64b --- /dev/null +++ b/Core_DOM/classes/ObjectClass.thy @@ -0,0 +1,191 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Object\ +text\In this theory, we introduce the definition of the class Object. This class is the +common superclass of our class model.\ + +theory ObjectClass + imports + BaseClass + "../pointers/ObjectPointer" +begin + +record RObject = + nothing :: unit +register_default_tvars "'Object RObject_ext" +type_synonym 'Object Object = "'Object RObject_scheme" +register_default_tvars "'Object Object" + +datatype ('object_ptr, 'Object) heap = Heap (the_heap: "((_) object_ptr, (_) Object) fmap") +register_default_tvars "('object_ptr, 'Object) heap" + +definition object_ptr_kinds :: "(_) heap \ (_) object_ptr fset" + where + "object_ptr_kinds = fmdom \ the_heap" + +lemma object_ptr_kinds_simp [simp]: + "object_ptr_kinds (Heap (fmupd object_ptr object (the_heap h))) + = {|object_ptr|} |\| object_ptr_kinds h" + by(auto simp add: object_ptr_kinds_def) + +definition get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t :: "(_) object_ptr \ (_) heap \ (_) Object option" + where + "get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h = fmlookup (the_heap h) ptr" +adhoc_overloading get get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + +locale l_type_wf_def\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t +begin +definition a_type_wf :: "(_) heap \ bool" + where + "a_type_wf h = True" +end +global_interpretation l_type_wf_def\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t defines type_wf = a_type_wf . +lemmas type_wf_defs = a_type_wf_def + +locale l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t = l_type_wf type_wf for type_wf :: "((_) heap \ bool)" + + assumes type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: "type_wf h \ ObjectClass.type_wf h" + +locale l_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas = l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t +begin +lemma get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf: + assumes "type_wf h" + shows "object_ptr |\| object_ptr_kinds h \ get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr h \ None" + using l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_axioms assms + apply(simp add: type_wf_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def) + by (simp add: fmlookup_dom_iff object_ptr_kinds_def) +end + +global_interpretation l_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas type_wf + by (simp add: l_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas.intro l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t.intro) + +definition put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t :: "(_) object_ptr \ (_) Object \ (_) heap \ (_) heap" + where + "put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h = Heap (fmupd ptr obj (the_heap h))" +adhoc_overloading put put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + +lemma put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ptr_in_heap: + assumes "put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr object h = h'" + shows "object_ptr |\| object_ptr_kinds h'" + using assms + unfolding put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def + by auto + +lemma put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_put_ptrs: + assumes "put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr object h = h'" + shows "object_ptr_kinds h' = object_ptr_kinds h |\| {|object_ptr|}" + using assms + by (metis comp_apply fmdom_fmupd funion_finsert_right heap.sel object_ptr_kinds_def + sup_bot.right_neutral put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def) + +lemma object_more_extend_id [simp]: "more (extend x y) = y" + by(simp add: extend_def) + +lemma object_empty [simp]: "\nothing = (), \ = more x\ = x" + by simp + +locale l_known_ptr\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t +begin +definition a_known_ptr :: "(_) object_ptr \ bool" + where + "a_known_ptr ptr = False" + +lemma known_ptr_not_object_ptr: + "a_known_ptr ptr \ \is_object_ptr ptr \ known_ptr ptr" + by(simp add: a_known_ptr_def) +end +global_interpretation l_known_ptr\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t defines known_ptr = a_known_ptr . +lemmas known_ptr_defs = a_known_ptr_def + +locale l_known_ptrs = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr \ bool" + + fixes known_ptrs :: "(_) heap \ bool" + assumes known_ptrs_known_ptr: "known_ptrs h \ ptr |\| object_ptr_kinds h \ known_ptr ptr" + assumes known_ptrs_preserved: "object_ptr_kinds h = object_ptr_kinds h' \ known_ptrs h = known_ptrs h'" + assumes known_ptrs_subset: "object_ptr_kinds h' |\| object_ptr_kinds h \ known_ptrs h \ known_ptrs h'" + +locale l_known_ptrs\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr \ bool" +begin +definition a_known_ptrs :: "(_) heap \ bool" + where + "a_known_ptrs h = (\ptr. ptr |\| object_ptr_kinds h \ known_ptr ptr)" + +lemma known_ptrs_known_ptr: + "a_known_ptrs h \ ptr |\| object_ptr_kinds h \ known_ptr ptr" + by(simp add: a_known_ptrs_def) + +lemma known_ptrs_preserved: "object_ptr_kinds h = object_ptr_kinds h' \ a_known_ptrs h = a_known_ptrs h'" + by(auto simp add: a_known_ptrs_def) +lemma known_ptrs_subset: "object_ptr_kinds h' |\| object_ptr_kinds h \ a_known_ptrs h \ a_known_ptrs h'" + by(auto simp add: a_known_ptrs_def) +end +global_interpretation l_known_ptrs\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t known_ptr defines known_ptrs = a_known_ptrs . +lemmas known_ptrs_defs = a_known_ptrs_def + +lemma known_ptrs_is_l_known_ptrs: "l_known_ptrs known_ptr known_ptrs" + using known_ptrs_known_ptr known_ptrs_preserved l_known_ptrs_def known_ptrs_subset by blast + + +lemma get_object_ptr_simp1 [simp]: "get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr object h) = Some object" + by(simp add: get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def) +lemma get_object_ptr_simp2 [simp]: + "object_ptr \ object_ptr' + \ get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr' object h) = get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr h" + by(simp add: get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def) + + +subsection\Limited Heap Modifications\ + +definition heap_unchanged_except :: "(_) object_ptr set \ (_) heap \ (_) heap \ bool" + where + "heap_unchanged_except S h h' = (\ptr \ (fset (object_ptr_kinds h) + \ (fset (object_ptr_kinds h'))) - S. get ptr h = get ptr h')" + +definition delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t :: "(_) object_ptr \ (_) heap \ (_) heap option" where + "delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h = (if ptr |\| object_ptr_kinds h then Some (Heap (fmdrop ptr (the_heap h))) + else None)" + +lemma delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_pointer_removed: + assumes "delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h = Some h'" + shows "ptr |\| object_ptr_kinds h'" + using assms + by(auto simp add: delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def object_ptr_kinds_def split: if_splits) + +lemma delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_pointer_ptr_in_heap: + assumes "delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h = Some h'" + shows "ptr |\| object_ptr_kinds h" + using assms + by(auto simp add: delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def object_ptr_kinds_def split: if_splits) + +lemma delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ok: + assumes "ptr |\| object_ptr_kinds h" + shows "delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h \ None" + using assms + by(auto simp add: delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def object_ptr_kinds_def split: if_splits) + +end diff --git a/Core_DOM/document/root.bib b/Core_DOM/document/root.bib new file mode 100644 index 0000000..3c39e52 --- /dev/null +++ b/Core_DOM/document/root.bib @@ -0,0 +1,508 @@ +@STRING{j-fac = "Formal Aspects of Computing" } +@STRING{pub-springer={Springer-Verlag} } +@STRING{pub-springer:adr={Heidelberg} } +@STRING{s-lncs = "Lecture Notes in Computer Science" } + +@Book{ nipkow.ea:isabelle:2002, + author = {Tobias Nipkow and Lawrence C. Paulson and Markus Wenzel}, + title = {Isabelle/HOL---A Proof Assistant for Higher-Order Logic}, + publisher = pub-springer, + address = pub-springer:adr, + series = s-lncs, + volume = 2283, + doi = {10.1007/3-540-45949-9}, + abstract = {This book is a self-contained introduction to interactive + proof in higher-order logic (HOL), using the proof + assistant Isabelle2002. It is a tutorial for potential + users rather than a monograph for researchers. The book has + three parts. + + 1. Elementary Techniques shows how to model functional + programs in higher-order logic. Early examples involve + lists and the natural numbers. Most proofs are two steps + long, consisting of induction on a chosen variable followed + by the auto tactic. But even this elementary part covers + such advanced topics as nested and mutual recursion. 2. + Logic and Sets presents a collection of lower-level tactics + that you can use to apply rules selectively. It also + describes Isabelle/HOL's treatment of sets, functions and + relations and explains how to define sets inductively. One + of the examples concerns the theory of model checking, and + another is drawn from a classic textbook on formal + languages. 3. Advanced Material describes a variety of + other topics. Among these are the real numbers, records and + overloading. Advanced techniques are described involving + induction and recursion. A whole chapter is devoted to an + extended example: the verification of a security protocol. + }, + year = 2002, + acknowledgement={brucker, 2007-02-19}, + bibkey = {nipkow.ea:isabelle:2002} +} + +@Misc{ dom-specification, + year = 2016, + month = {DOM Living Standard -- Last Updated 20 October 2016}, + day = 20, + url = {https://dom.spec.whatwg.org/}, + organization = {Web Hypertext Application Technology Working Group + (WHATWG)}, + note = {An archived copy of the version from 20 October 2016 is + available at + \url{https://git.logicalhacking.com/BrowserSecurity/fDOM-idl/}.} +} + +@InProceedings{ brucker.ea:core-dom:2018, + author = {Achim D. Brucker and Michael Herzberg}, + title = {A Formal Semantics of the Core {DOM} in {Isabelle/HOL}}, + booktitle = {Proceedings of the Web Programming, Design, Analysis, And + Implementation (WPDAI) track at WWW 2018}, + location = {Lyon, France}, + url = {https://www.brucker.ch/bibliography/abstract/brucker.ea-fdom-2018}, + year = {2018}, + abstract = {At its core, the Document Object Model (DOM) defines a + tree-like data structure for representing documents in + general and HTML documents in particular. It forms the + heart of any rendering engine of modern web browsers. + Formalizing the key concepts of the DOM is a pre-requisite + for the formal reasoning over client-side JavaScript + programs as well as for the analysis of security concepts + in modern web browsers. In this paper, we present a + formalization of the core DOM, with focus on the node-tree + and the operations defined on node-trees, in Isabelle/HOL. + We use the formalization to verify the functional + correctness of the most important functions defined in the + DOM standard. Moreover, our formalization is (1) + extensible, i.e., can be extended without the need of + re-proving already proven properties and (2) executable, + i.e., we can generate executable code from our + specification. }, + keywords = {Document Object Model, DOM, Formal Semantics, + Isabelle/HOL}, + classification= {conference}, + areas = {formal methods, software}, + public = {yes} +} +@Article{ klein:operating:2009, + author = {Gerwin Klein}, + title = {Operating System Verification --- An Overview}, + journal = {S\={a}dhan\={a}}, + publisher = pub-springer, + year = 2009, + volume = 34, + number = 1, + month = feb, + pages = {27--69}, + abstract = {This paper gives a high-level introduction to the topic of + formal, interactive, machine-checked software verification + in general, and the verification of operating systems code + in particular. We survey the state of the art, the + advantages and limitations of machine-checked code proofs, + and describe two specific ongoing larger-scale verification + projects in more detail.} +} + + +@InProceedings{ gardner.ea:securing:2009, + author = {Ryan W. Gardner and Sujata Garera and Matthew W. Pagano + and Matthew Green and Aviel D. Rubin}, + title = {Securing medical records on smart phones}, + booktitle = {ACM workshop on Security and privacy in medical and + home-care systems (SPIMACS)}, + year = 2009, + isbn = {978-1-60558-790-5}, + pages = {31--40}, + location = {Chicago, Illinois, USA}, + doi = {10.1145/1655084.1655090}, + address = pub-acm:adr, + publisher = pub-acm, + abstract = {There is an inherent conflict between the desire to + maintain privacy of one's medical records and the need to + make those records available during an emergency. To + satisfy both objectives, we introduce a flexible + architecture for the secure storage of medical records on + smart phones. In our system, a person can view her records + at any time, and emergency medical personnel can view the + records as long as the person is present (even if she is + unconscious). Our solution allows for efficient revocation + of access rights and is robust against adversaries who can + access the phone's storage offline.} +} + +@InProceedings{ raad.ea:dom:2016, + author = {Azalea Raad and Jos{\'{e}} Fragoso Santos and Philippa + Gardner}, + title = {{DOM:} Specification and Client Reasoning}, + booktitle = {Programming Languages and Systems - 14th Asian Symposium, + {APLAS} 2016, Hanoi, Vietnam, November 21-23, 2016, + Proceedings}, + pages = {401--422}, + year = 2016, + crossref = {igarashi:programming:2016}, + doi = {10.1007/978-3-319-47958-3_21}, + abstract = {We present an axiomatic specification of a key fragment of + DOM using structural separation logic. This specification + allows us to develop modular reasoning about client + programs that call the DOM.} +} + + +@InProceedings{ bohannon.ea:featherweight:2010, + author = {Aaron Bohannon and Benjamin C. Pierce}, + title = {Featherweight {F}irefox: {F}ormalizing the Core of a Web + Browser}, + booktitle = {Usenix Conference on Web Application Development + (WebApps)}, + year = 2010, + month = jun, + url = {http://www.cis.upenn.edu/~bohannon/browser-model/}, + abstract = {We offer a formal specification of the core functionality + of a web browser in the form of a small-step operational + semantics. The specification accurately models the asyn- + chronous nature of web browsers and covers the basic as- + pects of windows, DOM trees, cookies, HTTP requests and + responses, user input, and a minimal scripting lan- guage + with first-class functions, dynamic evaluation, and AJAX + requests. No security enforcement mechanisms are + included{\^a}instead, the model is intended to serve as a + basis for formalizing and experimenting with different + security policies and mechanisms. We survey the most + interesting design choices and discuss how our model re- + lates to real web browsers.} +} + +@Proceedings{ joyce.ea:higher:1994, + editor = {Jeffrey J. Joyce and Carl-Johan H. Seger}, + title = {Higher Order Logic Theorem Proving and Its Applications + (HUG)}, + booktitle = {Higher Order Logic Theorem Proving and Its Applications + (HUG)}, + publisher = pub-springer, + address = pub-springer:adr, + series = s-lncs, + abstract = {Theorem proving based techniques for formal hardware + verification have been evolving constantly and researchers + are getting able to reason about more complex issues than + it was possible or practically feasible in the past. It is + often the case that a model of a system is built in a + formal logic and then reasoning about this model is carried + out in the logic. Concern is growing on how to consistently + interface a model built in a formal logic with an informal + CAD environment. Researchers have been investigating how to + define the formal semantics of hardware description + languages so that one can formally reason about models + informally dealt with in a CAD environment. At the + University of Cambridge, the embedding of hardware + description languages in a logic is classified in two + categories: deep embedding and shallow embedding. In this + paper we argue that there are degrees of formality in + shallow embedding a language in a logic. The choice of the + degree of formality is a trade-off between the security of + the embedding and the amount and complexity of the proof + effort in the logic. We also argue that the design of a + language could consider this verifiability issue. There are + choices in the design of a language that can make it easier + to improve the degree of formality, without implying + serious drawbacks for the CAD environment.}, + volume = 780, + year = 1994, + doi = {10.1007/3-540-57826-9}, + isbn = {3-540-57826-9}, + acknowledgement={brucker, 2007-02-19} +} + + +@Misc{ whatwg:dom:2017, + key={whatwg}, + author={{WHATWG}}, + url={https://dom.spec.whatwg.org/commit-snapshots/6253e53af2fbfaa6d25ad09fd54280d8083b2a97/}, + month=mar, + year=2017, + day=24, + title={{DOM} -- Living Standard}, + note={Last Updated 24 {March} 2017}, + institution = {WHATWG}, +} + +@Misc{ whatwg:html:2017, + key={whatwg}, + author={{WHATWG}}, + url={https://html.spec.whatwg.org/}, + month=apr, + year=2017, + day=13, + title={{HTML} -- Living Standard}, + note={Last Updated 13 {April} 2017}, + institution = {WHATWG}, +} + + +@Misc{ w3c:dom:2015, + key={w3c}, + author={{W3C}}, + url={https://www.w3.org/TR/dom/}, + month=nov, + year=2015, + day=19, + title={{W3C} {DOM4}}, + institution = {W3C}, +} + + +@Proceedings{ igarashi:programming:2016, + editor = {Atsushi Igarashi}, + title = {Programming Languages and Systems - 14th Asian Symposium, + {APLAS} 2016, Hanoi, Vietnam, November 21-23, 2016, + Proceedings}, + series = {Lecture Notes in Computer Science}, + volume = 10017, + year = 2016, + doi = {10.1007/978-3-319-47958-3}, + isbn = {978-3-319-47957-6} +} + + + + + + +@InProceedings{ gardner.ea:dom:2008, + author = {Philippa Gardner and Gareth Smith and Mark J. Wheelhouse + and Uri Zarfaty}, + title = {{DOM:} Towards a Formal Specification}, + booktitle = {{PLAN-X} 2008, Programming Language Technologies for XML, + An {ACM} {SIGPLAN} Workshop colocated with {POPL} 2008, San + Francisco, California, USA, January 9, 2008}, + year = 2008, + crossref = {plan-x:2008}, + url = {http://gemo.futurs.inria.fr/events/PLANX2008/papers/p18.pdf}, + abstract = {The W3C Document Object Model (DOM) specifies an XML up- + date library. DOM is written in English, and is therefore + not compo- sitional and not complete. We provide a first + step towards a compo- sitional specification of DOM. Unlike + DOM, we are able to work with a minimal set of commands and + obtain a complete reason- ing for straight-line code. Our + work transfers O{\^a}Hearn, Reynolds and Yang{\^a}s + local Hoare reasoning for analysing heaps to XML, viewing + XML as an in-place memory store as does DOM. In par- + ticular, we apply recent work by Calcagno, Gardner and + Zarfaty on local Hoare reasoning about a simple tree-update + language to DOM, showing that our reasoning scales to DOM. + Our reasoning not only formally specifies a significant + subset of DOM Core Level 1, but can also be used to verify + e.g. invariant properties of simple Javascript programs.} +} + + + +@InProceedings{ jang.ea:establishing:2012, + author = {Dongseok Jang and Zachary Tatlock and Sorin Lerner}, + title = {Establishing Browser Security Guarantees through Formal + Shim Verification}, + booktitle = {Proceedings of the 21th {USENIX} Security Symposium, + Bellevue, WA, USA, August 8-10, 2012}, + pages = {113--128}, + year = 2012, + crossref = {kohno:proceedings:2012}, + url = {https://www.usenix.org/conference/usenixsecurity12/technical-sessions/presentation/jang}, + abstract = { Web browsers mediate access to valuable private data in + domains ranging from health care to banking. Despite this + critical role, attackers routinely exploit browser + vulnerabilities to exfiltrate private data and take over + the un- derlying system. We present Q UARK , a browser + whose kernel has been implemented and verified in Coq. We + give a specification of our kernel, show that the + implementation satisfies the specification, and finally + show that the specification implies several security + properties, including tab non-interference, cookie + integrity and confidentiality, and address bar integrity. + } +} + +@Proceedings{ kohno:proceedings:2012, + editor = {Tadayoshi Kohno}, + title = {Proceedings of the 21th {USENIX} Security Symposium, + Bellevue, WA, USA, August 8-10, 2012}, + publisher = {{USENIX} Association}, + year = 2012, + timestamp = {Thu, 15 May 2014 09:12:27 +0200} +} + + + +@Proceedings{ plan-x:2008, + title = {{PLAN-X} 2008, Programming Language Technologies for XML, + An {ACM} {SIGPLAN} Workshop colocated with {POPL} 2008, San + Francisco, California, USA, January 9, 2008}, + year = 2008, + timestamp = {Fri, 18 Jan 2008 13:01:04 +0100} +} + + +@Article{ brucker.ea:extensible:2008-b, + abstract = {We present an extensible encoding of object-oriented data models into HOL. Our encoding is supported by a datatype package that leverages the use of the shallow embedding technique to object-oriented specification and programming languages. The package incrementally compiles an object-oriented data model, i.e., a class model, to a theory containing object-universes, constructors, accessor functions, coercions (casts) between dynamic and static types, characteristic sets, and co-inductive class invariants. The package is conservative, i.e., all properties are derived entirely from constant definitions, including the constraints over object structures. As an application, we use the package for an object-oriented core-language called IMP++, for which we formally prove the correctness of a Hoare-Logic with respect to a denotational semantics.}, + address = {Heidelberg}, + author = {Achim D. Brucker and Burkhart Wolff}, + doi = {10.1007/s10817-008-9108-3}, + issn = {0168-7433}, + issue = {3}, + journal = {Journal of Automated Reasoning}, + keywords = {object-oriented data models, HOL, theorem proving, verification}, + language = {USenglish}, + pages = {219--249}, + pdf = {https://www.brucker.ch/bibliography/download/2008/brucker.ea-extensible-2008-b.pdf}, + publisher = {Springer-Verlag}, + title = {An Extensible Encoding of Object-oriented Data Models in HOL}, + url = {https://www.brucker.ch/bibliography/abstract/brucker.ea-extensible-2008-b}, + volume = {41}, + year = {2008}, +} + +@PhDThesis{ brucker:interactive:2007, + abstract = {We present a semantic framework for object-oriented specification languages. We develop this framework as a conservative shallow embedding in Isabelle/HOL. Using only conservative extensions guarantees by construction the consistency of our formalization. Moreover, we show how our framework can be used to build an interactive proof environment, called HOL-OCL, for object-oriented specifications in general and for UML/OCL in particular.\\\\Our main contributions are an extensible encoding of object-oriented data structures in HOL, a datatype package for object-oriented specifications, and the development of several equational and tableaux calculi for object-oriented specifications. Further, we show that our formal framework can be the basis of a formal machine-checked semantics for OCL that is compliant to the OCL 2.0 standard.}, + abstract_de = {In dieser Arbeit wird ein semantisches Rahmenwerk f{\"u}r objektorientierte Spezifikationen vorgestellt. Das Rahmenwerk ist als konservative, flache Einbettung in Isabelle/HOL realisiert. Durch die Beschr{\"a}nkung auf konservative Erweiterungen kann die logische Konsistenz der Einbettung garantiert werden. Das semantische Rahmenwerk wird verwendet, um das interaktives Beweissystem HOL-OCL f{\"u}r objektorientierte Spezifikationen im Allgemeinen und insbesondere f{\"u}r UML/OCL zu entwickeln.\\\\Die Hauptbeitr{\"a}ge dieser Arbeit sind die Entwicklung einer erweiterbaren Kodierung objektorientierter Datenstrukturen in HOL, ein Datentyp-Paket f{\"u}r objektorientierte Spezifikationen und die Entwicklung verschiedener Kalk{\"u}le f{\"u}r objektorientierte Spezifikationen. Zudem zeigen wir, wie das formale Rahmenwerk verwendet werden kann, um eine formale, maschinell gepr{\"u}fte Semantik f{\"u}r OCL anzugeben, die konform zum Standard f{\"u}r OCL 2.0 ist.}, + author = {Achim D. Brucker}, + keywords = {OCL, UML, formal semantics, theorem proving, Isabelle, HOL-OCL}, + month = {mar}, + note = {ETH Dissertation No. 17097.}, + pdf = {https://www.brucker.ch/bibliography/download/2007/brucker-interactive-2007.pdf}, + school = {ETH Zurich}, + title = {An Interactive Proof Environment for Object-oriented Specifications}, + url = {https://www.brucker.ch/bibliography/abstract/brucker-interactive-2007}, + year = {2007}, +} + +@InCollection{ brucker.ea:standard-compliance-testing:2018, + talk = {talk:brucker.ea:standard-compliance-testing:2018}, + abstract = {Most popular technologies are based on informal or + semiformal standards that lack a rigid formal semantics. + Typical examples include web technologies such as the DOM + or HTML, which are defined by the Web Hypertext Application + Technology Working Group (WHATWG) and the World Wide Web + Consortium (W3C). While there might be API specifications + and test cases meant to assert the compliance of a certain + implementation, the actual standard is rarely accompanied + by a formal model that would lend itself for, e.g., + verifying the security or safety properties of real + systems. + + Even when such a formalization of a standard exists, two + important questions arise: first, to what extend does the + formal model comply to the standard and, second, to what + extend does the implementation comply to the formal model + and the assumptions made during the verification? In this + paper, we present an approach that brings all three + involved artifacts - the (semi-)formal standard, the + formalization of the standard, and the implementations - + closer together by combining verification, symbolic + execution, and specification based testing.}, + keywords = {standard compliance, compliance tests, DOM}, + location = {Toulouse, France}, + author = {Achim D. Brucker and Michael Herzberg}, + booktitle = {{TAP} 2018: Tests And Proofs}, + language = {USenglish}, + publisher = pub-springer, + address = pub-springer:adr, + series = s-lncs, + number = 10889, + editor = {Cathrine Dubois and Burkhart Wolff}, + title = {Formalizing (Web) Standards: An Application of Test and + Proof}, + categories = {holtestgen, websecurity}, + classification= {conference}, + areas = {formal methods, software engineering}, + public = {yes}, + year = 2018, + doi = {10.1007/978-3-319-92994-1_9}, + pages = {159--166}, + isbn = {978-3-642-38915-3}, + pdf = {http://www.brucker.ch/bibliography/download/2018/brucker.ea-standard-compliance-testing-2018.pdf}, + url = {http://www.brucker.ch/bibliography/abstract/brucker.ea-standard-compliance-testing-2018} +} + + +@InCollection{ brucker.ea:interactive:2005, + keywords = {symbolic test case generations, black box testing, white + box testing, theorem proving, interactive testing}, + abstract = {HOL-TestGen is a test environment for specification-based + unit testing build upon the proof assistant Isabelle/HOL\@. + While there is considerable skepticism with regard to + interactive theorem provers in testing communities, we + argue that they are a natural choice for (automated) + symbolic computations underlying systematic tests. This + holds in particular for the development on non-trivial + formal test plans of complex software, where some parts of + the overall activity require inherently guidance by a test + engineer. In this paper, we present the underlying methods + for both black box and white box testing in interactive + unit test scenarios. HOL-TestGen can also be understood as + a unifying technical and conceptual framework for + presenting and investigating the variety of unit test + techniques in a logically consistent way. }, + location = {Edinburgh}, + author = {Achim D. Brucker and Burkhart Wolff}, + booktitle = {Formal Approaches to Testing of Software}, + language = {USenglish}, + publisher = pub-springer, + address = pub-springer:adr, + series = s-lncs, + number = 3997, + doi = {10.1007/11759744_7}, + isbn = {3-540-25109-X}, + editor = {Wolfgang Grieskamp and Carsten Weise}, + pdf = {http://www.brucker.ch/bibliography/download/2005/brucker.ea-interactive-2005.pdf}, + project = {CSFMDOS}, + title = {Interactive Testing using {HOL}-{TestGen}}, + classification= {workshop}, + areas = {formal methods, software}, + categories = {holtestgen}, + year = 2005, + public = {yes}, + url = {http://www.brucker.ch/bibliography/abstract/brucker.ea-interactive-2005} +} + + +@Article{ brucker.ea:theorem-prover:2012, + author = {Achim D. Brucker and Burkhart Wolff}, + journal = j-fac, + publisher = pub-springer, + address = pub-springer:adr, + language = {USenglish}, + categories = {holtestgen}, + title = {On Theorem Prover-based Testing}, + year = 2013, + issn = {0934-5043}, + pages = {683--721}, + volume = 25, + number = 5, + classification= {journal}, + areas = {formal methods, software}, + public = {yes}, + doi = {10.1007/s00165-012-0222-y}, + keywords = {test case generation, domain partitioning, test sequence, + theorem proving, HOL-TestGen}, + abstract = {HOL-TestGen is a specification and test case generation + environment extending the interactive theorem prover + Isabelle/HOL. As such, HOL-TestGen allows for an integrated + workflow supporting interactive theorem proving, test case + generation, and test data generation. + + The HOL-TestGen method is two-staged: first, the original + formula is partitioned into test cases by transformation + into a normal form called test theorem. Second, the test + cases are analyzed for ground instances (the test data) + satisfying the constraints of the test cases. Particular + emphasis is put on the control of explicit test-hypotheses + which can be proven over concrete programs. + + Due to the generality of the underlying framework, our + system can be used for black-box unit, sequence, reactive + sequence and white-box test scenarios. Although based on + particularly clean theoretical foundations, the system can + be applied for substantial case-studies. }, + pdf = {http://www.brucker.ch/bibliography/download/2012/brucker.ea-theorem-prover-2012.pdf}, + url = {http://www.brucker.ch/bibliography/abstract/brucker.ea-theorem-prover-2012} +} + + + diff --git a/Core_DOM/document/root.tex b/Core_DOM/document/root.tex new file mode 100644 index 0000000..451e8e0 --- /dev/null +++ b/Core_DOM/document/root.tex @@ -0,0 +1,266 @@ +\documentclass[10pt,DIV16,a4paper,abstract=true,twoside=semi,openright] +{scrreprt} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Overrides the (rightfully issued) warnings by Koma Script that \rm +%%% etc. should not be used (they are deprecated since more than a +%%% decade) + \DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm} + \DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf} + \DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt} + \DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf} + \DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[USenglish]{babel} +\usepackage[numbers, sort&compress]{natbib} +\usepackage{isabelle,isabellesym} +\usepackage{booktabs} +\usepackage{paralist} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{xspace} +\usepackage{xcolor} +\usepackage{listings} +\lstloadlanguages{HTML} +\usepackage[]{mathtools} +\usepackage[pdfpagelabels, pageanchor=false, plainpages=false]{hyperref} +\lstdefinestyle{html}{language=XML, + basicstyle=\ttfamily, + commentstyle=\itshape, + keywordstyle=\color{blue}, + ndkeywordstyle=\color{blue}, +} +\lstdefinestyle{displayhtml}{style=html, + floatplacement={tbp}, + captionpos=b, + framexleftmargin=0pt, + basicstyle=\ttfamily\scriptsize, + backgroundcolor=\color{black!2}, + frame=lines, +} +\lstnewenvironment{html}[1][]{\lstset{style=displayhtml, #1}}{} +\def\inlinehtml{\lstinline[style=html, columns=fullflexible]} + +\pagestyle{headings} +\isabellestyle{default} +\setcounter{tocdepth}{1} +\newcommand{\ie}{i.\,e.\xspace} +\newcommand{\eg}{e.\,g.\xspace} +\newcommand{\thy}{\isabellecontext} +\renewcommand{\isamarkupsection}[1]{% + \begingroup% + \def\isacharunderscore{\textunderscore}% + \section{#1 (\thy)}% + \def\isacharunderscore{-}% + \expandafter\label{sec:\isabellecontext}% + \endgroup% +} + +\title{Core DOM\\\medskip \Large + A Formal Model of the Document Object Model}% +\author{Achim~D.~Brucker \and Michael~Herzberg}% +\publishers{ + Department of Computer Science\\ + The University of Sheffield\\ + Sheffield, UK\\ + \texttt{\{\href{mailto:a.brucker@sheffield.ac.uk}{a.brucker}, + \href{mailto:msherzberg1@sheffield.ac.uk}{msherzberg1}\}@sheffield.ac.uk} +} +\begin{document} + \maketitle + \begin{abstract} + \begin{quote} + In this AFP entry, we formalize the core of the Document Object + Model (DOM). At its core, the DOM defines a tree-like data + structure for representing documents in general and HTML documents + in particular. It is the heart of any modern web browser. + + Formalizing the key concepts of the DOM is a prerequisite for the + formal reasoning over client-side JavaScript programs and for the + analysis of security concepts in modern web browsers. + + + We present a formalization of the core DOM, with focus on the + \emph{node-tree} and the operations defined on node-trees, in + Isabelle/HOL\@. We use the formalization to verify the functional + correctness of the most important functions defined in the DOM + standard. Moreover, our formalization is + \begin{inparaenum} + \item \emph{extensible}, i.e., can be extended without the need of + re-proving already proven properties and + \item \emph{executable}, i.e., we can generate executable code + from our specification. + \end{inparaenum} + + \bigskip + \noindent{\textbf{Keywords:}} + Document Object Model, DOM, Formal Semantics, Isabelle/HOL + \end{quote} + \end{abstract} + + +\tableofcontents +\cleardoublepage + +\chapter{Introduction} +In a world in which more and more applications are offered as services +on the internet, web browsers start to take on a similarly central +role in our daily IT infrastructure as operating systems. Thus, web +browsers should be developed as rigidly and formally as operating +systems. While formal methods are a well-established technique in the +development of operating systems (see, +\eg,~\citet{klein:operating:2009} for an overview of formal +verification of operating systems), there are few proposals for +improving the development of web browsers using formal +approaches~\cite{gardner.ea:dom:2008,raad.ea:dom:2016,jang.ea:establishing:2012,bohannon.ea:featherweight:2010}. + +As a first step towards a verified client-side web application stack, +we model and formally verify the Document Object Model (DOM) in +Isabelle/HOL\@. The DOM~\cite{whatwg:dom:2017,w3c:dom:2015} is +\emph{the} central data structure of all modern web browsers. At its +core, the Document Object Model (DOM), defines a tree-like data +structure for representing documents in general and HTML documents in +particular. Thus, the correctness of a DOM implementation is crucial +for ensuring that a web browser displays web pages correctly. +Moreover, the DOM is the core data structure underlying client-side +JavaScript programs, \ie, client-side JavaScript programs are mostly +programs that read, write, and update the DOM. + +In more detail, we formalize the core DOM as a shallow +embedding~\cite{joyce.ea:higher:1994} in Isabelle/HOL\@. Our +formalization is based on a typed data model for the \emph{node-tree}, +\ie, a data structure for representing XML-like documents in a tree +structure. Furthermore, we formalize a typed heap for storing +(partial) node-trees together with the necessary consistency +constraints. Finally, we formalize the operations (as described in the +DOM standard~\cite{whatwg:dom:2017}) on this heap that allow +manipulating node-trees. + +Our machine-checked formalization of the DOM node +tree~\cite{whatwg:dom:2017} has the following desirable properties: +\begin{itemize} +\item It provides a \emph{consistency guarantee.} Since all + definitions in our formal semantics are conservative and all rules + are derived, the logical consistency of the DOM node-tree is reduced + to the consistency of HOL. +\item It serves as a \emph{technical basis for a proof system.} Based + on the derived rules and specific setup of proof tactics over + node-trees, our formalization provides a generic proof environment + for the verification of programs manipulating node-trees. +\item It is \emph{executable}, which allows to validate its compliance + to the standard by evaluating the compliance test suite on the + formal model and +\item It is \emph{extensible} in the sense + of~\cite{brucker.ea:extensible:2008-b,brucker:interactive:2007}, + \ie, properties proven over the core DOM do not need to be re-proven + for object-oriented extensions such as the HTML document model. +\end{itemize} + +The rest of this document is automatically generated from the +formalization in Isabelle/HOL, i.e., all content is checked by +Isabelle.\footnote{For a brief overview of the work, we refer the + reader to~\cite{brucker.ea:core-dom:2018}.} The structure follows +the theory dependencies (see \autoref{fig:session-graph}): we start +with introducing the technical preliminaries of our formalization +(\autoref{cha:perliminaries}). Next, we introduce the concepts of +pointers (\autoref{cha:pointers}) and classes (\autoref{cha:classes}), +i.e., the core object-oriented datatypes of the DOM. On top of this +data model, we define the functional behavior of the DOM classes, +i.e., their methods (\autoref{cha:monads}). In \autoref{cha:dom}, we +introduce the formalization of the functionality of the core DOM, +i.e., the \emph{main entry point for users} that want to use this AFP +entry. Finally, we formalize the relevant compliance test cases in +\autoref{cha:tests}. + +\begin{figure} + \centering + \includegraphics[width=.8\textwidth]{session_graph} + \caption{The Dependency Graph of the Isabelle Theories.\label{fig:session-graph}} +\end{figure} + +\clearpage + +\chapter{Preliminaries} +\label{cha:perliminaries} +In this chapter, we introduce the technical preliminaries of our +formalization of the core DOM, namely a mechanism for hiding type +variables and the heap error monad. +\input{Hiding_Type_Variables} +\input{Heap_Error_Monad} + +\chapter{References and Pointers} +\label{cha:pointers} +In this chapter, we introduce a generic type for object-oriented +references and typed pointers for each class type defined in the DOM +standard. +\input{Ref} +\input{ObjectPointer} +\input{NodePointer} +\input{ElementPointer} +\input{CharacterDataPointer} +\input{DocumentPointer} +\input{ShadowRootPointer} + +\chapter{Classes} +\label{cha:classes} +In this chapter, we introduce the classes of our DOM model. +The definition of the class types follows closely the one of the +pointer types. Instead of datatypes, we use records for our classes. +a generic type for object-oriented references and typed pointers for +each class type defined in the DOM standard. +\input{BaseClass} +\input{ObjectClass} +\input{NodeClass} +\input{ElementClass} +\input{CharacterDataClass} +\input{DocumentClass} + +\chapter{Monadic Object Constructors and Accessors} +\label{cha:monads} +In this chapter, we introduce the moandic method definitions for the +classes of our DOM formalization. Again the overall structure follows +the same structure as for the class types and the pointer types. +\input{BaseMonad} +\input{ObjectMonad} +\input{NodeMonad} +\input{ElementMonad} +\input{CharacterDataMonad} +\input{DocumentMonad} + +\chapter{The Core DOM} +\label{cha:dom} +In this chapter, we introduce the formalization of the core DOM, i.e., +the most important algorithms for querying or modifying the DOM, as +defined int he standard. For more details, we refer the reader to +\cite{brucker.ea:core-dom:2018}. +\input{Core_DOM_Basic_Datatypes} +\input{Core_DOM_Functions} +\input{Core_DOM_Heap_WF} +\input{Core_DOM} + +\chapter{Test Suite} +\label{cha:tests} +In this chapter, we present the formalized compliance test cases for +the core DOM. As our formalization is executable, we can +(symbolically) execute the test cases on top of our model. Executing +these test cases successfully shows that our model is compliant to the +official DOM standard. As future work, we plan to generate test cases +from our formal model (e.g., +using~\cite{brucker.ea:interactive:2005,brucker.ea:theorem-prover:2012}) +to improve the quality of the official compliance test suite. For more +details on the relation of test and proof in the context of web +standards, we refer the reader to +\cite{brucker.ea:standard-compliance-testing:2018}. +\input{Core_DOM_BaseTest} \input{Document_adoptNode} +\input{Document_getElementById} \input{Node_insertBefore} +\input{Node_removeChild} \input{Core_DOM_Tests} + +{\small + \bibliographystyle{abbrvnat} + \bibliography{root} +} +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: diff --git a/Core_DOM/monads/BaseMonad.thy b/Core_DOM/monads/BaseMonad.thy new file mode 100644 index 0000000..346c768 --- /dev/null +++ b/Core_DOM/monads/BaseMonad.thy @@ -0,0 +1,376 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\The Monad Infrastructure\ +text\In this theory, we introduce the basic infrastructure for our monadic class encoding.\ +theory BaseMonad + imports + "../classes/BaseClass" + "../preliminaries/Heap_Error_Monad" +begin +subsection\Datatypes\ + +datatype exception = NotFoundError | SegmentationFault | HierarchyRequestError | AssertException + | NonTerminationException | InvokeError | TypeError | DebugException nat + +lemma finite_set_in [simp]: "x \ fset FS \ x |\| FS" + by (meson notin_fset) + +consts put_M :: 'a +consts get_M :: 'a +consts delete_M :: 'a + +lemma sorted_list_of_set_eq [dest]: + "sorted_list_of_set (fset x) = sorted_list_of_set (fset y) \ x = y" + by (metis finite_fset fset_inject sorted_list_of_set(1)) + + +locale l_ptr_kinds_M = + fixes ptr_kinds :: "'heap \ 'ptr::linorder fset" +begin +definition a_ptr_kinds_M :: "('heap, exception, 'ptr list) prog" + where + "a_ptr_kinds_M = do { + h \ get_heap; + return (sorted_list_of_set (fset (ptr_kinds h))) + }" + +lemma ptr_kinds_M_ok [simp]: "h \ ok a_ptr_kinds_M" + by(simp add: a_ptr_kinds_M_def) + +lemma ptr_kinds_M_pure [simp]: "pure a_ptr_kinds_M h" + by (auto simp add: a_ptr_kinds_M_def intro: bind_pure_I) + +lemma ptr_kinds_ptr_kinds_M [simp]: "ptr \ set |h \ a_ptr_kinds_M|\<^sub>r \ ptr |\| ptr_kinds h" + by(simp add: a_ptr_kinds_M_def) + +lemma ptr_kinds_M_ptr_kinds [simp]: + "h \ a_ptr_kinds_M \\<^sub>r xa \ xa = sorted_list_of_set (fset (ptr_kinds h))" + by(auto simp add: a_ptr_kinds_M_def) +lemma ptr_kinds_M_ptr_kinds_returns_result [simp]: + "h \ a_ptr_kinds_M \ f \\<^sub>r x \ h \ f (sorted_list_of_set (fset (ptr_kinds h))) \\<^sub>r x" + by(auto simp add: a_ptr_kinds_M_def) +lemma ptr_kinds_M_ptr_kinds_returns_heap [simp]: + "h \ a_ptr_kinds_M \ f \\<^sub>h h' \ h \ f (sorted_list_of_set (fset (ptr_kinds h))) \\<^sub>h h'" + by(auto simp add: a_ptr_kinds_M_def) +end + +locale l_get_M = + fixes get :: "'ptr \ 'heap \ 'obj option" + fixes type_wf :: "'heap \ bool" + fixes ptr_kinds :: "'heap \ 'ptr fset" + assumes "type_wf h \ ptr |\| ptr_kinds h \ get ptr h \ None" + assumes "get ptr h \ None \ ptr |\| ptr_kinds h" +begin + +definition a_get_M :: "'ptr \ ('obj \ 'result) \ ('heap, exception, 'result) prog" + where + "a_get_M ptr getter = (do { + h \ get_heap; + (case get ptr h of + Some res \ return (getter res) + | None \ error SegmentationFault) + })" + +lemma get_M_pure [simp]: "pure (a_get_M ptr getter) h" + by(auto simp add: a_get_M_def bind_pure_I split: option.splits) + +lemma get_M_ok: + "type_wf h \ ptr |\| ptr_kinds h \ h \ ok (a_get_M ptr getter)" + apply(simp add: a_get_M_def) + by (metis l_get_M_axioms l_get_M_def option.case_eq_if return_ok) +lemma get_M_ptr_in_heap: + "h \ ok (a_get_M ptr getter) \ ptr |\| ptr_kinds h" + apply(simp add: a_get_M_def) + by (metis error_returns_result is_OK_returns_result_E l_get_M_axioms l_get_M_def option.simps(4)) + +end + +locale l_put_M = l_get_M get for get :: "'ptr \ 'heap \ 'obj option" + + fixes put :: "'ptr \ 'obj \ 'heap \ 'heap" +begin +definition a_put_M :: "'ptr \ (('v \ 'v) \ 'obj \ 'obj) \ 'v \ ('heap, exception, unit) prog" + where + "a_put_M ptr setter v = (do { + obj \ a_get_M ptr id; + h \ get_heap; + return_heap (put ptr (setter (\_. v) obj) h) + })" + +lemma put_M_ok: + "type_wf h \ ptr |\| ptr_kinds h \ h \ ok (a_put_M ptr setter v)" + by(auto simp add: a_put_M_def intro!: bind_is_OK_I2 dest: get_M_ok elim!: bind_is_OK_E) + +lemma put_M_ptr_in_heap: + "h \ ok (a_put_M ptr setter v) \ ptr |\| ptr_kinds h" + by(auto simp add: a_put_M_def intro!: bind_is_OK_I2 elim: get_M_ptr_in_heap + dest: is_OK_returns_result_I elim!: bind_is_OK_E) + +end + +subsection \Setup for Defining Partial Functions\ + +lemma execute_admissible: + "ccpo.admissible (fun_lub (flat_lub (Inl (e::'e)))) (fun_ord (flat_ord (Inl e))) + ((\a. \(h::'heap) h2 (r::'result). h \ a = Inr (r, h2) \ P h h2 r) \ Prog)" +proof (unfold comp_def, rule ccpo.admissibleI, clarify) + fix A :: "('heap \ 'e + 'result \ 'heap) set" + let ?lub = "Prog (fun_lub (flat_lub (Inl e)) A)" + fix h h2 r + assume 1: "Complete_Partial_Order.chain (fun_ord (flat_ord (Inl e))) A" + and 2: "\xa\A. \h h2 r. h \ Prog xa = Inr (r, h2) \ P h h2 r" + and 4: "h \ Prog (fun_lub (flat_lub (Inl e)) A) = Inr (r, h2)" + have h1:"\a. Complete_Partial_Order.chain (flat_ord (Inl e)) {y. \f\A. y = f a}" + by (rule chain_fun[OF 1]) + show "P h h2 r" + using chain_fun[OF 1] flat_lub_in_chain[OF chain_fun[OF 1]] 2 4 unfolding execute_def fun_lub_def + by force +qed + +lemma execute_admissible2: + "ccpo.admissible (fun_lub (flat_lub (Inl (e::'e)))) (fun_ord (flat_ord (Inl e))) + ((\a. \(h::'heap) h' h2 h2' (r::'result) r'. + h \ a = Inr (r, h2) \ h' \ a = Inr (r', h2') \ P h h' h2 h2' r r') \ Prog)" +proof (unfold comp_def, rule ccpo.admissibleI, clarify) + fix A :: "('heap \ 'e + 'result \ 'heap) set" + let ?lub = "Prog (fun_lub (flat_lub (Inl e)) A)" + fix h h' h2 h2' r r' + assume 1: "Complete_Partial_Order.chain (fun_ord (flat_ord (Inl e))) A" + and 2 [rule_format]: "\xa\A. \h h' h2 h2' r r'. h \ Prog xa = Inr (r, h2) + \ h' \ Prog xa = Inr (r', h2') \ P h h' h2 h2' r r'" + and 4: "h \ Prog (fun_lub (flat_lub (Inl e)) A) = Inr (r, h2)" + and 5: "h' \ Prog (fun_lub (flat_lub (Inl e)) A) = Inr (r', h2')" + have h1:"\a. Complete_Partial_Order.chain (flat_ord (Inl e)) {y. \f\A. y = f a}" + by (rule chain_fun[OF 1]) + have "h \ ?lub \ {y. \f\A. y = f h}" + using flat_lub_in_chain[OF h1] 4 + unfolding execute_def fun_lub_def + by auto + moreover have "h' \ ?lub \ {y. \f\A. y = f h'}" + using flat_lub_in_chain[OF h1] 5 + unfolding execute_def fun_lub_def + by auto + ultimately obtain f where + "f \ A" and + "h \ Prog f = Inr (r, h2)" and + "h' \ Prog f = Inr (r', h2')" + using 1 4 5 + apply(auto simp add: chain_def fun_ord_def flat_ord_def execute_def)[1] + by (metis Inl_Inr_False) + then show "P h h' h2 h2' r r'" + by(fact 2) +qed + +definition dom_prog_ord :: + "('heap, exception, 'result) prog \ ('heap, exception, 'result) prog \ bool" where + "dom_prog_ord = img_ord (\a b. execute b a) (fun_ord (flat_ord (Inl NonTerminationException)))" + +definition dom_prog_lub :: + "('heap, exception, 'result) prog set \ ('heap, exception, 'result) prog" where + "dom_prog_lub = img_lub (\a b. execute b a) Prog (fun_lub (flat_lub (Inl NonTerminationException)))" + +lemma dom_prog_lub_empty: "dom_prog_lub {} = error NonTerminationException" + by(simp add: dom_prog_lub_def img_lub_def fun_lub_def flat_lub_def error_def) + +lemma dom_prog_interpretation: "partial_function_definitions dom_prog_ord dom_prog_lub" +proof - + have "partial_function_definitions (fun_ord (flat_ord (Inl NonTerminationException))) + (fun_lub (flat_lub (Inl NonTerminationException)))" + by (rule partial_function_lift) (rule flat_interpretation) + then show ?thesis + apply (simp add: dom_prog_lub_def dom_prog_ord_def flat_interpretation execute_def) + using partial_function_image prog.expand prog.sel by blast +qed + +interpretation dom_prog: partial_function_definitions dom_prog_ord dom_prog_lub + rewrites "dom_prog_lub {} \ error NonTerminationException" + by (fact dom_prog_interpretation)(simp add: dom_prog_lub_empty) + +lemma admissible_dom_prog: + "dom_prog.admissible (\f. \x h h' r. h \ f x \\<^sub>r r \ h \ f x \\<^sub>h h' \ P x h h' r)" +proof (rule admissible_fun[OF dom_prog_interpretation]) + fix x + show "ccpo.admissible dom_prog_lub dom_prog_ord (\a. \h h' r. h \ a \\<^sub>r r \ h \ a \\<^sub>h h' + \ P x h h' r)" + unfolding dom_prog_ord_def dom_prog_lub_def + proof (intro admissible_image partial_function_lift flat_interpretation) + show "ccpo.admissible (fun_lub (flat_lub (Inl NonTerminationException))) + (fun_ord (flat_ord (Inl NonTerminationException))) + ((\a. \h h' r. h \ a \\<^sub>r r \ h \ a \\<^sub>h h' \ P x h h' r) \ Prog)" + by(auto simp add: execute_admissible returns_result_def returns_heap_def split: sum.splits) + next + show "\x y. (\b. b \ x) = (\b. b \ y) \ x = y" + by(simp add: execute_def prog.expand) + next + show "\x. (\b. b \ Prog x) = x" + by(simp add: execute_def) + qed +qed + +lemma admissible_dom_prog2: + "dom_prog.admissible (\f. \x h h2 h' h2' r r2. h \ f x \\<^sub>r r \ h \ f x \\<^sub>h h' + \ h2 \ f x \\<^sub>r r2 \ h2 \ f x \\<^sub>h h2' \ P x h h2 h' h2' r r2)" +proof (rule admissible_fun[OF dom_prog_interpretation]) + fix x + show "ccpo.admissible dom_prog_lub dom_prog_ord (\a. \h h2 h' h2' r r2. h \ a \\<^sub>r r + \ h \ a \\<^sub>h h' \ h2 \ a \\<^sub>r r2 \ h2 \ a \\<^sub>h h2' \ P x h h2 h' h2' r r2)" + unfolding dom_prog_ord_def dom_prog_lub_def + proof (intro admissible_image partial_function_lift flat_interpretation) + show "ccpo.admissible (fun_lub (flat_lub (Inl NonTerminationException))) + (fun_ord (flat_ord (Inl NonTerminationException))) + ((\a. \h h2 h' h2' r r2. h \ a \\<^sub>r r \ h \ a \\<^sub>h h' \ h2 \ a \\<^sub>r r2 \ h2 \ a \\<^sub>h h2' + \ P x h h2 h' h2' r r2) \ Prog)" + by(auto simp add: returns_result_def returns_heap_def intro!: ccpo.admissibleI + dest!: ccpo.admissibleD[OF execute_admissible2[where P="P x"]] + split: sum.splits) + next + show "\x y. (\b. b \ x) = (\b. b \ y) \ x = y" + by(simp add: execute_def prog.expand) + next + show "\x. (\b. b \ Prog x) = x" + by(simp add: execute_def) + qed +qed + +lemma fixp_induct_dom_prog: + fixes F :: "'c \ 'c" and + U :: "'c \ 'b \ ('heap, exception, 'result) prog" and + C :: "('b \ ('heap, exception, 'result) prog) \ 'c" and + P :: "'b \ 'heap \ 'heap \ 'result \ bool" + assumes mono: "\x. monotone (fun_ord dom_prog_ord) dom_prog_ord (\f. U (F (C f)) x)" + assumes eq: "f \ C (ccpo.fixp (fun_lub dom_prog_lub) (fun_ord dom_prog_ord) (\f. U (F (C f))))" + assumes inverse2: "\f. U (C f) = f" + assumes step: "\f x h h' r. (\x h h' r. h \ (U f x) \\<^sub>r r \ h \ (U f x) \\<^sub>h h' \ P x h h' r) + \ h \ (U (F f) x) \\<^sub>r r \ h \ (U (F f) x) \\<^sub>h h' \ P x h h' r" + assumes defined: "h \ (U f x) \\<^sub>r r" and "h \ (U f x) \\<^sub>h h'" + shows "P x h h' r" + using step defined dom_prog.fixp_induct_uc[of U F C, OF mono eq inverse2 admissible_dom_prog, of P] + by (metis assms(6) error_returns_heap) + +declaration \Partial_Function.init "dom_prog" @{term dom_prog.fixp_fun} + @{term dom_prog.mono_body} @{thm dom_prog.fixp_rule_uc} @{thm dom_prog.fixp_induct_uc} + (SOME @{thm fixp_induct_dom_prog})\ + + +abbreviation "mono_dom_prog \ monotone (fun_ord dom_prog_ord) dom_prog_ord" + +lemma dom_prog_ordI: + assumes "\h. h \ f \\<^sub>e NonTerminationException \ h \ f = h \ g" + shows "dom_prog_ord f g" +proof(auto simp add: dom_prog_ord_def img_ord_def fun_ord_def flat_ord_def)[1] + fix x + assume "x \ f \ x \ g" + then show "x \ f = Inl NonTerminationException" + using assms[where h=x] + by(auto simp add: returns_error_def split: sum.splits) +qed + +lemma dom_prog_ordE: + assumes "dom_prog_ord x y" + obtains "h \ x \\<^sub>e NonTerminationException" | " h \ x = h \ y" + using assms unfolding dom_prog_ord_def img_ord_def fun_ord_def flat_ord_def + using returns_error_def by force + + +lemma bind_mono [partial_function_mono]: + fixes B :: "('a \ ('heap, exception, 'result) prog) \ ('heap, exception, 'result2) prog" + assumes mf: "mono_dom_prog B" and mg: "\y. mono_dom_prog (\f. C y f)" + shows "mono_dom_prog (\f. B f \ (\y. C y f))" +proof (rule monotoneI) + fix f g :: "'a \ ('heap, exception, 'result) prog" + assume fg: "dom_prog.le_fun f g" + from mf + have 1: "dom_prog_ord (B f) (B g)" by (rule monotoneD) (rule fg) + from mg + have 2: "\y'. dom_prog_ord (C y' f) (C y' g)" by (rule monotoneD) (rule fg) + + have "dom_prog_ord (B f \ (\y. C y f)) (B g \ (\y. C y f))" + (is "dom_prog_ord ?L ?R") + proof (rule dom_prog_ordI) + fix h + from 1 show "h \ ?L \\<^sub>e NonTerminationException \ h \ ?L = h \ ?R" + apply(rule dom_prog_ordE) + apply(auto)[1] + using bind_cong by fastforce + qed + also + have h1: "dom_prog_ord (B g \ (\y'. C y' f)) (B g \ (\y'. C y' g))" + (is "dom_prog_ord ?L ?R") + proof (rule dom_prog_ordI) + (* { *) + fix h + show "h \ ?L \\<^sub>e NonTerminationException \ h \ ?L = h \ ?R" + proof (cases "h \ ok (B g)") + case True + then obtain x h' where x: "h \ B g \\<^sub>r x" and h': "h \ B g \\<^sub>h h'" + by blast + then have "dom_prog_ord (C x f) (C x g)" + using 2 by simp + then show ?thesis + using x h' + apply(auto intro!: bind_returns_error_I3 dest: returns_result_eq dest!: dom_prog_ordE)[1] + apply(auto simp add: execute_bind_simp)[1] + using "2" dom_prog_ordE by metis + next + case False + then obtain e where e: "h \ B g \\<^sub>e e" + by(simp add: is_OK_def returns_error_def split: sum.splits) + have "h \ B g \ (\y'. C y' f) \\<^sub>e e" + using e by(auto) + moreover have "h \ B g \ (\y'. C y' g) \\<^sub>e e" + using e by auto + ultimately show ?thesis + using bind_returns_error_eq by metis + qed + qed + finally (dom_prog.leq_trans) + show "dom_prog_ord (B f \ (\y. C y f)) (B g \ (\y'. C y' g))" . +qed + +lemma mono_dom_prog1 [partial_function_mono]: + fixes g :: "('a \ ('heap, exception, 'result) prog) \ 'b \ ('heap, exception, 'result) prog" + assumes "\x. (mono_dom_prog (\f. g f x))" + shows "mono_dom_prog (\f. map_M (g f) xs)" + using assms + apply (induct xs) + by(auto simp add: call_mono dom_prog.const_mono intro!: bind_mono) + +lemma mono_dom_prog2 [partial_function_mono]: + fixes g :: "('a \ ('heap, exception, 'result) prog) \ 'b \ ('heap, exception, 'result) prog" + assumes "\x. (mono_dom_prog (\f. g f x))" + shows "mono_dom_prog (\f. forall_M (g f) xs)" + using assms + apply (induct xs) + by(auto simp add: call_mono dom_prog.const_mono intro!: bind_mono) + +lemma sorted_list_set_cong [simp]: + "sorted_list_of_set (fset FS) = sorted_list_of_set (fset FS') \ FS = FS'" + by auto + +end diff --git a/Core_DOM/monads/CharacterDataMonad.thy b/Core_DOM/monads/CharacterDataMonad.thy new file mode 100644 index 0000000..e34b42e --- /dev/null +++ b/Core_DOM/monads/CharacterDataMonad.thy @@ -0,0 +1,539 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\CharacterData\ +text\In this theory, we introduce the monadic method setup for the CharacterData class.\ +theory CharacterDataMonad + imports + ElementMonad + "../classes/CharacterDataClass" +begin + +type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, + 'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData, 'result) dom_prog + = "((_) heap, exception, 'result) prog" +register_default_tvars + "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, 'shadow_root_ptr, + 'Object, 'Node, 'Element, 'CharacterData, 'result) dom_prog" + + +global_interpretation l_ptr_kinds_M character_data_ptr_kinds + defines character_data_ptr_kinds_M = a_ptr_kinds_M . +lemmas character_data_ptr_kinds_M_defs = a_ptr_kinds_M_def + +lemma character_data_ptr_kinds_M_eq: + assumes "|h \ node_ptr_kinds_M|\<^sub>r = |h' \ node_ptr_kinds_M|\<^sub>r" + shows "|h \ character_data_ptr_kinds_M|\<^sub>r = |h' \ character_data_ptr_kinds_M|\<^sub>r" + using assms + by(auto simp add: character_data_ptr_kinds_M_defs node_ptr_kinds_M_defs + character_data_ptr_kinds_def) + +lemma character_data_ptr_kinds_M_reads: + "reads (\node_ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t node_ptr RObject.nothing)}) character_data_ptr_kinds_M h h'" + using node_ptr_kinds_M_reads + apply(simp add: reads_def node_ptr_kinds_M_defs character_data_ptr_kinds_M_defs + character_data_ptr_kinds_def preserved_def) + by (metis (mono_tags, hide_lams) node_ptr_kinds_small old.unit.exhaust preserved_def) + +global_interpretation l_dummy defines get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a = "l_get_M.a_get_M get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a" . +lemma get_M_is_l_get_M: "l_get_M get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a type_wf character_data_ptr_kinds" + apply(simp add: get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_type_wf l_get_M_def) + by (metis (no_types, hide_lams) NodeMonad.get_M_is_l_get_M bind_eq_Some_conv + character_data_ptr_kinds_commutes get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def l_get_M_def option.distinct(1)) +lemmas get_M_defs = get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def[unfolded l_get_M.a_get_M_def[OF get_M_is_l_get_M]] + +adhoc_overloading get_M get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a + +locale l_get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_lemmas = l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a +begin +sublocale l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas by unfold_locales + +interpretation l_get_M get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a type_wf character_data_ptr_kinds + apply(unfold_locales) + apply (simp add: get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_type_wf local.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a) + by (meson CharacterDataMonad.get_M_is_l_get_M l_get_M_def) +lemmas get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ok = get_M_ok[folded get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def] +end + +global_interpretation l_get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_lemmas type_wf by unfold_locales + + +global_interpretation l_put_M type_wf character_data_ptr_kinds get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a + rewrites "a_get_M = get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a" defines put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a = a_put_M + apply (simp add: get_M_is_l_get_M l_put_M_def) + by (simp add: get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def) + +lemmas put_M_defs = a_put_M_def +adhoc_overloading put_M put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a + + +locale l_put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_lemmas = l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a +begin +sublocale l_put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas by unfold_locales + +interpretation l_put_M type_wf character_data_ptr_kinds get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a + apply(unfold_locales) + using get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_type_wf l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a local.l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_axioms + apply blast + by (meson CharacterDataMonad.get_M_is_l_get_M l_get_M_def) +lemmas put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ok = put_M_ok[folded put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def] +end + +global_interpretation l_put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_lemmas type_wf by unfold_locales + + + +lemma CharacterData_simp1 [simp]: + "(\x. getter (setter (\_. v) x) = v) \ h \ put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr setter v \\<^sub>h h' + \ h' \ get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr getter \\<^sub>r v" + by(auto simp add: put_M_defs get_M_defs split: option.splits) +lemma CharacterData_simp2 [simp]: + "character_data_ptr \ character_data_ptr' + \ h \ put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr' getter) h h'" + by(auto simp add: put_M_defs get_M_defs preserved_def split: option.splits dest: get_heap_E) +lemma CharacterData_simp3 [simp]: " + (\x. getter (setter (\_. v) x) = getter x) + \ h \ put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr' getter) h h'" + apply(cases "character_data_ptr = character_data_ptr'") + by(auto simp add: put_M_defs get_M_defs preserved_def split: option.splits dest: get_heap_E) +lemma CharacterData_simp4 [simp]: + "h \ put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr getter) h h'" + by(auto simp add: put_M_defs ElementMonad.get_M_defs preserved_def + split: option.splits dest: get_heap_E) +lemma CharacterData_simp5 [simp]: + "h \ put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr getter) h h'" + by(auto simp add: ElementMonad.put_M_defs get_M_defs preserved_def + split: option.splits dest: get_heap_E) +lemma CharacterData_simp6 [simp]: + "(\x. getter (cast (setter (\_. v) x)) = getter (cast x)) + \ h \ put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'" + apply (cases "cast character_data_ptr = object_ptr") + by(auto simp add: put_M_defs get_M_defs ObjectMonad.get_M_defs NodeMonad.get_M_defs + get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def preserved_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + bind_eq_Some_conv split: option.splits) +lemma CharacterData_simp7 [simp]: + "(\x. getter (cast (setter (\_. v) x)) = getter (cast x)) + \ h \ put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr getter) h h'" + apply(cases "cast character_data_ptr = node_ptr") + by(auto simp add: put_M_defs get_M_defs ObjectMonad.get_M_defs NodeMonad.get_M_defs + get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def preserved_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + bind_eq_Some_conv split: option.splits) + +lemma CharacterData_simp8 [simp]: + "cast character_data_ptr \ node_ptr + \ h \ put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr getter) h h'" + by(auto simp add: put_M_defs get_M_defs get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def NodeMonad.get_M_defs + preserved_def split: option.splits dest: get_heap_E) +lemma CharacterData_simp9 [simp]: + "h \ put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr setter v \\<^sub>h h' + \ (\x. getter (cast (setter (\_. v) x)) = getter (cast x)) + \ preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr getter) h h'" + apply(cases "cast character_data_ptr \ node_ptr") + by(auto simp add: put_M_defs get_M_defs get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def + NodeMonad.get_M_defs preserved_def split: option.splits bind_splits + dest: get_heap_E) +lemma CharacterData_simp10 [simp]: + "cast character_data_ptr \ node_ptr + \ h \ put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr getter) h h'" + by(auto simp add: NodeMonad.put_M_defs get_M_defs get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def NodeMonad.get_M_defs + preserved_def split: option.splits dest: get_heap_E) + +lemma CharacterData_simp11 [simp]: + "cast character_data_ptr \ object_ptr + \ h \ put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'" + by(auto simp add: put_M_defs get_M_defs get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def + ObjectMonad.get_M_defs preserved_def + split: option.splits dest: get_heap_E) + +lemma CharacterData_simp12 [simp]: + "h \ put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr setter v \\<^sub>h h' + \ (\x. getter (cast (setter (\_. v) x)) = getter (cast x)) + \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'" + apply(cases "cast character_data_ptr \ object_ptr") + apply(auto simp add: put_M_defs get_M_defs get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def + get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def ObjectMonad.get_M_defs preserved_def + split: option.splits bind_splits dest: get_heap_E)[1] + by(auto simp add: put_M_defs get_M_defs get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def + get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def ObjectMonad.get_M_defs preserved_def + split: option.splits bind_splits dest: get_heap_E)[1] + +lemma CharacterData_simp13 [simp]: + "cast character_data_ptr \ object_ptr \ h \ put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr getter) h h'" + by(auto simp add: ObjectMonad.put_M_defs get_M_defs get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + ObjectMonad.get_M_defs preserved_def split: option.splits dest: get_heap_E) + +lemma new_element_get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a: + "h \ new_element \\<^sub>h h' \ preserved (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr getter) h h'" + by(auto simp add: new_element_def get_M_defs preserved_def split: prod.splits option.splits + elim!: bind_returns_result_E bind_returns_heap_E) + + +subsection\Creating CharacterData\ + +definition new_character_data :: "(_, (_) character_data_ptr) dom_prog" + where + "new_character_data = do { + h \ get_heap; + (new_ptr, h') \ return (new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h); + return_heap h'; + return new_ptr + }" + +lemma new_character_data_ok [simp]: + "h \ ok new_character_data" + by(auto simp add: new_character_data_def split: prod.splits) + +lemma new_character_data_ptr_in_heap: + assumes "h \ new_character_data \\<^sub>h h'" + and "h \ new_character_data \\<^sub>r new_character_data_ptr" + shows "new_character_data_ptr |\| character_data_ptr_kinds h'" + using assms + unfolding new_character_data_def + by(auto simp add: new_character_data_def new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def Let_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ptr_in_heap + is_OK_returns_result_I + elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_character_data_ptr_not_in_heap: + assumes "h \ new_character_data \\<^sub>h h'" + and "h \ new_character_data \\<^sub>r new_character_data_ptr" + shows "new_character_data_ptr |\| character_data_ptr_kinds h" + using assms new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ptr_not_in_heap + by(auto simp add: new_character_data_def split: prod.splits + elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_character_data_new_ptr: + assumes "h \ new_character_data \\<^sub>h h'" + and "h \ new_character_data \\<^sub>r new_character_data_ptr" + shows "object_ptr_kinds h' = object_ptr_kinds h |\| {|cast new_character_data_ptr|}" + using assms new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_new_ptr + by(auto simp add: new_character_data_def split: prod.splits + elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_character_data_is_character_data_ptr: + assumes "h \ new_character_data \\<^sub>r new_character_data_ptr" + shows "is_character_data_ptr new_character_data_ptr" + using assms new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_is_character_data_ptr + by(auto simp add: new_character_data_def elim!: bind_returns_result_E split: prod.splits) + +lemma new_character_data_child_nodes: + assumes "h \ new_character_data \\<^sub>h h'" + assumes "h \ new_character_data \\<^sub>r new_character_data_ptr" + shows "h' \ get_M new_character_data_ptr val \\<^sub>r ''''" + using assms + by(auto simp add: get_M_defs new_character_data_def new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def Let_def + split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_character_data_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: + "h \ new_character_data \\<^sub>h h' \ h \ new_character_data \\<^sub>r new_character_data_ptr + \ ptr \ cast new_character_data_ptr \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'" + by(auto simp add: new_character_data_def ObjectMonad.get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) +lemma new_character_data_get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e: + "h \ new_character_data \\<^sub>h h' \ h \ new_character_data \\<^sub>r new_character_data_ptr + \ ptr \ cast new_character_data_ptr \ preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr getter) h h'" + by(auto simp add: new_character_data_def NodeMonad.get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) +lemma new_character_data_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: + "h \ new_character_data \\<^sub>h h' \ h \ new_character_data \\<^sub>r new_character_data_ptr + \ preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'" + by(auto simp add: new_character_data_def ElementMonad.get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) +lemma new_character_data_get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a: + "h \ new_character_data \\<^sub>h h' \ h \ new_character_data \\<^sub>r new_character_data_ptr + \ ptr \ new_character_data_ptr \ preserved (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr getter) h h'" + by(auto simp add: new_character_data_def get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) + + + +subsection\Modified Heaps\ + +lemma get_CharacterData_ptr_simp [simp]: + "get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h) + = (if ptr = cast character_data_ptr then cast obj else get character_data_ptr h)" + by(auto simp add: get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def split: option.splits Option.bind_splits) + +lemma Character_data_ptr_kinds_simp [simp]: + "character_data_ptr_kinds (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h) = character_data_ptr_kinds h |\| + (if is_character_data_ptr_kind ptr then {|the (cast ptr)|} else {||})" + by(auto simp add: character_data_ptr_kinds_def is_node_ptr_kind_def split: option.splits) + +lemma type_wf_put_I: + assumes "type_wf h" + assumes "ElementClass.type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "is_character_data_ptr_kind ptr \ is_character_data_kind obj" + shows "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + using assms + by(auto simp add: type_wf_defs split: option.splits) + +lemma type_wf_put_ptr_not_in_heap_E: + assumes "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "ptr |\| object_ptr_kinds h" + shows "type_wf h" + using assms + by(auto simp add: type_wf_defs elim!: ElementMonad.type_wf_put_ptr_not_in_heap_E + split: option.splits if_splits) + +lemma type_wf_put_ptr_in_heap_E: + assumes "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "ptr |\| object_ptr_kinds h" + assumes "ElementClass.type_wf h" + assumes "is_character_data_ptr_kind ptr \ is_character_data_kind (the (get ptr h))" + shows "type_wf h" + using assms + apply(auto simp add: type_wf_defs split: option.splits if_splits)[1] + apply(case_tac "x2 = cast character_data_ptr") + apply(auto)[1] + apply(drule_tac x=character_data_ptr in allE) + apply(simp) + apply (metis (no_types, lifting) ElementClass.get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf assms(2) bind.bind_lunit + cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_inv cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inv get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def + get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def option.exhaust_sel) + by(blast) + +subsection\Preserving Types\ + +lemma new_element_type_wf_preserved [simp]: + assumes "h \ new_element \\<^sub>h h'" + shows "type_wf h = type_wf h'" + using assms + apply(auto simp add: new_element_def new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + elim!: bind_returns_heap_E type_wf_put_ptr_not_in_heap_E + intro!: type_wf_put_I split: if_splits)[1] + using CharacterDataClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t assms new_element_type_wf_preserved apply blast + using element_ptrs_def apply fastforce + using CharacterDataClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t assms new_element_type_wf_preserved apply blast + by (metis Suc_n_not_le_n element_ptr.sel(1) element_ptrs_def fMax_ge ffmember_filter + fimage_eqI is_element_ptr_ref) + +lemma new_element_is_l_new_element: "l_new_element type_wf" + using l_new_element.intro new_element_type_wf_preserved + by blast + +lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_tag_type_type_wf_preserved [simp]: + "h \ put_M element_ptr tag_type_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: ElementMonad.put_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + dest!: get_heap_E + elim!: bind_returns_heap_E2 + intro!: type_wf_put_I ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I + ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs split: option.splits)[1] + using ObjectMonad.type_wf_put_ptr_in_heap_E ObjectMonad.type_wf_put_ptr_not_in_heap_E apply blast + apply (metis (mono_tags, lifting) ElementMonad.a_get_M_def bind_eq_Some_conv error_returns_result + get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get_heap_returns_result option.exhaust_sel option.simps(4)) + by (metis (no_types, lifting) ElementMonad.a_get_M_def error_returns_result + get_heap_returns_result option.exhaust_sel option.simps(4)) + + +lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_child_nodes_type_wf_preserved [simp]: + "h \ put_M element_ptr child_nodes_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: ElementMonad.put_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + dest!: get_heap_E elim!: bind_returns_heap_E2 + intro!: type_wf_put_I ElementMonad.type_wf_put_I + NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs + split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs + split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs + split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs + split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs + split: option.splits)[1] + using ObjectMonad.type_wf_put_ptr_in_heap_E ObjectMonad.type_wf_put_ptr_not_in_heap_E apply blast + apply (metis (mono_tags, lifting) ElementMonad.a_get_M_def bind_eq_Some_conv error_returns_result + get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get_heap_returns_result option.exhaust_sel option.simps(4)) + by (metis (no_types, lifting) ElementMonad.a_get_M_def error_returns_result get_heap_returns_result + option.exhaust_sel option.simps(4)) + +lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_attrs_type_wf_preserved [simp]: + "h \ put_M element_ptr attrs_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: ElementMonad.put_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + dest!: get_heap_E + elim!: bind_returns_heap_E2 + intro!: type_wf_put_I ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I + ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs + ElementMonad.get_M_defs split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs + ElementMonad.get_M_defs split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs + ElementMonad.get_M_defs split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs + ElementMonad.get_M_defs split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs + ElementMonad.get_M_defs split: option.splits)[1] + using ObjectMonad.type_wf_put_ptr_in_heap_E ObjectMonad.type_wf_put_ptr_not_in_heap_E apply blast + apply (metis (mono_tags, lifting) ElementMonad.a_get_M_def bind_eq_Some_conv error_returns_result get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get_heap_returns_result option.exhaust_sel option.simps(4)) + by (metis (no_types, lifting) ElementMonad.a_get_M_def error_returns_result get_heap_returns_result option.exhaust_sel option.simps(4)) + +lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_shadow_root_opt_type_wf_preserved [simp]: + "h \ put_M element_ptr shadow_root_opt_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: ElementMonad.put_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + dest!: get_heap_E + elim!: bind_returns_heap_E2 + intro!: type_wf_put_I ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I + ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs + ElementMonad.get_M_defs split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs + ElementMonad.get_M_defs split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs + ElementMonad.get_M_defs split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs + ElementMonad.get_M_defs split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs + ElementMonad.get_M_defs split: option.splits)[1] + using ObjectMonad.type_wf_put_ptr_in_heap_E ObjectMonad.type_wf_put_ptr_not_in_heap_E apply blast + apply (metis (mono_tags, lifting) bind_eq_Some_conv get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + by (metis (no_types, lifting) ElementMonad.a_get_M_def error_returns_result get_heap_returns_result + option.exhaust_sel option.simps(4)) + + +lemma new_character_data_type_wf_preserved [simp]: + "h \ new_character_data \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: new_character_data_def new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def Let_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + elim!: bind_returns_heap_E type_wf_put_ptr_not_in_heap_E + intro!: type_wf_put_I ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I + split: if_splits)[1] + apply(simp_all add: type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs is_node_kind_def) + by (meson new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ptr_not_in_heap) + +locale l_new_character_data = l_type_wf + + assumes new_character_data_types_preserved: "h \ new_character_data \\<^sub>h h' \ type_wf h = type_wf h'" + +lemma new_character_data_is_l_new_character_data: "l_new_character_data type_wf" + using l_new_character_data.intro new_character_data_type_wf_preserved + by blast + +lemma put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_val_type_wf_preserved [simp]: + "h \ put_M character_data_ptr val_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: CharacterDataMonad.put_M_defs put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + CharacterDataClass.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e CharacterDataClass.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + is_node_kind_def + dest!: get_heap_E + elim!: bind_returns_heap_E2 + intro!: type_wf_put_I ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I + ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs CharacterDataMonad.get_M_defs + split: option.splits)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs CharacterDataMonad.get_M_defs + ObjectClass.a_type_wf_def + split: option.splits)[1] + apply (metis (no_types, lifting) bind_eq_Some_conv get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def) + by metis + +lemma character_data_ptr_kinds_small: + assumes "\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + shows "character_data_ptr_kinds h = character_data_ptr_kinds h'" + by(simp add: character_data_ptr_kinds_def node_ptr_kinds_def preserved_def + object_ptr_kinds_preserved_small[OF assms]) + +lemma character_data_ptr_kinds_preserved: + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "\h h'. \w \ SW. h \ w \\<^sub>h h' + \ (\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h')" + shows "character_data_ptr_kinds h = character_data_ptr_kinds h'" + using writes_small_big[OF assms] + apply(simp add: reflp_def transp_def preserved_def character_data_ptr_kinds_def) + by (metis assms node_ptr_kinds_preserved) + + +lemma type_wf_preserved_small: + assumes "\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + assumes "\node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'" + assumes "\element_ptr. preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr RElement.nothing) h h'" + assumes "\character_data_ptr. preserved (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr + RCharacterData.nothing) h h'" + shows "type_wf h = type_wf h'" + using type_wf_preserved_small[OF assms(1) assms(2) assms(3)] + allI[OF assms(4), of id, simplified] character_data_ptr_kinds_small[OF assms(1)] + apply(auto simp add: type_wf_defs preserved_def get_M_defs character_data_ptr_kinds_small[OF assms(1)] + split: option.splits)[1] + apply(force) + by force + +lemma type_wf_preserved: + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \element_ptr. preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr RElement.nothing) h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \character_data_ptr. preserved (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr + RCharacterData.nothing) h h'" + shows "type_wf h = type_wf h'" +proof - + have "\h h' w. w \ SW \ h \ w \\<^sub>h h' \ type_wf h = type_wf h'" + using assms type_wf_preserved_small by fast + with assms(1) assms(2) show ?thesis + apply(rule writes_small_big) + by(auto simp add: reflp_def transp_def) +qed + +lemma type_wf_drop: "type_wf h \ type_wf (Heap (fmdrop ptr (the_heap h)))" + apply(auto simp add: type_wf_def ElementMonad.type_wf_drop + l_type_wf_def\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a.a_type_wf_def)[1] + using type_wf_drop + by (metis (no_types, lifting) ElementClass.get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf character_data_ptr_kinds_commutes + fmlookup_drop get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def heap.sel + node_ptr_kinds_commutes) + +end diff --git a/Core_DOM/monads/DocumentMonad.thy b/Core_DOM/monads/DocumentMonad.thy new file mode 100644 index 0000000..3ed400d --- /dev/null +++ b/Core_DOM/monads/DocumentMonad.thy @@ -0,0 +1,605 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Document\ +text\In this theory, we introduce the monadic method setup for the Document class.\ + +theory DocumentMonad + imports + CharacterDataMonad + "../classes/DocumentClass" +begin + +type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, + 'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData, 'Document, 'result) dom_prog + = "((_) heap, exception, 'result) prog" +register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, + 'shadow_root_ptr, 'Object, 'Node, 'Element, 'CharacterData, 'Document, 'result) dom_prog" + + +global_interpretation l_ptr_kinds_M document_ptr_kinds defines document_ptr_kinds_M = a_ptr_kinds_M . +lemmas document_ptr_kinds_M_defs = a_ptr_kinds_M_def + +lemma document_ptr_kinds_M_eq: + assumes "|h \ object_ptr_kinds_M|\<^sub>r = |h' \ object_ptr_kinds_M|\<^sub>r" + shows "|h \ document_ptr_kinds_M|\<^sub>r = |h' \ document_ptr_kinds_M|\<^sub>r" + using assms + by(auto simp add: document_ptr_kinds_M_defs object_ptr_kinds_M_defs document_ptr_kinds_def) + +lemma document_ptr_kinds_M_reads: + "reads (\object_ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing)}) document_ptr_kinds_M h h'" + using object_ptr_kinds_M_reads + + apply(simp add: reads_def object_ptr_kinds_M_defs document_ptr_kinds_M_defs + document_ptr_kinds_def preserved_def) + by (metis (mono_tags, hide_lams) object_ptr_kinds_preserved_small old.unit.exhaust preserved_def) + +global_interpretation l_dummy defines get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t = "l_get_M.a_get_M get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t" . +lemma get_M_is_l_get_M: "l_get_M get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t type_wf document_ptr_kinds" + apply(simp add: get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_type_wf l_get_M_def) + by (metis ObjectClass.get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf ObjectClass.type_wf_defs bind_eq_None_conv + document_ptr_kinds_commutes get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def option.simps(3)) +lemmas get_M_defs = get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def[unfolded l_get_M.a_get_M_def[OF get_M_is_l_get_M]] + +adhoc_overloading get_M get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t + +locale l_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas = l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t +begin +sublocale l_get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_lemmas by unfold_locales + +interpretation l_get_M get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t type_wf document_ptr_kinds + apply(unfold_locales) + apply (simp add: get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_type_wf local.type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t) + by (meson DocumentMonad.get_M_is_l_get_M l_get_M_def) +lemmas get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok = get_M_ok[folded get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def] +end + +global_interpretation l_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf by unfold_locales + + +global_interpretation l_put_M type_wf document_ptr_kinds get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t + rewrites "a_get_M = get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t" defines put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t = a_put_M + apply (simp add: get_M_is_l_get_M l_put_M_def) + by (simp add: get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + +lemmas put_M_defs = a_put_M_def +adhoc_overloading put_M put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t + + +locale l_put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas = l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t +begin +sublocale l_put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_lemmas by unfold_locales + +interpretation l_put_M type_wf document_ptr_kinds get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t + apply(unfold_locales) + apply (simp add: get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_type_wf local.type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t) + by (meson DocumentMonad.get_M_is_l_get_M l_get_M_def) +lemmas put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok = put_M_ok[folded put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def] +end + +global_interpretation l_put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf by unfold_locales + + +lemma document_put_get [simp]: + "h \ put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr setter v \\<^sub>h h' + \ (\x. getter (setter (\_. v) x) = v) + \ h' \ get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr getter \\<^sub>r v" + by(auto simp add: put_M_defs get_M_defs split: option.splits) +lemma get_M_Mdocument_preserved1 [simp]: + "document_ptr \ document_ptr' + \ h \ put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr' getter) h h'" + by(auto simp add: put_M_defs get_M_defs preserved_def split: option.splits dest: get_heap_E) +lemma document_put_get_preserved [simp]: + "h \ put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr setter v \\<^sub>h h' + \ (\x. getter (setter (\_. v) x) = getter x) + \ preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr' getter) h h'" + apply(cases "document_ptr = document_ptr'") + by(auto simp add: put_M_defs get_M_defs preserved_def split: option.splits dest: get_heap_E) + +lemma get_M_Mdocument_preserved2 [simp]: + "h \ put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr setter v \\<^sub>h h' \ preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr getter) h h'" + by(auto simp add: put_M_defs get_M_defs NodeMonad.get_M_defs get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def preserved_def split: option.splits dest: get_heap_E) + +lemma get_M_Mdocument_preserved3 [simp]: + "cast document_ptr \ object_ptr + \ h \ put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'" + by(auto simp add: put_M_defs get_M_defs get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def ObjectMonad.get_M_defs + preserved_def split: option.splits dest: get_heap_E) +lemma get_M_Mdocument_preserved4 [simp]: + "h \ put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr setter v \\<^sub>h h' + \ (\x. getter (cast (setter (\_. v) x)) = getter (cast x)) + \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'" + apply(cases "cast document_ptr \ object_ptr")[1] + by(auto simp add: put_M_defs get_M_defs get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + ObjectMonad.get_M_defs preserved_def + split: option.splits bind_splits dest: get_heap_E) + +lemma get_M_Mdocument_preserved5 [simp]: + "cast document_ptr \ object_ptr + \ h \ put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr getter) h h'" + by(auto simp add: ObjectMonad.put_M_defs get_M_defs get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def ObjectMonad.get_M_defs + preserved_def split: option.splits dest: get_heap_E) + +lemma get_M_Mdocument_preserved6 [simp]: + "h \ put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr setter v \\<^sub>h h' \ preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr getter) h h'" + by(auto simp add: put_M_defs ElementMonad.get_M_defs preserved_def + split: option.splits dest: get_heap_E) +lemma get_M_Mdocument_preserved7 [simp]: + "h \ put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \\<^sub>h h' \ preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr getter) h h'" + by(auto simp add: ElementMonad.put_M_defs get_M_defs preserved_def + split: option.splits dest: get_heap_E) +lemma get_M_Mdocument_preserved8 [simp]: + "h \ put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr getter) h h'" + by(auto simp add: put_M_defs CharacterDataMonad.get_M_defs preserved_def + split: option.splits dest: get_heap_E) +lemma get_M_Mdocument_preserved9 [simp]: + "h \ put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr getter) h h'" + by(auto simp add: CharacterDataMonad.put_M_defs get_M_defs preserved_def + split: option.splits dest: get_heap_E) +lemma get_M_Mdocument_preserved10 [simp]: + "(\x. getter (cast (setter (\_. v) x)) = getter (cast x)) + \ h \ put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr setter v \\<^sub>h h' \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'" + apply(cases "cast document_ptr = object_ptr") + by(auto simp add: put_M_defs get_M_defs ObjectMonad.get_M_defs NodeMonad.get_M_defs get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def preserved_def put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def bind_eq_Some_conv + split: option.splits) + +lemma new_element_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: + "h \ new_element \\<^sub>h h' \ preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'" + by(auto simp add: new_element_def get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_character_data_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: + "h \ new_character_data \\<^sub>h h' \ preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'" + by(auto simp add: new_character_data_def get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) + + +subsection \Creating Documents\ + +definition new_document :: "(_, (_) document_ptr) dom_prog" + where + "new_document = do { + h \ get_heap; + (new_ptr, h') \ return (new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h); + return_heap h'; + return new_ptr + }" + +lemma new_document_ok [simp]: + "h \ ok new_document" + by(auto simp add: new_document_def split: prod.splits) + +lemma new_document_ptr_in_heap: + assumes "h \ new_document \\<^sub>h h'" + and "h \ new_document \\<^sub>r new_document_ptr" + shows "new_document_ptr |\| document_ptr_kinds h'" + using assms + unfolding new_document_def + by(auto simp add: new_document_def new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap is_OK_returns_result_I + elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_document_ptr_not_in_heap: + assumes "h \ new_document \\<^sub>h h'" + and "h \ new_document \\<^sub>r new_document_ptr" + shows "new_document_ptr |\| document_ptr_kinds h" + using assms new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_not_in_heap + by(auto simp add: new_document_def split: prod.splits elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_document_new_ptr: + assumes "h \ new_document \\<^sub>h h'" + and "h \ new_document \\<^sub>r new_document_ptr" + shows "object_ptr_kinds h' = object_ptr_kinds h |\| {|cast new_document_ptr|}" + using assms new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_new_ptr + by(auto simp add: new_document_def split: prod.splits elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_document_is_document_ptr: + assumes "h \ new_document \\<^sub>r new_document_ptr" + shows "is_document_ptr new_document_ptr" + using assms new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_is_document_ptr + by(auto simp add: new_document_def elim!: bind_returns_result_E split: prod.splits) + +lemma new_document_doctype: + assumes "h \ new_document \\<^sub>h h'" + assumes "h \ new_document \\<^sub>r new_document_ptr" + shows "h' \ get_M new_document_ptr doctype \\<^sub>r ''''" + using assms + by(auto simp add: get_M_defs new_document_def new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def + split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_document_document_element: + assumes "h \ new_document \\<^sub>h h'" + assumes "h \ new_document \\<^sub>r new_document_ptr" + shows "h' \ get_M new_document_ptr document_element \\<^sub>r None" + using assms + by(auto simp add: get_M_defs new_document_def new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def + split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_document_disconnected_nodes: + assumes "h \ new_document \\<^sub>h h'" + assumes "h \ new_document \\<^sub>r new_document_ptr" + shows "h' \ get_M new_document_ptr disconnected_nodes \\<^sub>r []" + using assms + by(auto simp add: get_M_defs new_document_def new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def + split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E) + + +lemma new_document_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: + "h \ new_document \\<^sub>h h' \ h \ new_document \\<^sub>r new_document_ptr + \ ptr \ cast new_document_ptr \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'" + by(auto simp add: new_document_def ObjectMonad.get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) +lemma new_document_get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e: + "h \ new_document \\<^sub>h h' \ h \ new_document \\<^sub>r new_document_ptr + \ preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr getter) h h'" + by(auto simp add: new_document_def NodeMonad.get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) +lemma new_document_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: + "h \ new_document \\<^sub>h h' \ h \ new_document \\<^sub>r new_document_ptr + \ preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'" + by(auto simp add: new_document_def ElementMonad.get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) +lemma new_document_get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a: + "h \ new_document \\<^sub>h h' \ h \ new_document \\<^sub>r new_document_ptr + \ preserved (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr getter) h h'" + by(auto simp add: new_document_def CharacterDataMonad.get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) +lemma new_document_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: + "h \ new_document \\<^sub>h h' + \ h \ new_document \\<^sub>r new_document_ptr \ ptr \ new_document_ptr + \ preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'" + by(auto simp add: new_document_def get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) + + + +subsection \Modified Heaps\ + +lemma get_document_ptr_simp [simp]: + "get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h) + = (if ptr = cast document_ptr then cast obj else get document_ptr h)" + by(auto simp add: get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def split: option.splits Option.bind_splits) + +lemma document_ptr_kidns_simp [simp]: + "document_ptr_kinds (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h) + = document_ptr_kinds h |\| (if is_document_ptr_kind ptr then {|the (cast ptr)|} else {||})" + by(auto simp add: document_ptr_kinds_def split: option.splits) + +lemma type_wf_put_I: + assumes "type_wf h" + assumes "CharacterDataClass.type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "is_document_ptr_kind ptr \ is_document_kind obj" + shows "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + using assms + by(auto simp add: type_wf_defs is_document_kind_def split: option.splits) + +lemma type_wf_put_ptr_not_in_heap_E: + assumes "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "ptr |\| object_ptr_kinds h" + shows "type_wf h" + using assms + by(auto simp add: type_wf_defs elim!: CharacterDataMonad.type_wf_put_ptr_not_in_heap_E + split: option.splits if_splits) + +lemma type_wf_put_ptr_in_heap_E: + assumes "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "ptr |\| object_ptr_kinds h" + assumes "CharacterDataClass.type_wf h" + assumes "is_document_ptr_kind ptr \ is_document_kind (the (get ptr h))" + shows "type_wf h" + using assms + apply(auto simp add: type_wf_defs elim!: CharacterDataMonad.type_wf_put_ptr_in_heap_E + split: option.splits if_splits)[1] + by (metis (no_types, lifting) CharacterDataClass.get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf bind.bind_lunit get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + is_document_kind_def option.collapse) + + + +subsection \Preserving Types\ + +lemma new_element_type_wf_preserved [simp]: + "h \ new_element \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: new_element_def new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + DocumentClass.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a DocumentClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + DocumentClass.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e DocumentClass.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + is_node_kind_def element_ptrs_def + elim!: bind_returns_heap_E type_wf_put_ptr_not_in_heap_E + intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I + NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I + split: if_splits)[1] + apply fastforce + by (metis Suc_n_not_le_n element_ptr.sel(1) element_ptrs_def fMax_ge ffmember_filter + fimage_eqI is_element_ptr_ref) + +lemma new_element_is_l_new_element [instances]: + "l_new_element type_wf" + using l_new_element.intro new_element_type_wf_preserved + by blast + +lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_tag_type_type_wf_preserved [simp]: + "h \ put_M element_ptr tag_type_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: ElementMonad.put_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + DocumentClass.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a DocumentClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + DocumentClass.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e DocumentClass.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + is_node_kind_def + dest!: get_heap_E + elim!: bind_returns_heap_E2 + intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I + NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs NodeClass.type_wf_defs + ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply (metis (no_types, lifting) Option.bind_cong bind_rzero + get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def option.distinct(1)) + by metis + +lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_child_nodes_type_wf_preserved [simp]: + "h \ put_M element_ptr child_nodes_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: ElementMonad.put_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + DocumentClass.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a DocumentClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + DocumentClass.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e DocumentClass.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + is_node_kind_def + dest!: get_heap_E + elim!: bind_returns_heap_E2 + intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I + NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply (metis (no_types, lifting) Option.bind_cong bind_rzero get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def option.distinct(1)) + by metis + +lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_attrs_type_wf_preserved [simp]: + "h \ put_M element_ptr attrs_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: ElementMonad.put_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + DocumentClass.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a DocumentClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + DocumentClass.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e DocumentClass.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + is_node_kind_def + dest!: get_heap_E + elim!: bind_returns_heap_E2 + intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I + NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply (metis (no_types, lifting) Option.bind_cong bind_rzero get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def option.distinct(1)) + by metis + +lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_shadow_root_opt_type_wf_preserved [simp]: + "h \ put_M element_ptr shadow_root_opt_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: ElementMonad.put_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + DocumentClass.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a DocumentClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + DocumentClass.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e DocumentClass.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + is_node_kind_def + dest!: get_heap_E + elim!: bind_returns_heap_E2 + intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I + NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply (metis (no_types, lifting) Option.bind_cong bind_rzero get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def option.distinct(1)) + by metis + +lemma new_character_data_type_wf_preserved [simp]: + "h \ new_character_data \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: ElementMonad.put_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + DocumentClass.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a DocumentClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + DocumentClass.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e DocumentClass.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + is_node_kind_def + new_character_data_def new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def Let_def put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + dest!: get_heap_E + elim!: bind_returns_heap_E2 bind_returns_heap_E type_wf_put_ptr_not_in_heap_E + intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I + NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1] + by (meson new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ptr_not_in_heap) + +lemma new_character_data_is_l_new_character_data [instances]: + "l_new_character_data type_wf" + using l_new_character_data.intro new_character_data_type_wf_preserved + by blast + +lemma put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_val_type_wf_preserved [simp]: + "h \ put_M character_data_ptr val_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: CharacterDataMonad.put_M_defs put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + DocumentClass.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a DocumentClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + DocumentClass.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e DocumentClass.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t is_node_kind_def + dest!: get_heap_E elim!: bind_returns_heap_E2 + intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I ElementMonad.type_wf_put_I + NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: is_node_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply (metis (no_types, lifting) CharacterDataMonad.a_get_M_def bind_eq_None_conv + error_returns_result get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def get_heap_returns_result option.exhaust_sel + option.simps(4)) + by (metis (no_types, lifting) CharacterDataMonad.a_get_M_def error_returns_result + get_heap_returns_result option.exhaust_sel option.simps(4)) + +lemma new_document_type_wf_preserved [simp]: "h \ new_document \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: new_document_def new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + DocumentClass.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a DocumentClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + DocumentClass.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e DocumentClass.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + is_node_ptr_kind_none + elim!: bind_returns_heap_E type_wf_put_ptr_not_in_heap_E + intro!: type_wf_put_I ElementMonad.type_wf_put_I CharacterDataMonad.type_wf_put_I + NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I + split: if_splits)[1] + apply(auto simp add: type_wf_defs ElementClass.type_wf_defs CharacterDataClass.type_wf_defs + NodeClass.type_wf_defs ObjectClass.type_wf_defs is_document_kind_def + split: option.splits)[1] + by (meson new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_not_in_heap) + +locale l_new_document = l_type_wf + + assumes new_document_types_preserved: "h \ new_document \\<^sub>h h' \ type_wf h = type_wf h'" + +lemma new_document_is_l_new_document [instances]: "l_new_document type_wf" + using l_new_document.intro new_document_type_wf_preserved + by blast + +lemma put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_doctype_type_wf_preserved [simp]: + "h \ put_M document_ptr doctype_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: put_M_defs put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def dest!: get_heap_E + elim!: bind_returns_heap_E2 + intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I + ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply(auto simp add: is_document_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + apply(auto simp add: get_M_defs) + by (metis (no_types, lifting) error_returns_result option.exhaust_sel option.simps(4)) + +lemma put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_document_element_type_wf_preserved [simp]: + "h \ put_M document_ptr document_element_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: put_M_defs put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + DocumentClass.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a + DocumentClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + DocumentClass.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e + DocumentClass.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t is_node_ptr_kind_none + cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_none is_document_kind_def + dest!: get_heap_E + elim!: bind_returns_heap_E2 + intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I + ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I + ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: get_M_defs is_document_kind_def type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs + split: option.splits)[1] + by (metis) + +lemma put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_disconnected_nodes_type_wf_preserved [simp]: + "h \ put_M document_ptr disconnected_nodes_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: put_M_defs put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + DocumentClass.type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a + DocumentClass.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + DocumentClass.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e + DocumentClass.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + is_node_ptr_kind_none + cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_none is_document_kind_def + dest!: get_heap_E + elim!: bind_returns_heap_E2 + intro!: type_wf_put_I CharacterDataMonad.type_wf_put_I + ElementMonad.type_wf_put_I NodeMonad.type_wf_put_I + ObjectMonad.type_wf_put_I)[1] + apply(auto simp add: is_document_kind_def get_M_defs type_wf_defs ElementClass.type_wf_defs + NodeClass.type_wf_defs ElementMonad.get_M_defs ObjectClass.type_wf_defs + CharacterDataClass.type_wf_defs split: option.splits)[1] + by (metis) + +lemma document_ptr_kinds_small: + assumes "\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + shows "document_ptr_kinds h = document_ptr_kinds h'" + by(simp add: document_ptr_kinds_def preserved_def object_ptr_kinds_preserved_small[OF assms]) + +lemma document_ptr_kinds_preserved: + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "\h h'. \w \ SW. h \ w \\<^sub>h h' + \ (\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h')" + shows "document_ptr_kinds h = document_ptr_kinds h'" + using writes_small_big[OF assms] + apply(simp add: reflp_def transp_def preserved_def document_ptr_kinds_def) + by (metis assms object_ptr_kinds_preserved) + +lemma type_wf_preserved_small: + assumes "\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + assumes "\node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'" + assumes "\element_ptr. preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr RElement.nothing) h h'" + assumes "\character_data_ptr. preserved + (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr RCharacterData.nothing) h h'" + assumes "\document_ptr. preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr RDocument.nothing) h h'" + shows "DocumentClass.type_wf h = DocumentClass.type_wf h'" + using type_wf_preserved_small[OF assms(1) assms(2) assms(3) assms(4)] + allI[OF assms(5), of id, simplified] document_ptr_kinds_small[OF assms(1)] + apply(auto simp add: type_wf_defs )[1] + apply(auto simp add: type_wf_defs preserved_def get_M_defs document_ptr_kinds_small[OF assms(1)] + split: option.splits)[1] + apply force + apply(auto simp add: type_wf_defs preserved_def get_M_defs document_ptr_kinds_small[OF assms(1)] + split: option.splits)[1] + by force + +lemma type_wf_preserved: + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \element_ptr. preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr RElement.nothing) h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \character_data_ptr. preserved + (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr RCharacterData.nothing) h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \document_ptr. preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr RDocument.nothing) h h'" + shows "DocumentClass.type_wf h = DocumentClass.type_wf h'" +proof - + have "\h h' w. w \ SW \ h \ w \\<^sub>h h' \ DocumentClass.type_wf h = DocumentClass.type_wf h'" + using assms type_wf_preserved_small by fast + with assms(1) assms(2) show ?thesis + apply(rule writes_small_big) + by(auto simp add: reflp_def transp_def) +qed + +lemma type_wf_drop: "type_wf h \ type_wf (Heap (fmdrop ptr (the_heap h)))" + apply(auto simp add: type_wf_defs)[1] + using type_wf_drop + apply blast + by (metis (mono_tags, lifting) comp_apply document_ptr_kinds_commutes ffmember_filter fmdom_filter + fmfilter_alt_defs(1) fmlookup_drop get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def heap.sel object_ptr_kinds_def) +end diff --git a/Core_DOM/monads/ElementMonad.thy b/Core_DOM/monads/ElementMonad.thy new file mode 100644 index 0000000..89d028c --- /dev/null +++ b/Core_DOM/monads/ElementMonad.thy @@ -0,0 +1,456 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Element\ +text\In this theory, we introduce the monadic method setup for the Element class.\ +theory ElementMonad + imports + NodeMonad + "../classes/ElementClass" +begin + +type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr, + 'shadow_root_ptr, 'Object, 'Node, 'Element,'result) dom_prog + = "((_) heap, exception, 'result) prog" +register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, + 'document_ptr, 'shadow_root_ptr, 'Object, 'Node, 'Element,'result) dom_prog" + + +global_interpretation l_ptr_kinds_M element_ptr_kinds defines element_ptr_kinds_M = a_ptr_kinds_M . +lemmas element_ptr_kinds_M_defs = a_ptr_kinds_M_def + + +lemma element_ptr_kinds_M_eq: + assumes "|h \ node_ptr_kinds_M|\<^sub>r = |h' \ node_ptr_kinds_M|\<^sub>r" + shows "|h \ element_ptr_kinds_M|\<^sub>r = |h' \ element_ptr_kinds_M|\<^sub>r" + using assms + by(auto simp add: element_ptr_kinds_M_defs node_ptr_kinds_M_defs element_ptr_kinds_def) + +lemma element_ptr_kinds_M_reads: + "reads (\element_ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t element_ptr RObject.nothing)}) element_ptr_kinds_M h h'" + apply(simp add: reads_def node_ptr_kinds_M_defs element_ptr_kinds_M_defs element_ptr_kinds_def + node_ptr_kinds_M_reads preserved_def) + by (metis (mono_tags, hide_lams) node_ptr_kinds_small old.unit.exhaust preserved_def) + +global_interpretation l_dummy defines get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t = "l_get_M.a_get_M get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t" . +lemma get_M_is_l_get_M: "l_get_M get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t type_wf element_ptr_kinds" + apply(simp add: get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_type_wf l_get_M_def) + by (metis (no_types, lifting) ObjectClass.get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf ObjectClass.type_wf_defs + bind_eq_Some_conv bind_eq_Some_conv element_ptr_kinds_commutes get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def node_ptr_kinds_commutes option.simps(3)) +lemmas get_M_defs = get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def[unfolded l_get_M.a_get_M_def[OF get_M_is_l_get_M]] + +adhoc_overloading get_M get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + +locale l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas = l_type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t +begin +sublocale l_get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas by unfold_locales + +interpretation l_get_M get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t type_wf element_ptr_kinds + apply(unfold_locales) + apply (simp add: get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_type_wf local.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t) + by (meson ElementMonad.get_M_is_l_get_M l_get_M_def) +lemmas get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok = get_M_ok[folded get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def] +lemmas get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap = get_M_ptr_in_heap[folded get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def] +end + +global_interpretation l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf by unfold_locales + + +global_interpretation l_put_M type_wf element_ptr_kinds get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + rewrites "a_get_M = get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t" + defines put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t = a_put_M + apply (simp add: get_M_is_l_get_M l_put_M_def) + by (simp add: get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def) + +lemmas put_M_defs = a_put_M_def +adhoc_overloading put_M put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + + +locale l_put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas = l_type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t +begin +sublocale l_put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas by unfold_locales + +interpretation l_put_M type_wf element_ptr_kinds get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t + apply(unfold_locales) + apply (simp add: get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_type_wf local.type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t) + by (meson ElementMonad.get_M_is_l_get_M l_get_M_def) + +lemmas put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok = put_M_ok[folded put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def] +end + +global_interpretation l_put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf by unfold_locales + + +lemma element_put_get [simp]: + "h \ put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \\<^sub>h h' \ (\x. getter (setter (\_. v) x) = v) + \ h' \ get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr getter \\<^sub>r v" + by(auto simp add: put_M_defs get_M_defs split: option.splits) +lemma get_M_Element_preserved1 [simp]: + "element_ptr \ element_ptr' \ h \ put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr' getter) h h'" + by(auto simp add: put_M_defs get_M_defs preserved_def split: option.splits dest: get_heap_E) +lemma element_put_get_preserved [simp]: + "(\x. getter (setter (\_. v) x) = getter x) \ h \ put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr' getter) h h'" + apply(cases "element_ptr = element_ptr'") + by(auto simp add: put_M_defs get_M_defs preserved_def + split: option.splits dest: get_heap_E) +lemma get_M_Element_preserved3 [simp]: + "(\x. getter (cast (setter (\_. v) x)) = getter (cast x)) + \ h \ put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \\<^sub>h h' \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'" + apply(cases "cast element_ptr = object_ptr") + by (auto simp add: put_M_defs get_M_defs ObjectMonad.get_M_defs NodeMonad.get_M_defs get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def preserved_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def bind_eq_Some_conv + split: option.splits) +lemma get_M_Element_preserved4 [simp]: + "(\x. getter (cast (setter (\_. v) x)) = getter (cast x)) + \ h \ put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \\<^sub>h h' \ preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr getter) h h'" + apply(cases "cast element_ptr = node_ptr") + by(auto simp add: put_M_defs get_M_defs ObjectMonad.get_M_defs NodeMonad.get_M_defs get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def preserved_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def bind_eq_Some_conv + split: option.splits) + +lemma get_M_Element_preserved5 [simp]: + "cast element_ptr \ node_ptr \ h \ put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr getter) h h'" + by(auto simp add: put_M_defs get_M_defs get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def NodeMonad.get_M_defs preserved_def + split: option.splits dest: get_heap_E) +lemma get_M_Element_preserved6 [simp]: + "h \ put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \\<^sub>h h' + \ (\x. getter (cast (setter (\_. v) x)) = getter (cast x)) + \ preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr getter) h h'" + apply(cases "cast element_ptr \ node_ptr") + by(auto simp add: put_M_defs get_M_defs get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def NodeMonad.get_M_defs preserved_def + split: option.splits bind_splits dest: get_heap_E) + +lemma get_M_Element_preserved7 [simp]: + "cast element_ptr \ node_ptr \ h \ put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr getter) h h'" + by(auto simp add: NodeMonad.put_M_defs get_M_defs get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def NodeMonad.get_M_defs preserved_def + split: option.splits dest: get_heap_E) + +lemma get_M_Element_preserved8 [simp]: + "cast element_ptr \ object_ptr \ h \ put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'" + by(auto simp add: put_M_defs get_M_defs get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + ObjectMonad.get_M_defs preserved_def + split: option.splits dest: get_heap_E) +lemma get_M_Element_preserved9 [simp]: + "h \ put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \\<^sub>h h' + \ (\x. getter (cast (setter (\_. v) x)) = getter (cast x)) + \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'" + apply(cases "cast element_ptr \ object_ptr") + by(auto simp add: put_M_defs get_M_defs get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + ObjectMonad.get_M_defs preserved_def + split: option.splits bind_splits dest: get_heap_E) + +lemma get_M_Element_preserved10 [simp]: + "cast element_ptr \ object_ptr \ h \ put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr getter) h h'" + by(auto simp add: ObjectMonad.put_M_defs get_M_defs get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + ObjectMonad.get_M_defs preserved_def + split: option.splits dest: get_heap_E) + +subsection\Creating Elements\ + +definition new_element :: "(_, (_) element_ptr) dom_prog" + where + "new_element = do { + h \ get_heap; + (new_ptr, h') \ return (new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h); + return_heap h'; + return new_ptr + }" + +lemma new_element_ok [simp]: + "h \ ok new_element" + by(auto simp add: new_element_def split: prod.splits) + +lemma new_element_ptr_in_heap: + assumes "h \ new_element \\<^sub>h h'" + and "h \ new_element \\<^sub>r new_element_ptr" + shows "new_element_ptr |\| element_ptr_kinds h'" + using assms + unfolding new_element_def + by(auto simp add: new_element_def new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap is_OK_returns_result_I + elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_element_ptr_not_in_heap: + assumes "h \ new_element \\<^sub>h h'" + and "h \ new_element \\<^sub>r new_element_ptr" + shows "new_element_ptr |\| element_ptr_kinds h" + using assms new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_not_in_heap + by(auto simp add: new_element_def split: prod.splits elim!: bind_returns_result_E + bind_returns_heap_E) + +lemma new_element_new_ptr: + assumes "h \ new_element \\<^sub>h h'" + and "h \ new_element \\<^sub>r new_element_ptr" + shows "object_ptr_kinds h' = object_ptr_kinds h |\| {|cast new_element_ptr|}" + using assms new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_new_ptr + by(auto simp add: new_element_def split: prod.splits elim!: bind_returns_result_E + bind_returns_heap_E) + +lemma new_element_is_element_ptr: + assumes "h \ new_element \\<^sub>r new_element_ptr" + shows "is_element_ptr new_element_ptr" + using assms new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_is_element_ptr + by(auto simp add: new_element_def elim!: bind_returns_result_E split: prod.splits) + +lemma new_element_child_nodes: + assumes "h \ new_element \\<^sub>h h'" + assumes "h \ new_element \\<^sub>r new_element_ptr" + shows "h' \ get_M new_element_ptr child_nodes \\<^sub>r []" + using assms + by(auto simp add: get_M_defs new_element_def new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def + split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_element_tag_type: + assumes "h \ new_element \\<^sub>h h'" + assumes "h \ new_element \\<^sub>r new_element_ptr" + shows "h' \ get_M new_element_ptr tag_type \\<^sub>r ''''" + using assms + by(auto simp add: get_M_defs new_element_def new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def + split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_element_attrs: + assumes "h \ new_element \\<^sub>h h'" + assumes "h \ new_element \\<^sub>r new_element_ptr" + shows "h' \ get_M new_element_ptr attrs \\<^sub>r fmempty" + using assms + by(auto simp add: get_M_defs new_element_def new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def + split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_element_shadow_root_opt: + assumes "h \ new_element \\<^sub>h h'" + assumes "h \ new_element \\<^sub>r new_element_ptr" + shows "h' \ get_M new_element_ptr shadow_root_opt \\<^sub>r None" + using assms + by(auto simp add: get_M_defs new_element_def new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def Let_def + split: option.splits prod.splits elim!: bind_returns_result_E bind_returns_heap_E) + +lemma new_element_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: + "h \ new_element \\<^sub>h h' \ h \ new_element \\<^sub>r new_element_ptr \ ptr \ cast new_element_ptr + \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'" + by(auto simp add: new_element_def ObjectMonad.get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) +lemma new_element_get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e: + "h \ new_element \\<^sub>h h' \ h \ new_element \\<^sub>r new_element_ptr \ ptr \ cast new_element_ptr + \ preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr getter) h h'" + by(auto simp add: new_element_def NodeMonad.get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) +lemma new_element_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: + "h \ new_element \\<^sub>h h' \ h \ new_element \\<^sub>r new_element_ptr \ ptr \ new_element_ptr + \ preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'" + by(auto simp add: new_element_def get_M_defs preserved_def + split: prod.splits option.splits elim!: bind_returns_result_E bind_returns_heap_E) + +subsection\Modified Heaps\ + +lemma get_Element_ptr_simp [simp]: + "get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h) + = (if ptr = cast element_ptr then cast obj else get element_ptr h)" + by(auto simp add: get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def split: option.splits Option.bind_splits) + + +lemma element_ptr_kinds_simp [simp]: + "element_ptr_kinds (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h) + = element_ptr_kinds h |\| (if is_element_ptr_kind ptr then {|the (cast ptr)|} else {||})" + by(auto simp add: element_ptr_kinds_def is_node_ptr_kind_def split: option.splits) + +lemma type_wf_put_I: + assumes "type_wf h" + assumes "NodeClass.type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "is_element_ptr_kind ptr \ is_element_kind obj" + shows "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + using assms + by(auto simp add: type_wf_defs split: option.splits) + +lemma type_wf_put_ptr_not_in_heap_E: + assumes "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "ptr |\| object_ptr_kinds h" + shows "type_wf h" + using assms + by(auto simp add: type_wf_defs elim!: NodeMonad.type_wf_put_ptr_not_in_heap_E + split: option.splits if_splits) + +lemma type_wf_put_ptr_in_heap_E: + assumes "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "ptr |\| object_ptr_kinds h" + assumes "NodeClass.type_wf h" + assumes "is_element_ptr_kind ptr \ is_element_kind (the (get ptr h))" + shows "type_wf h" + using assms + apply(auto simp add: type_wf_defs split: option.splits if_splits)[1] + apply(case_tac "x2 = cast element_ptr") + apply(drule_tac x=element_ptr in allE) + apply(auto)[1] + apply(metis (no_types, lifting) NodeClass.get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf assms(2) bind.bind_lunit + cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_inv cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inv get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def option.exhaust_sel) + by(auto) + +subsection\Preserving Types\ + +lemma new_element_type_wf_preserved [simp]: "h \ new_element \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: type_wf_defs NodeClass.type_wf_defs ObjectClass.type_wf_defs new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + new_element_def Let_def put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def + split: prod.splits if_splits elim!: bind_returns_heap_E)[1] + apply (metis element_ptr_kinds_commutes element_ptrs_def fempty_iff ffmember_filter + is_element_ptr_ref) + using element_ptrs_def apply fastforce + apply (metis (mono_tags, hide_lams) Suc_n_not_le_n element_ptr.sel(1) element_ptr_kinds_commutes + element_ptrs_def fMax_ge ffmember_filter fimageI is_element_ptr_ref) + by (metis (no_types, lifting) fMax_finsert fempty_iff fimage_is_fempty max_0L new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_not_in_heap) + +locale l_new_element = l_type_wf + + assumes new_element_types_preserved: "h \ new_element \\<^sub>h h' \ type_wf h = type_wf h'" + +lemma new_element_is_l_new_element: "l_new_element type_wf" + using l_new_element.intro new_element_type_wf_preserved + by blast + +lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_tag_type_type_wf_preserved [simp]: + "h \ put_M element_ptr tag_type_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: type_wf_defs NodeClass.type_wf_defs ObjectClass.type_wf_defs + Let_def put_M_defs get_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def + get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def + split: prod.splits option.splits Option.bind_splits elim!: bind_returns_heap_E)[1] + apply (metis option.distinct(1)) + apply (metis bind.bind_lunit cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_none) + apply (metis option.distinct(1)) + apply (metis bind.bind_lunit cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_none) + by (metis bind.bind_lunit cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_none cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inv) + + +lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_child_nodes_type_wf_preserved [simp]: + "h \ put_M element_ptr child_nodes_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: type_wf_defs NodeClass.type_wf_defs ObjectClass.type_wf_defs + Let_def put_M_defs get_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def + get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def + split: prod.splits option.splits Option.bind_splits elim!: bind_returns_heap_E)[1] + apply (metis option.distinct(1)) + apply (metis bind.bind_lunit cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_none) + apply (metis option.distinct(1)) + apply (metis bind.bind_lunit cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_none) + by (metis bind.bind_lunit cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_none cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inv) + +lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_attrs_type_wf_preserved [simp]: + "h \ put_M element_ptr attrs_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: type_wf_defs NodeClass.type_wf_defs ObjectClass.type_wf_defs Let_def + put_M_defs get_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def + get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def + split: prod.splits option.splits Option.bind_splits elim!: bind_returns_heap_E)[1] + apply (metis option.distinct(1)) + apply (metis bind.bind_lunit cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_none) + apply (metis option.distinct(1)) + apply (metis bind.bind_lunit cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_none) + by (metis bind.bind_lunit cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_none cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inv) + +lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_shadow_root_opt_type_wf_preserved [simp]: + "h \ put_M element_ptr shadow_root_opt_update v \\<^sub>h h' \ type_wf h = type_wf h'" + apply(auto simp add: type_wf_defs NodeClass.type_wf_defs ObjectClass.type_wf_defs + Let_def put_M_defs get_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def + get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def + split: prod.splits option.splits Option.bind_splits elim!: bind_returns_heap_E)[1] + apply (metis option.distinct(1)) + apply (metis bind.bind_lunit cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_none) + apply (metis option.distinct(1)) + apply (metis bind.bind_lunit cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_none) + by (metis bind.bind_lunit cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_none cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inv) + +lemma put_M_pointers_preserved: + assumes "h \ put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \\<^sub>h h'" + shows "object_ptr_kinds h = object_ptr_kinds h'" + using assms + apply(auto simp add: put_M_defs put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def + elim!: bind_returns_heap_E2 dest!: get_heap_E)[1] + by (meson get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap is_OK_returns_result_I) + +lemma element_ptr_kinds_preserved: + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "\h h'. \w \ SW. h \ w \\<^sub>h h' + \ (\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h')" + shows "element_ptr_kinds h = element_ptr_kinds h'" + using writes_small_big[OF assms] + apply(simp add: reflp_def transp_def preserved_def element_ptr_kinds_def) + by (metis assms node_ptr_kinds_preserved) + + +lemma element_ptr_kinds_small: + assumes "\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + shows "element_ptr_kinds h = element_ptr_kinds h'" + by(simp add: element_ptr_kinds_def node_ptr_kinds_def preserved_def + object_ptr_kinds_preserved_small[OF assms]) + +lemma type_wf_preserved_small: + assumes "\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + assumes "\node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'" + assumes "\element_ptr. preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr RElement.nothing) h h'" + shows "type_wf h = type_wf h'" + using type_wf_preserved_small[OF assms(1) assms(2)] allI[OF assms(3), of id, simplified] + apply(auto simp add: type_wf_defs )[1] + apply(auto simp add: preserved_def get_M_defs element_ptr_kinds_small[OF assms(1)] + split: option.splits,force)[1] + by(auto simp add: preserved_def get_M_defs element_ptr_kinds_small[OF assms(1)] + split: option.splits,force) + +lemma type_wf_preserved: + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \element_ptr. preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr RElement.nothing) h h'" + shows "type_wf h = type_wf h'" +proof - + have "\h h' w. w \ SW \ h \ w \\<^sub>h h' \ type_wf h = type_wf h'" + using assms type_wf_preserved_small by fast + with assms(1) assms(2) show ?thesis + apply(rule writes_small_big) + by(auto simp add: reflp_def transp_def) +qed + +lemma type_wf_drop: "type_wf h \ type_wf (Heap (fmdrop ptr (the_heap h)))" + apply(auto simp add: type_wf_defs NodeClass.type_wf_defs ObjectClass.type_wf_defs + node_ptr_kinds_def object_ptr_kinds_def is_node_ptr_kind_def + get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def)[1] + apply (metis (mono_tags, lifting) comp_apply ffmember_filter fimage_eqI + is_node_ptr_kind_cast node_ptr_casts_commute2 option.sel) + apply (metis (no_types, lifting) comp_apply element_ptr_kinds_commutes ffmember_filter + fmdom_filter fmfilter_alt_defs(1) heap.sel node_ptr_kinds_commutes object_ptr_kinds_def) + by (metis comp_eq_dest_lhs element_ptr_kinds_commutes fmdom_notI fmdrop_lookup heap.sel + node_ptr_kinds_commutes object_ptr_kinds_def) + +end diff --git a/Core_DOM/monads/NodeMonad.thy b/Core_DOM/monads/NodeMonad.thy new file mode 100644 index 0000000..42c764b --- /dev/null +++ b/Core_DOM/monads/NodeMonad.thy @@ -0,0 +1,217 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Node\ +text\In this theory, we introduce the monadic method setup for the Node class.\ +theory NodeMonad + imports + ObjectMonad + "../classes/NodeClass" +begin + +type_synonym ('object_ptr, 'node_ptr, 'Object, 'Node, 'result) dom_prog + = "((_) heap, exception, 'result) prog" +register_default_tvars "('object_ptr, 'node_ptr, 'Object, 'Node, 'result) dom_prog" + + +global_interpretation l_ptr_kinds_M node_ptr_kinds defines node_ptr_kinds_M = a_ptr_kinds_M . +lemmas node_ptr_kinds_M_defs = a_ptr_kinds_M_def + +lemma node_ptr_kinds_M_eq: + assumes "|h \ object_ptr_kinds_M|\<^sub>r = |h' \ object_ptr_kinds_M|\<^sub>r" + shows "|h \ node_ptr_kinds_M|\<^sub>r = |h' \ node_ptr_kinds_M|\<^sub>r" + using assms + by(auto simp add: node_ptr_kinds_M_defs object_ptr_kinds_M_defs node_ptr_kinds_def) + + +global_interpretation l_dummy defines get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e = "l_get_M.a_get_M get\<^sub>N\<^sub>o\<^sub>d\<^sub>e" . +lemma get_M_is_l_get_M: "l_get_M get\<^sub>N\<^sub>o\<^sub>d\<^sub>e type_wf node_ptr_kinds" + apply(simp add: get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_type_wf l_get_M_def) + by (metis ObjectClass.a_type_wf_def ObjectClass.get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf bind_eq_None_conv get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + node_ptr_kinds_commutes option.simps(3)) +lemmas get_M_defs = get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def[unfolded l_get_M.a_get_M_def[OF get_M_is_l_get_M]] + +adhoc_overloading get_M get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e + +locale l_get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas = l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e +begin +sublocale l_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas by unfold_locales + +interpretation l_get_M get\<^sub>N\<^sub>o\<^sub>d\<^sub>e type_wf node_ptr_kinds + apply(unfold_locales) + apply (simp add: get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_type_wf local.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e) + by (meson NodeMonad.get_M_is_l_get_M l_get_M_def) +lemmas get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_ok = get_M_ok[folded get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def] +end + +global_interpretation l_get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas type_wf by unfold_locales + +lemma node_ptr_kinds_M_reads: + "reads (\object_ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing)}) node_ptr_kinds_M h h'" + using object_ptr_kinds_M_reads + apply(simp add: reads_def node_ptr_kinds_M_defs node_ptr_kinds_M_defs node_ptr_kinds_def + object_ptr_kinds_M_reads preserved_def) + by (metis (mono_tags, hide_lams) object_ptr_kinds_preserved_small old.unit.exhaust preserved_def) + +global_interpretation l_put_M type_wf node_ptr_kinds get\<^sub>N\<^sub>o\<^sub>d\<^sub>e put\<^sub>N\<^sub>o\<^sub>d\<^sub>e + rewrites "a_get_M = get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e" + defines put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e = a_put_M + apply (simp add: get_M_is_l_get_M l_put_M_def) + by (simp add: get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def) + +lemmas put_M_defs = a_put_M_def +adhoc_overloading put_M put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e + + +locale l_put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas = l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e +begin +sublocale l_put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas by unfold_locales + +interpretation l_put_M type_wf node_ptr_kinds get\<^sub>N\<^sub>o\<^sub>d\<^sub>e put\<^sub>N\<^sub>o\<^sub>d\<^sub>e + apply(unfold_locales) + apply (simp add: get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_type_wf local.type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e) + by (meson NodeMonad.get_M_is_l_get_M l_get_M_def) +lemmas put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_ok = put_M_ok[folded put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def] +end + +global_interpretation l_put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas type_wf by unfold_locales + +lemma get_M_Object_preserved1 [simp]: + "(\x. getter (cast (setter (\_. v) x)) = getter (cast x)) \ h \ put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'" + apply(cases "cast node_ptr = object_ptr") + by(auto simp add: put_M_defs get_M_defs ObjectMonad.get_M_defs get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def preserved_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def + bind_eq_Some_conv + split: option.splits) + +lemma get_M_Object_preserved2 [simp]: + "cast node_ptr \ object_ptr \ h \ put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'" + by(auto simp add: put_M_defs get_M_defs get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def ObjectMonad.get_M_defs preserved_def + split: option.splits dest: get_heap_E) +lemma get_M_Object_preserved3 [simp]: + "h \ put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr setter v \\<^sub>h h' \ (\x. getter (cast (setter (\_. v) x)) = getter (cast x)) + \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'" + apply(cases "cast node_ptr \ object_ptr") + by(auto simp add: put_M_defs get_M_defs get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def ObjectMonad.get_M_defs preserved_def + split: option.splits bind_splits dest: get_heap_E) + +lemma get_M_Object_preserved4 [simp]: + "cast node_ptr \ object_ptr \ h \ put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr setter v \\<^sub>h h' + \ preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr getter) h h'" + by(auto simp add: ObjectMonad.put_M_defs get_M_defs get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def ObjectMonad.get_M_defs preserved_def + split: option.splits dest: get_heap_E) + +subsection\Modified Heaps\ + +lemma get_node_ptr_simp [simp]: + "get\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h) = (if ptr = cast node_ptr then cast obj else get node_ptr h)" + by(auto simp add: get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def) + +lemma node_ptr_kinds_simp [simp]: + "node_ptr_kinds (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h) + = node_ptr_kinds h |\| (if is_node_ptr_kind ptr then {|the (cast ptr)|} else {||})" + by(auto simp add: node_ptr_kinds_def) + +lemma type_wf_put_I: + assumes "type_wf h" + assumes "ObjectClass.type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "is_node_ptr_kind ptr \ is_node_kind obj" + shows "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + using assms + apply(auto simp add: type_wf_defs split: option.splits)[1] + using cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_none is_node_kind_def apply blast + using cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_none is_node_kind_def apply blast + done + +lemma type_wf_put_ptr_not_in_heap_E: + assumes "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "ptr |\| object_ptr_kinds h" + shows "type_wf h" + using assms + by(auto simp add: type_wf_defs elim!: ObjectMonad.type_wf_put_ptr_not_in_heap_E + split: option.splits if_splits) + +lemma type_wf_put_ptr_in_heap_E: + assumes "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "ptr |\| object_ptr_kinds h" + assumes "ObjectClass.type_wf h" + assumes "is_node_ptr_kind ptr \ is_node_kind (the (get ptr h))" + shows "type_wf h" + using assms + apply(auto simp add: type_wf_defs split: option.splits if_splits) + by (metis ObjectClass.get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf bind.bind_lunit get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def is_node_kind_def option.collapse) + + +subsection\Preserving Types\ + +lemma node_ptr_kinds_small: + assumes "\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + shows "node_ptr_kinds h = node_ptr_kinds h'" + by(simp add: node_ptr_kinds_def preserved_def object_ptr_kinds_preserved_small[OF assms]) + +lemma node_ptr_kinds_preserved: + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "\h h'. \w \ SW. h \ w \\<^sub>h h' + \ (\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h')" + shows "node_ptr_kinds h = node_ptr_kinds h'" + using writes_small_big[OF assms] + apply(simp add: reflp_def transp_def preserved_def node_ptr_kinds_def) + by (metis assms object_ptr_kinds_preserved) + + +lemma type_wf_preserved_small: + assumes "\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + assumes "\node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'" + shows "type_wf h = type_wf h'" + using type_wf_preserved allI[OF assms(2), of id, simplified] + apply(auto simp add: type_wf_defs) + apply(auto simp add: preserved_def get_M_defs node_ptr_kinds_small[OF assms(1)] + split: option.splits, force)[1] + by(auto simp add: preserved_def get_M_defs node_ptr_kinds_small[OF assms(1)] + split: option.splits, force)[1] + +lemma type_wf_preserved: + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' + \ \node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'" + shows "type_wf h = type_wf h'" +proof - + have "\h h' w. w \ SW \ h \ w \\<^sub>h h' \ type_wf h = type_wf h'" + using assms type_wf_preserved_small by fast + with assms(1) assms(2) show ?thesis + apply(rule writes_small_big) + by(auto simp add: reflp_def transp_def) +qed +end + diff --git a/Core_DOM/monads/ObjectMonad.thy b/Core_DOM/monads/ObjectMonad.thy new file mode 100644 index 0000000..038bf37 --- /dev/null +++ b/Core_DOM/monads/ObjectMonad.thy @@ -0,0 +1,247 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Object\ +text\In this theory, we introduce the monadic method setup for the Object class.\ +theory ObjectMonad + imports + BaseMonad + "../classes/ObjectClass" +begin + +type_synonym ('object_ptr, 'Object, 'result) dom_prog + = "((_) heap, exception, 'result) prog" +register_default_tvars "('object_ptr, 'Object, 'result) dom_prog" + +global_interpretation l_ptr_kinds_M object_ptr_kinds defines object_ptr_kinds_M = a_ptr_kinds_M . +lemmas object_ptr_kinds_M_defs = a_ptr_kinds_M_def + + +global_interpretation l_dummy defines get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t = "l_get_M.a_get_M get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t" . +lemma get_M_is_l_get_M: "l_get_M get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t type_wf object_ptr_kinds" + by (simp add: a_type_wf_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf l_get_M_def) +lemmas get_M_defs = get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def[unfolded l_get_M.a_get_M_def[OF get_M_is_l_get_M]] + +adhoc_overloading get_M get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + +locale l_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas = l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t +begin +interpretation l_get_M get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t type_wf object_ptr_kinds + apply(unfold_locales) + apply (simp add: get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf local.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t) + by (simp add: a_type_wf_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf) +lemmas get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ok = get_M_ok[folded get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def] +lemmas get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ptr_in_heap = get_M_ptr_in_heap[folded get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def] +end + +global_interpretation l_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas type_wf + by (simp add: l_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas_def l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_axioms) + +lemma object_ptr_kinds_M_reads: + "reads (\object_ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing)}) object_ptr_kinds_M h h'" + apply(auto simp add: object_ptr_kinds_M_defs get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf type_wf_defs reads_def + preserved_def get_M_defs + split: option.splits)[1] + using a_type_wf_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf by blast+ + + +global_interpretation l_put_M type_wf object_ptr_kinds get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + rewrites "a_get_M = get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t" + defines put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t = a_put_M + apply (simp add: get_M_is_l_get_M l_put_M_def) + by (simp add: get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def) +lemmas put_M_defs = a_put_M_def +adhoc_overloading put_M put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + + +locale l_put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas = l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t +begin +interpretation l_put_M type_wf object_ptr_kinds get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t + apply(unfold_locales) + using get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t.type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t local.l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_axioms apply blast + by (simp add: a_type_wf_def get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf) +lemmas put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ok = put_M_ok[folded put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def] +lemmas put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ptr_in_heap = put_M_ptr_in_heap[folded put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def] +end + +global_interpretation l_put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas type_wf + by (simp add: l_put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas_def l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_axioms) + + +definition check_in_heap :: "(_) object_ptr \ (_, unit) dom_prog" + where + "check_in_heap ptr = do { + h \ get_heap; + (if ptr |\| object_ptr_kinds h then + return () + else + error SegmentationFault + )}" + +lemma check_in_heap_ptr_in_heap: "ptr |\| object_ptr_kinds h \ h \ ok (check_in_heap ptr)" + by(auto simp add: check_in_heap_def) +lemma check_in_heap_pure [simp]: "pure (check_in_heap ptr) h" + by(auto simp add: check_in_heap_def intro!: bind_pure_I) +lemma check_in_heap_is_OK [simp]: + "ptr |\| object_ptr_kinds h \ h \ ok (check_in_heap ptr \ f) = h \ ok (f ())" + by(simp add: check_in_heap_def) +lemma check_in_heap_returns_result [simp]: + "ptr |\| object_ptr_kinds h \ h \ (check_in_heap ptr \ f) \\<^sub>r x = h \ f () \\<^sub>r x" + by(simp add: check_in_heap_def) +lemma check_in_heap_returns_heap [simp]: + "ptr |\| object_ptr_kinds h \ h \ (check_in_heap ptr \ f) \\<^sub>h h' = h \ f () \\<^sub>h h'" + by(simp add: check_in_heap_def) + +lemma check_in_heap_reads: + "reads {preserved (get_M object_ptr nothing)} (check_in_heap object_ptr) h h'" + apply(simp add: check_in_heap_def reads_def preserved_def) + by (metis a_type_wf_def get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ok get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ptr_in_heap is_OK_returns_result_E + is_OK_returns_result_I unit_all_impI) + +subsection\Invoke\ + +fun invoke_rec :: "(((_) object_ptr \ bool) \ ((_) object_ptr \ 'args + \ (_, 'result) dom_prog)) list \ (_) object_ptr \ 'args + \ (_, 'result) dom_prog" + where + "invoke_rec ((P, f)#xs) ptr args = (if P ptr then f ptr args else invoke_rec xs ptr args)" + | "invoke_rec [] ptr args = error InvokeError" + +definition invoke :: "(((_) object_ptr \ bool) \ ((_) object_ptr \ 'args + \ (_, 'result) dom_prog)) list + \ (_) object_ptr \ 'args \ (_, 'result) dom_prog" + where + "invoke xs ptr args = do { check_in_heap ptr; invoke_rec xs ptr args}" + +lemma invoke_split: "P (invoke ((Pred, f) # xs) ptr args) = + ((\(Pred ptr) \ P (invoke xs ptr args)) + \ (Pred ptr \ P (do {check_in_heap ptr; f ptr args})))" + by(simp add: invoke_def) + +lemma invoke_split_asm: "P (invoke ((Pred, f) # xs) ptr args) = + (\((\(Pred ptr) \ (\ P (invoke xs ptr args))) + \ (Pred ptr \ (\ P (do {check_in_heap ptr; f ptr args})))))" + by(simp add: invoke_def) +lemmas invoke_splits = invoke_split invoke_split_asm + +lemma invoke_ptr_in_heap: "h \ ok (invoke xs ptr args) \ ptr |\| object_ptr_kinds h" + by (metis bind_is_OK_E check_in_heap_ptr_in_heap invoke_def is_OK_returns_heap_I) + +lemma invoke_pure [simp]: "pure (invoke [] ptr args) h" + by(auto simp add: invoke_def intro!: bind_pure_I) + +lemma invoke_is_OK [simp]: + "ptr |\| object_ptr_kinds h \ Pred ptr + \ h \ ok (invoke ((Pred, f) # xs) ptr args) = h \ ok (f ptr args)" + by(simp add: invoke_def) +lemma invoke_returns_result [simp]: + "ptr |\| object_ptr_kinds h \ Pred ptr + \ h \ (invoke ((Pred, f) # xs) ptr args) \\<^sub>r x = h \ f ptr args \\<^sub>r x" + by(simp add: invoke_def) +lemma invoke_returns_heap [simp]: + "ptr |\| object_ptr_kinds h \ Pred ptr + \ h \ (invoke ((Pred, f) # xs) ptr args) \\<^sub>h h' = h \ f ptr args \\<^sub>h h'" + by(simp add: invoke_def) + +lemma invoke_not [simp]: "\Pred ptr \ invoke ((Pred, f) # xs) ptr args = invoke xs ptr args" + by(auto simp add: invoke_def) + +lemma invoke_empty [simp]: "\h \ ok (invoke [] ptr args)" + by(auto simp add: invoke_def check_in_heap_def) + +lemma invoke_empty_reads [simp]: "\P \ S. reflp P \ transp P \ reads S (invoke [] ptr args) h h'" + apply(simp add: invoke_def reads_def preserved_def) + by (meson bind_returns_result_E error_returns_result) + + +subsection\Modified Heaps\ + +lemma get_object_ptr_simp [simp]: + "get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h) = (if ptr = object_ptr then Some obj else get object_ptr h)" + by(auto simp add: get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def split: option.splits Option.bind_splits) + +lemma object_ptr_kinds_simp [simp]: "object_ptr_kinds (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h) = object_ptr_kinds h |\| {|ptr|}" + by(auto simp add: object_ptr_kinds_def put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def split: option.splits) + +lemma type_wf_put_I: + assumes "type_wf h" + shows "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + using assms + by(auto simp add: type_wf_defs split: option.splits) + +lemma type_wf_put_ptr_not_in_heap_E: + assumes "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "ptr |\| object_ptr_kinds h" + shows "type_wf h" + using assms + by(auto simp add: type_wf_defs split: option.splits if_splits) + +lemma type_wf_put_ptr_in_heap_E: + assumes "type_wf (put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr obj h)" + assumes "ptr |\| object_ptr_kinds h" + shows "type_wf h" + using assms + by(auto simp add: type_wf_defs split: option.splits if_splits) + + +subsection\Preserving Types\ + +lemma type_wf_preserved: "type_wf h = type_wf h'" + by(auto simp add: type_wf_defs) + + +lemma object_ptr_kinds_preserved_small: + assumes "\object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + shows "object_ptr_kinds h = object_ptr_kinds h'" + using assms + apply(auto simp add: object_ptr_kinds_def preserved_def get_M_defs get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def + split: option.splits)[1] + apply (metis (mono_tags, lifting) domIff error_returns_result fmdom.rep_eq fmember.rep_eq + old.unit.exhaust option.case_eq_if return_returns_result) + by (metis (mono_tags, lifting) domIff error_returns_result fmdom.rep_eq fmember.rep_eq + old.unit.exhaust option.case_eq_if return_returns_result) + +lemma object_ptr_kinds_preserved: + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "\h h' w object_ptr. w \ SW \ h \ w \\<^sub>h h' + \ preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + shows "object_ptr_kinds h = object_ptr_kinds h'" +proof - + { + fix object_ptr w + have "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" + apply(rule writes_small_big[OF assms]) + by auto + } + then show ?thesis + using object_ptr_kinds_preserved_small by blast +qed + +end diff --git a/Core_DOM/pointers/CharacterDataPointer.thy b/Core_DOM/pointers/CharacterDataPointer.thy new file mode 100644 index 0000000..147eb15 --- /dev/null +++ b/Core_DOM/pointers/CharacterDataPointer.thy @@ -0,0 +1,199 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\CharacterData\ +text\In this theory, we introduce the typed pointers for the class CharacterData.\ +theory CharacterDataPointer + imports + ElementPointer +begin + +datatype 'character_data_ptr character_data_ptr = Ref (the_ref: ref) | Ext 'character_data_ptr +register_default_tvars "'character_data_ptr character_data_ptr" +type_synonym ('node_ptr, 'element_ptr, 'character_data_ptr) node_ptr + = "('character_data_ptr character_data_ptr + 'node_ptr, 'element_ptr) node_ptr" +register_default_tvars "('node_ptr, 'element_ptr, 'character_data_ptr) node_ptr" +type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr) object_ptr + = "('object_ptr, 'character_data_ptr character_data_ptr + 'node_ptr, 'element_ptr) object_ptr" +register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr) object_ptr" + +definition cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) character_data_ptr \ (_) node_ptr" + where + "cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = node_ptr.Ext (Inr (Inl ptr))" + +abbreviation cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) character_data_ptr \ (_) object_ptr" + where + "cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)" + +definition cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) node_ptr \ (_) character_data_ptr option" + where + "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr = (case node_ptr of + node_ptr.Ext (Inr (Inl character_data_ptr)) \ Some character_data_ptr + | _ \ None)" + +abbreviation cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) object_ptr \ (_) character_data_ptr option" + where + "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of + Some node_ptr \ cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr + | None \ None)" + +adhoc_overloading cast cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r + cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r + +consts is_character_data_ptr_kind :: 'a +definition is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) node_ptr \ bool" + where + "is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr + of Some _ \ True | _ \ False)" + +abbreviation is_character_data_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) object_ptr \ bool" + where + "is_character_data_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast ptr of + Some node_ptr \ is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr + | None \ False)" + +adhoc_overloading is_character_data_ptr_kind is_character_data_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r + is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r +lemmas is_character_data_ptr_kind_def = is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + +consts is_character_data_ptr :: 'a +definition is_character_data_ptr\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) character_data_ptr \ bool" + where + "is_character_data_ptr\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case ptr + of character_data_ptr.Ref _ \ True | _ \ False)" + +abbreviation is_character_data_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) node_ptr \ bool" + where + "is_character_data_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast ptr of + Some character_data_ptr \ is_character_data_ptr\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r character_data_ptr + | _ \ False)" + +abbreviation is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) object_ptr \ bool" + where + "is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of + Some node_ptr \ is_character_data_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr + | None \ False)" + +adhoc_overloading is_character_data_ptr + is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r is_character_data_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r is_character_data_ptr\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r +lemmas is_character_data_ptr_def = is_character_data_ptr\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + +consts is_character_data_ptr_ext :: 'a +abbreviation + "is_character_data_ptr_ext\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ \ is_character_data_ptr\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr" + +abbreviation "is_character_data_ptr_ext\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of + Some character_data_ptr \ is_character_data_ptr_ext\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r character_data_ptr +| None \ False)" + +abbreviation "is_character_data_ptr_ext\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of + Some node_ptr \ is_character_data_ptr_ext\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr +| None \ False)" + +adhoc_overloading is_character_data_ptr_ext + is_character_data_ptr_ext\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r is_character_data_ptr_ext\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r is_character_data_ptr_ext\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r + +instantiation character_data_ptr :: (linorder) linorder +begin +definition + less_eq_character_data_ptr :: "(_::linorder) character_data_ptr \ (_) character_data_ptr \ bool" + where + "less_eq_character_data_ptr x y \ (case x of Ext i \ (case y of Ext j \ i \ j | Ref _ \ False) + | Ref i \ (case y of Ext _ \ True | Ref j \ i \ j))" +definition + less_character_data_ptr :: "(_::linorder) character_data_ptr \ (_) character_data_ptr \ bool" + where "less_character_data_ptr x y \ x \ y \ \ y \ x" +instance + apply(standard) + by(auto simp add: less_eq_character_data_ptr_def less_character_data_ptr_def + split: character_data_ptr.splits) +end + +lemma is_character_data_ptr_ref [simp]: "is_character_data_ptr (character_data_ptr.Ref n)" + by(simp add: is_character_data_ptr\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma cast_element_ptr_not_character_data_ptr [simp]: + "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr \ cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r character_data_ptr)" + "(cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r character_data_ptr \ cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr)" + unfolding cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by(auto) + +lemma is_character_data_ptr_kind_not_element_ptr [simp]: + "\ is_character_data_ptr_kind (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr)" + unfolding is_character_data_ptr_kind_def cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by auto +lemma is_element_ptr_kind_not_character_data_ptr [simp]: + "\ is_element_ptr_kind (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r character_data_ptr)" + using is_element_ptr_kind_obtains by fastforce + +lemma is_character_data_ptr_kind\<^sub>_cast [simp]: + "is_character_data_ptr_kind (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r character_data_ptr)" + by (simp add: cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma character_data_ptr_casts_commute [simp]: + "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr = Some character_data_ptr + \ cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r character_data_ptr = node_ptr" + unfolding cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by(auto split: node_ptr.splits sum.splits) + +lemma character_data_ptr_casts_commute2 [simp]: + "(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r character_data_ptr) = Some character_data_ptr)" + by simp + +lemma character_data_ptr_casts_commute3 [simp]: + assumes "is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr" + shows "cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r (the (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) = node_ptr" + using assms + by(auto simp add: is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + split: node_ptr.splits sum.splits) + +lemma is_character_data_ptr_kind_obtains: + assumes "is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr" + obtains character_data_ptr where "cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r character_data_ptr = node_ptr" + by (metis assms is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def case_optionE + character_data_ptr_casts_commute) + +lemma is_character_data_ptr_kind_none: + assumes "\is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr" + shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr = None" + using assms + unfolding is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by(auto split: node_ptr.splits sum.splits) + +lemma cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject [simp]: + "cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r x = cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r y \ x = y" + by(simp add: cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_ext_none [simp]: + "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r (node_ptr.Ext (Inr (Inr node_ext_ptr))) = None" + by(simp add: cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +end diff --git a/Core_DOM/pointers/DocumentPointer.thy b/Core_DOM/pointers/DocumentPointer.thy new file mode 100644 index 0000000..f207887 --- /dev/null +++ b/Core_DOM/pointers/DocumentPointer.thy @@ -0,0 +1,154 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Document\ +text\In this theory, we introduce the typed pointers for the class Document.\ +theory DocumentPointer + imports + CharacterDataPointer +begin + +datatype 'document_ptr document_ptr = Ref (the_ref: ref) | Ext 'document_ptr +register_default_tvars "'document_ptr document_ptr" +type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr) object_ptr + = "('document_ptr document_ptr + 'object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr) object_ptr" +register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr) object_ptr" + +definition cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_)document_ptr \ (_) object_ptr" + where + "cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = object_ptr.Ext (Inr (Inl ptr))" + +definition cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) object_ptr \ (_) document_ptr option" + where + "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case ptr of + object_ptr.Ext (Inr (Inl document_ptr)) \ Some document_ptr + | _ \ None)" + +adhoc_overloading cast cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r + + +definition is_document_ptr_kind :: "(_) object_ptr \ bool" + where + "is_document_ptr_kind ptr = (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of + Some _ \ True | None \ False)" + +consts is_document_ptr :: 'a +definition is_document_ptr\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) document_ptr \ bool" + where + "is_document_ptr\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case ptr of document_ptr.Ref _ \ True | _ \ False)" + +abbreviation is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) object_ptr \ bool" + where + "is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of + Some document_ptr \ is_document_ptr\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr + | None \ False)" +adhoc_overloading is_document_ptr is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r is_document_ptr\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r +lemmas is_document_ptr_def = is_document_ptr\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + +consts is_document_ptr_ext :: 'a +abbreviation "is_document_ptr_ext\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ \ is_document_ptr\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr" + +abbreviation "is_document_ptr_ext\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of + Some document_ptr \ is_document_ptr_ext\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr +| None \ False)" +adhoc_overloading is_document_ptr_ext is_document_ptr_ext\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r is_document_ptr_ext\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r + +instantiation document_ptr :: (linorder) linorder +begin +definition less_eq_document_ptr :: "(_::linorder) document_ptr \ (_) document_ptr \ bool" + where "less_eq_document_ptr x y \ (case x of Ext i \ (case y of Ext j \ i \ j | Ref _ \ False) + | Ref i \ (case y of Ext _ \ True | Ref j \ i \ j))" +definition less_document_ptr :: "(_::linorder) document_ptr \ (_) document_ptr \ bool" + where "less_document_ptr x y \ x \ y \ \ y \ x" +instance + apply(standard) + by(auto simp add: less_eq_document_ptr_def less_document_ptr_def split: document_ptr.splits) +end + +lemma is_document_ptr_ref [simp]: "is_document_ptr (document_ptr.Ref n)" + by(simp add: is_document_ptr\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma cast_document_ptr_not_node_ptr [simp]: + "cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr \ cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr" + "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr \ cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr" + unfolding cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by auto + +lemma document_ptr_no_node_ptr_cast [simp]: + "\ is_document_ptr_kind (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)" + by(simp add: cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def is_document_ptr_kind_def) +lemma node_ptr_no_document_ptr_cast [simp]: + "\ is_node_ptr_kind (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)" + using is_node_ptr_kind_obtains by fastforce + +lemma document_ptr_document_ptr_cast [simp]: + "is_document_ptr_kind (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)" + by (simp add: cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def is_document_ptr_kind_def) + +lemma document_ptr_casts_commute [simp]: + "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = Some document_ptr \ cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr = ptr" + unfolding cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by(auto split: object_ptr.splits sum.splits) + +lemma document_ptr_casts_commute2 [simp]: + "(cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr) = Some document_ptr)" + by simp + +lemma document_ptr_casts_commute3 [simp]: + assumes "is_document_ptr_kind ptr" + shows "cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) = ptr" + using assms + by(auto simp add: is_document_ptr_kind_def cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + split: object_ptr.splits sum.splits) + +lemma is_document_ptr_kind_obtains: + assumes "is_document_ptr_kind ptr" + obtains document_ptr where "ptr = cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr" + using assms is_document_ptr_kind_def + by (metis case_optionE document_ptr_casts_commute) + +lemma is_document_ptr_kind_none: + assumes "\is_document_ptr_kind ptr" + shows "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = None" + using assms + unfolding is_document_ptr_kind_def cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by (auto split: object_ptr.splits sum.splits) + +lemma cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject [simp]: + "cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r x = cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r y \ x = y" + by(simp add: cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_ext_none [simp]: + "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (object_ptr.Ext (Inr (Inr (Inr object_ext_ptr)))) = None" + by(simp add: cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma is_document_ptr_kind_not_element_ptr_kind [dest]: + "is_document_ptr_kind ptr \ \ is_element_ptr_kind ptr" + by(auto simp add: split: option.splits) +end diff --git a/Core_DOM/pointers/ElementPointer.thy b/Core_DOM/pointers/ElementPointer.thy new file mode 100644 index 0000000..99be418 --- /dev/null +++ b/Core_DOM/pointers/ElementPointer.thy @@ -0,0 +1,178 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Element\ +text\In this theory, we introduce the typed pointers for the class Element.\ +theory ElementPointer + imports + NodePointer +begin + +datatype 'element_ptr element_ptr = Ref (the_ref: ref) | Ext 'element_ptr +register_default_tvars "'element_ptr element_ptr" + +type_synonym ('node_ptr, 'element_ptr) node_ptr + = "('element_ptr element_ptr + 'node_ptr) node_ptr" +register_default_tvars "('node_ptr, 'element_ptr) node_ptr" +type_synonym ('object_ptr, 'node_ptr, 'element_ptr) object_ptr + = "('object_ptr, 'element_ptr element_ptr + 'node_ptr) object_ptr" +register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr) object_ptr" + + +definition cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) element_ptr \ (_) element_ptr" + where + "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r = id" + +definition cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) element_ptr \ (_) node_ptr" + where + "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = node_ptr.Ext (Inl ptr)" + +abbreviation cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) element_ptr \ (_) object_ptr" + where + "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)" + +definition cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) node_ptr \ (_) element_ptr option" + where + "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr = (case node_ptr of node_ptr.Ext (Inl element_ptr) + \ Some element_ptr | _ \ None)" + +abbreviation cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) object_ptr \ (_) element_ptr option" + where + "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of + Some node_ptr \ cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr + | None \ None)" + +adhoc_overloading cast cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r + cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r + +consts is_element_ptr_kind :: 'a +definition is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) node_ptr \ bool" + where + "is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of Some _ \ True | _ \ False)" + +abbreviation is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) object_ptr \ bool" + where + "is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast ptr of + Some node_ptr \ is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr + | None \ False)" + +adhoc_overloading is_element_ptr_kind is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r +lemmas is_element_ptr_kind_def = is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + +consts is_element_ptr :: 'a +definition is_element_ptr\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) element_ptr \ bool" + where + "is_element_ptr\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case ptr of element_ptr.Ref _ \ True | _ \ False)" + +abbreviation is_element_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) node_ptr \ bool" + where + "is_element_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast ptr of + Some element_ptr \ is_element_ptr\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr + | _ \ False)" + +abbreviation is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) object_ptr \ bool" + where + "is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast ptr of + Some node_ptr \ is_element_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr + | None \ False)" + +adhoc_overloading is_element_ptr is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r is_element_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r is_element_ptr\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r +lemmas is_element_ptr_def = is_element_ptr\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + +consts is_element_ptr_ext :: 'a +abbreviation "is_element_ptr_ext\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ \ is_element_ptr\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr" + +abbreviation "is_element_ptr_ext\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ is_element_ptr_kind ptr \ (\ is_element_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)" + +abbreviation "is_element_ptr_ext\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ is_element_ptr_kind ptr \ (\ is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)" +adhoc_overloading is_element_ptr_ext is_element_ptr_ext\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r is_element_ptr_ext\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r + + +instantiation element_ptr :: (linorder) linorder +begin +definition + less_eq_element_ptr :: "(_::linorder) element_ptr \ (_)element_ptr \ bool" + where + "less_eq_element_ptr x y \ (case x of Ext i \ (case y of Ext j \ i \ j | Ref _ \ False) + | Ref i \ (case y of Ext _ \ True | Ref j \ i \ j))" +definition + less_element_ptr :: "(_::linorder) element_ptr \ (_) element_ptr \ bool" + where "less_element_ptr x y \ x \ y \ \ y \ x" +instance + apply(standard) + by(auto simp add: less_eq_element_ptr_def less_element_ptr_def split: element_ptr.splits) +end + +lemma is_element_ptr_ref [simp]: "is_element_ptr (element_ptr.Ref n)" + by(simp add: is_element_ptr\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma element_ptr_casts_commute [simp]: + "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr = Some element_ptr \ cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr = node_ptr" + unfolding cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by(auto split: node_ptr.splits sum.splits) + +lemma element_ptr_casts_commute2 [simp]: + "(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr) = Some element_ptr)" + by simp + +lemma element_ptr_casts_commute3 [simp]: + assumes "is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr" + shows "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r (the (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) = node_ptr" + using assms + by(auto simp add: is_element_ptr_kind_def cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + split: node_ptr.splits sum.splits) + +lemma is_element_ptr_kind_obtains: + assumes "is_element_ptr_kind node_ptr" + obtains element_ptr where "node_ptr = cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr" + by (metis assms is_element_ptr_kind_def case_optionE element_ptr_casts_commute) + +lemma is_element_ptr_kind_none: + assumes "\is_element_ptr_kind node_ptr" + shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr = None" + using assms + unfolding is_element_ptr_kind_def cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by(auto split: node_ptr.splits sum.splits) + +lemma is_element_ptr_kind_cast [simp]: + "is_element_ptr_kind (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr)" + by (metis element_ptr_casts_commute is_element_ptr_kind_none option.distinct(1)) + +lemma cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject [simp]: + "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r x = cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r y \ x = y" + by(simp add: cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_ext_none [simp]: + "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (node_ptr.Ext (Inr (Inr node_ext_ptr))) = None" + by(simp add: cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma is_element_ptr_implies_kind [dest]: "is_element_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr" + by(auto split: option.splits) + +end diff --git a/Core_DOM/pointers/NodePointer.thy b/Core_DOM/pointers/NodePointer.thy new file mode 100644 index 0000000..f3bd2ca --- /dev/null +++ b/Core_DOM/pointers/NodePointer.thy @@ -0,0 +1,111 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Node\ +text\In this theory, we introduce the typed pointers for the class Node.\ +theory NodePointer + imports + ObjectPointer +begin + +datatype 'node_ptr node_ptr = Ext 'node_ptr +register_default_tvars "'node_ptr node_ptr" + +type_synonym ('object_ptr, 'node_ptr) object_ptr = "('node_ptr node_ptr + 'object_ptr) object_ptr" +register_default_tvars "('object_ptr, 'node_ptr) object_ptr" + +definition cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) node_ptr \ (_) object_ptr" + where + "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = object_ptr.Ext (Inl ptr)" + +definition cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) object_ptr \ (_) node_ptr option" + where + "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r object_ptr = (case object_ptr of object_ptr.Ext (Inl node_ptr) + \ Some node_ptr | _ \ None)" + +adhoc_overloading cast cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r + +definition is_node_ptr_kind :: "(_) object_ptr \ bool" + where + "is_node_ptr_kind ptr = (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ None)" + +instantiation node_ptr :: (linorder) linorder +begin +definition less_eq_node_ptr :: "(_::linorder) node_ptr \ (_) node_ptr \ bool" + where "less_eq_node_ptr x y \ (case x of Ext i \ (case y of Ext j \ i \ j))" +definition less_node_ptr :: "(_::linorder) node_ptr \ (_) node_ptr \ bool" + where "less_node_ptr x y \ x \ y \ \ y \ x" +instance + apply(standard) + by(auto simp add: less_eq_node_ptr_def less_node_ptr_def split: node_ptr.splits) +end + +lemma node_ptr_casts_commute [simp]: + "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = Some node_ptr \ cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr = ptr" + unfolding cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by(auto split: object_ptr.splits sum.splits) + +lemma node_ptr_casts_commute2 [simp]: + "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) = Some node_ptr" + by simp + +lemma node_ptr_casts_commute3 [simp]: + assumes "is_node_ptr_kind ptr" + shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) = ptr" + using assms + by(auto simp add: is_node_ptr_kind_def cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + split: object_ptr.splits sum.splits) + +lemma is_node_ptr_kind_obtains: + assumes "is_node_ptr_kind ptr" + obtains node_ptr where "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = Some node_ptr" + using assms is_node_ptr_kind_def by auto + +lemma is_node_ptr_kind_none: + assumes "\is_node_ptr_kind ptr" + shows "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = None" + using assms + unfolding is_node_ptr_kind_def cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by auto + +lemma is_node_ptr_kind_cast [simp]: "is_node_ptr_kind (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)" + unfolding is_node_ptr_kind_def by simp + +lemma cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject [simp]: + "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r x = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r y \ x = y" + by(simp add: cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_ext_none [simp]: + "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r (object_ptr.Ext (Inr (Inr (Inr object_ext_ptr)))) = None" + by(simp add: cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma node_ptr_inclusion [simp]: + "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr \ cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ` node_ptrs \ node_ptr \ node_ptrs" + by auto +end diff --git a/Core_DOM/pointers/ObjectPointer.thy b/Core_DOM/pointers/ObjectPointer.thy new file mode 100644 index 0000000..c4168c2 --- /dev/null +++ b/Core_DOM/pointers/ObjectPointer.thy @@ -0,0 +1,51 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Object\ +text\In this theory, we introduce the typed pointer for the class Object. This class is the +common superclass of our class model.\ +theory ObjectPointer + imports + Ref +begin + +datatype 'object_ptr object_ptr = Ext 'object_ptr +register_default_tvars "'object_ptr object_ptr" + +instantiation object_ptr :: (linorder) linorder +begin +definition less_eq_object_ptr :: "'object_ptr::linorder object_ptr \ 'object_ptr object_ptr \ bool" + where "less_eq_object_ptr x y \ (case x of Ext i \ (case y of Ext j \ i \ j))" +definition less_object_ptr :: "'object_ptr::linorder object_ptr \ 'object_ptr object_ptr \ bool" + where "less_object_ptr x y \ x \ y \ \ y \ x" +instance by(standard, auto simp add: less_eq_object_ptr_def less_object_ptr_def + split: object_ptr.splits) +end + +end diff --git a/Core_DOM/pointers/Ref.thy b/Core_DOM/pointers/Ref.thy new file mode 100644 index 0000000..fd29f5e --- /dev/null +++ b/Core_DOM/pointers/Ref.thy @@ -0,0 +1,62 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\References\ +text\ + This theory, we introduce a generic reference. All our typed pointers include such + a reference, which allows us to distinguish pointers of the same type, but also to + iterate over all pointers in a set.\ +theory + Ref + imports + "HOL-Library.Adhoc_Overloading" + "../preliminaries/Hiding_Type_Variables" +begin + +instantiation sum :: (linorder, linorder) linorder +begin +definition less_eq_sum :: "'a + 'b \ 'a + 'b \ bool" + where + "less_eq_sum t t' = (case t of + Inl l \ (case t' of + Inl l' \ l \ l' + | Inr r' \ True) + | Inr r \ (case t' of + Inl l' \ False + | Inr r' \ r \ r'))" +definition less_sum :: "'a + 'b \ 'a + 'b \ bool" + where + "less_sum t t' \ t \ t' \ \ t' \ t" +instance by(standard) (auto simp add: less_eq_sum_def less_sum_def split: sum.splits) +end + +type_synonym ref = nat +consts cast :: 'a + +end diff --git a/Core_DOM/pointers/ShadowRootPointer.thy b/Core_DOM/pointers/ShadowRootPointer.thy new file mode 100644 index 0000000..83f719b --- /dev/null +++ b/Core_DOM/pointers/ShadowRootPointer.thy @@ -0,0 +1,182 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\ShadowRoot\ +text\In this theory, we introduce the typed pointers for the class ShadowRoot. Note that, in +this document, we will not make use of ShadowRoots nor will we discuss their particular properties. +We only include them here, as they are required for future work and they cannot be added alter +following the object-oriented extensibility of our data model.\ +theory ShadowRootPointer + imports + DocumentPointer +begin + +datatype 'shadow_root_ptr shadow_root_ptr = Ref (the_ref: ref) | Ext 'shadow_root_ptr +register_default_tvars "'shadow_root_ptr shadow_root_ptr" +type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, + 'document_ptr, 'shadow_root_ptr) object_ptr + = "('shadow_root_ptr shadow_root_ptr + 'object_ptr, 'node_ptr, 'element_ptr, + 'character_data_ptr, 'document_ptr) object_ptr" +register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, + 'document_ptr, 'shadow_root_ptr) object_ptr" + +definition cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_)shadow_root_ptr \ (_) object_ptr" + where + "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = object_ptr.Ext (Inr (Inr (Inl ptr)))" + +definition cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) object_ptr \ (_) shadow_root_ptr option" + where + "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case ptr of + object_ptr.Ext (Inr (Inr (Inl shadow_root_ptr))) \ Some shadow_root_ptr + | _ \ None)" + +adhoc_overloading cast cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r + + +definition is_shadow_root_ptr_kind :: "(_) object_ptr \ bool" + where + "is_shadow_root_ptr_kind ptr = (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of Some _ \ True + | None \ False)" + +consts is_shadow_root_ptr :: 'a +definition is_shadow_root_ptr\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) shadow_root_ptr \ bool" + where + "is_shadow_root_ptr\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case ptr of shadow_root_ptr.Ref _ \ True + | _ \ False)" + +abbreviation is_shadow_root_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r :: "(_) object_ptr \ bool" + where + "is_shadow_root_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of + Some shadow_root_ptr \ is_shadow_root_ptr\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr + | None \ False)" +adhoc_overloading is_shadow_root_ptr is_shadow_root_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r is_shadow_root_ptr\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r +lemmas is_shadow_root_ptr_def = is_shadow_root_ptr\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + +consts is_shadow_root_ptr_ext :: 'a +abbreviation "is_shadow_root_ptr_ext\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ \ is_shadow_root_ptr\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr" + +abbreviation "is_shadow_root_ptr_ext\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \ (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of + Some shadow_root_ptr \ is_shadow_root_ptr_ext\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr +| None \ False)" +adhoc_overloading is_shadow_root_ptr_ext is_shadow_root_ptr_ext\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r is_shadow_root_ptr_ext\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r + +instantiation shadow_root_ptr :: (linorder) linorder +begin +definition + less_eq_shadow_root_ptr :: "(_::linorder) shadow_root_ptr \ (_) shadow_root_ptr \ bool" + where + "less_eq_shadow_root_ptr x y \ (case x of Ext i \ (case y of Ext j \ i \ j | Ref _ \ False) + | Ref i \ (case y of Ext _ \ True | Ref j \ i \ j))" +definition less_shadow_root_ptr :: "(_::linorder) shadow_root_ptr \ (_) shadow_root_ptr \ bool" + where "less_shadow_root_ptr x y \ x \ y \ \ y \ x" +instance + apply(standard) + by(auto simp add: less_eq_shadow_root_ptr_def less_shadow_root_ptr_def + split: shadow_root_ptr.splits) +end + + +lemma is_shadow_root_ptr_ref [simp]: "is_shadow_root_ptr (shadow_root_ptr.Ref n)" + by(simp add: is_shadow_root_ptr\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma is_shadow_root_ptr_not_node_ptr[simp]: "\is_shadow_root_ptr (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)" + by(simp add: is_shadow_root_ptr_def cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma cast_shadow_root_ptr_not_node_ptr [simp]: + "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr \ cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr" + "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr \ cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr" + unfolding cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def by auto + +lemma cast_shadow_root_ptr_not_document_ptr [simp]: + "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr \ cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr" + "cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr \ cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr" + unfolding cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def by auto + +lemma shadow_root_ptr_no_node_ptr_cast [simp]: + "\ is_shadow_root_ptr_kind (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)" + by(simp add: cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def is_shadow_root_ptr_kind_def) +lemma node_ptr_no_shadow_root_ptr_cast [simp]: + "\ is_node_ptr_kind (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr)" + using is_node_ptr_kind_obtains by fastforce + +lemma shadow_root_ptr_no_document_ptr_cast [simp]: + "\ is_shadow_root_ptr_kind (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)" + by(simp add: cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def is_shadow_root_ptr_kind_def) +lemma document_ptr_no_shadow_root_ptr_cast [simp]: + "\ is_document_ptr_kind (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr)" + using is_document_ptr_kind_obtains by fastforce + +lemma shadow_root_ptr_shadow_root_ptr_cast [simp]: + "is_shadow_root_ptr_kind (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr)" + by (simp add: cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def is_shadow_root_ptr_kind_def) + +lemma shadow_root_ptr_casts_commute [simp]: + "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = Some shadow_root_ptr \ cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr = ptr" + unfolding cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by(auto split: object_ptr.splits sum.splits) + +lemma shadow_root_ptr_casts_commute2 [simp]: + "(cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr) = Some shadow_root_ptr)" + by simp + +lemma shadow_root_ptr_casts_commute3 [simp]: + assumes "is_shadow_root_ptr_kind ptr" + shows "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) = ptr" + using assms + by(auto simp add: is_shadow_root_ptr_kind_def cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + split: object_ptr.splits sum.splits) + +lemma is_shadow_root_ptr_kind_obtains: + assumes "is_shadow_root_ptr_kind ptr" + obtains shadow_root_ptr where "ptr = cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr" + using assms is_shadow_root_ptr_kind_def + by (metis case_optionE shadow_root_ptr_casts_commute) + +lemma is_shadow_root_ptr_kind_none: + assumes "\is_shadow_root_ptr_kind ptr" + shows "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = None" + using assms + unfolding is_shadow_root_ptr_kind_def cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def + by (auto split: object_ptr.splits sum.splits) + +lemma cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject [simp]: + "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r x = cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r y \ x = y" + by(simp add: cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_ext_none [simp]: + "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (object_ptr.Ext (Inr (Inr (Inr object_ext_ptr)))) = None" + by(simp add: cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def) + +lemma is_shadow_root_ptr_kind_simp1 [dest]: "is_document_ptr_kind ptr \ \is_shadow_root_ptr_kind ptr" + by (metis document_ptr_no_shadow_root_ptr_cast shadow_root_ptr_casts_commute3) + +lemma is_shadow_root_ptr_kind_simp2 [dest]: "is_node_ptr_kind ptr \ \is_shadow_root_ptr_kind ptr" + by (metis node_ptr_no_shadow_root_ptr_cast shadow_root_ptr_casts_commute3) + +end diff --git a/Core_DOM/preliminaries/Heap_Error_Monad.thy b/Core_DOM/preliminaries/Heap_Error_Monad.thy new file mode 100644 index 0000000..2f8f1d7 --- /dev/null +++ b/Core_DOM/preliminaries/Heap_Error_Monad.thy @@ -0,0 +1,906 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\The Heap Error Monad\ +text \In this theory, we define a heap and error monad for modeling exceptions. +This allows us to define composite methods similar to stateful programming in Haskell, +but also to stay close to the official DOM specification.\ +theory + Heap_Error_Monad + imports + Hiding_Type_Variables + "HOL-Library.Monad_Syntax" +begin + +subsection \The Program Data Type\ + +datatype ('heap, 'e, 'result) prog = Prog (the_prog: "'heap \ 'e + 'result \ 'heap") +register_default_tvars "('heap, 'e, 'result) prog" (print, parse) + +subsection \Basic Functions\ + +definition + bind :: "(_, 'result) prog \ ('result \ (_, 'result2) prog) \ (_, 'result2) prog" + where + "bind f g = Prog (\h. (case (the_prog f) h of Inr (x, h') \ (the_prog (g x)) h' + | Inl exception \ Inl exception))" + +adhoc_overloading Monad_Syntax.bind bind + +definition + execute :: "'heap \ ('heap, 'e, 'result) prog \ ('e + 'result \ 'heap)" + ("((_)/ \ (_))" [51, 52] 55) + where + "execute h p = (the_prog p) h" + +definition + returns_result :: "'heap \ ('heap, 'e, 'result) prog \ 'result \ bool" + ("((_)/ \ (_)/ \\<^sub>r (_))" [60, 35, 61] 65) + where + "returns_result h p r \ (case h \ p of Inr (r', _) \ r = r' | Inl _ \ False)" + +fun select_result ("|(_)|\<^sub>r") + where + "select_result (Inr (r, _)) = r" + | "select_result (Inl _) = undefined" + +lemma returns_result_eq [elim]: "h \ f \\<^sub>r y \ h \ f \\<^sub>r y' \ y = y'" + by(auto simp add: returns_result_def split: sum.splits) + +definition + returns_heap :: "'heap \ ('heap, 'e, 'result) prog \ 'heap \ bool" + ("((_)/ \ (_)/ \\<^sub>h (_))" [60, 35, 61] 65) + where + "returns_heap h p h' \ (case h \ p of Inr (_ , h'') \ h' = h'' | Inl _ \ False)" + +lemma returns_heap_eq [elim]: "h \ f \\<^sub>h h' \ h \ f \\<^sub>h h'' \ h' = h''" + by(auto simp add: returns_heap_def split: sum.splits) + +definition + returns_error :: "'heap \ ('heap, 'e, 'result) prog \ 'e \ bool" + ("((_)/ \ (_)/ \\<^sub>e (_))" [60, 35, 61] 65) + where + "returns_error h p e = (case h \ p of Inr _ \ False | Inl e' \ e = e')" + +definition is_OK :: "'heap \ ('heap, 'e, 'result) prog \ bool" ("((_)/ \ ok (_))" [75, 75]) + where + "is_OK h p = (case h \ p of Inr _ \ True | Inl _ \ False)" + +lemma is_OK_returns_result_I [intro]: "h \ f \\<^sub>r y \ h \ ok f" + by(auto simp add: is_OK_def returns_result_def split: sum.splits) + +lemma is_OK_returns_result_E [elim]: + assumes "h \ ok f" + obtains x where "h \ f \\<^sub>r x" + using assms by(auto simp add: is_OK_def returns_result_def split: sum.splits) + +lemma is_OK_returns_heap_I [intro]: "h \ f \\<^sub>h h' \ h \ ok f" + by(auto simp add: is_OK_def returns_heap_def split: sum.splits) + +lemma is_OK_returns_heap_E [elim]: + assumes "h \ ok f" + obtains h' where "h \ f \\<^sub>h h'" + using assms by(auto simp add: is_OK_def returns_heap_def split: sum.splits) + +lemma select_result_I: + assumes "h \ ok f" + and "\x. h \ f \\<^sub>r x \ P x" + shows "P |h \ f|\<^sub>r" + using assms + by(auto simp add: is_OK_def returns_result_def split: sum.splits) + +lemma select_result_I2 [simp]: + assumes "h \ f \\<^sub>r x" + shows "|h \ f|\<^sub>r = x" + using assms + by(auto simp add: is_OK_def returns_result_def split: sum.splits) + +lemma returns_result_select_result [simp]: + assumes "h \ ok f" + shows "h \ f \\<^sub>r |h \ f|\<^sub>r" + using assms + by (simp add: select_result_I) + +lemma select_result_E: + assumes "P |h \ f|\<^sub>r" and "h \ ok f" + obtains x where "h \ f \\<^sub>r x" and "P x" + using assms + by(auto simp add: is_OK_def returns_result_def split: sum.splits) + +lemma select_result_eq: "(\x .h \ f \\<^sub>r x = h' \ f \\<^sub>r x) \ |h \ f|\<^sub>r = |h' \ f|\<^sub>r" + by (metis (no_types, lifting) is_OK_def old.sum.simps(6) select_result.elims + select_result_I select_result_I2) + +definition error :: "'e \ ('heap, 'e, 'result) prog" + where + "error exception = Prog (\h. Inl exception)" + +lemma error_bind [iff]: "(error e \ g) = error e" + unfolding error_def bind_def by auto + +lemma error_returns_result [simp]: "\ (h \ error e \\<^sub>r y)" + unfolding returns_result_def error_def execute_def by auto + +lemma error_returns_heap [simp]: "\ (h \ error e \\<^sub>h h')" + unfolding returns_heap_def error_def execute_def by auto + +lemma error_returns_error [simp]: "h \ error e \\<^sub>e e" + unfolding returns_error_def error_def execute_def by auto + +definition return :: "'result \ ('heap, 'e, 'result) prog" + where + "return result = Prog (\h. Inr (result, h))" + +lemma return_ok [simp]: "h \ ok (return x)" + by(simp add: return_def is_OK_def execute_def) + +lemma return_bind [iff]: "(return x \ g) = g x" + unfolding return_def bind_def by auto + +lemma return_id [simp]: "f \ return = f" + by (induct f) (auto simp add: return_def bind_def split: sum.splits prod.splits) + +lemma return_returns_result [iff]: "(h \ return x \\<^sub>r y) = (x = y)" + unfolding returns_result_def return_def execute_def by auto + +lemma return_returns_heap [iff]: "(h \ return x \\<^sub>h h') = (h = h')" + unfolding returns_heap_def return_def execute_def by auto + +lemma return_returns_error [iff]: "\ h \ return x \\<^sub>e e" + unfolding returns_error_def execute_def return_def by auto + +definition noop :: "('heap, 'e, unit) prog" + where + "noop = return ()" + +lemma noop_returns_heap [simp]: "h \ noop \\<^sub>h h' \ h = h'" + by(simp add: noop_def) + +definition get_heap :: "('heap, 'e, 'heap) prog" + where + "get_heap = Prog (\h. h \ return h)" + +lemma get_heap_ok [simp]: "h \ ok (get_heap)" + by (simp add: get_heap_def execute_def is_OK_def return_def) + +lemma get_heap_returns_result [simp]: "(h \ get_heap \ (\h'. f h') \\<^sub>r x) = (h \ f h \\<^sub>r x)" + by(simp add: get_heap_def returns_result_def bind_def return_def execute_def) + +lemma get_heap_returns_heap [simp]: "(h \ get_heap \ (\h'. f h') \\<^sub>h h'') = (h \ f h \\<^sub>h h'')" + by(simp add: get_heap_def returns_heap_def bind_def return_def execute_def) + +lemma get_heap_is_OK [simp]: "(h \ ok (get_heap \ (\h'. f h'))) = (h \ ok (f h))" + by(auto simp add: get_heap_def is_OK_def bind_def return_def execute_def) + +lemma get_heap_E [elim]: "(h \ get_heap \\<^sub>r x) \ x = h" + by(simp add: get_heap_def returns_result_def return_def execute_def) + +definition return_heap :: "'heap \ ('heap, 'e, unit) prog" + where + "return_heap h = Prog (\_. h \ return ())" + +lemma return_heap_E [iff]: "(h \ return_heap h' \\<^sub>h h'') = (h'' = h')" + by(simp add: return_heap_def returns_heap_def return_def execute_def) + +lemma return_heap_returns_result [simp]: "h \ return_heap h' \\<^sub>r ()" + by(simp add: return_heap_def execute_def returns_result_def return_def) + + +subsection \Pure Heaps\ + +definition pure :: "('heap, 'e, 'result) prog \ 'heap \ bool" + where "pure f h \ h \ ok f \ h \ f \\<^sub>h h" + +lemma return_pure [simp]: "pure (return x) h" + by(simp add: pure_def return_def is_OK_def returns_heap_def execute_def) + +lemma error_pure [simp]: "pure (error e) h" + by(simp add: pure_def error_def is_OK_def returns_heap_def execute_def) + +lemma noop_pure [simp]: "pure (noop) h" + by (simp add: noop_def) + +lemma get_pure [simp]: "pure get_heap h" + by(simp add: pure_def get_heap_def is_OK_def returns_heap_def return_def execute_def) + +lemma pure_returns_heap_eq: + "h \ f \\<^sub>h h' \ pure f h \ h = h'" + by (meson pure_def is_OK_returns_heap_I returns_heap_eq) + +lemma pure_eq_iff: + "(\h' x. h \ f \\<^sub>r x \ h \ f \\<^sub>h h' \ h = h') \ pure f h" + by(auto simp add: pure_def) + +subsection \Bind\ + +lemma bind_assoc [simp]: + "((bind f g) \ h) = (f \ (\x. (g x \ h)))" + by(auto simp add: bind_def split: sum.splits) + +lemma bind_returns_result_E: + assumes "h \ f \ g \\<^sub>r y" + obtains x h' where "h \ f \\<^sub>r x" and "h \ f \\<^sub>h h'" and "h' \ g x \\<^sub>r y" + using assms by(auto simp add: bind_def returns_result_def returns_heap_def execute_def + split: sum.splits) + +lemma bind_returns_result_E2: + assumes "h \ f \ g \\<^sub>r y" and "pure f h" + obtains x where "h \ f \\<^sub>r x" and "h \ g x \\<^sub>r y" + using assms pure_returns_heap_eq bind_returns_result_E by metis + +lemma bind_returns_result_E3: + assumes "h \ f \ g \\<^sub>r y" and "h \ f \\<^sub>r x" and "pure f h" + shows "h \ g x \\<^sub>r y" + using assms returns_result_eq bind_returns_result_E2 by metis + +lemma bind_returns_result_E4: + assumes "h \ f \ g \\<^sub>r y" and "h \ f \\<^sub>r x" + obtains h' where "h \ f \\<^sub>h h'" and "h' \ g x \\<^sub>r y" + using assms returns_result_eq bind_returns_result_E by metis + +lemma bind_returns_heap_E: + assumes "h \ f \ g \\<^sub>h h''" + obtains x h' where "h \ f \\<^sub>r x" and "h \ f \\<^sub>h h'" and "h' \ g x \\<^sub>h h''" + using assms by(auto simp add: bind_def returns_result_def returns_heap_def execute_def + split: sum.splits) + +lemma bind_returns_heap_E2 [elim]: + assumes "h \ f \ g \\<^sub>h h'" and "pure f h" + obtains x where "h \ f \\<^sub>r x" and "h \ g x \\<^sub>h h'" + using assms pure_returns_heap_eq by (fastforce elim: bind_returns_heap_E) + +lemma bind_returns_heap_E3 [elim]: + assumes "h \ f \ g \\<^sub>h h'" and "h \ f \\<^sub>r x" and "pure f h" + shows "h \ g x \\<^sub>h h'" + using assms pure_returns_heap_eq returns_result_eq by (fastforce elim: bind_returns_heap_E) + +lemma bind_returns_heap_E4: + assumes "h \ f \ g \\<^sub>h h''" and "h \ f \\<^sub>h h'" + obtains x where "h \ f \\<^sub>r x" and "h' \ g x \\<^sub>h h''" + using assms + by (metis bind_returns_heap_E returns_heap_eq) + +lemma bind_returns_error_I [intro]: + assumes "h \ f \\<^sub>e e" + shows "h \ f \ g \\<^sub>e e" + using assms + by(auto simp add: returns_error_def bind_def execute_def split: sum.splits) + +lemma bind_returns_error_I3: + assumes "h \ f \\<^sub>r x" and "h \ f \\<^sub>h h'" and "h' \ g x \\<^sub>e e" + shows "h \ f \ g \\<^sub>e e" + using assms + by(auto simp add: returns_error_def bind_def execute_def returns_heap_def returns_result_def + split: sum.splits) + +lemma bind_returns_error_I2 [intro]: + assumes "pure f h" and "h \ f \\<^sub>r x" and "h \ g x \\<^sub>e e" + shows "h \ f \ g \\<^sub>e e" + using assms + by (meson bind_returns_error_I3 is_OK_returns_result_I pure_def) + +lemma bind_is_OK_E [elim]: + assumes "h \ ok (f \ g)" + obtains x h' where "h \ f \\<^sub>r x" and "h \ f \\<^sub>h h'" and "h' \ ok (g x)" + using assms + by(auto simp add: bind_def returns_result_def returns_heap_def is_OK_def execute_def + split: sum.splits) + +lemma bind_is_OK_E2: + assumes "h \ ok (f \ g)" and "h \ f \\<^sub>r x" + obtains h' where "h \ f \\<^sub>h h'" and "h' \ ok (g x)" + using assms + by(auto simp add: bind_def returns_result_def returns_heap_def is_OK_def execute_def + split: sum.splits) + +lemma bind_returns_result_I [intro]: + assumes "h \ f \\<^sub>r x" and "h \ f \\<^sub>h h'" and "h' \ g x \\<^sub>r y" + shows "h \ f \ g \\<^sub>r y" + using assms + by(auto simp add: bind_def returns_result_def returns_heap_def execute_def + split: sum.splits) + +lemma bind_pure_returns_result_I [intro]: + assumes "pure f h" and "h \ f \\<^sub>r x" and "h \ g x \\<^sub>r y" + shows "h \ f \ g \\<^sub>r y" + using assms + by (meson bind_returns_result_I pure_def is_OK_returns_result_I) + +lemma bind_pure_returns_result_I2 [intro]: + assumes "pure f h" and "h \ ok f" and "\x. h \ f \\<^sub>r x \ h \ g x \\<^sub>r y" + shows "h \ f \ g \\<^sub>r y" + using assms by auto + +lemma bind_returns_heap_I [intro]: + assumes "h \ f \\<^sub>r x" and "h \ f \\<^sub>h h'" and "h' \ g x \\<^sub>h h''" + shows "h \ f \ g \\<^sub>h h''" + using assms + by(auto simp add: bind_def returns_result_def returns_heap_def execute_def + split: sum.splits) + +lemma bind_returns_heap_I2 [intro]: + assumes "h \ f \\<^sub>h h'" and "\x. h \ f \\<^sub>r x \ h' \ g x \\<^sub>h h''" + shows "h \ f \ g \\<^sub>h h''" + using assms + by (meson bind_returns_heap_I is_OK_returns_heap_I is_OK_returns_result_E) + +lemma bind_is_OK_I [intro]: + assumes "h \ f \\<^sub>r x" and "h \ f \\<^sub>h h'" and "h' \ ok (g x)" + shows "h \ ok (f \ g)" + by (meson assms(1) assms(2) assms(3) bind_returns_heap_I is_OK_returns_heap_E + is_OK_returns_heap_I) + +lemma bind_is_OK_I2 [intro]: + assumes "h \ ok f" and "\x h'. h \ f \\<^sub>r x \ h \ f \\<^sub>h h' \ h' \ ok (g x)" + shows "h \ ok (f \ g)" + using assms by blast + +lemma bind_is_OK_pure_I [intro]: + assumes "pure f h" and "h \ ok f" and "\x. h \ f \\<^sub>r x \ h \ ok (g x)" + shows "h \ ok (f \ g)" + using assms by blast + +lemma bind_pure_I: + assumes "pure f h" and "\x. h \ f \\<^sub>r x \ pure (g x) h" + shows "pure (f \ g) h" + using assms + by (metis bind_returns_heap_E2 pure_def pure_returns_heap_eq is_OK_returns_heap_E) + +lemma pure_pure: + assumes "h \ ok f" and "pure f h" + shows "h \ f \\<^sub>h h" + using assms returns_heap_eq + unfolding pure_def + by auto + +lemma bind_returns_error_eq: + assumes "h \ f \\<^sub>e e" + and "h \ g \\<^sub>e e" + shows "h \ f = h \ g" + using assms + by(auto simp add: returns_error_def split: sum.splits) + +subsection \Map\ + +fun map_M :: "('x \ ('heap, 'e, 'result) prog) \ 'x list \ ('heap, 'e, 'result list) prog" + where + "map_M f [] = return []" + | "map_M f (x#xs) = do { + y \ f x; + ys \ map_M f xs; + return (y # ys) + }" + +lemma map_M_ok_I [intro]: + "(\x. x \ set xs \ h \ ok (f x)) \ (\x. x \ set xs \ pure (f x) h) \ h \ ok (map_M f xs)" + apply(induct xs) + by (simp_all add: bind_is_OK_I2 bind_is_OK_pure_I) + +lemma map_M_pure_I : "\h. (\x. x \ set xs \ pure (f x) h) \ pure (map_M f xs) h" + apply(induct xs) + apply(simp) + by(auto intro!: bind_pure_I) + +lemma map_M_pure_E : + assumes "h \ map_M g xs \\<^sub>r ys" and "x \ set xs" and "\x h. x \ set xs \ pure (g x) h" + obtains y where "h \ g x \\<^sub>r y" and "y \ set ys" + apply(insert assms, induct xs arbitrary: ys) + apply(simp) + apply(auto elim!: bind_returns_result_E)[1] + by (metis (full_types) pure_returns_heap_eq) + +lemma map_M_pure_E2: + assumes "h \ map_M g xs \\<^sub>r ys" and "y \ set ys" and "\x h. x \ set xs \ pure (g x) h" + obtains x where "h \ g x \\<^sub>r y" and "x \ set xs" + apply(insert assms, induct xs arbitrary: ys) + apply(simp) + apply(auto elim!: bind_returns_result_E)[1] + by (metis (full_types) pure_returns_heap_eq) + + +subsection \Forall\ + +fun forall_M :: "('y \ ('heap, 'e, 'result) prog) \ 'y list \ ('heap, 'e, unit) prog" + where + "forall_M P [] = return ()" + | "forall_M P (x # xs) = do { + P x; + forall_M P xs + }" + (* +lemma forall_M_elim: + assumes "h \ forall_M P xs \\<^sub>r True" and "\x h. x \ set xs \ pure (P x) h" + shows "\x \ set xs. h \ P x \\<^sub>r True" + apply(insert assms, induct xs) + apply(simp) + apply(auto elim!: bind_returns_result_E)[1] + by (metis (full_types) pure_returns_heap_eq) *) + +lemma pure_forall_M_I: "(\x. x \ set xs \ pure (P x) h) \ pure (forall_M P xs) h" + apply(induct xs) + by(auto intro!: bind_pure_I) + (* +lemma forall_M_pure_I: + assumes "\x. x \ set xs \ h \ P x \\<^sub>r True" and "\x h. x \ set xs \ pure (P x)h" + shows "h \ forall_M P xs \\<^sub>r True" + apply(insert assms, induct xs) + apply(simp) + by(fastforce) + +lemma forall_M_pure_eq: + assumes "\x. x \ set xs \ h \ P x \\<^sub>r True \ h' \ P x \\<^sub>r True" + and "\x h. x \ set xs \ pure (P x) h" + shows "(h \ forall_M P xs \\<^sub>r True) \ h' \ forall_M P xs \\<^sub>r True" + using assms + by(auto intro!: forall_M_pure_I dest!: forall_M_elim) *) + +subsection \Fold\ + +fun fold_M :: "('result \ 'y \ ('heap, 'e, 'result) prog) \ 'result \ 'y list + \ ('heap, 'e, 'result) prog" + where + "fold_M f d [] = return d" | + "fold_M f d (x # xs) = do { y \ f d x; fold_M f y xs }" + +lemma fold_M_pure_I : "(\d x. pure (f d x) h) \ (\d. pure (fold_M f d xs) h)" + apply(induct xs) + by(auto intro: bind_pure_I) + +subsection \Filter\ + +fun filter_M :: "('x \ ('heap, 'e, bool) prog) \ 'x list \ ('heap, 'e, 'x list) prog" + where + "filter_M P [] = return []" + | "filter_M P (x#xs) = do { + p \ P x; + ys \ filter_M P xs; + return (if p then x # ys else ys) + }" + +lemma filter_M_pure_I [intro]: "(\x. x \ set xs \ pure (P x) h) \ pure (filter_M P xs)h" + apply(induct xs) + by(auto intro!: bind_pure_I) + +lemma filter_M_is_OK_I [intro]: "(\x. x \ set xs \ h \ ok (P x)) \ (\x. x \ set xs \ pure (P x) h) \ h \ ok (filter_M P xs)" + apply(induct xs) + apply(simp) + by(auto intro!: bind_is_OK_pure_I) + +lemma filter_M_not_more_elements: + assumes "h \ filter_M P xs \\<^sub>r ys" and "\x. x \ set xs \ pure (P x) h" and "x \ set ys" + shows "x \ set xs" + apply(insert assms, induct xs arbitrary: ys) + by(auto elim!: bind_returns_result_E2 split: if_splits intro!: set_ConsD) + +lemma filter_M_in_result_if_ok: + assumes "h \ filter_M P xs \\<^sub>r ys" and "\h x. x \ set xs \ pure (P x) h" and "x \ set xs" and "h \ P x \\<^sub>r True" + shows "x \ set ys" + apply(insert assms, induct xs arbitrary: ys) + apply(simp) + apply(auto elim!: bind_returns_result_E2)[1] + by (metis returns_result_eq) + +lemma filter_M_holds_for_result: + assumes "h \ filter_M P xs \\<^sub>r ys" and "x \ set ys" and "\x h. x \ set xs \ pure (P x) h" + shows "h \ P x \\<^sub>r True" + apply(insert assms, induct xs arbitrary: ys) + by(auto elim!: bind_returns_result_E2 split: if_splits intro!: set_ConsD) + +lemma filter_M_empty_I: + assumes "\x. pure (P x) h" + and "\x \ set xs. h \ P x \\<^sub>r False" + shows "h \ filter_M P xs \\<^sub>r []" + using assms + apply(induct xs) + by(auto intro!: bind_pure_returns_result_I) + +lemma filter_M_subset_2: "h \ filter_M P xs \\<^sub>r ys \ h' \ filter_M P xs \\<^sub>r ys' + \ (\x. pure (P x) h) \ (\x. pure (P x) h') + \ (\b. \x \ set xs. h \ P x \\<^sub>r True \ h' \ P x \\<^sub>r b \ b) + \ set ys \ set ys'" +proof - + assume 1: "h \ filter_M P xs \\<^sub>r ys" and 2: "h' \ filter_M P xs \\<^sub>r ys'" + and 3: "(\x. pure (P x) h)" and "(\x. pure (P x) h')" + and 4: "\b. \x\set xs. h \ P x \\<^sub>r True \ h' \ P x \\<^sub>r b \ b" + have h1: "\x \ set xs. h' \ ok (P x)" + using 2 3 \(\x. pure (P x) h')\ + apply(induct xs arbitrary: ys') + by(auto elim!: bind_returns_result_E2) + then have 5: "\x\set xs. h \ P x \\<^sub>r True \ h' \ P x \\<^sub>r True" + using 4 + apply(auto)[1] + by (metis is_OK_returns_result_E) + show ?thesis + using 1 2 3 5 \(\x. pure (P x) h')\ + apply(induct xs arbitrary: ys ys') + apply(auto)[1] + apply(auto elim!: bind_returns_result_E2 split: if_splits)[1] + apply auto[1] + apply auto[1] + apply(metis returns_result_eq) + apply auto[1] + apply auto[1] + apply auto[1] + by(auto) +qed + +lemma filter_M_subset: "h \ filter_M P xs \\<^sub>r ys \ set ys \ set xs" + apply(induct xs arbitrary: h ys) + apply(auto)[1] + apply(auto elim!: bind_returns_result_E split: if_splits)[1] + apply blast + by blast + +lemma filter_M_distinct: "h \ filter_M P xs \\<^sub>r ys \ distinct xs \ distinct ys" + apply(induct xs arbitrary: h ys) + apply(auto)[1] + using filter_M_subset + apply(auto elim!: bind_returns_result_E)[1] + by fastforce + +lemma filter_M_filter: "h \ filter_M P xs \\<^sub>r ys \ (\x. x \ set xs \ pure (P x) h) + \ (\x \ set xs. h \ ok P x) \ ys = filter (\x. |h \ P x|\<^sub>r) xs" + apply(induct xs arbitrary: ys) + by(auto elim!: bind_returns_result_E2) + +lemma filter_M_filter2: "(\x. x \ set xs \ pure (P x) h \ h \ ok P x) + \ filter (\x. |h \ P x|\<^sub>r) xs = ys \ h \ filter_M P xs \\<^sub>r ys" + apply(induct xs arbitrary: ys) + by(auto elim!: bind_returns_result_E2 intro!: bind_pure_returns_result_I) + +lemma filter_ex1: "\!x \ set xs. P x \ P x \ x \ set xs \ distinct xs + \ filter P xs = [x]" + apply(auto)[1] + apply(induct xs) + apply(auto)[1] + apply(auto)[1] + using filter_empty_conv by fastforce + +lemma filter_M_ex1: + assumes "h \ filter_M P xs \\<^sub>r ys" + and "x \ set xs" + and "\!x \ set xs. h \ P x \\<^sub>r True" + and "\x. x \ set xs \ pure (P x) h" + and "distinct xs" + and "h \ P x \\<^sub>r True" + shows "ys = [x]" +proof - + have "\!x \ set xs. |h \ P x|\<^sub>r" + apply(insert assms(1) assms(3) assms(4)) + apply(drule filter_M_filter) + apply(simp) + apply(auto simp add: select_result_I2)[1] + by (metis (full_types) is_OK_returns_result_E select_result_I2) + then show ?thesis + apply(insert assms(1) assms(4)) + apply(drule filter_M_filter) + apply(auto)[1] + by (metis \\!x. x \ set xs \ |h \ P x|\<^sub>r\ assms(2) assms(5) assms(6) distinct_filter + distinct_length_2_or_more filter_empty_conv filter_set list.exhaust + list.set_intros(1) list.set_intros(2) member_filter select_result_I2) +qed + +lemma filter_M_eq: + assumes "\x. pure (P x) h" and "\x. pure (P x) h'" + and "\b x. x \ set xs \ h \ P x \\<^sub>r b = h' \ P x \\<^sub>r b" + shows "h \ filter_M P xs \\<^sub>r ys \ h' \ filter_M P xs \\<^sub>r ys" + using assms + apply (induct xs arbitrary: ys) + by(auto elim!: bind_returns_result_E2 intro!: bind_pure_returns_result_I + dest: returns_result_eq) + + +subsection \Map Filter\ + +definition map_filter_M :: "('x \ ('heap, 'e, 'y option) prog) \ 'x list + \ ('heap, 'e, 'y list) prog" + where + "map_filter_M f xs = do { + ys_opts \ map_M f xs; + ys_no_opts \ filter_M (\x. return (x \ None)) ys_opts; + map_M (\x. return (the x)) ys_no_opts + }" + +lemma map_filter_M_pure: "(\x h. x \ set xs \ pure (f x) h) \ pure (map_filter_M f xs) h" + by(auto simp add: map_filter_M_def map_M_pure_I intro!: bind_pure_I) + +lemma map_filter_M_pure_E: + assumes "h \ (map_filter_M::('x \ ('heap, 'e, 'y option) prog) \ 'x list + \ ('heap, 'e, 'y list) prog) f xs \\<^sub>r ys" and "y \ set ys" and "\x h. x \ set xs \ pure (f x) h" + obtains x where "h \ f x \\<^sub>r Some y" and "x \ set xs" +proof - + obtain ys_opts ys_no_opts where + ys_opts: "h \ map_M f xs \\<^sub>r ys_opts" and + ys_no_opts: "h \ filter_M (\x. (return (x \ None)::('heap, 'e, bool) prog)) ys_opts \\<^sub>r ys_no_opts" and + ys: "h \ map_M (\x. (return (the x)::('heap, 'e, 'y) prog)) ys_no_opts \\<^sub>r ys" + using assms + by(auto simp add: map_filter_M_def map_M_pure_I elim!: bind_returns_result_E2) + have "\y \ set ys_no_opts. y \ None" + using ys_no_opts filter_M_holds_for_result + by fastforce + then have "Some y \ set ys_no_opts" + using map_M_pure_E2 ys \y \ set ys\ + by (metis (no_types, lifting) option.collapse return_pure return_returns_result) + then have "Some y \ set ys_opts" + using filter_M_subset ys_no_opts by fastforce + then show "(\x. h \ f x \\<^sub>r Some y \ x \ set xs \ thesis) \ thesis" + by (metis assms(3) map_M_pure_E2 ys_opts) +qed + + +subsection \Iterate\ + +fun iterate_M :: "('heap, 'e, 'result) prog list \ ('heap, 'e, 'result) prog" + where + "iterate_M [] = return undefined" + | "iterate_M (x # xs) = x \ (\_. iterate_M xs)" + + +lemma iterate_M_concat: + assumes "h \ iterate_M xs \\<^sub>h h'" + and "h' \ iterate_M ys \\<^sub>h h''" + shows "h \ iterate_M (xs @ ys) \\<^sub>h h''" + using assms + apply(induct "xs" arbitrary: h h'') + apply(simp) + apply(auto)[1] + by (meson bind_returns_heap_E bind_returns_heap_I) + +subsection\Miscellaneous Rules\ + +lemma execute_bind_simp: + assumes "h \ f \\<^sub>r x" and "h \ f \\<^sub>h h'" + shows "h \ f \ g = h' \ g x" + using assms + by(auto simp add: returns_result_def returns_heap_def bind_def execute_def + split: sum.splits) + +lemma bind_cong [fundef_cong]: + fixes f1 f2 :: "('heap, 'e, 'result) prog" + and g1 g2 :: "'result \ ('heap, 'e, 'result2) prog" + assumes "h \ f1 = h \ f2" + and "\y h'. h \ f1 \\<^sub>r y \ h \ f1 \\<^sub>h h' \ h' \ g1 y = h' \ g2 y" + shows "h \ (f1 \ g1) = h \ (f2 \ g2)" + apply(insert assms, cases "h \ f1") + by(auto simp add: bind_def returns_result_def returns_heap_def execute_def + split: sum.splits) + +lemma bind_cong_2: + assumes "pure f h" and "pure f h'" + and "\x. h \ f \\<^sub>r x = h' \ f \\<^sub>r x" + and "\x. h \ f \\<^sub>r x \ h \ g x \\<^sub>r y = h' \ g x \\<^sub>r y'" + shows "h \ f \ g \\<^sub>r y = h' \ f \ g \\<^sub>r y'" + using assms + by(auto intro!: bind_pure_returns_result_I elim!: bind_returns_result_E2) + +lemma bind_case_cong [fundef_cong]: + assumes "x = x'" and "\a. x = Some a \ f a h = f' a h" + shows "(case x of Some a \ f a | None \ g) h = (case x' of Some a \ f' a | None \ g) h" + by (insert assms, simp add: option.case_eq_if) + + +subsection \Reasoning About Reads and Writes\ + +definition preserved :: "('heap, 'e, 'result) prog \ 'heap \ 'heap \ bool" + where + "preserved f h h' \ (\x. h \ f \\<^sub>r x \ h' \ f \\<^sub>r x)" + +lemma reflp_preserved_f [simp]: "reflp (preserved f)" + by(auto simp add: preserved_def reflp_def) +lemma transp_preserved_f [simp]: "transp (preserved f)" + by(auto simp add: preserved_def transp_def) + + +definition + all_args :: "('a \ ('heap, 'e, 'result) prog) \ ('heap, 'e, 'result) prog set" + where + "all_args f = (\arg. {f arg})" + + +definition + reads :: "('heap \ 'heap \ bool) set \ ('heap, 'e, 'result) prog \ 'heap + \ 'heap \ bool" + where + "reads S getter h h' \ (\P \ S. reflp P \ transp P) \ ((\P \ S. P h h') + \ preserved getter h h')" + +lemma reads_singleton [simp]: "reads {preserved f} f h h'" + by(auto simp add: reads_def) + +lemma reads_bind_pure: + assumes "pure f h" and "pure f h'" + and "reads S f h h'" + and "\x. h \ f \\<^sub>r x \ reads S (g x) h h'" + shows "reads S (f \ g) h h'" + using assms + by(auto simp add: reads_def pure_pure preserved_def + intro!: bind_pure_returns_result_I is_OK_returns_result_I + dest: pure_returns_heap_eq + elim!: bind_returns_result_E) + +lemma reads_insert_writes_set_left: "\P \ S. reflp P \ transp P \ reads {getter} f h h' \ reads (insert getter S) f h h'" + unfolding reads_def by simp + +lemma reads_insert_writes_set_right: "reflp getter \ transp getter \ reads S f h h' \ reads (insert getter S) f h h'" + unfolding reads_def by blast + +lemma reads_subset: "reads S f h h' \ \P \ S' - S. reflp P \ transp P \ S \ S' \ reads S' f h h'" + by(auto simp add: reads_def) + +lemma return_reads [simp]: "reads {} (return x) h h'" + by(simp add: reads_def preserved_def) + +lemma error_reads [simp]: "reads {} (error e) h h'" + by(simp add: reads_def preserved_def) + +lemma noop_reads [simp]: "reads {} noop h h'" + by(simp add: reads_def noop_def preserved_def) + +lemma filter_M_reads: + assumes "\x. x \ set xs \ pure (P x) h" and "\x. x \ set xs \ pure (P x) h'" + and "\x. x \ set xs \ reads S (P x) h h'" + and "\P \ S. reflp P \ transp P" + shows "reads S (filter_M P xs) h h'" + using assms + apply(induct xs) + by(auto intro: reads_subset[OF return_reads] intro!: reads_bind_pure) + +definition writes :: + "('heap, 'e, 'result) prog set \ ('heap, 'e, 'result2) prog \ 'heap \ 'heap \ bool" + where + "writes S setter h h' + \ (h \ setter \\<^sub>h h' \ (\progs. set progs \ S \ h \ iterate_M progs \\<^sub>h h'))" + +lemma writes_singleton [simp]: "writes (all_args f) (f a) h h'" + apply(auto simp add: writes_def all_args_def)[1] + apply(rule exI[where x="[f a]"]) + by(auto) + +lemma writes_singleton2 [simp]: "writes {f} f h h'" + apply(auto simp add: writes_def all_args_def)[1] + apply(rule exI[where x="[f]"]) + by(auto) + +lemma writes_union_left_I: + assumes "writes S f h h'" + shows "writes (S \ S') f h h'" + using assms + by(auto simp add: writes_def) + +lemma writes_union_right_I: + assumes "writes S' f h h'" + shows "writes (S \ S') f h h'" + using assms + by(auto simp add: writes_def) + +lemma writes_union_minus_split: + assumes "writes (S - S2) f h h'" + and "writes (S' - S2) f h h'" + shows "writes ((S \ S') - S2) f h h'" + using assms + by(auto simp add: writes_def) + +lemma writes_subset: "writes S f h h' \ S \ S' \ writes S' f h h'" + by(auto simp add: writes_def) + +lemma writes_error [simp]: "writes S (error e) h h'" + by(simp add: writes_def) + +lemma writes_not_ok [simp]: "\h \ ok f \ writes S f h h'" + by(auto simp add: writes_def) + +lemma writes_pure [simp]: + assumes "pure f h" + shows "writes S f h h'" + using assms + apply(auto simp add: writes_def)[1] + by (metis bot.extremum iterate_M.simps(1) list.set(1) pure_returns_heap_eq return_returns_heap) + +lemma writes_bind: + assumes "\h2. writes S f h h2" + assumes "\x h2. h \ f \\<^sub>r x \ h \ f \\<^sub>h h2 \ writes S (g x) h2 h'" + shows "writes S (f \ g) h h'" + using assms + apply(auto simp add: writes_def elim!: bind_returns_heap_E)[1] + by (metis iterate_M_concat le_supI set_append) + +lemma writes_bind_pure: + assumes "pure f h" + assumes "\x. h \ f \\<^sub>r x \ writes S (g x) h h'" + shows "writes S (f \ g) h h'" + using assms + by(auto simp add: writes_def elim!: bind_returns_heap_E2) + +lemma writes_small_big: + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "\h h' w. w \ SW \ h \ w \\<^sub>h h' \ P h h'" + assumes "reflp P" + assumes "transp P" + shows "P h h'" +proof - + obtain progs where "set progs \ SW" and iterate: "h \ iterate_M progs \\<^sub>h h'" + by (meson assms(1) assms(2) writes_def) + then have "\h h'. \prog \ set progs. h \ prog \\<^sub>h h' \ P h h'" + using assms(3) by auto + with iterate assms(4) assms(5) have "h \ iterate_M progs \\<^sub>h h' \ P h h'" + proof(induct progs arbitrary: h) + case Nil + then show ?case + using reflpE by force + next + case (Cons a progs) + then show ?case + apply(auto elim!: bind_returns_heap_E)[1] + by (metis (full_types) transpD) + qed + then show ?thesis + using assms(1) iterate by blast +qed + +lemma reads_writes_preserved: + assumes "reads SR getter h h'" + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "\h h'. \w \ SW. h \ w \\<^sub>h h' \ (\r \ SR. r h h')" + shows "h \ getter \\<^sub>r x \ h' \ getter \\<^sub>r x" +proof - + obtain progs where "set progs \ SW" and iterate: "h \ iterate_M progs \\<^sub>h h'" + by (meson assms(2) assms(3) writes_def) + then have "\h h'. \prog \ set progs. h \ prog \\<^sub>h h' \ (\r \ SR. r h h')" + using assms(4) by blast + with iterate have "\r \ SR. r h h'" + using writes_small_big assms(1) unfolding reads_def + by (metis assms(2) assms(3) assms(4)) + then show ?thesis + using assms(1) + by (simp add: preserved_def reads_def) +qed + +lemma reads_writes_separate_forwards: + assumes "reads SR getter h h'" + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "h \ getter \\<^sub>r x" + assumes "\h h'. \w \ SW. h \ w \\<^sub>h h' \ (\r \ SR. r h h')" + shows "h' \ getter \\<^sub>r x" + using reads_writes_preserved[OF assms(1) assms(2) assms(3) assms(5)] assms(4) + by(auto simp add: preserved_def) + +lemma reads_writes_separate_backwards: + assumes "reads SR getter h h'" + assumes "writes SW setter h h'" + assumes "h \ setter \\<^sub>h h'" + assumes "h' \ getter \\<^sub>r x" + assumes "\h h'. \w \ SW. h \ w \\<^sub>h h' \ (\r \ SR. r h h')" + shows "h \ getter \\<^sub>r x" + using reads_writes_preserved[OF assms(1) assms(2) assms(3) assms(5)] assms(4) + by(auto simp add: preserved_def) + +end diff --git a/Core_DOM/preliminaries/Hiding_Type_Variables.thy b/Core_DOM/preliminaries/Hiding_Type_Variables.thy new file mode 100644 index 0000000..3c593ef --- /dev/null +++ b/Core_DOM/preliminaries/Hiding_Type_Variables.thy @@ -0,0 +1,584 @@ +(*********************************************************************************** + * Copyright (c) 2018 Achim D. Brucker + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + * Repository: https://git.logicalhacking.com/adbrucker/isabelle-hacks/ + * Dependencies: None (assert.thy is used for testing the theory but it is + * not required for providing the functionality of this hack) + ***********************************************************************************) + +(* + This file is based on commit 8a5e95421521c36ab71ab2711435a9bc0fa2c5cc from upstream + (https://git.logicalhacking.com/adbrucker/isabelle-hacks/). Merely the dependency to + Assert.thy has been removed by disabling the example section (which include assert + checks). +*) + +section\Hiding Type Variables\ +text\ This theory\footnote{This theory can be used ``stand-alone,'' i.e., this theory is + not specific to the DOM formalization. The latest version is part of the ``Isabelle Hacks'' + repository: \url{https://git.logicalhacking.com/adbrucker/isabelle-hacks/}.} implements + a mechanism for declaring default type variables for data types. This comes handy for complex + data types with many type variables.\ +theory + "Hiding_Type_Variables" +imports + Main +keywords + "register_default_tvars" + "update_default_tvars_mode"::thy_decl +begin +(*<*) +section\Implementation\ +subsection\Theory Managed Data Structure\ +ML\ +signature HIDE_TVAR = sig + datatype print_mode = print_all | print | noprint + datatype tvar_subst = right | left + datatype parse_mode = parse | noparse + type hide_varT = { + name: string, + tvars: typ list, + typ_syn_tab : (string * typ list*string) Symtab.table, + print_mode: print_mode, + parse_mode: parse_mode + } + val parse_print_mode : string -> print_mode + val parse_parse_mode : string -> parse_mode + val register : string -> print_mode option -> parse_mode option -> + theory -> theory + val update_mode : string -> print_mode option -> parse_mode option -> + theory -> theory + val lookup : theory -> string -> hide_varT option + val hide_tvar_tr' : string -> Proof.context -> term list -> term + val hide_tvar_ast_tr : Proof.context -> Ast.ast list -> Ast.ast + val hide_tvar_subst_ast_tr : tvar_subst -> Proof.context -> Ast.ast list + -> Ast.ast + val hide_tvar_subst_return_ast_tr : tvar_subst -> Proof.context + -> Ast.ast list -> Ast.ast +end + +structure Hide_Tvar : HIDE_TVAR = struct + datatype print_mode = print_all | print | noprint + datatype tvar_subst = right | left + datatype parse_mode = parse | noparse + type hide_varT = { + name: string, + tvars: typ list, + typ_syn_tab : (string * typ list*string) Symtab.table, + print_mode: print_mode, + parse_mode: parse_mode + } + type hide_tvar_tab = (hide_varT) Symtab.table + fun hide_tvar_eq (a, a') = (#name a) = (#name a') + fun merge_tvar_tab (tab,tab') = Symtab.merge hide_tvar_eq (tab,tab') + + structure Data = Generic_Data + ( + type T = hide_tvar_tab + val empty = Symtab.empty:hide_tvar_tab + val extend = I + fun merge(t1,t2) = merge_tvar_tab (t1, t2) + ); + + + fun parse_print_mode "print_all" = print_all + | parse_print_mode "print" = print + | parse_print_mode "noprint" = noprint + | parse_print_mode s = error("Print mode not supported: "^s) + + fun parse_parse_mode "parse" = parse + | parse_parse_mode "noparse" = noparse + | parse_parse_mode s = error("Parse mode not supported: "^s) + + fun update_mode typ_str print_mode parse_mode thy = + let + val ctx = Toplevel.context_of(Toplevel.theory_toplevel thy) + val typ = Syntax.parse_typ ctx typ_str (* no type checking *) + val name = case typ of + Type(name,_) => name + | _ => error("Complex type not (yet) supported.") + fun update tab = + let + val old_entry = (case Symtab.lookup tab name of + SOME t => t + | NONE => error ("Type shorthand not registered: "^name)) + val print_m = case print_mode of + SOME m => m + | NONE => #print_mode old_entry + val parse_m = case parse_mode of + SOME m => m + | NONE => #parse_mode old_entry + val entry = { + name = name, + tvars = #tvars old_entry, + typ_syn_tab = #typ_syn_tab old_entry, + print_mode = print_m, + parse_mode = parse_m + } + in + Symtab.update (name,entry) tab + end + in + Context.theory_of ( (Data.map update) (Context.Theory thy)) + end + + fun lookup thy name = + let + val tab = (Data.get o Context.Theory) thy + in + Symtab.lookup tab name + end + + fun obtain_normalized_vname lookup_table vname = + case List.find (fn e => fst e = vname) lookup_table of + SOME (_,idx) => (lookup_table, Int.toString idx) + | NONE => let + fun max_idx [] = 0 + | max_idx ((_,idx)::lt) = Int.max(idx,max_idx lt) + + val idx = (max_idx lookup_table ) + 1 + in + ((vname,idx)::lookup_table, Int.toString idx) end + + fun normalize_typvar_type lt (Type (a, Ts)) = + let + fun switch (a,b) = (b,a) + val (Ts', lt') = fold_map (fn t => fn lt => switch (normalize_typvar_type lt t)) Ts lt + in + (lt', Type (a, Ts')) + end + | normalize_typvar_type lt (TFree (vname, S)) = + let + val (lt, vname) = obtain_normalized_vname lt (vname) + in + (lt, TFree( vname, S)) + end + | normalize_typvar_type lt (TVar (xi, S)) = + let + val (lt, vname) = obtain_normalized_vname lt (Term.string_of_vname xi) + in + (lt, TFree( vname, S)) + end + + fun normalize_typvar_type' t = snd ( normalize_typvar_type [] t) + + fun mk_p s = s (* "("^s^")" *) + + fun key_of_type (Type(a, TS)) = mk_p (a^String.concat(map key_of_type TS)) + | key_of_type (TFree (vname, _)) = mk_p vname + | key_of_type (TVar (xi, _ )) = mk_p (Term.string_of_vname xi) + val key_of_type' = key_of_type o normalize_typvar_type' + + + fun normalize_typvar_term lt (Const (a, t)) = (lt, Const(a, t)) + | normalize_typvar_term lt (Free (a, t)) = let + val (lt, vname) = obtain_normalized_vname lt a + in + (lt, Free(vname,t)) + end + | normalize_typvar_term lt (Var (xi, t)) = + let + val (lt, vname) = obtain_normalized_vname lt (Term.string_of_vname xi) + in + (lt, Free(vname,t)) + end + | normalize_typvar_term lt (Bound (i)) = (lt, Bound(i)) + | normalize_typvar_term lt (Abs(s,ty,tr)) = + let + val (lt,tr) = normalize_typvar_term lt tr + in + (lt, Abs(s,ty,tr)) + end + | normalize_typvar_term lt (t1$t2) = + let + val (lt,t1) = normalize_typvar_term lt t1 + val (lt,t2) = normalize_typvar_term lt t2 + in + (lt, t1$t2) + end + + + fun normalize_typvar_term' t = snd(normalize_typvar_term [] t) + + fun key_of_term (Const(s,_)) = if String.isPrefix "\<^type>" s + then Lexicon.unmark_type s + else "" + | key_of_term (Free(s,_)) = s + | key_of_term (Var(xi,_)) = Term.string_of_vname xi + | key_of_term (Bound(_)) = error("Bound() not supported in key_of_term") + | key_of_term (Abs(_,_,_)) = error("Abs() not supported in key_of_term") + | key_of_term (t1$t2) = (key_of_term t1)^(key_of_term t2) + + val key_of_term' = key_of_term o normalize_typvar_term' + + + fun hide_tvar_tr' tname ctx terms = + let + + val mtyp = Syntax.parse_typ ctx tname (* no type checking *) + + val (fq_name, _) = case mtyp of + Type(s,ts) => (s,ts) + | _ => error("Complex type not (yet) supported.") + + val local_name_of = hd o rev o String.fields (fn c => c = #".") + + fun hide_type tname = Syntax.const("(_) "^tname) + + val reg_type_as_term = Term.list_comb(Const(Lexicon.mark_type tname,dummyT),terms) + val key = key_of_term' reg_type_as_term + val actual_tvars_key = key_of_term reg_type_as_term + + in + case lookup (Proof_Context.theory_of ctx) fq_name of + NONE => raise Match + | SOME e => let + val (tname,default_tvars_key) = + case Symtab.lookup (#typ_syn_tab e) key of + NONE => (local_name_of tname, "") + | SOME (s,_,tv) => (local_name_of s,tv) + in + case (#print_mode e) of + print_all => hide_type tname + | print => if default_tvars_key=actual_tvars_key + then hide_type tname + else raise Match + | noprint => raise Match + end + end + + fun hide_tvar_ast_tr ctx ast= + let + val thy = Proof_Context.theory_of ctx + + fun parse_ast ((Ast.Constant const)::[]) = (const,NONE) + | parse_ast ((Ast.Constant sort)::(Ast.Constant const)::[]) + = (const,SOME sort) + | parse_ast _ = error("AST type not supported.") + + val (decorated_name, decorated_sort) = parse_ast ast + + val name = Lexicon.unmark_type decorated_name + val default_info = case lookup thy name of + NONE => error("No default type vars registered: "^name) + | SOME e => e + val _ = if #parse_mode default_info = noparse + then error("Default type vars disabled (option noparse): "^name) + else () + fun name_of_tvar tvar = case tvar of (TFree(n,_)) => n + | _ => error("Unsupported type structure.") + val type_vars_ast = + let fun mk_tvar n = + case decorated_sort of + NONE => Ast.Variable(name_of_tvar n) + | SOME sort => Ast.Appl([Ast.Constant("_ofsort"), + Ast.Variable(name_of_tvar n), + Ast.Constant(sort)]) + in + map mk_tvar (#tvars default_info) + end + in + Ast.Appl ((Ast.Constant decorated_name)::type_vars_ast) + end + + fun register typ_str print_mode parse_mode thy = + let + val ctx = Toplevel.context_of(Toplevel.theory_toplevel thy) + val typ = Syntax.parse_typ ctx typ_str + val (name,tvars) = case typ of Type(name,tvars) => (name,tvars) + | _ => error("Unsupported type structure.") + + val base_typ = Syntax.read_typ ctx typ_str + val (base_name,base_tvars) = case base_typ of Type(name,tvars) => (name,tvars) + | _ => error("Unsupported type structure.") + + val base_key = key_of_type' base_typ + val base_tvar_key = key_of_type base_typ + + val print_m = case print_mode of + SOME m => m + | NONE => print_all + val parse_m = case parse_mode of + SOME m => m + | NONE => parse + val entry = { + name = name, + tvars = tvars, + typ_syn_tab = Symtab.empty:((string * typ list * string) Symtab.table), + print_mode = print_m, + parse_mode = parse_m + } + + val base_entry = if name = base_name + then + { + name = "", + tvars = [], + typ_syn_tab = Symtab.empty:((string * typ list * string) Symtab.table), + print_mode = noprint, + parse_mode = noparse + } + else case lookup thy base_name of + SOME e => e + | NONE => error ("No entry found for "^base_name^ + " (via "^name^")") + + val base_entry = { + name = #name base_entry, + tvars = #tvars base_entry, + typ_syn_tab = Symtab.update (base_key, (name, base_tvars, base_tvar_key)) + (#typ_syn_tab (base_entry)), + print_mode = #print_mode base_entry, + parse_mode = #parse_mode base_entry + } + + fun reg tab = let + val tab = Symtab.update_new(name, entry) tab + val tab = if name = base_name + then tab + else Symtab.update(base_name, base_entry) tab + in + tab + end + + val thy = Sign.print_translation + [(Lexicon.mark_type name, hide_tvar_tr' name)] thy + + in + Context.theory_of ( (Data.map reg) (Context.Theory thy)) + handle Symtab.DUP _ => error("Type shorthand already registered: "^name) + end + + fun hide_tvar_subst_ast_tr hole ctx (ast::[]) = + let + + val thy = Proof_Context.theory_of ctx + val (decorated_name, args) = case ast + of (Ast.Appl ((Ast.Constant s)::args)) => (s, args) + | _ => error "Error in obtaining type constructor." + + val name = Lexicon.unmark_type decorated_name + val default_info = case lookup thy name of + NONE => error("No default type vars registered: "^name) + | SOME e => e + val _ = if #parse_mode default_info = noparse + then error("Default type vars disabled (option noparse): "^name) + else () + fun name_of_tvar tvar = case tvar of (TFree(n,_)) => n + | _ => error("Unsupported type structure.") + val type_vars_ast = map (fn n => Ast.Variable(name_of_tvar n)) (#tvars default_info) + val type_vars_ast = case hole of + right => (List.rev(List.drop(List.rev type_vars_ast, List.length args)))@args + | left => args@List.drop(type_vars_ast, List.length args) + in + Ast.Appl ((Ast.Constant decorated_name)::type_vars_ast) + end + | hide_tvar_subst_ast_tr _ _ _ = error("hide_tvar_subst_ast_tr: empty AST.") + + fun hide_tvar_subst_return_ast_tr hole ctx (retval::constructor::[]) = + hide_tvar_subst_ast_tr hole ctx [Ast.Appl (constructor::retval::[])] + | hide_tvar_subst_return_ast_tr _ _ _ = + error("hide_tvar_subst_return_ast_tr: error in parsing AST") + + +end +\ + + + +subsection\Register Parse Translations\ +syntax "_tvars_wildcard" :: "type \ type" ("'('_') _") +syntax "_tvars_wildcard_retval" :: "type \ type \ type" ("'('_, _') _") +syntax "_tvars_wildcard_sort" :: "sort \ type \ type" ("'('_::_') _") +syntax "_tvars_wildcard_right" :: "type \ type" ("_ '_..") +syntax "_tvars_wildcard_left" :: "type \ type" ("_ ..'_") + +parse_ast_translation\ + [ + (@{syntax_const "_tvars_wildcard_sort"}, Hide_Tvar.hide_tvar_ast_tr), + (@{syntax_const "_tvars_wildcard"}, Hide_Tvar.hide_tvar_ast_tr), + (@{syntax_const "_tvars_wildcard_retval"}, Hide_Tvar.hide_tvar_subst_return_ast_tr Hide_Tvar.right), + (@{syntax_const "_tvars_wildcard_right"}, Hide_Tvar.hide_tvar_subst_ast_tr Hide_Tvar.right), + (@{syntax_const "_tvars_wildcard_left"}, Hide_Tvar.hide_tvar_subst_ast_tr Hide_Tvar.left) + ] +\ + +subsection\Register Top-Level Isar Commands\ +ML\ + val modeP = (Parse.$$$ "(" + |-- (Parse.name --| Parse.$$$ "," + -- Parse.name --| + Parse.$$$ ")")) + val typ_modeP = Parse.typ -- (Scan.optional modeP ("print_all","parse")) + + val _ = Outer_Syntax.command @{command_keyword "register_default_tvars"} + "Register default variables (and hiding mechanims) for a type." + (typ_modeP >> (fn (typ,(print_m,parse_m)) => + (Toplevel.theory + (Hide_Tvar.register typ + (SOME (Hide_Tvar.parse_print_mode print_m)) + (SOME (Hide_Tvar.parse_parse_mode parse_m)))))); + + val _ = Outer_Syntax.command @{command_keyword "update_default_tvars_mode"} + "Update print and/or parse mode or the default type variables for a certain type." + (typ_modeP >> (fn (typ,(print_m,parse_m)) => + (Toplevel.theory + (Hide_Tvar.update_mode typ + (SOME (Hide_Tvar.parse_print_mode print_m)) + (SOME (Hide_Tvar.parse_parse_mode parse_m)))))); +\ +(* +section\Examples\ +subsection\Print Translation\ +datatype ('a, 'b) hide_tvar_foobar = hide_tvar_foo 'a | hide_tvar_bar 'b +type_synonym ('a, 'b, 'c, 'd) hide_tvar_baz = "('a+'b, 'a \ 'b) hide_tvar_foobar" + +definition hide_tvar_f::"('a, 'b) hide_tvar_foobar \ ('a, 'b) hide_tvar_foobar \ ('a, 'b) hide_tvar_foobar" + where "hide_tvar_f a b = a" +definition hide_tvar_g::"('a, 'b, 'c, 'd) hide_tvar_baz \ ('a, 'b, 'c, 'd) hide_tvar_baz \ ('a, 'b, 'c, 'd) hide_tvar_baz" + where "hide_tvar_g a b = a" + +assert[string_of_thm_equal, + thm_def="hide_tvar_f_def", + str="hide_tvar_f (a::('a, 'b) hide_tvar_foobar) (b::('a, 'b) hide_tvar_foobar) = a"] +assert[string_of_thm_equal, + thm_def="hide_tvar_g_def", + str="hide_tvar_g (a::('a + 'b, 'a \ 'b) hide_tvar_foobar) (b::('a + 'b, 'a \ 'b) hide_tvar_foobar) = a"] + +register_default_tvars "('alpha, 'beta) hide_tvar_foobar" (print_all,parse) +register_default_tvars "('alpha, 'beta, 'gamma, 'delta) hide_tvar_baz" (print_all,parse) + +update_default_tvars_mode "_ hide_tvar_foobar" (noprint,noparse) +assert[string_of_thm_equal, + thm_def="hide_tvar_f_def", + str="hide_tvar_f (a::('a, 'b) hide_tvar_foobar) (b::('a, 'b) hide_tvar_foobar) = a"] +assert[string_of_thm_equal, + thm_def="hide_tvar_g_def", + str="hide_tvar_g (a::('a + 'b, 'a \ 'b) hide_tvar_foobar) (b::('a + 'b, 'a \ 'b) hide_tvar_foobar) = a"] + +update_default_tvars_mode "_ hide_tvar_foobar" (print_all,noparse) + +assert[string_of_thm_equal, + thm_def="hide_tvar_f_def", str="hide_tvar_f (a::(_) hide_tvar_foobar) (b::(_) hide_tvar_foobar) = a"] +assert[string_of_thm_equal, + thm_def="hide_tvar_g_def", str="hide_tvar_g (a::(_) hide_tvar_baz) (b::(_) hide_tvar_baz) = a"] + +subsection\Parse Translation\ +update_default_tvars_mode "_ hide_tvar_foobar" (print_all,parse) + +declare [[show_types]] +definition hide_tvar_A :: "'x \ (('x::linorder) hide_tvar_foobar) .._" + where "hide_tvar_A x = hide_tvar_foo x" +assert[string_of_thm_equal, + thm_def="hide_tvar_A_def", str="hide_tvar_A (x::'x) = hide_tvar_foo x"] + +definition hide_tvar_A' :: "'x \ (('x,'b) hide_tvar_foobar) .._" + where "hide_tvar_A' x = hide_tvar_foo x" +assert[string_of_thm_equal, + thm_def="hide_tvar_A'_def", str="hide_tvar_A' (x::'x) = hide_tvar_foo x"] + +definition hide_tvar_B' :: "(_) hide_tvar_foobar \ (_) hide_tvar_foobar \ (_) hide_tvar_foobar" + where "hide_tvar_B' x y = x" +assert[string_of_thm_equal, + thm_def="hide_tvar_A'_def", str="hide_tvar_A' (x::'x) = hide_tvar_foo x"] + + +definition hide_tvar_B :: "(_) hide_tvar_foobar \ (_) hide_tvar_foobar \ (_) hide_tvar_foobar" + where "hide_tvar_B x y = x" +assert[string_of_thm_equal, + thm_def="hide_tvar_B_def", str="hide_tvar_B (x::(_) hide_tvar_foobar) (y::(_) hide_tvar_foobar) = x"] + +definition hide_tvar_C :: "(_) hide_tvar_baz \ (_) hide_tvar_foobar \ (_) hide_tvar_baz" + where "hide_tvar_C x y = x" +assert[string_of_thm_equal, + thm_def="hide_tvar_C_def", str="hide_tvar_C (x::(_) hide_tvar_baz) (y::(_) hide_tvar_foobar) = x"] + +definition hide_tvar_E :: "(_::linorder) hide_tvar_baz \ (_::linorder) hide_tvar_foobar \ (_::linorder) hide_tvar_baz" + where "hide_tvar_E x y = x" +assert[string_of_thm_equal, + thm_def="hide_tvar_C_def", str="hide_tvar_C (x::(_) hide_tvar_baz) (y::(_) hide_tvar_foobar) = x"] + +definition hide_tvar_X :: "(_, 'retval::linorder) hide_tvar_baz + \ (_,'retval) hide_tvar_foobar + \ (_,'retval) hide_tvar_baz" + where "hide_tvar_X x y = x" +*) +(*>*) + +subsection\Introduction\ +text\ + When modelling object-oriented data models in HOL with the goal of preserving \<^emph>\extensibility\ + (e.g., as described in~\cite{brucker.ea:extensible:2008-b,brucker:interactive:2007}) one needs + to define type constructors with a large number of type variables. This can reduce the readability + of the overall formalization. Thus, we use a short-hand notation in cases were the names of + the type variables are known from the context. In more detail, this theory sets up both + configurable print and parse translations that allows for replacing @{emph \all\} type variables + by \(_)\, e.g., a five-ary constructor \('a, 'b, 'c, 'd, 'e) hide_tvar_foo\ can + be shorted to \(_) hide_tvar_foo\. The use of this shorthand in output (printing) and + input (parsing) is, on a per-type basis, user-configurable using the top-level commands + \register_default_tvars\ (for registering the names of the default type variables and + the print/parse mode) and \update_default_tvars_mode\ (for changing the print/parse mode + dynamically). + + The input also supports short-hands for declaring default sorts (e.g., \(_::linorder)\ + specifies that all default variables need to be instances of the sort (type class) + @{class \linorder\} and short-hands of overriding a suffice (or prefix) of the default type + variables. For example, \('state) hide_tvar_foo _.\ is a short-hand for + \('a, 'b, 'c, 'd, 'state) hide_tvar_foo\. In this document, we omit the implementation + details (we refer the interested reader to theory file) and continue directly with a few + examples. +\ + +subsection\Example\ +text\Given the following type definition:\ +datatype ('a, 'b) hide_tvar_foobar = hide_tvar_foo 'a | hide_tvar_bar 'b +type_synonym ('a, 'b, 'c, 'd) hide_tvar_baz = "('a+'b, 'a \ 'b) hide_tvar_foobar" +text\We can register default values for the type variables for the abstract +data type as well as the type synonym:\ +register_default_tvars "('alpha, 'beta) hide_tvar_foobar" (print_all,parse) +register_default_tvars "('alpha, 'beta, 'gamma, 'delta) hide_tvar_baz" (print_all,parse) +text\This allows us to write\ +definition hide_tvar_f::"(_) hide_tvar_foobar \ (_) hide_tvar_foobar \ (_) hide_tvar_foobar" + where "hide_tvar_f a b = a" +definition hide_tvar_g::"(_) hide_tvar_baz \ (_) hide_tvar_baz \ (_) hide_tvar_baz" + where "hide_tvar_g a b = a" + +text\Instead of specifying the type variables explicitely. This makes, in particular +for type constructors with a large number of type variables, definitions much +more concise. This syntax is also used in the output of antiquotations, e.g., +@{term[show_types] "x = hide_tvar_g"}. Both the print translation and the parse +translation can be disabled for each type individually:\ + +update_default_tvars_mode "_ hide_tvar_foobar" (noprint,noparse) +update_default_tvars_mode "_ hide_tvar_foobar" (noprint,noparse) + +text\ Now, Isabelle's interactive output and the antiquotations will show +all type variables, e.g., @{term[show_types] "x = hide_tvar_g"}.\ + + + +end diff --git a/Core_DOM/preliminaries/Testing_Utils.thy b/Core_DOM/preliminaries/Testing_Utils.thy new file mode 100644 index 0000000..e8280ba --- /dev/null +++ b/Core_DOM/preliminaries/Testing_Utils.thy @@ -0,0 +1,39 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +theory Testing_Utils + imports Main +begin +ML \ +val _ = Theory.setup + (Method.setup @{binding timed_code_simp} + (Scan.succeed (SIMPLE_METHOD' o (CHANGED_PROP oo (fn a => fn b => Timeout.apply (seconds 3600.0) (Code_Simp.dynamic_tac a b))))) + "simplification with code equations, aborts after 1 hour"); +\ +end diff --git a/Core_DOM/tests/Core_DOM_BaseTest.thy b/Core_DOM/tests/Core_DOM_BaseTest.thy new file mode 100644 index 0000000..19839f4 --- /dev/null +++ b/Core_DOM/tests/Core_DOM_BaseTest.thy @@ -0,0 +1,306 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Common Test Setup\ +text\This theory provides the common test setup that is used by all formalized test cases.\ + +theory Core_DOM_BaseTest + imports + (*<*) + "../preliminaries/Testing_Utils" + (*>*) + "../Core_DOM" +begin + +definition "assert_throws e p = do { + h \ get_heap; + (if (h \ p \\<^sub>e e) then return () else error AssertException) +}" +notation assert_throws ("assert'_throws'(_, _')") + +definition "test p h \ h \ ok p" + + +definition field_access :: "(string \ (_, (_) object_ptr option) dom_prog) \ string + \ (_, (_) object_ptr option) dom_prog" (infix "." 80) + where + "field_access m field = m field" + +definition assert_equals :: "'a \ 'a \ (_, unit) dom_prog" + where + "assert_equals l r = (if l = r then return () else error AssertException)" +definition assert_equals_with_message :: "'a \ 'a \ 'b \ (_, unit) dom_prog" + where + "assert_equals_with_message l r _ = (if l = r then return () else error AssertException)" +notation assert_equals ("assert'_equals'(_, _')") +notation assert_equals_with_message ("assert'_equals'(_, _, _')") +notation assert_equals ("assert'_array'_equals'(_, _')") +notation assert_equals_with_message ("assert'_array'_equals'(_, _, _')") + +definition assert_not_equals :: "'a \ 'a \ (_, unit) dom_prog" + where + "assert_not_equals l r = (if l \ r then return () else error AssertException)" +definition assert_not_equals_with_message :: "'a \ 'a \ 'b \ (_, unit) dom_prog" + where + "assert_not_equals_with_message l r _ = (if l \ r then return () else error AssertException)" +notation assert_not_equals ("assert'_not'_equals'(_, _')") +notation assert_not_equals_with_message ("assert'_not'_equals'(_, _, _')") +notation assert_not_equals ("assert'_array'_not'_equals'(_, _')") +notation assert_not_equals_with_message ("assert'_array'_not'_equals'(_, _, _')") + +definition removeWhiteSpaceOnlyTextNodes :: "((_) object_ptr option) \ (_, unit) dom_prog" + where + "removeWhiteSpaceOnlyTextNodes _ = return ()" + + +subsection \create\_heap\ + +(* We use this construction because partially applied functions such as "map_of xs" don't play + well together with the code generator. *) +definition "create_heap xs = Heap (fmap_of_list xs)" + +code_datatype ObjectClass.heap.Heap create_heap + +lemma object_ptr_kinds_code1 [code]: + "object_ptr_kinds (Heap (fmap_of_list xs)) = object_ptr_kinds (create_heap xs)" + by(simp add: create_heap_def) + +lemma object_ptr_kinds_code2 [code]: + "object_ptr_kinds (create_heap xs) = fset_of_list (map fst xs)" + by (simp add: object_ptr_kinds_def create_heap_def dom_map_of_conv_image_fst) + +lemma object_ptr_kinds_code3 [code]: + "fmlookup (the_heap (create_heap xs)) x = map_of xs x" + by(auto simp add: create_heap_def fmlookup_of_list) + +lemma object_ptr_kinds_code4 [code]: + "the_heap (create_heap xs) = fmap_of_list xs" + by(simp add: create_heap_def) + +lemma object_ptr_kinds_code5 [code]: + "the_heap (Heap x) = x" + by simp + +lemma object_ptr_kinds_code6 [code]: + "noop = return ()" + by(simp add: noop_def) + + +subsection \Making the functions under test compatible with untyped languages such as JavaScript\ + +fun set_attribute_with_null :: "((_) object_ptr option) \ attr_key \ attr_value \ (_, unit) dom_prog" + where + "set_attribute_with_null (Some ptr) k v = (case cast ptr of + Some element_ptr \ set_attribute element_ptr k (Some v))" +fun set_attribute_with_null2 :: "((_) object_ptr option) \ attr_key \ attr_value option \ (_, unit) dom_prog" + where + "set_attribute_with_null2 (Some ptr) k v = (case cast ptr of + Some element_ptr \ set_attribute element_ptr k v)" +notation set_attribute_with_null ("_ . setAttribute'(_, _')") +notation set_attribute_with_null2 ("_ . setAttribute'(_, _')") + +fun get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_with_null :: "((_) object_ptr option) \ (_, (_) object_ptr option list) dom_prog" + where + "get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_with_null (Some ptr) = do { + children \ get_child_nodes ptr; + return (map (Some \ cast) children) + }" +notation get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_with_null ("_ . childNodes") + +fun create_element_with_null :: "((_) object_ptr option) \ string \ (_, ((_) object_ptr option)) dom_prog" + where + "create_element_with_null (Some owner_document_obj) tag = (case cast owner_document_obj of + Some owner_document \ do { + element_ptr \ create_element owner_document tag; + return (Some (cast element_ptr))})" +notation create_element_with_null ("_ . createElement'(_')") + +fun create_character_data_with_null :: "((_) object_ptr option) \ string \ (_, ((_) object_ptr option)) dom_prog" + where + "create_character_data_with_null (Some owner_document_obj) tag = (case cast owner_document_obj of + Some owner_document \ do { + character_data_ptr \ create_character_data owner_document tag; + return (Some (cast character_data_ptr))})" +notation create_character_data_with_null ("_ . createTextNode'(_')") + +definition create_document_with_null :: "string \ (_, ((_::linorder) object_ptr option)) dom_prog" + where + "create_document_with_null title = do { + new_document_ptr \ create_document; + html \ create_element new_document_ptr ''html''; + append_child (cast new_document_ptr) (cast html); + heap \ create_element new_document_ptr ''heap''; + append_child (cast html) (cast heap); + body \ create_element new_document_ptr ''body''; + append_child (cast html) (cast body); + return (Some (cast new_document_ptr)) + }" +abbreviation "create_document_with_null2 _ _ _ \ create_document_with_null ''''" +notation create_document_with_null ("createDocument'(_')") +notation create_document_with_null2 ("createDocument'(_, _, _')") + +fun get_element_by_id_with_null :: "((_::linorder) object_ptr option) \ string \ (_, ((_) object_ptr option)) dom_prog" + where + "get_element_by_id_with_null (Some ptr) id' = do { + element_ptr_opt \ get_element_by_id ptr id'; + (case element_ptr_opt of + Some element_ptr \ return (Some (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr)) + | None \ return None)}" + | "get_element_by_id_with_null _ _ = error SegmentationFault" +notation get_element_by_id_with_null ("_ . getElementById'(_')") + +fun get_elements_by_class_name_with_null :: "((_::linorder) object_ptr option) \ string \ (_, ((_) object_ptr option) list) dom_prog" + where + "get_elements_by_class_name_with_null (Some ptr) class_name = + get_elements_by_class_name ptr class_name \ map_M (return \ Some \ cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r)" +notation get_elements_by_class_name_with_null ("_ . getElementsByClassName'(_')") + +fun get_elements_by_tag_name_with_null :: "((_::linorder) object_ptr option) \ string \ (_, ((_) object_ptr option) list) dom_prog" + where + "get_elements_by_tag_name_with_null (Some ptr) tag_name = + get_elements_by_tag_name ptr tag_name \ map_M (return \ Some \ cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r)" +notation get_elements_by_tag_name_with_null ("_ . getElementsByTagName'(_')") + +fun insert_before_with_null :: "((_::linorder) object_ptr option) \ ((_) object_ptr option) \ ((_) object_ptr option) \ (_, ((_) object_ptr option)) dom_prog" + where + "insert_before_with_null (Some ptr) (Some child_obj) ref_child_obj_opt = (case cast child_obj of + Some child \ do { + (case ref_child_obj_opt of + Some ref_child_obj \ insert_before ptr child (cast ref_child_obj) + | None \ insert_before ptr child None); + return (Some child_obj)} + | None \ error HierarchyRequestError)" +notation insert_before_with_null ("_ . insertBefore'(_, _')") + +fun append_child_with_null :: "((_::linorder) object_ptr option) \ ((_) object_ptr option) \ (_, unit) dom_prog" + where + "append_child_with_null (Some ptr) (Some child_obj) = (case cast child_obj of + Some child \ append_child ptr child + | None \ error SegmentationFault)" +notation append_child_with_null ("_ . appendChild'(_')") + +fun get_body :: "((_::linorder) object_ptr option) \ (_, ((_) object_ptr option)) dom_prog" + where + "get_body ptr = do { + ptrs \ ptr . getElementsByTagName(''body''); + return (hd ptrs) + }" +notation get_body ("_ . body") + +fun get_document_element_with_null :: "((_::linorder) object_ptr option) \ (_, ((_) object_ptr option)) dom_prog" + where + "get_document_element_with_null (Some ptr) = (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of + Some document_ptr \ do { + element_ptr_opt \ get_M document_ptr document_element; + return (case element_ptr_opt of + Some element_ptr \ Some (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr) + | None \ None)})" +notation get_document_element_with_null ("_ . documentElement") + +fun get_owner_document_with_null :: "((_::linorder) object_ptr option) \ (_, ((_) object_ptr option)) dom_prog" + where + "get_owner_document_with_null (Some ptr) = (do { + document_ptr \ get_owner_document ptr; + return (Some (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr))})" +notation get_owner_document_with_null ("_ . ownerDocument") + +fun remove_with_null :: "((_::linorder) object_ptr option) \ ((_) object_ptr option) \ (_, ((_) object_ptr option)) dom_prog" + where + "remove_with_null (Some ptr) (Some child) = (case cast child of + Some child_node \ do { + remove child_node; + return (Some child)} + | None \ error NotFoundError)" + | "remove_with_null None _ = error TypeError" + | "remove_with_null _ None = error TypeError" +notation remove_with_null ("_ . remove'(')") + +fun remove_child_with_null :: "((_::linorder) object_ptr option) \ ((_) object_ptr option) \ (_, ((_) object_ptr option)) dom_prog" + where + "remove_child_with_null (Some ptr) (Some child) = (case cast child of + Some child_node \ do { + remove_child ptr child_node; + return (Some child)} + | None \ error NotFoundError)" + | "remove_child_with_null None _ = error TypeError" + | "remove_child_with_null _ None = error TypeError" +notation remove_child_with_null ("_ . removeChild") + +fun get_tag_name_with_null :: "((_) object_ptr option) \ (_, attr_value) dom_prog" + where + "get_tag_name_with_null (Some ptr) = (case cast ptr of + Some element_ptr \ get_M element_ptr tag_type)" +notation get_tag_name_with_null ("_ . tagName") + +abbreviation "remove_attribute_with_null ptr k \ set_attribute_with_null2 ptr k None" +notation remove_attribute_with_null ("_ . removeAttribute'(_')") + +fun get_attribute_with_null :: "((_) object_ptr option) \ attr_key \ (_, attr_value option) dom_prog" + where + "get_attribute_with_null (Some ptr) k = (case cast ptr of + Some element_ptr \ get_attribute element_ptr k)" +fun get_attribute_with_null2 :: "((_) object_ptr option) \ attr_key \ (_, attr_value) dom_prog" + where + "get_attribute_with_null2 (Some ptr) k = (case cast ptr of + Some element_ptr \ do { + a \ get_attribute element_ptr k; + return (the a)})" +notation get_attribute_with_null ("_ . getAttribute'(_')") +notation get_attribute_with_null2 ("_ . getAttribute'(_')") + +fun get_parent_with_null :: "((_::linorder) object_ptr option) \ (_, (_) object_ptr option) dom_prog" + where + "get_parent_with_null (Some ptr) = (case cast ptr of + Some node_ptr \ get_parent node_ptr)" +notation get_parent_with_null ("_ . parentNode") + +fun first_child_with_null :: "((_) object_ptr option) \ (_, ((_) object_ptr option)) dom_prog" + where + "first_child_with_null (Some ptr) = do { + child_opt \ first_child ptr; + return (case child_opt of + Some child \ Some (cast child) + | None \ None)}" +notation first_child_with_null ("_ . firstChild") + +fun adopt_node_with_null :: "((_::linorder) object_ptr option) \ ((_) object_ptr option) \ (_, ((_) object_ptr option)) dom_prog" + where + "adopt_node_with_null (Some ptr) (Some child) = (case cast ptr of + Some document_ptr \ (case cast child of + Some child_node \ do { + adopt_node document_ptr child_node; + return (Some child)}))" +notation adopt_node_with_null ("_ . adoptNode'(_')") + + +definition createTestTree :: "((_::linorder) object_ptr option) \ (_, (string \ (_, ((_) object_ptr option)) dom_prog)) dom_prog" + where + "createTestTree ref = return (\id. get_element_by_id_with_null ref id)" + +end diff --git a/Core_DOM/tests/Document-adoptNode.html b/Core_DOM/tests/Document-adoptNode.html new file mode 100644 index 0000000..75d4531 --- /dev/null +++ b/Core_DOM/tests/Document-adoptNode.html @@ -0,0 +1,36 @@ + + +Document.adoptNode + + + +
+x + diff --git a/Core_DOM/tests/Document-adoptNode.html.orig b/Core_DOM/tests/Document-adoptNode.html.orig new file mode 100644 index 0000000..584d5d9 --- /dev/null +++ b/Core_DOM/tests/Document-adoptNode.html.orig @@ -0,0 +1,50 @@ + + +Document.adoptNode + + + +
+x + diff --git a/Core_DOM/tests/Document-getElementById.html b/Core_DOM/tests/Document-getElementById.html new file mode 100644 index 0000000..d565ef0 --- /dev/null +++ b/Core_DOM/tests/Document-getElementById.html @@ -0,0 +1,251 @@ + + +Document.getElementById + + + + + +
+ +
+ +
+ +
+

P

+ +
+ +
+
+
+
+
+ + + + diff --git a/Core_DOM/tests/Document-getElementById.html.orig b/Core_DOM/tests/Document-getElementById.html.orig new file mode 100644 index 0000000..1dec4c0 --- /dev/null +++ b/Core_DOM/tests/Document-getElementById.html.orig @@ -0,0 +1,350 @@ + + +Document.getElementById + + + + + +
+ + +
+ + +
+ + +
+

P

+ +
+ + +
+
+
+
+
+ + + + diff --git a/Core_DOM/tests/Document_adoptNode.thy b/Core_DOM/tests/Document_adoptNode.thy new file mode 100644 index 0000000..4adb19b --- /dev/null +++ b/Core_DOM/tests/Document_adoptNode.thy @@ -0,0 +1,111 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Testing adoptNode\ +text\This theory contains the test cases for adoptNode.\ + +theory Document_adoptNode +imports + "Core_DOM_BaseTest" +begin + +definition Document_adoptNode_heap :: "(unit, unit, unit, unit, unit, unit, unit, unit, unit, unit, unit) heap" where + "Document_adoptNode_heap = create_heap [(cast (document_ptr.Ref 1), cast (create_document_obj html (Some (cast (element_ptr.Ref 1))) [])), + (cast (element_ptr.Ref 1), cast (create_element_obj ''html'' [cast (element_ptr.Ref 2), cast (element_ptr.Ref 8)] fmempty None)), + (cast (element_ptr.Ref 2), cast (create_element_obj ''head'' [cast (element_ptr.Ref 3), cast (element_ptr.Ref 4), cast (element_ptr.Ref 5), cast (element_ptr.Ref 6), cast (element_ptr.Ref 7)] fmempty None)), + (cast (element_ptr.Ref 3), cast (create_element_obj ''meta'' [] (fmap_of_list [(''charset'', ''utf-8'')]) None)), + (cast (element_ptr.Ref 4), cast (create_element_obj ''title'' [cast (character_data_ptr.Ref 1)] fmempty None)), + (cast (character_data_ptr.Ref 1), cast (create_character_data_obj ''Document.adoptNode'')), + (cast (element_ptr.Ref 5), cast (create_element_obj ''link'' [] (fmap_of_list [(''rel'', ''help''), (''href'', ''https://dom.spec.whatwg.org/#dom-document-adoptnode'')]) None)), + (cast (element_ptr.Ref 6), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharness.js'')]) None)), + (cast (element_ptr.Ref 7), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharnessreport.js'')]) None)), + (cast (element_ptr.Ref 8), cast (create_element_obj ''body'' [cast (element_ptr.Ref 9), cast (element_ptr.Ref 10), cast (element_ptr.Ref 11)] fmempty None)), + (cast (element_ptr.Ref 9), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', ''log'')]) None)), + (cast (element_ptr.Ref 10), cast (create_element_obj ''x<'' [cast (character_data_ptr.Ref 2)] fmempty None)), + (cast (character_data_ptr.Ref 2), cast (create_character_data_obj ''x'')), + (cast (element_ptr.Ref 11), cast (create_element_obj ''script'' [cast (character_data_ptr.Ref 3)] fmempty None)), + (cast (character_data_ptr.Ref 3), cast (create_character_data_obj ''%3C%3Cscript%3E%3E''))]" + +definition document :: "(unit, unit, unit, unit, unit, unit) object_ptr option" where "document = Some (cast (document_ptr.Ref 1))" + + +text \"Adopting an Element called 'x<' should work."\ + +lemma "test (do { + tmp0 \ document . getElementsByTagName(''x<''); + y \ return (tmp0 ! 0); + child \ y . firstChild; + tmp1 \ y . parentNode; + tmp2 \ document . body; + assert_equals(tmp1, tmp2); + tmp3 \ y . ownerDocument; + assert_equals(tmp3, document); + tmp4 \ document . adoptNode(y); + assert_equals(tmp4, y); + tmp5 \ y . parentNode; + assert_equals(tmp5, None); + tmp6 \ y . firstChild; + assert_equals(tmp6, child); + tmp7 \ y . ownerDocument; + assert_equals(tmp7, document); + tmp8 \ child . ownerDocument; + assert_equals(tmp8, document); + doc \ createDocument(None, None, None); + tmp9 \ doc . adoptNode(y); + assert_equals(tmp9, y); + tmp10 \ y . parentNode; + assert_equals(tmp10, None); + tmp11 \ y . firstChild; + assert_equals(tmp11, child); + tmp12 \ y . ownerDocument; + assert_equals(tmp12, doc); + tmp13 \ child . ownerDocument; + assert_equals(tmp13, doc) +}) Document_adoptNode_heap" + by eval + + +text \"Adopting an Element called ':good:times:' should work."\ + +lemma "test (do { + x \ document . createElement('':good:times:''); + tmp0 \ document . adoptNode(x); + assert_equals(tmp0, x); + doc \ createDocument(None, None, None); + tmp1 \ doc . adoptNode(x); + assert_equals(tmp1, x); + tmp2 \ x . parentNode; + assert_equals(tmp2, None); + tmp3 \ x . ownerDocument; + assert_equals(tmp3, doc) +}) Document_adoptNode_heap" + by eval + + +end diff --git a/Core_DOM/tests/Document_getElementById.thy b/Core_DOM/tests/Document_getElementById.thy new file mode 100644 index 0000000..9ece29a --- /dev/null +++ b/Core_DOM/tests/Document_getElementById.thy @@ -0,0 +1,277 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Testing getElementById\ +text\This theory contains the test cases for getElementById.\ + +theory Document_getElementById +imports + "Core_DOM_BaseTest" +begin + +definition Document_getElementById_heap :: "(unit, unit, unit, unit, unit, unit, unit, unit, unit, unit, unit) heap" where + "Document_getElementById_heap = create_heap [(cast (document_ptr.Ref 1), cast (create_document_obj html (Some (cast (element_ptr.Ref 1))) [])), + (cast (element_ptr.Ref 1), cast (create_element_obj ''html'' [cast (element_ptr.Ref 2), cast (element_ptr.Ref 9)] fmempty None)), + (cast (element_ptr.Ref 2), cast (create_element_obj ''head'' [cast (element_ptr.Ref 3), cast (element_ptr.Ref 4), cast (element_ptr.Ref 5), cast (element_ptr.Ref 6), cast (element_ptr.Ref 7), cast (element_ptr.Ref 8)] fmempty None)), + (cast (element_ptr.Ref 3), cast (create_element_obj ''meta'' [] (fmap_of_list [(''charset'', ''utf-8'')]) None)), + (cast (element_ptr.Ref 4), cast (create_element_obj ''title'' [cast (character_data_ptr.Ref 1)] fmempty None)), + (cast (character_data_ptr.Ref 1), cast (create_character_data_obj ''Document.getElementById'')), + (cast (element_ptr.Ref 5), cast (create_element_obj ''link'' [] (fmap_of_list [(''rel'', ''author''), (''title'', ''Tetsuharu OHZEKI''), (''href'', ''mailto:saneyuki.snyk@gmail.com'')]) None)), + (cast (element_ptr.Ref 6), cast (create_element_obj ''link'' [] (fmap_of_list [(''rel'', ''help''), (''href'', ''https://dom.spec.whatwg.org/#dom-document-getelementbyid'')]) None)), + (cast (element_ptr.Ref 7), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharness.js'')]) None)), + (cast (element_ptr.Ref 8), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharnessreport.js'')]) None)), + (cast (element_ptr.Ref 9), cast (create_element_obj ''body'' [cast (element_ptr.Ref 10), cast (element_ptr.Ref 11), cast (element_ptr.Ref 12), cast (element_ptr.Ref 13), cast (element_ptr.Ref 16), cast (element_ptr.Ref 19)] fmempty None)), + (cast (element_ptr.Ref 10), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', ''log'')]) None)), + (cast (element_ptr.Ref 11), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', '''')]) None)), + (cast (element_ptr.Ref 12), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', ''test1'')]) None)), + (cast (element_ptr.Ref 13), cast (create_element_obj ''div'' [cast (element_ptr.Ref 14), cast (element_ptr.Ref 15)] (fmap_of_list [(''id'', ''test5''), (''data-name'', ''1st'')]) None)), + (cast (element_ptr.Ref 14), cast (create_element_obj ''p'' [cast (character_data_ptr.Ref 2)] (fmap_of_list [(''id'', ''test5''), (''data-name'', ''2nd'')]) None)), + (cast (character_data_ptr.Ref 2), cast (create_character_data_obj ''P'')), + (cast (element_ptr.Ref 15), cast (create_element_obj ''input'' [] (fmap_of_list [(''id'', ''test5''), (''type'', ''submit''), (''value'', ''Submit''), (''data-name'', ''3rd'')]) None)), + (cast (element_ptr.Ref 16), cast (create_element_obj ''div'' [cast (element_ptr.Ref 17)] (fmap_of_list [(''id'', ''outer'')]) None)), + (cast (element_ptr.Ref 17), cast (create_element_obj ''div'' [cast (element_ptr.Ref 18)] (fmap_of_list [(''id'', ''middle'')]) None)), + (cast (element_ptr.Ref 18), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', ''inner'')]) None)), + (cast (element_ptr.Ref 19), cast (create_element_obj ''script'' [cast (character_data_ptr.Ref 3)] fmempty None)), + (cast (character_data_ptr.Ref 3), cast (create_character_data_obj ''%3C%3Cscript%3E%3E''))]" + +definition document :: "(unit, unit, unit, unit, unit, unit) object_ptr option" where "document = Some (cast (document_ptr.Ref 1))" + + +text \"Document.getElementById with a script-inserted element"\ + +lemma "test (do { + gBody \ document . body; + TEST_ID \ return ''test2''; + test \ document . createElement(''div''); + test . setAttribute(''id'', TEST_ID); + gBody . appendChild(test); + result \ document . getElementById(TEST_ID); + assert_not_equals(result, None, ''should not be null.''); + tmp0 \ result . tagName; + assert_equals(tmp0, ''div'', ''should have appended element's tag name''); + gBody . removeChild(test); + removed \ document . getElementById(TEST_ID); + assert_equals(removed, None, ''should not get removed element.'') +}) Document_getElementById_heap" + by eval + + +text \"update `id` attribute via setAttribute/removeAttribute"\ + +lemma "test (do { + gBody \ document . body; + TEST_ID \ return ''test3''; + test \ document . createElement(''div''); + test . setAttribute(''id'', TEST_ID); + gBody . appendChild(test); + UPDATED_ID \ return ''test3-updated''; + test . setAttribute(''id'', UPDATED_ID); + e \ document . getElementById(UPDATED_ID); + assert_equals(e, test, ''should get the element with id.''); + old \ document . getElementById(TEST_ID); + assert_equals(old, None, ''shouldn't get the element by the old id.''); + test . removeAttribute(''id''); + e2 \ document . getElementById(UPDATED_ID); + assert_equals(e2, None, ''should return null when the passed id is none in document.'') +}) Document_getElementById_heap" + by eval + + +text \"Ensure that the id attribute only affects elements present in a document"\ + +lemma "test (do { + TEST_ID \ return ''test4-should-not-exist''; + e \ document . createElement(''div''); + e . setAttribute(''id'', TEST_ID); + tmp0 \ document . getElementById(TEST_ID); + assert_equals(tmp0, None, ''should be null''); + tmp1 \ document . body; + tmp1 . appendChild(e); + tmp2 \ document . getElementById(TEST_ID); + assert_equals(tmp2, e, ''should be the appended element'') +}) Document_getElementById_heap" + by eval + + +text \"in tree order, within the context object's tree"\ + +lemma "test (do { + gBody \ document . body; + TEST_ID \ return ''test5''; + target \ document . getElementById(TEST_ID); + assert_not_equals(target, None, ''should not be null''); + tmp0 \ target . getAttribute(''data-name''); + assert_equals(tmp0, ''1st'', ''should return the 1st''); + element4 \ document . createElement(''div''); + element4 . setAttribute(''id'', TEST_ID); + element4 . setAttribute(''data-name'', ''4th''); + gBody . appendChild(element4); + target2 \ document . getElementById(TEST_ID); + assert_not_equals(target2, None, ''should not be null''); + tmp1 \ target2 . getAttribute(''data-name''); + assert_equals(tmp1, ''1st'', ''should be the 1st''); + tmp2 \ target2 . parentNode; + tmp2 . removeChild(target2); + target3 \ document . getElementById(TEST_ID); + assert_not_equals(target3, None, ''should not be null''); + tmp3 \ target3 . getAttribute(''data-name''); + assert_equals(tmp3, ''4th'', ''should be the 4th'') +}) Document_getElementById_heap" + by eval + + +text \"Modern browsers optimize this method with using internal id cache. + This test checks that their optimization should effect only append to + `Document`, not append to `Node`."\ + +lemma "test (do { + TEST_ID \ return ''test6''; + s \ document . createElement(''div''); + s . setAttribute(''id'', TEST_ID); + tmp0 \ document . createElement(''div''); + tmp0 . appendChild(s); + tmp1 \ document . getElementById(TEST_ID); + assert_equals(tmp1, None, ''should be null'') +}) Document_getElementById_heap" + by eval + + +text \"changing attribute's value via `Attr` gotten from `Element.attribute`."\ + +lemma "test (do { + gBody \ document . body; + TEST_ID \ return ''test7''; + element \ document . createElement(''div''); + element . setAttribute(''id'', TEST_ID); + gBody . appendChild(element); + target \ document . getElementById(TEST_ID); + assert_equals(target, element, ''should return the element before changing the value''); + element . setAttribute(''id'', (TEST_ID @ ''-updated'')); + target2 \ document . getElementById(TEST_ID); + assert_equals(target2, None, ''should return null after updated id via Attr.value''); + target3 \ document . getElementById((TEST_ID @ ''-updated'')); + assert_equals(target3, element, ''should be equal to the updated element.'') +}) Document_getElementById_heap" + by eval + + +text \"update `id` attribute via element.id"\ + +lemma "test (do { + gBody \ document . body; + TEST_ID \ return ''test12''; + test \ document . createElement(''div''); + test . setAttribute(''id'', TEST_ID); + gBody . appendChild(test); + UPDATED_ID \ return (TEST_ID @ ''-updated''); + test . setAttribute(''id'', UPDATED_ID); + e \ document . getElementById(UPDATED_ID); + assert_equals(e, test, ''should get the element with id.''); + old \ document . getElementById(TEST_ID); + assert_equals(old, None, ''shouldn't get the element by the old id.''); + test . setAttribute(''id'', ''''); + e2 \ document . getElementById(UPDATED_ID); + assert_equals(e2, None, ''should return null when the passed id is none in document.'') +}) Document_getElementById_heap" + by eval + + +text \"where insertion order and tree order don't match"\ + +lemma "test (do { + gBody \ document . body; + TEST_ID \ return ''test13''; + container \ document . createElement(''div''); + container . setAttribute(''id'', (TEST_ID @ ''-fixture'')); + gBody . appendChild(container); + element1 \ document . createElement(''div''); + element1 . setAttribute(''id'', TEST_ID); + element2 \ document . createElement(''div''); + element2 . setAttribute(''id'', TEST_ID); + element3 \ document . createElement(''div''); + element3 . setAttribute(''id'', TEST_ID); + element4 \ document . createElement(''div''); + element4 . setAttribute(''id'', TEST_ID); + container . appendChild(element2); + container . appendChild(element4); + container . insertBefore(element3, element4); + container . insertBefore(element1, element2); + test \ document . getElementById(TEST_ID); + assert_equals(test, element1, ''should return 1st element''); + container . removeChild(element1); + test \ document . getElementById(TEST_ID); + assert_equals(test, element2, ''should return 2nd element''); + container . removeChild(element2); + test \ document . getElementById(TEST_ID); + assert_equals(test, element3, ''should return 3rd element''); + container . removeChild(element3); + test \ document . getElementById(TEST_ID); + assert_equals(test, element4, ''should return 4th element''); + container . removeChild(element4) +}) Document_getElementById_heap" + by eval + + +text \"Inserting an id by inserting its parent node"\ + +lemma "test (do { + gBody \ document . body; + TEST_ID \ return ''test14''; + a \ document . createElement(''a''); + b \ document . createElement(''b''); + a . appendChild(b); + b . setAttribute(''id'', TEST_ID); + tmp0 \ document . getElementById(TEST_ID); + assert_equals(tmp0, None); + gBody . appendChild(a); + tmp1 \ document . getElementById(TEST_ID); + assert_equals(tmp1, b) +}) Document_getElementById_heap" + by eval + + +text \"Document.getElementById must not return nodes not present in document"\ + +lemma "test (do { + TEST_ID \ return ''test15''; + outer \ document . getElementById(''outer''); + middle \ document . getElementById(''middle''); + inner \ document . getElementById(''inner''); + tmp0 \ document . getElementById(''middle''); + outer . removeChild(tmp0); + new_el \ document . createElement(''h1''); + new_el . setAttribute(''id'', ''heading''); + inner . appendChild(new_el); + tmp1 \ document . getElementById(''heading''); + assert_equals(tmp1, None) +}) Document_getElementById_heap" + by eval + + +end diff --git a/Core_DOM/tests/Node-insertBefore.html b/Core_DOM/tests/Node-insertBefore.html new file mode 100644 index 0000000..db2675b --- /dev/null +++ b/Core_DOM/tests/Node-insertBefore.html @@ -0,0 +1,288 @@ + +Node.insertBefore + + + +
+ + diff --git a/Core_DOM/tests/Node-insertBefore.html.orig b/Core_DOM/tests/Node-insertBefore.html.orig new file mode 100644 index 0000000..a9fc83b --- /dev/null +++ b/Core_DOM/tests/Node-insertBefore.html.orig @@ -0,0 +1,306 @@ + +Node.insertBefore + + +
+ diff --git a/Core_DOM/tests/Node-removeChild.html b/Core_DOM/tests/Node-removeChild.html new file mode 100644 index 0000000..83c4c3d --- /dev/null +++ b/Core_DOM/tests/Node-removeChild.html @@ -0,0 +1,66 @@ + +Node.removeChild + + + + +
+ + + diff --git a/Core_DOM/tests/Node-removeChild.html.orig b/Core_DOM/tests/Node-removeChild.html.orig new file mode 100644 index 0000000..fb22583 --- /dev/null +++ b/Core_DOM/tests/Node-removeChild.html.orig @@ -0,0 +1,54 @@ + +Node.removeChild + + + +
+ + diff --git a/Core_DOM/tests/Node_insertBefore.thy b/Core_DOM/tests/Node_insertBefore.thy new file mode 100644 index 0000000..48d1b19 --- /dev/null +++ b/Core_DOM/tests/Node_insertBefore.thy @@ -0,0 +1,126 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Testing insertBefore\ +text\This theory contains the test cases for insertBefore.\ + +theory Node_insertBefore +imports + "Core_DOM_BaseTest" +begin + +definition Node_insertBefore_heap :: "(unit, unit, unit, unit, unit, unit, unit, unit, unit, unit, unit) heap" where + "Node_insertBefore_heap = create_heap [(cast (document_ptr.Ref 1), cast (create_document_obj html (Some (cast (element_ptr.Ref 1))) [])), + (cast (element_ptr.Ref 1), cast (create_element_obj ''html'' [cast (element_ptr.Ref 2), cast (element_ptr.Ref 6)] fmempty None)), + (cast (element_ptr.Ref 2), cast (create_element_obj ''head'' [cast (element_ptr.Ref 3), cast (element_ptr.Ref 4), cast (element_ptr.Ref 5)] fmempty None)), + (cast (element_ptr.Ref 3), cast (create_element_obj ''title'' [cast (character_data_ptr.Ref 1)] fmempty None)), + (cast (character_data_ptr.Ref 1), cast (create_character_data_obj ''Node.insertBefore'')), + (cast (element_ptr.Ref 4), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharness.js'')]) None)), + (cast (element_ptr.Ref 5), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharnessreport.js'')]) None)), + (cast (element_ptr.Ref 6), cast (create_element_obj ''body'' [cast (element_ptr.Ref 7), cast (element_ptr.Ref 8)] fmempty None)), + (cast (element_ptr.Ref 7), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', ''log'')]) None)), + (cast (element_ptr.Ref 8), cast (create_element_obj ''script'' [cast (character_data_ptr.Ref 2)] fmempty None)), + (cast (character_data_ptr.Ref 2), cast (create_character_data_obj ''%3C%3Cscript%3E%3E''))]" + +definition document :: "(unit, unit, unit, unit, unit, unit) object_ptr option" where "document = Some (cast (document_ptr.Ref 1))" + + +text \"Calling insertBefore an a leaf node Text must throw HIERARCHY\_REQUEST\_ERR."\ + +lemma "test (do { + node \ document . createTextNode(''Foo''); + tmp0 \ document . createTextNode(''fail''); + assert_throws(HierarchyRequestError, node . insertBefore(tmp0, None)) +}) Node_insertBefore_heap" + by eval + + +text \"Calling insertBefore with an inclusive ancestor of the context object must throw HIERARCHY\_REQUEST\_ERR."\ + +lemma "test (do { + tmp1 \ document . body; + tmp2 \ document . getElementById(''log''); + tmp0 \ document . body; + assert_throws(HierarchyRequestError, tmp0 . insertBefore(tmp1, tmp2)); + tmp4 \ document . documentElement; + tmp5 \ document . getElementById(''log''); + tmp3 \ document . body; + assert_throws(HierarchyRequestError, tmp3 . insertBefore(tmp4, tmp5)) +}) Node_insertBefore_heap" + by eval + + +text \"Calling insertBefore with a reference child whose parent is not the context node must throw a NotFoundError."\ + +lemma "test (do { + a \ document . createElement(''div''); + b \ document . createElement(''div''); + c \ document . createElement(''div''); + assert_throws(NotFoundError, a . insertBefore(b, c)) +}) Node_insertBefore_heap" + by eval + + +text \"If the context node is a document, inserting a document or text node should throw a HierarchyRequestError."\ + +lemma "test (do { + doc \ createDocument(''title''); + doc2 \ createDocument(''title2''); + tmp0 \ doc . documentElement; + assert_throws(HierarchyRequestError, doc . insertBefore(doc2, tmp0)); + tmp1 \ doc . createTextNode(''text''); + tmp2 \ doc . documentElement; + assert_throws(HierarchyRequestError, doc . insertBefore(tmp1, tmp2)) +}) Node_insertBefore_heap" + by eval + + +text \"Inserting a node before itself should not move the node"\ + +lemma "test (do { + a \ document . createElement(''div''); + b \ document . createElement(''div''); + c \ document . createElement(''div''); + a . appendChild(b); + a . appendChild(c); + tmp0 \ a . childNodes; + assert_array_equals(tmp0, [b, c]); + tmp1 \ a . insertBefore(b, b); + assert_equals(tmp1, b); + tmp2 \ a . childNodes; + assert_array_equals(tmp2, [b, c]); + tmp3 \ a . insertBefore(c, c); + assert_equals(tmp3, c); + tmp4 \ a . childNodes; + assert_array_equals(tmp4, [b, c]) +}) Node_insertBefore_heap" + by eval + + +end diff --git a/Core_DOM/tests/Node_removeChild.thy b/Core_DOM/tests/Node_removeChild.thy new file mode 100644 index 0000000..74dfa2c --- /dev/null +++ b/Core_DOM/tests/Node_removeChild.thy @@ -0,0 +1,157 @@ +(*********************************************************************************** + * Copyright (c) 2016-2018 The University of Sheffield, UK + * + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * + * * Redistributions of source code must retain the above copyright notice, this + * list of conditions and the following disclaimer. + * + * * Redistributions in binary form must reproduce the above copyright notice, + * this list of conditions and the following disclaimer in the documentation + * and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" + * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + * SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + * OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * SPDX-License-Identifier: BSD-2-Clause + ***********************************************************************************) + +section\Testing removeChild\ +text\This theory contains the test cases for removeChild.\ + +theory Node_removeChild +imports + "Core_DOM_BaseTest" +begin + +definition Node_removeChild_heap :: "(unit, unit, unit, unit, unit, unit, unit, unit, unit, unit, unit) heap" where + "Node_removeChild_heap = create_heap [(cast (document_ptr.Ref 1), cast (create_document_obj html (Some (cast (element_ptr.Ref 1))) [])), + (cast (element_ptr.Ref 1), cast (create_element_obj ''html'' [cast (element_ptr.Ref 2), cast (element_ptr.Ref 7)] fmempty None)), + (cast (element_ptr.Ref 2), cast (create_element_obj ''head'' [cast (element_ptr.Ref 3), cast (element_ptr.Ref 4), cast (element_ptr.Ref 5), cast (element_ptr.Ref 6)] fmempty None)), + (cast (element_ptr.Ref 3), cast (create_element_obj ''title'' [cast (character_data_ptr.Ref 1)] fmempty None)), + (cast (character_data_ptr.Ref 1), cast (create_character_data_obj ''Node.removeChild'')), + (cast (element_ptr.Ref 4), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharness.js'')]) None)), + (cast (element_ptr.Ref 5), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''/resources/testharnessreport.js'')]) None)), + (cast (element_ptr.Ref 6), cast (create_element_obj ''script'' [] (fmap_of_list [(''src'', ''creators.js'')]) None)), + (cast (element_ptr.Ref 7), cast (create_element_obj ''body'' [cast (element_ptr.Ref 8), cast (element_ptr.Ref 9), cast (element_ptr.Ref 10)] fmempty None)), + (cast (element_ptr.Ref 8), cast (create_element_obj ''div'' [] (fmap_of_list [(''id'', ''log'')]) None)), + (cast (element_ptr.Ref 9), cast (create_element_obj ''iframe'' [] (fmap_of_list [(''src'', ''about:blank'')]) None)), + (cast (element_ptr.Ref 10), cast (create_element_obj ''script'' [cast (character_data_ptr.Ref 2)] fmempty None)), + (cast (character_data_ptr.Ref 2), cast (create_character_data_obj ''%3C%3Cscript%3E%3E''))]" + +definition document :: "(unit, unit, unit, unit, unit, unit) object_ptr option" where "document = Some (cast (document_ptr.Ref 1))" + + +text \"Passing a detached Element to removeChild should not affect it."\ + +lemma "test (do { + doc \ return document; + s \ doc . createElement(''div''); + tmp0 \ s . ownerDocument; + assert_equals(tmp0, doc); + tmp1 \ document . body; + assert_throws(NotFoundError, tmp1 . removeChild(s)); + tmp2 \ s . ownerDocument; + assert_equals(tmp2, doc) +}) Node_removeChild_heap" + by eval + + +text \"Passing a non-detached Element to removeChild should not affect it."\ + +lemma "test (do { + doc \ return document; + s \ doc . createElement(''div''); + tmp0 \ doc . documentElement; + tmp0 . appendChild(s); + tmp1 \ s . ownerDocument; + assert_equals(tmp1, doc); + tmp2 \ document . body; + assert_throws(NotFoundError, tmp2 . removeChild(s)); + tmp3 \ s . ownerDocument; + assert_equals(tmp3, doc) +}) Node_removeChild_heap" + by eval + + +text \"Calling removeChild on an Element with no children should throw NOT\_FOUND\_ERR."\ + +lemma "test (do { + doc \ return document; + s \ doc . createElement(''div''); + tmp0 \ doc . body; + tmp0 . appendChild(s); + tmp1 \ s . ownerDocument; + assert_equals(tmp1, doc); + assert_throws(NotFoundError, s . removeChild(doc)) +}) Node_removeChild_heap" + by eval + + +text \"Passing a detached Element to removeChild should not affect it."\ + +lemma "test (do { + doc \ createDocument(''''); + s \ doc . createElement(''div''); + tmp0 \ s . ownerDocument; + assert_equals(tmp0, doc); + tmp1 \ document . body; + assert_throws(NotFoundError, tmp1 . removeChild(s)); + tmp2 \ s . ownerDocument; + assert_equals(tmp2, doc) +}) Node_removeChild_heap" + by eval + + +text \"Passing a non-detached Element to removeChild should not affect it."\ + +lemma "test (do { + doc \ createDocument(''''); + s \ doc . createElement(''div''); + tmp0 \ doc . documentElement; + tmp0 . appendChild(s); + tmp1 \ s . ownerDocument; + assert_equals(tmp1, doc); + tmp2 \ document . body; + assert_throws(NotFoundError, tmp2 . removeChild(s)); + tmp3 \ s . ownerDocument; + assert_equals(tmp3, doc) +}) Node_removeChild_heap" + by eval + + +text \"Calling removeChild on an Element with no children should throw NOT\_FOUND\_ERR."\ + +lemma "test (do { + doc \ createDocument(''''); + s \ doc . createElement(''div''); + tmp0 \ doc . body; + tmp0 . appendChild(s); + tmp1 \ s . ownerDocument; + assert_equals(tmp1, doc); + assert_throws(NotFoundError, s . removeChild(doc)) +}) Node_removeChild_heap" + by eval + + +text \"Passing a value that is not a Node reference to removeChild should throw TypeError."\ + +lemma "test (do { + tmp0 \ document . body; + assert_throws(TypeError, tmp0 . removeChild(None)) +}) Node_removeChild_heap" + by eval + + +end