446 lines
31 KiB
Plaintext
446 lines
31 KiB
Plaintext
(***********************************************************************************
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* Copyright (c) 2016-2018 The University of Sheffield, UK
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*
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions are met:
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*
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* * Redistributions of source code must retain the above copyright notice, this
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* list of conditions and the following disclaimer.
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*
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* * Redistributions in binary form must reproduce the above copyright notice,
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* this list of conditions and the following disclaimer in the documentation
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* and/or other materials provided with the distribution.
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*
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* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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* AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
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* DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
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* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
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* SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
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* CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
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* OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
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* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*
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* SPDX-License-Identifier: BSD-2-Clause
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***********************************************************************************)
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section\<open>Element\<close>
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text\<open>In this theory, we introduce the monadic method setup for the Element class.\<close>
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theory ElementMonad
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imports
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NodeMonad
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"ElementClass"
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begin
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type_synonym ('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr, 'document_ptr,
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'shadow_root_ptr, 'Object, 'Node, 'Element,'result) dom_prog
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= "((_) heap, exception, 'result) prog"
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register_default_tvars "('object_ptr, 'node_ptr, 'element_ptr, 'character_data_ptr,
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'document_ptr, 'shadow_root_ptr, 'Object, 'Node, 'Element,'result) dom_prog"
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global_interpretation l_ptr_kinds_M element_ptr_kinds defines element_ptr_kinds_M = a_ptr_kinds_M .
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lemmas element_ptr_kinds_M_defs = a_ptr_kinds_M_def
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lemma element_ptr_kinds_M_eq:
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assumes "|h \<turnstile> node_ptr_kinds_M|\<^sub>r = |h' \<turnstile> node_ptr_kinds_M|\<^sub>r"
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shows "|h \<turnstile> element_ptr_kinds_M|\<^sub>r = |h' \<turnstile> element_ptr_kinds_M|\<^sub>r"
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using assms
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by(auto simp add: element_ptr_kinds_M_defs node_ptr_kinds_M_defs element_ptr_kinds_def)
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lemma element_ptr_kinds_M_reads:
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"reads (\<Union>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'"
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apply (simp add: reads_def node_ptr_kinds_M_defs element_ptr_kinds_M_defs element_ptr_kinds_def
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node_ptr_kinds_M_reads preserved_def cong del: image_cong_simp)
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apply (metis (mono_tags, hide_lams) node_ptr_kinds_small old.unit.exhaust preserved_def)
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done
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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" .
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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"
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apply(simp add: get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_type_wf l_get_M_def)
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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
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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
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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))
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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]]
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adhoc_overloading get_M get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t
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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
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begin
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sublocale l_get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas by unfold_locales
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interpretation l_get_M get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t type_wf element_ptr_kinds
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apply(unfold_locales)
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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)
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by (meson ElementMonad.get_M_is_l_get_M l_get_M_def)
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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]
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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]
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end
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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
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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
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rewrites "a_get_M = get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t"
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defines put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t = a_put_M
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apply (simp add: get_M_is_l_get_M l_put_M_def)
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by (simp add: get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def)
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lemmas put_M_defs = a_put_M_def
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adhoc_overloading put_M put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t
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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
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begin
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sublocale l_put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas by unfold_locales
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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
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apply(unfold_locales)
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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)
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by (meson ElementMonad.get_M_is_l_get_M l_get_M_def)
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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]
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end
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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
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lemma element_put_get [simp]:
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"h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h' \<Longrightarrow> (\<And>x. getter (setter (\<lambda>_. v) x) = v)
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\<Longrightarrow> h' \<turnstile> get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr getter \<rightarrow>\<^sub>r v"
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by(auto simp add: put_M_defs get_M_defs split: option.splits)
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lemma get_M_Element_preserved1 [simp]:
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"element_ptr \<noteq> element_ptr' \<Longrightarrow> h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'
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\<Longrightarrow> preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr' getter) h h'"
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by(auto simp add: put_M_defs get_M_defs preserved_def split: option.splits dest: get_heap_E)
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lemma element_put_get_preserved [simp]:
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"(\<And>x. getter (setter (\<lambda>_. v) x) = getter x) \<Longrightarrow> h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'
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\<Longrightarrow> preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr' getter) h h'"
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apply(cases "element_ptr = element_ptr'")
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by(auto simp add: put_M_defs get_M_defs preserved_def
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split: option.splits dest: get_heap_E)
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lemma get_M_Element_preserved3 [simp]:
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"(\<And>x. getter (cast (setter (\<lambda>_. v) x)) = getter (cast x))
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\<Longrightarrow> h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h' \<Longrightarrow> preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'"
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apply(cases "cast element_ptr = object_ptr")
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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
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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
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split: option.splits)
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lemma get_M_Element_preserved4 [simp]:
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"(\<And>x. getter (cast (setter (\<lambda>_. v) x)) = getter (cast x))
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\<Longrightarrow> h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h' \<Longrightarrow> preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr getter) h h'"
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apply(cases "cast element_ptr = node_ptr")
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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
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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
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split: option.splits)
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lemma get_M_Element_preserved5 [simp]:
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"cast element_ptr \<noteq> node_ptr \<Longrightarrow> h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'
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\<Longrightarrow> preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr getter) h h'"
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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
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split: option.splits dest: get_heap_E)
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lemma get_M_Element_preserved6 [simp]:
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"h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'
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\<Longrightarrow> (\<And>x. getter (cast (setter (\<lambda>_. v) x)) = getter (cast x))
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\<Longrightarrow> preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr getter) h h'"
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apply(cases "cast element_ptr \<noteq> node_ptr")
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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
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split: option.splits bind_splits dest: get_heap_E)
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lemma get_M_Element_preserved7 [simp]:
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"cast element_ptr \<noteq> node_ptr \<Longrightarrow> h \<turnstile> put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr setter v \<rightarrow>\<^sub>h h'
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\<Longrightarrow> preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr getter) h h'"
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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
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split: option.splits dest: get_heap_E)
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lemma get_M_Element_preserved8 [simp]:
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"cast element_ptr \<noteq> object_ptr \<Longrightarrow> h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'
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\<Longrightarrow> preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'"
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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
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ObjectMonad.get_M_defs preserved_def
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split: option.splits dest: get_heap_E)
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lemma get_M_Element_preserved9 [simp]:
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"h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'
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\<Longrightarrow> (\<And>x. getter (cast (setter (\<lambda>_. v) x)) = getter (cast x))
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\<Longrightarrow> preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'"
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apply(cases "cast element_ptr \<noteq> object_ptr")
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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
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ObjectMonad.get_M_defs preserved_def
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split: option.splits bind_splits dest: get_heap_E)
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lemma get_M_Element_preserved10 [simp]:
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"cast element_ptr \<noteq> object_ptr \<Longrightarrow> h \<turnstile> put_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr setter v \<rightarrow>\<^sub>h h'
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\<Longrightarrow> preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr getter) h h'"
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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
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ObjectMonad.get_M_defs preserved_def
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split: option.splits dest: get_heap_E)
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subsection\<open>Creating Elements\<close>
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definition new_element :: "(_, (_) element_ptr) dom_prog"
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where
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"new_element = do {
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h \<leftarrow> get_heap;
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(new_ptr, h') \<leftarrow> return (new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h);
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return_heap h';
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return new_ptr
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}"
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lemma new_element_ok [simp]:
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"h \<turnstile> ok new_element"
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by(auto simp add: new_element_def split: prod.splits)
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lemma new_element_ptr_in_heap:
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assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"
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and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"
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shows "new_element_ptr |\<in>| element_ptr_kinds h'"
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using assms
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unfolding new_element_def
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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
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elim!: bind_returns_result_E bind_returns_heap_E)
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lemma new_element_ptr_not_in_heap:
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assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"
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and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"
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shows "new_element_ptr |\<notin>| element_ptr_kinds h"
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using assms new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_not_in_heap
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by(auto simp add: new_element_def split: prod.splits elim!: bind_returns_result_E
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bind_returns_heap_E)
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lemma new_element_new_ptr:
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assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"
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and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"
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shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast new_element_ptr|}"
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using assms new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_new_ptr
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by(auto simp add: new_element_def split: prod.splits elim!: bind_returns_result_E
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bind_returns_heap_E)
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lemma new_element_is_element_ptr:
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assumes "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"
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shows "is_element_ptr new_element_ptr"
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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 \<turnstile> new_element \<rightarrow>\<^sub>h h'"
|
|
assumes "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"
|
|
shows "h' \<turnstile> get_M new_element_ptr child_nodes \<rightarrow>\<^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_name:
|
|
assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"
|
|
assumes "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"
|
|
shows "h' \<turnstile> get_M new_element_ptr tag_name \<rightarrow>\<^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 \<turnstile> new_element \<rightarrow>\<^sub>h h'"
|
|
assumes "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"
|
|
shows "h' \<turnstile> get_M new_element_ptr attrs \<rightarrow>\<^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 \<turnstile> new_element \<rightarrow>\<^sub>h h'"
|
|
assumes "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"
|
|
shows "h' \<turnstile> get_M new_element_ptr shadow_root_opt \<rightarrow>\<^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 \<turnstile> new_element \<rightarrow>\<^sub>h h' \<Longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr \<Longrightarrow> ptr \<noteq> cast new_element_ptr
|
|
\<Longrightarrow> 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 \<turnstile> new_element \<rightarrow>\<^sub>h h' \<Longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr \<Longrightarrow> ptr \<noteq> cast new_element_ptr
|
|
\<Longrightarrow> 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 \<turnstile> new_element \<rightarrow>\<^sub>h h' \<Longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr \<Longrightarrow> ptr \<noteq> new_element_ptr
|
|
\<Longrightarrow> 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\<open>Modified Heaps\<close>
|
|
|
|
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 |\<union>| (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 \<Longrightarrow> 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 |\<notin>| object_ptr_kinds h"
|
|
shows "type_wf h"
|
|
using assms
|
|
apply(auto simp add: type_wf_defs elim!: NodeMonad.type_wf_put_ptr_not_in_heap_E
|
|
split: option.splits if_splits)[1]
|
|
using assms(2) node_ptr_kinds_commutes by blast
|
|
|
|
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 |\<in>| object_ptr_kinds h"
|
|
assumes "NodeClass.type_wf h"
|
|
assumes "is_element_ptr_kind ptr \<Longrightarrow> 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]
|
|
by (metis (no_types, lifting) NodeClass.l_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas_axioms 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 finite_set_in 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
|
|
l_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas.get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf option.collapse)
|
|
|
|
subsection\<open>Preserving Types\<close>
|
|
|
|
lemma new_element_type_wf_preserved [simp]: "h \<turnstile> new_element \<rightarrow>\<^sub>h h' \<Longrightarrow> 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 finite_set_in
|
|
is_element_ptr_ref)
|
|
apply (metis element_ptrs_def fempty_iff ffmember_filter finite_set_in is_element_ptr_ref)
|
|
apply (metis (no_types, lifting) Suc_n_not_le_n element_ptr.sel(1) element_ptr_kinds_commutes
|
|
element_ptrs_def fMax_ge ffmember_filter fimage_eqI is_element_ptr_ref notin_fset)
|
|
apply (metis (no_types, lifting) Suc_n_not_le_n element_ptr.sel(1) element_ptrs_def
|
|
fMax_ge ffmember_filter fimage_eqI finite_set_in is_element_ptr_ref)
|
|
done
|
|
|
|
locale l_new_element = l_type_wf +
|
|
assumes new_element_types_preserved: "h \<turnstile> new_element \<rightarrow>\<^sub>h h' \<Longrightarrow> 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_name_type_wf_preserved [simp]:
|
|
"h \<turnstile> put_M element_ptr tag_name_update v \<rightarrow>\<^sub>h h' \<Longrightarrow> 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 finite_set_in option.inject)
|
|
apply (metis cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inv finite_set_in option.sel)
|
|
done
|
|
|
|
lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_child_nodes_type_wf_preserved [simp]:
|
|
"h \<turnstile> put_M element_ptr child_nodes_update v \<rightarrow>\<^sub>h h' \<Longrightarrow> 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 finite_set_in option.inject)
|
|
apply (metis cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inv finite_set_in option.sel)
|
|
done
|
|
|
|
lemma put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_attrs_type_wf_preserved [simp]:
|
|
"h \<turnstile> put_M element_ptr attrs_update v \<rightarrow>\<^sub>h h' \<Longrightarrow> 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 finite_set_in option.inject)
|
|
apply (metis cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inv finite_set_in option.sel)
|
|
done
|
|
|
|
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 \<turnstile> put_M element_ptr shadow_root_opt_update v \<rightarrow>\<^sub>h h' \<Longrightarrow> 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 finite_set_in option.inject)
|
|
apply (metis cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e_inv finite_set_in option.sel)
|
|
done
|
|
|
|
lemma put_M_pointers_preserved:
|
|
assumes "h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^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 \<turnstile> setter \<rightarrow>\<^sub>h h'"
|
|
assumes "\<And>h h'. \<forall>w \<in> SW. h \<turnstile> w \<rightarrow>\<^sub>h h'
|
|
\<longrightarrow> (\<forall>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 "\<And>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 "\<And>object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'"
|
|
assumes "\<And>node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'"
|
|
assumes "\<And>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 \<turnstile> setter \<rightarrow>\<^sub>h h'"
|
|
assumes "\<And>h h' w. w \<in> SW \<Longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h'
|
|
\<Longrightarrow> \<forall>object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'"
|
|
assumes "\<And>h h' w. w \<in> SW \<Longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h'
|
|
\<Longrightarrow> \<forall>node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'"
|
|
assumes "\<And>h h' w. w \<in> SW \<Longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h'
|
|
\<Longrightarrow> \<forall>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 "\<And>h h' w. w \<in> SW \<Longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<Longrightarrow> 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 \<Longrightarrow> 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 (no_types, lifting) element_ptr_kinds_commutes finite_set_in fmdom_notD fmdom_notI
|
|
fmlookup_drop heap.sel node_ptr_kinds_commutes o_apply object_ptr_kinds_def)
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by (metis element_ptr_kinds_commutes fmdom_notI fmdrop_lookup heap.sel node_ptr_kinds_commutes
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o_apply object_ptr_kinds_def)
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end
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