210 lines
12 KiB
Plaintext
210 lines
12 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>Node\<close>
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text\<open>In this theory, we introduce the types for the Node class.\<close>
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theory NodeClass
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imports
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ObjectClass
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"../pointers/NodePointer"
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begin
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subsubsection\<open>Node\<close>
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record RNode = RObject
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+ nothing :: unit
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register_default_tvars "'Node RNode_ext"
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type_synonym 'Node Node = "'Node RNode_scheme"
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register_default_tvars "'Node Node"
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type_synonym ('Object, 'Node) Object = "('Node RNode_ext + 'Object) Object"
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register_default_tvars "('Object, 'Node) Object"
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type_synonym ('object_ptr, 'node_ptr, 'Object, 'Node) heap
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= "('node_ptr node_ptr + 'object_ptr, 'Node RNode_ext + 'Object) heap"
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register_default_tvars
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"('object_ptr, 'node_ptr, 'Object, 'Node) heap"
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type_synonym heap\<^sub>f\<^sub>i\<^sub>n\<^sub>a\<^sub>l = "(unit, unit, unit, unit) heap"
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definition node_ptr_kinds :: "(_) heap \<Rightarrow> (_) node_ptr fset"
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where
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"node_ptr_kinds heap =
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(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))))"
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lemma node_ptr_kinds_simp [simp]:
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"node_ptr_kinds (Heap (fmupd (cast node_ptr) node (the_heap h)))
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= {|node_ptr|} |\<union>| node_ptr_kinds h"
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apply(auto simp add: node_ptr_kinds_def)[1]
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by force
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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 \<Rightarrow> (_) Node option"
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where
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"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
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\<Rightarrow> Some (RObject.extend (RObject.truncate obj) node) | _ \<Rightarrow> None)"
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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
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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 \<Rightarrow> (_) Object"
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where
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"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)))"
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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
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definition is_node_kind :: "(_) Object \<Rightarrow> bool"
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where
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"is_node_kind ptr \<longleftrightarrow> cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr \<noteq> None"
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definition get\<^sub>N\<^sub>o\<^sub>d\<^sub>e :: "(_) node_ptr \<Rightarrow> (_) heap \<Rightarrow> (_) Node option"
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where
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"get\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr h = Option.bind (get (cast node_ptr) h) cast"
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adhoc_overloading get get\<^sub>N\<^sub>o\<^sub>d\<^sub>e
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locale l_type_wf_def\<^sub>N\<^sub>o\<^sub>d\<^sub>e
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begin
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definition a_type_wf :: "(_) heap \<Rightarrow> bool"
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where
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"a_type_wf h = (ObjectClass.type_wf h
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\<and> (\<forall>node_ptr \<in> fset( node_ptr_kinds h). get\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr h \<noteq> None))"
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end
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global_interpretation l_type_wf_def\<^sub>N\<^sub>o\<^sub>d\<^sub>e defines type_wf = a_type_wf .
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lemmas type_wf_defs = a_type_wf_def
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locale l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e = l_type_wf type_wf for type_wf :: "((_) heap \<Rightarrow> bool)" +
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assumes type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e: "type_wf h \<Longrightarrow> NodeClass.type_wf h"
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sublocale l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e \<subseteq> l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t
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apply(unfold_locales)
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using ObjectClass.a_type_wf_def by auto
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locale l_get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas = l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e
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begin
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sublocale l_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_lemmas by unfold_locales
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lemma get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_type_wf:
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assumes "type_wf h"
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shows "node_ptr |\<in>| node_ptr_kinds h \<longleftrightarrow> get\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr h \<noteq> None"
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using l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e_axioms assms
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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)
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by (metis bind_eq_None_conv ffmember_filter fimage_eqI fmember.rep_eq is_node_ptr_kind_cast
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get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf node_ptr_casts_commute2 node_ptr_kinds_def option.sel option.simps(3))
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end
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global_interpretation l_get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_lemmas type_wf
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by unfold_locales
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definition put\<^sub>N\<^sub>o\<^sub>d\<^sub>e :: "(_) node_ptr \<Rightarrow> (_) Node \<Rightarrow> (_) heap \<Rightarrow> (_) heap"
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where
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"put\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr node = put (cast node_ptr) (cast node)"
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adhoc_overloading put put\<^sub>N\<^sub>o\<^sub>d\<^sub>e
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lemma put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_ptr_in_heap:
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assumes "put\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr node h = h'"
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shows "node_ptr |\<in>| node_ptr_kinds h'"
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using assms
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unfolding put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def node_ptr_kinds_def
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by (metis ffmember_filter fimage_eqI is_node_ptr_kind_cast node_ptr_casts_commute2
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option.sel put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ptr_in_heap)
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lemma put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_put_ptrs:
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assumes "put\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr node h = h'"
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shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast node_ptr|}"
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using assms
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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)
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lemma node_ptr_kinds_commutes [simp]:
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"cast node_ptr |\<in>| object_ptr_kinds h \<longleftrightarrow> node_ptr |\<in>| node_ptr_kinds h"
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apply(auto simp add: node_ptr_kinds_def split: option.splits)[1]
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by (metis (no_types, lifting) ffmember_filter fimage_eqI fset.map_comp
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is_node_ptr_kind_none node_ptr_casts_commute2
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option.distinct(1) option.sel)
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lemma node_empty [simp]:
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"\<lparr>RObject.nothing = (), RNode.nothing = (), \<dots> = RNode.more node\<rparr> = node"
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by simp
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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 \<longleftrightarrow> x = y"
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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)
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by (metis (full_types) RObject.surjective old.unit.exhaust)
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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]:
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"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 \<longleftrightarrow> \<not> (\<exists>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)"
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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]
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by (metis (full_types) RObject.select_convs(2) RObject.surjective old.unit.exhaust)
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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 \<longleftrightarrow> cast node = obj"
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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)
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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"
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by simp
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locale l_known_ptr\<^sub>N\<^sub>o\<^sub>d\<^sub>e
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begin
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definition a_known_ptr :: "(_) object_ptr \<Rightarrow> bool"
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where
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"a_known_ptr ptr = False"
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end
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global_interpretation l_known_ptr\<^sub>N\<^sub>o\<^sub>d\<^sub>e defines known_ptr = a_known_ptr .
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lemmas known_ptr_defs = a_known_ptr_def
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locale l_known_ptrs\<^sub>N\<^sub>o\<^sub>d\<^sub>e = l_known_ptr known_ptr for known_ptr :: "(_) object_ptr \<Rightarrow> bool"
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begin
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definition a_known_ptrs :: "(_) heap \<Rightarrow> bool"
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where
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"a_known_ptrs h = (\<forall>ptr \<in> fset (object_ptr_kinds h). known_ptr ptr)"
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lemma known_ptrs_known_ptr: "a_known_ptrs h \<Longrightarrow> ptr |\<in>| object_ptr_kinds h \<Longrightarrow> known_ptr ptr"
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apply(simp add: a_known_ptrs_def)
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using notin_fset by fastforce
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lemma known_ptrs_preserved:
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"object_ptr_kinds h = object_ptr_kinds h' \<Longrightarrow> a_known_ptrs h = a_known_ptrs h'"
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by(auto simp add: a_known_ptrs_def)
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lemma known_ptrs_subset:
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"object_ptr_kinds h' |\<subseteq>| object_ptr_kinds h \<Longrightarrow> a_known_ptrs h \<Longrightarrow> a_known_ptrs h'"
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by(simp add: a_known_ptrs_def less_eq_fset.rep_eq subsetD)
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lemma known_ptrs_new_ptr:
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"object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|new_ptr|} \<Longrightarrow> known_ptr new_ptr \<Longrightarrow>
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a_known_ptrs h \<Longrightarrow> a_known_ptrs h'"
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by(simp add: a_known_ptrs_def)
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end
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global_interpretation l_known_ptrs\<^sub>N\<^sub>o\<^sub>d\<^sub>e known_ptr defines known_ptrs = a_known_ptrs .
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lemmas known_ptrs_defs = a_known_ptrs_def
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lemma known_ptrs_is_l_known_ptrs: "l_known_ptrs known_ptr known_ptrs"
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using known_ptrs_known_ptr known_ptrs_preserved l_known_ptrs_def known_ptrs_subset
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known_ptrs_new_ptr
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by blast
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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"
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by(auto simp add: get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def)
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lemma get_node_ptr_simp2 [simp]:
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"node_ptr \<noteq> node_ptr' \<Longrightarrow> 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"
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by(auto simp add: get\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def)
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end
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