Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
| (* Title: BDD | |
| Author: Veronika Ortner and Norbert Schirmer, 2004 | |
| Maintainer: Norbert Schirmer, norbert.schirmer at web de | |
| License: LGPL | |
| *) | |
| (* | |
| ShareRepProof.thy | |
| Copyright (C) 2004-2008 Veronika Ortner and Norbert Schirmer | |
| Some rights reserved, TU Muenchen | |
| This library is free software; you can redistribute it and/or modify | |
| it under the terms of the GNU Lesser General Public License as | |
| published by the Free Software Foundation; either version 2.1 of the | |
| License, or (at your option) any later version. | |
| This library is distributed in the hope that it will be useful, but | |
| WITHOUT ANY WARRANTY; without even the implied warranty of | |
| MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| Lesser General Public License for more details. | |
| You should have received a copy of the GNU Lesser General Public | |
| License along with this library; if not, write to the Free Software | |
| Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 | |
| USA | |
| *) | |
| section \<open>Proof of Procedure ShareRep\<close> | |
| theory ShareRepProof imports ProcedureSpecs Simpl.HeapList begin | |
| lemma (in ShareRep_impl) ShareRep_modifies: | |
| shows "\<forall>\<sigma>. \<Gamma>\<turnstile>{\<sigma>} PROC ShareRep (\<acute>nodeslist, \<acute>p) | |
| {t. t may_only_modify_globals \<sigma> in [rep]}" | |
| apply (hoare_rule HoarePartial.ProcRec1) | |
| apply (vcg spec=modifies) | |
| done | |
| lemma hd_filter_cons: | |
| "\<And> i. \<lbrakk> P (xs ! i) p; i < length xs; \<forall> no \<in> set (take i xs). \<not> P no p; \<forall> a b. P a b = P b a\<rbrakk> | |
| \<Longrightarrow> xs ! i = hd (filter (P p) xs)" | |
| apply (induct xs) | |
| apply simp | |
| apply (case_tac "P a p") | |
| apply simp | |
| apply (case_tac i) | |
| apply simp | |
| apply simp | |
| apply (case_tac i) | |
| apply simp | |
| apply auto | |
| done | |
| lemma (in ShareRep_impl) ShareRep_spec_total: | |
| shows | |
| "\<forall>\<sigma> ns. \<Gamma>,\<Theta>\<turnstile>\<^sub>t | |
| \<lbrace>\<sigma>. List \<acute>nodeslist \<acute>next ns \<and> | |
| (\<forall>no \<in> set ns. no \<noteq> Null \<and> | |
| ((no\<rightarrow>\<acute>low = Null) = (no\<rightarrow>\<acute>high = Null)) \<and> | |
| (isLeaf_pt \<acute>p \<acute>low \<acute>high \<longrightarrow> isLeaf_pt no \<acute>low \<acute>high) \<and> | |
| no\<rightarrow>\<acute>var = \<acute>p\<rightarrow>\<acute>var) \<and> | |
| \<acute>p \<in> set ns\<rbrace> | |
| PROC ShareRep (\<acute>nodeslist, \<acute>p) | |
| \<lbrace> (\<^bsup>\<sigma>\<^esup>p \<rightarrow> \<acute>rep = hd (filter (\<lambda> sn. repNodes_eq sn \<^bsup>\<sigma>\<^esup>p \<^bsup>\<sigma>\<^esup>low \<^bsup>\<sigma>\<^esup>high \<^bsup>\<sigma>\<^esup>rep) ns)) \<and> | |
| (\<forall>pt. pt \<noteq> \<^bsup>\<sigma>\<^esup>p \<longrightarrow> pt\<rightarrow>\<^bsup>\<sigma>\<^esup>rep = pt\<rightarrow>\<acute>rep) \<and> | |
| (\<^bsup>\<sigma>\<^esup>p\<rightarrow>\<acute>rep\<rightarrow>\<^bsup>\<sigma>\<^esup>var = \<^bsup>\<sigma>\<^esup>p \<rightarrow> \<^bsup>\<sigma>\<^esup>var)\<rbrace>" | |
| apply (hoare_rule HoareTotal.ProcNoRec1) | |
| apply (hoare_rule anno= | |
| "IF (isLeaf_pt \<acute>p \<acute>low \<acute>high) | |
| THEN \<acute>p \<rightarrow> \<acute>rep :== \<acute>nodeslist | |
| ELSE | |
| WHILE (\<acute>nodeslist \<noteq> Null) | |
| INV \<lbrace>\<exists>prx sfx. List \<acute>nodeslist \<acute>next sfx \<and> ns=prx@sfx \<and> | |
| \<not> isLeaf_pt \<acute>p \<acute>low \<acute>high \<and> | |
| (\<forall>no \<in> set ns. no \<noteq> Null \<and> | |
| ((no\<rightarrow>\<^bsup>\<sigma>\<^esup>low = Null) = (no\<rightarrow>\<^bsup>\<sigma>\<^esup>high = Null)) \<and> | |
| (isLeaf_pt \<^bsup>\<sigma>\<^esup>p \<^bsup>\<sigma>\<^esup>low \<^bsup>\<sigma>\<^esup>high \<longrightarrow> isLeaf_pt no \<^bsup>\<sigma>\<^esup>low \<^bsup>\<sigma>\<^esup>high) \<and> | |
| no\<rightarrow>\<^bsup>\<sigma>\<^esup>var = \<^bsup>\<sigma>\<^esup>p\<rightarrow>\<^bsup>\<sigma>\<^esup>var) \<and> | |
| \<^bsup>\<sigma>\<^esup>p \<in> set ns \<and> | |
| ((\<exists>pt \<in> set prx. repNodes_eq pt \<^bsup>\<sigma>\<^esup>p \<^bsup>\<sigma>\<^esup>low \<^bsup>\<sigma>\<^esup>high \<^bsup>\<sigma>\<^esup>rep) | |
| \<longrightarrow> \<acute>rep \<^bsup>\<sigma>\<^esup>p = hd (filter (\<lambda> sn. repNodes_eq sn \<^bsup>\<sigma>\<^esup>p \<^bsup>\<sigma>\<^esup>low \<^bsup>\<sigma>\<^esup>high \<^bsup>\<sigma>\<^esup>rep) prx) \<and> | |
| (\<forall>pt. pt \<noteq> \<^bsup>\<sigma>\<^esup>p \<longrightarrow> pt\<rightarrow>\<^bsup>\<sigma>\<^esup>rep = pt\<rightarrow>\<acute>rep)) \<and> | |
| ((\<forall>pt \<in> set prx. \<not> repNodes_eq pt \<^bsup>\<sigma>\<^esup>p \<^bsup>\<sigma>\<^esup>low \<^bsup>\<sigma>\<^esup>high \<^bsup>\<sigma>\<^esup>rep) \<longrightarrow> \<^bsup>\<sigma>\<^esup>rep = \<acute>rep) \<and> | |
| (\<acute>nodeslist \<noteq> Null \<longrightarrow> | |
| (\<forall>pt \<in> set prx. \<not> repNodes_eq pt \<^bsup>\<sigma>\<^esup>p \<^bsup>\<sigma>\<^esup>low \<^bsup>\<sigma>\<^esup>high \<^bsup>\<sigma>\<^esup>rep)) \<and> | |
| (\<acute>p = \<^bsup>\<sigma>\<^esup>p \<and> \<acute>high = \<^bsup>\<sigma>\<^esup>high \<and> \<acute>low = \<^bsup>\<sigma>\<^esup>low)\<rbrace> | |
| VAR MEASURE (length (list \<acute>nodeslist \<acute>next)) | |
| DO | |
| IF (repNodes_eq \<acute>nodeslist \<acute>p \<acute>low \<acute>high \<acute>rep) | |
| THEN \<acute>p\<rightarrow>\<acute>rep :== \<acute>nodeslist;; \<acute>nodeslist :== Null | |
| ELSE \<acute>nodeslist :== \<acute>nodeslist\<rightarrow>\<acute>next | |
| FI | |
| OD | |
| FI" in HoareTotal.annotateI) | |
| apply vcg | |
| using [[simp_depth_limit = 2]] | |
| apply (rule conjI) | |
| apply clarify | |
| apply (simp (no_asm_use)) | |
| prefer 2 | |
| apply clarify | |
| apply (rule_tac x="[]" in exI) | |
| apply (rule_tac x=ns in exI) | |
| apply (simp (no_asm_use)) | |
| prefer 2 | |
| apply clarify | |
| apply (rule conjI) | |
| apply clarify | |
| apply (rule conjI) | |
| apply (clarsimp simp add: List_list) (* solving termination contraint *) | |
| apply (simp (no_asm_use)) | |
| apply (rule conjI) | |
| apply assumption | |
| prefer 2 | |
| apply clarify | |
| apply (simp (no_asm_use)) | |
| apply (rule conjI) | |
| apply (clarsimp simp add: List_list) (* solving termination constraint *) | |
| apply (simp only: List_not_Null simp_thms triv_forall_equality) | |
| apply clarify | |
| apply (simp only: triv_forall_equality) | |
| apply (rename_tac sfx) | |
| apply (rule_tac x="prx@[nodeslist]" in exI) | |
| apply (rule_tac x="sfx" in exI) | |
| apply (rule conjI) | |
| apply assumption | |
| apply (rule conjI) | |
| apply simp | |
| prefer 4 | |
| apply (elim exE conjE) | |
| apply (simp (no_asm_use)) | |
| apply hypsubst | |
| using [[simp_depth_limit = 100]] | |
| proof - | |
| (* IF-THEN to postcondition *) | |
| fix ns var low high rep "next" p nodeslist | |
| assume ns: "List nodeslist next ns" | |
| assume no_prop: "\<forall>no\<in>set ns. | |
| no \<noteq> Null \<and> | |
| (low no = Null) = (high no = Null) \<and> | |
| (isLeaf_pt p low high \<longrightarrow> isLeaf_pt no low high) \<and> var no = var p" | |
| assume p_in_ns: "p \<in> set ns" | |
| assume p_Leaf: "isLeaf_pt p low high" | |
| show "nodeslist = hd [sn\<leftarrow>ns . repNodes_eq sn p low high rep] \<and> | |
| var nodeslist = var p" | |
| proof - | |
| from p_in_ns no_prop have p_not_Null: "p\<noteq>Null" | |
| using [[simp_depth_limit=2]] | |
| by auto | |
| from p_in_ns have "ns \<noteq> []" | |
| by (cases ns) auto | |
| with ns obtain ns' where ns': "ns = nodeslist#ns'" | |
| by (cases "nodeslist=Null") auto | |
| with no_prop p_Leaf obtain | |
| "isLeaf_pt nodeslist low high" and | |
| var_eq: "var nodeslist = var p" and | |
| "nodeslist\<noteq>Null" | |
| using [[simp_depth_limit=2]] | |
| by auto | |
| with p_not_Null p_Leaf have "repNodes_eq nodeslist p low high rep" | |
| by (simp add: repNodes_eq_def isLeaf_pt_def null_comp_def) | |
| with ns' var_eq | |
| show ?thesis | |
| by simp | |
| qed | |
| next | |
| (* From invariant to postcondition *) | |
| fix var::"ref\<Rightarrow>nat" and low high rep repa p prx sfx "next" | |
| assume sfx: "List Null next sfx" | |
| assume p_in_ns: "p \<in> set (prx @ sfx)" | |
| assume no_props: "\<forall>no\<in>set (prx @ sfx). | |
| no \<noteq> Null \<and> | |
| (low no = Null) = (high no = Null) \<and> | |
| (isLeaf_pt p low high \<longrightarrow> isLeaf_pt no low high) \<and> var no = var p" | |
| assume match_prx: "(\<exists>pt\<in>set prx. repNodes_eq pt p low high rep) \<longrightarrow> | |
| repa p = hd [sn\<leftarrow>prx . repNodes_eq sn p low high rep] \<and> | |
| (\<forall>pt. pt \<noteq> p \<longrightarrow> rep pt = repa pt)" | |
| show "repa p = hd [sn\<leftarrow>prx @ sfx . repNodes_eq sn p low high rep] \<and> | |
| (\<forall>pt. pt \<noteq> p \<longrightarrow> rep pt = repa pt) \<and> var (repa p) = var p" | |
| proof - | |
| from sfx | |
| have sfx_Nil: "sfx=[]" | |
| by simp | |
| with p_in_ns have ex_match: "(\<exists>pt\<in>set prx. repNodes_eq pt p low high rep)" | |
| apply - | |
| apply (rule_tac x=p in bexI) | |
| apply (simp add: repNodes_eq_def) | |
| apply simp | |
| done | |
| hence not_empty: "[sn\<leftarrow>prx . repNodes_eq sn p low high rep] \<noteq> []" | |
| apply - | |
| apply (erule bexE) | |
| apply (rule filter_not_empty) | |
| apply auto | |
| done | |
| from ex_match match_prx obtain | |
| found: "repa p = hd [sn\<leftarrow>prx . repNodes_eq sn p low high rep]" and | |
| unmodif: "\<forall>pt. pt \<noteq> p \<longrightarrow> rep pt = repa pt" | |
| by blast | |
| from hd_filter_in_list [OF not_empty] found | |
| have "repa p \<in> set prx" | |
| by simp | |
| with no_props | |
| have "var (repa p) = var p" | |
| using [[simp_depth_limit=2]] | |
| by simp | |
| with found unmodif sfx_Nil | |
| show ?thesis | |
| by simp | |
| qed | |
| next | |
| (* Invariant to invariant; ELSE part *) | |
| fix var low high p repa "next" nodeslist prx sfx | |
| assume nodeslist_not_Null: "nodeslist \<noteq> Null" | |
| assume p_no_Leaf: "\<not> isLeaf_pt p low high" | |
| assume no_props: "\<forall>no\<in>set prx \<union> set (nodeslist # sfx). | |
| no \<noteq> Null \<and> (low no = Null) = (high no = Null) \<and> var no = var p" | |
| assume p_in_ns: "p \<in> set prx \<or> p \<in> set (nodeslist # sfx)" | |
| assume match_prx: "(\<exists>pt\<in>set prx. repNodes_eq pt p low high repa) \<longrightarrow> | |
| repa p = hd [sn\<leftarrow>prx . repNodes_eq sn p low high repa]" | |
| assume nomatch_prx: "\<forall>pt\<in>set prx. \<not> repNodes_eq pt p low high repa" | |
| assume nomatch_nodeslist: "\<not> repNodes_eq nodeslist p low high repa" | |
| assume sfx: "List (next nodeslist) next sfx" | |
| show "(\<forall>no\<in>set prx \<union> set (nodeslist # sfx). | |
| no \<noteq> Null \<and> (low no = Null) = (high no = Null) \<and> var no = var p) \<and> | |
| ((\<exists>pt\<in>set (prx @ [nodeslist]). repNodes_eq pt p low high repa) \<longrightarrow> | |
| repa p = hd [sn\<leftarrow>prx @ [nodeslist] . repNodes_eq sn p low high repa]) \<and> | |
| (next nodeslist \<noteq> Null \<longrightarrow> | |
| (\<forall>pt\<in>set (prx @ [nodeslist]). \<not> repNodes_eq pt p low high repa))" | |
| proof - | |
| from nomatch_prx nomatch_nodeslist | |
| have "((\<exists>pt\<in>set (prx @ [nodeslist]). repNodes_eq pt p low high repa) \<longrightarrow> | |
| repa p = hd [sn\<leftarrow>prx @ [nodeslist] . repNodes_eq sn p low high repa])" | |
| by auto | |
| moreover | |
| from nomatch_prx nomatch_nodeslist | |
| have "(next nodeslist \<noteq> Null \<longrightarrow> | |
| (\<forall>pt\<in>set (prx @ [nodeslist]). \<not> repNodes_eq pt p low high repa))" | |
| by auto | |
| ultimately show ?thesis | |
| using no_props | |
| by (intro conjI) | |
| qed | |
| next | |
| (* Invariant to invariant: THEN part *) | |
| fix var low high p repa "next" nodeslist prx sfx | |
| assume nodeslist_not_Null: "nodeslist \<noteq> Null" | |
| assume sfx: "List nodeslist next sfx" | |
| assume p_not_Leaf: "\<not> isLeaf_pt p low high" | |
| assume no_props: "\<forall>no\<in>set prx \<union> set sfx. | |
| no \<noteq> Null \<and> | |
| (low no = Null) = (high no = Null) \<and> | |
| (isLeaf_pt p low high \<longrightarrow> isLeaf_pt no low high) \<and> var no = var p" | |
| assume p_in_ns: "p \<in> set prx \<or> p \<in> set sfx" | |
| assume match_prx: "(\<exists>pt\<in>set prx. repNodes_eq pt p low high repa) \<longrightarrow> | |
| repa p = hd [sn\<leftarrow>prx . repNodes_eq sn p low high repa]" | |
| assume nomatch_prx: "\<forall>pt\<in>set prx. \<not> repNodes_eq pt p low high repa" | |
| assume match: "repNodes_eq nodeslist p low high repa" | |
| show "(\<forall>no\<in>set prx \<union> set sfx. | |
| no \<noteq> Null \<and> | |
| (low no = Null) = (high no = Null) \<and> | |
| (isLeaf_pt p low high \<longrightarrow> isLeaf_pt no low high) \<and> var no = var p) \<and> | |
| (p \<in> set prx \<or> p \<in> set sfx) \<and> | |
| ((\<exists>pt\<in>set prx \<union> set sfx. repNodes_eq pt p low high repa) \<longrightarrow> | |
| nodeslist = | |
| hd ([sn\<leftarrow>prx . repNodes_eq sn p low high repa] @ | |
| [sn\<leftarrow>sfx . repNodes_eq sn p low high repa])) \<and> | |
| ((\<forall>pt\<in>set prx \<union> set sfx. \<not> repNodes_eq pt p low high repa) \<longrightarrow> | |
| repa = repa(p := nodeslist))" | |
| proof - | |
| from nodeslist_not_Null sfx | |
| obtain sfx' where sfx': "sfx=nodeslist#sfx'" | |
| by (cases "nodeslist=Null") auto | |
| from nomatch_prx match sfx' | |
| have hd: "hd ([sn\<leftarrow>prx . repNodes_eq sn p low high repa] @ | |
| [sn\<leftarrow>sfx . repNodes_eq sn p low high repa]) = nodeslist" | |
| by simp | |
| from match sfx' | |
| have triv: "((\<forall>pt\<in>set prx \<union> set sfx. \<not> repNodes_eq pt p low high repa) \<longrightarrow> | |
| repa = repa(p := nodeslist))" | |
| by simp | |
| show ?thesis | |
| apply (rule conjI) | |
| apply (rule no_props) | |
| apply (intro conjI) | |
| apply (rule p_in_ns) | |
| apply (simp add: hd) | |
| apply (rule triv) | |
| done | |
| qed | |
| qed | |
| end | |