author | wenzelm |
Wed, 23 Dec 2020 22:25:22 +0100 | |
changeset 72990 | db8f94656024 |
parent 72987 | b1be35908165 |
child 74503 | 403ce50e6a2a |
permissions | -rw-r--r-- |
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(* Title: HOL/Hoare/Hoare_Logic.thy |
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Author: Leonor Prensa Nieto & Tobias Nipkow |
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Copyright 1998 TUM |
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Author: Walter Guttmann (extension to total-correctness proofs) |
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*) |
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section \<open>Hoare logic\<close> |
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theory Hoare_Logic |
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imports Hoare_Syntax Hoare_Tac |
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begin |
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subsection \<open>Sugared semantic embedding of Hoare logic\<close> |
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text \<open> |
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Strictly speaking a shallow embedding (as implemented by Norbert Galm |
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following Mike Gordon) would suffice. Maybe the datatype com comes in useful |
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later. |
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\<close> |
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type_synonym 'a bexp = "'a set" |
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type_synonym 'a assn = "'a set" |
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type_synonym 'a var = "'a \<Rightarrow> nat" |
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datatype 'a com = |
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Basic "'a \<Rightarrow> 'a" |
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| Seq "'a com" "'a com" |
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| Cond "'a bexp" "'a com" "'a com" |
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| While "'a bexp" "'a assn" "'a var" "'a com" |
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abbreviation annskip ("SKIP") where "SKIP == Basic id" |
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type_synonym 'a sem = "'a => 'a => bool" |
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inductive Sem :: "'a com \<Rightarrow> 'a sem" |
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where |
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"Sem (Basic f) s (f s)" |
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| "Sem c1 s s'' \<Longrightarrow> Sem c2 s'' s' \<Longrightarrow> Sem (Seq c1 c2) s s'" |
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| "s \<in> b \<Longrightarrow> Sem c1 s s' \<Longrightarrow> Sem (Cond b c1 c2) s s'" |
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| "s \<notin> b \<Longrightarrow> Sem c2 s s' \<Longrightarrow> Sem (Cond b c1 c2) s s'" |
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| "s \<notin> b \<Longrightarrow> Sem (While b x y c) s s" |
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| "s \<in> b \<Longrightarrow> Sem c s s'' \<Longrightarrow> Sem (While b x y c) s'' s' \<Longrightarrow> |
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Sem (While b x y c) s s'" |
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definition Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool" |
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where "Valid p c q \<equiv> \<forall>s s'. Sem c s s' \<longrightarrow> s \<in> p \<longrightarrow> s' \<in> q" |
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definition ValidTC :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool" |
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where "ValidTC p c q \<equiv> \<forall>s. s \<in> p \<longrightarrow> (\<exists>t. Sem c s t \<and> t \<in> q)" |
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inductive_cases [elim!]: |
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"Sem (Basic f) s s'" "Sem (Seq c1 c2) s s'" |
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"Sem (Cond b c1 c2) s s'" |
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lemma Sem_deterministic: |
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assumes "Sem c s s1" |
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and "Sem c s s2" |
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shows "s1 = s2" |
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proof - |
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have "Sem c s s1 \<Longrightarrow> (\<forall>s2. Sem c s s2 \<longrightarrow> s1 = s2)" |
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by (induct rule: Sem.induct) (subst Sem.simps, blast)+ |
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thus ?thesis |
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using assms by simp |
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qed |
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lemma tc_implies_pc: |
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"ValidTC p c q \<Longrightarrow> Valid p c q" |
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by (metis Sem_deterministic Valid_def ValidTC_def) |
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lemma tc_extract_function: |
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"ValidTC p c q \<Longrightarrow> \<exists>f . \<forall>s . s \<in> p \<longrightarrow> f s \<in> q" |
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by (metis ValidTC_def) |
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lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q" |
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by (auto simp:Valid_def) |
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lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q" |
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by (auto simp:Valid_def) |
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lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (Seq c1 c2) R" |
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by (auto simp:Valid_def) |
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lemma CondRule: |
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"p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')} |
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\<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q" |
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by (auto simp:Valid_def) |
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lemma While_aux: |
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assumes "Sem (While b i v c) s s'" |
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shows "\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> I \<and> s \<in> b \<longrightarrow> s' \<in> I \<Longrightarrow> |
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s \<in> I \<Longrightarrow> s' \<in> I \<and> s' \<notin> b" |
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using assms |
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by (induct "While b i v c" s s') auto |
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lemma WhileRule: |
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"p \<subseteq> i \<Longrightarrow> Valid (i \<inter> b) c i \<Longrightarrow> i \<inter> (-b) \<subseteq> q \<Longrightarrow> Valid p (While b i v c) q" |
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apply (clarsimp simp:Valid_def) |
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apply(drule While_aux) |
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apply assumption |
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apply blast |
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apply blast |
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done |
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lemma SkipRuleTC: |
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assumes "p \<subseteq> q" |
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shows "ValidTC p (Basic id) q" |
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by (metis assms Sem.intros(1) ValidTC_def id_apply subsetD) |
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lemma BasicRuleTC: |
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assumes "p \<subseteq> {s. f s \<in> q}" |
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shows "ValidTC p (Basic f) q" |
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by (metis assms Ball_Collect Sem.intros(1) ValidTC_def) |
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lemma SeqRuleTC: |
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assumes "ValidTC p c1 q" |
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and "ValidTC q c2 r" |
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shows "ValidTC p (Seq c1 c2) r" |
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by (meson assms Sem.intros(2) ValidTC_def) |
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lemma CondRuleTC: |
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assumes "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}" |
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and "ValidTC w c1 q" |
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and "ValidTC w' c2 q" |
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shows "ValidTC p (Cond b c1 c2) q" |
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proof (unfold ValidTC_def, rule allI) |
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fix s |
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show "s \<in> p \<longrightarrow> (\<exists>t . Sem (Cond b c1 c2) s t \<and> t \<in> q)" |
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apply (cases "s \<in> b") |
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apply (metis (mono_tags, lifting) assms(1,2) Ball_Collect Sem.intros(3) ValidTC_def) |
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by (metis (mono_tags, lifting) assms(1,3) Ball_Collect Sem.intros(4) ValidTC_def) |
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qed |
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lemma WhileRuleTC: |
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assumes "p \<subseteq> i" |
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and "\<And>n::nat . ValidTC (i \<inter> b \<inter> {s . v s = n}) c (i \<inter> {s . v s < n})" |
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and "i \<inter> uminus b \<subseteq> q" |
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shows "ValidTC p (While b i v c) q" |
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proof - |
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{ |
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fix s n |
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have "s \<in> i \<and> v s = n \<longrightarrow> (\<exists>t . Sem (While b i v c) s t \<and> t \<in> q)" |
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proof (induction "n" arbitrary: s rule: less_induct) |
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fix n :: nat |
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fix s :: 'a |
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assume 1: "\<And>(m::nat) s::'a . m < n \<Longrightarrow> s \<in> i \<and> v s = m \<longrightarrow> (\<exists>t . Sem (While b i v c) s t \<and> t \<in> q)" |
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show "s \<in> i \<and> v s = n \<longrightarrow> (\<exists>t . Sem (While b i v c) s t \<and> t \<in> q)" |
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proof (rule impI, cases "s \<in> b") |
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assume 2: "s \<in> b" and "s \<in> i \<and> v s = n" |
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hence "s \<in> i \<inter> b \<inter> {s . v s = n}" |
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using assms(1) by auto |
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hence "\<exists>t . Sem c s t \<and> t \<in> i \<inter> {s . v s < n}" |
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by (metis assms(2) ValidTC_def) |
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from this obtain t where 3: "Sem c s t \<and> t \<in> i \<inter> {s . v s < n}" |
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by auto |
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hence "\<exists>u . Sem (While b i v c) t u \<and> u \<in> q" |
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using 1 by auto |
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thus "\<exists>t . Sem (While b i v c) s t \<and> t \<in> q" |
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using 2 3 Sem.intros(6) by force |
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next |
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assume "s \<notin> b" and "s \<in> i \<and> v s = n" |
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thus "\<exists>t . Sem (While b i v c) s t \<and> t \<in> q" |
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using Sem.intros(5) assms(3) by fastforce |
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qed |
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qed |
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} |
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thus ?thesis |
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using assms(1) ValidTC_def by force |
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qed |
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subsubsection \<open>Concrete syntax\<close> |
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|
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setup \<open> |
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Hoare_Syntax.setup |
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{Basic = \<^const_syntax>\<open>Basic\<close>, |
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Skip = \<^const_syntax>\<open>annskip\<close>, |
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Seq = \<^const_syntax>\<open>Seq\<close>, |
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Cond = \<^const_syntax>\<open>Cond\<close>, |
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While = \<^const_syntax>\<open>While\<close>, |
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Valid = \<^const_syntax>\<open>Valid\<close>, |
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ValidTC = \<^const_syntax>\<open>ValidTC\<close>} |
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\<close> |
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subsubsection \<open>Proof methods: VCG\<close> |
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declare BasicRule [Hoare_Tac.BasicRule] |
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and SkipRule [Hoare_Tac.SkipRule] |
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and SeqRule [Hoare_Tac.SeqRule] |
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and CondRule [Hoare_Tac.CondRule] |
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and WhileRule [Hoare_Tac.WhileRule] |
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|
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declare BasicRuleTC [Hoare_Tac.BasicRuleTC] |
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and SkipRuleTC [Hoare_Tac.SkipRuleTC] |
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and SeqRuleTC [Hoare_Tac.SeqRuleTC] |
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and CondRuleTC [Hoare_Tac.CondRuleTC] |
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and WhileRuleTC [Hoare_Tac.WhileRuleTC] |
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method_setup vcg = \<open> |
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Scan.succeed (fn ctxt => SIMPLE_METHOD' (Hoare_Tac.hoare_tac ctxt (K all_tac)))\<close> |
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"verification condition generator" |
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method_setup vcg_simp = \<open> |
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Scan.succeed (fn ctxt => |
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SIMPLE_METHOD' (Hoare_Tac.hoare_tac ctxt (asm_full_simp_tac ctxt)))\<close> |
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"verification condition generator plus simplification" |
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method_setup vcg_tc = \<open> |
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Scan.succeed (fn ctxt => SIMPLE_METHOD' (Hoare_Tac.hoare_tc_tac ctxt (K all_tac)))\<close> |
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"verification condition generator" |
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|
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method_setup vcg_tc_simp = \<open> |
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Scan.succeed (fn ctxt => |
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SIMPLE_METHOD' (Hoare_Tac.hoare_tc_tac ctxt (asm_full_simp_tac ctxt)))\<close> |
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"verification condition generator plus simplification" |
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|
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end |