author | kleing |
Tue, 15 Oct 2002 15:37:57 +0200 | |
changeset 13649 | 0f562a70c07d |
parent 13365 | a2c4faad4d35 |
child 15425 | 6356d2523f73 |
permissions | -rw-r--r-- |
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(* |
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ID: $Id$ |
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Author: Gerwin Klein |
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Copyright 1999 Technische Universitaet Muenchen |
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*) |
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header {* \isaheader{Correctness of the LBV} *} |
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theory LBVCorrect = LBVSpec + Typing_Framework: |
9757
1024a2d80ac0
functional LBV style, dead code, type safety -> Isar
kleing
parents:
9664
diff
changeset
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|
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locale (open) lbvs = lbv + |
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fixes s0 :: 'a ("s\<^sub>0") |
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fixes c :: "'a list" |
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fixes ins :: "'b list" |
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fixes phi :: "'a list" ("\<phi>") |
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defines phi_def: |
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"\<phi> \<equiv> map (\<lambda>pc. if c!pc = \<bottom> then wtl (take pc ins) c 0 s0 else c!pc) |
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[0..length ins(]" |
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||
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assumes bounded: "bounded step (length ins)" |
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assumes cert: "cert_ok c (length ins) \<top> \<bottom> A" |
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assumes pres: "pres_type step (length ins) A" |
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lemma (in lbvs) phi_None [intro?]: |
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"\<lbrakk> pc < length ins; c!pc = \<bottom> \<rbrakk> \<Longrightarrow> \<phi> ! pc = wtl (take pc ins) c 0 s0" |
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by (simp add: phi_def) |
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9757
1024a2d80ac0
functional LBV style, dead code, type safety -> Isar
kleing
parents:
9664
diff
changeset
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lemma (in lbvs) phi_Some [intro?]: |
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"\<lbrakk> pc < length ins; c!pc \<noteq> \<bottom> \<rbrakk> \<Longrightarrow> \<phi> ! pc = c ! pc" |
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by (simp add: phi_def) |
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lemma (in lbvs) phi_len [simp]: |
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"length \<phi> = length ins" |
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by (simp add: phi_def) |
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||
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lemma (in lbvs) wtl_suc_pc: |
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assumes all: "wtl ins c 0 s\<^sub>0 \<noteq> \<top>" |
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assumes pc: "pc+1 < length ins" |
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shows "wtl (take (pc+1) ins) c 0 s0 \<le>\<^sub>r \<phi>!(pc+1)" |
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proof - |
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from all pc |
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have "wtc c (pc+1) (wtl (take (pc+1) ins) c 0 s0) \<noteq> T" by (rule wtl_all) |
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with pc show ?thesis by (simp add: phi_def wtc split: split_if_asm) |
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qed |
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||
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lemma (in lbvs) wtl_stable: |
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assumes wtl: "wtl ins c 0 s0 \<noteq> \<top>" |
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assumes s0: "s0 \<in> A" |
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assumes pc: "pc < length ins" |
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shows "stable r step \<phi> pc" |
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proof (unfold stable_def, clarify) |
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fix pc' s' assume step: "(pc',s') \<in> set (step pc (\<phi> ! pc))" |
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(is "(pc',s') \<in> set (?step pc)") |
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||
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from bounded pc step have pc': "pc' < length ins" by (rule boundedD) |
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have tkpc: "wtl (take pc ins) c 0 s0 \<noteq> \<top>" (is "?s1 \<noteq> _") by (rule wtl_take) |
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have s2: "wtl (take (pc+1) ins) c 0 s0 \<noteq> \<top>" (is "?s2 \<noteq> _") by (rule wtl_take) |
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from wtl pc have wt_s1: "wtc c pc ?s1 \<noteq> \<top>" by (rule wtl_all) |
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have c_Some: "\<forall>pc t. pc < length ins \<longrightarrow> c!pc \<noteq> \<bottom> \<longrightarrow> \<phi>!pc = c!pc" |
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by (simp add: phi_def) |
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have c_None: "c!pc = \<bottom> \<Longrightarrow> \<phi>!pc = ?s1" .. |
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from wt_s1 pc c_None c_Some |
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have inst: "wtc c pc ?s1 = wti c pc (\<phi>!pc)" |
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by (simp add: wtc split: split_if_asm) |
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have "?s1 \<in> A" by (rule wtl_pres) |
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with pc c_Some cert c_None |
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have "\<phi>!pc \<in> A" by (cases "c!pc = \<bottom>") (auto dest: cert_okD1) |
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with pc pres |
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have step_in_A: "snd`set (?step pc) \<subseteq> A" by (auto dest: pres_typeD2) |
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show "s' <=_r \<phi>!pc'" |
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proof (cases "pc' = pc+1") |
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case True |
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with pc' cert |
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have cert_in_A: "c!(pc+1) \<in> A" by (auto dest: cert_okD1) |
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from True pc' have pc1: "pc+1 < length ins" by simp |
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with tkpc have "?s2 = wtc c pc ?s1" by - (rule wtl_Suc) |
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with inst |
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have merge: "?s2 = merge c pc (?step pc) (c!(pc+1))" by (simp add: wti) |
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also |
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from s2 merge have "\<dots> \<noteq> \<top>" (is "?merge \<noteq> _") by simp |
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with cert_in_A step_in_A |
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have "?merge = (map snd [(p',t')\<in>?step pc. p'=pc+1] ++_f (c!(pc+1)))" |
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by (rule merge_not_top_s) |
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finally |
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have "s' <=_r ?s2" using step_in_A cert_in_A True step |
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by (auto intro: pp_ub1') |
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also |
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from wtl pc1 have "?s2 <=_r \<phi>!(pc+1)" by (rule wtl_suc_pc) |
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also note True [symmetric] |
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finally show ?thesis by simp |
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next |
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case False |
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from wt_s1 inst |
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have "merge c pc (?step pc) (c!(pc+1)) \<noteq> \<top>" by (simp add: wti) |
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with step_in_A |
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have "\<forall>(pc', s')\<in>set (?step pc). pc'\<noteq>pc+1 \<longrightarrow> s' <=_r c!pc'" |
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by - (rule merge_not_top) |
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with step False |
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have ok: "s' <=_r c!pc'" by blast |
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moreover |
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from ok |
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have "c!pc' = \<bottom> \<Longrightarrow> s' = \<bottom>" by simp |
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moreover |
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from c_Some pc' |
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have "c!pc' \<noteq> \<bottom> \<Longrightarrow> \<phi>!pc' = c!pc'" by auto |
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ultimately |
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show ?thesis by (cases "c!pc' = \<bottom>") auto |
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qed |
9549
40d64cb4f4e6
BV and LBV specified in terms of app and step functions
kleing
parents:
9376
diff
changeset
|
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qed |
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lemma (in lbvs) phi_not_top: |
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assumes wtl: "wtl ins c 0 s0 \<noteq> \<top>" |
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assumes pc: "pc < length ins" |
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shows "\<phi>!pc \<noteq> \<top>" |
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proof (cases "c!pc = \<bottom>") |
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case False with pc |
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have "\<phi>!pc = c!pc" .. |
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also from cert pc have "\<dots> \<noteq> \<top>" by (rule cert_okD4) |
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finally show ?thesis . |
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next |
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case True with pc |
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have "\<phi>!pc = wtl (take pc ins) c 0 s0" .. |
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also from wtl have "\<dots> \<noteq> \<top>" by (rule wtl_take) |
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finally show ?thesis . |
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qed |
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lemma (in lbvs) phi_in_A: |
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assumes wtl: "wtl ins c 0 s0 \<noteq> \<top>" |
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assumes s0: "s0 \<in> A" |
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shows "\<phi> \<in> list (length ins) A" |
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proof - |
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{ fix x assume "x \<in> set \<phi>" |
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then obtain xs ys where "\<phi> = xs @ x # ys" |
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by (auto simp add: in_set_conv_decomp) |
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then obtain pc where pc: "pc < length \<phi>" and x: "\<phi>!pc = x" |
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by (simp add: that [of "length xs"] nth_append) |
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from wtl s0 pc |
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have "wtl (take pc ins) c 0 s0 \<in> A" by (auto intro!: wtl_pres) |
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moreover |
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from pc have "pc < length ins" by simp |
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with cert have "c!pc \<in> A" .. |
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ultimately |
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have "\<phi>!pc \<in> A" using pc by (simp add: phi_def) |
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hence "x \<in> A" using x by simp |
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} |
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hence "set \<phi> \<subseteq> A" .. |
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thus ?thesis by (unfold list_def) simp |
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qed |
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lemma (in lbvs) phi0: |
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assumes wtl: "wtl ins c 0 s0 \<noteq> \<top>" |
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assumes 0: "0 < length ins" |
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shows "s0 <=_r \<phi>!0" |
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proof (cases "c!0 = \<bottom>") |
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case True |
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with 0 have "\<phi>!0 = wtl (take 0 ins) c 0 s0" .. |
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moreover have "wtl (take 0 ins) c 0 s0 = s0" by simp |
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ultimately have "\<phi>!0 = s0" by simp |
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thus ?thesis by simp |
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next |
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case False |
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with 0 have "phi!0 = c!0" .. |
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moreover |
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have "wtl (take 1 ins) c 0 s0 \<noteq> \<top>" by (rule wtl_take) |
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with 0 False |
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have "s0 <=_r c!0" by (auto simp add: neq_Nil_conv wtc split: split_if_asm) |
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ultimately |
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show ?thesis by simp |
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qed |
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theorem (in lbvs) wtl_sound: |
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assumes "wtl ins c 0 s0 \<noteq> \<top>" |
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assumes "s0 \<in> A" |
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shows "\<exists>ts. wt_step r \<top> step ts" |
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proof - |
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have "wt_step r \<top> step \<phi>" |
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proof (unfold wt_step_def, intro strip conjI) |
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fix pc assume "pc < length \<phi>" |
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then obtain "pc < length ins" by simp |
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show "\<phi>!pc \<noteq> \<top>" by (rule phi_not_top) |
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show "stable r step \<phi> pc" by (rule wtl_stable) |
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qed |
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thus ?thesis .. |
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qed |
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theorem (in lbvs) wtl_sound_strong: |
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assumes "wtl ins c 0 s0 \<noteq> \<top>" |
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assumes "s0 \<in> A" |
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assumes "0 < length ins" |
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shows "\<exists>ts \<in> list (length ins) A. wt_step r \<top> step ts \<and> s0 <=_r ts!0" |
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proof - |
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have "\<phi> \<in> list (length ins) A" by (rule phi_in_A) |
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moreover |
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have "wt_step r \<top> step \<phi>" |
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proof (unfold wt_step_def, intro strip conjI) |
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fix pc assume "pc < length \<phi>" |
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then obtain "pc < length ins" by simp |
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show "\<phi>!pc \<noteq> \<top>" by (rule phi_not_top) |
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show "stable r step \<phi> pc" by (rule wtl_stable) |
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qed |
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moreover |
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have "s0 <=_r \<phi>!0" by (rule phi0) |
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ultimately |
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show ?thesis by fast |
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9549
40d64cb4f4e6
BV and LBV specified in terms of app and step functions
kleing
parents:
9376
diff
changeset
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qed |
40d64cb4f4e6
BV and LBV specified in terms of app and step functions
kleing
parents:
9376
diff
changeset
|
220 |
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end |