author  haftmann 
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parent 51272  9c8d63b4b6be 
child 58272  61d94335ef6c 
permissions  rwrr 
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more explicit HOLProofs sessions, including former ex/Hilbert_Classical.thy which works in parallel mode without the antiquotation option "margin" (which is still critical);
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(* Title: HOL/Proofs/Extraction/Warshall.thy 
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Author: Stefan Berghofer, TU Muenchen 
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*) 

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header {* Warshall's algorithm *} 

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theory Warshall 
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imports Main 

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begin 

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text {* 

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Derivation of Warshall's algorithm using program extraction, 

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based on Berger, Schwichtenberg and Seisenberger \cite{BergerJAR2001}. 

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*} 

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datatype b = T  F 

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primrec 
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is_path' :: "('a \<Rightarrow> 'a \<Rightarrow> b) \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a \<Rightarrow> bool" 
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where 
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"is_path' r x [] z = (r x z = T)" 

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 "is_path' r x (y # ys) z = (r x y = T \<and> is_path' r y ys z)" 

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definition 
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is_path :: "(nat \<Rightarrow> nat \<Rightarrow> b) \<Rightarrow> (nat * nat list * nat) \<Rightarrow> 
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nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" 

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where 
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"is_path r p i j k \<longleftrightarrow> fst p = j \<and> snd (snd p) = k \<and> 

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list_all (\<lambda>x. x < i) (fst (snd p)) \<and> 
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is_path' r (fst p) (fst (snd p)) (snd (snd p))" 

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definition 
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conc :: "('a * 'a list * 'a) \<Rightarrow> ('a * 'a list * 'a) \<Rightarrow> ('a * 'a list * 'a)" 
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where 
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"conc p q = (fst p, fst (snd p) @ fst q # fst (snd q), snd (snd q))" 

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theorem is_path'_snoc [simp]: 

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"\<And>x. is_path' r x (ys @ [y]) z = (is_path' r x ys y \<and> r y z = T)" 

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by (induct ys) simp+ 

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theorem list_all_scoc [simp]: "list_all P (xs @ [x]) \<longleftrightarrow> P x \<and> list_all P xs" 
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by (induct xs, simp+, iprover) 
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theorem list_all_lemma: 

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"list_all P xs \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> list_all Q xs" 

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proof  

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assume PQ: "\<And>x. P x \<Longrightarrow> Q x" 

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show "list_all P xs \<Longrightarrow> list_all Q xs" 

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proof (induct xs) 

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case Nil 

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show ?case by simp 

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next 

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case (Cons y ys) 

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hence Py: "P y" by simp 

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from Cons have Pys: "list_all P ys" by simp 

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show ?case 

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by simp (rule conjI PQ Py Cons Pys)+ 

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qed 

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qed 

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theorem lemma1: "\<And>p. is_path r p i j k \<Longrightarrow> is_path r p (Suc i) j k" 

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apply (unfold is_path_def) 

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apply (simp cong add: conj_cong add: split_paired_all) 

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apply (erule conjE)+ 

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apply (erule list_all_lemma) 

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apply simp 

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done 

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theorem lemma2: "\<And>p. is_path r p 0 j k \<Longrightarrow> r j k = T" 

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apply (unfold is_path_def) 

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apply (simp cong add: conj_cong add: split_paired_all) 

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apply (case_tac "aa") 

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apply simp+ 

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done 

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theorem is_path'_conc: "is_path' r j xs i \<Longrightarrow> is_path' r i ys k \<Longrightarrow> 

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is_path' r j (xs @ i # ys) k" 

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proof  

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assume pys: "is_path' r i ys k" 

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show "\<And>j. is_path' r j xs i \<Longrightarrow> is_path' r j (xs @ i # ys) k" 

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proof (induct xs) 

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case (Nil j) 

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hence "r j i = T" by simp 

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with pys show ?case by simp 

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next 

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case (Cons z zs j) 

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hence jzr: "r j z = T" by simp 

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from Cons have pzs: "is_path' r z zs i" by simp 

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show ?case 

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by simp (rule conjI jzr Cons pzs)+ 

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qed 

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qed 

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theorem lemma3: 

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"\<And>p q. is_path r p i j i \<Longrightarrow> is_path r q i i k \<Longrightarrow> 

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is_path r (conc p q) (Suc i) j k" 

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apply (unfold is_path_def conc_def) 

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apply (simp cong add: conj_cong add: split_paired_all) 

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apply (erule conjE)+ 

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apply (rule conjI) 

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apply (erule list_all_lemma) 

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apply simp 

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apply (rule conjI) 

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apply (erule list_all_lemma) 

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apply simp 

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apply (rule is_path'_conc) 

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apply assumption+ 

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done 

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theorem lemma5: 

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"\<And>p. is_path r p (Suc i) j k \<Longrightarrow> ~ is_path r p i j k \<Longrightarrow> 

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(\<exists>q. is_path r q i j i) \<and> (\<exists>q'. is_path r q' i i k)" 

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proof (simp cong add: conj_cong add: split_paired_all is_path_def, (erule conjE)+) 

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fix xs 

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assume asms: 
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"list_all (\<lambda>x. x < Suc i) xs" 

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"is_path' r j xs k" 

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"\<not> list_all (\<lambda>x. x < i) xs" 

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show "(\<exists>ys. list_all (\<lambda>x. x < i) ys \<and> is_path' r j ys i) \<and> 
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(\<exists>ys. list_all (\<lambda>x. x < i) ys \<and> is_path' r i ys k)" 

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proof 

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show "\<And>j. list_all (\<lambda>x. x < Suc i) xs \<Longrightarrow> is_path' r j xs k \<Longrightarrow> 

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\<not> list_all (\<lambda>x. x < i) xs \<Longrightarrow> 

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\<exists>ys. list_all (\<lambda>x. x < i) ys \<and> is_path' r j ys i" (is "PROP ?ih xs") 

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proof (induct xs) 

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case Nil 

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thus ?case by simp 

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next 

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case (Cons a as j) 

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show ?case 

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proof (cases "a=i") 

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case True 
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show ?thesis 
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proof 
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from True and Cons have "r j i = T" by simp 
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thus "list_all (\<lambda>x. x < i) [] \<and> is_path' r j [] i" by simp 
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qed 
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next 
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case False 
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have "PROP ?ih as" by (rule Cons) 
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then obtain ys where ys: "list_all (\<lambda>x. x < i) ys \<and> is_path' r a ys i" 
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proof 
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from Cons show "list_all (\<lambda>x. x < Suc i) as" by simp 
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from Cons show "is_path' r a as k" by simp 
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from Cons and False show "\<not> list_all (\<lambda>x. x < i) as" by (simp) 
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qed 
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show ?thesis 
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proof 
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from Cons False ys 
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show "list_all (\<lambda>x. x<i) (a#ys) \<and> is_path' r j (a#ys) i" by simp 
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qed 
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qed 
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qed 

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show "\<And>k. list_all (\<lambda>x. x < Suc i) xs \<Longrightarrow> is_path' r j xs k \<Longrightarrow> 

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\<not> list_all (\<lambda>x. x < i) xs \<Longrightarrow> 

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\<exists>ys. list_all (\<lambda>x. x < i) ys \<and> is_path' r i ys k" (is "PROP ?ih xs") 

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proof (induct xs rule: rev_induct) 

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case Nil 

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thus ?case by simp 

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next 

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case (snoc a as k) 

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show ?case 

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proof (cases "a=i") 

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case True 
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show ?thesis 
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proof 
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from True and snoc have "r i k = T" by simp 
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thus "list_all (\<lambda>x. x < i) [] \<and> is_path' r i [] k" by simp 
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qed 
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next 
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case False 
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have "PROP ?ih as" by (rule snoc) 
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then obtain ys where ys: "list_all (\<lambda>x. x < i) ys \<and> is_path' r i ys a" 
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proof 
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from snoc show "list_all (\<lambda>x. x < Suc i) as" by simp 
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from snoc show "is_path' r j as a" by simp 
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from snoc and False show "\<not> list_all (\<lambda>x. x < i) as" by simp 
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qed 
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show ?thesis 
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proof 
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from snoc False ys 
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show "list_all (\<lambda>x. x < i) (ys @ [a]) \<and> is_path' r i (ys @ [a]) k" 
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by simp 
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qed 
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qed 
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qed 

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qed (rule asms)+ 
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qed 
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theorem lemma5': 

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"\<And>p. is_path r p (Suc i) j k \<Longrightarrow> \<not> is_path r p i j k \<Longrightarrow> 

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\<not> (\<forall>q. \<not> is_path r q i j i) \<and> \<not> (\<forall>q'. \<not> is_path r q' i i k)" 

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by (iprover dest: lemma5) 
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theorem warshall: 

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"\<And>j k. \<not> (\<exists>p. is_path r p i j k) \<or> (\<exists>p. is_path r p i j k)" 

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proof (induct i) 

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case (0 j k) 

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show ?case 

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proof (cases "r j k") 

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assume "r j k = T" 

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hence "is_path r (j, [], k) 0 j k" 

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by (simp add: is_path_def) 

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hence "\<exists>p. is_path r p 0 j k" .. 

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thus ?thesis .. 

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next 

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assume "r j k = F" 

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hence "r j k ~= T" by simp 

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hence "\<not> (\<exists>p. is_path r p 0 j k)" 

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by (iprover dest: lemma2) 
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thus ?thesis .. 
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qed 

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next 

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case (Suc i j k) 

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thus ?case 

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proof 

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assume h1: "\<not> (\<exists>p. is_path r p i j k)" 

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from Suc show ?case 

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proof 

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assume "\<not> (\<exists>p. is_path r p i j i)" 

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with h1 have "\<not> (\<exists>p. is_path r p (Suc i) j k)" 

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by (iprover dest: lemma5') 
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thus ?case .. 
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next 

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assume "\<exists>p. is_path r p i j i" 

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then obtain p where h2: "is_path r p i j i" .. 

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from Suc show ?case 

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proof 

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assume "\<not> (\<exists>p. is_path r p i i k)" 
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with h1 have "\<not> (\<exists>p. is_path r p (Suc i) j k)" 
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by (iprover dest: lemma5') 
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thus ?case .. 
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next 
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assume "\<exists>q. is_path r q i i k" 
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then obtain q where "is_path r q i i k" .. 
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with h2 have "is_path r (conc p q) (Suc i) j k" 
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by (rule lemma3) 
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hence "\<exists>pq. is_path r pq (Suc i) j k" .. 
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thus ?case .. 
13405  240 
qed 
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qed 

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next 

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assume "\<exists>p. is_path r p i j k" 

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hence "\<exists>p. is_path r p (Suc i) j k" 

17604  245 
by (iprover intro: lemma1) 
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thus ?case .. 
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qed 

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qed 

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extract warshall 

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text {* 

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The program extracted from the above proof looks as follows 

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@{thm [display, eta_contract=false] warshall_def [no_vars]} 
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The corresponding correctness theorem is 
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@{thm [display] warshall_correctness [no_vars]} 

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*} 

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ML_val "@{code warshall}" 
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end 