--- a/src/HOL/Extraction/Warshall.thy Mon Sep 06 13:22:11 2010 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,261 +0,0 @@
-(* Title: HOL/Extraction/Warshall.thy
- Author: Stefan Berghofer, TU Muenchen
-*)
-
-header {* Warshall's algorithm *}
-
-theory Warshall
-imports Main
-begin
-
-text {*
- Derivation of Warshall's algorithm using program extraction,
- based on Berger, Schwichtenberg and Seisenberger \cite{Berger-JAR-2001}.
-*}
-
-datatype b = T | F
-
-primrec
- is_path' :: "('a \<Rightarrow> 'a \<Rightarrow> b) \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a \<Rightarrow> bool"
-where
- "is_path' r x [] z = (r x z = T)"
- | "is_path' r x (y # ys) z = (r x y = T \<and> is_path' r y ys z)"
-
-definition
- is_path :: "(nat \<Rightarrow> nat \<Rightarrow> b) \<Rightarrow> (nat * nat list * nat) \<Rightarrow>
- nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"
-where
- "is_path r p i j k \<longleftrightarrow> fst p = j \<and> snd (snd p) = k \<and>
- list_all (\<lambda>x. x < i) (fst (snd p)) \<and>
- is_path' r (fst p) (fst (snd p)) (snd (snd p))"
-
-definition
- conc :: "('a * 'a list * 'a) \<Rightarrow> ('a * 'a list * 'a) \<Rightarrow> ('a * 'a list * 'a)"
-where
- "conc p q = (fst p, fst (snd p) @ fst q # fst (snd q), snd (snd q))"
-
-theorem is_path'_snoc [simp]:
- "\<And>x. is_path' r x (ys @ [y]) z = (is_path' r x ys y \<and> r y z = T)"
- by (induct ys) simp+
-
-theorem list_all_scoc [simp]: "list_all P (xs @ [x]) \<longleftrightarrow> P x \<and> list_all P xs"
- by (induct xs, simp+, iprover)
-
-theorem list_all_lemma:
- "list_all P xs \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> list_all Q xs"
-proof -
- assume PQ: "\<And>x. P x \<Longrightarrow> Q x"
- show "list_all P xs \<Longrightarrow> list_all Q xs"
- proof (induct xs)
- case Nil
- show ?case by simp
- next
- case (Cons y ys)
- hence Py: "P y" by simp
- from Cons have Pys: "list_all P ys" by simp
- show ?case
- by simp (rule conjI PQ Py Cons Pys)+
- qed
-qed
-
-theorem lemma1: "\<And>p. is_path r p i j k \<Longrightarrow> is_path r p (Suc i) j k"
- apply (unfold is_path_def)
- apply (simp cong add: conj_cong add: split_paired_all)
- apply (erule conjE)+
- apply (erule list_all_lemma)
- apply simp
- done
-
-theorem lemma2: "\<And>p. is_path r p 0 j k \<Longrightarrow> r j k = T"
- apply (unfold is_path_def)
- apply (simp cong add: conj_cong add: split_paired_all)
- apply (case_tac "aa")
- apply simp+
- done
-
-theorem is_path'_conc: "is_path' r j xs i \<Longrightarrow> is_path' r i ys k \<Longrightarrow>
- is_path' r j (xs @ i # ys) k"
-proof -
- assume pys: "is_path' r i ys k"
- show "\<And>j. is_path' r j xs i \<Longrightarrow> is_path' r j (xs @ i # ys) k"
- proof (induct xs)
- case (Nil j)
- hence "r j i = T" by simp
- with pys show ?case by simp
- next
- case (Cons z zs j)
- hence jzr: "r j z = T" by simp
- from Cons have pzs: "is_path' r z zs i" by simp
- show ?case
- by simp (rule conjI jzr Cons pzs)+
- qed
-qed
-
-theorem lemma3:
- "\<And>p q. is_path r p i j i \<Longrightarrow> is_path r q i i k \<Longrightarrow>
- is_path r (conc p q) (Suc i) j k"
- apply (unfold is_path_def conc_def)
- apply (simp cong add: conj_cong add: split_paired_all)
- apply (erule conjE)+
- apply (rule conjI)
- apply (erule list_all_lemma)
- apply simp
- apply (rule conjI)
- apply (erule list_all_lemma)
- apply simp
- apply (rule is_path'_conc)
- apply assumption+
- done
-
-theorem lemma5:
- "\<And>p. is_path r p (Suc i) j k \<Longrightarrow> ~ is_path r p i j k \<Longrightarrow>
- (\<exists>q. is_path r q i j i) \<and> (\<exists>q'. is_path r q' i i k)"
-proof (simp cong add: conj_cong add: split_paired_all is_path_def, (erule conjE)+)
- fix xs
- assume asms:
- "list_all (\<lambda>x. x < Suc i) xs"
- "is_path' r j xs k"
- "\<not> list_all (\<lambda>x. x < i) xs"
- show "(\<exists>ys. list_all (\<lambda>x. x < i) ys \<and> is_path' r j ys i) \<and>
- (\<exists>ys. list_all (\<lambda>x. x < i) ys \<and> is_path' r i ys k)"
- proof
- show "\<And>j. list_all (\<lambda>x. x < Suc i) xs \<Longrightarrow> is_path' r j xs k \<Longrightarrow>
- \<not> list_all (\<lambda>x. x < i) xs \<Longrightarrow>
- \<exists>ys. list_all (\<lambda>x. x < i) ys \<and> is_path' r j ys i" (is "PROP ?ih xs")
- proof (induct xs)
- case Nil
- thus ?case by simp
- next
- case (Cons a as j)
- show ?case
- proof (cases "a=i")
- case True
- show ?thesis
- proof
- from True and Cons have "r j i = T" by simp
- thus "list_all (\<lambda>x. x < i) [] \<and> is_path' r j [] i" by simp
- qed
- next
- case False
- have "PROP ?ih as" by (rule Cons)
- then obtain ys where ys: "list_all (\<lambda>x. x < i) ys \<and> is_path' r a ys i"
- proof
- from Cons show "list_all (\<lambda>x. x < Suc i) as" by simp
- from Cons show "is_path' r a as k" by simp
- from Cons and False show "\<not> list_all (\<lambda>x. x < i) as" by (simp)
- qed
- show ?thesis
- proof
- from Cons False ys
- show "list_all (\<lambda>x. x<i) (a#ys) \<and> is_path' r j (a#ys) i" by simp
- qed
- qed
- qed
- show "\<And>k. list_all (\<lambda>x. x < Suc i) xs \<Longrightarrow> is_path' r j xs k \<Longrightarrow>
- \<not> list_all (\<lambda>x. x < i) xs \<Longrightarrow>
- \<exists>ys. list_all (\<lambda>x. x < i) ys \<and> is_path' r i ys k" (is "PROP ?ih xs")
- proof (induct xs rule: rev_induct)
- case Nil
- thus ?case by simp
- next
- case (snoc a as k)
- show ?case
- proof (cases "a=i")
- case True
- show ?thesis
- proof
- from True and snoc have "r i k = T" by simp
- thus "list_all (\<lambda>x. x < i) [] \<and> is_path' r i [] k" by simp
- qed
- next
- case False
- have "PROP ?ih as" by (rule snoc)
- then obtain ys where ys: "list_all (\<lambda>x. x < i) ys \<and> is_path' r i ys a"
- proof
- from snoc show "list_all (\<lambda>x. x < Suc i) as" by simp
- from snoc show "is_path' r j as a" by simp
- from snoc and False show "\<not> list_all (\<lambda>x. x < i) as" by simp
- qed
- show ?thesis
- proof
- from snoc False ys
- show "list_all (\<lambda>x. x < i) (ys @ [a]) \<and> is_path' r i (ys @ [a]) k"
- by simp
- qed
- qed
- qed
- qed (rule asms)+
-qed
-
-theorem lemma5':
- "\<And>p. is_path r p (Suc i) j k \<Longrightarrow> \<not> is_path r p i j k \<Longrightarrow>
- \<not> (\<forall>q. \<not> is_path r q i j i) \<and> \<not> (\<forall>q'. \<not> is_path r q' i i k)"
- by (iprover dest: lemma5)
-
-theorem warshall:
- "\<And>j k. \<not> (\<exists>p. is_path r p i j k) \<or> (\<exists>p. is_path r p i j k)"
-proof (induct i)
- case (0 j k)
- show ?case
- proof (cases "r j k")
- assume "r j k = T"
- hence "is_path r (j, [], k) 0 j k"
- by (simp add: is_path_def)
- hence "\<exists>p. is_path r p 0 j k" ..
- thus ?thesis ..
- next
- assume "r j k = F"
- hence "r j k ~= T" by simp
- hence "\<not> (\<exists>p. is_path r p 0 j k)"
- by (iprover dest: lemma2)
- thus ?thesis ..
- qed
-next
- case (Suc i j k)
- thus ?case
- proof
- assume h1: "\<not> (\<exists>p. is_path r p i j k)"
- from Suc show ?case
- proof
- assume "\<not> (\<exists>p. is_path r p i j i)"
- with h1 have "\<not> (\<exists>p. is_path r p (Suc i) j k)"
- by (iprover dest: lemma5')
- thus ?case ..
- next
- assume "\<exists>p. is_path r p i j i"
- then obtain p where h2: "is_path r p i j i" ..
- from Suc show ?case
- proof
- assume "\<not> (\<exists>p. is_path r p i i k)"
- with h1 have "\<not> (\<exists>p. is_path r p (Suc i) j k)"
- by (iprover dest: lemma5')
- thus ?case ..
- next
- assume "\<exists>q. is_path r q i i k"
- then obtain q where "is_path r q i i k" ..
- with h2 have "is_path r (conc p q) (Suc i) j k"
- by (rule lemma3)
- hence "\<exists>pq. is_path r pq (Suc i) j k" ..
- thus ?case ..
- qed
- qed
- next
- assume "\<exists>p. is_path r p i j k"
- hence "\<exists>p. is_path r p (Suc i) j k"
- by (iprover intro: lemma1)
- thus ?case ..
- qed
-qed
-
-extract warshall
-
-text {*
- The program extracted from the above proof looks as follows
- @{thm [display, eta_contract=false] warshall_def [no_vars]}
- The corresponding correctness theorem is
- @{thm [display] warshall_correctness [no_vars]}
-*}
-
-ML "@{code warshall}"
-
-end