src/HOL/Proofs/Extraction/Warshall.thy
 changeset 39157 b98909faaea8 parent 37599 b8e3400dab19 child 51272 9c8d63b4b6be
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Proofs/Extraction/Warshall.thy	Mon Sep 06 14:18:16 2010 +0200
@@ -0,0 +1,261 @@
+(*  Title:      HOL/Proofs/Extraction/Warshall.thy
+    Author:     Stefan Berghofer, TU Muenchen
+*)
+
+
+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 (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 (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 (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)"
+  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"
+    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