
diff -r c8ff97d9f8da -r 7edbd5657702 thys/turing_hoare.thy
--- a/thys/turing_hoare.thy	Sun Jan 20 16:01:16 2013 +0000
+++ b/thys/turing_hoare.thy	Tue Jan 22 14:38:56 2013 +0000
@@ -2,11 +2,6 @@
 imports turing_basic
 begin
 
-declare step.simps[simp del]
-declare steps.simps[simp del]
-declare tm_comp.simps [simp del] adjust.simps[simp del] shift.simps[simp del]
-declare tm_wf.simps[simp del]
-
 
 type_synonym assert = "tape \<Rightarrow> bool"
 
@@ -20,9 +15,35 @@
 where
   "P holds_for (s, l, r) = P (l, r)"  
 
+(* halting case *)
+definition
+  Hoare :: "assert \<Rightarrow> tprog0 \<Rightarrow> assert \<Rightarrow> bool" ("({(1_)}/ (_)/ {(1_)})" [50, 49] 50)
+where
+  "{P} p {Q} \<equiv> 
+     (\<forall>tp. P tp \<longrightarrow> (\<exists>n. is_final (steps0 (1, tp) p n) \<and> Q holds_for (steps0 (1, tp) p n)))"
+
+(* not halting case *)
+definition
+  Hoare_unhalt :: "assert \<Rightarrow> tprog0 \<Rightarrow> bool" ("({(1_)}/ (_)) \<up>" [50, 49] 50)
+where
+  "{P} p \<up> \<equiv> (\<forall>tp. P tp \<longrightarrow> (\<forall> n . \<not> (is_final (steps0 (1, tp) p n))))"
+
+
+lemma HoareI:
+  assumes "\<And>l r. P (l, r) \<Longrightarrow> \<exists>n. is_final (steps0 (1, (l, r)) p n) \<and> Q holds_for (steps0 (1, (l, r)) p n)"
+  shows "{P} p {Q}"
+unfolding Hoare_def 
+using assms by auto
+
+lemma Hoare_unhalt_I:
+  assumes "\<And>l r n. P (l, r) \<Longrightarrow> \<not> is_final (steps0 (1, (l, r)) p n)"
+  shows "{P} p \<up>"
+unfolding Hoare_unhalt_def 
+using assms by auto
+
 lemma is_final_holds[simp]:
   assumes "is_final c"
-  shows "Q holds_for (steps c (p, off) n) = Q holds_for c"
+  shows "Q holds_for (steps c p n) = Q holds_for c"
 using assms 
 apply(induct n)
 apply(auto)
@@ -30,24 +51,6 @@
 apply(auto)
 done
 
-lemma holds_for_imp:
-  assumes "P holds_for c"
-  and "P \<mapsto> Q"
-  shows "Q holds_for c"
-using assms unfolding assert_imp_def 
-by (case_tac c) (auto)
-
-definition
-  Hoare :: "assert \<Rightarrow> tprog0 \<Rightarrow> assert \<Rightarrow> bool" ("({(1_)}/ (_)/ {(1_)})" 50)
-where
-  "{P} p {Q} \<equiv> 
-     (\<forall>l r. P (l, r) \<longrightarrow> (\<exists>n. is_final (steps0 (1, (l, r)) p n) \<and> Q holds_for (steps0 (1, (l, r)) p n)))"
-
-lemma HoareI:
-  assumes "\<And>l r. P (l, r) \<Longrightarrow> \<exists>n. is_final (steps0 (1, (l, r)) p n) \<and> Q holds_for (steps0 (1, (l, r)) p n)"
-  shows "{P} p {Q}"
-unfolding Hoare_def using assms by auto
-
 
 text {*
   {P1} A {Q1}   {P2} B {Q2}  Q1 \<mapsto> P2   A, B well-formed
@@ -57,118 +60,103 @@
 
 
 lemma Hoare_plus_halt: 
-  assumes aimpb: "Q1 \<mapsto> P2"
+  assumes a_imp: "Q1 \<mapsto> P2"
   and A_wf : "tm_wf (A, 0)"
-  and B_wf : "tm_wf (B, 0)"
   and A_halt : "{P1} A {Q1}"
   and B_halt : "{P2} B {Q2}"
   shows "{P1} A |+| B {Q2}"
 proof(rule HoareI)
   fix l r
   assume h: "P1 (l, r)"
-  then obtain n1 
-    where "is_final (steps0 (1, l, r) A n1)" and "Q1 holds_for (steps0 (1, l, r) A n1)"
-    using A_halt unfolding Hoare_def by auto
-  then obtain l' r' where "steps0 (1, l, r) A n1 = (0, l', r')" and c: "Q1 holds_for (0, l', r')"
-    by(case_tac "steps0 (1, l, r) A n1") (auto)
-  then obtain stpa where d: "steps0 (1, l, r) (A |+| B) stpa = (Suc (length A div 2), l', r')"
-    using A_wf by(rule_tac tm_comp_pre_halt_same) (auto)
+  then obtain n1 l' r' 
+    where "is_final (steps0 (1, l, r) A n1)"  
+      and a1: "Q1 holds_for (steps0 (1, l, r) A n1)"
+      and a2: "steps0 (1, l, r) A n1 = (0, l', r')"
+    using A_halt unfolding Hoare_def
+    by (metis is_final_eq surj_pair) 
+  then obtain n2 
+    where "steps0 (1, l, r) (A |+| B) n2 = (Suc (length A div 2), l', r')"
+    using A_wf by (rule_tac tm_comp_pre_halt_same) 
   moreover
-  from c aimpb have "P2 holds_for (0, l', r')"
-    by (rule holds_for_imp)
-  then have "P2 (l', r')" by auto
-  then obtain n2 
-    where "is_final (steps0 (1, l', r') B n2)" and "Q2 holds_for (steps0 (1, l', r') B n2)"
-    using B_halt unfolding Hoare_def by auto
-  then obtain l'' r'' where "steps0 (1, l', r') B n2 = (0, l'', r'')" and g: "Q2 holds_for (0, l'', r'')"
-    by (case_tac "steps0 (1, l', r') B n2") (auto)
-  then have "steps0 (Suc (length A div 2), l', r')  (A |+| B) n2 = (0, l'', r'')"
-    by (rule_tac t_merge_second_halt_same) (auto simp: A_wf B_wf)
+  from a1 a2 a_imp have "P2 (l', r')" by (simp add: assert_imp_def)
+  then obtain n3 l'' r''
+    where "is_final (steps0 (1, l', r') B n3)" 
+    and b1: "Q2 holds_for (steps0 (1, l', r') B n3)"
+    and b2: "steps0 (1, l', r') B n3 = (0, l'', r'')"
+    using B_halt unfolding Hoare_def 
+    by (metis is_final_eq surj_pair) 
+  then have "steps0 (Suc (length A div 2), l', r')  (A |+| B) n3 = (0, l'', r'')"
+    using A_wf by (rule_tac t_merge_second_halt_same) 
   ultimately show 
     "\<exists>n. is_final (steps0 (1, l, r) (A |+| B) n) \<and> Q2 holds_for (steps0 (1, l, r) (A |+| B) n)"
-    using g 
-    apply(rule_tac x = "stpa + n2" in exI)
-    apply(simp add: steps_add)
-    done
+    using b1 b2 by (rule_tac x = "n2 + n3" in exI) (simp)
 qed
 
-definition
-  Hoare_unhalt :: "assert \<Rightarrow> tprog0 \<Rightarrow> bool" ("({(1_)}/ (_))" 50)
-where
-  "{P} p \<equiv> (\<forall>l r. P (l, r) \<longrightarrow> (\<forall> n . \<not> (is_final (steps0 (1, (l, r)) p n))))"
 
-lemma Hoare_unhalt_I:
-  assumes "\<And>l r. P (l, r) \<Longrightarrow> \<forall> n. \<not> is_final (steps0 (1, (l, r)) p n)"
-  shows "{P} p"
-unfolding Hoare_unhalt_def using assms by auto
+text {*
+  {P1} A {Q1}   {P2} B loops    Q1 \<mapsto> P2   A well-formed
+  ------------------------------------------------------
+          {P1} A |+| B  loops
+*}
 
 lemma Hoare_plus_unhalt:
-  fixes A B :: tprog0 
-  assumes aimpb: "Q1 \<mapsto> P2"
+  assumes a_imp: "Q1 \<mapsto> P2"
   and A_wf : "tm_wf (A, 0)"
-  and B_wf : "tm_wf (B, 0)"  (* probably not needed *)
   and A_halt : "{P1} A {Q1}"
-  and B_uhalt : "{P2} B"
-  shows "{P1} (A |+| B)"
+  and B_uhalt : "{P2} B \<up>"
+  shows "{P1} (A |+| B) \<up>"
 proof(rule_tac Hoare_unhalt_I)
-  fix l r
+  fix n l r 
   assume h: "P1 (l, r)"
-  then obtain n1 where a: "is_final (steps0 (1, l, r) A n1)" and b: "Q1 holds_for (steps0 (1, l, r) A n1)"
-    using A_halt unfolding Hoare_def by auto
-  then obtain l' r' where "steps0 (1, l, r) A n1 = (0, l', r')" and c: "Q1 holds_for (0, l', r')"
-    by(case_tac "steps0 (1, l, r) A n1", auto)
-  then obtain stpa where d: "steps0 (1, l, r) (A |+| B) stpa = (Suc (length A div 2), l', r')"
-    using A_wf
-    by(rule_tac tm_comp_pre_halt_same, auto)
-  from c aimpb have "P2 holds_for (0, l', r')"
-    by(rule holds_for_imp)
-  from this have "P2 (l', r')" by auto
-  from this have e: "\<forall> n. \<not> is_final (steps0 (Suc 0, l', r') B n)  "
-    using B_uhalt unfolding Hoare_unhalt_def
-    by auto
-  from e show "\<forall>n. \<not> is_final (steps0 (1, l, r) (A |+| B) n)"
-  proof(rule_tac allI, case_tac "n > stpa")
-    fix n
-    assume h2: "stpa < n"
-    hence "\<not> is_final (steps0 (Suc 0, l', r') B (n - stpa))"
-      using e
-      apply(erule_tac x = "n - stpa" in allE) by simp
-    then obtain s'' l'' r'' where f: "steps0 (Suc 0, l', r') B (n - stpa) = (s'', l'', r'')" and g: "s'' \<noteq> 0"
-      apply(case_tac "steps0 (Suc 0, l', r') B (n - stpa)", auto)
-      done
-    have k: "steps0 (Suc (length A div 2), l', r') (A |+| B) (n - stpa) = (s''+ length A div 2, l'', r'') "
-      using A_wf B_wf f g
-      apply(drule_tac t_merge_second_same, auto)
-      done
-    show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"
-    proof -
-      have "\<not> is_final (steps0 (1, l, r) (A |+| B) (stpa + (n  - stpa)))"
-        using d k A_wf
-        apply(simp only: steps_add d, simp add: tm_wf.simps)
-        done
-      thus "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"
-        using h2 by simp
-    qed
-  next
-    fix n
-    assume h2: "\<not> stpa < n"
-    with d show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"
-      apply(auto)
-      apply(subgoal_tac "\<exists> d. stpa = n + d", auto)
-      apply(case_tac "(steps0 (Suc 0, l, r) (A |+| B) n)", simp)
-      apply(rule_tac x = "stpa - n" in exI, simp)
-      done
+  then obtain n1 l' r'
+    where a: "is_final (steps0 (1, l, r) A n1)" 
+    and b: "Q1 holds_for (steps0 (1, l, r) A n1)"
+    and c: "steps0 (1, l, r) A n1 = (0, l', r')"
+    using A_halt unfolding Hoare_def 
+    by (metis is_final_eq surj_pair) 
+  then obtain n2 where eq: "steps0 (1, l, r) (A |+| B) n2 = (Suc (length A div 2), l', r')"
+    using A_wf by (rule_tac tm_comp_pre_halt_same)
+  then show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"
+  proof(cases "n2 \<le> n")
+    case True
+    from b c a_imp have "P2 (l', r')" by (simp add: assert_imp_def)
+    then have "\<forall> n. \<not> is_final (steps0 (1, l', r') B n)  "
+      using B_uhalt unfolding Hoare_unhalt_def by simp
+    then have "\<not> is_final (steps0 (Suc 0, l', r') B (n - n2))" by auto
+    then obtain s'' l'' r'' 
+      where "steps0 (Suc 0, l', r') B (n - n2) = (s'', l'', r'')" 
+      and "\<not> is_final (s'', l'', r'')" by (metis surj_pair)
+    then have "steps0 (Suc (length A div 2), l', r') (A |+| B) (n - n2) = (s''+ length A div 2, l'', r'')"
+      using A_wf by (auto dest: t_merge_second_same simp del: steps.simps)
+    then have "\<not> is_final (steps0 (1, l, r) (A |+| B) (n2 + (n  - n2)))"
+      using A_wf by (simp only: steps_add eq) (simp add: tm_wf.simps)
+    then show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)" 
+      using `n2 \<le> n` by simp
+  next 
+    case False
+    then obtain n3 where "n = n2 - n3"
+      by (metis diff_le_self le_imp_diff_is_add nat_add_commute nat_le_linear)
+    moreover
+    with eq show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"
+      by (simp add: not_is_final[where ?n1.0="n2"])
   qed
 qed
 
 lemma Hoare_weak:
-  fixes p::tprog0
   assumes a: "{P} p {Q}"
   and b: "P' \<mapsto> P" 
   and c: "Q \<mapsto> Q'"
   shows "{P'} p {Q'}"
 using assms
 unfolding Hoare_def assert_imp_def
-by (blast intro: holds_for_imp[simplified assert_imp_def])
+by (metis holds_for.simps surj_pair)
+
+
+declare step.simps[simp del]
+declare steps.simps[simp del]
+declare tm_comp.simps [simp del] adjust.simps[simp del] shift.simps[simp del]
+declare tm_wf.simps[simp del]
+
+
 
 end
\ No newline at end of file




