thys/turing_hoare.thy
changeset 163 67063c5365e1
parent 162 a63c3f8d7234
child 164 8a3e63163910
--- a/thys/turing_hoare.thy	Thu Feb 07 06:39:06 2013 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,159 +0,0 @@
-theory turing_hoare
-imports turing_basic
-begin
-
-
-type_synonym assert = "tape \<Rightarrow> bool"
-
-definition 
-  assert_imp :: "assert \<Rightarrow> assert \<Rightarrow> bool" ("_ \<mapsto> _" [0, 0] 100)
-where
-  "P \<mapsto> Q \<equiv> \<forall>l r. P (l, r) \<longrightarrow> Q (l, r)"
-
-lemma [intro, simp]:
-  "P \<mapsto> P"
-unfolding assert_imp_def by simp
-
-fun 
-  holds_for :: "(tape \<Rightarrow> bool) \<Rightarrow> config \<Rightarrow> bool" ("_ holds'_for _" [100, 99] 100)
-where
-  "P holds_for (s, l, r) = P (l, r)"  
-
-lemma is_final_holds[simp]:
-  assumes "is_final c"
-  shows "Q holds_for (steps c p n) = Q holds_for c"
-using assms 
-apply(induct n)
-apply(auto)
-apply(case_tac [!] c)
-apply(auto)
-done
-
-(* Hoare Rules *)
-
-(* halting case *)
-definition
-  Hoare_halt :: "assert \<Rightarrow> tprog0 \<Rightarrow> assert \<Rightarrow> bool" ("({(1_)}/ (_)/ {(1_)})" 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)
-where
-  "{P} p \<up> \<equiv> \<forall>tp. P tp \<longrightarrow> (\<forall> n . \<not> (is_final (steps0 (1, tp) p n)))"
-
-
-lemma Hoare_haltI:
-  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_halt_def 
-using assms by auto
-
-lemma Hoare_unhaltI:
-  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
-
-
-
-
-text {*
-  {P} A {Q}   {Q} B {S}  A well-formed
-  -----------------------------------------
-  {P} A |+| B {S}
-*}
-
-
-lemma Hoare_plus_halt [case_names A_halt B_halt A_wf]: 
-  assumes A_halt : "{P} A {Q}"
-  and B_halt : "{Q} B {S}"
-  and A_wf : "tm_wf (A, 0)"
-  shows "{P} A |+| B {S}"
-proof(rule Hoare_haltI)
-  fix l r
-  assume h: "P (l, r)"
-  then obtain n1 l' r' 
-    where "is_final (steps0 (1, l, r) A n1)"  
-      and a1: "Q holds_for (steps0 (1, l, r) A n1)"
-      and a2: "steps0 (1, l, r) A n1 = (0, l', r')"
-    using A_halt unfolding Hoare_halt_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 a1 a2 have "Q (l', r')" by (simp)
-  then obtain n3 l'' r''
-    where "is_final (steps0 (1, l', r') B n3)" 
-    and b1: "S holds_for (steps0 (1, l', r') B n3)"
-    and b2: "steps0 (1, l', r') B n3 = (0, l'', r'')"
-    using B_halt unfolding Hoare_halt_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 tm_comp_second_halt_same) 
-  ultimately show 
-    "\<exists>n. is_final (steps0 (1, l, r) (A |+| B) n) \<and> S holds_for (steps0 (1, l, r) (A |+| B) n)"
-    using b1 b2 by (rule_tac x = "n2 + n3" in exI) (simp)
-qed
-
-text {*
-  {P} A {Q}   {Q} B loops   A well-formed
-  ------------------------------------------
-          {P} A |+| B  loops
-*}
-
-lemma Hoare_plus_unhalt [case_names A_halt B_unhalt A_wf]:
-  assumes A_halt: "{P} A {Q}"
-  and B_uhalt: "{Q} B \<up>"
-  and A_wf : "tm_wf (A, 0)"
-  shows "{P} (A |+| B) \<up>"
-proof(rule_tac Hoare_unhaltI)
-  fix n l r 
-  assume h: "P (l, r)"
-  then obtain n1 l' r'
-    where a: "is_final (steps0 (1, l, r) A n1)" 
-    and b: "Q holds_for (steps0 (1, l, r) A n1)"
-    and c: "steps0 (1, l, r) A n1 = (0, l', r')"
-    using A_halt unfolding Hoare_halt_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 have "Q (l', r')" by simp
-    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 (1, l', r') B (n - n2))" by auto
-    then obtain s'' l'' r'' 
-      where "steps0 (1, 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: tm_comp_second_same simp del: tm_wf.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_consequence:
-  assumes "P' \<mapsto> P" "{P} p {Q}" "Q \<mapsto> Q'"
-  shows "{P'} p {Q'}"
-using assms
-unfolding Hoare_halt_def assert_imp_def
-by (metis holds_for.simps surj_pair)
-
-
-
-end
\ No newline at end of file