--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys/turing_hoare.thy Sat Jan 19 14:44:07 2013 +0000
@@ -0,0 +1,173 @@
+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)"
+
+
+
+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, off) n) = Q holds_for c"
+using assms
+apply(induct n)
+apply(auto)
+apply(case_tac [!] c)
+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
+ -----------------------------------
+ {P1} A |+| B {Q2}
+*}
+
+
+lemma Hoare_plus_halt:
+ assumes aimpb: "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 t_merge_pre_halt_same) (auto)
+ 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)
+ 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
+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
+
+lemma Hoare_plus_unhalt:
+ fixes A B :: tprog0
+ assumes aimpb: "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_uhalt : "{P2} B"
+ shows "{P1} (A |+| B)"
+proof(rule_tac Hoare_unhalt_I)
+ fix 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 t_merge_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
+ 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])
+
+end
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