--- a/thys/turing_basic.thy Sat Jan 19 14:29:56 2013 +0000
+++ b/thys/turing_basic.thy Sat Jan 19 14:44:07 2013 +0000
@@ -147,6 +147,14 @@
where
"tm_comp p1 p2 = ((adjust p1) @ (shift p2 (length p1 div 2)))"
+lemma step_0 [simp]:
+ shows "step (0, (l, r)) p = (0, (l, r))"
+by (case_tac p, simp)
+
+lemma steps_0 [simp]:
+ shows "steps (0, (l, r)) p n = (0, (l, r))"
+by (induct n) (simp_all)
+
fun
is_final :: "config \<Rightarrow> bool"
where
@@ -157,54 +165,7 @@
shows "is_final (steps (s, l, r) (p, off) n)"
using assms by (induct n) (auto)
-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
-
-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 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
-
-
-lemma step_0 [simp]:
- shows "step (0, (l, r)) p = (0, (l, r))"
-by (case_tac p, simp)
-
-lemma steps_0 [simp]:
- shows "steps (0, (l, r)) p n = (0, (l, r))"
-by (induct n) (simp_all)
(* if the machine is in the halting state, there must have
been a state just before the halting state *)
@@ -480,119 +441,6 @@
apply(auto)
done
-
-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
end