--- a/src/HOL/IMP/Hoare.thy Thu Mar 11 17:52:15 2010 +0100
+++ b/src/HOL/IMP/Hoare.thy Thu Mar 11 19:06:03 2010 +0100
@@ -36,76 +36,62 @@
wrt denotational semantics
*)
-lemma hoare_conseq1: "[| !s. P' s --> P s; |- {P}c{Q} |] ==> |- {P'}c{Q}"
-apply (erule hoare.conseq)
-apply assumption
-apply fast
-done
+lemma strengthen_pre: "[| !s. P' s --> P s; |- {P}c{Q} |] ==> |- {P'}c{Q}"
+by (blast intro: conseq)
-lemma hoare_conseq2: "[| |- {P}c{Q}; !s. Q s --> Q' s |] ==> |- {P}c{Q'}"
-apply (rule hoare.conseq)
-prefer 2 apply (assumption)
-apply fast
-apply fast
-done
+lemma weaken_post: "[| |- {P}c{Q}; !s. Q s --> Q' s |] ==> |- {P}c{Q'}"
+by (blast intro: conseq)
lemma hoare_sound: "|- {P}c{Q} ==> |= {P}c{Q}"
-apply (unfold hoare_valid_def)
-apply (induct set: hoare)
- apply (simp_all (no_asm_simp))
- apply fast
- apply fast
-apply (rule allI, rule allI, rule impI)
-apply (erule lfp_induct2)
- apply (rule Gamma_mono)
-apply (unfold Gamma_def)
-apply fast
-done
+proof(induct rule: hoare.induct)
+ case (While P b c)
+ { fix s t
+ let ?G = "Gamma b (C c)"
+ assume "(s,t) \<in> lfp ?G"
+ hence "P s \<longrightarrow> P t \<and> \<not> b t"
+ proof(rule lfp_induct2)
+ show "mono ?G" by(rule Gamma_mono)
+ next
+ fix s t assume "(s,t) \<in> ?G (lfp ?G \<inter> {(s,t). P s \<longrightarrow> P t \<and> \<not> b t})"
+ thus "P s \<longrightarrow> P t \<and> \<not> b t" using While.hyps
+ by(auto simp: hoare_valid_def Gamma_def)
+ qed
+ }
+ thus ?case by(simp add:hoare_valid_def)
+qed (auto simp: hoare_valid_def)
+
lemma wp_SKIP: "wp \<SKIP> Q = Q"
-apply (unfold wp_def)
-apply (simp (no_asm))
-done
+by (simp add: wp_def)
lemma wp_Ass: "wp (x:==a) Q = (%s. Q(s[x\<mapsto>a s]))"
-apply (unfold wp_def)
-apply (simp (no_asm))
-done
+by (simp add: wp_def)
lemma wp_Semi: "wp (c;d) Q = wp c (wp d Q)"
-apply (unfold wp_def)
-apply (simp (no_asm))
-apply (rule ext)
-apply fast
-done
+by (rule ext) (auto simp: wp_def)
lemma wp_If:
"wp (\<IF> b \<THEN> c \<ELSE> d) Q = (%s. (b s --> wp c Q s) & (~b s --> wp d Q s))"
-apply (unfold wp_def)
-apply (simp (no_asm))
-apply (rule ext)
-apply fast
-done
+by (rule ext) (auto simp: wp_def)
-lemma wp_While_True:
- "b s ==> wp (\<WHILE> b \<DO> c) Q s = wp (c;\<WHILE> b \<DO> c) Q s"
-apply (unfold wp_def)
-apply (subst C_While_If)
-apply (simp (no_asm_simp))
-done
-
-lemma wp_While_False: "~b s ==> wp (\<WHILE> b \<DO> c) Q s = Q s"
-apply (unfold wp_def)
-apply (subst C_While_If)
-apply (simp (no_asm_simp))
-done
-
-lemmas [simp] = wp_SKIP wp_Ass wp_Semi wp_If wp_While_True wp_While_False
+lemma wp_While_If:
+ "wp (\<WHILE> b \<DO> c) Q s =
+ wp (IF b THEN c;\<WHILE> b \<DO> c ELSE SKIP) Q s"
+by(simp only: wp_def C_While_If)
(*Not suitable for rewriting: LOOPS!*)
lemma wp_While_if:
"wp (\<WHILE> b \<DO> c) Q s = (if b s then wp (c;\<WHILE> b \<DO> c) Q s else Q s)"
- by simp
+by(simp add:wp_While_If wp_If wp_SKIP)
+
+lemma wp_While_True: "b s ==>
+ wp (\<WHILE> b \<DO> c) Q s = wp (c;\<WHILE> b \<DO> c) Q s"
+by(simp add: wp_While_if)
+
+lemma wp_While_False: "~b s ==> wp (\<WHILE> b \<DO> c) Q s = Q s"
+by(simp add: wp_While_if)
+
+lemmas [simp] = wp_SKIP wp_Ass wp_Semi wp_If wp_While_True wp_While_False
lemma wp_While: "wp (\<WHILE> b \<DO> c) Q s =
(s : gfp(%S.{s. if b s then wp c (%s. s:S) s else Q s}))"
@@ -132,23 +118,48 @@
lemmas [intro!] = hoare.skip hoare.ass hoare.semi hoare.If
lemma wp_is_pre: "|- {wp c Q} c {Q}"
-apply (induct c arbitrary: Q)
- apply (simp_all (no_asm))
- apply fast+
- apply (blast intro: hoare_conseq1)
-apply (rule hoare_conseq2)
- apply (rule hoare.While)
- apply (rule hoare_conseq1)
- prefer 2 apply fast
- apply safe
- apply simp
-apply simp
-done
+proof(induct c arbitrary: Q)
+ case SKIP show ?case by auto
+next
+ case Assign show ?case by auto
+next
+ case Semi thus ?case by auto
+next
+ case (Cond b c1 c2)
+ let ?If = "IF b THEN c1 ELSE c2"
+ show ?case
+ proof(rule If)
+ show "|- {\<lambda>s. wp ?If Q s \<and> b s} c1 {Q}"
+ proof(rule strengthen_pre[OF _ Cond(1)])
+ show "\<forall>s. wp ?If Q s \<and> b s \<longrightarrow> wp c1 Q s" by auto
+ qed
+ show "|- {\<lambda>s. wp ?If Q s \<and> \<not> b s} c2 {Q}"
+ proof(rule strengthen_pre[OF _ Cond(2)])
+ show "\<forall>s. wp ?If Q s \<and> \<not> b s \<longrightarrow> wp c2 Q s" by auto
+ qed
+ qed
+next
+ case (While b c)
+ let ?w = "WHILE b DO c"
+ have "|- {wp ?w Q} ?w {\<lambda>s. wp ?w Q s \<and> \<not> b s}"
+ proof(rule hoare.While)
+ show "|- {\<lambda>s. wp ?w Q s \<and> b s} c {wp ?w Q}"
+ proof(rule strengthen_pre[OF _ While(1)])
+ show "\<forall>s. wp ?w Q s \<and> b s \<longrightarrow> wp c (wp ?w Q) s" by auto
+ qed
+ qed
+ thus ?case
+ proof(rule weaken_post)
+ show "\<forall>s. wp ?w Q s \<and> \<not> b s \<longrightarrow> Q s" by auto
+ qed
+qed
-lemma hoare_relative_complete: "|= {P}c{Q} ==> |- {P}c{Q}"
-apply (rule hoare_conseq1 [OF _ wp_is_pre])
-apply (unfold hoare_valid_def wp_def)
-apply fast
-done
+lemma hoare_relative_complete: assumes "|= {P}c{Q}" shows "|- {P}c{Q}"
+proof(rule conseq)
+ show "\<forall>s. P s \<longrightarrow> wp c Q s" using assms
+ by (auto simp: hoare_valid_def wp_def)
+ show "|- {wp c Q} c {Q}" by(rule wp_is_pre)
+ show "\<forall>s. Q s \<longrightarrow> Q s" by auto
+qed
end