--- a/src/HOL/IMP/VC.thy Fri May 31 07:30:23 2013 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,146 +0,0 @@
-(* Author: Tobias Nipkow *)
-
-theory VC imports Hoare begin
-
-subsection "Verification Conditions"
-
-text{* Annotated commands: commands where loops are annotated with
-invariants. *}
-
-datatype acom =
- Askip ("SKIP") |
- Aassign vname aexp ("(_ ::= _)" [1000, 61] 61) |
- Aseq acom acom ("_;;/ _" [60, 61] 60) |
- Aif bexp acom acom ("(IF _/ THEN _/ ELSE _)" [0, 0, 61] 61) |
- Awhile assn bexp acom ("({_}/ WHILE _/ DO _)" [0, 0, 61] 61)
-
-text{* Weakest precondition from annotated commands: *}
-
-fun pre :: "acom \<Rightarrow> assn \<Rightarrow> assn" where
-"pre Askip Q = Q" |
-"pre (Aassign x a) Q = (\<lambda>s. Q(s(x := aval a s)))" |
-"pre (Aseq c\<^isub>1 c\<^isub>2) Q = pre c\<^isub>1 (pre c\<^isub>2 Q)" |
-"pre (Aif b c\<^isub>1 c\<^isub>2) Q =
- (\<lambda>s. (bval b s \<longrightarrow> pre c\<^isub>1 Q s) \<and>
- (\<not> bval b s \<longrightarrow> pre c\<^isub>2 Q s))" |
-"pre (Awhile I b c) Q = I"
-
-text{* Verification condition: *}
-
-fun vc :: "acom \<Rightarrow> assn \<Rightarrow> assn" where
-"vc Askip Q = (\<lambda>s. True)" |
-"vc (Aassign x a) Q = (\<lambda>s. True)" |
-"vc (Aseq c\<^isub>1 c\<^isub>2) Q = (\<lambda>s. vc c\<^isub>1 (pre c\<^isub>2 Q) s \<and> vc c\<^isub>2 Q s)" |
-"vc (Aif b c\<^isub>1 c\<^isub>2) Q = (\<lambda>s. vc c\<^isub>1 Q s \<and> vc c\<^isub>2 Q s)" |
-"vc (Awhile I b c) Q =
- (\<lambda>s. (I s \<and> \<not> bval b s \<longrightarrow> Q s) \<and>
- (I s \<and> bval b s \<longrightarrow> pre c I s) \<and>
- vc c I s)"
-
-text{* Strip annotations: *}
-
-fun strip :: "acom \<Rightarrow> com" where
-"strip Askip = com.SKIP" |
-"strip (Aassign x a) = (x::=a)" |
-"strip (Aseq c\<^isub>1 c\<^isub>2) = (strip c\<^isub>1;; strip c\<^isub>2)" |
-"strip (Aif b c\<^isub>1 c\<^isub>2) = (IF b THEN strip c\<^isub>1 ELSE strip c\<^isub>2)" |
-"strip (Awhile I b c) = (WHILE b DO strip c)"
-
-text {* Soundness: *}
-
-lemma vc_sound: "\<forall>s. vc c Q s \<Longrightarrow> \<turnstile> {pre c Q} strip c {Q}"
-proof(induction c arbitrary: Q)
- case (Awhile I b c)
- show ?case
- proof(simp, rule While')
- from `\<forall>s. vc (Awhile I b c) Q s`
- have vc: "\<forall>s. vc c I s" and IQ: "\<forall>s. I s \<and> \<not> bval b s \<longrightarrow> Q s" and
- pre: "\<forall>s. I s \<and> bval b s \<longrightarrow> pre c I s" by simp_all
- have "\<turnstile> {pre c I} strip c {I}" by(rule Awhile.IH[OF vc])
- with pre show "\<turnstile> {\<lambda>s. I s \<and> bval b s} strip c {I}"
- by(rule strengthen_pre)
- show "\<forall>s. I s \<and> \<not>bval b s \<longrightarrow> Q s" by(rule IQ)
- qed
-qed (auto intro: hoare.conseq)
-
-corollary vc_sound':
- "(\<forall>s. vc c Q s) \<and> (\<forall>s. P s \<longrightarrow> pre c Q s) \<Longrightarrow> \<turnstile> {P} strip c {Q}"
-by (metis strengthen_pre vc_sound)
-
-
-text{* Completeness: *}
-
-lemma pre_mono:
- "\<forall>s. P s \<longrightarrow> P' s \<Longrightarrow> pre c P s \<Longrightarrow> pre c P' s"
-proof (induction c arbitrary: P P' s)
- case Aseq thus ?case by simp metis
-qed simp_all
-
-lemma vc_mono:
- "\<forall>s. P s \<longrightarrow> P' s \<Longrightarrow> vc c P s \<Longrightarrow> vc c P' s"
-proof(induction c arbitrary: P P')
- case Aseq thus ?case by simp (metis pre_mono)
-qed simp_all
-
-lemma vc_complete:
- "\<turnstile> {P}c{Q} \<Longrightarrow> \<exists>c'. strip c' = c \<and> (\<forall>s. vc c' Q s) \<and> (\<forall>s. P s \<longrightarrow> pre c' Q s)"
- (is "_ \<Longrightarrow> \<exists>c'. ?G P c Q c'")
-proof (induction rule: hoare.induct)
- case Skip
- show ?case (is "\<exists>ac. ?C ac")
- proof show "?C Askip" by simp qed
-next
- case (Assign P a x)
- show ?case (is "\<exists>ac. ?C ac")
- proof show "?C(Aassign x a)" by simp qed
-next
- case (Seq P c1 Q c2 R)
- from Seq.IH obtain ac1 where ih1: "?G P c1 Q ac1" by blast
- from Seq.IH obtain ac2 where ih2: "?G Q c2 R ac2" by blast
- show ?case (is "\<exists>ac. ?C ac")
- proof
- show "?C(Aseq ac1 ac2)"
- using ih1 ih2 by (fastforce elim!: pre_mono vc_mono)
- qed
-next
- case (If P b c1 Q c2)
- from If.IH obtain ac1 where ih1: "?G (\<lambda>s. P s \<and> bval b s) c1 Q ac1"
- by blast
- from If.IH obtain ac2 where ih2: "?G (\<lambda>s. P s \<and> \<not>bval b s) c2 Q ac2"
- by blast
- show ?case (is "\<exists>ac. ?C ac")
- proof
- show "?C(Aif b ac1 ac2)" using ih1 ih2 by simp
- qed
-next
- case (While P b c)
- from While.IH obtain ac where ih: "?G (\<lambda>s. P s \<and> bval b s) c P ac" by blast
- show ?case (is "\<exists>ac. ?C ac")
- proof show "?C(Awhile P b ac)" using ih by simp qed
-next
- case conseq thus ?case by(fast elim!: pre_mono vc_mono)
-qed
-
-
-text{* An Optimization: *}
-
-fun vcpre :: "acom \<Rightarrow> assn \<Rightarrow> assn \<times> assn" where
-"vcpre Askip Q = (\<lambda>s. True, Q)" |
-"vcpre (Aassign x a) Q = (\<lambda>s. True, \<lambda>s. Q(s[a/x]))" |
-"vcpre (Aseq c\<^isub>1 c\<^isub>2) Q =
- (let (vc\<^isub>2,wp\<^isub>2) = vcpre c\<^isub>2 Q;
- (vc\<^isub>1,wp\<^isub>1) = vcpre c\<^isub>1 wp\<^isub>2
- in (\<lambda>s. vc\<^isub>1 s \<and> vc\<^isub>2 s, wp\<^isub>1))" |
-"vcpre (Aif b c\<^isub>1 c\<^isub>2) Q =
- (let (vc\<^isub>2,wp\<^isub>2) = vcpre c\<^isub>2 Q;
- (vc\<^isub>1,wp\<^isub>1) = vcpre c\<^isub>1 Q
- in (\<lambda>s. vc\<^isub>1 s \<and> vc\<^isub>2 s, \<lambda>s. (bval b s \<longrightarrow> wp\<^isub>1 s) \<and> (\<not>bval b s \<longrightarrow> wp\<^isub>2 s)))" |
-"vcpre (Awhile I b c) Q =
- (let (vcc,wpc) = vcpre c I
- in (\<lambda>s. (I s \<and> \<not> bval b s \<longrightarrow> Q s) \<and>
- (I s \<and> bval b s \<longrightarrow> wpc s) \<and> vcc s, I))"
-
-lemma vcpre_vc_pre: "vcpre c Q = (vc c Q, pre c Q)"
-by (induct c arbitrary: Q) (simp_all add: Let_def)
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/IMP/VCG.thy Fri May 31 07:55:40 2013 +0200
@@ -0,0 +1,126 @@
+(* Author: Tobias Nipkow *)
+
+theory VCG imports Hoare begin
+
+subsection "Verification Conditions"
+
+text{* Annotated commands: commands where loops are annotated with
+invariants. *}
+
+datatype acom =
+ Askip ("SKIP") |
+ Aassign vname aexp ("(_ ::= _)" [1000, 61] 61) |
+ Aseq acom acom ("_;;/ _" [60, 61] 60) |
+ Aif bexp acom acom ("(IF _/ THEN _/ ELSE _)" [0, 0, 61] 61) |
+ Awhile assn bexp acom ("({_}/ WHILE _/ DO _)" [0, 0, 61] 61)
+
+
+text{* Strip annotations: *}
+
+fun strip :: "acom \<Rightarrow> com" where
+"strip SKIP = com.SKIP" |
+"strip (x ::= a) = (x ::= a)" |
+"strip (c\<^isub>1;; c\<^isub>2) = (strip c\<^isub>1;; strip c\<^isub>2)" |
+"strip (IF b THEN c\<^isub>1 ELSE c\<^isub>2) = (IF b THEN strip c\<^isub>1 ELSE strip c\<^isub>2)" |
+"strip ({_} WHILE b DO c) = (WHILE b DO strip c)"
+
+text{* Weakest precondition from annotated commands: *}
+
+fun pre :: "acom \<Rightarrow> assn \<Rightarrow> assn" where
+"pre SKIP Q = Q" |
+"pre (x ::= a) Q = (\<lambda>s. Q(s(x := aval a s)))" |
+"pre (c\<^isub>1;; c\<^isub>2) Q = pre c\<^isub>1 (pre c\<^isub>2 Q)" |
+"pre (IF b THEN c\<^isub>1 ELSE c\<^isub>2) Q =
+ (\<lambda>s. (bval b s \<longrightarrow> pre c\<^isub>1 Q s) \<and>
+ (\<not> bval b s \<longrightarrow> pre c\<^isub>2 Q s))" |
+"pre ({I} WHILE b DO c) Q = I"
+
+text{* Verification condition: *}
+
+fun vc :: "acom \<Rightarrow> assn \<Rightarrow> assn" where
+"vc SKIP Q = (\<lambda>s. True)" |
+"vc (x ::= a) Q = (\<lambda>s. True)" |
+"vc (c\<^isub>1;; c\<^isub>2) Q = (\<lambda>s. vc c\<^isub>1 (pre c\<^isub>2 Q) s \<and> vc c\<^isub>2 Q s)" |
+"vc (IF b THEN c\<^isub>1 ELSE c\<^isub>2) Q = (\<lambda>s. vc c\<^isub>1 Q s \<and> vc c\<^isub>2 Q s)" |
+"vc ({I} WHILE b DO c) Q =
+ (\<lambda>s. (I s \<and> \<not> bval b s \<longrightarrow> Q s) \<and>
+ (I s \<and> bval b s \<longrightarrow> pre c I s) \<and>
+ vc c I s)"
+
+
+text {* Soundness: *}
+
+lemma vc_sound: "\<forall>s. vc c Q s \<Longrightarrow> \<turnstile> {pre c Q} strip c {Q}"
+proof(induction c arbitrary: Q)
+ case (Awhile I b c)
+ show ?case
+ proof(simp, rule While')
+ from `\<forall>s. vc (Awhile I b c) Q s`
+ have vc: "\<forall>s. vc c I s" and IQ: "\<forall>s. I s \<and> \<not> bval b s \<longrightarrow> Q s" and
+ pre: "\<forall>s. I s \<and> bval b s \<longrightarrow> pre c I s" by simp_all
+ have "\<turnstile> {pre c I} strip c {I}" by(rule Awhile.IH[OF vc])
+ with pre show "\<turnstile> {\<lambda>s. I s \<and> bval b s} strip c {I}"
+ by(rule strengthen_pre)
+ show "\<forall>s. I s \<and> \<not>bval b s \<longrightarrow> Q s" by(rule IQ)
+ qed
+qed (auto intro: hoare.conseq)
+
+corollary vc_sound':
+ "(\<forall>s. vc c Q s) \<and> (\<forall>s. P s \<longrightarrow> pre c Q s) \<Longrightarrow> \<turnstile> {P} strip c {Q}"
+by (metis strengthen_pre vc_sound)
+
+
+text{* Completeness: *}
+
+lemma pre_mono:
+ "\<forall>s. P s \<longrightarrow> P' s \<Longrightarrow> pre c P s \<Longrightarrow> pre c P' s"
+proof (induction c arbitrary: P P' s)
+ case Aseq thus ?case by simp metis
+qed simp_all
+
+lemma vc_mono:
+ "\<forall>s. P s \<longrightarrow> P' s \<Longrightarrow> vc c P s \<Longrightarrow> vc c P' s"
+proof(induction c arbitrary: P P')
+ case Aseq thus ?case by simp (metis pre_mono)
+qed simp_all
+
+lemma vc_complete:
+ "\<turnstile> {P}c{Q} \<Longrightarrow> \<exists>c'. strip c' = c \<and> (\<forall>s. vc c' Q s) \<and> (\<forall>s. P s \<longrightarrow> pre c' Q s)"
+ (is "_ \<Longrightarrow> \<exists>c'. ?G P c Q c'")
+proof (induction rule: hoare.induct)
+ case Skip
+ show ?case (is "\<exists>ac. ?C ac")
+ proof show "?C Askip" by simp qed
+next
+ case (Assign P a x)
+ show ?case (is "\<exists>ac. ?C ac")
+ proof show "?C(Aassign x a)" by simp qed
+next
+ case (Seq P c1 Q c2 R)
+ from Seq.IH obtain ac1 where ih1: "?G P c1 Q ac1" by blast
+ from Seq.IH obtain ac2 where ih2: "?G Q c2 R ac2" by blast
+ show ?case (is "\<exists>ac. ?C ac")
+ proof
+ show "?C(Aseq ac1 ac2)"
+ using ih1 ih2 by (fastforce elim!: pre_mono vc_mono)
+ qed
+next
+ case (If P b c1 Q c2)
+ from If.IH obtain ac1 where ih1: "?G (\<lambda>s. P s \<and> bval b s) c1 Q ac1"
+ by blast
+ from If.IH obtain ac2 where ih2: "?G (\<lambda>s. P s \<and> \<not>bval b s) c2 Q ac2"
+ by blast
+ show ?case (is "\<exists>ac. ?C ac")
+ proof
+ show "?C(Aif b ac1 ac2)" using ih1 ih2 by simp
+ qed
+next
+ case (While P b c)
+ from While.IH obtain ac where ih: "?G (\<lambda>s. P s \<and> bval b s) c P ac" by blast
+ show ?case (is "\<exists>ac. ?C ac")
+ proof show "?C(Awhile P b ac)" using ih by simp qed
+next
+ case conseq thus ?case by(fast elim!: pre_mono vc_mono)
+qed
+
+end
--- a/src/HOL/ROOT Fri May 31 07:30:23 2013 +0200
+++ b/src/HOL/ROOT Fri May 31 07:55:40 2013 +0200
@@ -129,7 +129,7 @@
Live
Live_True
Hoare_Examples
- VC
+ VCG
HoareT
Collecting1
Collecting_Examples