avoid session qualification because no tex is generated when used;
tuned sectioning
(* Author: Tobias Nipkow *)
subsection "Verification Condition Generation"
theory VCG imports Hoare begin
subsubsection "Annotated Commands"
text\<open>Commands where loops are annotated with invariants.\<close>
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)
notation com.SKIP ("SKIP")
text\<open>Strip annotations:\<close>
fun strip :: "acom \<Rightarrow> com" where
"strip SKIP = SKIP" |
"strip (x ::= a) = (x ::= a)" |
"strip (C\<^sub>1;; C\<^sub>2) = (strip C\<^sub>1;; strip C\<^sub>2)" |
"strip (IF b THEN C\<^sub>1 ELSE C\<^sub>2) = (IF b THEN strip C\<^sub>1 ELSE strip C\<^sub>2)" |
"strip ({_} WHILE b DO C) = (WHILE b DO strip C)"
subsubsection "Weeakest Precondistion and Verification Condition"
text\<open>Weakest precondition:\<close>
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\<^sub>1;; C\<^sub>2) Q = pre C\<^sub>1 (pre C\<^sub>2 Q)" |
"pre (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q =
(\<lambda>s. if bval b s then pre C\<^sub>1 Q s else pre C\<^sub>2 Q s)" |
"pre ({I} WHILE b DO C) Q = I"
text\<open>Verification condition:\<close>
fun vc :: "acom \<Rightarrow> assn \<Rightarrow> bool" where
"vc SKIP Q = True" |
"vc (x ::= a) Q = True" |
"vc (C\<^sub>1;; C\<^sub>2) Q = (vc C\<^sub>1 (pre C\<^sub>2 Q) \<and> vc C\<^sub>2 Q)" |
"vc (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q = (vc C\<^sub>1 Q \<and> vc C\<^sub>2 Q)" |
"vc ({I} WHILE b DO C) Q =
((\<forall>s. (I s \<and> bval b s \<longrightarrow> pre C I s) \<and>
(I s \<and> \<not> bval b s \<longrightarrow> Q s)) \<and>
vc C I)"
subsubsection "Soundness"
lemma vc_sound: "vc C Q \<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 \<open>vc (Awhile I b C) Q\<close>
have vc: "vc C I" 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':
"\<lbrakk> vc C Q; \<forall>s. P s \<longrightarrow> pre C Q s \<rbrakk> \<Longrightarrow> \<turnstile> {P} strip C {Q}"
by (metis strengthen_pre vc_sound)
subsubsection "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 \<Longrightarrow> vc C P'"
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> vc C Q \<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>C. ?C C")
proof show "?C Askip" by simp qed
next
case (Assign P a x)
show ?case (is "\<exists>C. ?C C")
proof show "?C(Aassign x a)" by simp qed
next
case (Seq P c1 Q c2 R)
from Seq.IH obtain C1 where ih1: "?G P c1 Q C1" by blast
from Seq.IH obtain C2 where ih2: "?G Q c2 R C2" by blast
show ?case (is "\<exists>C. ?C C")
proof
show "?C(Aseq C1 C2)"
using ih1 ih2 by (fastforce elim!: pre_mono vc_mono)
qed
next
case (If P b c1 Q c2)
from If.IH obtain C1 where ih1: "?G (\<lambda>s. P s \<and> bval b s) c1 Q C1"
by blast
from If.IH obtain C2 where ih2: "?G (\<lambda>s. P s \<and> \<not>bval b s) c2 Q C2"
by blast
show ?case (is "\<exists>C. ?C C")
proof
show "?C(Aif b C1 C2)" using ih1 ih2 by simp
qed
next
case (While P b c)
from While.IH obtain C where ih: "?G (\<lambda>s. P s \<and> bval b s) c P C" by blast
show ?case (is "\<exists>C. ?C C")
proof show "?C(Awhile P b C)" using ih by simp qed
next
case conseq thus ?case by(fast elim!: pre_mono vc_mono)
qed
end