(* Author: Tobias Nipkow *)
subsubsection "VCG for Total Correctness With Logical Variables"
theory VCG_Total_EX2
imports Hoare_Total_EX2
begin
text \<open>
Theory \<open>VCG_Total_EX\<close> conatins a VCG built on top of a Hoare logic without logical variables.
As a result the completeness proof runs into a problem. This theory uses a Hoare logic
with logical variables and proves soundness and completeness.
\<close>
text\<open>Annotated commands: 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 assn2 lvname bexp acom
("({_'/_}/ WHILE _/ DO _)" [0, 0, 0, 61] 61)
notation com.SKIP ("SKIP")
text\<open>Strip annotations:\<close>
fun strip :: "acom \ 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)"
text\<open>Weakest precondition from annotated commands:\<close>
fun pre :: "acom \ assn2 \ assn2" where
"pre SKIP Q = Q" |
"pre (x ::= a) Q = (\l s. Q l (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>l s. if bval b s then pre C\<^sub>1 Q l s else pre C\<^sub>2 Q l s)" |
"pre ({I/x} WHILE b DO C) Q = (\l s. \n. I (l(x:=n)) s)"
text\<open>Verification condition:\<close>
fun vc :: "acom \ assn2 \ 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) \ vc C\<^sub>2 Q)" |
"vc (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q = (vc C\<^sub>1 Q \ vc C\<^sub>2 Q)" |
"vc ({I/x} WHILE b DO C) Q =
(\<forall>l s. (I (l(x:=Suc(l x))) s \<longrightarrow> pre C I l s) \<and>
(l x > 0 \<and> I l s \<longrightarrow> bval b s) \<and>
(I (l(x := 0)) s \<longrightarrow> \<not> bval b s \<and> Q l s) \<and>
vc C I)"
lemma vc_sound: "vc C Q \ \\<^sub>t {pre C Q} strip C {Q}"
proof(induction C arbitrary: Q)
case (Awhile I x b C)
show ?case
proof(simp, rule weaken_post[OF While[of I x]], goal_cases)
case 1 show ?case
using Awhile.IH[of "I"] Awhile.prems by (auto intro: strengthen_pre)
next
case 3 show ?case
using Awhile.prems by (simp) (metis fun_upd_triv)
qed (insert Awhile.prems, auto)
qed (auto intro: conseq Seq If simp: Skip Assign)
text\<open>Completeness:\<close>
lemma pre_mono:
"\l s. P l s \ P' l s \ pre C P l s \ pre C P' l s"
proof (induction C arbitrary: P P' l s)
case Aseq thus ?case by simp metis
qed simp_all
lemma vc_mono:
"\l s. P l s \ P' l s \ vc C P \ vc C P'"
proof(induction C arbitrary: P P')
case Aseq thus ?case by simp (metis pre_mono)
qed simp_all
lemma vc_complete:
"\\<^sub>t {P}c{Q} \ \C. strip C = c \ vc C Q \ (\l s. P l s \ pre C Q l s)"
(is "_ \ \C. ?G P c Q C")
proof (induction rule: hoaret.induct)
case Skip
show ?case (is "\C. ?C C")
proof show "?C Askip" by simp qed
next
case (Assign P a x)
show ?case (is "\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 "\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 (\l s. P l s \ bval b s) c1 Q C1"
by blast
from If.IH obtain C2 where ih2: "?G (\l s. P l s \ \bval b s) c2 Q C2"
by blast
show ?case (is "\C. ?C C")
proof
show "?C(Aif b C1 C2)" using ih1 ih2 by simp
qed
next
case (While P x c b)
from While.IH obtain C where
ih: "?G (\l s. P (l(x:=Suc(l x))) s \ bval b s) c P C"
by blast
show ?case (is "\C. ?C C")
proof
have "vc ({P/x} WHILE b DO C) (\l. P (l(x := 0)))"
using ih While.hyps(2,3)
by simp (metis fun_upd_same zero_less_Suc)
thus "?C(Awhile P x b C)" using ih by simp
qed
next
case conseq thus ?case by(fast elim!: pre_mono vc_mono)
qed
end
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