src/HOL/IMP/VCG_Total_EX.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/IMP/VCG_Total_EX.thy	Sat Jul 23 13:25:44 2016 +0200
@@ -0,0 +1,71 @@
+(* Author: Tobias Nipkow *)
+
+theory VCG_Total_EX
+imports "~~/src/HOL/IMP/Hoare_Total_EX"
+begin
+
+subsection "Verification Conditions for Total Correctness"
+
+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 "nat \<Rightarrow> assn" bexp acom
+    ("({_}/ WHILE _/ DO _)"  [0, 0, 61] 61)
+
+notation com.SKIP ("SKIP")
+
+text{* Strip annotations: *}
+
+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)"
+
+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\<^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 = (\<lambda>s. EX n. I n s)"
+
+text{* Verification condition: *}
+
+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 n. (I (Suc n) s \<longrightarrow> pre C (I n) s) \<and>
+       (I (Suc n) s \<longrightarrow> bval b s) \<and>
+       (I 0 s \<longrightarrow> \<not> bval b s \<and> Q s) \<and>
+       vc C (I n))"
+
+lemma vc_sound: "vc C Q \<Longrightarrow> \<turnstile>\<^sub>t {pre C Q} strip C {Q}"
+proof(induction C arbitrary: Q)
+  case (Awhile I b C)
+  show ?case
+  proof(simp, rule conseq[OF _ While[of I]], goal_cases)
+    case (2 n) show ?case
+      using Awhile.IH[of "I n"] Awhile.prems
+      by (auto intro: strengthen_pre)
+  qed (insert Awhile.prems, auto)
+qed (auto intro: conseq Seq If simp: Skip Assign)
+
+text\<open>When trying to extend the completeness proof of the VCG for partial correctness
+to total correctness one runs into the following problem.
+In the case of the while-rule, the universally quantified \<open>n\<close> in the first premise
+means that for that premise the induction hypothesis does not yield a single
+annotated command \<open>C\<close> but merely that for every \<open>n\<close> such a \<open>C\<close> exists.\<close>
+
+end