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Rewrite hoare_provability inductively

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Théophile Bastian 2017-12-05 14:19:43 +01:00
parent d1b62ad2b5
commit 06f655c1c1
1 changed files with 32 additions and 34 deletions
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wp.v
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@ -163,41 +163,39 @@ Definition assertImplLogical : Assert -> Assert -> Prop :=
(***** Hoare provability *****************************************************)
Parameter hoare_provability : Assert -> Instr -> Assert -> Prop.
Notation "|- [| pre |] instr [| post |]" :=
(hoare_provability pre instr post) (at level 30): assert.
Reserved Notation "|- [| x |] y [| z |]" (at level 30).
Axiom h_skip: forall (asser: Assert), ( |- [|asser|] skip [|asser|]) % assert.
Axiom h_abort: forall (a1:Assert), forall (a2: Assert),
(|- [|a1|] abort [|a2|]) % assert.
Axiom h_assign: forall (asser: Assert), forall (x: Var), forall (e: Expr),
(|- [| asser [[ x <- expr e ]] |] (assign x e) [|asser|]) % assert.
Axiom h_conseq:
forall (a1: Assert), forall (a1': Assert),
forall (a2: Assert), forall (a2': Assert),
forall (s: Instr),
(|- [| a1' |] s [| a2' |]) % assert ->
(assertImplLogical a1 a1') ->
(assertImplLogical a2' a2) ->
(|- [| a1 |] s [| a2 |]) % assert.
Axiom h_seq:
forall (a1: Assert), forall (a2: Assert), forall (a3: Assert),
forall (s1: Instr), forall (s2: Instr),
(|- [|a1|] s1 [|a2|]) % assert -> (|- [|a2|] s2 [|a3|]) % assert ->
(|- [|a1|] (seq s1 s2) [|a3|]) % assert.
Axiom h_if:
forall (a1: Assert), forall (a2: Assert),
forall (e: Expr),
forall (s1: Instr), forall (s2: Instr),
(|- [| a1 /\ (assertOfExpr e) |] s1 [| a2 |]) % assert ->
(|- [| a1 /\ ~ (assertOfExpr e) |] s2 [| a2 |]) % assert ->
(|- [| a1 |] (ifelse e s1 s2) [| a2 |]) % assert.
Axiom h_while:
forall (inv: Assert),
forall (e: Expr),
forall (s: Instr),
(|- [| inv /\ (assertOfExpr e) |] s [| inv |]) % assert ->
(|- [| inv |] (while e s) [| inv /\ ~ (assertOfExpr e) |]) % assert.
Inductive hoare_provability : Assert -> Instr -> Assert -> Prop :=
| H_skip: forall pre, hoare_provability pre skip pre
| H_abort: forall pre, forall post, hoare_provability pre abort post
| H_assign: forall post, forall (x: Var), forall (e: Expr),
(|- [| post [[ x <- expr e ]] |] (assign x e) [| post |]) % assert
| H_conseq:
forall pre, forall post,
forall pre', forall post',
forall s,
(|- [| pre' |] s [| post' |]) % assert ->
(assertImplLogical pre pre') ->
(assertImplLogical post' post) ->
(|- [| pre |] s [| post |]) % assert
| H_seq:
forall pre, forall mid, forall post, forall s1, forall s2,
(|- [|pre|] s1 [|mid|]) % assert ->
(|- [|mid|] s2 [|post|]) % assert ->
(|- [|pre|] (seq s1 s2) [|post|]) % assert
| H_if:
forall pre, forall post, forall expr, forall sIf, forall sElse,
(|- [| pre /\ (assertOfExpr expr) |] sIf [| post |]) % assert ->
(|- [| pre /\ ~(assertOfExpr expr) |] sElse [| post |]) % assert ->
(|- [| pre |] (ifelse expr sIf sElse) [| post |]) % assert
| H_while:
forall inv, forall expr, forall sBody,
(|- [| inv /\ (assertOfExpr expr) |] sBody [| inv |]) % assert ->
(|- [| inv |] (while expr sBody)
[| inv /\ ~ (assertOfExpr expr) |]) % assert
where "|- [| pre |] instr [| post |]" :=
(hoare_provability pre instr post) : assert.
(***** Hoare: provability implies consequence ********************************)