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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 *****************************************************) (***** Hoare provability *****************************************************)
Parameter hoare_provability : Assert -> Instr -> Assert -> Prop. Reserved Notation "|- [| x |] y [| z |]" (at level 30).
Notation "|- [| pre |] instr [| post |]" :=
(hoare_provability pre instr post) (at level 30): assert.
Axiom h_skip: forall (asser: Assert), ( |- [|asser|] skip [|asser|]) % assert. Inductive hoare_provability : Assert -> Instr -> Assert -> Prop :=
Axiom h_abort: forall (a1:Assert), forall (a2: Assert), | H_skip: forall pre, hoare_provability pre skip pre
(|- [|a1|] abort [|a2|]) % assert. | H_abort: forall pre, forall post, hoare_provability pre abort post
Axiom h_assign: forall (asser: Assert), forall (x: Var), forall (e: Expr), | H_assign: forall post, forall (x: Var), forall (e: Expr),
(|- [| asser [[ x <- expr e ]] |] (assign x e) [|asser|]) % assert. (|- [| post [[ x <- expr e ]] |] (assign x e) [| post |]) % assert
Axiom h_conseq: | H_conseq:
forall (a1: Assert), forall (a1': Assert), forall pre, forall post,
forall (a2: Assert), forall (a2': Assert), forall pre', forall post',
forall (s: Instr), forall s,
(|- [| a1' |] s [| a2' |]) % assert ->
(assertImplLogical a1 a1') -> (|- [| pre' |] s [| post' |]) % assert ->
(assertImplLogical a2' a2) -> (assertImplLogical pre pre') ->
(|- [| a1 |] s [| a2 |]) % assert. (assertImplLogical post' post) ->
Axiom h_seq: (|- [| pre |] s [| post |]) % assert
forall (a1: Assert), forall (a2: Assert), forall (a3: Assert), | H_seq:
forall (s1: Instr), forall (s2: Instr), forall pre, forall mid, forall post, forall s1, forall s2,
(|- [|a1|] s1 [|a2|]) % assert -> (|- [|a2|] s2 [|a3|]) % assert -> (|- [|pre|] s1 [|mid|]) % assert ->
(|- [|a1|] (seq s1 s2) [|a3|]) % assert. (|- [|mid|] s2 [|post|]) % assert ->
Axiom h_if: (|- [|pre|] (seq s1 s2) [|post|]) % assert
forall (a1: Assert), forall (a2: Assert), | H_if:
forall (e: Expr), forall pre, forall post, forall expr, forall sIf, forall sElse,
forall (s1: Instr), forall (s2: Instr), (|- [| pre /\ (assertOfExpr expr) |] sIf [| post |]) % assert ->
(|- [| a1 /\ (assertOfExpr e) |] s1 [| a2 |]) % assert -> (|- [| pre /\ ~(assertOfExpr expr) |] sElse [| post |]) % assert ->
(|- [| a1 /\ ~ (assertOfExpr e) |] s2 [| a2 |]) % assert -> (|- [| pre |] (ifelse expr sIf sElse) [| post |]) % assert
(|- [| a1 |] (ifelse e s1 s2) [| a2 |]) % assert. | H_while:
Axiom h_while: forall inv, forall expr, forall sBody,
forall (inv: Assert), (|- [| inv /\ (assertOfExpr expr) |] sBody [| inv |]) % assert ->
forall (e: Expr), (|- [| inv |] (while expr sBody)
forall (s: Instr), [| inv /\ ~ (assertOfExpr expr) |]) % assert
(|- [| inv /\ (assertOfExpr e) |] s [| inv |]) % assert -> where "|- [| pre |] instr [| post |]" :=
(|- [| inv |] (while e s) [| inv /\ ~ (assertOfExpr e) |]) % assert. (hoare_provability pre instr post) : assert.
(***** Hoare: provability implies consequence ********************************) (***** Hoare: provability implies consequence ********************************)