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Axiomatize Hoare provability |-

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Théophile Bastian 2017-12-04 23:37:38 +01:00
parent 90ca9bc5bb
commit 7e2c2e08dc
1 changed files with 47 additions and 2 deletions
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49
wp.v
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@ -147,9 +147,54 @@ Notation "'exists_m' x" := (existsMemForall x)
Definition substAssert : Assert -> Var -> Z -> Assert :=
fun asser ident val mem => asser (mem [ident <- val]).
Notation "a < x <- z >" := (substAssert a x z)
Notation "a [[ x <- z ]]" := (substAssert a x z)
(at level 50, left associativity).
Definition substAssertExpr : Assert -> Var -> Expr -> Assert :=
fun asser ident expr mem => asser (mem [ident <- (expr mem)]).
Notation "a < x <- 'expr' z >" := (substAssertExpr a x z)
Notation "a [[ x <- 'expr' z ]]" := (substAssertExpr a x z)
(at level 50, left associativity).
Definition assertOfExpr : Expr -> Assert :=
fun expr mem => expr mem <> 0%Z.
Definition assertImplLogical : Assert -> Assert -> Prop :=
fun a1 a2 => forall (m : Mem), (a1 m) -> (a2 m).
(***** Hoare provability *****************************************************)
Parameter hoare_provability : Assert -> Instr -> Assert -> Prop.
Notation "[| pre |] instr [| post |]" := (hoare_provability pre instr post)
(at level 30): assert.
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.