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Modify notation, state theorem 2.3

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Théophile Bastian 2017-12-05 00:27:35 +01:00
parent 7e2c2e08dc
commit d1b62ad2b5
1 changed files with 37 additions and 14 deletions
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@ -164,37 +164,60 @@ Definition assertImplLogical : Assert -> Assert -> Prop :=
(***** Hoare provability *****************************************************) (***** Hoare provability *****************************************************)
Parameter hoare_provability : Assert -> Instr -> Assert -> Prop. Parameter hoare_provability : Assert -> Instr -> Assert -> Prop.
Notation "[| pre |] instr [| post |]" := (hoare_provability pre instr post) Notation "|- [| pre |] instr [| post |]" :=
(at level 30): assert. (hoare_provability pre instr post) (at level 30): assert.
Axiom h_skip: forall (asser: Assert), ( [|asser|] skip [|asser|]) % assert. Axiom h_skip: forall (asser: Assert), ( |- [|asser|] skip [|asser|]) % assert.
Axiom h_abort: forall (a1:Assert), forall (a2: Assert), Axiom h_abort: forall (a1:Assert), forall (a2: Assert),
([|a1|] abort [|a2|]) % assert. (|- [|a1|] abort [|a2|]) % assert.
Axiom h_assign: forall (asser: Assert), forall (x: Var), forall (e: Expr), Axiom h_assign: forall (asser: Assert), forall (x: Var), forall (e: Expr),
([| asser [[ x <- expr e ]] |] (assign x e) [|asser|]) % assert. (|- [| asser [[ x <- expr e ]] |] (assign x e) [|asser|]) % assert.
Axiom h_conseq: Axiom h_conseq:
forall (a1: Assert), forall (a1': Assert), forall (a1: Assert), forall (a1': Assert),
forall (a2: Assert), forall (a2': Assert), forall (a2: Assert), forall (a2': Assert),
forall (s: Instr), forall (s: Instr),
([| a1' |] s [| a2' |]) % assert -> (|- [| a1' |] s [| a2' |]) % assert ->
(assertImplLogical a1 a1') -> (assertImplLogical a1 a1') ->
(assertImplLogical a2' a2) -> (assertImplLogical a2' a2) ->
([| a1 |] s [| a2 |]) % assert. (|- [| a1 |] s [| a2 |]) % assert.
Axiom h_seq: Axiom h_seq:
forall (a1: Assert), forall (a2: Assert), forall (a3: Assert), forall (a1: Assert), forall (a2: Assert), forall (a3: Assert),
forall (s1: Instr), forall (s2: Instr), forall (s1: Instr), forall (s2: Instr),
([|a1|] s1 [|a2|]) % assert -> ([|a2|] s2 [|a3|]) % assert -> (|- [|a1|] s1 [|a2|]) % assert -> (|- [|a2|] s2 [|a3|]) % assert ->
([|a1|] (seq s1 s2) [|a3|]) % assert. (|- [|a1|] (seq s1 s2) [|a3|]) % assert.
Axiom h_if: Axiom h_if:
forall (a1: Assert), forall (a2: Assert), forall (a1: Assert), forall (a2: Assert),
forall (e: Expr), forall (e: Expr),
forall (s1: Instr), forall (s2: Instr), forall (s1: Instr), forall (s2: Instr),
([| a1 /\ (assertOfExpr e) |] s1 [| a2 |]) % assert -> (|- [| a1 /\ (assertOfExpr e) |] s1 [| a2 |]) % assert ->
([| a1 /\ ~ (assertOfExpr e) |] s2 [| a2 |]) % assert -> (|- [| a1 /\ ~ (assertOfExpr e) |] s2 [| a2 |]) % assert ->
([| a1 |] (ifelse e s1 s2) [| a2 |]) % assert. (|- [| a1 |] (ifelse e s1 s2) [| a2 |]) % assert.
Axiom h_while: Axiom h_while:
forall (inv: Assert), forall (inv: Assert),
forall (e: Expr), forall (e: Expr),
forall (s: Instr), forall (s: Instr),
([| inv /\ (assertOfExpr e) |] s [| inv |]) % assert -> (|- [| inv /\ (assertOfExpr e) |] s [| inv |]) % assert ->
([| inv |] (while e s) [| inv /\ ~ (assertOfExpr e) |]) % assert. (|- [| inv |] (while e s) [| inv /\ ~ (assertOfExpr e) |]) % assert.
(***** Hoare: provability implies consequence ********************************)
Definition conseq_or_bottom (y: Assert) (m: MemCpo) :=
match m with
| CpoError _ => True
| CpoElem _ m0 => y m0
end.
Definition hoare_consequence (pre: Assert) (instr: Instr) (post: Assert) :=
forall mem: Mem,
(pre mem) -> (conseq_or_bottom post (interp instr (MemElem mem))).
Notation "|= [| pre |] instr [| post |]" :=
(hoare_consequence pre instr post) (at level 30): assert.
Theorem hoare_provability_implies_consequence :
forall (pre: Assert), forall (s: Instr), forall (post: Assert),
( |- [| pre |] s [| post |] ) % assert
-> ( |= [| pre |] s [| post |] ) % assert.
Proof.
(* TODO *)
Admitted.