mpri-coq-project/wp.v

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(* Projet Coq - WP - MPRI 2.7.1 *)
(***** Partie 1 : definition de While ****************************************)
Require Import ZArith.BinInt.
Import Z.
Definition Var := nat.
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Definition Mem := Var -> Z.
Definition Expr:= Mem -> Z.
Inductive Instr : Type :=
| skip: Instr
| abort: Instr
| assign: Var -> Expr -> Instr
| seq: Instr -> Instr -> Instr
| ifelse: Expr -> Instr -> Instr -> Instr
| while: Expr -> Instr -> Instr.
Definition ifonly (exp: Expr) (inst: Instr) : Instr :=
ifelse exp inst skip.
(***** CPO *******************************************************************)
Fixpoint nat_eq x y :=
match x, y with
| 0, 0 => true
| 0, S _ => false
| S _, 0 => false
| S x0, S y0 => nat_eq x0 y0
end.
Definition Sequence (S: Type) := nat -> S.
Inductive cpo (T: Type): Type :=
| CpoError: (cpo T)
| CpoElem: T -> (cpo T).
Definition cpo_leq: forall (T: Type), cpo T -> cpo T -> Prop :=
fun T x y => match x, y with
| CpoError _, _ => True
| CpoElem _ x0, CpoElem _ y0 => x0 = y0
| _, _ => False
end.
Arguments cpo_leq {T} _ _.
Infix "cpo<=" := cpo_leq (at level 100).
Definition is_chain: forall (T: Type), Sequence (cpo T) -> Prop :=
fun T chain => forall (n: nat), (chain n) cpo<= (chain (S n)).
Arguments is_chain {T} _.
Definition is_lub_of: forall (T: Type), Sequence (cpo T) -> cpo T -> Prop :=
fun T chain elt => forall (n: nat), (chain n) cpo<= elt.
Arguments is_lub_of {T} _ _.
Axiom find_lub: forall (T: Type), Sequence (cpo T) -> cpo T.
Arguments find_lub {T} _.
Axiom find_lub_correct:
forall (T: Type), forall (chain: Sequence (cpo T)),
is_chain chain -> is_lub_of chain (find_lub chain).
Arguments find_lub_correct {T} {chain} _.
(***** Interpretation ********************************************************)
Definition subst: Mem -> Var -> Z -> Mem :=
fun (m: Mem) (v: Var) (z: Z) (v2: Var) =>
if nat_eq v v2 then z else m v2.
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Notation "m [ x <- z ]" := (subst m x z) (at level 50, left associativity).
Definition MemCpo := cpo Mem.
Definition MemError := CpoError Mem.
Definition MemElem := CpoElem Mem.
Fixpoint interp (inst: Instr) (mem: MemCpo) : MemCpo :=
match mem with
| CpoError _ => MemError
| CpoElem _ mem0 =>
match inst with
| skip => MemElem mem0
| abort => MemError
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| assign v e => (MemElem (mem0 [v <- (e mem0)]))
| seq instr1 instr2 => interp instr2 (interp instr1 (MemElem mem0))
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| ifelse guard instrIf instrElse =>
if ((guard mem0) =? 0) % Z
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then interp instrElse mem
else interp instrIf mem
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| while guard body =>
let fix while_chain (mem: MemCpo) (n: nat): MemCpo :=
match n with
| 0 => mem
| S m =>
match while_chain (MemElem mem0) m with
| CpoError _ => MemError
| CpoElem _ submem =>
if ((guard submem) =? 0) % Z
then interp body (MemElem submem)
else mem
end
end
in find_lub (fun n =>
match while_chain (MemElem mem0) n with
| CpoError _ => MemError
| CpoElem _ submem =>
if ((guard submem) =? 0) % Z
then MemError
else MemElem submem
end)
end
end.
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Fixpoint nth_iterate (instr: Instr) (n: nat) : Instr :=
match n with
| 0 => skip
| S m => seq instr (nth_iterate instr m)
end.
Definition satisfies_expr (mem: MemCpo) (expr: Expr) : Prop := match mem with
| CpoError _ => False
| CpoElem _ mem0 => (expr mem0 <> 0) % Z
end.
Infix "|=e" := satisfies_expr (at level 32).
Definition expr_neg (expr: Expr) : Expr :=
fun mem => match expr mem with
| 0%Z => 1%Z
| _ => 0%Z
end.
Lemma certain_termination body guard mem :
interp (while guard body) mem <> MemError ->
exists n: nat,
(interp (nth_iterate body n) mem) |=e (expr_neg guard)
/\ forall p, p < n -> (interp (nth_iterate body p) mem) |=e guard
/\ interp (while guard body) mem = interp (nth_iterate body n) mem.
Proof.
intros notError.
Admitted.
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(***** Validite, prouvabilite pour Hoare *************************************)
Definition Assert := Mem -> Prop.
Delimit Scope assert with assert.
Definition assertTop : Assert := fun _ => True.
Definition assertBot : Assert := fun _ => False.
Definition assertNot : Assert -> Assert :=
fun orig mem => ~ (orig mem).
Notation "~ x" := (assertNot x) (at level 75, right associativity) : assert.
Definition assertAnd : Assert -> Assert -> Assert :=
fun x1 x2 mem => (x1 mem) /\ (x2 mem).
Infix "/\" := assertAnd : assert.
Definition assertOr : Assert -> Assert -> Assert :=
fun x1 x2 mem => (x1 mem) \/ (x2 mem).
Infix "\/" := assertOr : assert.
Definition assertImpl : Assert -> Assert -> Assert :=
fun x1 x2 => (~x1 \/ x2) % assert.
Infix "->" := assertImpl : assert.
Definition assertForall : Var -> Assert -> Assert :=
fun ident asser mem => forall (z: Z), asser (mem [ident <- z]).
Notation "\-/ x" := (assertForall x) (at level 90, no associativity) : assert.
Definition existsForall : Var -> Assert -> Assert :=
fun ident asser mem => exists (z: Z), asser (mem [ident <- z]).
Notation "'exists' x" := (existsForall x)
(at level 87, no associativity): assert.
Definition assertMemForall : Assert -> Assert :=
fun asser mem => forall (mem: Mem), asser mem.
Notation "'\-/m' x" := (assertMemForall x)
(at level 90, no associativity): assert.
Definition existsMemForall : Assert -> Assert :=
fun asser mem => exists (mem: Mem), asser mem.
Notation "'exists_m' x" := (existsMemForall x)
(at level 87, no associativity): assert.
Definition substAssert : Assert -> Var -> Z -> Assert :=
fun asser ident val mem => asser (mem [ident <- val]).
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Notation "a [[ x <- z ]]" := (substAssert a x z)
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(at level 50, left associativity).
Definition substAssertExpr : Assert -> Var -> Expr -> Assert :=
fun asser ident expr mem => asser (mem [ident <- (expr mem)]).
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Notation "a [[ x <- 'expr' z ]]" := (substAssertExpr a x z)
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(at level 50, left associativity).
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Definition assertOfExpr : Expr -> Assert :=
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fun expr mem => (expr mem <> 0)%Z.
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Definition assertImplLogical (a1 a2: Assert): Prop :=
forall (m: Mem), (a1 m) -> (a2 m).
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(***** Hoare provability *****************************************************)
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Reserved Notation "|- [| x |] y [| z |]" (at level 30).
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.
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(***** 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.
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Lemma weaken_in_conseq:
forall a1, forall a2, forall m,
assertImplLogical a1 a2 -> conseq_or_bottom a1 m ->
conseq_or_bottom a2 m.
Proof.
intros a1 a2 m impl conseq. destruct m.
- apply conseq.
- simpl. unfold assertImplLogical in impl. apply (impl m). apply conseq.
Qed.
Lemma interp_of_error (s: Instr):
interp s (MemError) = MemError.
Proof.
unfold MemError. destruct s; cbv; trivial.
Qed.
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Theorem hoare_provability_implies_consequence :
forall (pre: Assert), forall (s: Instr), forall (post: Assert),
( |- [| pre |] s [| post |] ) % assert
-> ( |= [| pre |] s [| post |] ) % assert.
Proof.
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intros pre instr post. intros deduction.
induction deduction; intros mem preInMem.
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- exact preInMem.
- simpl; trivial.
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- exact preInMem.
- apply (weaken_in_conseq post' post (interp s (MemElem mem)) H0).
apply IHdeduction. apply H. exact preInMem.
- simpl; destruct (interp s1 (MemElem mem)) eqn:mRel.
* fold MemError. rewrite (interp_of_error s2); simpl; trivial.
* apply (IHdeduction2 m). unfold hoare_consequence in IHdeduction1.
specialize IHdeduction1 with mem as IH1_mem.
rewrite mRel in IH1_mem. apply IH1_mem. assumption.
- simpl; destruct (expr mem =? 0)%Z eqn:branchEqn.
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* apply (IHdeduction2 mem). unfold assertOfExpr.
unfold assertAnd. split.
+ assumption.
+ unfold assertNot. rewrite <- Z.eqb_eq. congruence.
* apply (IHdeduction1 mem). unfold assertOfExpr.
unfold assertAnd. split.
+ assumption.
+ rewrite <- Z.eqb_eq. congruence.
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- unfold conseq_or_bottom.
destruct (interp (while expr sBody) (MemElem mem)) eqn:interpRel.
* trivial.
* unfold assertAnd. split.
+ specialize (IHdeduction m) as IHdeduction_m.
unfold conseq_or_bottom in IHdeduction_m.
Admitted.
(***** Weakest precondition **************************************************)
Fixpoint wp (instr: Instr) (cond: Assert) : Assert := match instr with
| skip =>
cond
| abort =>
assertTop
| assign x expr =>
cond [[ x <- expr expr ]]
| seq s1 s2 =>
wp s1 (wp s2 cond)
| ifelse guard sIf sElse =>
(assertOfExpr guard -> wp sIf cond
/\ (~ (assertOfExpr guard) -> wp sElse cond)) % assert
| while guard body => assertTop
end.
Theorem wp_correctness (instr: Instr) (post: Assert) :
( |= [| wp instr post |] instr [| post |] ) % assert.
Proof.
(* TODO *)
Admitted.