(* Projet Coq - WP - MPRI 2.7.1 *)

(***** Partie 1 : definition de While ****************************************)

Require Import ZArith.BinInt.
Import Z.

Definition Var := nat.
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.
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
        | assign v e => (MemElem (mem0 [v <- (e mem0)]))
        | seq instr1 instr2 => interp instr2 (interp instr1 (MemElem mem0))
        | ifelse guard instrIf instrElse =>
            if ((guard mem0) =? 0) % Z
                then interp instrIf mem
                else interp instrElse mem
        | 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.

(***** 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]).
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)
    (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.

(***** 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.