Require Import Facts States GCSyntax.
Set Implicit Arguments.

Module GCSemantics (Sigma : State).
Module GCSyn := GCSyntax.GCSyntax Sigma.
Export GCSyn.

Implicit Types (P Q : Pred state) (x y z : state).
Implicit Types (a : action) (b : guard) (G : gc) (s t : cmd).

Definition gtest G : state → bool :=
  fun x ⇒ has (fun p : guard ⇒ p x) (unzip1 G).
Coercion gtest : gc >-> Funclass.

Inductive wps Q : cmd → Pred state :=
| wps_skip x :
    Q x →
    wps Q Skip x
| wps_assn a x :
    Q (a x) →
    wps Q (Assn a) x
| wps_seq s t x P :
    wps P s x →
    P <<= wps Q t →
    wps Q (Seq s t) x
| wps_case G x :
    G x →
    (∀ b s, (b,s) \in G → b x → wps Q s x) →
    wps Q (Case G) x
| wps_loop_true G x P :
    G x →
    (∀ b s, (b,s) \in G → b x → wps P s x) →
    P <<= wps Q (Do G) →
    wps Q (Do G) x
| wps_loop_false G x :
    ~~G x →
    Q x →
    wps Q (Do G) x.

Definition wpG´ (wp : Pred state → cmd → Pred state) Q : gc → Pred state :=
  fix rec G x := match G with
  | (b,s) :: G ⇒ (b x → wp Q s x) ∧ rec G x
  | [::] ⇒ True
  end.

Fixpoint wpg Q s : Pred state :=
  match s with
  | Skip ⇒ Q
  | Assn a ⇒ Q \o a
  | Seq s t ⇒ wpg (wpg Q t) s
  | Case G ⇒ fun x ⇒ gtest G x ∧ wpG´ wpg Q G x
  | Do G ⇒ Fix (fun P x ⇒ if gtest G x then wpG´ wpg P G x else Q x)
  end.

Notation wpG := (wpG´ wpg).

Lemma gtest_cons (G : gc) b s x :
  gtest ((b,s) :: G) x = b x || G x.

Lemma gtestP (G : gc) x :
  reflect (∃ (b:guard) (s:cmd), (b,s) \in G ∧ b x) (G x).

Lemma gtest_contra (G : gc) b s x :
  (b,s) \in G → ~~G x → ~~b x.

Lemma wpgG_mono :
  (∀ s, monotone (wpg^~ s)) ∧ (∀ G, monotone (wpG^~ G)).

Lemma wpg_mono s : monotone (wpg^~ s).
Lemma wpG_mono G : monotone (wpG^~ G).

Lemma wpgG_wps :
  (∀ s Q, wpg Q s <<= wps Q s) ∧
  (∀ G Q x,
      wpG Q G x → ∀ b s, (b,s) \in G → b x → wps Q s x).

Lemma wpg_wps Q s : wpg Q s <<= wps Q s.

Lemma wps_wpG Q (G:gc) x :
  (∀ b s, (b,s) \in G → b x → wpg Q s x) →
  wpG Q G x.

Lemma wps_wpg Q (s : cmd) : wps Q s <<= wpg Q s.

End GCSemantics.