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From iris.heap_lang Require Export lifting.
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Import uPred.

(** Define some derived forms, and derived lemmas about them. *)
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Notation Lam x e := (Rec BAnon x e).
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Notation Let x e1 e2 := (App (Lam x e2) e1).
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Notation Seq e1 e2 := (Let BAnon e1 e2).
Notation LamV x e := (RecV BAnon x e).
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Notation LetCtx x e2 := (AppRCtx (LamV x e2)).
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Notation SeqCtx e2 := (LetCtx BAnon e2).
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Notation Skip := (Seq (Lit LitUnit) (Lit LitUnit)).
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Notation Match e0 x1 e1 x2 e2 := (Case e0 (Lam x1 e1) (Lam x2 e2)).
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Section derived.
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Context {Σ : iFunctor}.
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Implicit Types P Q : iProp heap_lang Σ.
Implicit Types Φ : val  iProp heap_lang Σ.
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(** Proof rules for the sugar *)
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Lemma wp_lam E x ef e v Φ :
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  to_val e = Some v 
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  Closed (x :b: []) ef 
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   WP subst' x e ef @ E {{ Φ }}  WP App (Lam x ef) e @ E {{ Φ }}.
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Proof. intros. by rewrite -(wp_rec _ BAnon) //. Qed.
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Lemma wp_let E x e1 e2 v Φ :
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  to_val e1 = Some v 
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  Closed (x :b: []) e2 
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   WP subst' x e1 e2 @ E {{ Φ }}  WP Let x e1 e2 @ E {{ Φ }}.
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Proof. apply wp_lam. Qed.
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Lemma wp_seq E e1 e2 v Φ :
  to_val e1 = Some v 
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  Closed [] e2 
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   WP e2 @ E {{ Φ }}  WP Seq e1 e2 @ E {{ Φ }}.
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Proof. intros ??. by rewrite -wp_let. Qed.
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Lemma wp_skip E Φ :  Φ (LitV LitUnit)  WP Skip @ E {{ Φ }}.
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Proof. rewrite -wp_seq // -wp_value //. Qed.
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Lemma wp_match_inl E e0 v0 x1 e1 x2 e2 Φ :
  to_val e0 = Some v0 
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  Closed (x1 :b: []) e1 
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   WP subst' x1 e0 e1 @ E {{ Φ }}  WP Match (InjL e0) x1 e1 x2 e2 @ E {{ Φ }}.
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Proof. intros. by rewrite -wp_case_inl // -[X in _  X]later_intro -wp_let. Qed.
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Lemma wp_match_inr E e0 v0 x1 e1 x2 e2 Φ :
  to_val e0 = Some v0 
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  Closed (x2 :b: []) e2 
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   WP subst' x2 e0 e2 @ E {{ Φ }}  WP Match (InjR e0) x1 e1 x2 e2 @ E {{ Φ }}.
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Proof. intros. by rewrite -wp_case_inr // -[X in _  X]later_intro -wp_let. Qed.
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Lemma wp_le E (n1 n2 : Z) P Φ :
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  (n1  n2  P   |={E}=> Φ (LitV (LitBool true))) 
  (n2 < n1  P   |={E}=> Φ (LitV (LitBool false))) 
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  P  WP BinOp LeOp (Lit (LitInt n1)) (Lit (LitInt n2)) @ E {{ Φ }}.
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Proof.
  intros. rewrite -wp_bin_op //; [].
  destruct (bool_decide_reflect (n1  n2)); by eauto with omega.
Qed.

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Lemma wp_lt E (n1 n2 : Z) P Φ :
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  (n1 < n2  P   |={E}=> Φ (LitV (LitBool true))) 
  (n2  n1  P   |={E}=> Φ (LitV (LitBool false))) 
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  P  WP BinOp LtOp (Lit (LitInt n1)) (Lit (LitInt n2)) @ E {{ Φ }}.
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Proof.
  intros. rewrite -wp_bin_op //; [].
  destruct (bool_decide_reflect (n1 < n2)); by eauto with omega.
Qed.

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Lemma wp_eq E (n1 n2 : Z) P Φ :
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  (n1 = n2  P   |={E}=> Φ (LitV (LitBool true))) 
  (n1  n2  P   |={E}=> Φ (LitV (LitBool false))) 
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  P  WP BinOp EqOp (Lit (LitInt n1)) (Lit (LitInt n2)) @ E {{ Φ }}.
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Proof.
  intros. rewrite -wp_bin_op //; [].
  destruct (bool_decide_reflect (n1 = n2)); by eauto with omega.
Qed.
End derived.