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From heap_lang Require Export lifting.
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Import uPred.

(** Define some derived forms, and derived lemmas about them. *)
Notation Lam x e := (Rec "" x e).
Notation Let x e1 e2 := (App (Lam x e2) e1).
Notation Seq e1 e2 := (Let "" e1 e2).
Notation LamV x e := (RecV "" x e).
Notation LetCtx x e2 := (AppRCtx (LamV x e2)).
Notation SeqCtx e2 := (LetCtx "" e2).
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Notation Skip := (Seq (Lit LitUnit) (Lit LitUnit)).
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Section derived.
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 
   || subst ef x v @ E {{ Φ }}  || App (Lam x ef) e @ E {{ Φ }}.
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Proof. intros. by rewrite -wp_rec ?subst_empty. Qed.
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Lemma wp_let E x e1 e2 v Φ :
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  to_val e1 = Some v 
   || subst e2 x v @ E {{ Φ }}  || 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 
   || e2 @ E {{ Φ }}  || Seq e1 e2 @ E {{ Φ }}.
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Proof. intros ?. rewrite -wp_let // subst_empty //. Qed.
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Lemma wp_skip E Φ :  Φ (LitV LitUnit)  || Skip @ E {{ Φ }}.
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Proof. rewrite -wp_seq // -wp_value //. Qed.
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Lemma wp_le E (n1 n2 : Z) P Φ :
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  (n1  n2  P   Φ (LitV (LitBool true))) 
  (n2 < n1  P   Φ (LitV (LitBool false))) 
  P  || 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   Φ (LitV (LitBool true))) 
  (n2  n1  P   Φ (LitV (LitBool false))) 
  P  || 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   Φ (LitV (LitBool true))) 
  (n1  n2  P   Φ (LitV (LitBool false))) 
  P  || 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.