Merge branch 'master' of gitlab.mpi-sws.org:FP/iris-coq

heap_lang/heap_lang.v

@@ -6,18 +6,24 @@ Module heap_lang.
 (** Expressions and vals. *)
 Definition loc := positive. (* Really, any countable type. *)
 
+Inductive base_lit : Set :=
+  | LitNat (n : nat) | LitBool (b : bool) | LitUnit.
+Inductive un_op : Set :=
+  | NegOp.
+Inductive bin_op : Set :=
+  | PlusOp | MinusOp | LeOp | LtOp | EqOp.
+
 Inductive expr :=
   (* Base lambda calculus *)
   | Var (x : var)
   | Rec (e : {bind 2 of expr}) (* These are recursive lambdas.
                                   The *inner* binder is the recursive call! *)
   | App (e1 e2 : expr)
-  (* Natural numbers *)
-  | LitNat (n : nat)
-  | Plus (e1 e2 : expr)
-  | Le (e1 e2 : expr)
-  (* Unit *)
-  | LitUnit
+  (* Base types and their operations *)
+  | Lit (l : base_lit)
+  | UnOp (op : un_op) (e : expr)
+  | BinOp (op : bin_op) (e1 e2 : expr)
+  | If (e0 e1 e2 : expr)
   (* Products *)
   | Pair (e1 e2 : expr)
   | Fst (e : expr)
@@ -40,29 +46,19 @@ Instance Rename_expr : Rename expr. derive. Defined.
 Instance Subst_expr : Subst expr. derive. Defined.
 Instance SubstLemmas_expr : SubstLemmas expr. derive. Qed.
 
-(* This sugar is used by primitive reduction riles (<=, CAS) and hence
-defined here. *)
-Notation LitTrue := (InjL LitUnit).
-Notation LitFalse := (InjR LitUnit).
-
 Inductive val :=
   | RecV (e : {bind 2 of expr}) (* These are recursive lambdas.
                                    The *inner* binder is the recursive call! *)
-  | LitNatV (n : nat)
-  | LitUnitV
+  | LitV (l : base_lit)
   | PairV (v1 v2 : val)
   | InjLV (v : val)
   | InjRV (v : val)
   | LocV (l : loc).
 
-Definition LitTrueV := InjLV LitUnitV.
-Definition LitFalseV := InjRV LitUnitV.
-
 Fixpoint of_val (v : val) : expr :=
   match v with
   | RecV e => Rec e
-  | LitNatV n => LitNat n
-  | LitUnitV => LitUnit
+  | LitV l => Lit l
   | PairV v1 v2 => Pair (of_val v1) (of_val v2)
   | InjLV v => InjL (of_val v)
   | InjRV v => InjR (of_val v)
@@ -71,8 +67,7 @@ Fixpoint of_val (v : val) : expr :=
 Fixpoint to_val (e : expr) : option val :=
   match e with
   | Rec e => Some (RecV e)
-  | LitNat n => Some (LitNatV n)
-  | LitUnit => Some LitUnitV
+  | Lit l => Some (LitV l)
   | Pair e1 e2 => v1 ← to_val e1; v2 ← to_val e2; Some (PairV v1 v2)
   | InjL e => InjLV <$> to_val e
   | InjR e => InjRV <$> to_val e
@@ -87,10 +82,10 @@ Definition state := gmap loc val.
 Inductive ectx_item :=
   | AppLCtx (e2 : expr)
   | AppRCtx (v1 : val)
-  | PlusLCtx (e2 : expr)
-  | PlusRCtx (v1 : val)
-  | LeLCtx (e2 : expr)
-  | LeRCtx (v1 : val)
+  | UnOpCtx (op : un_op)
+  | BinOpLCtx (op : bin_op) (e2 : expr)
+  | BinOpRCtx (op : bin_op) (v1 : val)
+  | IfCtx (e1 e2 : expr)
   | PairLCtx (e2 : expr)
   | PairRCtx (v1 : val)
   | FstCtx
@@ -112,10 +107,10 @@ Definition fill_item (Ki : ectx_item) (e : expr) : expr :=
   match Ki with
   | AppLCtx e2 => App e e2
   | AppRCtx v1 => App (of_val v1) e
-  | PlusLCtx e2 => Plus e e2
-  | PlusRCtx v1 => Plus (of_val v1) e
-  | LeLCtx e2 => Le e e2
-  | LeRCtx v1 => Le (of_val v1) e
+  | UnOpCtx op => UnOp op e
+  | BinOpLCtx op e2 => BinOp op e e2
+  | BinOpRCtx op v1 => BinOp op (of_val v1) e
+  | IfCtx e1 e2 => If e e1 e2
   | PairLCtx e2 => Pair e e2
   | PairRCtx v1 => Pair (of_val v1) e
   | FstCtx => Fst e
@@ -134,18 +129,43 @@ Definition fill_item (Ki : ectx_item) (e : expr) : expr :=
 Definition fill (K : ectx) (e : expr) : expr := fold_right fill_item e K.
 
 (** The stepping relation *)
+Definition un_op_eval (op : un_op) (l : base_lit) : option base_lit :=
+  match op, l with
+  | NegOp, LitBool b => Some $ LitBool (negb b)
+  | _, _ => None
+  end.
+
+(* FIXME RJ I am *sure* this already exists somewhere... but I can't find it. *)
+Definition sum2bool {A B} (x : { A } + { B }) : bool :=
+  match x with
+  | left _ => true
+  | right _ => false
+  end.
+
+Definition bin_op_eval (op : bin_op) (l1 l2 : base_lit) : option base_lit :=
+  match op, l1, l2 with
+  | PlusOp, LitNat n1, LitNat n2 => Some $ LitNat (n1 + n2)
+  | MinusOp, LitNat n1, LitNat n2 => Some $ LitNat (n1 - n2)
+  | LeOp, LitNat n1, LitNat n2 => Some $ LitBool $ sum2bool $ decide (n1 ≤ n2)
+  | LtOp, LitNat n1, LitNat n2 => Some $ LitBool $ sum2bool $ decide (n1 < n2)
+  | EqOp, LitNat n1, LitNat n2 => Some $ LitBool $ sum2bool $ decide (n1 = n2)
+  | _, _, _ => None
+  end.
+
 Inductive head_step : expr -> state -> expr -> state -> option expr -> Prop :=
   | BetaS e1 e2 v2 σ :
      to_val e2 = Some v2 →
      head_step (App (Rec e1) e2) σ e1.[(Rec e1),e2/] σ None
-  | PlusS n1 n2 σ:
-     head_step (Plus (LitNat n1) (LitNat n2)) σ (LitNat (n1 + n2)) σ None
-  | LeTrueS n1 n2 σ :
-     n1 ≤ n2 →
-     head_step (Le (LitNat n1) (LitNat n2)) σ LitTrue σ None
-  | LeFalseS n1 n2 σ :
-     n1 > n2 →
-     head_step (Le (LitNat n1) (LitNat n2)) σ LitFalse σ None
+  | UnOpS op l l' σ: 
+     un_op_eval op l = Some l' → 
+     head_step (UnOp op (Lit l)) σ (Lit l') σ None
+  | BinOpS op l1 l2 l' σ: 
+     bin_op_eval op l1 l2 = Some l' → 
+     head_step (BinOp op (Lit l1) (Lit l2)) σ (Lit l') σ None
+  | IfTrueS e1 e2 σ :
+     head_step (If (Lit $ LitBool true) e1 e2) σ e1 σ None
+  | IfFalseS e1 e2 σ :
+     head_step (If (Lit $ LitBool false) e1 e2) σ e2 σ None
   | FstS e1 v1 e2 v2 σ :
      to_val e1 = Some v1 → to_val e2 = Some v2 →
      head_step (Fst (Pair e1 e2)) σ e1 σ None
@@ -159,7 +179,7 @@ Inductive head_step : expr -> state -> expr -> state -> option expr -> Prop :=
      to_val e0 = Some v0 →
      head_step (Case (InjR e0) e1 e2) σ e2.[e0/] σ None
   | ForkS e σ:
-     head_step (Fork e) σ LitUnit σ (Some e)
+     head_step (Fork e) σ (Lit LitUnit) σ (Some e)
   | AllocS e v σ l :
      to_val e = Some v → σ !! l = None →
      head_step (Alloc e) σ (Loc l) (<[l:=v]>σ) None
@@ -168,15 +188,15 @@ Inductive head_step : expr -> state -> expr -> state -> option expr -> Prop :=
      head_step (Load (Loc l)) σ (of_val v) σ None
   | StoreS l e v σ :
      to_val e = Some v → is_Some (σ !! l) →
-     head_step (Store (Loc l) e) σ LitUnit (<[l:=v]>σ) None
+     head_step (Store (Loc l) e) σ (Lit LitUnit) (<[l:=v]>σ) None
   | CasFailS l e1 v1 e2 v2 vl σ :
      to_val e1 = Some v1 → to_val e2 = Some v2 →
      σ !! l = Some vl → vl ≠ v1 →
-     head_step (Cas (Loc l) e1 e2) σ LitFalse σ None
+     head_step (Cas (Loc l) e1 e2) σ (Lit $ LitBool false)  σ None
   | CasSucS l e1 v1 e2 v2 σ :
      to_val e1 = Some v1 → to_val e2 = Some v2 →
      σ !! l = Some v1 →
-     head_step (Cas (Loc l) e1 e2) σ LitTrue (<[l:=v2]>σ) None.
+     head_step (Cas (Loc l) e1 e2) σ (Lit $ LitBool true) (<[l:=v2]>σ) None.
 
 (** Atomic expressions *)
 Definition atomic (e: expr) :=
@@ -263,7 +283,7 @@ Lemma fill_item_no_val_inj Ki1 Ki2 e1 e2 :
   fill_item Ki1 e1 = fill_item Ki2 e2 → Ki1 = Ki2.
 Proof.
   destruct Ki1, Ki2; intros; try discriminate; simplify_equality';
-    try match goal with
+    repeat match goal with
     | H : to_val (of_val _) = None |- _ => by rewrite to_of_val in H
     end; auto.
 Qed.

heap_lang/heap_lang_tactics.v

@@ -38,14 +38,13 @@ Ltac reshape_expr e tac :=
      lazymatch e1 with
      | of_val ?v1 => go (AppRCtx v1 :: K) e2 | _ => go (AppLCtx e2 :: K) e1
      end
-  | Plus ?e1 ?e2 =>
+  | UnOp ?op ?e =>
+     go (UnOpCtx op :: K) e
+  | BinOp ?op ?e1 ?e2 =>
      lazymatch e1 with
-     | of_val ?v1 => go (PlusRCtx v1 :: K) e2 | _ => go (PlusLCtx e2 :: K) e1
-     end
-  | Le ?e1 ?e2 =>
-     lazymatch e1 with
-     | of_val ?v1 => go (LeRCtx v1 :: K) e2 | _ => go (LeLCtx e2 :: K) e1
+     | of_val ?v1 => go (BinOpRCtx op v1 :: K) e2 | _ => go (BinOpLCtx op e2 :: K) e1
      end
+  | If ?e0 ?e1 ?e2 => go (IfCtx e1 e2 :: K) e0
   | Pair ?e1 ?e2 =>
      lazymatch e1 with
      | of_val ?v1 => go (PairRCtx v1 :: K) e2 | _ => go (PairLCtx e2 :: K) e1

heap_lang/lifting.v

@@ -48,36 +48,36 @@ Qed.
 
 Lemma wp_store_pst E σ l e v v' Q :
   to_val e = Some v → σ !! l = Some v' →
-  (ownP σ ★ ▷ (ownP (<[l:=v]>σ) -★ Q LitUnitV)) ⊑ wp E (Store (Loc l) e) Q.
+  (ownP σ ★ ▷ (ownP (<[l:=v]>σ) -★ Q $ LitV LitUnit)) ⊑ wp E (Store (Loc l) e) Q.
 Proof.
-  intros. rewrite -(wp_lift_atomic_det_step σ LitUnitV (<[l:=v]>σ) None)
+  intros. rewrite -(wp_lift_atomic_det_step σ (LitV LitUnit) (<[l:=v]>σ) None)
     ?right_id //; last by intros; inv_step; eauto.
 Qed.
 
 Lemma wp_cas_fail_pst E σ l e1 v1 e2 v2 v' Q :
   to_val e1 = Some v1 → to_val e2 = Some v2 → σ !! l = Some v' → v' ≠ v1 →
-  (ownP σ ★ ▷ (ownP σ -★ Q LitFalseV)) ⊑ wp E (Cas (Loc l) e1 e2) Q.
+  (ownP σ ★ ▷ (ownP σ -★ Q $ LitV $ LitBool false)) ⊑ wp E (Cas (Loc l) e1 e2) Q.
 Proof.
-  intros. rewrite -(wp_lift_atomic_det_step σ LitFalseV σ None) ?right_id //;
+  intros. rewrite -(wp_lift_atomic_det_step σ (LitV $ LitBool false) σ None) ?right_id //;
     last by intros; inv_step; eauto.
 Qed.
 
 Lemma wp_cas_suc_pst E σ l e1 v1 e2 v2 Q :
   to_val e1 = Some v1 → to_val e2 = Some v2 → σ !! l = Some v1 →
-  (ownP σ ★ ▷ (ownP (<[l:=v2]>σ) -★ Q LitTrueV)) ⊑ wp E (Cas (Loc l) e1 e2) Q.
+  (ownP σ ★ ▷ (ownP (<[l:=v2]>σ) -★ Q $ LitV $ LitBool true)) ⊑ wp E (Cas (Loc l) e1 e2) Q.
 Proof.
-  intros. rewrite -(wp_lift_atomic_det_step σ LitTrueV (<[l:=v2]>σ) None)
+  intros. rewrite -(wp_lift_atomic_det_step σ (LitV $ LitBool true) (<[l:=v2]>σ) None)
     ?right_id //; last by intros; inv_step; eauto.
 Qed.
 
 (** Base axioms for core primitives of the language: Stateless reductions *)
 Lemma wp_fork E e :
-  ▷ wp (Σ:=Σ) coPset_all e (λ _, True) ⊑ wp E (Fork e) (λ v, ■(v = LitUnitV)).
+  ▷ wp (Σ:=Σ) coPset_all e (λ _, True) ⊑ wp E (Fork e) (λ v, ■(v = LitV $ LitUnit)).
 Proof.
-  rewrite -(wp_lift_pure_det_step (Fork e) LitUnit (Some e)) //=;
+  rewrite -(wp_lift_pure_det_step (Fork e) (Lit LitUnit) (Some e)) //=;
     last by intros; inv_step; eauto.
   apply later_mono, sep_intro_True_l; last done.
-  by rewrite -(wp_value' _ _ LitUnit) //; apply const_intro.
+  by rewrite -(wp_value' _ _ (Lit _)) //; apply const_intro.
 Qed.
 
 Lemma wp_rec E erec e v Q :
@@ -88,30 +88,36 @@ Proof.
     ?right_id //=; last by intros; inv_step; eauto.
 Qed.
 
-Lemma wp_plus E n1 n2 Q :
-  ▷ Q (LitNatV (n1 + n2)) ⊑ wp E (Plus (LitNat n1) (LitNat n2)) Q.
+Lemma wp_un_op E op l l' Q :
+  un_op_eval op l = Some l' →
+  ▷ Q (LitV l') ⊑ wp E (UnOp op (Lit l)) Q.
 Proof.
-  rewrite -(wp_lift_pure_det_step (Plus _ _) (LitNat (n1 + n2)) None)
+  intros Heval. rewrite -(wp_lift_pure_det_step (UnOp op _) (Lit l') None)
     ?right_id //; last by intros; inv_step; eauto.
   by rewrite -wp_value'.
 Qed.
 
-Lemma wp_le_true E n1 n2 Q :
-  n1 ≤ n2 →
-  ▷ Q LitTrueV ⊑ wp E (Le (LitNat n1) (LitNat n2)) Q.
+Lemma wp_bin_op E op l1 l2 l' Q :
+  bin_op_eval op l1 l2 = Some l' →
+  ▷ Q (LitV l') ⊑ wp E (BinOp op (Lit l1) (Lit l2)) Q.
 Proof.
-  intros. rewrite -(wp_lift_pure_det_step (Le _ _) LitTrue None) ?right_id //;
-    last by intros; inv_step; eauto with omega.
+  intros Heval. rewrite -(wp_lift_pure_det_step (BinOp op _ _) (Lit l') None)
+    ?right_id //; last by intros; inv_step; eauto.
   by rewrite -wp_value'.
 Qed.
 
-Lemma wp_le_false E n1 n2 Q :
-  n1 > n2 →
-  ▷ Q LitFalseV ⊑ wp E (Le (LitNat n1) (LitNat n2)) Q.
+Lemma wp_if_true E e1 e2 Q :
+  ▷ wp E e1 Q ⊑ wp E (If (Lit $ LitBool true) e1 e2) Q.
 Proof.
-  intros. rewrite -(wp_lift_pure_det_step (Le _ _) LitFalse None) ?right_id //;
-    last by intros; inv_step; eauto with omega.
-  by rewrite -wp_value'.
+  rewrite -(wp_lift_pure_det_step (If _ _ _) e1 None)
+    ?right_id //; last by intros; inv_step; eauto.
+Qed.
+
+Lemma wp_if_false E e1 e2 Q :
+  ▷ wp E e2 Q ⊑ wp E (If (Lit $ LitBool false) e1 e2) Q.
+Proof.
+  rewrite -(wp_lift_pure_det_step (If _ _ _) e2 None)
+    ?right_id //; last by intros; inv_step; eauto.
 Qed.
 
 Lemma wp_fst E e1 v1 e2 v2 Q :
@@ -148,15 +154,4 @@ Proof.
     ?right_id //; last by intros; inv_step; eauto.
 Qed.
 
-(** Some derived stateless axioms *)
-Lemma wp_le E n1 n2 P Q :
-  (n1 ≤ n2 → P ⊑ ▷ Q LitTrueV) →
-  (n1 > n2 → P ⊑ ▷ Q LitFalseV) →
-  P ⊑ wp E (Le (LitNat n1) (LitNat n2)) Q.
-Proof.
-  intros; destruct (decide (n1 ≤ n2)).
-  * rewrite -wp_le_true; auto.
-  * rewrite -wp_le_false; auto with omega.
-Qed.
-
 End lifting.

heap_lang/sugar.v

@@ -5,11 +5,6 @@ Import uPred heap_lang.
 Definition Lam (e : {bind expr}) := Rec e.[ren(+1)].
 Definition Let (e1 : expr) (e2: {bind expr}) := App (Lam e2) e1.
 Definition Seq (e1 e2 : expr) := Let e1 e2.[ren(+1)].
-Definition If (e0 e1 e2 : expr) := Case e0 e1.[ren(+1)] e2.[ren(+1)].
-Definition Lt e1 e2 := Le (Plus e1 $ LitNat 1) e2.
-Definition Eq e1 e2 :=
-  Let e1 (Let e2.[ren(+1)]
-         (If (Le (Var 0) (Var 1)) (Le (Var 1) (Var 0)) LitFalse)).
 
 Definition LamV (e : {bind expr}) := RecV e.[ren(+1)].
 
@@ -21,8 +16,10 @@ Module notations.
   Bind Scope lang_scope with expr.
   Arguments wp {_ _} _ _%L _.
 
-  Coercion LitNat : nat >-> expr.
-  Coercion LitNatV : nat >-> val.
+  Coercion LitNat : nat >-> base_lit.
+  Coercion LitBool : bool >-> base_lit.
+  (* No coercion from base_lit to expr. This makes is slightly easier to tell
+     apart language and Coq expressions. *)
   Coercion Loc : loc >-> expr.
   Coercion LocV : loc >-> val.
   Coercion App : expr >-> Funclass.
@@ -35,10 +32,13 @@ Module notations.
   Notation "# n" := (ids (term:=expr) n) (at level 1, format "# n") : lang_scope.
   Notation "! e" := (Load e%L) (at level 10, format "! e") : lang_scope.
   Notation "'ref' e" := (Alloc e%L) (at level 30) : lang_scope.
-  Notation "e1 + e2" := (Plus e1%L e2%L)
+  Notation "e1 + e2" := (BinOp PlusOp e1%L e2%L)
     (at level 50, left associativity) : lang_scope.
-  Notation "e1 ≤ e2" := (Le e1%L e2%L) (at level 70) : lang_scope.
-  Notation "e1 < e2" := (Lt e1%L e2%L) (at level 70) : lang_scope.
+  Notation "e1 - e2" := (BinOp MinusOp e1%L e2%L)
+    (at level 50, left associativity) : lang_scope.
+  Notation "e1 ≤ e2" := (BinOp LeOp e1%L e2%L) (at level 70) : lang_scope.
+  Notation "e1 < e2" := (BinOp LtOp e1%L e2%L) (at level 70) : lang_scope.
+  Notation "e1 = e2" := (BinOp EqOp e1%L e2%L) (at level 70) : lang_scope.
   (* The unicode ← is already part of the notation "_ ← _; _" for bind. *)
   Notation "e1 <- e2" := (Store e1%L e2%L) (at level 80) : lang_scope.
   Notation "e1 ; e2" := (Seq e1%L e2%L)
@@ -65,35 +65,39 @@ Proof.
      to talk to the Autosubst guys. *)
   by asimpl.
 Qed.
+
 Lemma wp_let E e1 e2 Q :
   wp E e1 (λ v, ▷wp E (e2.[of_val v/]) Q) ⊑ wp E (Let e1 e2) Q.
 Proof.
   rewrite -(wp_bind [LetCtx e2]). apply wp_mono=>v.
   by rewrite -wp_lam //= to_of_val.
 Qed.
-Lemma wp_if_true E e1 e2 Q : ▷ wp E e1 Q ⊑ wp E (If LitTrue e1 e2) Q.
-Proof. rewrite -wp_case_inl //. by asimpl. Qed.
-Lemma wp_if_false E e1 e2 Q : ▷ wp E e2 Q ⊑ wp E (If LitFalse e1 e2) Q.
-Proof. rewrite -wp_case_inr //. by asimpl. Qed.
-Lemma wp_lt E n1 n2 P Q :
-  (n1 < n2 → P ⊑ ▷ Q LitTrueV) →
-  (n1 ≥ n2 → P ⊑ ▷ Q LitFalseV) →
-  P ⊑ wp E (Lt (LitNat n1) (LitNat n2)) Q.
+
+Lemma wp_le E (n1 n2 : nat) P Q :
+  (n1 ≤ n2 → P ⊑ ▷ Q (LitV true)) →
+  (n1 > n2 → P ⊑ ▷ Q (LitV false)) →
+  P ⊑ wp E (BinOp LeOp (Lit n1) (Lit n2)) Q.
+Proof.
+  intros ? ?. rewrite -wp_bin_op //; [].
+  destruct (decide _); by eauto with omega.
+Qed.
+
+Lemma wp_lt E (n1 n2 : nat) P Q :
+  (n1 < n2 → P ⊑ ▷ Q (LitV true)) →
+  (n1 ≥ n2 → P ⊑ ▷ Q (LitV false)) →
+  P ⊑ wp E (BinOp LtOp (Lit n1) (Lit n2)) Q.
 Proof.
-  intros; rewrite -(wp_bind [LeLCtx _]) -wp_plus -later_intro /=.
-  auto using wp_le with lia.
+  intros ? ?. rewrite -wp_bin_op //; [].
+  destruct (decide _); by eauto with omega.
 Qed.
-Lemma wp_eq E n1 n2 P Q :
-  (n1 = n2 → P ⊑ ▷ Q LitTrueV) →
-  (n1 ≠ n2 → P ⊑ ▷ Q LitFalseV) →
-  P ⊑ wp E (Eq (LitNat n1) (LitNat n2)) Q.
+
+Lemma wp_eq E (n1 n2 : nat) P Q :
+  (n1 = n2 → P ⊑ ▷ Q (LitV true)) →
+  (n1 ≠ n2 → P ⊑ ▷ Q (LitV false)) →
+  P ⊑ wp E (BinOp EqOp (Lit n1) (Lit n2)) Q.
 Proof.
-  intros HPeq HPne.
-  rewrite -wp_let -wp_value' // -later_intro; asimpl.
-  rewrite -wp_rec //; asimpl.
-  rewrite -(wp_bind [CaseCtx _ _]) -later_intro; asimpl.
-  apply wp_le; intros; asimpl.
-  * rewrite -wp_case_inl // -!later_intro. apply wp_le; auto with lia.
-  * rewrite -wp_case_inr // -later_intro -wp_value' //; auto with lia.
+  intros ? ?. rewrite -wp_bin_op //; [].
+  destruct (decide _); by eauto with omega.
 Qed.
+
 End suger.

heap_lang/tests.v

@@ -4,15 +4,14 @@ Require Import heap_lang.lifting heap_lang.sugar.
 Import heap_lang uPred notations.
 
 Module LangTests.
-  Definition add := (21 + 21)%L.
-  Goal ∀ σ, prim_step add σ 42 σ None.
+  Definition add := (Lit 21 + Lit 21)%L.
+  Goal ∀ σ, prim_step add σ (Lit 42) σ None.
   Proof. intros; do_step done. Qed.
-  Definition rec : expr := rec:: #0 #1. (* fix f x => f x *)
-  Definition rec_app : expr := rec 0.
+  Definition rec_app : expr := (rec:: #0 #1) (Lit 0).
   Goal ∀ σ, prim_step rec_app σ rec_app σ None.
   Proof. Set Printing All. intros; do_step done. Qed.
-  Definition lam : expr := λ: #0 + 21.
-  Goal ∀ σ, prim_step (App lam (LitNat 21)) σ add σ None.
+  Definition lam : expr := λ: #0 + Lit 21.
+  Goal ∀ σ, prim_step (lam (Lit 21)) σ add σ None.
   Proof. intros; do_step done. Qed.
 End LangTests.
 
@@ -22,8 +21,8 @@ Module LiftingTests.
   Implicit Types Q : val → iProp heap_lang Σ.
 
   (* FIXME: Fix levels so that we do not need the parenthesis here. *)
-  Definition e  : expr := let: ref 1 in #0 <- !#0 + 1; !#0.
-  Goal ∀ σ E, (ownP σ : iProp heap_lang Σ) ⊑ (wp E e (λ v, ■(v = 2))).
+  Definition e  : expr := let: ref (Lit 1) in #0 <- !#0 + Lit 1; !#0.
+  Goal ∀ σ E, (ownP σ : iProp heap_lang Σ) ⊑ (wp E e (λ v, ■(v = LitV 2))).
   Proof.
     move=> σ E. rewrite /e.
     rewrite -wp_let. rewrite -wp_alloc_pst; last done.
@@ -35,11 +34,11 @@ Module LiftingTests.
        all so nicely. *)
     rewrite -(wp_bindi (SeqCtx (Load (Loc _)))).
     rewrite -(wp_bindi (StoreRCtx (LocV _))).
-    rewrite -(wp_bindi (PlusLCtx _)).
+    rewrite -(wp_bindi (BinOpLCtx PlusOp _)).
     rewrite -wp_load_pst; first (apply sep_intro_True_r; first done); last first.
     { by rewrite lookup_insert. } (* RJ FIXME: figure out why apply and eapply fail. *)
     rewrite -later_intro. apply wand_intro_l. rewrite right_id.
-    rewrite -wp_plus -later_intro.
+    rewrite -wp_bin_op // -later_intro.
     rewrite -wp_store_pst; first (apply sep_intro_True_r; first done); last first.
     { by rewrite lookup_insert. }
     { done. }
@@ -57,12 +56,12 @@ Module LiftingTests.
   Definition FindPred' (n1 Sn1 n2 f : expr) : expr :=
     if Sn1 < n2 then f Sn1 else n1.
   Definition FindPred (n2 : expr) : expr :=
-    rec:: let: #1 + 1 in FindPred' #2 #0 n2.[ren(+3)] #1.
+    rec:: (let: #1 + Lit 1 in FindPred' #2 #0 n2.[ren(+3)] #1)%L.
   Definition Pred : expr :=
-    λ: if #0 ≤ 0 then 0 else FindPred #0 0.
+    λ: (if #0 ≤ Lit 0 then Lit 0 else FindPred #0 (Lit 0))%L.
 
   Lemma FindPred_spec n1 n2 E Q :
-    (■ (n1 < n2) ∧ Q (pred n2)) ⊑ wp E (FindPred n2 n1) Q.
+    (■ (n1 < n2) ∧ Q (LitV $ pred n2)) ⊑ wp E (FindPred (Lit n2) (Lit n1)) Q.
   Proof.
     revert n1. apply löb_all_1=>n1.
     rewrite -wp_rec //. asimpl.
@@ -70,45 +69,44 @@ Module LiftingTests.
     etransitivity; first (etransitivity; last eapply equiv_spec, later_and).
     { apply and_mono; first done. by rewrite -later_intro. }
     apply later_mono.
-    (* "rewrite -(wp_let _ _ (FindPred' n1 #0 n2 (FindPred n2)))" started to
+    (* Go on *)
+    (* FIXME "rewrite -(wp_let _ _ (FindPred' n1 #0 n2 (FindPred n2)))" started to
     fail after we changed # n to use ids instead of Var *)
-    pose proof (wp_let (Σ:=Σ) E (n1 + 1)%L (FindPred' n1 #0 n2 (FindPred n2)) Q).
+    pose proof (wp_let (Σ:=Σ) E (Lit n1 + Lit 1)%L
+                   (FindPred' (Lit n1) #0 (Lit n2) (FindPred (Lit n2))) Q).
     unfold Let, Lam in H; rewrite -H.
-    rewrite -wp_plus. asimpl.
-    rewrite -(wp_bindi (CaseCtx _ _)).
+    rewrite -wp_bin_op //. asimpl.
+    rewrite -(wp_bindi (IfCtx _ _)).
     rewrite -!later_intro /=.
     apply wp_lt; intros Hn12.
-    * (* TODO RJ: It would be better if we could use wp_if_true here
-         (and below). But we cannot, because the substitutions in there
-         got already unfolded. *)
-      rewrite -wp_case_inl //.
+    * rewrite -wp_if_true.
       rewrite -!later_intro. asimpl.
       rewrite (forall_elim (S n1)).
       eapply impl_elim; first by eapply and_elim_l. apply and_intro.
       + apply const_intro; omega.
       + by rewrite !and_elim_r.
-    * rewrite -wp_case_inr //.
+    * rewrite -wp_if_false.
       rewrite -!later_intro -wp_value' //.
       rewrite and_elim_r. apply const_elim_l=>Hle.
       by replace n1 with (pred n2) by omega.
   Qed.
 
   Lemma Pred_spec n E Q :
-    ▷Q (pred n) ⊑ wp E (Pred n) Q.
+    ▷Q (LitV $ pred n) ⊑ wp E (Pred $ Lit n) Q.
   Proof.
     rewrite -wp_lam //. asimpl.
-    rewrite -(wp_bindi (CaseCtx _ _)).
+    rewrite -(wp_bindi (IfCtx _ _)).
     apply later_mono, wp_le=> Hn.
-    - rewrite -wp_case_inl //.
+    - rewrite -wp_if_true.