Twiggle the notation

- Use the type of literals in `val` - Notation for `match` - "Better" coercions

Twiggle the notation
- Use the type of literals in `val` - Notation for `match` - "Better" coercions
11948e6e · Dan Frumin · 722c917b · 11948e6e · 11948e6e · 11948e6e
Commit 11948e6e authored Aug 17, 2017 by Dan Frumin
16 changed files
--- a/F_mu_ref_conc/examples/bit.v
+++ b/F_mu_ref_conc/examples/bit.v
@@ -7,14 +7,14 @@ Definition bitτ : type :=
                         (TVar 0))).

 Definition bit_bool : val :=
-  PackV ((λ: "b", "b"), (λ: "b", BinOp Xor "b" (#♭v true)), #♭v true).
+  PackV ((λ: "b", "b"), (λ: "b", "b" ⊕ true), #♭v true).

 Definition flip_nat : val := λ: "n",
-  if: BinOp Eq "n" (#n 0)
-  then (#n 1)
-  else (#n 0).
+  if: "n" = 0
+  then 1
+  else 0.
 Definition bit_nat : val :=
-  PackV ((λ: "b", BinOp Eq "b" (#n 0)), flip_nat, #nv 0).
+  PackV ((λ: "b", "b" = 0), flip_nat, #nv 0).

 Lemma bit_bool_type Γ :
    typed Γ bit_bool bitτ.

--- a/F_mu_ref_conc/examples/counter.v
+++ b/F_mu_ref_conc/examples/counter.v
@@ -9,29 +9,30 @@ From iris_logrel.F_mu_ref_conc Require Import hax.

 Definition CG_increment : val := λ: "x" "l" <>,
  acquire "l";;
-  "x" <- BinOp Add (#n 1) (! "x");;
+  "x" <- 1 + !"x";;
  release "l".

 Definition counter_read : val := λ: "x" <>, !"x".

 Definition CG_counter : val := λ: <>,
-  let: "l" := newlock #() in
-  let: "x" := ref (#n 0) in
+  let: "l" := newlock () in
+  let: "x" := ref 0 in
  (CG_increment "x" "l", counter_read "x").

 Definition FG_increment : val := λ: "v", rec: "inc" <> :=
  let: "c" := !"v" in
-  if: CAS "v" "c" (BinOp Add (#n 1) "c")
-  then #()
-  else "inc" #().
+  if: CAS "v" "c" (1 + "c")
+  then ()
+  else "inc" ().

 Definition FG_counter : val := λ: <>,
-  let: "x" := Alloc (#n 0) in
+  let: "x" := ref 0 in
  (FG_increment "x", counter_read "x").

 Section CG_Counter.
  Context `{logrelG Σ}.
  Variable (Δ : list (prodC valC valC -n> iProp Σ)).
+  Open Scope expr_scope.
  
  (* Coarse-grained increment *)  
  Lemma CG_increment_type Γ :
@@ -40,16 +41,16 @@ Section CG_Counter.

  Hint Resolve CG_increment_type : typeable.

-  Lemma bin_log_related_CG_increment_r Γ K E1 E2 t τ x n l :
+  Lemma bin_log_related_CG_increment_r Γ K E1 E2 t τ x (n : nat) l :
    nclose specN ⊆ E1 →
    (x ↦ₛ (#nv n) -∗ l ↦ₛ (#♭v false) -∗
    (x ↦ₛ (#nv S n) -∗ l ↦ₛ (#♭v false) -∗
-      ({E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (Lit Unit) : τ)) -∗
-    {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K ((CG_increment $/ LocV x $/ LocV l) Unit)%E : τ)%I.
+      ({E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (Lit ()) : τ)) -∗
+    {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K ((CG_increment $/ LocV x $/ LocV l) ())%E : τ)%I.
  Proof.
    iIntros (?) "Hx Hl Hlog".
    unfold CG_increment. unlock. simpl_subst/=.
-    rel_rec_r.
+    rel_rec_r.    
    rel_bind_r (acquire #l).
    iApply (bin_log_related_acquire_r with "Hl"); eauto. iIntros "Hl". simpl.
    rel_rec_r.
@@ -82,8 +83,8 @@ Section CG_Counter.

  Lemma bin_log_FG_increment_l Γ K E x n t τ :
    x ↦ᵢ (#nv n) -∗
-    (x ↦ᵢ (#nv (S n)) -∗ {E,E;Δ;Γ} ⊨ fill K (Lit Unit) ≤log≤ t : τ) -∗
-    {E,E;Δ;Γ} ⊨ fill K (FG_increment (Loc x) Unit) ≤log≤ t : τ.
+    (x ↦ᵢ (#nv (S n)) -∗ {E,E;Δ;Γ} ⊨ fill K (Lit ()) ≤log≤ t : τ) -∗
+    {E,E;Δ;Γ} ⊨ fill K (FG_increment #x ()) ≤log≤ t : τ.
  Proof.
    iIntros "Hx Hlog".
    iApply bin_log_related_wp_l.
@@ -125,7 +126,7 @@ Section CG_Counter.
    x ↦ₛ (#nv n)
    -∗ (x ↦ₛ (#nv n)
         -∗ {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (#n n)%E : τ)%I
-    -∗ {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (lamsubst counter_read (LocV x) #())%E : τ.
+    -∗ {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K ((counter_read $/ LocV x) ())%E : τ.
  Proof.
    iIntros "Hx Hlog".
    unfold counter_read. unlock. simpl.
@@ -141,8 +142,8 @@ Section CG_Counter.
    □ (|={E1,E2}=> ∃ n, x ↦ᵢ (#nv n) ∗ R(n) ∗
       ((∃ n, x ↦ᵢ (#nv n) ∗ R(n)) ={E2,E1}=∗ True) ∧
        (∀ m, x ↦ᵢ (#nv (S m)) ∗ R(m) -∗ 
-            {E2,E1;Δ;Γ} ⊨ fill K (Lit Unit) ≤log≤ t : τ))
-    -∗ ({E1,E1;Δ;Γ} ⊨ fill K ((FG_increment $/ LocV x) Unit)%E ≤log≤ t : τ).
+            {E2,E1;Δ;Γ} ⊨ fill K (Lit ()) ≤log≤ t : τ))
+    -∗ ({E1,E1;Δ;Γ} ⊨ fill K ((FG_increment $/ LocV x) ())%E ≤log≤ t : τ).
  Proof.
    iIntros "#H".
    unfold FG_increment. unlock. simpl.
@@ -186,7 +187,7 @@ Section CG_Counter.
       ((∃ n, x ↦ᵢ (#nv n) ∗ R(n)) ={E2,E1}=∗ True) ∧
        (∀ m, x ↦ᵢ (#nv m) ∗ R(m) -∗
            {E2,E1;Δ;Γ} ⊨ fill K (#n m) ≤log≤ t : τ))
-    -∗ {E1,E1;Δ;Γ} ⊨ fill K ((counter_read $/ LocV x) #())%E ≤log≤ t : τ.
+    -∗ {E1,E1;Δ;Γ} ⊨ fill K ((counter_read $/ LocV x) ())%E ≤log≤ t : τ.
  Proof.
    iIntros "#H".
    unfold counter_read. unlock. simpl.
@@ -233,7 +234,7 @@ Section CG_Counter.

  Lemma counter_read_refinement l cnt cnt' Γ :
    inv counterN (counter_inv l cnt cnt') -∗
-    {⊤,⊤;Δ;Γ} ⊨ counter_read $/ #cnt ≤log≤ counter_read $/ #cnt' : TArrow TUnit TNat.
+    {⊤,⊤;Δ;Γ} ⊨ counter_read $/ LocV cnt ≤log≤ counter_read $/ LocV cnt' : TArrow TUnit TNat.
  Proof.
    iIntros "#Hinv".
    iApply bin_log_related_arrow.
@@ -266,7 +267,7 @@ Section CG_Counter.
    iApply bin_log_related_arrow; try by (unlock; eauto).
    iAlways. iIntros ([? ?]) "/= [% %]"; simplify_eq/=.
    unlock. rel_rec_l. rel_rec_r.
-    rel_bind_r (newlock #())%E.
+    rel_bind_r (newlock ())%E.
    iApply (bin_log_related_newlock_r); auto; simpl.
    iIntros (l) "Hl".
    rel_rec_r.
@@ -287,7 +288,7 @@ Section CG_Counter.
      rel_rec_r.
      unfold CG_increment. unlock; simpl_subst/=.
      rel_rec_r.
-      replace (λ: <>, acquire #l ;; #cnt' <- BinOp Add (Nat 1) (! #cnt');; release #l)%E
+      replace (λ: <>, acquire #l ;; #cnt' <- 1 + (! #cnt');; release #l)%E
        with (CG_increment $/ LocV cnt' $/ LocV l)%E; last first.
      { unfold CG_increment. unlock; simpl_subst/=. reflexivity. }
      iApply (FG_CG_increment_refinement with "Hinv").

--- a/F_mu_ref_conc/examples/lateearlychoice.v
+++ b/F_mu_ref_conc/examples/lateearlychoice.v
@@ -8,13 +8,13 @@ From iris.program_logic Require Import adequacy.
 From iris_logrel.F_mu_ref_conc Require Import hax.

 Definition rand : val := λ: <>,
-  let: "y" := (ref (#♭ false))
-  in Fork ("y" <- #♭ true);;
+  let: "y" := ref false
+  in Fork ("y" <- true);;
     !"y".
 Definition lateChoice : val := λ: "x",
-  "x" <- #n 0;; rand #().
+  "x" <- 0;; rand ().
 Definition earlyChoice : val := λ: "x",
-  let: "r" := rand #() in "x" <- #n 0;; "r".
+  let: "r" := rand () in "x" <- 0;; "r".

 Section Refinement.
  Context `{logrelG Σ}.
@@ -25,7 +25,7 @@ Section Refinement.
    (∃ f, y ↦ᵢ (#♭v f) ∗ y' ↦ₛ (#♭v f))%I.

  Lemma wp_rand :
-    (WP rand #() {{ v, ⌜v = #♭v true⌝ ∨ ⌜v = #♭v false⌝}})%I.
+    (WP rand () {{ v, ⌜v = #♭v true⌝ ∨ ⌜v = #♭v false⌝}})%I.
  Proof.
    iStartProof.
    unfold rand. unlock.
@@ -45,7 +45,7 @@ Section Refinement.
  Lemma rand_l Δ Γ E1 K ρ t τ :
    ↑choiceN ⊆ E1 →
    spec_ctx ρ -∗ (∀ b, {E1,E1;Δ;Γ} ⊨ fill K (#♭ b) ≤log≤ t : τ)
-    -∗ {E1,E1;Δ;Γ} ⊨ fill K (rand #())%E ≤log≤ t : τ.
+    -∗ {E1,E1;Δ;Γ} ⊨ fill K (rand ())%E ≤log≤ t : τ.
  Proof.
    iIntros (?) "#Hs Hlog".
    unfold rand. unlock. simpl.
@@ -79,7 +79,7 @@ Section Refinement.
    ↑choiceN ⊆ E1 →
    spec_ctx ρ -∗
    {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (#♭ b) : τ -∗
-    {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (rand #())%E : τ.
+    {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (rand ())%E : τ.
  Proof.
    iIntros (??) "#Hs Hlog".
    unfold rand. unlock.
@@ -104,7 +104,7 @@ Section Refinement.
    rel_rec_l.
    rel_store_l. simpl.
    rel_rec_l.
-    rel_bind_l (rand #())%E.
+    rel_bind_l (rand ())%E.
    iApply (rand_l with "Hs"); eauto. simpl.
    by iApply "Hlog".
  Qed.
@@ -119,7 +119,7 @@ Section Refinement.
    unfold earlyChoice. unlock.
    rel_rec_r.

-    rel_bind_r (rand #())%E.
+    rel_bind_r (rand ())%E.
    iApply (rand_r b with "Hspec"); eauto. simpl.
    rel_rec_r.
    rel_store_r. simpl.
@@ -134,7 +134,7 @@ Section Refinement.
    iIntros "#Hspec Hx Hx'".
    unfold earlyChoice. unlock.
    rel_rec_l.
-    rel_bind_l (rand #())%E.
+    rel_bind_l (rand ())%E.
    iApply (rand_l with "Hspec"); eauto. simpl. iIntros (b).
    rel_rec_l.

@@ -142,12 +142,12 @@ Section Refinement.
    rel_rec_r.
    rel_store_r. simpl.
    rel_rec_r.
-    rel_bind_r (rand #())%E.
+    rel_bind_r (rand ())%E.
    iApply (rand_r b with "Hspec"); eauto. simpl.

    rel_store_l. simpl.
    rel_rec_l.
    rel_vals; simpl; eauto.
  Qed.
-    
+
 End Refinement.
--- a/F_mu_ref_conc/examples/lock.v
+++ b/F_mu_ref_conc/examples/lock.v
@@ -3,13 +3,13 @@ From iris_logrel.F_mu_ref_conc Require Import tactics rel_tactics.
 From iris_logrel.F_mu_ref_conc Require Export rules_binary typing fundamental_binary relational_properties notation.
 From iris.base_logic Require Import namespaces.

-Definition newlock : val := λ: <>, ref (#♭ false).
+Definition newlock : val := λ: <>, ref false.
 Definition acquire : val := rec: "acquire" "x" :=
-  if: CAS "x" (#♭ false) (#♭ true)
-  then #()
+  if: CAS "x" false true
+  then ()
  else "acquire" "x".

-Definition release : val := λ: "x", "x" <- #♭ false.
+Definition release : val := λ: "x", "x" <- false.
 Definition with_lock : val := λ: "e" "l" "x",
  acquire "l";;
  let: "v" := "e" "x" in
@@ -47,7 +47,7 @@ Section proof.

  Lemma steps_newlock ρ j K
    (Hcl : nclose specN ⊆ E1) :
-    spec_ctx ρ -∗ j ⤇ fill K (newlock #())%E
+    spec_ctx ρ -∗ j ⤇ fill K (newlock ())%E
    ={E1}=∗ ∃ l, j ⤇ fill K (Loc l) ∗ l ↦ₛ (#♭v false).
  Proof.
    iIntros "#Hspec Hj".
@@ -60,7 +60,7 @@ Section proof.
  Lemma bin_log_related_newlock_r Γ K t τ 
    (Hcl : nclose specN ⊆ E1) :
    (∀ l : loc, l ↦ₛ (#♭v false) -∗ {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (Loc l) : τ)%I
-    -∗ {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (newlock #())%E: τ.
+    -∗ {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (newlock ())%E: τ.
  Proof.
    iIntros "Hlog".
    unfold newlock. unlock.

--- a/F_mu_ref_conc/examples/par.v
+++ b/F_mu_ref_conc/examples/par.v
@@ -8,14 +8,15 @@ From iris.program_logic Require Import adequacy.
 From iris_logrel.F_mu_ref_conc Require Import hax.

 Definition par : val := λ: "e1" "e2",
-  let: "x1" := ref InjL #() in
-  Fork ("x1" <- InjR ("e1" #()));;
-  let: "x2" := "e2" #() in
-  let: "f" := rec: "f" <> := case: !"x1" of
-                              InjL => λ: <>, "f" #()
-                            | InjR => λ: "v", "v"
-                            end
-  in ("f" #(), "x2").
+  let: "x1" := ref InjL () in
+  Fork ("x1" <- InjR ("e1" ()));;
+  let: "x2" := "e2" () in
+  let: "f" := rec: "f" <> := 
+    match: !"x1" with
+      InjL <>  => "f" ()
+    | InjR "v" => "v"
+    end
+  in ("f" (), "x2").

 Lemma par_type Γ τ1 τ2 :
  typed Γ par

--- a/F_mu_ref_conc/fundamental_binary.v
+++ b/F_mu_ref_conc/fundamental_binary.v
@@ -51,7 +51,7 @@ Section masked.
    rewrite bin_log_related_eq.
    iIntros (vvs ρ) "#Hs HΓ"; iIntros (j K) "Hj /=".
    value_case. 
-    iExists UnitV; eauto.
+    iExists (LitV ()); eauto.
  Qed.

  Lemma bin_log_related_nat Δ Γ n : {E,E;Δ;Γ} ⊨ #n n ≤log≤ #n n : TNat.
@@ -513,7 +513,7 @@ Section masked.
    iIntros (vvs ρ) "#Hs HΓ"; iIntros (j K) "Hj /=".
    tp_fork j as i "Hi". fold subst_p.  iModIntro.
    iApply wp_fork. fold subst_p. iNext. iSplitL "Hj".
-    - iExists UnitV; eauto.
+    - iExists (LitV ()); eauto.
    - iSpecialize ("IH" with "Hs HΓ"). 
      iSpecialize ("IH" $! i []). simpl.
      iSpecialize ("IH" with "Hi"). 
@@ -583,7 +583,7 @@ Section masked.
    tp_store j.
    iMod ("Hclose" with "[Hw2 Hv2]").
    { iNext; iExists (_, _); simpl; iFrame. iFrame "IH2". }
-    iExists UnitV; iFrame; auto.
+    iExists (LitV ()); iFrame; auto.
  Qed.

  Lemma bin_log_related_CAS Δ Γ e1 e2 e3 e1' e2' e3' τ

--- a/F_mu_ref_conc/hax.v
+++ b/F_mu_ref_conc/hax.v
@@ -51,7 +51,7 @@ Section hax.
  Qed.

  Lemma weird_bind e Q :
-    WP App e #() {{ Q }} ⊢ WP e {{ v, WP (App v #()) {{ Q }} }}.
+    WP App e () {{ Q }} ⊢ WP e {{ v, WP (App v ()) {{ Q }} }}.
  Proof.
    (* ugh, turns out this is just the inverse bind:
       Check (wp_bind_inv (fun v => App v #())). *)
@@ -73,11 +73,11 @@ Section hax.
      iSplitR; eauto.
      iNext.
      iIntros (e2 σ2 efs) "Hpst". iDestruct "Hpst" as %Hpst.
-      iSpecialize ("wp" $! (App e2 #()) σ2 efs).
-      iAssert (⌜prim_step (e #()%E) σ1 (e2 #()%E) σ2 efs⌝)%I as "Hprim'".
+      iSpecialize ("wp" $! (App e2 ()) σ2 efs).
+      iAssert (⌜prim_step (e (Lit Unit))%E σ1 (e2 (Lit Unit)%E) σ2 efs⌝)%I as "Hprim'".
      { iPureIntro.
        inversion Hpst; simpl in *; subst; clear Hpst.
-        eapply (Ectx_step _ σ1 _ σ2 efs (K++[AppLCtx (#())%E])); simpl; eauto.
+        eapply (Ectx_step _ σ1 _ σ2 efs (K++[AppLCtx (())%E])); simpl; eauto.
          by rewrite fill_app.
          by rewrite fill_app. }
      iMod ("wp" with "Hprim'") as "[$ [wp $]]". iModIntro.

--- a/F_mu_ref_conc/lang.v
+++ b/F_mu_ref_conc/lang.v
@@ -96,9 +96,7 @@ Module lang.
  Inductive val :=
  | RecV (f x : binder) (e : expr) `{!Closed (x :b: f :b: ∅) e}
  | TLamV (e : expr) `{!Closed ∅ e} (* only closed lambdas are values *)
-  | UnitV
-  | NatV (n : nat)
-  | BoolV (b : bool)
+  | LitV (l : Literal)
  | PairV (v1 v2 : val)
  | InjLV (v : val)
  | InjRV (v : val)
@@ -108,8 +106,8 @@ Module lang.
  Bind Scope val_scope with val.

  (* Notation for bool and nat *)
-  Notation "'#♭v' b" := (BoolV b) (at level 20).
-  Notation "'#nv' n" := (NatV n) (at level 20).
+  Notation "'#♭v' b" := (LitV (Bool b)) (at level 20).
+  Notation "'#nv' n" := (LitV (Nat n)) (at level 20).

  Fixpoint binop_eval (op : binop) (v1 v2 : val) : option val :=
    match op,v1,v2 with
@@ -122,15 +120,13 @@ Module lang.
    | Xor,#♭v a,#♭v b => Some $ #♭v(xorb a b)
    | _,_,_ => None
    end.
-  Instance val_inh : Inhabited val := populate UnitV.
+  Instance val_inh : Inhabited val := populate (LitV Unit).

  Fixpoint of_val (v : val) : expr :=
    match v with
    | RecV f x e _ => Rec f x e
    | TLamV e _ => TLam e
-    | UnitV => Lit Unit
-    | NatV v => Lit $ Nat v
-    | BoolV v => Lit $ Bool v
+    | 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)
@@ -149,9 +145,7 @@ Module lang.
      if decide (Closed ∅ e)
      then Some (TLamV e)
      else None
-    | Lit Unit => Some UnitV
-    | Lit (Nat n) => Some (NatV n)
-    | Lit (Bool b) => Some (BoolV b)
+    | 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
@@ -169,7 +163,6 @@ Module lang.
  Lemma of_to_val e v : to_val e = Some v → of_val v = e.
  Proof.
    revert v; induction e; intros v ?; simplify_option_eq; auto with f_equal.
-    destruct l; simplify_option_eq; auto.
  Qed.

  Instance of_val_inj : Inj (=) (=) of_val.

--- a/F_mu_ref_conc/logrel_binary.v
+++ b/F_mu_ref_conc/logrel_binary.v
@@ -57,7 +57,7 @@ Section logrel.
  Solve Obligations with solve_proper.

  Program Definition interp_unit : listC D -n> D := λne Δ ww,
-    (⌜ww.1 = UnitV⌝ ∧ ⌜ww.2 = UnitV⌝)%I.
+    (⌜ww.1 = LitV Unit⌝ ∧ ⌜ww.2 = LitV Unit⌝)%I.
  Solve Obligations with solve_proper_alt.

  Program Definition interp_nat : listC D -n> D := λne Δ ww,
@@ -387,10 +387,10 @@ Section bin_log_def.
 End bin_log_def.

 Notation "'{' E1 ',' E2 ';' Δ ';' Γ '}' ⊨ e '≤log≤' e' : τ" :=
-  (bin_log_related E1 E2 Δ Γ e e' τ) (at level 74, E1,E2, e, e', τ at next level).
+  (bin_log_related E1 E2 Δ Γ e%E e'%E τ) (at level 74, E1,E2, e, e', τ at next level).
 Notation "'{' E1 ',' E2 ';' Γ '}' ⊨ e '≤log≤' e' : τ" :=
-  (bin_log_related E1 E2 [] Γ e e' τ) (at level 74, E1,E2, e, e', τ at next level).
+  (bin_log_related E1 E2 [] Γ e%E e'%E τ) (at level 74, E1,E2, e, e', τ at next level).
 (* Notation "Δ ',' Γ ⊨ e '≤log≤' e' : τ" := *)
 (*   (bin_log_related ⊤ ⊤ [] Γ e e' τ) (at level 74, e, e', τ at next level). *)
 Notation "Γ ⊨ e '≤log≤' e' : τ" :=
-  (bin_log_related ⊤ ⊤ [] Γ e e' τ) (at level 74, e, e', τ at next level).
+  (bin_log_related ⊤ ⊤ [] Γ e%E e'%E τ) (at level 74, e, e', τ at next level).
--- a/F_mu_ref_conc/notation.v
+++ b/F_mu_ref_conc/notation.v
@@ -4,10 +4,19 @@ From stdpp Require Import strings.

 Set Default Proof Using "Type".
 Import lang.
+
 Coercion App : expr >-> Funclass.
 Coercion Var : string >-> expr.
-Coercion BNamed : string >-> binder.
+Coercion of_val : val >-> expr.
+
+Coercion Nat : nat >-> Literal.
+Coercion Bool : bool >-> Literal.
+Definition litunit : unit -> Literal := fun _ => Unit.
+Coercion litunit : unit >-> Literal.
+
 Coercion Lit : Literal >-> expr.
+
+Coercion BNamed : string >-> binder.
 Notation "<>" := BAnon : binder_scope.

 Notation Lam x e := (Rec BAnon x e).
@@ -27,16 +36,34 @@ Notation "# l" := (Loc l%Z%V) (at level 8, format "# l") : expr_scope.
    first. *)
 Notation "( e1 , e2 , .. , en )" := (Pair .. (Pair e1 e2) .. en) : expr_scope.
 Notation "( e1 , e2 , .. , en )" := (PairV .. (PairV e1 e2) .. en) : val_scope.
+
+Notation Match e0 x1 e1 x2 e2 := (Case e0 (Lam x1 e1) (Lam x2 e2)).
+
+Notation "'match:' e0 'with' 'InjL' x1 => e1 | 'InjR' x2 => e2 'end'" :=
+  (Match e0 x1%bind e1 x2%bind e2)
+  (e0, x1, e1, x2, e2 at level 200,
+   format "'[' 'match:'  e0  'with'  '/' '[hv' 'InjL'  x1  =>  '[' e1 ']'  '/' |  'InjR'  x2  =>  '[' e2 ']'  '/' 'end' ']' ']'") : expr_scope.
+Notation "'match:' e0 'with' 'InjR' x1 => e1 | 'InjL' x2 => e2 'end'" :=
+  (Match e0 x2%bind e2 x1%bind e1)
+  (e0, x1, e1, x2, e2 at level 200, only parsing) : expr_scope.
+
 Notation "'case:' e0 'of' 'InjL' => e1 | 'InjR' => e2 'end'" :=
  (Case e0 e1 e2)
  (e0, e1, e2 at level 200) : expr_scope.
-Notation "#()" := Unit : expr_scope.
-Notation "#()" := UnitV : val_scope.

 Notation "! e" := (Load e%E) (at level 9, right associativity) : expr_scope.
 Notation "'ref' e" := (Alloc e%E)
  (at level 30, right associativity) : expr_scope.

+Notation "e1 + e2" := (BinOp Add e1%E e2%E)
+  (at level 50, left associativity) : expr_scope.
+Notation "e1 - e2" := (BinOp Sub e1%E e2%E)
+  (at level 50, left associativity) : expr_scope.
+Notation "e1 ≤ e2" := (BinOp Le e1%E e2%E) (at level 70) : expr_scope.
+Notation "e1 < e2" := (BinOp Lt e1%E e2%E) (at level 70) : expr_scope.
+Notation "e1 = e2" := (BinOp Eq e1%E e2%E) (at level 70) : expr_scope.
+Notation "e1 ⊕ e2" := (BinOp Xor e1%E e2%E) (at level 70) : expr_scope.
+
 (* The unicode ← is already part of the notation "_ ← _; _" for bind. *)
 Notation "e1 <- e2" := (Store e1%E e2%E) (at level 80) : expr_scope.
 Notation "'rec:' f x := e" := (Rec f%bind x%bind e%E)
@@ -71,4 +98,3 @@ Notation "'let:' x := e1 'in' e2" := (Let x%bind e1%E e2%E)
  (at level 102, x at level 1, e1, e2 at level 200,
   format "'[' '[hv' 'let:'  x  :=  '[' e1 ']'  'in'  ']' '/' e2 ']'") : expr_scope.

-Coercion of_val : val >-> expr.
--- a/F_mu_ref_conc/reflection.v
+++ b/F_mu_ref_conc/reflection.v
@@ -202,9 +202,7 @@ Fixpoint to_val (e : expr) : option val :=
  | TLam e =>
     if decide (is_closed ∅ e) is left H
     then Some (@TLamV (to_expr e) (is_closed_correct _ _ H)) else None
-  | Lit Unit => Some UnitV
-  | Lit (Nat n) => Some (NatV n)
-  | Lit (Bool b) => Some (BoolV b)
+  | Lit l => Some (LitV l)
  | Loc l => Some (LocV l)
  | Pair e1 e2 => v1 ← to_val e1; v2 ← to_val e2; Some (PairV v1 v2)
  | InjL e => InjLV <$> to_val e

--- a/F_mu_ref_conc/rel_tactics.v
+++ b/F_mu_ref_conc/rel_tactics.v
@@ -1050,7 +1050,7 @@ Section test.

  Lemma test_store Γ y y' :
    inv choiceN (choice_inv y y')
-    -∗ Γ ⊨ storeFalse #y ≤log≤ storeFalse #y' : TUnit.
+    -∗ Γ ⊨ (storeFalse #y) ≤log≤ (storeFalse #y') : TUnit.
  Proof.
    iIntros "#Hinv".
    unfold storeFalse. unlock.

--- a/F_mu_ref_conc/relational_properties.v
+++ b/F_mu_ref_conc/relational_properties.v
@@ -714,11 +714,11 @@ Section properties.
   {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (Fork e) : τ.
  Proof.
    iIntros "Hlog".
-    pose (Φ := (fun (v : val) => ∃ i, i ⤇ e ∗ ⌜v = #()⌝%V)%I).
+    pose (Φ := (fun (v : val) => ∃ i, i ⤇ e ∗ ⌜v = LitV tt⌝%V)%I).
    iApply (bin_log_related_step_r Φ with "[]"); cbv[Φ].
    { iIntros (ρ j K') "#Hspec Hj".
      tp_fork j as i "Hi".
-      iModIntro. iExists #()%V. iFrame. eauto.
+      iModIntro. iExists (LitV tt). iFrame. eauto.
    }
    iIntros (v). iDestruct 1 as (i) "[Hi %]"; subst.
    by iApply "Hlog".
@@ -768,11 +768,10 @@ Section properties.
    -∗ {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (Store (Loc l) e) : τ.
  Proof.
    iIntros (?) "Hl Hlog".
-    pose (Φ := (fun w => ⌜w = UnitV⌝ ∗ l ↦ₛ v')%I).
+    pose (Φ := (fun w => ⌜w = LitV tt⌝ ∗ l ↦ₛ v')%I).
    iApply (bin_log_related_step_r Φ with "[Hl]"); eauto.
    { cbv[Φ].
-      iIntros (ρ j K') "#Hs Hj /=". iExists UnitV.
-      Ltac solve_to_val ::= idtac.
+      iIntros (ρ j K') "#Hs Hj /=". iExists (LitV tt).
      tp_store j.
      iFrame. eauto. }
    iIntros (w) "[% Hl]"; subst.
@@ -790,12 +789,12 @@ Section properties.
    -∗ {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (CAS (Loc l) e1 e2) : τ.
  Proof.
    iIntros (????) "Hl Hlog".
-    pose (Φ := (fun w => ⌜w = BoolV false⌝ ∗ l ↦ₛ v)%I).
+    pose (Φ := (fun (w : val) => ⌜w = LitV false⌝ ∗ l ↦ₛ v)%I).
    iApply (bin_log_related_step_r Φ with "[Hl]"); eauto.
    { cbv[Φ].
      iIntros (ρ j K') "#Hs Hj /=".
      tp_cas_fail j; auto.
-      iExists (BoolV false). simpl.
+      iExists (LitV false). simpl.
      iFrame. eauto. }
    iIntros (w) "[% Hl]"; subst.
    iApply "Hlog".
@@ -812,12 +811,12 @@ Section properties.
    -∗ {E1,E2;Δ;Γ} ⊨ t ≤log≤ fill K (CAS (Loc l) e1 e2) : τ.
  Proof.
    iIntros (????) "Hl Hlog".
-    pose (Φ := (fun w => ⌜w = BoolV true⌝ ∗ l ↦ₛ v2)%I).
+    pose (Φ := (fun w => ⌜w = LitV true⌝ ∗ l ↦ₛ v2)%I).
    iApply (bin_log_related_step_r Φ with "[Hl]"); eauto.
    { cbv[Φ].
      iIntros (ρ j K') "#Hs Hj /=".
      tp_cas_suc j; auto.
-      iExists (BoolV true). simpl.
+      iExists (LitV true). simpl.
      iFrame. eauto. }
    iIntros (w) "[% Hl]"; subst.
    iApply "Hlog".

--- a/F_mu_ref_conc/rules.v
+++ b/F_mu_ref_conc/rules.v
@@ -62,7 +62,7 @@ Section lang_rules.
  Lemma wp_store E l v' e v :
    to_val e = Some v →
    {{{ ▷ l ↦ᵢ v' }}} Store (Loc l) e @ E
-    {{{ RET UnitV; l ↦ᵢ v }}}.
+    {{{ RET (LitV ()); l ↦ᵢ v }}}.
  Proof.
    iIntros (<-%of_to_val Φ) ">Hl HΦ".
    iApply wp_lift_atomic_head_step_no_fork; auto.
@@ -75,7 +75,7 @@ Section lang_rules.
  Lemma wp_cas_fail E l q v' e1 v1 e2 v2 :
    to_val e1 = Some v1 → to_val e2 = Some v2 → v' ≠ v1 →
    {{{ ▷ l ↦ᵢ{q} v' }}} CAS (Loc l) e1 e2 @ E
-    {{{ RET (BoolV false); l ↦ᵢ{q} v' }}}.
+    {{{ RET (LitV false); l ↦ᵢ{q} v' }}}.
  Proof.
    iIntros (<-%of_to_val <-%of_to_val ? Φ) ">Hl HΦ".
    iApply wp_lift_atomic_head_step_no_fork; auto.
@@ -88,7 +88,7 @@ Section lang_rules.
  Lemma wp_cas_suc E l e1 v1 e2 v2 :
    to_val e1 = Some v1 → to_val e2 = Some v2 →
    {{{ ▷ l ↦ᵢ v1 }}} CAS (Loc l) e1 e2 @ E
-    {{{ RET (BoolV true); l ↦ᵢ v2 }}}.
+    {{{ RET (LitV true); l ↦ᵢ v2 }}}.
  Proof.
    iIntros (<-%of_to_val <-%of_to_val Φ) ">Hl HΦ".
    iApply wp_lift_atomic_head_step_no_fork; auto.
@@ -175,7 +175,7 @@ Section lang_rules.
  Qed.

  Lemma wp_fork E e Φ :
-    ▷ (|={E}=> Φ UnitV) ∗ ▷ WP e {{ _, True }} ⊢ WP Fork e @ E {{ Φ }}.
+    ▷ (|={E}=> Φ (LitV ())) ∗ ▷ WP e {{ _, True }} ⊢ WP Fork e @ E {{ Φ }}.
  Proof.
  rewrite -(wp_lift_pure_det_head_step (Fork e) Unit [e]) //=; eauto.
  - rewrite -(wp_value_fupd _ _ (Lit Unit)); auto.
@@ -184,14 +184,14 @@ Section lang_rules.
  Qed.

  Lemma wp_if_true E e1 e2 Φ :
-    ▷ WP e1 @ E {{ Φ }} ⊢ WP If (#♭ true) e1 e2 @ E {{ Φ }}.
+    ▷ WP e1 @ E {{ Φ }} ⊢ WP If true e1 e2 @ E {{ Φ }}.
  Proof.
    rewrite -(wp_lift_pure_det_head_step_no_fork (If _ _ _) e1); eauto.
    intros; inv_head_step; eauto.
  Qed.

  Lemma wp_if_false E e1 e2 Φ :
-    ▷ WP e2 @ E {{ Φ }} ⊢ WP If (#♭ false) e1 e2 @ E {{ Φ }}.
+    ▷ WP e2 @ E {{ Φ }} ⊢ WP If false e1 e2 @ E {{ Φ }}.
  Proof.
    rewrite -(wp_lift_pure_det_head_step_no_fork (If _ _ _) e2); eauto.
    intros; inv_head_step; eauto.

--- a/F_mu_ref_conc/rules_binary.v
+++ b/F_mu_ref_conc/rules_binary.v
@@ -214,7 +214,7 @@ Section cfg.
  Lemma step_store E ρ j K l v' e v:
    to_val e = Some v → nclose specN ⊆ E →
    spec_ctx ρ ∗ j ⤇ fill K (Store (Loc l) e) ∗ l ↦ₛ v'
-    ={E}=∗ j ⤇ (fill K (Lit #())%E) ∗ l ↦ₛ v.
+    ={E}=∗ j ⤇ (fill K (Lit ())%E) ∗ l ↦ₛ v.
  Proof.
    iIntros (??) "(#Hinv & Hj & Hl)".
    rewrite /spec_ctx tpool_mapsto_eq /tpool_mapsto_def heapS_mapsto_eq /heapS_mapsto_def.
@@ -225,13 +225,13 @@ Section cfg.
      as %[[_ Hl%gen_heap_singleton_included]%prod_included _]%auth_valid_discrete_2.
    iMod (own_update_2 with "Hown Hj") as "[Hown Hj]".
    { by eapply auth_update, prod_local_update_1, singleton_local_update,
-        (exclusive_local_update _ (Excl (fill K (Lit #())%E))). }
+        (exclusive_local_update _ (Excl (fill K (Lit ())%E))). }
    iMod (own_update_2 with "Hown Hl") as "[Hown Hl]".
    { eapply auth_update, prod_local_update_2, singleton_local_update,
        (exclusive_local_update _ (1%Qp, to_agree v)); last done.
      by rewrite /to_gen_heap lookup_fmap Hl. }
    iFrame "Hj Hl". iApply "Hclose". iNext.
-    iExists (<[j:=fill K (Lit #())%E]> tp), (<[l:=v]>σ).
+    iExists (<[j:=fill K (Lit ())%E]> tp), (<[l:=v]>σ).
    rewrite to_gen_heap_insert to_tpool_insert'; last eauto. iFrame. iPureIntro.
    eapply rtc_r, step_insert_no_fork; eauto. econstructor; eauto.
  Qed.
@@ -352,7 +352,7 @@ Section cfg.

  Lemma step_fork E ρ j K e :
    nclose specN ⊆ E →
-    spec_ctx ρ ∗ j ⤇ fill K (Fork e) ={E}=∗ ∃ j', j ⤇ fill K (Lit #())%E ∗ j' ⤇ e.
+    spec_ctx ρ ∗ j ⤇ fill K (Fork e) ={E}=∗ ∃ j', j ⤇ fill K (Lit ())%E ∗ j' ⤇ e.
  Proof.
    iIntros (?) "[#Hspec Hj]". rewrite /spec_ctx tpool_mapsto_eq /tpool_mapsto_def.
    iInv specN as (tp σ) ">[Hown %]" "Hclose".

--- a/F_mu_ref_conc/tactics.v
+++ b/F_mu_ref_conc/tactics.v
@@ -16,9 +16,7 @@ Ltac reshape_val e tac :=
  | of_val ?v => v
  | Rec ?f ?x ?e => constr:(RecV f x e)
  | TLam ?e => constr:(TLamV e)
-  | Lit Unit => constr:(UnitV)
-  | Lit (Nat ?n) => constr:(NatV n)
-  | Lit (Bool ?b) => constr:(BoolV b)
+  | Lit ?l => constr:(LitV l)
  | Pair ?e1 ?e2 =>
    let v1 := go e1 in let v2 := go e2 in constr:(PairV v1 v2)
  | InjL ?e => let v := go e in constr:(InjLV v)
@@ -987,7 +985,7 @@ Lemma test1 E1 j K l n ρ :
  nclose specN ⊆ E1 →
  j ⤇ fill K (App (Lam "x" (Store (Loc l) (BinOp Add (Nat 1) (Var "x")))) (Load (Loc l))) -∗
  spec_ctx ρ -∗
-  l ↦ₛ (NatV n) ={E1}=∗ True.
+  l ↦ₛ (#nv n) ={E1}=∗ True.
 Proof.
  iIntros (?) "Hj #? Hl".
  tp_load j. tp_normalise j.
@@ -1005,7 +1003,7 @@ Lemma test2 E j K l n ρ :
       (App (If (CAS (Loc l) (Nat n) (Nat (n+1))) (Lam "x" (Fs