Semantic typing for heap_lang.

Makefile.coq.local

@@ -17,6 +17,9 @@ concurrent_stacks: $(filter theories/concurrent_stacks/%,$(VOFILES))
 logrel: $(filter theories/logrel/%,$(VOFILES))
 .PHONY: logrel
 
+logrel_heaplang: $(filter theories/logrel_heaplang/%,$(VOFILES))
+.PHONY: logrel_heaplang
+
 hocap: $(filter theories/hocap/%,$(VOFILES))
 .PHONY: hocap
 

_CoqProject

@@ -72,6 +72,10 @@ theories/logrel/F_mu_ref_conc/examples/stack/FG_stack.v
 theories/logrel/F_mu_ref_conc/examples/stack/refinement.v
 theories/logrel/F_mu_ref_conc/examples/fact.v
 
+theories/logrel_heaplang/ltyping.v
+theories/logrel_heaplang/ltyping_safety.v
+theories/logrel_heaplang/lib/symbol_adt.v
+
 theories/hocap/abstract_bag.v
 theories/hocap/cg_bag.v
 theories/hocap/fg_bag.v

opam

@@ -9,6 +9,6 @@ build: [make "-j%{jobs}%"]
 install: [make "install"]
 remove: ["rm" "-rf" "%{lib}%/coq/user-contrib/iris_examples"]
 depends: [
-  "coq-iris" { (= "dev.2018-11-08.2.9eee9408") | (= "dev") }
+  "coq-iris" { (= "dev.2018-12-12.0.b1270b7d") | (= "dev") }
   "coq-autosubst" { = "dev.coq86" }
 ]

theories/logrel_heaplang/lib/symbol_adt.v

+From iris_examples.logrel_heaplang Require Export ltyping.
+From iris.heap_lang.lib Require Import assert.
+From iris.algebra Require Import auth.
+From iris.base_logic.lib Require Import invariants.
+From iris.heap_lang Require Import notation proofmode.
+
+(* Semantic typing of a symbol ADT (taken from Dreyer's POPL'18 talk) *)
+Definition symbol_adt_inc : val := λ: "x" <>, FAA "x" #1.
+Definition symbol_adt_check : val := λ: "x" "y", assert: "y" < !"x".
+Definition symbol_adt : val := λ: <>,
+  let: "x" := Alloc #0 in (symbol_adt_inc "x", symbol_adt_check "x").
+Definition symbol_adt_ty `{heapG Σ} : lty :=
+  (() → ∃ A, (() → A) * (A → ()))%lty.
+
+(* The required ghost theory *)
+Class symbolG Σ := { symbol_inG :> inG Σ (authR mnatUR) }.
+Definition symbolΣ : gFunctors := #[GFunctor (authR mnatUR)].
+
+Instance subG_symbolΣ {Σ} : subG symbolΣ Σ → symbolG Σ.
+Proof. solve_inG. Qed.
+
+Section symbol_ghosts.
+  Context `{!symbolG Σ}.
+
+  Definition counter (γ : gname) (n : nat) : iProp Σ := own γ (● (n : mnat)).
+  Definition symbol (γ : gname) (n : nat) : iProp Σ := own γ (◯ (S n : mnat)).
+
+  Global Instance counter_timeless γ n : Timeless (counter γ n).
+  Proof. apply _. Qed.
+  Global Instance symbol_timeless γ n : Timeless (symbol γ n).
+  Proof. apply _. Qed.
+
+  Lemma counter_exclusive γ n1 n2 : counter γ n1 -∗ counter γ n2 -∗ False.
+  Proof. apply bi.wand_intro_r. by rewrite -own_op own_valid auth_validI. Qed.
+  Global Instance symbol_persistent γ n : Persistent (symbol γ n).
+  Proof. apply _. Qed.
+
+  Lemma counter_alloc n : (|==> ∃ γ, counter γ n)%I.
+  Proof.
+    iMod (own_alloc (● (n:mnat) ⋅ ◯ (n:mnat))) as (γ) "[Hγ Hγf]"; first done.
+    iExists γ. by iFrame.
+  Qed.
+
+  Lemma counter_inc γ n : counter γ n ==∗ counter γ (S n) ∗ symbol γ n.
+  Proof.
+    rewrite -own_op.
+    apply own_update, auth_update_alloc, mnat_local_update. omega.
+  Qed.
+
+  Lemma symbol_obs γ s n : counter γ n -∗ symbol γ s -∗ ⌜(s < n)%nat⌝.
+  Proof.
+    iIntros "Hc Hs".
+    iDestruct (own_valid_2 with "Hc Hs") as %[?%mnat_included _]%auth_valid_discrete_2.
+    iPureIntro. omega.
+  Qed.
+End symbol_ghosts.
+
+Typeclasses Opaque counter symbol.
+
+Section ltyped_symbol_adt.
+  Context `{heapG Σ, symbolG Σ}.
+
+  Definition symbol_adtN := nroot .@ "symbol_adt".
+
+  Definition symbol_inv (γ : gname) (l : loc) : iProp Σ :=
+    (∃ n : nat, l ↦ #n ∗ counter γ n)%I.
+
+  Definition symbol_ctx (γ : gname) (l : loc) : iProp Σ :=
+    inv symbol_adtN (symbol_inv γ l).
+
+  Definition lty_symbol (γ : gname) : lty := Lty (λ w,
+    ∃ n : nat, ⌜w = #n⌝ ∧ symbol γ n)%I.
+
+  Lemma ltyped_symbol_adt Γ : Γ ⊨ symbol_adt : symbol_adt_ty.
+  Proof.
+    iIntros (vs) "!# _ /=". iApply wp_value.
+    iIntros "!#" (v ->). wp_lam. wp_alloc l as "Hl"; wp_pures.
+    iMod (counter_alloc 0) as (γ) "Hc".
+    iMod (inv_alloc symbol_adtN _ (symbol_inv γ l) with "[Hl Hc]") as "#?".
+    { iExists 0%nat. by iFrame. }
+    do 2 (wp_lam; wp_pures).
+    iExists (lty_symbol γ), _, _; repeat iSplit=> //.
+    - repeat rewrite /lty_car /=. iIntros "!#" (? ->). wp_pures.
+      iInv symbol_adtN as (n) ">[Hl Hc]". wp_faa.
+      iMod (counter_inc with "Hc") as "[Hc #Hs]".
+      iModIntro; iSplitL; last eauto.
+      iExists (S n). rewrite Nat2Z.inj_succ -Z.add_1_r. iFrame.
+    - repeat rewrite /lty_car /=. iIntros "!#" (v).
+      iDestruct 1 as (n ->) "#Hs". wp_pures. iApply wp_assert.
+      wp_bind (!_)%E. iInv symbol_adtN as (n') ">[Hl Hc]". wp_load.
+      iDestruct (symbol_obs with "Hc Hs") as %?. iModIntro. iSplitL.
+      { iExists n'. iFrame. }
+      wp_op. rewrite bool_decide_true; last lia. eauto.
+  Qed.
+End ltyped_symbol_adt.

theories/logrel_heaplang/ltyping.v

+From iris.heap_lang Require Export lifting metatheory.
+From iris.base_logic.lib Require Import invariants.
+From iris.heap_lang Require Import notation proofmode.
+
+(* The domain of semantic types: persistent Iris predicates over values *)
+Record lty `{heapG Σ} := Lty {
+  lty_car :> val → iProp Σ;
+  lty_persistent v : Persistent (lty_car v)
+}.
+Arguments Lty {_ _} _%I {_}.
+Arguments lty_car {_ _} _ _ : simpl never.
+Bind Scope lty_scope with lty.
+Delimit Scope lty_scope with lty.
+Existing Instance lty_persistent.
+
+(* The COFE structure on semantic types *)
+Section lty_ofe.
+  Context `{heapG Σ}.
+
+  Instance lty_equiv : Equiv lty := λ A B, ∀ w, A w ≡ B w.
+  Instance lty_dist : Dist lty := λ n A B, ∀ w, A w ≡{n}≡ B w.
+  Lemma lty_ofe_mixin : OfeMixin lty.
+  Proof. by apply (iso_ofe_mixin (lty_car : _ → val -c> _)). Qed.
+  Canonical Structure ltyC := OfeT lty lty_ofe_mixin.
+  Global Instance lty_cofe : Cofe ltyC.
+  Proof.
+    apply (iso_cofe_subtype' (λ A : val -c> iProp Σ, ∀ w, Persistent (A w))
+      (@Lty _ _) lty_car)=> //.
+    - apply _.
+    - apply limit_preserving_forall=> w.
+      by apply bi.limit_preserving_Persistent=> n ??.
+  Qed.
+
+  Global Instance lty_inhabited : Inhabited lty := populate (Lty inhabitant).
+
+  Global Instance lty_car_ne n : Proper (dist n ==> (=) ==> dist n) lty_car.
+  Proof. by intros A A' ? w ? <-. Qed.
+  Global Instance lty_car_proper : Proper ((≡) ==> (=) ==> (≡)) lty_car.
+  Proof. by intros A A' ? w ? <-. Qed.
+End lty_ofe.
+
+(* Typing for operators *)
+Class LTyUnboxed `{heapG Σ} (A : lty) :=
+  lty_unboxed v : A v -∗ ⌜ val_is_unboxed v ⌝.
+
+Class LTyUnOp `{heapG Σ} (op : un_op) (A B : lty) :=
+  lty_un_op v : A v -∗ ∃ w, ⌜ un_op_eval op v = Some w ⌝ ∧ B w.
+
+Class LTyBinOp `{heapG Σ} (op : bin_op) (A1 A2 B : lty) :=
+  lty_bin_op v1 v2 : A1 v1 -∗ A2 v2 -∗ ∃ w, ⌜ bin_op_eval op v1 v2 = Some w ⌝ ∧ B w.
+
+(* The type formers *)
+Section types.
+  Context `{heapG Σ}.
+
+  Definition lty_unit : lty := Lty (λ w, ⌜ w = #() ⌝%I).
+  Definition lty_bool : lty := Lty (λ w, ∃ b : bool, ⌜ w = #b ⌝)%I.
+  Definition lty_int : lty := Lty (λ w, ∃ n : Z, ⌜ w = #n ⌝)%I.
+
+  Definition lty_arr (A1 A2 : lty) : lty := Lty (λ w,
+    □ ∀ v, A1 v -∗ WP App w v {{ A2 }})%I.
+
+  Definition lty_prod (A1 A2 : lty) : lty := Lty (λ w,
+    ∃ w1 w2, ⌜w = PairV w1 w2⌝ ∧ A1 w1 ∧ A2 w2)%I.
+
+  Definition lty_sum (A1 A2 : lty) : lty := Lty (λ w,
+    (∃ w1, ⌜w = InjLV w1⌝ ∧ A1 w1) ∨ (∃ w2, ⌜w = InjRV w2⌝ ∧ A2 w2))%I.
+
+  Definition lty_forall (C : lty → lty) : lty := Lty (λ w,
+    □ ∀ A : lty, WP w #() {{ w, C A w }})%I.
+  Definition lty_exist (C : lty → lty) : lty := Lty (λ w,
+    ∃ A : lty, C A w)%I.
+
+  Definition lty_rec1 (C : ltyC -n> ltyC) (rec : lty) : lty := Lty (λ w,
+    ▷ C rec w)%I.
+  Instance lty_rec1_contractive C : Contractive (lty_rec1 C).
+  Proof. solve_contractive. Qed.
+  Definition lty_rec (C : ltyC -n> ltyC) : lty := fixpoint (lty_rec1 C).
+
+  Definition tyN := nroot .@ "ty".
+  Definition lty_ref (A : lty) : lty := Lty (λ w,
+    ∃ l : loc, ⌜w = #l⌝ ∧ inv (tyN .@ l) (∃ v, l ↦ v ∗ A v))%I.
+End types.
+
+(* Nice notations *)
+Notation "()" := lty_unit : lty_scope.
+Infix "→" := lty_arr : lty_scope.
+Infix "*" := lty_prod : lty_scope.
+Infix "+" := lty_sum : lty_scope.
+Notation "∀ A1 .. An , C" :=
+  (lty_forall (λ A1, .. (lty_forall (λ An, C%lty)) ..)) : lty_scope.
+Notation "∃ A1 .. An , C" :=
+  (lty_exist (λ A1, .. (lty_exist (λ An, C%lty)) ..)) : lty_scope.
+Notation "'ref' A" := (lty_ref A) : lty_scope.
+
+(* The semantic typing judgment *)
+Definition env_ltyped `{heapG Σ} (Γ : gmap string lty)
+    (vs : gmap string val) : iProp Σ :=
+  (⌜ ∀ x, is_Some (Γ !! x) ↔ is_Some (vs !! x) ⌝ ∧
+  [∗ map] i ↦ Av ∈ map_zip Γ vs, lty_car Av.1 Av.2)%I.
+Definition ltyped  `{heapG Σ}
+    (Γ : gmap string lty) (e : expr) (A : lty) : iProp Σ :=
+  (□ ∀ vs, env_ltyped Γ vs -∗ WP subst_map vs e {{ A }})%I.
+Notation "Γ ⊨ e : A" := (ltyped Γ e A)
+  (at level 100, e at next level, A at level 200).
+
+(* To unfold a recursive type, we need to take a step. We thus define the
+unfold operator to be the identity function. *)
+Definition rec_unfold : val := λ: "x", "x".
+
+Section types_properties.
+  Context `{heapG Σ}.
+  Implicit Types A B : lty.
+  Implicit Types C : lty → lty.
+
+  (* Boring stuff: all type formers are non-expansive *)
+  Global Instance lty_prod_ne : NonExpansive2 lty_prod.
+  Proof. solve_proper. Qed.
+  Global Instance lty_sum_ne : NonExpansive2 lty_sum.
+  Proof. solve_proper. Qed.
+  Global Instance lty_arr_ne : NonExpansive2 lty_arr.
+  Proof. solve_proper. Qed.
+  Global Instance lty_forall_ne n : Proper (pointwise_relation _ (dist n) ==> dist n) lty_forall.
+  Proof. solve_proper. Qed.
+  Global Instance lty_exist_ne n : Proper (pointwise_relation _ (dist n) ==> dist n) lty_exist.
+  Proof. solve_proper. Qed.
+  Global Instance lty_rec_ne n : Proper (dist n ==> dist n) lty_rec.
+  Proof. intros C C' HC. apply fixpoint_ne. solve_proper. Qed.
+  Global Instance lty_ref_ne : NonExpansive2 lty_ref.
+  Proof. solve_proper. Qed.
+
+  Lemma lty_rec_unfold (C : ltyC -n> ltyC) : lty_rec C ≡ lty_rec1 C (lty_rec C).
+  Proof. apply fixpoint_unfold. Qed.
+
+  (* Environments *)
+  Lemma env_ltyped_lookup Γ vs x A :
+    Γ !! x = Some A →
+    env_ltyped Γ vs -∗ ∃ v, ⌜ vs !! x = Some v ⌝ ∧ A v.
+  Proof.
+    iIntros (HΓx) "[Hlookup HΓ]". iDestruct "Hlookup" as %Hlookup.
+    destruct (proj1 (Hlookup x)) as [v Hx]; eauto.
+    iExists v. iSplit; first done. iApply (big_sepM_lookup _ _ x (A,v) with "HΓ").
+    by rewrite map_lookup_zip_with HΓx /= Hx.
+  Qed.
+  Lemma env_ltyped_insert Γ vs x A v :
+    A v -∗ env_ltyped Γ vs -∗
+    env_ltyped (binder_insert x A Γ) (binder_insert x v vs).
+  Proof.
+    destruct x as [|x]=> /=; first by auto.
+    iIntros "#HA [Hlookup #HΓ]". iDestruct "Hlookup" as %Hlookup. iSplit.
+    - iPureIntro=> y. rewrite !lookup_insert_is_Some'. naive_solver.
+    - rewrite -map_insert_zip_with. by iApply big_sepM_insert_2.
+  Qed.
+
+  (* Unboxed types *)
+  Global Instance lty_unit_unboxed : LTyUnboxed ().
+  Proof. by iIntros (v ->). Qed.
+  Global Instance lty_bool_unboxed : LTyUnboxed lty_bool.
+  Proof. iIntros (v). by iDestruct 1 as (b) "->". Qed.
+  Global Instance lty_int_unboxed : LTyUnboxed lty_int.
+  Proof. iIntros (v). by iDestruct 1 as (i) "->". Qed.
+  Global Instance lty_ref_unboxed A : LTyUnboxed (ref A).
+  Proof. iIntros (v). by iDestruct 1 as (i ->) "?". Qed.
+
+  (* Operator typing *)
+  Global Instance lty_bin_op_eq A : LTyBinOp EqOp A A lty_bool.
+  Proof. iIntros (v1 v2) "_ _". rewrite /bin_op_eval /lty_car /=. eauto. Qed.
+  Global Instance lty_bin_op_arith op :
+    TCElemOf op [PlusOp; MinusOp; MultOp; QuotOp; RemOp;
+                 AndOp; OrOp; XorOp; ShiftLOp; ShiftROp] →
+    LTyBinOp op lty_int lty_int lty_int.
+  Proof.
+    iIntros (? v1 v2); iDestruct 1 as (i1) "->"; iDestruct 1 as (i2) "->".
+    repeat match goal with H : TCElemOf _ _ |- _ => inversion_clear H end;
+      rewrite /lty_car /=; eauto.
+  Qed.
+  Global Instance lty_bin_op_compare op :
+    TCElemOf op [LeOp; LtOp] →
+    LTyBinOp op lty_int lty_int lty_bool.
+  Proof.
+    iIntros (? v1 v2); iDestruct 1 as (i1) "->"; iDestruct 1 as (i2) "->".
+    repeat match goal with H : TCElemOf _ _ |- _ => inversion_clear H end;
+      rewrite /lty_car /=; eauto.
+  Qed.
+  Global Instance lty_bin_op_bool op :
+    TCElemOf op [AndOp; OrOp; XorOp] →
+    LTyBinOp op lty_bool lty_bool lty_bool.
+  Proof.
+    iIntros (? v1 v2); iDestruct 1 as (i1) "->"; iDestruct 1 as (i2) "->".
+    repeat match goal with H : TCElemOf _ _ |- _ => inversion_clear H end;
+      rewrite /lty_car /=; eauto.
+  Qed.
+
+  (* The semantic typing rules *)
+  Lemma ltyped_var Γ (x : string) A : Γ !! x = Some A → Γ ⊨ x : A.
+  Proof.
+    iIntros (HΓx vs) "!# #HΓ /=".
+    iDestruct (env_ltyped_lookup with "HΓ") as (v ->) "HA"; first done.
+    by iApply wp_value.
+  Qed.
+
+  Lemma ltyped_unit Γ : Γ ⊨ #() : ().
+  Proof. iIntros (vs) "!# _ /=". by iApply wp_value. Qed.
+  Lemma ltyped_bool Γ (b : bool) : Γ ⊨ #b : lty_bool.
+  Proof. iIntros (vs) "!# _ /=". iApply wp_value. rewrite /lty_car /=. eauto. Qed.
+  Lemma ltyped_nat Γ (n : Z) : Γ ⊨ #n : lty_int.
+  Proof. iIntros (vs) "!# _ /=". iApply wp_value. rewrite /lty_car /=. eauto. Qed.
+
+  Lemma ltyped_rec Γ f x e A1 A2 :
+    (binder_insert f (A1 → A2)%lty (binder_insert x A1 Γ) ⊨ e : A2) -∗
+    Γ ⊨ (rec: f x := e) : A1 → A2.
+  Proof.
+    iIntros "#H" (vs) "!# #HΓ /=". wp_pures. iLöb as "IH".
+    iIntros "!#" (v) "#HA1". wp_pures. set (r := RecV f x _).
+    iSpecialize ("H" $! (binder_insert f r (binder_insert x v vs)) with "[#]").
+    { iApply (env_ltyped_insert with "IH"). by iApply env_ltyped_insert. }
+    destruct x as [|x], f as [|f]; rewrite /= -?subst_map_insert //.
+    destruct (decide (x = f)) as [->|].
+    - by rewrite subst_subst delete_idemp insert_insert -subst_map_insert.
+    - rewrite subst_subst_ne // -subst_map_insert.
+      by rewrite -delete_insert_ne // -subst_map_insert.
+  Qed.
+
+  Lemma ltyped_app Γ e1 e2 A1 A2 :
+    (Γ ⊨ e1 : A1 → A2) -∗ (Γ ⊨ e2 : A1) -∗ Γ ⊨ e1 e2 : A2.
+  Proof.
+    iIntros "#H1 #H2" (vs) "!# #HΓ /=".
+    iSpecialize ("H1" with "HΓ"). iSpecialize ("H2" with "HΓ").
+    wp_apply (wp_wand with "H2"); iIntros (w) "HA1".
+    wp_apply (wp_wand with "H1"); iIntros (v) "#HA". by iApply "HA".
+  Qed.
+
+  Lemma ltyped_fold Γ e (B : ltyC -n> ltyC) :
+    (Γ ⊨ e : B (lty_rec B)) -∗ Γ ⊨ e : lty_rec B.
+  Proof.
+    iIntros "#H" (vs) "!# #HΓ /=". iSpecialize ("H" with "HΓ").
+    wp_apply (wp_wand with "H"); iIntros (w) "#HB".
+    by iEval (rewrite lty_rec_unfold /lty_car /=).
+  Qed.
+  Lemma ltyped_unfold Γ e (B : ltyC -n> ltyC) :
+    (Γ ⊨ e : lty_rec B) -∗ Γ ⊨ rec_unfold e : B (lty_rec B).
+  Proof.
+    iIntros "#H" (vs) "!# #HΓ /=". iSpecialize ("H" with "HΓ").
+    wp_apply (wp_wand with "H"); iIntros (w) "HB".
+    iEval (rewrite lty_rec_unfold /lty_car /=) in "HB". by wp_lam.
+  Qed.
+
+  Lemma ltyped_tlam Γ e C : (∀ A, Γ ⊨ e : C A) -∗ Γ ⊨ (λ: <>, e) : ∀ A, C A.
+  Proof.
+    iIntros "#H" (vs) "!# #HΓ /=". wp_pures.
+    iIntros "!#" (A) "/=". wp_pures. by iApply ("H" $! A).
+  Qed.
+  Lemma ltyped_tapp Γ e C A : (Γ ⊨ e : ∀ A, C A) -∗ Γ ⊨ e #() : C A.
+  Proof.
+    iIntros "#H" (vs) "!# #HΓ /=". iSpecialize ("H" with "HΓ").
+    wp_apply (wp_wand with "H"); iIntros (w) "#HB". by iApply ("HB" $! A).
+  Qed.
+
+  Lemma ltyped_pack Γ e C A : (Γ ⊨ e : C A) -∗ Γ ⊨ e : ∃ A, C A.
+  Proof.
+    iIntros "#H" (vs) "!# #HΓ /=". iSpecialize ("H" with "HΓ").
+    wp_apply (wp_wand with "H"); iIntros (w) "#HB". by iExists A.
+  Qed.
+  Lemma ltyped_unpack Γ e1 e2 C B :
+    (Γ ⊨ e1 : ∃ A, C A) -∗ (∀ A, Γ ⊨ e2 : C A → B) -∗ Γ ⊨ e2 e1 : B.
+  Proof.
+    iIntros "#H1 #H2" (vs) "!# #HΓ /=". iSpecialize ("H1" with "HΓ").
+    wp_apply (wp_wand with "H1"); iIntros (v); iDestruct 1 as (A) "#HC".
+    iSpecialize ("H2" $! A with "HΓ").
+    wp_apply (wp_wand with "H2"); iIntros (w) "HCB". by iApply "HCB".
+  Qed.
+
+  Lemma ltyped_pair Γ e1 e2 A1 A2 :
+    (Γ ⊨ e1 : A1) -∗ (Γ ⊨ e2 : A2) -∗ Γ ⊨ (e1,e2) : A1 * A2.
+  Proof.
+    iIntros "#H1 #H2" (vs) "!# #HΓ /=".
+    iSpecialize ("H1" with "HΓ"). iSpecialize ("H2" with "HΓ").
+    wp_apply (wp_wand with "H2"); iIntros (w2) "#HA2".
+    wp_apply (wp_wand with "H1"); iIntros (w1) "#HA1".
+    wp_pures. iExists w1, w2. auto.
+  Qed.
+  Lemma ltyped_fst Γ e A1 A2 : (Γ ⊨ e : A1 * A2) -∗ Γ ⊨ Fst e : A1.
+  Proof.
+    iIntros "#H" (vs) "!# #HΓ /=". iSpecialize ("H" with "HΓ").
+    wp_apply (wp_wand with "H"); iIntros (w).
+    iDestruct 1 as (w1 w2 ->) "[??]". by wp_pures.
+  Qed.
+  Lemma ltyped_snd Γ e A1 A2 : (Γ ⊨ e : A1 * A2) -∗ Γ ⊨ Snd e : A2.
+  Proof.
+    iIntros "#H" (vs) "!# #HΓ /=". iSpecialize ("H" with "HΓ").
+    wp_apply (wp_wand with "H"); iIntros (w).
+    iDestruct 1 as (w1 w2 ->) "[??]". by wp_pures.
+  Qed.
+
+  Lemma ltyped_injl Γ e A1 A2 : (Γ ⊨ e : A1) -∗ Γ ⊨ InjL e : A1 + A2.
+  Proof.
+    iIntros "#H" (vs) "!# HΓ /=". iSpecialize ("H" with "HΓ").
+    wp_apply (wp_wand with "H"); iIntros (w) "#HA".
+    wp_pures. iLeft. iExists w. auto.
+  Qed.
+  Lemma ltyped_injr Γ e A1 A2 : (Γ ⊨ e : A2) -∗ Γ ⊨ InjR e : A1 + A2.
+  Proof.
+    iIntros "#H" (vs) "!# HΓ /=". iSpecialize ("H" with "HΓ").
+    wp_apply (wp_wand with "H"); iIntros (w) "#HA".
+    wp_pures. iRight. iExists w. auto.
+  Qed.
+  Lemma ltyped_case Γ e e1 e2 A1 A2 B :
+    (Γ ⊨ e : A1 + A2) -∗ (Γ ⊨ e1 : A1 → B) -∗ (Γ ⊨ e2 : A2 → B) -∗
+    Γ ⊨ Case e e1 e2 : B.
+  Proof.
+    iIntros "#H #H1 #H2" (vs) "!# #HΓ /=". iSpecialize ("H" with "HΓ").
+    iSpecialize ("H1" with "HΓ"). iSpecialize ("H2" with "HΓ").
+    wp_apply (wp_wand with "H"); iIntros (w) "#[HA|HA]".
+    - iDestruct "HA" as (w1 ->) "HA". wp_pures.
+      wp_apply (wp_wand with "H1"); iIntros (v) "#HAB". by iApply "HAB".
+    - iDestruct "HA" as (w2 ->) "HA". wp_pures.
+      wp_apply (wp_wand with "H2"); iIntros (v) "#HAB". by iApply "HAB".
+  Qed.
+
+  Lemma ltyped_un_op Γ e op A B :
+    LTyUnOp op A B → (Γ ⊨ e : A) -∗ Γ ⊨ UnOp op e : B.
+  Proof.
+    intros ?. iIntros "#H" (vs) "!# #HΓ /=". iSpecialize ("H" with "HΓ").
+    wp_apply (wp_wand with "H"); iIntros (v) "#HA".
+    iDestruct (lty_un_op with "HA") as (w ?) "#HB". by wp_unop.
+  Qed.
+  Lemma ltyped_bin_op Γ e1 e2 op A1 A2 B :
+    LTyBinOp op A1 A2 B → (Γ ⊨ e1 : A1) -∗ (Γ ⊨ e2 : A2) -∗ Γ ⊨ BinOp op e1 e2 : B.
+  Proof.
+    intros. iIntros "#H1 #H2" (vs) "!# #HΓ /=".
+    iSpecialize ("H1" with "HΓ"). iSpecialize ("H2" with "HΓ").
+    wp_apply (wp_wand with "H2"); iIntros (v2) "#HA2".
+    wp_apply (wp_wand with "H1"); iIntros (v1) "#HA1".
+    iDestruct (lty_bin_op with "HA1 HA2") as (w ?) "#HB". by wp_binop.
+  Qed.
+
+  Lemma ltyped_if Γ e e1 e2 B :
+    (Γ ⊨ e : lty_bool) -∗ (Γ ⊨ e1 : B) -∗ (Γ ⊨ e2 : B) -∗
+    Γ ⊨ (if: e then e1 else e2) : B.
+  Proof.
+    iIntros "#H #H1 #H2" (vs) "!# #HΓ /=". iSpecialize ("H" with "HΓ").
+    iSpecialize ("H1" with "HΓ"). iSpecialize ("H2" with "HΓ").
+    wp_apply (wp_wand with "H"); iIntros (w). iDestruct 1 as ([]) "->"; by wp_if.
+  Qed.
+
+  Lemma ltyped_fork Γ e : (Γ ⊨ e : ()) -∗ Γ ⊨ Fork e : ().
+  Proof.
+    iIntros "#H" (vs) "!# #HΓ /=". iSpecialize ("H" with "HΓ").
+    wp_apply wp_fork; last done. by iApply (wp_wand with "H").
+  Qed.
+
+  Lemma ltyped_alloc Γ e A : (Γ ⊨ e : A) -∗ Γ ⊨ ref e : ref A.
+  Proof.
+    iIntros "#H" (vs) "!# #HΓ /=". iSpecialize ("H" with "HΓ").
+    wp_bind (subst_map _ e). iApply (wp_wand with "H"); iIntros (w) "HA".
+    iApply wp_fupd. wp_alloc l as "Hl".
+    iMod (inv_alloc (tyN .@ l) _ (∃ v, l ↦ v ∗ A v)%I with "[Hl HA]") as "#?".
+    { iExists w. iFrame. }
+    iModIntro. iExists l; auto.
+  Qed.
+  Lemma ltyped_load Γ e A : (Γ ⊨ e : ref A) -∗ Γ ⊨ ! e : A.
+  Proof.
+    iIntros "#H" (vs) "!# #HΓ /=". iSpecialize ("H" with "HΓ").
+    wp_bind (subst_map _ e). iApply (wp_wand with "H"); iIntros (w).
+    iDestruct 1 as (l ->) "#?".
+    iInv (tyN.@l) as (v) "[>Hl HA]". wp_load. eauto 10.
+  Qed.
+  Lemma ltyped_store Γ e1 e2 A :
+    (Γ ⊨ e1 : ref A) -∗ (Γ ⊨ e2 : A) -∗ Γ ⊨ (e1 <- e2) : ().
+  Proof.
+    iIntros "#H1 #H2" (vs) "!# #HΓ /=".
+    iSpecialize ("H1" with "HΓ"). iSpecialize ("H2" with "HΓ").
+    wp_apply (wp_wand with "H2"); iIntros (w2) "#HA".
+    wp_apply (wp_wand with "H1"); iIntros (w1); iDestruct 1 as (l ->) "#?".
+    iInv (tyN.@l) as (v) "[>Hl _]". wp_store. eauto 10.
+  Qed.
+  Lemma ltyped_faa Γ e1 e2 :
+    (Γ ⊨ e1 : ref lty_int) -∗ (Γ ⊨ e2 : lty_int) -∗ Γ ⊨ FAA e1 e2 : lty_int.
+  Proof.
+    iIntros "#H1 #H2" (vs) "!# #HΓ /=".
+    iSpecialize ("H1" with "HΓ"). iSpecialize ("H2" with "HΓ").
+    wp_apply (wp_wand with "H2"); iIntros (w2); iDestruct 1 as (n) "->".
+    wp_apply (wp_wand with "H1"); iIntros (w1); iDestruct 1 as (l ->) "#?".
+    iInv (tyN.@l) as (v) "[>Hl Hv]"; iDestruct "Hv" as (n') "> ->".
+    wp_faa. iModIntro. eauto 10.
+  Qed.
+  Lemma ltyped_cas Γ A e1 e2 e3 :
+    LTyUnboxed A →
+    (Γ ⊨ e1 : ref A) -∗ (Γ ⊨ e2 : A) -∗ (Γ ⊨ e3 : A) -∗ Γ ⊨ CAS e1 e2 e3 : lty_bool.
+  Proof.
+    intros. iIntros "#H1 #H2 #H3" (vs) "!# #HΓ /=". iSpecialize ("H1" with "HΓ").
+    iSpecialize ("H2" with "HΓ"). iSpecialize ("H3" with "HΓ").
+    wp_apply (wp_wand with "H3"); iIntros (w3) "HA3".
+    wp_apply (wp_wand with "H2"); iIntros (w2) "HA2".