LogrelCC.logrel_unary
From iris.proofmode Require Import tactics.
From LogrelCC.program_logic Require Export weakestpre.
From LogrelCC Require Export lang rules_unary typing.
From iris.algebra Require Import list big_op.
From iris.base_logic.lib Require Import invariants.
Import uPred.
Definition logN : namespace := nroot .@ "logN".
From LogrelCC.program_logic Require Export weakestpre.
From LogrelCC Require Export lang rules_unary typing.
From iris.algebra Require Import list big_op.
From iris.base_logic.lib Require Import invariants.
Import uPred.
Definition logN : namespace := nroot .@ "logN".
interp : is a unary logical relation.
Section logrel.
Context `{heapG Σ}.
Notation D := (valC -n> iProp Σ).
Implicit Types τi : D.
Implicit Types Δ : listC D.
Implicit Types interp : listC D → D.
Program Definition interp_ectx (interp : listC D -n> D) (K : ectx)
: listC D -n> iProp Σ :=
λne Δ, (□ ∀ v, interp Δ v -∗ WP (fill K (of_val v)) {{_, True}})%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Program Definition interp_cont (interp : listC D -n> D)
: listC D -n> D :=
λne Δ w, (∃ K, ⌜w = ContV K⌝ ∧ interp_ectx interp K Δ)%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Program Definition interp_expr (interp : listC D -n> D) :
listC D -n> (exprC -n> iProp Σ) :=
λne Δ e, (∀ K, interp_ectx interp K Δ -∗ WP (fill K e) {{_, True}})%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Program Definition env_lookup (x : var) : listC D -n> D := λne Δ,
from_option id (cconst True)%I (Δ !! x).
Solve Obligations with repeat intros ?; simpl; solve_proper.
Definition interp_unit : listC D -n> D := λne Δ w, ⌜w = UnitV⌝%I.
Definition interp_nat : listC D -n> D := λne Δ w, ⌜∃ n, w = #nv n⌝%I.
Definition interp_bool : listC D -n> D := λne Δ w, ⌜∃ n, w = #♭v n⌝%I.
Program Definition interp_prod
(interp1 interp2 : listC D -n> D) : listC D -n> D := λne Δ w,
(∃ w1 w2, ⌜w = PairV w1 w2⌝ ∧ interp1 Δ w1 ∧ interp2 Δ w2)%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Program Definition interp_sum
(interp1 interp2 : listC D -n> D) : listC D -n> D := λne Δ w,
((∃ w1, ⌜w = InjLV w1⌝ ∧ interp1 Δ w1) ∨
(∃ w2, ⌜w = InjRV w2⌝ ∧ interp2 Δ w2))%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Program Definition interp_arrow
(interp1 interp2 : listC D -n> D) : listC D -n> D := λne Δ w,
(□ ∀ v, interp1 Δ v -∗
interp_expr interp2 Δ (App (of_val w) (of_val v)))%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Program Definition interp_forall
(interp : listC D -n> D) : listC D -n> D := λne Δ w,
(□ ∀ τi : D,
⌜∀ v, Persistent (τi v)⌝ -∗
interp_expr interp (τi :: Δ) (TApp (of_val w)))%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Definition interp_rec1
(interp : listC D -n> D) (Δ : listC D) (τi : D) : D := λne w,
(□ (∃ v, ⌜w = FoldV v⌝ ∧ ▷ interp (τi :: Δ) v))%I.
Global Instance interp_rec1_contractive
(interp : listC D -n> D) (Δ : listC D) : Contractive (interp_rec1 interp Δ).
Proof. by solve_contractive. Qed.
Program Definition interp_rec (interp : listC D -n> D) : listC D -n> D :=
λne Δ, fixpoint (interp_rec1 interp Δ).
Next Obligation.
intros interp n Δ1 Δ2 HΔ; apply fixpoint_ne => τi w. solve_proper.
Qed.
Program Definition interp_ref_inv (l : loc) : D -n> iProp Σ := λne τi,
(∃ v, l ↦ᵢ v ∗ τi v)%I.
Solve Obligations with solve_proper.
Program Definition interp_ref
(interp : listC D -n> D) : listC D -n> D := λne Δ w,
(∃ l, ⌜w = LocV l⌝ ∧ inv (logN .@ l) (interp_ref_inv l (interp Δ)))%I.
Solve Obligations with solve_proper.
Fixpoint interp (τ : type) : listC D -n> D :=
match τ return _ with
| TUnit => interp_unit
| TNat => interp_nat
| TBool => interp_bool
| TProd τ1 τ2 => interp_prod (interp τ1) (interp τ2)
| TSum τ1 τ2 => interp_sum (interp τ1) (interp τ2)
| TArrow τ1 τ2 => interp_arrow (interp τ1) (interp τ2)
| TVar x => env_lookup x
| TForall τ' => interp_forall (interp τ')
| TRec τ' => interp_rec (interp τ')
| Tref τ' => interp_ref (interp τ')
| TCont τ' => interp_cont (interp τ')
end.
Notation "⟦ τ ⟧" := (interp τ).
Definition interp_env (Γ : list type)
(Δ : listC D) (vs : list val) : iProp Σ :=
(⌜length Γ = length vs⌝ ∗ [∗] zip_with (λ τ, ⟦ τ ⟧ Δ) Γ vs)%I.
Notation "⟦ Γ ⟧*" := (interp_env Γ).
Class env_Persistent Δ :=
env_persistent : Forall (λ τi, ∀ v, Persistent (τi v)) Δ.
Global Instance env_persistent_nil : env_Persistent [].
Proof. by constructor. Qed.
Global Instance env_persistent_cons τi Δ :
(∀ v, Persistent (τi v)) → env_Persistent Δ → env_Persistent (τi :: Δ).
Proof. by constructor. Qed.
Global Instance env_persistent_lookup Δ x v :
env_Persistent Δ → Persistent (env_lookup x Δ v).
Proof. intros HΔ; revert x; induction HΔ=>-[|?] /=; apply _. Qed.
Global Instance interp_persistent τ Δ v :
env_Persistent Δ → Persistent (interp τ Δ v).
Proof.
revert v Δ; induction τ=> v Δ HΔ; simpl; try apply _.
rewrite /Persistent /interp_rec
(fixpoint_unfold (interp_rec1 ⟦ τ ⟧ Δ) v) /interp_rec1 /=
intuitionistically_into_persistently.
by apply persistently_intro'.
Qed.
Global Instance interp_env_persistent Γ Δ vs :
env_Persistent Δ → Persistent (⟦ Γ ⟧* Δ vs).
Proof.
intros HΔ.
apply sep_persistent; first apply _.
apply big_sepL_persistent_id.
revert vs. induction Γ; simpl; first econstructor.
destruct vs; first econstructor; simpl.
econstructor; first apply _.
eapply TCForall_ind; eauto; first econstructor.
intros; econstructor; eauto.
Qed.
Lemma interp_weaken Δ1 Π Δ2 τ :
⟦ τ.[upn (length Δ1) (ren (+ length Π))] ⟧ (Δ1 ++ Π ++ Δ2)
≡ ⟦ τ ⟧ (Δ1 ++ Δ2).
Proof.
revert Δ1 Π Δ2. induction τ=> Δ1 Π Δ2; simpl; auto.
- intros w; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
- intros w; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
- intros w; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
- apply fixpoint_proper=> τi w /=.
properness; auto. apply (IHτ (_ :: _)).
- rewrite iter_up; destruct lt_dec as [Hl | Hl]; simpl.
{ by rewrite !lookup_app_l. }
(* going around a weird bug in lia or perhaps rewriting system!
The following equalities are not solved by lia when they are
generated by as side conditions for rewriting lemmas. *)
assert (length Δ1 ≤ x) by lia.
assert (length Π ≤ length Δ1 + (length Π + (x - length Δ1)) - length Δ1)
by lia.
assert (length Δ1 ≤ length Δ1 + (length Π + (x - length Δ1))) by lia.
assert (length Δ1 + (length Π + (x - length Δ1)) - length Δ1 - length Π = x - length Δ1) by lia.
rewrite !lookup_app_r //.
by do 2 f_equiv.
- intros w; simpl; properness; auto. apply (IHτ (_ :: _)).
- intros w; simpl; properness; auto. by apply IHτ.
- intros w; simpl; properness; auto. by apply IHτ.
Qed.
Lemma interp_subst_up Δ1 Δ2 τ τ' :
⟦ τ ⟧ (Δ1 ++ interp τ' Δ2 :: Δ2)
≡ ⟦ τ.[upn (length Δ1) (τ' .: ids)] ⟧ (Δ1 ++ Δ2).
Proof.
revert Δ1 Δ2; induction τ=> Δ1 Δ2; simpl; auto.
- intros w; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
- intros w; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
- intros w; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
- apply fixpoint_proper=> τi w /=.
properness; auto. apply (IHτ (_ :: _)).
- rewrite iter_up; destruct lt_dec as [Hl | Hl]; simpl.
{ by rewrite !lookup_app_l. }
(* going around a weird bug in lia or perhaps rewriting system!
The following equalities are not solved by lia when they are
generated by as side conditions for rewriting lemmas. *)
assert (length Δ1 ≤ x) by lia.
rewrite lookup_app_r //.
destruct (x - length Δ1) as [|n] eqn:Heqn; rewrite /= Heqn.
{ symmetry. asimpl. apply (interp_weaken [] Δ1 Δ2 τ'). }
(* going around a weird bug in lia or perhaps rewriting system!
The following equalities are not solved by lia when they are
generated by as side conditions for rewriting lemmas. *)
assert (length Δ1 ≤ length Δ1 + n) by lia.
assert (n = length Δ1 + n - length Δ1) by lia.
rewrite !lookup_app_r //. by do 2 f_equiv /=.
- intros w; simpl; properness; auto. apply (IHτ (_ :: _)).
- intros w; simpl; properness; auto. by apply IHτ.
- intros w; simpl; properness; auto. by apply IHτ.
Qed.
Lemma interp_subst Δ2 τ τ' : ⟦ τ ⟧ (⟦ τ' ⟧ Δ2 :: Δ2) ≡ ⟦ τ.[τ'/] ⟧ Δ2.
Proof. apply (interp_subst_up []). Qed.
Lemma interp_env_length Δ Γ vs : ⟦ Γ ⟧* Δ vs ⊢ ⌜length Γ = length vs⌝.
Proof. by iIntros "[% ?]". Qed.
Lemma interp_env_Some_l Δ Γ vs x τ :
Γ !! x = Some τ → ⟦ Γ ⟧* Δ vs ⊢ ∃ v, ⌜vs !! x = Some v⌝ ∧ ⟦ τ ⟧ Δ v.
Proof.
iIntros (?) "[Hlen HΓ]"; iDestruct "Hlen" as %Hlen.
destruct (lookup_lt_is_Some_2 vs x) as [v Hv].
{ by rewrite -Hlen; apply lookup_lt_Some with τ. }
iExists v; iSplit. done. iApply (big_sepL_elem_of with "HΓ").
apply elem_of_list_lookup_2 with x.
rewrite lookup_zip_with; by simplify_option_eq.
Qed.
Lemma interp_env_nil Δ : True ⊢ ⟦ [] ⟧* Δ [].
Proof. iIntros "_"; iSplit; simpl; auto. Qed.
Lemma interp_env_cons Δ Γ vs τ v :
⟦ τ :: Γ ⟧* Δ (v :: vs) ⊣⊢ ⟦ τ ⟧ Δ v ∗ ⟦ Γ ⟧* Δ vs.
Proof.
rewrite /interp_env /= (assoc _ (⟦ _ ⟧ _ _)) -(comm _ ⌜(_ = _)⌝%I) -assoc.
by apply sep_proper; [apply pure_proper; omega|].
Qed.
Lemma interp_env_ren Δ (Γ : list type) (vs : list val) τi :
⟦ subst (ren (+1)) <$> Γ ⟧* (τi :: Δ) vs ⊣⊢ ⟦ Γ ⟧* Δ vs.
Proof.
apply sep_proper; [apply pure_proper; by rewrite fmap_length|].
revert Δ vs τi; induction Γ=> Δ [|v vs] τi; csimpl; auto.
apply sep_proper; auto. apply (interp_weaken [] [τi] Δ).
Qed.
End logrel.
Typeclasses Opaque interp_env.
Notation "⟦ τ ⟧" := (interp τ).
Notation "⟦ τ ⟧ₑ" := (interp_expr (interp τ)).
Notation "⟦ Γ ⟧*" := (interp_env Γ).
Context `{heapG Σ}.
Notation D := (valC -n> iProp Σ).
Implicit Types τi : D.
Implicit Types Δ : listC D.
Implicit Types interp : listC D → D.
Program Definition interp_ectx (interp : listC D -n> D) (K : ectx)
: listC D -n> iProp Σ :=
λne Δ, (□ ∀ v, interp Δ v -∗ WP (fill K (of_val v)) {{_, True}})%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Program Definition interp_cont (interp : listC D -n> D)
: listC D -n> D :=
λne Δ w, (∃ K, ⌜w = ContV K⌝ ∧ interp_ectx interp K Δ)%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Program Definition interp_expr (interp : listC D -n> D) :
listC D -n> (exprC -n> iProp Σ) :=
λne Δ e, (∀ K, interp_ectx interp K Δ -∗ WP (fill K e) {{_, True}})%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Program Definition env_lookup (x : var) : listC D -n> D := λne Δ,
from_option id (cconst True)%I (Δ !! x).
Solve Obligations with repeat intros ?; simpl; solve_proper.
Definition interp_unit : listC D -n> D := λne Δ w, ⌜w = UnitV⌝%I.
Definition interp_nat : listC D -n> D := λne Δ w, ⌜∃ n, w = #nv n⌝%I.
Definition interp_bool : listC D -n> D := λne Δ w, ⌜∃ n, w = #♭v n⌝%I.
Program Definition interp_prod
(interp1 interp2 : listC D -n> D) : listC D -n> D := λne Δ w,
(∃ w1 w2, ⌜w = PairV w1 w2⌝ ∧ interp1 Δ w1 ∧ interp2 Δ w2)%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Program Definition interp_sum
(interp1 interp2 : listC D -n> D) : listC D -n> D := λne Δ w,
((∃ w1, ⌜w = InjLV w1⌝ ∧ interp1 Δ w1) ∨
(∃ w2, ⌜w = InjRV w2⌝ ∧ interp2 Δ w2))%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Program Definition interp_arrow
(interp1 interp2 : listC D -n> D) : listC D -n> D := λne Δ w,
(□ ∀ v, interp1 Δ v -∗
interp_expr interp2 Δ (App (of_val w) (of_val v)))%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Program Definition interp_forall
(interp : listC D -n> D) : listC D -n> D := λne Δ w,
(□ ∀ τi : D,
⌜∀ v, Persistent (τi v)⌝ -∗
interp_expr interp (τi :: Δ) (TApp (of_val w)))%I.
Solve Obligations with repeat intros ?; simpl; solve_proper.
Definition interp_rec1
(interp : listC D -n> D) (Δ : listC D) (τi : D) : D := λne w,
(□ (∃ v, ⌜w = FoldV v⌝ ∧ ▷ interp (τi :: Δ) v))%I.
Global Instance interp_rec1_contractive
(interp : listC D -n> D) (Δ : listC D) : Contractive (interp_rec1 interp Δ).
Proof. by solve_contractive. Qed.
Program Definition interp_rec (interp : listC D -n> D) : listC D -n> D :=
λne Δ, fixpoint (interp_rec1 interp Δ).
Next Obligation.
intros interp n Δ1 Δ2 HΔ; apply fixpoint_ne => τi w. solve_proper.
Qed.
Program Definition interp_ref_inv (l : loc) : D -n> iProp Σ := λne τi,
(∃ v, l ↦ᵢ v ∗ τi v)%I.
Solve Obligations with solve_proper.
Program Definition interp_ref
(interp : listC D -n> D) : listC D -n> D := λne Δ w,
(∃ l, ⌜w = LocV l⌝ ∧ inv (logN .@ l) (interp_ref_inv l (interp Δ)))%I.
Solve Obligations with solve_proper.
Fixpoint interp (τ : type) : listC D -n> D :=
match τ return _ with
| TUnit => interp_unit
| TNat => interp_nat
| TBool => interp_bool
| TProd τ1 τ2 => interp_prod (interp τ1) (interp τ2)
| TSum τ1 τ2 => interp_sum (interp τ1) (interp τ2)
| TArrow τ1 τ2 => interp_arrow (interp τ1) (interp τ2)
| TVar x => env_lookup x
| TForall τ' => interp_forall (interp τ')
| TRec τ' => interp_rec (interp τ')
| Tref τ' => interp_ref (interp τ')
| TCont τ' => interp_cont (interp τ')
end.
Notation "⟦ τ ⟧" := (interp τ).
Definition interp_env (Γ : list type)
(Δ : listC D) (vs : list val) : iProp Σ :=
(⌜length Γ = length vs⌝ ∗ [∗] zip_with (λ τ, ⟦ τ ⟧ Δ) Γ vs)%I.
Notation "⟦ Γ ⟧*" := (interp_env Γ).
Class env_Persistent Δ :=
env_persistent : Forall (λ τi, ∀ v, Persistent (τi v)) Δ.
Global Instance env_persistent_nil : env_Persistent [].
Proof. by constructor. Qed.
Global Instance env_persistent_cons τi Δ :
(∀ v, Persistent (τi v)) → env_Persistent Δ → env_Persistent (τi :: Δ).
Proof. by constructor. Qed.
Global Instance env_persistent_lookup Δ x v :
env_Persistent Δ → Persistent (env_lookup x Δ v).
Proof. intros HΔ; revert x; induction HΔ=>-[|?] /=; apply _. Qed.
Global Instance interp_persistent τ Δ v :
env_Persistent Δ → Persistent (interp τ Δ v).
Proof.
revert v Δ; induction τ=> v Δ HΔ; simpl; try apply _.
rewrite /Persistent /interp_rec
(fixpoint_unfold (interp_rec1 ⟦ τ ⟧ Δ) v) /interp_rec1 /=
intuitionistically_into_persistently.
by apply persistently_intro'.
Qed.
Global Instance interp_env_persistent Γ Δ vs :
env_Persistent Δ → Persistent (⟦ Γ ⟧* Δ vs).
Proof.
intros HΔ.
apply sep_persistent; first apply _.
apply big_sepL_persistent_id.
revert vs. induction Γ; simpl; first econstructor.
destruct vs; first econstructor; simpl.
econstructor; first apply _.
eapply TCForall_ind; eauto; first econstructor.
intros; econstructor; eauto.
Qed.
Lemma interp_weaken Δ1 Π Δ2 τ :
⟦ τ.[upn (length Δ1) (ren (+ length Π))] ⟧ (Δ1 ++ Π ++ Δ2)
≡ ⟦ τ ⟧ (Δ1 ++ Δ2).
Proof.
revert Δ1 Π Δ2. induction τ=> Δ1 Π Δ2; simpl; auto.
- intros w; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
- intros w; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
- intros w; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
- apply fixpoint_proper=> τi w /=.
properness; auto. apply (IHτ (_ :: _)).
- rewrite iter_up; destruct lt_dec as [Hl | Hl]; simpl.
{ by rewrite !lookup_app_l. }
(* going around a weird bug in lia or perhaps rewriting system!
The following equalities are not solved by lia when they are
generated by as side conditions for rewriting lemmas. *)
assert (length Δ1 ≤ x) by lia.
assert (length Π ≤ length Δ1 + (length Π + (x - length Δ1)) - length Δ1)
by lia.
assert (length Δ1 ≤ length Δ1 + (length Π + (x - length Δ1))) by lia.
assert (length Δ1 + (length Π + (x - length Δ1)) - length Δ1 - length Π = x - length Δ1) by lia.
rewrite !lookup_app_r //.
by do 2 f_equiv.
- intros w; simpl; properness; auto. apply (IHτ (_ :: _)).
- intros w; simpl; properness; auto. by apply IHτ.
- intros w; simpl; properness; auto. by apply IHτ.
Qed.
Lemma interp_subst_up Δ1 Δ2 τ τ' :
⟦ τ ⟧ (Δ1 ++ interp τ' Δ2 :: Δ2)
≡ ⟦ τ.[upn (length Δ1) (τ' .: ids)] ⟧ (Δ1 ++ Δ2).
Proof.
revert Δ1 Δ2; induction τ=> Δ1 Δ2; simpl; auto.
- intros w; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
- intros w; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
- intros w; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
- apply fixpoint_proper=> τi w /=.
properness; auto. apply (IHτ (_ :: _)).
- rewrite iter_up; destruct lt_dec as [Hl | Hl]; simpl.
{ by rewrite !lookup_app_l. }
(* going around a weird bug in lia or perhaps rewriting system!
The following equalities are not solved by lia when they are
generated by as side conditions for rewriting lemmas. *)
assert (length Δ1 ≤ x) by lia.
rewrite lookup_app_r //.
destruct (x - length Δ1) as [|n] eqn:Heqn; rewrite /= Heqn.
{ symmetry. asimpl. apply (interp_weaken [] Δ1 Δ2 τ'). }
(* going around a weird bug in lia or perhaps rewriting system!
The following equalities are not solved by lia when they are
generated by as side conditions for rewriting lemmas. *)
assert (length Δ1 ≤ length Δ1 + n) by lia.
assert (n = length Δ1 + n - length Δ1) by lia.
rewrite !lookup_app_r //. by do 2 f_equiv /=.
- intros w; simpl; properness; auto. apply (IHτ (_ :: _)).
- intros w; simpl; properness; auto. by apply IHτ.
- intros w; simpl; properness; auto. by apply IHτ.
Qed.
Lemma interp_subst Δ2 τ τ' : ⟦ τ ⟧ (⟦ τ' ⟧ Δ2 :: Δ2) ≡ ⟦ τ.[τ'/] ⟧ Δ2.
Proof. apply (interp_subst_up []). Qed.
Lemma interp_env_length Δ Γ vs : ⟦ Γ ⟧* Δ vs ⊢ ⌜length Γ = length vs⌝.
Proof. by iIntros "[% ?]". Qed.
Lemma interp_env_Some_l Δ Γ vs x τ :
Γ !! x = Some τ → ⟦ Γ ⟧* Δ vs ⊢ ∃ v, ⌜vs !! x = Some v⌝ ∧ ⟦ τ ⟧ Δ v.
Proof.
iIntros (?) "[Hlen HΓ]"; iDestruct "Hlen" as %Hlen.
destruct (lookup_lt_is_Some_2 vs x) as [v Hv].
{ by rewrite -Hlen; apply lookup_lt_Some with τ. }
iExists v; iSplit. done. iApply (big_sepL_elem_of with "HΓ").
apply elem_of_list_lookup_2 with x.
rewrite lookup_zip_with; by simplify_option_eq.
Qed.
Lemma interp_env_nil Δ : True ⊢ ⟦ [] ⟧* Δ [].
Proof. iIntros "_"; iSplit; simpl; auto. Qed.
Lemma interp_env_cons Δ Γ vs τ v :
⟦ τ :: Γ ⟧* Δ (v :: vs) ⊣⊢ ⟦ τ ⟧ Δ v ∗ ⟦ Γ ⟧* Δ vs.
Proof.
rewrite /interp_env /= (assoc _ (⟦ _ ⟧ _ _)) -(comm _ ⌜(_ = _)⌝%I) -assoc.
by apply sep_proper; [apply pure_proper; omega|].
Qed.
Lemma interp_env_ren Δ (Γ : list type) (vs : list val) τi :
⟦ subst (ren (+1)) <$> Γ ⟧* (τi :: Δ) vs ⊣⊢ ⟦ Γ ⟧* Δ vs.
Proof.
apply sep_proper; [apply pure_proper; by rewrite fmap_length|].
revert Δ vs τi; induction Γ=> Δ [|v vs] τi; csimpl; auto.
apply sep_proper; auto. apply (interp_weaken [] [τi] Δ).
Qed.
End logrel.
Typeclasses Opaque interp_env.
Notation "⟦ τ ⟧" := (interp τ).
Notation "⟦ τ ⟧ₑ" := (interp_expr (interp τ)).
Notation "⟦ Γ ⟧*" := (interp_env Γ).