Prove more compatibility lemmas without unfolding the definition

F_mu_ref_conc/context_refinement.v

 From iris_logrel.F_mu_ref_conc Require Export fundamental_binary.
 From iris.proofmode Require Import tactics classes.
+From Autosubst Require Import Autosubst_Classes. (* for [subst] *)
 
 Inductive ctx_item :=
   (* λ-rec *)

F_mu_ref_conc/logrel_binary.v

@@ -155,6 +155,7 @@ Section logrel.
     | Tref τ' => interp_ref (interp E1 E2 τ')
     end.
   Notation "⟦ τ ⟧" := (interp ⊤ ⊤ τ).
+  Notation "⟦ τ ⟧ₑ" := (interp_expr ⊤ ⊤ ⟦ τ ⟧).
 
   Definition interp_env (Γ : stringmap type) (E1 E2 : coPset)
       (Δ : listC D) (vvs : stringmap (val * val)) : iProp Σ :=
@@ -218,9 +219,17 @@ Section logrel.
   Qed.
 
   Lemma interp_subst Δ2 τ τ' :
-    interp ⊤ ⊤ τ ((interp ⊤ ⊤ τ' Δ2) :: Δ2) ≡ interp ⊤ ⊤ (τ.[τ'/]) Δ2.
+    ⟦ τ ⟧ ((⟦ τ' ⟧ Δ2) :: Δ2) ≡ ⟦ τ.[τ'/] ⟧ Δ2.
   Proof. apply (interp_subst_up []). Qed.
 
+  Lemma interp_expr_subst Δ2 τ τ' ww :
+    ⟦ τ ⟧ₑ ((⟦ τ' ⟧ Δ2) :: Δ2) ww ≡ ⟦ τ.[τ'/] ⟧ₑ Δ2 ww.
+  Proof.
+    unfold interp_expr.
+    properness; auto.
+    apply interp_subst.
+  Qed.
+
   Lemma interp_env_dom Δ Γ E1 E2 vvs : interp_env Γ E1 E2 Δ vvs ⊢ ⌜dom _ Γ = dom _ vvs⌝.
   Proof. by iIntros "[% ?]". Qed.
 
@@ -398,16 +407,19 @@ Notation "Γ ⊨ e '≤log≤' e' : τ" :=
 Section monadic.
   Context `{logrelG Σ}.
 
-  Lemma related_ret τ Δ e1 e2 Γ `{Closed ∅ e1} `{Closed ∅ e2} E :
-    interp_expr E E ⟦ τ ⟧ Δ (e1,e2) -∗ {E,E;Δ;Γ} ⊨ e1 ≤log≤ e2 : τ%I.
+  Lemma related_ret E1 E2 Δ Γ e1 e2 τ `{Closed ∅ e1} `{Closed ∅ e2} :
+    interp_expr E1 E1 ⟦ τ ⟧ Δ (e1,e2) -∗ {E2,E2;Δ;Γ} ⊨ e1 ≤log≤ e2 : τ%I.
   Proof.
     iIntros "Hτ".
     rewrite bin_log_related_eq /bin_log_related_def.
     iIntros (vvs ρ) "#Hs #HΓ".
-    by rewrite /env_subst !Closed_subst_p_id.
+    rewrite /env_subst !Closed_subst_p_id.
+    iIntros (j K) "Hj /=". iModIntro.
+    iApply fupd_wp. iApply (fupd_mask_mono E1); auto.
+    by iMod ("Hτ" with "Hj") as "Hτ".
   Qed.
 
-  Lemma interp_ret τ Δ e1 e2 v1 v2 E :
+  Lemma interp_ret E τ Δ e1 e2 v1 v2 :
     to_val e1 = Some v1 →
     to_val e2 = Some v2 →
     ⟦ τ ⟧ Δ (v1,v2) -∗ interp_expr E E ⟦ τ ⟧ Δ (e1,e2).
@@ -419,7 +431,7 @@ Section monadic.
     iApply wp_value; eauto.
   Qed.
 
-  Lemma related_bind Δ Γ (e1 e2 : expr) (τ τ' : type) (K K' : list ectx_item) E :
+  Lemma related_bind E Δ Γ (e1 e2 : expr) (τ τ' : type) (K K' : list ectx_item) :
     ({E,E;Δ;Γ} ⊨ e1 ≤log≤ e2 : τ) -∗
     (∀ vv, ⟦ τ ⟧ Δ vv -∗ {E,E;Δ;Γ} ⊨ fill K (of_val (vv.1)) ≤log≤ fill K' (of_val (vv.2)) : τ') -∗
     ({E,E;Δ;Γ} ⊨ fill K e1 ≤log≤ fill K' e2 : τ').