Simplify env_subst

theories/logrel/F_mu/fundamental.v

@@ -23,7 +23,7 @@ Section fundamental.
     induction 1; iIntros (Δ vs HΔ) "#HΓ"; cbn.
     - (* var *)
       iDestruct (interp_env_Some_l with "HΓ") as (v) "[% ?]"; first done.
-      rewrite /env_subst. simplify_option_eq. by iApply wp_value.
+      erewrite env_subst_lookup; eauto. by iApply wp_value.
     - (* unit *) by iApply wp_value.
     - (* pair *)
       smart_wp_bind (PairLCtx e2.[env_subst vs]) v "# Hv" IHtyped1.
@@ -48,16 +48,14 @@ Section fundamental.
       iDestruct (interp_env_length with "HΓ") as %?.
       iDestruct "Hv" as "[Hv|Hv]"; iDestruct "Hv" as (w) "[% Hw]"; simplify_eq/=.
       + iApply wp_pure_step_later; auto; asimpl. iNext.
-        erewrite typed_subst_head_simpl by naive_solver.
         iApply (IHtyped2 Δ (w :: vs)). iApply interp_env_cons; auto.
       + iApply wp_pure_step_later; auto 1 using to_of_val; asimpl. iNext.
-        erewrite typed_subst_head_simpl by naive_solver.
         iApply (IHtyped3 Δ (w :: vs)). iApply interp_env_cons; auto.
     - (* lam *)
       iApply wp_value; simpl; iAlways; iIntros (w) "#Hw".
       iDestruct (interp_env_length with "HΓ") as %?.
       iApply wp_pure_step_later; auto 1 using to_of_val. iNext.
-      asimpl. erewrite typed_subst_head_simpl by naive_solver.
+      asimpl.
       iApply (IHtyped Δ (w :: vs)). iApply interp_env_cons; auto.
     - (* app *)
       smart_wp_bind (AppLCtx (e2.[env_subst vs])) v "#Hv" IHtyped1.

theories/logrel/F_mu/soundness.v

@@ -10,7 +10,7 @@ Proof.
   intros Hlog ??. cut (adequate NotStuck e σ (λ _ _, True)); first (intros [_ ?]; eauto).
   eapply (wp_adequacy Σ); eauto.
   iIntros (Hinv ?). iModIntro. iExists (λ _ _, True%I). iSplit=> //.
-  rewrite -(empty_env_subst e).
+  replace e with e.[env_subst[]] by by asimpl.
   set (HΣ := IrisG _ _ Hinv (λ _ _ _, True)%I (λ _, True)%I).
   iApply (wp_wand with "[]"). iApply Hlog; eauto. by iApply interp_env_nil. auto.
 Qed.

theories/logrel/F_mu/typing.v

@@ -48,18 +48,18 @@ Proof.
   induction Htyped => s1 s2 Hs; f_equal/=; eauto using lookup_lt_Some with lia.
 Qed.
 
-Definition env_subst (vs : list val) (x : var) : expr :=
-  from_option id (Var x) (of_val <$> vs !! x).
+Fixpoint env_subst (vs : list val) : var → expr :=
+  match vs with
+  | [] => ids
+  | v :: vs' => #v .: env_subst vs'
+  end.
 
-Lemma typed_subst_head_simpl Δ τ e w ws :
-  Δ ⊢ₜ e : τ → length Δ = S (length ws) →
-  e.[# w .: env_subst ws] = e.[env_subst (w :: ws)].
+Lemma env_subst_lookup vs x v :
+  vs !! x = Some v → env_subst vs x = of_val v.
 Proof.
-  intros H1 H2. rewrite /env_subst.
-  eapply typed_subst_invariant; eauto=> /= -[|x] ? //=.
-  destruct (lookup_lt_is_Some_2 ws x) as [v' ?]; first lia.
-  by simplify_option_eq.
-Qed.
-
-Lemma empty_env_subst e : e.[env_subst []] = e.
-Proof. change (env_subst []) with (@ids expr _). by asimpl. Qed.
+  revert vs; induction x => vs.
+  - by destruct vs; inversion 1.
+  - destruct vs as [|w vs]; first by inversion 1.
+    rewrite -lookup_tail /=.
+    apply IHx.
+Qed.
\ No newline at end of file

theories/logrel/F_mu_ref/fundamental.v

@@ -22,7 +22,7 @@ Section fundamental.
     induction 1; iIntros (Δ vs HΔ) "#HΓ /=".
     - (* var *)
       iDestruct (interp_env_Some_l with "HΓ") as (v) "[% ?]"; first done.
-      rewrite /env_subst. simplify_option_eq. by iApply wp_value.
+      erewrite env_subst_lookup; eauto. by iApply wp_value.
     - (* unit *) by iApply wp_value.
     - (* pair *)
       smart_wp_bind (PairLCtx e2.[env_subst vs]) v "#Hv" IHtyped1.
@@ -47,16 +47,14 @@ Section fundamental.
       iDestruct (interp_env_length with "HΓ") as %?.
       iDestruct "Hv" as "[Hv|Hv]"; iDestruct "Hv" as (w) "[% Hw]"; simplify_eq/=.
       + iApply wp_pure_step_later; auto 1 using to_of_val; asimpl. iNext.
-        erewrite typed_subst_head_simpl by naive_solver.
         iApply (IHtyped2 Δ (w :: vs)). iApply interp_env_cons; auto.
       + iApply wp_pure_step_later; auto 1 using to_of_val; asimpl. iNext.
-        erewrite typed_subst_head_simpl by naive_solver.
         iApply (IHtyped3 Δ (w :: vs)). iApply interp_env_cons; auto.
     - (* lam *)
       iApply wp_value. simpl. iAlways. iIntros (w) "#Hw".
       iDestruct (interp_env_length with "HΓ") as %?.
       iApply wp_pure_step_later; auto 1 using to_of_val. iNext.
-      asimpl. erewrite typed_subst_head_simpl by naive_solver.
+      asimpl.
       iApply (IHtyped Δ (w :: vs)). iApply interp_env_cons; auto.
     - (* app *)
       smart_wp_bind (AppLCtx (e2.[env_subst vs])) v "#Hv" IHtyped1.

theories/logrel/F_mu_ref/fundamental_binary.v

@@ -48,8 +48,9 @@ Section fundamental.
     Γ !! x = Some τ → Γ ⊨ Var x ≤log≤ Var x : τ.
   Proof.
     iIntros (? Δ vvs ρ ?) "[#Hρ #HΓ]". iIntros (K) "Hj /=".
-    iDestruct (interp_env_Some_l with "HΓ") as ([v v']) "[% Hv]"; first done.
-    rewrite /env_subst !list_lookup_fmap; simplify_option_eq.
+    iDestruct (interp_env_Some_l with "HΓ") as ([v v']) "[Heq Hv]"; first done.
+    iDestruct "Heq" as %Heq.
+    erewrite !env_subst_lookup; rewrite ?list_lookup_fmap ?Heq; eauto.
     iApply wp_value; auto.
   Qed.
 
@@ -137,15 +138,15 @@ Section fundamental.
     iDestruct "Hiv" as "[Hiv|Hiv]".
     - iDestruct "Hiv" as ([w w']) "[% Hw]"; simplify_eq.
       iMod (step_case_inl _ _ K (of_val w') with "* [-]") as "Hz"; eauto.
+      simpl.
       iApply wp_pure_step_later; auto 1 using to_of_val. iNext.
-      asimpl. erewrite !n_closed_subst_head_simpl by (rewrite ?fmap_length; eauto).
-      iApply ('`IHHtyped2 _ ((w,w') :: vvs)); repeat iSplit; eauto.
+      asimpl. iApply ('`IHHtyped2 _ ((w,w') :: vvs)); repeat iSplit; eauto.
       iApply interp_env_cons; auto.
     - iDestruct "Hiv" as ([w w']) "[% Hw]"; simplify_eq.
       iMod (step_case_inr _ _ K (of_val w') with "* [-]") as "Hz"; eauto.
+      simpl.
       iApply wp_pure_step_later; auto 1 using to_of_val. iNext.
-      asimpl. erewrite !n_closed_subst_head_simpl by (rewrite ?fmap_length; eauto).
-      iApply ('`IHHtyped3 _ ((w,w') :: vvs)); repeat iSplit; eauto.
+      asimpl. iApply ('`IHHtyped3 _ ((w,w') :: vvs)); repeat iSplit; eauto.
       iApply interp_env_cons; auto.
   Qed.
 
@@ -161,8 +162,7 @@ Section fundamental.
     iDestruct (interp_env_length with "HΓ") as %?.
     iApply wp_pure_step_later; auto 1 using to_of_val. iNext.
     iMod (step_lam _ _ K' _ (of_val v') with "* [-]") as "Hz"; eauto.
-    asimpl. erewrite !n_closed_subst_head_simpl by (rewrite ?fmap_length; eauto).
-    iApply ('`IHHtyped _ ((v,v') :: vvs)); repeat iSplit; eauto.
+    asimpl. iApply ('`IHHtyped _ ((v,v') :: vvs)); repeat iSplit; eauto.
     iApply interp_env_cons; iSplit; auto.
   Qed.
 

theories/logrel/F_mu_ref/soundness.v

@@ -21,7 +21,7 @@ Proof.
   iModIntro. iExists (λ σ _, own γ (● to_gen_heap σ)); iFrame.
   set (HeapΣ := (HeapG Σ Hinv (GenHeapG _ _ Σ _ _ _ γ))).
   iApply (wp_wand with "[]").
-  - rewrite -(empty_env_subst e).
+  - replace e with e.[env_subst[]] by by asimpl.
     iApply (Hlog HeapΣ [] []). iApply (@interp_env_nil _ HeapΣ).
   - auto.
 Qed.
@@ -32,8 +32,8 @@ Corollary type_soundness e τ e' thp σ σ' :
   is_Some (to_val e') ∨ reducible e' σ'.
 Proof.
   intros ??. set (Σ := #[invΣ ; gen_heapΣ loc val]).
-  set (HG := HeapPreG Σ _ _). 
+  set (HG := HeapPreG Σ _ _).
   eapply (soundness Σ).
-  - intros ?. by apply fundamental. 
+  - intros ?. by apply fundamental.
   - eauto.
 Qed.

theories/logrel/F_mu_ref/soundness_binary.v

@@ -30,8 +30,9 @@ Proof.
   iApply wp_fupd. iApply (wp_wand with "[-]").
   - iPoseProof (Hlog _ _ with "[$Hcfg]") as "Hrel".
     { iApply (@logrel_binary.interp_env_nil Σ HeapΣ). }
-    rewrite (empty_env_subst e). iApply ("Hrel" $! []).
-    rewrite /tpool_mapsto (empty_env_subst e'). asimpl. iFrame.
+    replace e with e.[env_subst[]] at 2 by by asimpl.
+    iApply ("Hrel" $! []).
+    rewrite /tpool_mapsto. asimpl. iFrame.
   - iModIntro. iIntros (v'). iDestruct 1 as (v2) "[Hj #Hinterp]".
     iInv specN as ">Hinv" "Hclose".
     iDestruct "Hinv" as (e'' σ) "[Hown %]".

theories/logrel/F_mu_ref/typing.v

@@ -87,9 +87,21 @@ Proof.
       asimpl; rewrite H1; auto with lia.
 Qed.
 
-Definition env_subst (vs : list val) (x : var) : expr :=
-  from_option id (Var x) (of_val <$> vs !! x).
-
+Fixpoint env_subst (vs : list val) : var → expr :=
+  match vs with
+  | [] => ids
+  | v :: vs' => #v .: env_subst vs'
+  end.
+
+Lemma env_subst_lookup vs x v :
+  vs !! x = Some v → env_subst vs x = of_val v.
+Proof.
+  revert vs; induction x => vs.
+  - by destruct vs; inversion 1.
+  - destruct vs as [|w vs]; first by inversion 1.
+    rewrite -lookup_tail /=.
+    apply IHx.
+Qed.
 Lemma typed_n_closed Γ τ e : Γ ⊢ₜ e : τ → (∀ f, e.[upn (length Γ) f] = e).
 Proof.
   intros H. induction H => f; asimpl; simpl in *; auto with f_equal.
@@ -97,40 +109,6 @@ Proof.
   - by f_equal; rewrite map_length in IHtyped.
 Qed.
 
-Lemma n_closed_subst_head_simpl n e w ws :
-  (∀ f, e.[upn n f] = e) →
-  S (length ws) = n →
-  e.[of_val w .: env_subst ws] = e.[env_subst (w :: ws)].
-Proof.
-  intros H1 H2.
-  rewrite /env_subst. eapply n_closed_invariant; eauto=> /= -[|x] ? //=.
-  destruct (lookup_lt_is_Some_2 ws x) as [v' Hv]; first lia; simpl.
-  by rewrite Hv.
-Qed.
-
-Lemma typed_subst_head_simpl Δ τ e w ws :
-  Δ ⊢ₜ e : τ → length Δ = S (length ws) →
-  e.[of_val w .: env_subst ws] = e.[env_subst (w :: ws)].
-Proof. eauto using n_closed_subst_head_simpl, typed_n_closed. Qed.
-
-Lemma n_closed_subst_head_simpl_2 n e w w' ws :
-  (∀ f, e.[upn n f] = e) → (S (S (length ws))) = n →
-  e.[of_val w .: of_val w' .: env_subst ws] = e.[env_subst (w :: w' :: ws)].
-Proof.
-  intros H1 H2.
-  rewrite /env_subst. eapply n_closed_invariant; eauto => /= -[|[|x]] H3 //=.
-  destruct (lookup_lt_is_Some_2 ws x) as [v' Hv]; first lia; simpl.
-  by rewrite Hv.
-Qed.
-
-Lemma typed_subst_head_simpl_2 Δ τ e w w' ws :
-  Δ ⊢ₜ e : τ → length Δ = 2 + length ws →
-  e.[of_val w .: of_val w' .: env_subst ws] = e.[env_subst (w :: w' :: ws)].
-Proof. eauto using n_closed_subst_head_simpl_2, typed_n_closed. Qed.
-
-Lemma empty_env_subst e : e.[env_subst []] = e.
-Proof. change (env_subst []) with (@ids expr _). by asimpl. Qed.
-
 (** Weakening *)
 Lemma context_gen_weakening ξ Γ' Γ e τ :
   Γ' ++ Γ ⊢ₜ e : τ →

theories/logrel/F_mu_ref_conc/examples/fact.v

@@ -76,7 +76,6 @@ Section fact_equiv.
   Proof.
     iIntros (? vs ρ _) "[#HE HΔ]".
     iDestruct (interp_env_length with "HΔ") as %?; destruct vs; simplify_eq.
-    rewrite !empty_env_subst.
     iClear "HΔ". simpl.
     iIntros (j K) "Hj"; simpl.
     iApply wp_value; iExists (LamV _); iFrame.
@@ -155,7 +154,6 @@ Section fact_equiv.
   Proof.
     iIntros (? vs ρ _) "[#HE HΔ]".
     iDestruct (interp_env_length with "HΔ") as %?; destruct vs; simplify_eq.
-    rewrite !empty_env_subst.
     iClear "HΔ". simpl.
     iIntros (j K) "Hj"; simpl.
     iApply wp_value; iExists (RecV _); iFrame.

theories/logrel/F_mu_ref_conc/fundamental_binary.v

@@ -48,8 +48,10 @@ Section fundamental.
     Γ !! x = Some τ → Γ ⊨ Var x ≤log≤ Var x : τ.
   Proof.
     iIntros (? Δ vvs ρ ?) "#(Hs & HΓ)"; iIntros (j K) "Hj /=".
-    iDestruct (interp_env_Some_l with "HΓ") as ([v v']) "[% ?]"; first done.
-    rewrite /env_subst !list_lookup_fmap; simplify_option_eq. iApply wp_value; eauto.
+    iDestruct (interp_env_Some_l with "HΓ") as ([v v']) "[Heq ?]"; first done.
+    iDestruct "Heq" as %Heq.
+    erewrite !env_subst_lookup; rewrite ?list_lookup_fmap ?Heq; eauto.
+    iApply wp_value; eauto.
   Qed.
 
   Lemma bin_log_related_unit Γ : Γ ⊨ Unit ≤log≤ Unit : TUnit.
@@ -150,14 +152,12 @@ Section fundamental.
       iMod (step_case_inl with "[Hs Hv]") as "Hz"; eauto.
       iApply wp_pure_step_later; auto. fold of_val. iModIntro. iNext.
       asimpl.
-      erewrite !n_closed_subst_head_simpl by (rewrite ?fmap_length; eauto).
       iApply ('`IHHtyped2 _ ((w,w') :: vvs)). repeat iSplit; eauto.
       iApply interp_env_cons; auto.
     - iApply fupd_wp.
       iMod (step_case_inr with "[Hs Hv]") as "Hz"; eauto.
       iApply wp_pure_step_later; auto. fold of_val. iModIntro. iNext.
       asimpl.
-      erewrite !n_closed_subst_head_simpl by (rewrite ?fmap_length; eauto).
       iApply ('`IHHtyped3 _ ((w,w') :: vvs)); repeat iSplit; eauto.
       iApply interp_env_cons; auto.
   Qed.
@@ -214,7 +214,6 @@ Section fundamental.
     iApply fupd_wp.
     iMod (step_rec _ _ j' K' _ (of_val v') v' with "* [-]") as "Hz"; eauto.
     asimpl. change (Rec ?e) with (of_val (RecV e)).
-    erewrite !n_closed_subst_head_simpl_2 by (rewrite ?fmap_length; eauto).
     iApply ('`IHHtyped _ ((_,_) :: (v,v') :: vvs)); repeat iSplit; eauto.
     iModIntro.
     rewrite !interp_env_cons; iSplit; try iApply interp_env_cons; auto.
@@ -234,7 +233,6 @@ Section fundamental.
     iApply fupd_wp.
     iMod (step_lam _ _ j' K' _ (of_val v') v' with "* [-]") as "Hz"; eauto.
     asimpl. iFrame "#". change (Lam ?e) with (of_val (LamV e)).
-    erewrite !n_closed_subst_head_simpl by (rewrite ?fmap_length; eauto).
     iApply ('`IHHtyped _  ((v,v') :: vvs)); repeat iSplit; eauto.
     iModIntro.
     rewrite !interp_env_cons; iSplit; try iApply interp_env_cons; auto.
@@ -254,7 +252,6 @@ Section fundamental.
     iMod (step_letin _ _ j K with "[-]") as "Hz"; eauto.
     iApply wp_pure_step_later; auto. iModIntro.
     asimpl.
-    erewrite !n_closed_subst_head_simpl by (rewrite ?fmap_length; eauto).
     iApply ('`IHHtyped2 _ ((v, v') :: vvs)); repeat iSplit; eauto.
     rewrite !interp_env_cons; iSplit; try iApply interp_env_cons; auto.
   Qed.

theories/logrel/F_mu_ref_conc/fundamental_unary.v

@@ -23,7 +23,8 @@ Section typed_interp.
     induction 1; iIntros (Δ vs HΔ) "#HΓ /=".
     - (* var *)
       iDestruct (interp_env_Some_l with "HΓ") as (v) "[% ?]"; first done.
-      rewrite /env_subst. simplify_option_eq. by iApply wp_value.
+      erewrite env_subst_lookup; eauto.
+        by iApply wp_value.
     - (* unit *) iApply wp_value; trivial.
     - (* nat *) iApply wp_value; simpl; eauto.
     - (* bool *) iApply wp_value; simpl; eauto.
@@ -57,10 +58,8 @@ Section typed_interp.
       iDestruct (interp_env_length with "HΓ") as %?.
       iDestruct "Hv" as "[Hv|Hv]"; iDestruct "Hv" as (w) "[% Hw]"; simplify_eq/=.
       + iApply wp_pure_step_later; auto 1 using to_of_val; asimpl. iNext.
-        erewrite typed_subst_head_simpl by naive_solver.
         iApply (IHtyped2 Δ (w :: vs)). iApply interp_env_cons; auto.
       + iApply wp_pure_step_later; auto 1 using to_of_val; asimpl. iNext.
-        erewrite typed_subst_head_simpl by naive_solver.
         iApply (IHtyped3 Δ (w :: vs)). iApply interp_env_cons; auto.
     - (* If *)
       smart_wp_bind (IfCtx _ _) v "#Hv" IHtyped1; cbn.
@@ -72,7 +71,6 @@ Section typed_interp.
       iDestruct (interp_env_length with "HΓ") as %?.
       iApply wp_pure_step_later; auto 1 using to_of_val. iNext.
       asimpl. change (Rec _) with (of_val (RecV e.[upn 2 (env_subst vs)])) at 2.
-      erewrite typed_subst_head_simpl_2 by naive_solver.
       iApply (IHtyped Δ (_ :: w :: vs)).
       iApply interp_env_cons; iSplit; [|iApply interp_env_cons]; auto.
     - (* Lam *)
@@ -80,15 +78,13 @@ Section typed_interp.
       iDestruct (interp_env_length with "HΓ") as %?.
       iApply wp_pure_step_later; auto 1 using to_of_val. iNext.
       asimpl.
-      erewrite typed_subst_head_simpl by naive_solver.
       iApply (IHtyped Δ (w :: vs)); auto.
       iApply interp_env_cons; iSplit; auto.
     - (* LetIn *)
       smart_wp_bind (LetInCtx _) v "#Hv" IHtyped1; cbn.
       iDestruct (interp_env_length with "HΓ") as %?.
       iApply wp_pure_step_later; auto 1 using to_of_val. iNext.
-      asimpl. erewrite typed_subst_head_simpl by naive_solver.
-      iApply (IHtyped2 Δ (v :: vs)).
+      asimpl. iApply (IHtyped2 Δ (v :: vs)).
       iApply interp_env_cons; iSplit; eauto.
     - (* Seq *)
       smart_wp_bind (SeqCtx _) v "#Hv" IHtyped1; cbn.

theories/logrel/F_mu_ref_conc/soundness_binary.v

@@ -29,8 +29,9 @@ Proof.
   iPoseProof ((Hlog _ _ [] [] ([e'], ∅)) with "[$Hcfg]") as "Hrel".
   { iApply (@logrel_binary.interp_env_nil Σ HeapΣ). }
   simpl.
-  rewrite empty_env_subst empty_env_subst. iApply ("Hrel" $! 0 []).
-  { rewrite /tpool_mapsto. iAsimpl. by iFrame. }
+  replace e with e.[env_subst[]] at 2 by by asimpl.
+  iApply ("Hrel" $! 0 []).
+  { rewrite /tpool_mapsto. asimpl. by iFrame. }
   iModIntro. iIntros (v1); iDestruct 1 as (v2) "[Hj #Hinterp]".
   iInv specN as (tp σ) ">[Hown Hsteps]" "Hclose"; iDestruct "Hsteps" as %Hsteps'.
   rewrite /tpool_mapsto /=.

theories/logrel/F_mu_ref_conc/soundness_unary.v

@@ -20,7 +20,7 @@ Proof.
   iModIntro. iExists (λ σ _, own γ (● to_gen_heap σ)); iFrame.
   set (HeapΣ := (HeapIG Σ Hinv (GenHeapG _ _ Σ _ _ _ γ))).
   iApply (wp_wand with "[]").
-  - rewrite -(empty_env_subst e).
+  - replace e with e.[env_subst[]] by by asimpl.
     iApply (Hlog HeapΣ [] []). iApply (@interp_env_nil _ HeapΣ).
   - eauto.
 Qed.

theories/logrel/F_mu_ref_conc/typing.v

@@ -123,8 +123,21 @@ Proof.
       asimpl; rewrite H1; auto with lia.
 Qed.
 
-Definition env_subst (vs : list val) (x : var) : expr :=
-  from_option id (Var x) (of_val <$> vs !! x).
+Fixpoint env_subst (vs : list val) : var → expr :=
+  match vs with
+  | [] => ids
+  | v :: vs' => (of_val v) .: env_subst vs'
+  end.
+
+Lemma env_subst_lookup vs x v :
+  vs !! x = Some v → env_subst vs x = of_val v.
+Proof.
+  revert vs; induction x => vs.
+  - by destruct vs; inversion 1.
+  - destruct vs as [|w vs]; first by inversion 1.
+    rewrite -lookup_tail /=.
+    apply IHx.
+Qed.
 
 Lemma typed_n_closed Γ τ e : Γ ⊢ₜ e : τ → (∀ f, e.[upn (length Γ) f] = e).
 Proof.
@@ -134,40 +147,6 @@ Proof.
   - by f_equal; rewrite map_length in IHtyped.
 Qed.
 
-Lemma n_closed_subst_head_simpl n e w ws :
-  (∀ f, e.[upn n f] = e) →
-  S (length ws) = n →
-  e.[of_val w .: env_subst ws] = e.[env_subst (w :: ws)].
-Proof.
-  intros H1 H2.
-  rewrite /env_subst. eapply n_closed_invariant; eauto=> /= -[|x] ? //=.
-  destruct (lookup_lt_is_Some_2 ws x) as [v' Hv]; first lia; simpl.
-  by rewrite Hv.
-Qed.
-
-Lemma typed_subst_head_simpl Δ τ e w ws :
-  Δ ⊢ₜ e : τ → length Δ = S (length ws) →
-  e.[of_val w .: env_subst ws] = e.[env_subst (w :: ws)].
-Proof. eauto using n_closed_subst_head_simpl, typed_n_closed. Qed.
-
-Lemma n_closed_subst_head_simpl_2 n e w w' ws :
-  (∀ f, e.[upn n f] = e) → (S (S (length ws))) = n →
-  e.[of_val w .: of_val w' .: env_subst ws] = e.[env_subst (w :: w' :: ws)].
-Proof.
-  intros H1 H2.
-  rewrite /env_subst. eapply n_closed_invariant; eauto => /= -[|[|x]] H3 //=.
-  destruct (lookup_lt_is_Some_2 ws x) as [v' Hv]; first lia; simpl.
-  by rewrite Hv.
-Qed.
-
-Lemma typed_subst_head_simpl_2 Δ τ e w w' ws :
-  Δ ⊢ₜ e : τ → length Δ = 2 + length ws →
-  e.[of_val w .: of_val w' .: env_subst ws] = e.[env_subst (w :: w' :: ws)].
-Proof. eauto using n_closed_subst_head_simpl_2, typed_n_closed. Qed.
-
-Lemma empty_env_subst e : e.[env_subst []] = e.
-Proof. change (env_subst []) with (@ids expr _). by asimpl. Qed.
-
 (** Weakening *)
 Lemma context_gen_weakening ξ Γ' Γ e τ :
   Γ' ++ Γ ⊢ₜ e : τ →

theories/logrel/stlc/fundamental.v

@@ -18,7 +18,7 @@ Section typed_interp.
     induction Htyped; iIntros (vs) "#Hctx /=".
     - (* var *)
       iDestruct (interp_env_Some_l with "[]") as (v) "[#H1 #H2]"; eauto;
-        iDestruct "H1" as %Heq; rewrite /env_subst Heq /=.
+        iDestruct "H1" as %Heq. erewrite env_subst_lookup; eauto.
       by iApply wp_value.
     - (* unit *) by iApply wp_value.
     - (* pair *)
@@ -43,17 +43,15 @@ Section typed_interp.
       iDestruct "Hv" as "[Hv|Hv]"; iDestruct "Hv" as (w) "[% Hw]"; subst.
       + simpl. iApply wp_pure_step_later; auto. asimpl.
         specialize (IHHtyped2 (w::vs)).
-        erewrite <- ?typed_subst_head_simpl in *; eauto; simpl in *; auto.
         iNext. iApply (IHHtyped2). iApply interp_env_cons; by iSplit.
       + simpl. iApply wp_pure_step_later; auto. asimpl.
         specialize (IHHtyped3 (w::vs)).
-        erewrite <- ?typed_subst_head_simpl in *; eauto; simpl in *; auto.
         iNext. iApply (IHHtyped3). iApply interp_env_cons; by iSplit.
     - (* lam *)
       iDestruct (interp_env_length with "[]") as %Hlen; auto.
       iApply wp_value. simpl. iAlways; iIntros (w) "#Hw".
       iApply wp_pure_step_later; auto.
-      asimpl; erewrite typed_subst_head_simpl; [|eauto|cbn]; eauto.
+      asimpl.
       iNext; iApply (IHHtyped (w :: vs)). iApply interp_env_cons; by iSplit.
     - (* app *)
       smart_wp_bind (AppLCtx (e2.[env_subst vs])) v "#Hv" IHHtyped1.

theories/logrel/stlc/soundness.v

@@ -4,7 +4,8 @@ From iris.program_logic Require Import adequacy.
 
 Lemma wp_soundness `{irisG stlc_lang Σ} e τ : [] ⊢ₜ e : τ → (WP e {{ ⟦ τ ⟧ }})%I.
 Proof.
-  iIntros (?). rewrite -(empty_env_subst e).
+  iIntros (?).
+  replace e with e.[env_subst[]] by by asimpl.
   iApply fundamental; eauto. iApply interp_env_nil.
 Qed.
 

theories/logrel/stlc/typing.v

@@ -34,18 +34,18 @@ Proof.
   induction Htyped => s1 s2 Hs; f_equal/=; eauto using lookup_lt_Some with lia.
 Qed.
 
-Definition env_subst (vs : list val) (x : var) : expr :=
-  from_option id (Var x) (of_val <$> vs !! x).
+Fixpoint env_subst (vs : list val) : var → expr :=
+  match vs with
+  | [] => ids
+  | v :: vs' => #v .: env_subst vs'
+  end.
 
-Lemma typed_subst_head_simpl Δ τ e w ws :
-  Δ ⊢ₜ e : τ → length Δ = S (length ws) →
-  e.[# w .: env_subst ws] = e.[env_subst (w :: ws)].
+Lemma env_subst_lookup vs x v :
+  vs !! x = Some v → env_subst vs x = of_val v.
 Proof.
-  intros H1 H2.
-  rewrite /env_subst. eapply typed_subst_invariant; eauto=> /= -[|x] ? //=.
-  destruct (lookup_lt_is_Some_2 ws x) as [v' Hv]; first lia; simpl.
-  by rewrite Hv.
-Qed.
-
-Lemma empty_env_subst e : e.[env_subst []] = e.
-Proof. change (env_subst []) with ids. by asimpl. Qed.
+  revert vs; induction x => vs.
+  - by destruct vs; inversion 1.
+  - destruct vs as [|w vs]; first by inversion 1.
+    rewrite -lookup_tail /=.
+    apply IHx.
+Qed.
\ No newline at end of file