From ba61dfd69504ec6263a9dee9931d93adeb6f3142 Mon Sep 17 00:00:00 2001
From: Rutger Broekhoff
Date: Mon, 7 Jul 2025 21:52:08 +0200
Subject: Initialize repository

---
 theories/evallang/interp_proofs.v | 478 ++++++++++++++++++++++++++++++++++++++
 1 file changed, 478 insertions(+)
 create mode 100644 theories/evallang/interp_proofs.v

(limited to 'theories/evallang/interp_proofs.v')

diff --git a/theories/evallang/interp_proofs.v b/theories/evallang/interp_proofs.v
new file mode 100644
index 0000000..0a26dd1
--- /dev/null
+++ b/theories/evallang/interp_proofs.v
@@ -0,0 +1,478 @@
+From mininix Require Export evallang.interp.
+From stdpp Require Import options.
+
+Module Import evallang.
+Export evallang.
+
+Lemma interp_S n : interp (S n) = interp1 (interp n).
+Proof. done. Qed.
+
+Fixpoint thunk_size (t : thunk) : nat :=
+  S (map_sum_with thunk_size (thunk_env t)).
+Definition env_size (E : env) : nat :=
+  map_sum_with thunk_size E.
+
+Lemma env_ind (P : env → Prop) :
+  (∀ E, map_Forall (λ i, P ∘ thunk_env) E → P E) →
+  ∀ E : env, P E.
+Proof.
+  intros Pbs E.
+  induction (Nat.lt_wf_0_projected env_size E) as [E _ IH].
+  apply Pbs, map_Forall_lookup=> y [E' e'] Hy.
+  apply (map_sum_with_lookup_le thunk_size) in Hy.
+  apply IH. by rewrite -Nat.le_succ_l.
+Qed.
+
+(** Correspondence to operational semantics *)
+Definition subst_env' (thunk_to_expr : thunk → expr) (E : env) : expr → expr :=
+  subst (thunk_to_expr <$> E).
+Fixpoint thunk_to_expr (t : thunk) : expr :=
+  subst_env' thunk_to_expr (thunk_env t) (thunk_expr t).
+Notation subst_env := (subst_env' thunk_to_expr).
+
+Lemma subst_env_eq e E :
+  subst_env E e =
+    match e with
+    | EString s => EString s
+    | EId ds x => EId ((thunk_to_expr <$> E !! x) ∪ ds) x
+    | EEval ds e => EEval ((thunk_to_expr <$> E) ∪ ds) (subst_env E e)
+    | EAbs ex e => EAbs (subst_env E ex) (subst_env E e)
+    | EApp e1 e2 => EApp (subst_env E e1) (subst_env E e2)
+    end.
+Proof. destruct e; rewrite /subst_env' /= ?lookup_fmap //. Qed.
+
+Lemma subst_env_alt E e : subst_env E e = subst (thunk_to_expr <$> E) e.
+Proof. done. Qed.
+
+(* Use the unfolding lemmas, don't rely on conversion *)
+Opaque subst_env'.
+
+Definition val_to_expr (v : val) : expr :=
+  match v with
+  | VString s => EString s
+  | VClo x E e => EAbs (EString x) (subst_env E e)
+  end.
+
+Lemma final_val_to_expr v : final (val_to_expr v).
+Proof. by destruct v. Qed.
+Lemma step_not_val_to_expr v e : val_to_expr v --> e → False.
+Proof. intros []%step_not_final. apply final_val_to_expr. Qed.
+
+Lemma subst_empty e : subst ∅ e = e.
+Proof. induction e; f_equal/=; rewrite ?lookup_empty ?left_id_L //. Qed.
+
+Lemma subst_env_empty e : subst_env ∅ e = e.
+Proof. rewrite subst_env_alt. apply subst_empty. Qed.
+
+Lemma interp_le {n1 n2 E e mv} :
+  interp n1 E e = Res mv → n1 ≤ n2 → interp n2 E e = Res mv.
+Proof.
+  revert n2 E e mv.
+  induction n1 as [|n1 IH]; intros [|n2] E e mv He ?; [by (done || lia)..|].
+  rewrite interp_S in He; rewrite interp_S; destruct e;
+    repeat match goal with
+    | _ => case_match
+    | H : context [(_ <$> ?mx)] |- _ => destruct mx eqn:?; simplify_res
+    | H : ?r ≫= _ = _ |- _ => destruct r as [[]|] eqn:?; simplify_res
+    | H : _ <$> ?r = _ |- _ => destruct r as [[]|] eqn:?; simplify_res
+    | |- interp ?n ?E ?e ≫= _ = _ => erewrite (IH n E e) by (done || lia)
+    | _ => progress simplify_res
+    | _ => progress simplify_option_eq
+    end; eauto with lia.
+Qed.
+
+Lemma interp_agree {n1 n2 E e mv1 mv2} :
+  interp n1 E e = Res mv1 → interp n2 E e = Res mv2 → mv1 = mv2.
+Proof.
+  intros He1 He2. apply (inj Res). destruct (total (≤) n1 n2).
+  - rewrite -He2. symmetry. eauto using interp_le. 
+  - rewrite -He1. eauto using interp_le.
+Qed.
+
+Lemma subst_env_union E1 E2 e :
+  subst_env (E1 ∪ E2) e = subst_env E1 (subst_env E2 e).
+Proof.
+  revert E1 E2. induction e; intros E1 E2; rewrite subst_env_eq /=.
+  - done.
+  - rewrite !subst_env_eq lookup_union. by destruct (E1 !! _), (E2 !! _), ds.
+  - rewrite !(subst_env_eq (EEval _ _)) IHe. f_equal.
+    by rewrite assoc_L map_fmap_union.
+  - rewrite !(subst_env_eq (EAbs _ _)) /=. f_equal; auto.
+  - rewrite !(subst_env_eq (EApp _ _)) /=. f_equal; auto.
+Qed.
+
+Lemma subst_env_insert E x e t :
+  subst_env (<[x:=t]> E) e = subst {[x:=thunk_to_expr t]} (subst_env E e).
+Proof.
+  rewrite insert_union_singleton_l subst_env_union subst_env_alt.
+  by rewrite map_fmap_singleton.
+Qed.
+
+Lemma subst_env_insert_eq e1 e2 E1 E2 x E1' E2' e1' e2' :
+  subst_env E1 e1 = subst_env E2 e2 →
+  subst_env E1' e1' = subst_env E2' e2' →
+  subst_env (<[x:=Thunk E1' e1']> E1) e1 = subst_env (<[x:=Thunk E2' e2']> E2) e2.
+Proof. intros He He'. by rewrite !subst_env_insert //= He' He. Qed.
+
+Lemma option_fmap_thunk_to_expr_Thunk (me : option expr) :
+  thunk_to_expr <$> (Thunk ∅ <$> me) = me.
+Proof. destruct me; f_equal/=. by rewrite subst_env_empty. Qed.
+
+Lemma map_fmap_thunk_to_expr_Thunk (es : gmap string expr) :
+  thunk_to_expr <$> (Thunk ∅ <$> es) = es.
+Proof.
+  apply map_eq=> x. by rewrite !lookup_fmap option_fmap_thunk_to_expr_Thunk.
+Qed.
+
+Lemma subst_env_eval_eq E1 E2 ds1 ds2 e :
+  (thunk_to_expr <$> E1) ∪ ds1 = (thunk_to_expr <$> E2) ∪ ds2 →
+  subst_env (E1 ∪ (Thunk ∅ <$> ds1)) e = subst_env (E2 ∪ (Thunk ∅ <$> ds2)) e.
+Proof.
+  intros HE.
+  by rewrite !subst_env_alt !map_fmap_union !map_fmap_thunk_to_expr_Thunk HE.
+Qed.
+
+Lemma interp_proper n E1 E2 e1 e2 mv :
+  subst_env E1 e1 = subst_env E2 e2 →
+  interp n E1 e1 = Res mv →
+  ∃ mw m, interp m E2 e2 = Res mw ∧
+          val_to_expr <$> mv = val_to_expr <$> mw.
+Proof.
+  revert n E1 E2 e1 e2 mv. induction n as [|n IHn]; [done|].
+  intros E1 E2 e1 e2 mv Hsubst Hinterp.
+  rewrite 2!subst_env_eq in Hsubst.
+  rewrite interp_S in Hinterp. destruct e1, e2; simplify_res; try done.
+  - eexists (Some (VString _)), 1. by rewrite interp_S.
+  - assert (thunk_to_expr <$> E1 !! x0 ∪ (Thunk ∅ <$> ds) =
+      thunk_to_expr <$> E2 !! x0 ∪ (Thunk ∅ <$> ds0)).
+    { destruct (E1 !! _), (E2 !! _), ds, ds0; simplify_eq/=;
+        f_equal/=; by rewrite ?subst_env_empty. }
+    destruct (E1 !! x0 ∪ (Thunk ∅ <$> ds)) as [[E1' e1']|],
+      (E2 !! x0 ∪ (Thunk ∅ <$> ds0)) as [[E2' e2']|] eqn:HE2;
+      simplify_res; last first.
+    { exists None, 1. by rewrite interp_S /= HE2. }
+    eapply IHn in Hinterp as (mw & m & Hinterp2 & ?); [|by eauto..].
+    exists mw, (S m). split; [|done]. rewrite interp_S /= HE2 /=. done.
+  - destruct (interp n _ e1) as [mv1|] eqn:Hinterp'; simplify_eq/=.
+    eapply IHn in Hinterp' as (mw1 & m1 & Hinterp1 & ?); last done.
+    destruct mv1 as [v1|], mw1 as [w1|]; simplify_res; last first.
+    { exists None, (S m1). by rewrite interp_S /= Hinterp1. }
+    destruct (maybe VString v1) as [x|] eqn:Hv1;
+      simplify_res; last first.
+    { exists None, (S m1). split; [|done]. rewrite interp_S /= Hinterp1 /=.
+      destruct v1, w1; repeat destruct select base_lit; by simplify_eq/=. }
+    destruct v1, w1; repeat destruct select base_lit; simplify_eq/=.
+    destruct (parse _) as [e|] eqn:Hparse; simplify_res; last first.
+    { exists None, (S m1). split; [|done]. rewrite interp_S /= Hinterp1 /=.
+      by rewrite Hparse. }
+    eapply IHn in Hinterp
+      as (mw & m2 & Hinterp2 & ?); last by apply subst_env_eval_eq.
+    exists mw, (S (m1 `max` m2)). split; [|done]. rewrite interp_S /=.
+    rewrite (interp_le Hinterp1) /=; last lia. rewrite Hparse /=.
+    eauto using interp_le with lia.
+  - destruct (interp n _ _) as [mv1|] eqn:Hinterp'; simplify_eq/=.
+    eapply IHn in Hinterp' as (mw1 & m1 & Hinterp1 & ?); last done.
+    destruct mv1 as [v1|], mw1 as [w1|]; simplify_res; last first.
+    { exists None, (S m1). by rewrite interp_S /= Hinterp1. }
+    destruct (maybe VString _) eqn:Hstring; simplify_res; last first.
+    { exists None, (S m1). rewrite interp_S /= Hinterp1 /=.
+      by assert (maybe VString w1 = None) as -> by (by destruct v1, w1). }
+    destruct v1, w1; simplify_eq/=.
+    eexists (Some (VClo _ _ _)), (S m1).
+    rewrite interp_S /= Hinterp1 /=. split; [done|]. by do 2 f_equal/=.
+  - destruct (interp n _ _) as [mv'|] eqn:Hinterp'; simplify_res.
+    eapply IHn in Hinterp' as (mw' & m1 & Hinterp1 & ?); last done.
+    destruct mv' as [v'|], mw' as [w'|]; simplify_res; last first.
+    { exists None, (S m1). by rewrite interp_S /= Hinterp1. }
+    destruct (maybe3 VClo _) eqn:Hclo; simplify_res; last first.
+    { exists None, (S m1). rewrite interp_S /= Hinterp1 /=.
+      by assert (maybe3 VClo w' = None) as -> by (by destruct v', w'). }
+    destruct v', w'; simplify_eq/=.
+    eapply IHn with (E2 := <[x0:=Thunk E2 e2_2]> E0) in Hinterp
+      as (w & m2 & Hinterp2 & ?); last by apply subst_env_insert_eq.
+    exists w, (S (m1 `max` m2)). rewrite interp_S /=.
+    rewrite (interp_le Hinterp1) /=; last lia.
+    rewrite (interp_le Hinterp2) /=; last lia. done.
+Qed.
+
+Lemma subst_as_subst_env x e1 e2 :
+  subst {[x:=e2]} e1 = subst_env (<[x:=Thunk ∅ e2]> ∅) e1.
+Proof. rewrite subst_env_insert //= !subst_env_empty //. Qed.
+
+Lemma interp_subst_abs n x e1 e2 mv :
+  interp n ∅ (subst {[x:=e2]} e1) = Res mv →
+  ∃ mw m, interp m (<[x:=Thunk ∅ e2]> ∅) e1 = Res mw ∧
+          val_to_expr <$> mv = val_to_expr <$> mw.
+Proof.
+  apply interp_proper. by rewrite subst_env_empty subst_as_subst_env.
+Qed.
+
+Lemma interp_subst_eval n e ds mv :
+  interp n ∅ (subst ds e) = Res mv →
+  ∃ mw m, interp m (Thunk ∅ <$> ds) e = Res mw ∧
+          val_to_expr <$> mv = val_to_expr <$> mw.
+Proof.
+  apply interp_proper.
+  by rewrite subst_env_empty subst_env_alt map_fmap_thunk_to_expr_Thunk.
+Qed.
+
+Lemma interp_step e1 e2 n mv :
+  e1 --> e2 →
+  interp n ∅ e2 = Res mv →
+  ∃ mw m, interp m ∅ e1 = Res mw ∧ val_to_expr <$> mv = val_to_expr <$> mw.
+Proof.
+  intros Hstep. revert mv n.
+  induction Hstep; intros mv n Hinterp.
+  - apply interp_subst_abs in Hinterp as (mw & [|m] & Hinterp & Hv);
+      simplify_eq/=; [|done..].
+    exists mw, (S (S (S m))). rewrite !interp_S /= -!interp_S.
+    eauto using interp_le with lia.
+  - exists mv, (S n). rewrite !interp_S /=.
+    rewrite lookup_empty left_id_L /=. done.
+  - apply interp_subst_eval in Hinterp as (mw & [|m] & Hinterp & Hv);
+      simplify_eq/=; [|done..].
+    exists mw, (S (S m)). rewrite !interp_S /= -interp_S.
+    rewrite left_id_L H /=. done.
+  - destruct n as [|n]; [done|rewrite interp_S /= in Hinterp].
+    destruct (interp n _ _) as [mv'|] eqn:Hinterp'; simplify_res.
+    apply IHHstep in Hinterp' as (mw' & m1 & Hinterp1 & ?); simplify_res.
+    destruct mv' as [v'|], mw' as [w'|]; simplify_res; last first.
+    { exists None, (S m1). by rewrite interp_S /= Hinterp1. }
+    destruct (maybe VString _) eqn:Hstring; simplify_res; last first.
+    { exists None, (S m1). rewrite interp_S /= Hinterp1 /=.
+      by assert (maybe VString w' = None) as -> by (by destruct v', w'). }
+    destruct v', w'; simplify_eq/=.
+    eexists (Some (VClo _ _ _)), (S m1). rewrite !interp_S /=.
+    rewrite (interp_le Hinterp1) /=; last lia. done.
+  - destruct n as [|n]; [done|rewrite interp_S /= in Hinterp].
+    destruct (interp n _ _) as [mv'|] eqn:Hinterp'; simplify_res.
+    apply IHHstep in Hinterp' as (mw' & m1 & Hinterp1 & ?); simplify_res.
+    destruct mv' as [v'|], mw' as [w'|]; simplify_res; last first.
+    { exists None, (S m1). by rewrite interp_S /= Hinterp1. }
+    destruct (maybe3 VClo _) eqn:Hclo; simplify_res; last first.
+    { exists None, (S m1). rewrite interp_S /= Hinterp1 /=.
+      by assert (maybe3 VClo w' = None) as -> by (by destruct v', w'). }
+    destruct v', w'; simplify_eq/=.
+    eapply interp_proper in Hinterp as (mw & m2 & Hinterp2 & Hv);
+      last apply subst_env_insert_eq; try done.
+    exists mw, (S (m1 `max` m2)). rewrite !interp_S /=.
+    rewrite (interp_le Hinterp1) /=; last lia.
+    by rewrite (interp_le Hinterp2) /=; last lia.
+  - destruct n as [|n]; [done|rewrite interp_S /= in Hinterp].
+    destruct (interp n _ e1') as [mv1|] eqn:Hinterp1; simplify_eq/=.
+    apply IHHstep in Hinterp1 as (mw1 & m & Hinterp1 & Hw1).
+    destruct mv1 as [v1|], mw1 as [w1|]; simplify_res; last first.
+    { exists None, (S m). by rewrite interp_S /= Hinterp1. }
+    destruct (maybe VString _) eqn:Hstring; simplify_res; last first.
+    { exists None, (S m). rewrite interp_S /= Hinterp1 /=.
+      by assert (maybe VString w1 = None) as -> by (by destruct v1, w1). }
+    destruct v1, w1; simplify_eq/=.
+    exists mv, (S (n `max` m)). split; [|done]. rewrite interp_S /=.
+    rewrite (interp_le Hinterp1) /=; last lia.
+    destruct (parse _); simplify_res; eauto using interp_le with lia.
+Qed.
+
+Lemma final_interp e :
+  final e →
+  ∃ w m, interp m ∅ e = mret w ∧ e = val_to_expr w.
+Proof.
+  induction e as [| | |[]|]; inv 1.
+  - eexists (VString _), 1. by rewrite interp_S /=.
+  - eexists (VClo _ _ _), 2. rewrite interp_S /=. split; [done|].
+    by rewrite subst_env_empty.
+Qed.
+
+Lemma red_final_interp e :
+  red step e ∨ final e ∨ ∃ m, interp m ∅ e = mfail.
+Proof.
+  induction e.
+  - (* ENat *) right; left. constructor.
+  - (* EId *) destruct ds as [e|].
+    + left. by repeat econstructor.
+    + do 2 right. by exists 1.
+  - (* EEval *) destruct IHe as [[??]|[Hfinal|[m Hinterp]]].
+    + left. by repeat econstructor.
+    + apply final_interp in Hfinal as (w & m & Hinterp & ->).
+      destruct (maybe VString w) as [x|] eqn:Hw; last first.
+      { do 2 right. exists (S m). rewrite interp_S /= Hinterp /=.
+        by rewrite Hw. }
+      destruct w; simplify_eq/=.
+      destruct (parse x) as [e|] eqn:Hparse; last first.
+      { do 2 right. exists (S m). rewrite interp_S /= Hinterp /=.
+        by rewrite Hparse. }
+      left. by repeat econstructor.
+    + do 2 right. exists (S m). rewrite interp_S /= Hinterp. done.
+  - (* EAbs *) destruct IHe1 as [[??]|[Hfinal|[m Hinterp]]].
+    + left. by repeat econstructor.
+    + apply final_interp in Hfinal as (w & m & Hinterp & ->).
+      destruct (maybe VString w) as [x|] eqn:Hw; last first.
+      { do 2 right. exists (S m). rewrite interp_S /= Hinterp /=.
+        by rewrite Hw. }
+      destruct w; naive_solver.
+    + do 2 right. exists (S m). rewrite interp_S /= Hinterp. done.
+  - (* EApp *) destruct IHe1 as [[??]|[Hfinal|[m Hinterp]]].
+    + left. by repeat econstructor.
+    + apply final_interp in Hfinal as (w & m & Hinterp & ->).
+      destruct (maybe3 VClo w) eqn:Hw.
+      { destruct w; simplify_eq/=. left. by repeat econstructor. }
+      do 2 right. exists (S m). by rewrite interp_S /= Hinterp /= Hw.
+    + do 2 right. exists (S m). by rewrite interp_S /= Hinterp.
+Qed.
+
+Lemma interp_complete e1 e2 :
+  e1 -->* e2 →
+  nf step e2 →
+  ∃ mw m, interp m ∅ e1 = Res mw ∧
+    if mw is Some w then e2 = val_to_expr w else ¬final e2.
+Proof.
+  intros Hsteps Hnf. induction Hsteps as [e|e1 e2 e3 Hstep _ IH].
+  { destruct (red_final_interp e) as [?|[Hfinal|[m Hinterp]]]; [done|..].
+    - apply final_interp in Hfinal as (w & m & ? & ?).
+      by exists (Some w), m.
+    - exists None, m. split; [done|]. intros Hfinal.
+      apply final_interp in Hfinal as (w & m' & ? & _).
+      by assert (mfail = mret w) by eauto using interp_agree. }
+  destruct IH as (mw & m & Hinterp & ?); try done.
+  eapply interp_step in Hinterp as (mw' & m' & ? & ?); last done.
+  destruct mw, mw'; naive_solver.
+Qed.
+
+Lemma interp_complete_ret e1 e2 :
+  e1 -->* e2 → final e2 →
+  ∃ w m, interp m ∅ e1 = mret w ∧ e2 = val_to_expr w.
+Proof.
+  intros Hsteps Hfinal. apply interp_complete in Hsteps
+    as ([w|] & m & ? & ?); naive_solver eauto using final_nf.
+Qed.
+Lemma interp_complete_fail e1 e2 :
+  e1 -->* e2 → nf step e2 → ¬final e2 →
+  ∃ m, interp m ∅ e1 = mfail.
+Proof.
+  intros Hsteps Hnf Hforce.
+  apply interp_complete in Hsteps as ([w|] & m & ? & ?); simplify_eq/=; try by eauto.
+  destruct Hforce. apply final_val_to_expr.
+Qed.
+
+Lemma interp_sound_open E e n mv :
+  interp n E e = Res mv →
+  ∃ e', subst_env E e -->* e' ∧
+    if mv is Some v then e' = val_to_expr v else stuck e'.
+Proof.
+  revert E e mv.
+  induction n as [|n IH]; intros E e mv Hinterp; first done.
+  rewrite subst_env_eq. rewrite interp_S in Hinterp.
+  destruct e; simplify_res.
+  - (* EString *) by eexists.
+  - (* EId *)
+    assert (thunk_to_expr <$> (E !! x) ∪ (Thunk ∅ <$> ds)
+      = (thunk_to_expr <$> E !! x) ∪ ds).
+    { destruct (_ !! _), ds; f_equal/=. by rewrite subst_env_empty. }
+    destruct (_ ∪ (_ <$> _)) as [[E1 e1]|], (_ ∪ _) as [e2|]; simplify_res.
+    * apply IH in Hinterp as (e'' & Hsteps & He'').
+      exists e''; split; [|done].
+      eapply rtc_l; [|done]. by econstructor.
+    * eexists; split; [done|]. split; [|inv 1].
+      intros [? Hstep]. inv_step; simplify_eq/=; congruence.
+  - (* EEval *)
+    destruct (interp _ _ _) as [mv1|] eqn:Hinterp1; simplify_res.
+    apply IH in Hinterp1 as (e1' & Hsteps1 & He1').
+    destruct mv1 as [v1|]; simplify_res; last first.
+    { eexists; split; [by eapply SEval_rtc|]. split; [|inv 1].
+      intros [??]. destruct He1' as [Hnf []].
+      inv_step; simpl; eauto. destruct Hnf; eauto. }
+    destruct (maybe VString _) as [x|] eqn:Hv1; simplify_res; last first.
+    { eexists; split; [by eapply SEval_rtc|]. split; [|inv 1].
+      intros [??]. destruct v1; inv_step. }
+    destruct v1; simplify_eq/=.
+    destruct (parse x) as [ex|] eqn:Hparse; simplify_res; last first.
+    { eexists; split; [by eapply SEval_rtc|].
+      split; [|inv 1]. intros [??]. inv_step. }
+    apply IH in Hinterp as (e'' & Hsteps & He'').
+    exists e''; split; [|done]. etrans; [by eapply SEval_rtc|].
+    eapply rtc_l; [by econstructor|].
+    by rewrite subst_env_alt map_fmap_union
+      map_fmap_thunk_to_expr_Thunk in Hsteps.
+  - (* EAbs *)
+    destruct (interp _ _ _) as [mv1|] eqn:Hinterp1; simplify_res.
+    apply IH in Hinterp1 as (e1' & Hsteps1 & He1').
+    destruct mv1 as [v1|]; simplify_res; last first.
+    { eexists; split; [by eapply SAbsL_rtc|]. split.
+      + intros [??]. destruct He1' as [Hnf []].
+        inv_step; simpl; eauto. destruct Hnf; eauto.
+      + intros ?. destruct He1' as [_ []]. by destruct e1'. }
+    eexists; split; [by eapply SAbsL_rtc|].
+    destruct (maybe VString _) as [x|] eqn:Hv1; simplify_res; last first.
+    { split; [|destruct v1; inv 1]. intros [??]. destruct v1; inv_step. }
+    by destruct v1; simplify_eq/=.
+  - (* EApp *) destruct (interp _ _ _) as [mv'|] eqn:Hinterp'; simplify_res.
+    apply IH in Hinterp' as (e' & Hsteps & He'); try done.
+    destruct mv' as [v'|]; simplify_res; last first.
+    { eexists; repeat split; [by apply SAppL_rtc| |inv 1].
+      intros [e'' Hstep]. destruct He' as [Hnf Hfinal].
+      inv Hstep; [by destruct Hfinal; constructor|]. destruct Hnf. eauto. }
+    destruct (maybe3 VClo v') eqn:?; simplify_res; last first.
+    { eexists; repeat split; [by apply SAppL_rtc| |inv 1].
+      intros [e'' Hstep]. inv Hstep; destruct v'; by repeat inv_step. }
+    destruct v'; simplify_res.
+    apply IH in Hinterp as (e'' & Hsteps' & He'').
+    eexists; split; [|done]. etrans; [by apply SAppL_rtc|].
+    eapply rtc_l; first by constructor.
+    rewrite subst_env_insert // in Hsteps'.
+Qed.
+
+Lemma interp_sound n e mv :
+  interp n ∅ e = Res mv →
+  ∃ e', e -->* e' ∧ if mv is Some v then e' = val_to_expr v else stuck e'.
+Proof.
+  intros Hsteps%interp_sound_open; try done.
+  by rewrite subst_env_empty in Hsteps.
+Qed.
+
+(** Final theorems *)
+Theorem interp_sound_complete_ret e v :
+  (∃ w n, interp n ∅ e = mret w ∧ val_to_expr v = val_to_expr w)
+    ↔ e -->* val_to_expr v.
+Proof.
+  split.
+  - by intros (n & w & (e' & ? & ->)%interp_sound & ->).
+  - intros Hsteps. apply interp_complete in Hsteps as ([] & ? & ? & ?);
+      unfold nf, red;
+      naive_solver eauto using final_val_to_expr, step_not_val_to_expr.
+Qed.
+
+Theorem interp_sound_complete_ret_string e s :
+  (∃ n, interp n ∅ e = mret (VString s)) ↔ e -->* EString s.
+Proof.
+  split.
+  - by intros [n (e' & ? & ->)%interp_sound].
+  - intros Hsteps. apply interp_complete_ret in Hsteps as ([] & ? & ? & ?);
+      simplify_eq/=; eauto.
+Qed.
+
+Theorem interp_sound_complete_fail e :
+  (∃ n, interp n ∅ e = mfail) ↔ ∃ e', e -->* e' ∧ stuck e'.
+Proof.
+  split.
+  - by intros [n ?%interp_sound].
+  - intros (e' & Hsteps & Hnf & Hforced). by eapply interp_complete_fail.
+Qed.
+
+Theorem interp_sound_complete_no_fuel e :
+  (∀ n, interp n ∅ e = NoFuel) ↔ all_loop step e.
+Proof.
+  rewrite all_loop_alt. split.
+  - intros Hnofuel e' Hsteps.
+    destruct (red_final_interp e') as [|[|He']]; [done|..].
+    + apply interp_complete_ret in Hsteps as (w & m & Hinterp & _); last done.
+      by rewrite Hnofuel in Hinterp.
+    + apply interp_sound_complete_fail in He' as (e'' & ? & [Hnf _]).
+      destruct (interp_complete e e'') as (mv & n & Hinterp & _); [by etrans|done|].
+      by rewrite Hnofuel in Hinterp.
+  - intros Hred n. destruct (interp n ∅ e) as [mv|] eqn:Hinterp; [|done].
+    apply interp_sound in Hinterp as (e' & Hsteps%Hred & Hstuck).
+    destruct mv as [v|]; simplify_eq/=.
+    + apply final_nf in Hsteps as []. apply final_val_to_expr.
+    + by destruct Hstuck as [[] ?].
+Qed.
+
+End evallang.
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