path: root/theories/lambda/interp_proofs.v

From mininix Require Export lambda.interp.
From stdpp Require Import options.

Module Import lambda.
Export lambda.

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 x => if E !! x is Some t then thunk_to_expr t else EId x
    | EAbs x e => EAbs x (subst_env (delete x E) e)
    | EApp e1 e2 => EApp (subst_env E e1) (subst_env E e2)
    end.
Proof.
  rewrite /subst_env. destruct e; simpl; try done.
  - rewrite lookup_fmap. by destruct (E !! x) as [[]|].
  - by rewrite fmap_delete.
Qed.
Lemma subst_env_id x E :
  subst_env E (EId x) = if E !! x is Some t then thunk_to_expr t else EId x.
Proof. by rewrite subst_env_eq. 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 x (subst_env (delete x 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/=; auto. 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.

Definition is_not_id (e : expr) : Prop :=
  match e with EId _ => False | _ => True end.

Lemma id_or_not e : (∃ x, e = EId x) ∨ is_not_id e.
Proof. destruct e; naive_solver. Qed.

Lemma interp_not_id n E e v :
  interp n E e = mret v → is_not_id (subst_env E e).
Proof.
  revert E e v. induction n as [|n IH]; intros E e v; [done|].
  rewrite interp_S. destruct e; simpl; try done.
  rewrite subst_env_id. destruct (_ !! _) as [[[]]|]; naive_solver.
Qed.

Fixpoint closed (X : stringset) (e : expr) : Prop :=
  match e with
  | EString _ => True
  | EId x => x ∈ X
  | EAbs x e => closed ({[ x ]} ∪ X) e
  | EApp e1 e2 => closed X e1 ∧ closed X e2
  end.

Inductive closed_thunk (t : thunk) : Prop := {
  closed_thunk_env : map_Forall (λ _, closed_thunk) (thunk_env t);
  closed_thunk_expr : closed (dom (thunk_env t)) (thunk_expr t);
}.
Notation closed_env := (map_Forall (M:=env) (λ _, closed_thunk)).

Definition closed_val (v : val) : Prop :=
  match v with
  | VString _ => True
  | VClo x E e => closed_env E ∧ closed ({[x]} ∪ dom E) e
  end.

Lemma closed_thunk_eq E e :
  closed_thunk (Thunk E e) ↔ closed_env E ∧ closed (dom E) e.
Proof. split; inv 1; constructor; done. Qed.

Lemma closed_env_delete x E : closed_env E → closed_env (delete x E).
Proof. apply map_Forall_delete. Qed.

Lemma closed_env_insert x t E :
  closed_thunk t → closed_env E → closed_env (<[x:=t]> E).
Proof. apply: map_Forall_insert_2. Qed.

Lemma closed_env_lookup E x t :
  closed_env E → E !! x = Some t → closed_thunk t.
Proof. apply map_Forall_lookup_1. Qed.

Lemma closed_subst E ds e :
  dom ds ## E → closed E e → subst ds e = e.
Proof.
  revert E ds.
  induction e; intros E ds Hdisj Heclosed; simplify_eq/=; first done.
  - assert (Hxds : x ∉ dom ds) by set_solver.
    by rewrite (not_elem_of_dom_1 _ _ Hxds).
  - f_equal. by apply IHe with (E := {[x]} ∪ E); first set_solver.
  - f_equal; naive_solver.
Qed.

Lemma closed_weaken X Y e : closed X e → X ⊆ Y → closed Y e.
Proof. revert X Y; induction e; naive_solver eauto with set_solver. Qed.

Lemma subst_closed ds X e :
  map_Forall (λ _, closed ∅) ds →
  closed (dom ds ∪ X) e →
  closed X (subst ds e).
Proof.
  revert X ds. induction e; intros X ds; repeat (case_decide || simplify_eq/=).
  - done.
  - intros. case_match.
    + apply H in H1. by eapply closed_weaken.
    + apply not_elem_of_dom in H1. set_solver.
  - intros. apply IHe.
    + by apply map_Forall_delete.
    + by rewrite dom_delete_L assoc_L difference_union_L
                 [dom _ ∪ _]comm_L -assoc_L.
  - naive_solver.
Qed.

Lemma subst_env_delete_closed E X e x :
  closed_env E →
  closed ({[x]} ∪ X) (subst_env E e) →
  closed ({[x]} ∪ X) (subst_env (delete x E) e).
Proof.
  revert E X x.
  induction e as [s | z | z e IHe | e1 IHe1 e2 IHe2]; intros E X x.
  - rewrite !subst_env_eq //.
  - rewrite !subst_env_eq /=. case_match.
    + destruct (decide (x = z)) as [->|?].
      * rewrite lookup_delete. set_solver.
      * rewrite lookup_delete_ne // H //.
    + destruct (decide (x = z)) as [->|?].
      * rewrite delete_notin // H //.
      * rewrite lookup_delete_ne // H //.
  - intros HE.
    rewrite [subst_env (delete _ _) _]subst_env_eq subst_env_eq /=
            delete_commute comm_L -assoc_L.
    by apply IHe, map_Forall_delete.
  - rewrite [subst_env (delete _ _) _]subst_env_eq subst_env_eq /=.
    naive_solver.
Qed.

Lemma subst_env_closed E X e :
  closed_env E → closed (dom E ∪ X) e → closed X (subst_env E e).
Proof.
  revert e X. induction E using env_ind.
  induction e; intros X Hcenv Hclosed; simplify_eq/=.
  - done.
  - rewrite subst_env_eq. case_match.
    + destruct t as [Et et]; simpl.
      apply closed_env_lookup in H0 as Htclosed; last done.
      apply closed_thunk_eq in Htclosed as [HEtclosed Hetclosed].
      apply (H _ _ H0); simpl.
      * exact HEtclosed.
      * eapply closed_weaken; set_solver.
    + simpl in *. apply not_elem_of_dom in H0. set_solver.
  - rewrite subst_env_eq. simpl in *.
    rewrite comm_L -assoc_L in Hclosed.
    apply IHe in Hclosed; last exact Hcenv.
    apply subst_env_delete_closed; first done.
    by rewrite comm_L.
  - rewrite subst_env_eq. naive_solver.
Qed.

Lemma thunk_to_expr_closed t : closed_thunk t → closed ∅ (thunk_to_expr t).
Proof.
  destruct t as [E e]. intros [HEclosed Heclosed]%closed_thunk_eq.
  by apply subst_env_closed; last rewrite union_empty_r_L.
Qed.

Lemma subst_env_insert E x e t :
  closed_env E → 
  subst_env (<[x:=t]> E) e
  = subst {[x:=thunk_to_expr t]} (subst_env (delete x E) e).
Proof.
  revert E. induction e; intros E HEclosed; simpl.
  - done.
  - destruct (decide (x = x0)) as [->|?].
    + rewrite subst_env_eq lookup_insert subst_env_id
        lookup_delete /= lookup_singleton. done.
    + rewrite subst_env_eq lookup_insert_ne // subst_env_id.
      destruct (E !! x0) eqn:Elookup.
      * apply closed_env_lookup in Elookup as Hc0closed; last done.
        apply thunk_to_expr_closed in Hc0closed.
        rewrite lookup_delete_ne // Elookup.
        by erewrite closed_subst with (E := ∅).
      * by rewrite lookup_delete_ne // Elookup /= lookup_singleton_ne.
  - rewrite (subst_env_eq (EAbs x0 e)) (subst_env_eq (EAbs _ _)) /=. f_equal.
    destruct (decide (x0 = x)) as [->|?].
    + by rewrite delete_insert_delete delete_idemp
                 delete_singleton subst_empty.
    + rewrite delete_insert_ne // delete_singleton_ne // delete_commute.
      apply IHe. by apply closed_env_delete.
  - rewrite (subst_env_eq (EApp _ _)) [subst_env (delete x E) _]subst_env_eq /=.
    f_equal; auto.
Qed.

Lemma subst_env_insert_eq e1 e2 E1 E2 x E1' E2' e1' e2' :
  closed_env E1 → closed_env E2 →
  subst_env (delete x E1) e1 = subst_env (delete x 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 HE1closed HE2closed He' He.
  rewrite !subst_env_insert //=. by rewrite He' He.
Qed.

Lemma interp_closed n E e mv :
  closed_env E → closed (dom E) e → interp n E e = Res mv →
  if mv is Some v then closed_val v else True.
Proof.
  revert E e mv.
  induction n; first done; intros E e mv HEclosed Heclosed Hinterp.
  destruct e.
  - rewrite interp_S /= in Hinterp. by destruct mv; simplify_res.
  - rewrite interp_S /= in Hinterp. simplify_option_eq.
    destruct (E !! x) eqn:Hlookup; simplify_res; try done.
    apply closed_env_lookup in Hlookup; last assumption.
    destruct t as [E' e']. apply closed_thunk_eq in Hlookup as [Henv Hexpr].
    by apply IHn with (E := E') (e := e').
  - rewrite interp_S /= in Hinterp. simplify_option_eq.
    destruct mv as [v|]; simplify_res. split_and!.
    + set_solver.
    + done.
  - rewrite interp_S /= in Hinterp. simplify_option_eq.
    destruct Heclosed as [He1closed He2closed].
    destruct (interp n E e1) as [[[]|]|] eqn:Einterp; simplify_res; try done.
    apply IHn in Einterp; try done.
    simpl in Einterp. destruct Einterp as [Hinterp1 Hinterp2].
    apply IHn in Hinterp; first done.
    + rewrite <-insert_delete_insert.
      apply map_Forall_insert; first apply lookup_delete. split.
      * by split.
      * by apply closed_env_delete.
    + by rewrite dom_insert_L.
Qed.

Lemma interp_proper n E1 E2 e1 e2 mv :
  closed_env E1 → closed_env E2 →
  closed (dom E1) e1 → closed (dom E2) e2 →
  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 E2 E1 e1 e2 mv. induction n as [|n IHn]; [done|].
  intros E2. induction E2 as [E2 IH] using env_ind.
  intros E1 e1 e2 mv HE1closed HE2closed He1closed He2closed Hsubst Hinterp.
  destruct (id_or_not e1) as [[x ->]|?].
  { rewrite interp_S /= in Hinterp.
    destruct (E1 !! x) as [[E' e']|] eqn:Hx; simplify_eq/=;
      last by apply not_elem_of_dom in Hx.
    rewrite subst_env_id Hx in Hsubst.
    apply closed_env_lookup in Hx; last done.
    rewrite closed_thunk_eq in Hx.
    destruct Hx as [HE'close He'closed].
    eauto. }
  destruct (id_or_not e2) as [[x ->]|?].
  { rewrite subst_env_id in Hsubst.
    destruct (E2 !! x) as [[E' e']|] eqn:Hx; simplify_eq/=.
    - apply closed_env_lookup in Hx as Hclosed; last done.
      rewrite closed_thunk_eq in Hclosed.
      destruct Hclosed as [HE'closed He'closed].
      rewrite map_Forall_lookup in IH.
      odestruct (IH _ _ Hx) as (w & m & Hinterp' & Hw);
        first apply HE1closed; try done.
      exists w, (S m). by rewrite interp_S /= Hx /=.
    - destruct mv as [v|].
      + apply interp_not_id in Hinterp. by rewrite Hsubst in Hinterp.
      + exists None, 1. by rewrite interp_S /= Hx. }
  rewrite (subst_env_eq e1) (subst_env_eq e2) in Hsubst.
  rewrite interp_S in Hinterp. destruct e1, e2; simplify_res; try done.
  - eexists (Some (VString _)), 1. by rewrite interp_S.
  - eexists (Some (VClo _ _ _)), 1. split; first by rewrite interp_S.
    by do 2 f_equal/=.
  - destruct (interp n _ _) as [mv'|] eqn:Hinterp'; simplify_res.
    destruct He1closed as [He1_1closed He1_2closed],
             He2closed as [He2_1closed He2_2closed].
    apply interp_closed in Hinterp' as Hclosed; [|done..].
    eapply IHn with (e2 := e2_1) in Hinterp' as (mw' & m1 & Hinterp1 & ?);
      try 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 & ?).
    + exists w, (S (m1 `max` m2)). rewrite interp_S /=.
      rewrite (interp_le Hinterp1) /=; last lia.
      rewrite (interp_le Hinterp2) /=; last lia. done.
    + rewrite -insert_delete_insert.
      apply map_Forall_insert; first apply lookup_delete.
      split; first done. apply closed_env_delete. naive_solver.
    + apply interp_closed in Hinterp1; [|done..].
      rewrite /closed_val in Hinterp1. destruct Hinterp1 as [??].
      by apply map_Forall_insert_2.
    + rewrite dom_insert_L. naive_solver.
    + rewrite dom_insert_L.
      apply interp_closed in Hinterp1; [|done..].
      rewrite /closed_val in Hinterp1. by destruct Hinterp1 as [_ ?].
    + apply interp_closed in Hinterp1; [|done..].
      rewrite /closed_val in Hinterp1. destruct Hinterp1 as [? _].
      apply subst_env_insert_eq; try naive_solver.
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 n x e1 e2 mv :
  closed {[x]} e1 → closed ∅ e2 →
  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.
  intros He1 He2.
  apply interp_proper.
  - done.
  - by apply closed_env_insert.
  - apply subst_closed.
    + by apply map_Forall_singleton.
    + by rewrite dom_singleton_L dom_empty_L union_empty_r_L.
  - by rewrite insert_empty dom_singleton_L.
  - by rewrite subst_env_empty subst_as_subst_env.
Qed.

Lemma closed_step e1 e2 : closed ∅ e1 → e1 --> e2 → closed ∅ e2.
Proof.
  intros Hclosed Hstep. revert Hclosed.
  induction Hstep; intros He1closed.
  - simplify_eq/=. destruct He1closed.
    apply subst_closed.
    + by eapply map_Forall_singleton.
    + by rewrite dom_singleton_L.
  - simplify_eq/=. destruct He1closed. auto.
Qed.

Lemma closed_steps e1 e2 : closed ∅ e1 → e1 -->* e2 → closed ∅ e2.
Proof. induction 2; eauto using closed_step. Qed.

Lemma interp_step e1 e2 n v :
  closed ∅ e1 →
  e1 --> e2 →
  interp n ∅ e2 = Res v →
  ∃ w m, interp m ∅ e1 = Res w ∧ val_to_expr <$> v = val_to_expr <$> w.
Proof.
  intros He1closed Hstep. revert v n He1closed.
  induction Hstep as [|???? IH]; intros v n He1closed Hinterp.
  { rewrite /= union_empty_r_L in He1closed.
    destruct He1closed as [He1closed He2closed].
    apply interp_subst in Hinterp as (w & [|m] & Hinterp & Hv);
      simplify_eq/=; [|done..].
    exists w, (S (S m)). by rewrite !interp_S /= -interp_S. }
  simpl in He1closed. destruct He1closed as [He1closed He2closed].
  destruct n as [|n]; [done|rewrite interp_S /= in Hinterp].
  destruct (interp n _ _) eqn:Hinterp'; simplify_res.
  destruct x; simplify_res; last first.
  { apply IH in Hinterp' as (mw' & m1 & Hinterp1 & ?); simplify_res; last done.
    destruct mw'; try done. exists None, (S m1).
    by rewrite interp_S /= Hinterp1. }
  apply closed_step in Hstep as He1'closed; last done.
  apply interp_closed in Hinterp' as Hcloclosed;
    [|done|by rewrite dom_empty_L].
  apply IH in Hinterp' as ([] & m1 & Hinterp1 & ?); simplify_eq/=; last done.
  destruct (maybe3 VClo _) eqn:Hclo; simplify_res; last first.
  { exists None, (S m1). rewrite interp_S /= Hinterp1 /=.
    by assert (maybe3 VClo v1 = None) as -> by (by destruct v1, v0). }
  simplify_option_eq.
  simpl in Hcloclosed. destruct Hcloclosed as [HEclosed Heclosed].
  apply interp_closed in Hinterp1 as Hcloclosed;
    [|done|by rewrite dom_empty_L]. simpl in Hcloclosed.
  destruct v1; simplify_option_eq.
  destruct Hcloclosed as [HE0closed He0closed].
  eapply interp_proper with (E2 := <[x0:=Thunk ∅ e2]> E0) (e2 := e0)
    in Hinterp as (w & m2 & Hinterp2 & Hv); last apply subst_env_insert_eq.
  { exists w, (S (m1 `max` m2)). rewrite !interp_S /=.
    rewrite (interp_le Hinterp1) /=; last lia.
    by rewrite (interp_le Hinterp2) /=; last lia. }
  - by apply closed_env_insert.
  - by apply closed_env_insert.
  - by rewrite dom_insert_L.
  - by rewrite dom_insert_L.
  - done.
  - done.
  - done.
  - done.
Qed.

Lemma final_interp e :
  final e →
  ∃ w m, interp m ∅ e = mret w ∧ e = val_to_expr w.
Proof.
  induction e; inv 1.
  - eexists (VString _), 1. by rewrite interp_S /=.
  - eexists (VClo _ _ _), 1. rewrite interp_S /=. split; [done|].
    by rewrite delete_empty 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 *) do 2 right. by exists 1.
  - (* EAbs *) right; left. constructor.
  - (* 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 :
  closed ∅ e1 →
  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 He1 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. }
  apply closed_step in Hstep as He2; last assumption.
  destruct IH as (mw & m & Hinterp & ?); try done.
  eapply interp_step in Hinterp as (mw' & m' & ? & ?).
  - destruct mw, mw'; naive_solver.
  - done.
  - done.
Qed.

Lemma interp_complete_ret e1 e2 :
  closed ∅ e1 →
  e1 -->* e2 → final e2 →
  ∃ w m, interp m ∅ e1 = mret w ∧ e2 = val_to_expr w.
Proof.
  intros Hclosed Hsteps Hfinal. apply interp_complete in Hsteps
    as ([w|] & m & ? & ?); naive_solver eauto using final_nf.
Qed.
Lemma interp_complete_fail e1 e2 :
  closed ∅ e1 →
  e1 -->* e2 → nf step e2 → ¬final e2 →
  ∃ m, interp m ∅ e1 = mfail.
Proof.
  intros Hclosed 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 :
  closed_env E → closed (dom E) e →
  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 HEclosed Heclosed Hinterp; first done.
  rewrite subst_env_eq. rewrite interp_S in Hinterp.
  destruct e; simplify_res.
  - (* ENat *) by eexists.
  - (* EId *) destruct (_ !! _) as [[E' e]|] eqn:Hx; simplify_res.
    + apply closed_env_lookup in Hx as Hxclosed; last done.
      rewrite closed_thunk_eq in Hxclosed. destruct_and!.
      apply IH in Hinterp as (e' & Hsteps & He'); naive_solver.
    + eexists; repeat split; [done| |inv 1]. intros [? Hstep]. inv Hstep.
  - (* EAbs *) by eexists.
  - (* EApp *) destruct_and!.
    destruct (interp _ _ _) as [mv'|] eqn:Hinterp'; simplify_res.
    apply interp_closed in Hinterp' as Hvclosed; [|done..].
    apply IH in Hinterp' as (e' & Hsteps & He'); [|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. destruct_and!.
    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'.
    + by apply closed_env_insert.
    + by rewrite dom_insert_L.
Qed.

Lemma interp_sound n e mv :
  closed ∅ e →
  interp n ∅ e = Res mv →
  ∃ e', e -->* e' ∧ if mv is Some v then e' = val_to_expr v else stuck e'.
Proof.
  intros He Hsteps%interp_sound_open; try done.
  by rewrite subst_env_empty in Hsteps.
Qed.

(** Final theorems *)
Theorem interp_sound_complete_ret e v :
  closed ∅ e →
  (∃ 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 :
  closed ∅ e →
  (∃ 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 :
  closed ∅ 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 :
  closed ∅ 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 & _); [|done..].
      by rewrite Hnofuel in Hinterp.
    + apply interp_sound_complete_fail in He' as (e'' & ? & [Hnf _]);
        last by eauto using closed_steps.
      destruct (interp_complete e e'') as (mv & n & Hinterp & _); [done|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); [|done].
    destruct mv as [v|]; simplify_eq/=.
    + apply final_nf in Hsteps as []. apply final_val_to_expr.
    + by destruct Hstuck as [[] ?].
Qed.

End lambda.