path: root/theories/evallang/interp_proofs.v

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.