(* iris/algebra/cofe_solver.v *)
From iris.algebra Require Export ofe
From iris.algebra Require Export stepindex_finite
From iris.prelude Require Import options

Record solution(F : oFunctor) := Solution {
  solution_car :> ofe;
  solution_inhabited : Inhabited solution_car;
  solution_cofe : Cofe solution_car;
  solution_iso :> ofe_iso (oFunctor_apply F solution_car) solution_car;
}.

Global ExistingInstances solution_inhabited solution_cofe.

Notation map:= (oFunctor_map F).

Fixpoint A'(k : nat) : { C : ofe & Cofe C } :=
  match k with
  | 0 => existT (P:=Cofe) unitO _
  | S k => existT (P:=Cofe) (@oFunctor_apply _ F (projT1 (A' k)) (projT2 (A' k))) _
  end.

Notation Ak := (projT1 (A' k)).

Local InstanceA_cofe k : Cofe (A k) := projT2 (A' k).

Fixpoint f(k : nat) : A k -n> A (S k) :=
  match k with 0 => OfeMor (λ _, inhabitant) | S k => map (g k,f k) end
with g (k : nat) : A (S k) -n> A k :=
  match k with 0 => OfeMor (λ _, ()) | S k => map (f k,g k) end.

Definition f_Sk (x : A (S k)) : f (S k) x = map (g k,f k) x := eq_refl.

Definition g_Sk (x : A (S (S k))) : g (S k) x = map (f k,g k) x := eq_refl.

Global Argumentsf : simpl never.

Global Argumentsg : simpl never.

Lemma gf{k} (x : A k) : g k (f k x) ≡ x.
Proof using Fcontr.
  induction k as [|k IH]; simpl in *; [by destruct x|].
  rewrite -oFunctor_map_compose -{2}[x]oFunctor_map_id.
  by apply (contractive_proper map).
Qed.

Lemma fg{k} (x : A (S (S k))) : f (S k) (g (S k) x) ≡{k}≡ x.
Proof using Fcontr.
  induction k as [|k IH]; simpl.
  - rewrite f_S g_S -{2}[x]oFunctor_map_id -oFunctor_map_compose.
    apply (contractive_0 map).
  - rewrite f_S g_S -{2}[x]oFunctor_map_id -oFunctor_map_compose.
    by apply (contractive_S map).
Qed.

Record tower:= {
  tower_car k :> A k;
  g_tower k : g k (tower_car (S k)) ≡ tower_car k
}.

Global Instancetower_equiv : Equiv tower := λ X Y, ∀ k, X k ≡ Y k.

Global Instancetower_dist : Dist tower := λ n X Y, ∀ k, X k ≡{n}≡ Y k.

Definition tower_ofe_mixin: OfeMixin tower.
Proof.
  apply ofe_mixin_finite.
  - intros X Y; split; [by intros HXY n k; apply equiv_dist|].
    intros HXY k; apply equiv_dist; intros n; apply HXY.
  - intros k; split.
    + by intros X n.
    + by intros X Y ? n.
    + by intros X Y Z ?? n; trans (Y n).
  - intros k X Y HXY n. specialize (HXY (S n)).
    apply (dist_le _ k) in HXY; [|apply SIdx.le_succ_diag_r].
    by rewrite -(g_tower X) HXY g_tower.
Qed.

Definition T: ofe := Ofe tower tower_ofe_mixin.

Program Definitiontower_chain (c : chain T) (k : nat) : chain (A k) :=
  {| chain_car i := c i k |}.
Next Obligation. intros c k n i ?; by apply (chain_cauchy c n). Qed.

Program Definitiontower_compl : Compl T := λ c,
  {| tower_car n := compl (tower_chain c n) |}.
Next Obligation.
  intros c k; apply equiv_dist=> n.
  by rewrite (conv_compl n (tower_chain c k))
    (conv_compl n (tower_chain c (S k))) /= (g_tower (c _) k).
Qed.

Global ProgramInstance tower_cofe : Cofe T := cofe_finite tower_compl _.
Next Obligation.
  intros n c k; rewrite /= (conv_compl n (tower_chain c k)). done.
Qed.

Fixpoint ff{k} (i : nat) : A k -n> A (i + k) :=
  match i with 0 => cid | S i => f (i + k) ◎ ff i end.

Fixpoint gg{k} (i : nat) : A (i + k) -n> A k :=
  match i with 0 => cid | S i => gg i ◎ g (i + k) end.

Lemma ggff{k i} (x : A k) : gg i (ff i x) ≡ x.
Proof using Fcontr. induction i as [|i IH]; simpl; [done|by rewrite (gf (ff i x)) IH]. Qed.

Lemma f_towerk (X : tower) : f (S k) (X (S k)) ≡{k}≡ X (S (S k)).
Proof using Fcontr. intros. by rewrite -(fg (X (S (S k)))) -(g_tower X). Qed.

Lemma ff_towerk i (X : tower) : ff i (X (S k)) ≡{k}≡ X (i + S k).
Proof using Fcontr.
  intros; induction i as [|i IH]; simpl; [done|].
  by rewrite IH Nat.add_succ_r (dist_le _ _ _ _ (f_tower _ X)); last lia.
Qed.

Lemma gg_towerk i (X : tower) : gg i (X (i + k)) ≡ X k.
Proof. by induction i as [|i IH]; simpl; [done|rewrite g_tower IH]. Qed.

Global Instancetower_car_ne k : NonExpansive (λ X, tower_car X k).
Proof. by intros X Y HX. Qed.

Definition project(k : nat) : T -n> A k := OfeMor (λ X : T, tower_car X k).

Definition coerce{i j} (H : i = j) : A i -n> A j :=
  eq_rect _ (λ i', A i -n> A i') cid _ H.

Lemma coerce_id{i} (H : i = i) (x : A i) : coerce H x = x.
Proof. unfold coerce. by rewrite (proof_irrel H (eq_refl i)). Qed.

Lemma coerce_proper{i j} (x y : A i) (H1 H2 : i = j) :
  x = y → coerce H1 x = coerce H2 y.
Proof. by destruct H1; rewrite !coerce_id. Qed.

Lemma g_coerce{k j} (H : S k = S j) (x : A (S k)) :
  g j (coerce H x) = coerce (Nat.succ_inj _ _ H) (g k x).
Proof. by assert (k = j) by lia; subst; rewrite !coerce_id. Qed.

Lemma coerce_f{k j} (H : S k = S j) (x : A k) :
  coerce H (f k x) = f j (coerce (Nat.succ_inj _ _ H) x).
Proof. by assert (k = j) by lia; subst; rewrite !coerce_id. Qed.

Lemma gg_gg{k i i1 i2 j} : ∀ (H1: k = i + j) (H2: k = i2 + (i1 + j)) (x: A k),
  gg i (coerce H1 x) = gg i1 (gg i2 (coerce H2 x)).
Proof.
  intros Hij -> x. assert (i = i2 + i1) as -> by lia. revert j x Hij.
  induction i2 as [|i2 IH]; intros j X Hij; simplify_eq/=;
    [by rewrite coerce_id|by rewrite g_coerce IH].
Qed.

Lemma ff_ff{k i i1 i2 j} : ∀ (H1: i + k = j) (H2: i1 + (i2 + k) = j) (x: A k),
  coerce H1 (ff i x) = coerce H2 (ff i1 (ff i2 x)).
Proof.
  intros ? <- x. assert (i = i1 + i2) as -> by lia.
  induction i1 as [|i1 IH]; simplify_eq/=;
    [by rewrite coerce_id|by rewrite coerce_f IH].
Qed.

Definition embed_coerce{k} (i : nat) : A k -n> A i :=
  match le_lt_dec i k with
  | left H => gg (k-i) ◎ coerce (eq_sym (Nat.sub_add _ _ H))
  | right H => coerce (Nat.sub_add k i (Nat.lt_le_incl _ _ H)) ◎ ff (i-k)
  end.

Lemma g_embed_coerce{k i} (x : A k) :
  g i (embed_coerce (S i) x) ≡ embed_coerce i x.
Proof using Fcontr.
  unfold embed_coerce; destruct (le_lt_dec (S i) k), (le_lt_dec i k); simpl.
  - symmetry; by erewrite (@gg_gg _ _ 1 (k - S i)); simpl.
  - exfalso; lia.
  - assert (i = k) by lia; subst.
    rewrite (ff_ff _ (eq_refl (1 + (0 + k)))) /= gf.
    by rewrite (gg_gg _ (eq_refl (0 + (0 + k)))).
  - assert (H : 1 + ((i - k) + k) = S i) by lia.
    rewrite (ff_ff _ H) /= -{2}(gf (ff (i - k) x)) g_coerce.
    by erewrite coerce_proper by done.
Qed.

Program Definitionembed (k : nat) (x : A k) : T :=
  {| tower_car n := embed_coerce n x |}.
Next Obligation. intros k x i. apply g_embed_coerce. Qed.

Global Instance: Params (@embed) 1 := {}.

Global Instanceembed_ne k : NonExpansive (embed k).
Proof. by intros n x y Hxy i; rewrite /= Hxy. Qed.

Definition embed'(k : nat) : A k -n> T := OfeMor (embed k).

Lemma embed_fk (x : A k) : embed (S k) (f k x) ≡ embed k x.
Proof.
  rewrite equiv_dist=> n i; rewrite /embed /= /embed_coerce.
  destruct (le_lt_dec i (S k)), (le_lt_dec i k); simpl.
  - assert (H : S k = S (k - i) + (0 + i)) by lia; rewrite (gg_gg _ H) /=.
    by erewrite g_coerce, gf, coerce_proper by done.
  - assert (S k = 0 + (0 + i)) as H by lia.
    rewrite (gg_gg _ H); simplify_eq/=.
    by rewrite (ff_ff _ (eq_refl (1 + (0 + k)))).
  - exfalso; lia.
  - assert (H : (i - S k) + (1 + k) = i) by lia; rewrite (ff_ff _ H) /=.
    by erewrite coerce_proper by done.
Qed.

Lemma embed_towerk (X : T) : embed (S k) (X (S k)) ≡{k}≡ X.
Proof.
  intros i; rewrite /= /embed_coerce.
  destruct (le_lt_dec i (S k)) as [H|H]; simpl.
  - rewrite -(gg_tower i (S k - i) X).
    apply (_ : Proper (_ ==> _) (gg _)); by destruct (eq_sym _).
  - rewrite (ff_tower k (i - S k) X). by destruct (Nat.sub_add _ _ _).
Qed.

Global Instancetower_inhabited : Inhabited tower := populate (embed 0 ()).

Program Definitionunfold_chain (X : T) : chain (oFunctor_apply F T) :=
  {| chain_car n := map (project n,embed' n) (X (S n)) |}.
Next Obligation.
  simpl; intros X n i Hi.
  assert (∃ k, i = k + n) as [k ?] by (exists (i - n); lia); subst; clear Hi.
  induction k as [|k IH]; simpl; first done.
  rewrite -IH -(dist_le _ _ _ _ (f_tower (k + n) _)); last lia.
  rewrite f_S -oFunctor_map_compose.
  by apply (contractive_ne map); split=> Y /=; rewrite ?g_tower ?embed_f.
Qed.

Definition unfold(X : T) : oFunctor_apply F T := compl (unfold_chain X).

Global Instanceunfold_ne : NonExpansive unfold.
Proof.
  intros n X Y HXY. by rewrite /unfold (conv_compl n (unfold_chain X))
    (conv_compl n (unfold_chain Y)) /= (HXY (S n)).
Qed.

Program Definitionfold (X : oFunctor_apply F T) : T :=
  {| tower_car n := g n (map (embed' n,project n) X) |}.
Next Obligation.
  intros X k. apply (_ : Proper ((≡) ==> (≡)) (g k)).
  rewrite g_S -oFunctor_map_compose.
  apply (contractive_proper map); split=> Y; [apply embed_f|apply g_tower].
Qed.

Global Instancefold_ne : NonExpansive fold.
Proof. by intros n X Y HXY k; rewrite /fold /= HXY. Qed.

Theorem result: solution F.
Proof using Type*.
  refine (Solution F T _ _ (OfeIso (OfeMor fold) (OfeMor unfold) _ _)).
  - move=> X /=. rewrite equiv_dist=> n k; rewrite /unfold /fold /=.
    rewrite -g_tower -(gg_tower _ n); apply (_ : Proper (_ ==> _) (g _)).
    trans (map (ff n, gg n) (X (S (n + k)))).
    { rewrite /unfold (conv_compl n (unfold_chain X)).
      rewrite -(chain_cauchy (unfold_chain X) n (S (n + k))) /=; last lia.
      rewrite -(dist_le _ _ _ _ (f_tower (n + k) _)); last lia.
      rewrite f_S -!oFunctor_map_compose; apply (contractive_ne map); split=> Y.
      + rewrite /embed' /= /embed_coerce.
        destruct (le_lt_dec _ _); simpl; [exfalso; lia|].
        by rewrite (ff_ff _ (eq_refl (S n + (0 + k)))) /= gf.
      + rewrite /embed' /= /embed_coerce.
        destruct (le_lt_dec _ _); simpl; [|exfalso; lia].
        by rewrite (gg_gg _ (eq_refl (0 + (S n + k)))) /= gf. }
    assert (∀ i k (x : A (S i + k)) (H : S i + k = i + S k),
      map (ff i, gg i) x ≡ gg i (coerce H x)) as map_ff_gg.
    { intros i; induction i as [|i IH]; intros k' x H; simpl.
      { by rewrite coerce_id oFunctor_map_id. }
      rewrite oFunctor_map_compose g_coerce; apply IH. }
    assert (H: S n + k = n + S k) by lia.
    rewrite (map_ff_gg _ _ _ H).
    apply (_ : Proper (_ ==> _) (gg _)); by destruct H.
  - intros X; rewrite equiv_dist=> n /=.
    rewrite /unfold /= (conv_compl_S n (unfold_chain (fold X))) /=.
    rewrite g_S -!oFunctor_map_compose -{2}[X]oFunctor_map_id.
    apply (con
...