# Library iris.algebra.cofe_solver

From iris.algebra Require Export ofe.
From iris.prelude Require Import options.

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

Module solver. Section solver.
Context (F : oFunctor) `{Fcontr : !oFunctorContractive F}.
Context `{Fcofe : (T : ofe) `{!Cofe T}, Cofe (oFunctor_apply F T)}.
Context `{Finh : Inhabited (oFunctor_apply F unitO)}.
Notation map := (oFunctor_map F).

Fixpoint A' (k : nat) : { C : ofe & Cofe C } :=
match k with
| 0 ⇒ existT (P:=Cofe) unitO _
| S kexistT (P:=Cofe) (@oFunctor_apply F (projT1 (A' k)) (projT2 (A' k))) _
end.
Notation A k := (projT1 (A' k)).
Local Instance A_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 kmap (g k,f k) end
with g (k : nat) : A (S k) -n> A k :=
match k with 0 ⇒ OfeMor (λ _, ()) | S kmap (f k,g k) end.
Definition f_S k (x : A (S k)) : f (S k) x = map (g k,f k) x := eq_refl.
Definition g_S k (x : A (S (S k))) : g (S k) x = map (f k,g k) x := eq_refl.
Global Arguments f : simpl never.
Global Arguments g : 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 Instance tower_equiv : Equiv tower := λ X Y, k, X k Y k.
Global Instance tower_dist : Dist tower := λ n X Y, k, X k ≡{n}≡ Y k.
Definition tower_ofe_mixin : OfeMixin tower.
Proof.
split.
- 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 j X Y HXY Hlt n. apply (dist_le k); [|lia].
by rewrite -(g_tower X) (HXY (S n)) g_tower.
Qed.
Definition T : ofe := Ofe tower tower_ofe_mixin.

Program Definition tower_chain (c : chain T) (k : nat) : chain (A k) :=
{| chain_car i := c i k |}.
Next Obligation. intros c k n i ?; apply (chain_cauchy c n); lia. Qed.
Program Definition tower_compl : Compl T := λ c,
{| tower_car n := compl (tower_chain c n) |}.
Next Obligation.
intros c k; apply equiv_distn.
by rewrite (conv_compl n (tower_chain c k))
(conv_compl n (tower_chain c (S k))) /= (g_tower (c _) k).
Qed.
Global Program Instance tower_cofe : Cofe T := { compl := tower_compl }.
Next Obligation.
intros n c k; rewrite /= (conv_compl n (tower_chain c k)).
apply (chain_cauchy c); lia.
Qed.

Fixpoint ff {k} (i : nat) : A k -n> A (i + k) :=
match i with 0 ⇒ cid | S if (i + k) ff i end.
Fixpoint gg {k} (i : nat) : A (i + k) -n> A k :=
match i with 0 ⇒ cid | S igg 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_tower k (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_tower k 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_tower k 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 Instance tower_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 Hijx. assert (i = i2 + i1) asby 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) asby 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 Hgg (k-i) coerce (eq_sym (Nat.sub_add _ _ H))
| right Hcoerce (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 Definition embed (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 Instance embed_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_f k (x : A k) : embed (S k) (f k x) embed k x.
Proof.
rewrite equiv_distn 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_tower k (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.

Program Definition unfold_chain (X : T) : chain (oFunctor_apply F T) :=
{| chain_car n := map (project n,embed' n) (X (S n)) |}.
Next Obligation.
intros X n i Hi.
assert ( k, i = k + n) as [k ?] by ( (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); splitY /=; rewrite ?g_tower ?embed_f.
Qed.
Definition unfold (X : T) : oFunctor_apply F T := compl (unfold_chain X).
Global Instance unfold_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 Definition fold (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); splitY; [apply embed_f|apply g_tower].
Qed.
Global Instance fold_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) _ _)).
- moveX /=. rewrite equiv_distn 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); splitY.
+ 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_distn /=.
rewrite /unfold /= (conv_compl' n (unfold_chain (fold X))) /=.
rewrite g_S -!oFunctor_map_compose -{2}[X]oFunctor_map_id.
apply (contractive_ne map); splitY /=.
+ rewrite f_tower. apply dist_S. by rewrite embed_tower.
+ etrans; [apply embed_ne, equiv_dist, g_tower|apply embed_tower].
Qed.
End solver. End solver.