Require Export modures.cofe. Section solver. Context (F : cofeT → cofeT → cofeT). Context `{Finhab : Inhabited (F (CofeT unit) (CofeT unit))}. Context (map : ∀ {A1 A2 B1 B2 : cofeT}, ((A2 -n> A1) * (B1 -n> B2)) → (F A1 B1 -n> F A2 B2)). Arguments map {_ _ _ _} _. Instance: Params (@map) 4. Context (map_id : ∀ {A B : cofeT} (x : F A B), map (cid, cid) x ≡ x). Context (map_comp : ∀ {A1 A2 A3 B1 B2 B3 : cofeT} (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x, map (f ◎ g, g' ◎ f') x ≡ map (g,g') (map (f,f') x)). Context (map_contractive : ∀ {A1 A2 B1 B2}, Contractive (@map A1 A2 B1 B2)). Lemma map_ext {A1 A2 B1 B2 : cofeT} (f : A2 -n> A1) (f' : A2 -n> A1) (g : B1 -n> B2) (g' : B1 -n> B2) x x' : (∀ x, f x ≡ f' x) → (∀ y, g y ≡ g' y) → x ≡ x' → map (f,g) x ≡ map (f',g') x'. Proof. by rewrite -!cofe_mor_ext; intros -> -> ->. Qed. Fixpoint A (k : nat) : cofeT := match k with 0 => CofeT unit | S k => F (A k) (A k) end. Fixpoint f {k} : A k -n> A (S k) := match k with 0 => CofeMor (λ _, inhabitant) | S k => map (g,f) end with g {k} : A (S k) -n> A k := match k with 0 => CofeMor (λ _, () : CofeT ()) | S k => map (f,g) end. Definition f_S k (x : A (S k)) : f x = map (g,f) x := eq_refl. Definition g_S k (x : A (S (S k))) : g x = map (f,g) x := eq_refl. Lemma gf {k} (x : A k) : g (f x) ≡ x. Proof. induction k as [|k IH]; simpl in *; [by destruct x|]. rewrite -map_comp -{2}(map_id _ _ x); by apply map_ext. Qed. Lemma fg {n k} (x : A (S k)) : n ≤ k → f (g x) ={n}= x. Proof. intros Hnk; apply dist_le with k; auto; clear Hnk. induction k as [|k IH]; simpl; [apply dist_0|]. rewrite -{2}(map_id _ _ x) -map_comp; by apply map_contractive. Qed. Arguments A _ : simpl never. Arguments f {_} : simpl never. Arguments g {_} : simpl never. Record tower := { tower_car k :> A k; g_tower k : g (tower_car (S k)) ≡ tower_car k }. Instance tower_equiv : Equiv tower := λ X Y, ∀ k, X k ≡ Y k. Instance tower_dist : Dist tower := λ n X Y, ∀ k, X k ={n}= Y k. Program Definition tower_chain (c : chain tower) (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 Instance tower_compl : Compl tower := λ c, {| tower_car n := compl (tower_chain c n) |}. Next Obligation. intros c k; apply equiv_dist; intros n. rewrite (conv_compl (tower_chain c k) n). by rewrite (conv_compl (tower_chain c (S k)) n) /= (g_tower (c n) k). Qed. Instance tower_cofe : Cofe 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; transitivity (Y n). * intros k X Y HXY n; apply dist_S. by rewrite -(g_tower X) (HXY (S n)) g_tower. * intros X Y k; apply dist_0. * intros c n k; rewrite /= (conv_compl (tower_chain c k) n). apply (chain_cauchy c); lia. Qed. Definition T : cofeT := CofeT tower. Fixpoint ff {k} (i : nat) : A k -n> A (i + k) := match i with 0 => cid | S i => f ◎ ff i end. Fixpoint gg {k} (i : nat) : A (i + k) -n> A k := match i with 0 => cid | S i => gg i ◎ g end. Lemma ggff {k i} (x : A k) : gg i (ff i x) ≡ x. Proof. induction i as [|i IH]; simpl; [done|by rewrite (gf (ff i x)) IH]. Qed. Lemma f_tower {n k} (X : tower) : n ≤ k → f (X k) ={n}= X (S k). Proof. intros. by rewrite -(fg (X (S k))) // -(g_tower X). Qed. Lemma ff_tower {n} k i (X : tower) : n ≤ k → ff i (X k) ={n}= X (i + k). Proof. intros; induction i as [|i IH]; simpl; [done|]. by rewrite IH (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. Instance tower_car_ne n k : Proper (dist n ==> dist n) (λ X, tower_car X k). Proof. by intros X Y HX. Qed. Definition project (k : nat) : T -n> A k := CofeMor (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 (coerce H x) = coerce (Nat.succ_inj _ _ H) (g 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 x) = f (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. assert (i = i2 + i1) by lia; simplify_equality'. revert j x H1. induction i2 as [|i2 IH]; intros j X H1; simplify_equality'; [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. assert (i = i1 + i2) by lia; simplify_equality'. induction i1 as [|i1 IH]; simplify_equality'; [by rewrite coerce_id|by rewrite coerce_f IH]. Qed. Definition embed' {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' {k i} (x : A k) : g (embed' (S i) x) ≡ embed' i x. Proof. unfold embed'; 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_inf (k : nat) (x : A k) : T := {| tower_car n := embed' n x |}. Next Obligation. intros k x i. apply g_embed'. Qed. Instance embed_inf_ne k n : Proper (dist n ==> dist n) (embed_inf k). Proof. by intros x y Hxy i; simpl; rewrite Hxy. Qed. Definition embed (k : nat) : A k -n> T := CofeMor (embed_inf k). Lemma embed_f k (x : A k) : embed (S k) (f x) ≡ embed k x. Proof. rewrite equiv_dist; intros n i. unfold embed_inf, embed; simpl; unfold embed'. 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_equality'. 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 j n (X : T) : n ≤ j → embed j (X j) ={n}= X. Proof. intros Hn i; simpl; unfold embed'; destruct (le_lt_dec i j) as [H|H]; simpl. * rewrite -(gg_tower i (j - i) X). apply (_ : Proper (_ ==> _) (gg _)); by destruct (eq_sym _). * rewrite (ff_tower j (i-j) X); last lia. by destruct (Nat.sub_add _ _ _). Qed. Program Definition unfold_chain (X : T) : chain (F T T) := {| chain_car n := map (project n,embed n) (f (X n)) |}. Next Obligation. intros X n i Hi. assert (∃ k, i = k + n) as [k ?] by (exists (i - n); lia); subst; clear Hi. induction k as [|k Hk]; simpl; [done|]. rewrite Hk (f_tower X); last lia; rewrite f_S -map_comp. apply dist_S, map_contractive. split; intros Y; symmetry; apply equiv_dist; [apply g_tower|apply embed_f]. Qed. Definition unfold' (X : T) : F T T := compl (unfold_chain X). Instance unfold_ne : Proper (dist n ==> dist n) unfold'. Proof. by intros n X Y HXY; unfold unfold'; apply compl_ne; rewrite /= (HXY n). Qed. Definition unfold : T -n> F T T := CofeMor unfold'. Program Definition fold' (X : F T T) : T := {| tower_car n := g (map (embed n,project n) X) |}. Next Obligation. intros X k; apply (_ : Proper ((≡) ==> (≡)) g). rewrite -(map_comp _ _ _ _ _ _ (embed (S k)) f (project (S k)) g). apply map_ext; [apply embed_f|intros Y; apply g_tower|done]. Qed. Instance fold_ne : Proper (dist n ==> dist n) fold'. Proof. by intros n X Y HXY k; simpl; unfold fold'; simpl; rewrite HXY. Qed. Definition fold : F T T -n> T := CofeMor fold'. Definition fold_unfold X : fold (unfold X) ≡ X. Proof. assert (map_ff_gg : ∀ 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)). { intros i; induction i as [|i IH]; intros k x H; simpl. { by rewrite coerce_id map_id. } rewrite map_comp g_coerce; apply IH. } rewrite equiv_dist; intros n k; unfold unfold, fold; simpl. rewrite -g_tower -(gg_tower _ n); apply (_ : Proper (_ ==> _) g). transitivity (map (ff n, gg n) (X (S (n + k)))). { unfold unfold'; rewrite (conv_compl (unfold_chain X) n). rewrite (chain_cauchy (unfold_chain X) n (n + k)) /=; last lia. rewrite (f_tower X); last lia; rewrite -map_comp. apply dist_S. apply map_contractive; split; intros x; simpl; unfold embed'. * destruct (le_lt_dec _ _); simpl. { assert (n = 0) by lia; subst. apply dist_0. } by rewrite (ff_ff _ (eq_refl (n + (0 + k)))). * destruct (le_lt_dec _ _); [|exfalso; lia]; simpl. by rewrite (gg_gg _ (eq_refl (0 + (n + k)))). } assert (H: S n + k = n + S k) by lia; rewrite (map_ff_gg _ _ _ H). apply (_ : Proper (_ ==> _) (gg _)); by destruct H. Qed. Definition unfold_fold X : unfold (fold X) ≡ X. Proof. rewrite equiv_dist; intros n. rewrite /(unfold) /= /(unfold') (conv_compl (unfold_chain (fold X)) n) /=. rewrite (fg (map (embed _,project n) X)); last lia. rewrite -map_comp -{2}(map_id _ _ X). apply dist_S, map_contractive; split; intros Y i; apply embed_tower; lia. Qed. End solver. Global Opaque T fold unfold.