cofe_solver.v 10.3 KB
 Robbert Krebbers committed Feb 04, 2016 1 ``````Require Export algebra.cofe. `````` Robbert Krebbers committed Nov 11, 2015 2 `````` `````` Robbert Krebbers committed Jan 14, 2016 3 4 5 6 7 8 9 10 11 12 13 ``````Record solution (F : cofeT → cofeT → cofeT) := Solution { solution_car :> cofeT; solution_unfold : solution_car -n> F solution_car solution_car; solution_fold : F solution_car solution_car -n> solution_car; solution_fold_unfold X : solution_fold (solution_unfold X) ≡ X; solution_unfold_fold X : solution_unfold (solution_fold X) ≡ X }. Arguments solution_unfold {_} _. Arguments solution_fold {_} _. Module solver. Section solver. `````` Robbert Krebbers committed Nov 11, 2015 14 ``````Context (F : cofeT → cofeT → cofeT). `````` Robbert Krebbers committed Jan 14, 2016 15 ``````Context `{Finhab : Inhabited (F unitC unitC)}. `````` Robbert Krebbers committed Nov 11, 2015 16 17 18 19 20 21 22 23 24 25 26 ``````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)). Fixpoint A (k : nat) : cofeT := `````` Robbert Krebbers committed Jan 14, 2016 27 `````` match k with 0 => unitC | S k => F (A k) (A k) end. `````` Robbert Krebbers committed Feb 10, 2016 28 29 30 31 32 33 34 35 36 37 38 ``````Fixpoint f (k : nat) : A k -n> A (S k) := match k with 0 => CofeMor (λ _, inhabitant) | S k => map (g k,f k) end with g (k : nat) : A (S k) -n> A k := match k with 0 => CofeMor (λ _, ()) | S k => map (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. Arguments A : simpl never. Arguments f : simpl never. Arguments g : simpl never. Lemma gf {k} (x : A k) : g k (f k x) ≡ x. `````` Robbert Krebbers committed Nov 11, 2015 39 40 ``````Proof. induction k as [|k IH]; simpl in *; [by destruct x|]. `````` Robbert Krebbers committed Feb 12, 2016 41 `````` rewrite -map_comp -{2}(map_id _ _ x). by apply (contractive_proper map). `````` Robbert Krebbers committed Nov 11, 2015 42 ``````Qed. `````` Robbert Krebbers committed Feb 10, 2016 43 ``````Lemma fg {k} (x : A (S (S k))) : f (S k) (g (S k) x) ≡{k}≡ x. `````` Robbert Krebbers committed Nov 11, 2015 44 ``````Proof. `````` Robbert Krebbers committed Feb 10, 2016 45 `````` induction k as [|k IH]; simpl. `````` Robbert Krebbers committed Feb 12, 2016 46 47 `````` * rewrite f_S g_S -{2}(map_id _ _ x) -map_comp. apply (contractive_0 map). * rewrite f_S g_S -{2}(map_id _ _ x) -map_comp. by apply (contractive_S map). `````` Robbert Krebbers committed Nov 11, 2015 48 49 50 51 ``````Qed. Record tower := { tower_car k :> A k; `````` Robbert Krebbers committed Feb 10, 2016 52 `````` g_tower k : g k (tower_car (S k)) ≡ tower_car k `````` Robbert Krebbers committed Nov 11, 2015 53 54 ``````}. Instance tower_equiv : Equiv tower := λ X Y, ∀ k, X k ≡ Y k. `````` Ralf Jung committed Feb 10, 2016 55 ``````Instance tower_dist : Dist tower := λ n X Y, ∀ k, X k ≡{n}≡ Y k. `````` Robbert Krebbers committed Nov 11, 2015 56 57 58 59 60 61 ``````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. `````` Robbert Krebbers committed Feb 10, 2016 62 63 64 `````` intros c k; apply equiv_dist=> n. by rewrite (conv_compl (tower_chain c k) n) (conv_compl (tower_chain c (S k)) n) /= (g_tower (c (S n)) k). `````` Robbert Krebbers committed Nov 11, 2015 65 ``````Qed. `````` Robbert Krebbers committed Jan 14, 2016 66 ``````Definition tower_cofe_mixin : CofeMixin tower. `````` Robbert Krebbers committed Nov 11, 2015 67 68 69 70 71 72 73 74 75 ``````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. `````` Robbert Krebbers committed Jan 13, 2016 76 77 `````` by rewrite -(g_tower X) (HXY (S n)) g_tower. * intros c n k; rewrite /= (conv_compl (tower_chain c k) n). `````` Robbert Krebbers committed Nov 11, 2015 78 79 `````` apply (chain_cauchy c); lia. Qed. `````` Robbert Krebbers committed Jan 14, 2016 80 ``````Definition T : cofeT := CofeT tower_cofe_mixin. `````` Robbert Krebbers committed Nov 11, 2015 81 82 `````` Fixpoint ff {k} (i : nat) : A k -n> A (i + k) := `````` Robbert Krebbers committed Feb 10, 2016 83 `````` match i with 0 => cid | S i => f (i + k) ◎ ff i end. `````` Robbert Krebbers committed Nov 11, 2015 84 ``````Fixpoint gg {k} (i : nat) : A (i + k) -n> A k := `````` Robbert Krebbers committed Feb 10, 2016 85 `````` match i with 0 => cid | S i => gg i ◎ g (i + k) end. `````` Robbert Krebbers committed Nov 11, 2015 86 ``````Lemma ggff {k i} (x : A k) : gg i (ff i x) ≡ x. `````` Robbert Krebbers committed Jan 13, 2016 87 ``````Proof. induction i as [|i IH]; simpl; [done|by rewrite (gf (ff i x)) IH]. Qed. `````` Robbert Krebbers committed Feb 10, 2016 88 89 90 ``````Lemma f_tower k (X : tower) : f (S k) (X (S k)) ≡{k}≡ X (S (S k)). Proof. 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). `````` Robbert Krebbers committed Nov 11, 2015 91 92 ``````Proof. intros; induction i as [|i IH]; simpl; [done|]. `````` Robbert Krebbers committed Feb 10, 2016 93 `````` by rewrite IH Nat.add_succ_r (dist_le _ _ _ _ (f_tower _ X)); last omega. `````` Robbert Krebbers committed Nov 11, 2015 94 95 ``````Qed. Lemma gg_tower k i (X : tower) : gg i (X (i + k)) ≡ X k. `````` Robbert Krebbers committed Jan 13, 2016 96 ``````Proof. by induction i as [|i IH]; simpl; [done|rewrite g_tower IH]. Qed. `````` Robbert Krebbers committed Nov 11, 2015 97 98 99 100 101 102 103 104 105 106 107 108 109 `````` 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)) : `````` Robbert Krebbers committed Feb 10, 2016 110 `````` g j (coerce H x) = coerce (Nat.succ_inj _ _ H) (g k x). `````` Robbert Krebbers committed Nov 11, 2015 111 112 ``````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) : `````` Robbert Krebbers committed Feb 10, 2016 113 `````` coerce H (f k x) = f j (coerce (Nat.succ_inj _ _ H) x). `````` Robbert Krebbers committed Nov 11, 2015 114 115 116 117 118 119 ``````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'; `````` Robbert Krebbers committed Jan 13, 2016 120 `````` [by rewrite coerce_id|by rewrite g_coerce IH]. `````` Robbert Krebbers committed Nov 11, 2015 121 122 123 124 125 126 ``````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'; `````` Robbert Krebbers committed Jan 13, 2016 127 `````` [by rewrite coerce_id|by rewrite coerce_f IH]. `````` Robbert Krebbers committed Nov 11, 2015 128 129 ``````Qed. `````` Robbert Krebbers committed Feb 10, 2016 130 ``````Definition embed_coerce {k} (i : nat) : A k -n> A i := `````` Robbert Krebbers committed Nov 11, 2015 131 132 133 134 `````` 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. `````` Robbert Krebbers committed Feb 10, 2016 135 136 ``````Lemma g_embed_coerce {k i} (x : A k) : g i (embed_coerce (S i) x) ≡ embed_coerce i x. `````` Robbert Krebbers committed Nov 11, 2015 137 ``````Proof. `````` Robbert Krebbers committed Feb 10, 2016 138 `````` unfold embed_coerce; destruct (le_lt_dec (S i) k), (le_lt_dec i k); simpl. `````` Robbert Krebbers committed Nov 11, 2015 139 140 141 `````` * symmetry; by erewrite (@gg_gg _ _ 1 (k - S i)); simpl. * exfalso; lia. * assert (i = k) by lia; subst. `````` Robbert Krebbers committed Jan 13, 2016 142 `````` rewrite (ff_ff _ (eq_refl (1 + (0 + k)))) /= gf. `````` Robbert Krebbers committed Nov 11, 2015 143 `````` by rewrite (gg_gg _ (eq_refl (0 + (0 + k)))). `````` Robbert Krebbers committed Jan 13, 2016 144 145 `````` * assert (H : 1 + ((i - k) + k) = S i) by lia. rewrite (ff_ff _ H) /= -{2}(gf (ff (i - k) x)) g_coerce. `````` Robbert Krebbers committed Nov 11, 2015 146 147 `````` by erewrite coerce_proper by done. Qed. `````` Robbert Krebbers committed Feb 10, 2016 148 149 150 151 152 ``````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. Instance: Params (@embed) 1. Instance embed_ne k n : Proper (dist n ==> dist n) (embed k). `````` Robbert Krebbers committed Jan 14, 2016 153 ``````Proof. by intros x y Hxy i; rewrite /= Hxy. Qed. `````` Robbert Krebbers committed Feb 10, 2016 154 155 ``````Definition embed' (k : nat) : A k -n> T := CofeMor (embed k). Lemma embed_f k (x : A k) : embed (S k) (f k x) ≡ embed k x. `````` Robbert Krebbers committed Nov 11, 2015 156 ``````Proof. `````` Robbert Krebbers committed Feb 10, 2016 157 `````` rewrite equiv_dist=> n i; rewrite /embed /= /embed_coerce. `````` Robbert Krebbers committed Nov 11, 2015 158 `````` destruct (le_lt_dec i (S k)), (le_lt_dec i k); simpl. `````` Robbert Krebbers committed Jan 13, 2016 159 `````` * assert (H : S k = S (k - i) + (0 + i)) by lia; rewrite (gg_gg _ H) /=. `````` Robbert Krebbers committed Nov 11, 2015 160 161 162 163 164 `````` 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. `````` Robbert Krebbers committed Jan 13, 2016 165 `````` * assert (H : (i - S k) + (1 + k) = i) by lia; rewrite (ff_ff _ H) /=. `````` Robbert Krebbers committed Nov 11, 2015 166 167 `````` by erewrite coerce_proper by done. Qed. `````` Robbert Krebbers committed Feb 10, 2016 168 ``````Lemma embed_tower k (X : T) : embed (S k) (X (S k)) ≡{k}≡ X. `````` Robbert Krebbers committed Nov 11, 2015 169 ``````Proof. `````` Robbert Krebbers committed Feb 10, 2016 170 171 172 `````` intros i; rewrite /= /embed_coerce. destruct (le_lt_dec i (S k)) as [H|H]; simpl. * rewrite -(gg_tower i (S k - i) X). `````` Robbert Krebbers committed Nov 11, 2015 173 `````` apply (_ : Proper (_ ==> _) (gg _)); by destruct (eq_sym _). `````` Robbert Krebbers committed Feb 10, 2016 174 `````` * rewrite (ff_tower k (i - S k) X). by destruct (Nat.sub_add _ _ _). `````` Robbert Krebbers committed Nov 11, 2015 175 176 177 ``````Qed. Program Definition unfold_chain (X : T) : chain (F T T) := `````` Robbert Krebbers committed Feb 10, 2016 178 `````` {| chain_car n := map (project n,embed' n) (X (S n)) |}. `````` Robbert Krebbers committed Nov 11, 2015 179 180 181 ``````Next Obligation. intros X n i Hi. assert (∃ k, i = k + n) as [k ?] by (exists (i - n); lia); subst; clear Hi. `````` Robbert Krebbers committed Feb 10, 2016 182 183 `````` induction k as [|k IH]; simpl. { rewrite -f_tower f_S -map_comp. `````` Robbert Krebbers committed Feb 12, 2016 184 `````` by apply (contractive_ne map); split=> Y /=; rewrite ?g_tower ?embed_f. } `````` Robbert Krebbers committed Feb 10, 2016 185 186 `````` rewrite -IH -(dist_le _ _ _ _ (f_tower (k + n) _)); last lia. rewrite f_S -map_comp. `````` Robbert Krebbers committed Feb 12, 2016 187 `````` by apply (contractive_ne map); split=> Y /=; rewrite ?g_tower ?embed_f. `````` Robbert Krebbers committed Nov 11, 2015 188 ``````Qed. `````` Robbert Krebbers committed Jan 14, 2016 189 190 ``````Definition unfold (X : T) : F T T := compl (unfold_chain X). Instance unfold_ne : Proper (dist n ==> dist n) unfold. `````` Robbert Krebbers committed Feb 10, 2016 191 192 193 194 ``````Proof. intros n X Y HXY. by rewrite /unfold (conv_compl (unfold_chain X) n) (conv_compl (unfold_chain Y) n) /= (HXY (S (S n))). Qed. `````` Robbert Krebbers committed Nov 11, 2015 195 `````` `````` Robbert Krebbers committed Jan 14, 2016 196 ``````Program Definition fold (X : F T T) : T := `````` Robbert Krebbers committed Feb 10, 2016 197 `````` {| tower_car n := g n (map (embed' n,project n) X) |}. `````` Robbert Krebbers committed Nov 11, 2015 198 ``````Next Obligation. `````` Robbert Krebbers committed Feb 10, 2016 199 200 `````` intros X k. apply (_ : Proper ((≡) ==> (≡)) (g k)). rewrite g_S -map_comp. `````` Robbert Krebbers committed Feb 12, 2016 201 `````` apply (contractive_proper map); split=> Y; [apply embed_f|apply g_tower]. `````` Robbert Krebbers committed Nov 11, 2015 202 ``````Qed. `````` Robbert Krebbers committed Jan 14, 2016 203 204 ``````Instance fold_ne : Proper (dist n ==> dist n) fold. Proof. by intros n X Y HXY k; rewrite /fold /= HXY. Qed. `````` Robbert Krebbers committed Nov 11, 2015 205 `````` `````` Robbert Krebbers committed Jan 14, 2016 206 ``````Theorem result : solution F. `````` Robbert Krebbers committed Nov 11, 2015 207 ``````Proof. `````` Robbert Krebbers committed Jan 14, 2016 208 `````` apply (Solution F T (CofeMor unfold) (CofeMor fold)). `````` Robbert Krebbers committed Feb 10, 2016 209 `````` * move=> X /=. `````` Robbert Krebbers committed Jan 14, 2016 210 `````` rewrite equiv_dist; intros n k; unfold unfold, fold; simpl. `````` Robbert Krebbers committed Feb 10, 2016 211 `````` rewrite -g_tower -(gg_tower _ n); apply (_ : Proper (_ ==> _) (g _)). `````` Robbert Krebbers committed Jan 14, 2016 212 213 `````` transitivity (map (ff n, gg n) (X (S (n + k)))). { rewrite /unfold (conv_compl (unfold_chain X) n). `````` Robbert Krebbers committed Feb 10, 2016 214 215 `````` rewrite -(chain_cauchy (unfold_chain X) n (S (n + k))) /=; last lia. rewrite -(dist_le _ _ _ _ (f_tower (n + k) _)); last lia. `````` Robbert Krebbers committed Feb 12, 2016 216 `````` rewrite f_S -!map_comp; apply (contractive_ne map); split=> Y. `````` Robbert Krebbers committed Feb 10, 2016 217 218 219 220 221 222 223 224 225 226 227 228 229 `````` + 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 map_id. } rewrite map_comp g_coerce; apply IH. } assert (H: S n + k = n + S k) by lia. rewrite (map_ff_gg _ _ _ H). `````` Robbert Krebbers committed Jan 14, 2016 230 `````` apply (_ : Proper (_ ==> _) (gg _)); by destruct H. `````` Robbert Krebbers committed Feb 10, 2016 231 232 `````` * intros X; rewrite equiv_dist=> n /=. rewrite /unfold /= (conv_compl (unfold_chain (fold X)) n) /=. `````` Robbert Krebbers committed Feb 12, 2016 233 234 `````` rewrite g_S -!map_comp -{2}(map_id _ _ X). apply (contractive_ne map); split => Y /=. `````` Robbert Krebbers committed Feb 10, 2016 235 236 237 `````` + apply dist_le with n; last omega. rewrite f_tower. apply dist_S. by rewrite embed_tower. + etransitivity; [apply embed_ne, equiv_dist, g_tower|apply embed_tower]. `````` Robbert Krebbers committed Nov 11, 2015 238 ``````Qed. `````` Robbert Krebbers committed Jan 14, 2016 239 ``End solver. End solver.``