finite.v 12 KB
 Robbert Krebbers committed May 02, 2014 1 2 ``````(* Copyright (c) 2012-2014, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) `````` Robbert Krebbers committed Jun 17, 2013 3 4 5 6 7 ``````Require Export countable list. Obligation Tactic := idtac. Class Finite A `{∀ x y : A, Decision (x = y)} := { enum : list A; `````` Robbert Krebbers committed Jun 05, 2014 8 `````` NoDup_enum : NoDup enum; `````` Robbert Krebbers committed Jun 17, 2013 9 10 11 `````` elem_of_enum x : x ∈ enum }. Arguments enum _ {_ _} : clear implicits. `````` Robbert Krebbers committed Jun 05, 2014 12 ``````Arguments NoDup_enum _ {_ _} : clear implicits. `````` Robbert Krebbers committed Jun 17, 2013 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 ``````Definition card A `{Finite A} := length (enum A). Program Instance finite_countable `{Finite A} : Countable A := {| encode := λ x, Pos.of_nat \$ S \$ from_option 0 \$ list_find (x =) (enum A); decode := λ p, enum A !! pred (Pos.to_nat p) |}. Arguments Pos.of_nat _ : simpl never. Next Obligation. intros ?? [xs Hxs HA] x; unfold encode, decode; simpl. destruct (list_find_eq_elem_of xs x) as [i Hi]; auto. rewrite Nat2Pos.id by done; simpl. rewrite Hi; eauto using list_find_eq_Some. Qed. Definition find `{Finite A} P `{∀ x, Decision (P x)} : option A := list_find P (enum A) ≫= decode_nat. Lemma encode_lt_card `{finA: Finite A} x : encode_nat x < card A. Proof. destruct finA as [xs Hxs HA]; unfold encode_nat, encode, card; simpl. rewrite Nat2Pos.id by done; simpl. destruct (list_find _ xs) eqn:?; simpl. * eapply lookup_lt_Some, list_find_eq_Some; eauto. * destruct xs; simpl. exfalso; eapply not_elem_of_nil, (HA x). lia. Qed. Lemma encode_decode A `{finA: Finite A} i : i < card A → ∃ x, decode_nat i = Some x ∧ encode_nat x = i. Proof. destruct finA as [xs Hxs HA]. unfold encode_nat, decode_nat, encode, decode, card; simpl. intros Hi. apply lookup_lt_is_Some in Hi. destruct Hi as [x Hx]. exists x. rewrite !Nat2Pos.id by done; simpl. destruct (list_find_eq_elem_of xs x) as [j Hj]; auto. rewrite Hj. eauto using list_find_eq_Some, NoDup_lookup. Qed. Lemma find_Some `{finA: Finite A} P `{∀ x, Decision (P x)} x : find P = Some x → P x. Proof. destruct finA as [xs Hxs HA]; unfold find, decode_nat, decode; simpl. intros Hx. destruct (list_find _ _) as [i|] eqn:Hi; simplify_option_equality. rewrite !Nat2Pos.id in Hx by done. destruct (list_find_Some P xs i) as (?&?&?); simplify_option_equality; eauto. Qed. Lemma find_is_Some `{finA: Finite A} P `{∀ x, Decision (P x)} x : P x → ∃ y, find P = Some y ∧ P y. Proof. destruct finA as [xs Hxs HA]; unfold find, decode; simpl. intros Hx. destruct (list_find_elem_of P xs x) as [i Hi]; auto. rewrite Hi. destruct (list_find_Some P xs i) as (y&?&?); subst; auto. `````` Robbert Krebbers committed Jun 16, 2014 59 `````` exists y. csimpl. by rewrite !Nat2Pos.id by done. `````` Robbert Krebbers committed Jun 17, 2013 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 ``````Qed. Lemma card_0_inv P `{finA: Finite A} : card A = 0 → A → P. Proof. intros ? x. destruct finA as [[|??] ??]; simplify_equality. by destruct (not_elem_of_nil x). Qed. Lemma finite_inhabited A `{finA: Finite A} : 0 < card A → Inhabited A. Proof. unfold card. destruct finA as [[|x ?] ??]; simpl; auto with lia. constructor; exact x. Qed. Lemma finite_injective_contains `{finA: Finite A} `{finB: Finite B} (f: A → B) `{!Injective (=) (=) f} : f <\$> enum A `contains` enum B. Proof. `````` Robbert Krebbers committed Jun 05, 2014 75 `````` intros. destruct finA, finB. apply NoDup_contains; auto using NoDup_fmap_2. `````` Robbert Krebbers committed Jun 17, 2013 76 77 78 79 ``````Qed. Lemma finite_injective_Permutation `{Finite A} `{Finite B} (f : A → B) `{!Injective (=) (=) f} : card A = card B → f <\$> enum A ≡ₚ enum B. Proof. `````` Robbert Krebbers committed Aug 15, 2013 80 `````` intros. apply contains_Permutation_length_eq. `````` Robbert Krebbers committed Jun 17, 2013 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 `````` * by rewrite fmap_length. * by apply finite_injective_contains. Qed. Lemma finite_injective_surjective `{Finite A} `{Finite B} (f : A → B) `{!Injective (=) (=) f} : card A = card B → Surjective (=) f. Proof. intros HAB y. destruct (elem_of_list_fmap_2 f (enum A) y) as (x&?&?); eauto. rewrite finite_injective_Permutation; auto using elem_of_enum. Qed. Lemma finite_surjective A `{Finite A} B `{Finite B} : 0 < card A ≤ card B → ∃ g : B → A, Surjective (=) g. Proof. intros [??]. destruct (finite_inhabited A) as [x']; auto with lia. exists (λ y : B, from_option x' (decode_nat (encode_nat y))). intros x. destruct (encode_decode B (encode_nat x)) as (y&Hy1&Hy2). { pose proof (encode_lt_card x); lia. } exists y. by rewrite Hy2, decode_encode_nat. Qed. Lemma finite_injective A `{Finite A} B `{Finite B} : card A ≤ card B ↔ ∃ f : A → B, Injective (=) (=) f. Proof. split. * intros. destruct (decide (card A = 0)) as [HA|?]. { exists (card_0_inv B HA). intros y. apply (card_0_inv _ HA y). } destruct (finite_surjective A B) as (g&?); auto with lia. destruct (surjective_cancel g) as (f&?). exists f. apply cancel_injective. * intros [f ?]. unfold card. rewrite <-(fmap_length f). by apply contains_length, (finite_injective_contains f). Qed. Lemma finite_bijective A `{Finite A} B `{Finite B} : card A = card B ↔ ∃ f : A → B, Injective (=) (=) f ∧ Surjective (=) f. Proof. split. * intros; destruct (proj1 (finite_injective A B)) as [f ?]; auto with lia. exists f; auto using (finite_injective_surjective f). * intros (f&?&?). apply (anti_symmetric (≤)); apply finite_injective. + by exists f. + destruct (surjective_cancel f) as (g&?); eauto using cancel_injective. Qed. Lemma injective_card `{Finite A} `{Finite B} (f : A → B) `{!Injective (=) (=) f} : card A ≤ card B. Proof. apply finite_injective. eauto. Qed. Lemma surjective_card `{Finite A} `{Finite B} (f : A → B) `{!Surjective (=) f} : card B ≤ card A. Proof. destruct (surjective_cancel f) as (g&?). apply injective_card with g, cancel_injective. Qed. Lemma bijective_card `{Finite A} `{Finite B} (f : A → B) `{!Injective (=) (=) f} `{!Surjective (=) f} : card A = card B. Proof. apply finite_bijective. eauto. Qed. `````` Robbert Krebbers committed Aug 27, 2013 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 ``````(** Decidability of quantification over finite types *) Section forall_exists. Context `{Finite A} (P : A → Prop) `{∀ x, Decision (P x)}. Lemma Forall_finite : Forall P (enum A) ↔ (∀ x, P x). Proof. rewrite Forall_forall. intuition auto using elem_of_enum. Qed. Lemma Exists_finite : Exists P (enum A) ↔ (∃ x, P x). Proof. rewrite Exists_exists. naive_solver eauto using elem_of_enum. Qed. Global Instance forall_dec: Decision (∀ x, P x). Proof. refine (cast_if (decide (Forall P (enum A)))); abstract by rewrite <-Forall_finite. Defined. Global Instance exists_dec: Decision (∃ x, P x). Proof. refine (cast_if (decide (Exists P (enum A)))); abstract by rewrite <-Exists_finite. Defined. End forall_exists. `````` Robbert Krebbers committed Jun 17, 2013 155 156 157 158 159 160 161 162 163 164 165 166 ``````(** Instances *) Section enc_finite. Context `{∀ x y : A, Decision (x = y)}. Context (to_nat : A → nat) (of_nat : nat → A) (c : nat). Context (of_to_nat : ∀ x, of_nat (to_nat x) = x). Context (to_nat_c : ∀ x, to_nat x < c). Context (to_of_nat : ∀ i, i < c → to_nat (of_nat i) = i). Program Instance enc_finite : Finite A := {| enum := of_nat <\$> seq 0 c |}. Next Obligation. apply NoDup_alt. intros i j x. rewrite !list_lookup_fmap. intros Hi Hj. destruct (seq _ _ !! i) as [i'|] eqn:Hi', `````` Robbert Krebbers committed Sep 30, 2014 167 `````` (seq _ _ !! j) as [j'|] eqn:Hj'; simplify_equality'. `````` Robbert Krebbers committed Jun 17, 2013 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 `````` destruct (lookup_seq_inv _ _ _ _ Hi'), (lookup_seq_inv _ _ _ _ Hj'); subst. rewrite <-(to_of_nat i), <-(to_of_nat j) by done. by f_equal. Qed. Next Obligation. intros x. rewrite elem_of_list_fmap. exists (to_nat x). split; auto. by apply elem_of_list_lookup_2 with (to_nat x), lookup_seq. Qed. Lemma enc_finite_card : card A = c. Proof. unfold card. simpl. by rewrite fmap_length, seq_length. Qed. End enc_finite. Section bijective_finite. Context `{Finite A} `{∀ x y : B, Decision (x = y)} (f : A → B) (g : B → A). Context `{!Injective (=) (=) f} `{!Cancel (=) f g}. Program Instance bijective_finite: Finite B := {| enum := f <\$> enum A |}. `````` Robbert Krebbers committed Jun 05, 2014 184 `````` Next Obligation. apply (NoDup_fmap_2 _), NoDup_enum. Qed. `````` Robbert Krebbers committed Jun 17, 2013 185 186 187 188 189 190 191 192 193 194 `````` Next Obligation. intros y. rewrite elem_of_list_fmap. eauto using elem_of_enum. Qed. End bijective_finite. Program Instance option_finite `{Finite A} : Finite (option A) := {| enum := None :: Some <\$> enum A |}. Next Obligation. constructor. * rewrite elem_of_list_fmap. by intros (?&?&?). `````` Robbert Krebbers committed Jun 05, 2014 195 `````` * apply (NoDup_fmap_2 _); auto using NoDup_enum. `````` Robbert Krebbers committed Jun 17, 2013 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 ``````Qed. Next Obligation. intros ??? [x|]; [right|left]; auto. apply elem_of_list_fmap. eauto using elem_of_enum. Qed. Lemma option_cardinality `{Finite A} : card (option A) = S (card A). Proof. unfold card. simpl. by rewrite fmap_length. Qed. Program Instance unit_finite : Finite () := {| enum := [tt] |}. Next Obligation. apply NoDup_singleton. Qed. Next Obligation. intros []. by apply elem_of_list_singleton. Qed. Lemma unit_card : card unit = 1. Proof. done. Qed. Program Instance bool_finite : Finite bool := {| enum := [true; false] |}. Next Obligation. constructor. by rewrite elem_of_list_singleton. apply NoDup_singleton. Qed. Next Obligation. intros [|]. left. right; left. Qed. Lemma bool_card : card bool = 2. Proof. done. Qed. Program Instance sum_finite `{Finite A} `{Finite B} : Finite (A + B)%type := {| enum := (inl <\$> enum A) ++ (inr <\$> enum B) |}. Next Obligation. intros. apply NoDup_app; split_ands. `````` Robbert Krebbers committed Jun 05, 2014 222 `````` * apply (NoDup_fmap_2 _). by apply NoDup_enum. `````` Robbert Krebbers committed Jun 17, 2013 223 `````` * intro. rewrite !elem_of_list_fmap. intros (?&?&?) (?&?&?); congruence. `````` Robbert Krebbers committed Jun 05, 2014 224 `````` * apply (NoDup_fmap_2 _). by apply NoDup_enum. `````` Robbert Krebbers committed Jun 17, 2013 225 226 227 228 229 230 231 232 233 234 235 ``````Qed. Next Obligation. intros ?????? [x|y]; rewrite elem_of_app, !elem_of_list_fmap; eauto using @elem_of_enum. Qed. Lemma sum_card `{Finite A} `{Finite B} : card (A + B) = card A + card B. Proof. unfold card. simpl. by rewrite app_length, !fmap_length. Qed. Program Instance prod_finite `{Finite A} `{Finite B} : Finite (A * B)%type := {| enum := foldr (λ x, (pair x <\$> enum B ++)) [] (enum A) |}. Next Obligation. `````` Robbert Krebbers committed Jun 05, 2014 236 `````` intros ??????. induction (NoDup_enum A) as [|x xs Hx Hxs IH]; simpl. `````` Robbert Krebbers committed Jun 17, 2013 237 238 `````` { constructor. } apply NoDup_app; split_ands. `````` Robbert Krebbers committed Jun 05, 2014 239 `````` * by apply (NoDup_fmap_2 _), NoDup_enum. `````` Robbert Krebbers committed Jun 17, 2013 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 `````` * intros [? y]. rewrite elem_of_list_fmap. intros (?&?&?); simplify_equality. clear IH. induction Hxs as [|x' xs ?? IH]; simpl. { rewrite elem_of_nil. tauto. } rewrite elem_of_app, elem_of_list_fmap. intros [(?&?&?)|?]; simplify_equality. + destruct Hx. by left. + destruct IH. by intro; destruct Hx; right. auto. * done. Qed. Next Obligation. intros ?????? [x y]. induction (elem_of_enum x); simpl. * rewrite elem_of_app, !elem_of_list_fmap. eauto using @elem_of_enum. * rewrite elem_of_app; eauto. Qed. Lemma prod_card `{Finite A} `{Finite B} : card (A * B) = card A * card B. Proof. unfold card; simpl. induction (enum A); simpl; auto. rewrite app_length, fmap_length. auto. Qed. Let list_enum {A} (l : list A) : ∀ n, list { l : list A | length l = n } := fix go n := match n with | 0 => [[]↾eq_refl] | S n => foldr (λ x, (sig_map (x ::) (λ _ H, f_equal S H) <\$> (go n) ++)) [] l end. Program Instance list_finite `{Finite A} n : Finite { l | length l = n } := {| enum := list_enum (enum A) n |}. Next Obligation. intros ????. induction n as [|n IH]; simpl; [apply NoDup_singleton |]. revert IH. generalize (list_enum (enum A) n). intros l Hl. `````` Robbert Krebbers committed Jun 05, 2014 271 `````` induction (NoDup_enum A) as [|x xs Hx Hxs IH]; simpl; auto; [constructor |]. `````` Robbert Krebbers committed Jun 17, 2013 272 `````` apply NoDup_app; split_ands. `````` Robbert Krebbers committed Jun 05, 2014 273 `````` * by apply (NoDup_fmap_2 _). `````` Robbert Krebbers committed Jun 17, 2013 274 `````` * intros [k1 Hk1]. clear Hxs IH. rewrite elem_of_list_fmap. `````` Robbert Krebbers committed Sep 30, 2014 275 `````` intros ([k2 Hk2]&?&?) Hxk2; simplify_equality'. destruct Hx. revert Hxk2. `````` Robbert Krebbers committed Jun 17, 2013 276 277 `````` induction xs as [|x' xs IH]; simpl in *; [by rewrite elem_of_nil |]. rewrite elem_of_app, elem_of_list_fmap, elem_of_cons. `````` Robbert Krebbers committed Sep 30, 2014 278 `````` intros [([??]&?&?)|?]; simplify_equality'; auto. `````` Robbert Krebbers committed Jun 17, 2013 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 `````` * apply IH. Qed. Next Obligation. intros ???? [l Hl]. revert l Hl. induction n as [|n IH]; intros [|x l] ?; simpl; simplify_equality. { apply elem_of_list_singleton. by apply (sig_eq_pi _). } revert IH. generalize (list_enum (enum A) n). intros k Hk. induction (elem_of_enum x) as [x xs|x xs]; simpl in *. * rewrite elem_of_app, elem_of_list_fmap. left. injection Hl. intros Hl'. eexists (l↾Hl'). split. by apply (sig_eq_pi _). done. * rewrite elem_of_app. eauto. Qed. Lemma list_card `{Finite A} n : card { l | length l = n } = card A ^ n. Proof. unfold card; simpl. induction n as [|n IH]; simpl; auto. rewrite <-IH. clear IH. generalize (list_enum (enum A) n). induction (enum A) as [|x xs IH]; intros l; simpl; auto. by rewrite app_length, fmap_length, IH. Qed.``````