natmap.v 15.3 KB
 Robbert Krebbers committed Nov 11, 2015 1 2 3 4 5 ``````(* Copyright (c) 2012-2015, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** This files implements a type [natmap A] of finite maps whose keys range over Coq's data type of unary natural numbers [nat]. The implementation equips a list with a proof of canonicity. *) `````` Robbert Krebbers committed Mar 10, 2016 6 ``````From iris.prelude Require Import fin_maps mapset. `````` Robbert Krebbers committed Nov 11, 2015 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 `````` Notation natmap_raw A := (list (option A)). Definition natmap_wf {A} (l : natmap_raw A) := match last l with None => True | Some x => is_Some x end. Instance natmap_wf_pi {A} (l : natmap_raw A) : ProofIrrel (natmap_wf l). Proof. unfold natmap_wf. case_match; apply _. Qed. Lemma natmap_wf_inv {A} (o : option A) (l : natmap_raw A) : natmap_wf (o :: l) → natmap_wf l. Proof. by destruct l. Qed. Lemma natmap_wf_lookup {A} (l : natmap_raw A) : natmap_wf l → l ≠ [] → ∃ i x, mjoin (l !! i) = Some x. Proof. intros Hwf Hl. induction l as [|[x|] l IH]; simpl; [done| |]. { exists 0. simpl. eauto. } `````` Robbert Krebbers committed Nov 11, 2015 22 `````` destruct IH as (i&x&?); eauto using natmap_wf_inv; [|by exists (S i), x]. `````` Robbert Krebbers committed Nov 11, 2015 23 24 25 26 27 28 29 30 31 32 33 34 35 36 `````` intros ->. by destruct Hwf. Qed. Record natmap (A : Type) : Type := NatMap { natmap_car : natmap_raw A; natmap_prf : natmap_wf natmap_car }. Arguments NatMap {_} _ _. Arguments natmap_car {_} _. Arguments natmap_prf {_} _. Lemma natmap_eq {A} (m1 m2 : natmap A) : m1 = m2 ↔ natmap_car m1 = natmap_car m2. Proof. split; [by intros ->|intros]; destruct m1 as [t1 ?], m2 as [t2 ?]. `````` Robbert Krebbers committed Feb 17, 2016 37 `````` simplify_eq/=; f_equal; apply proof_irrel. `````` Robbert Krebbers committed Nov 11, 2015 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 ``````Qed. Global Instance natmap_eq_dec `{∀ x y : A, Decision (x = y)} (m1 m2 : natmap A) : Decision (m1 = m2) := match decide (natmap_car m1 = natmap_car m2) with | left H => left (proj2 (natmap_eq m1 m2) H) | right H => right (H ∘ proj1 (natmap_eq m1 m2)) end. Instance natmap_empty {A} : Empty (natmap A) := NatMap [] I. Instance natmap_lookup {A} : Lookup nat A (natmap A) := λ i m, let (l,_) := m in mjoin (l !! i). Fixpoint natmap_singleton_raw {A} (i : nat) (x : A) : natmap_raw A := match i with 0 => [Some x]| S i => None :: natmap_singleton_raw i x end. Lemma natmap_singleton_wf {A} (i : nat) (x : A) : natmap_wf (natmap_singleton_raw i x). `````` Robbert Krebbers committed Feb 17, 2016 54 ``````Proof. unfold natmap_wf. induction i as [|[]]; simplify_eq/=; eauto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 ``````Lemma natmap_lookup_singleton_raw {A} (i : nat) (x : A) : mjoin (natmap_singleton_raw i x !! i) = Some x. Proof. induction i; simpl; auto. Qed. Lemma natmap_lookup_singleton_raw_ne {A} (i j : nat) (x : A) : i ≠ j → mjoin (natmap_singleton_raw i x !! j) = None. Proof. revert j; induction i; intros [|?]; simpl; auto with congruence. Qed. Hint Rewrite @natmap_lookup_singleton_raw : natmap. Definition natmap_cons_canon {A} (o : option A) (l : natmap_raw A) := match o, l with None, [] => [] | _, _ => o :: l end. Lemma natmap_cons_canon_wf {A} (o : option A) (l : natmap_raw A) : natmap_wf l → natmap_wf (natmap_cons_canon o l). Proof. unfold natmap_wf, last. destruct o, l; simpl; eauto. Qed. Lemma natmap_cons_canon_O {A} (o : option A) (l : natmap_raw A) : mjoin (natmap_cons_canon o l !! 0) = o. Proof. by destruct o, l. Qed. Lemma natmap_cons_canon_S {A} (o : option A) (l : natmap_raw A) i : natmap_cons_canon o l !! S i = l !! i. Proof. by destruct o, l. Qed. Hint Rewrite @natmap_cons_canon_O @natmap_cons_canon_S : natmap. Definition natmap_alter_raw {A} (f : option A → option A) : nat → natmap_raw A → natmap_raw A := fix go i l {struct l} := match l with | [] => match f None with | Some x => natmap_singleton_raw i x | None => [] end | o :: l => match i with | 0 => natmap_cons_canon (f o) l | S i => natmap_cons_canon o (go i l) end end. Lemma natmap_alter_wf {A} (f : option A → option A) i l : natmap_wf l → natmap_wf (natmap_alter_raw f i l). Proof. revert i. induction l; [intro | intros [|?]]; simpl; repeat case_match; eauto using natmap_singleton_wf, natmap_cons_canon_wf, natmap_wf_inv. Qed. Instance natmap_alter {A} : PartialAlter nat A (natmap A) := λ f i m, let (l,Hl) := m in NatMap _ (natmap_alter_wf f i l Hl). Lemma natmap_lookup_alter_raw {A} (f : option A → option A) i l : mjoin (natmap_alter_raw f i l !! i) = f (mjoin (l !! i)). Proof. revert i. induction l; intros [|?]; simpl; repeat case_match; simpl; autorewrite with natmap; auto. Qed. Lemma natmap_lookup_alter_raw_ne {A} (f : option A → option A) i j l : i ≠ j → mjoin (natmap_alter_raw f i l !! j) = mjoin (l !! j). Proof. revert i j. induction l; intros [|?] [|?] ?; simpl; repeat case_match; simpl; autorewrite with natmap; auto with congruence. rewrite natmap_lookup_singleton_raw_ne; congruence. Qed. Definition natmap_omap_raw {A B} (f : A → option B) : natmap_raw A → natmap_raw B := fix go l := match l with [] => [] | o :: l => natmap_cons_canon (o ≫= f) (go l) end. Lemma natmap_omap_raw_wf {A B} (f : A → option B) l : natmap_wf l → natmap_wf (natmap_omap_raw f l). Proof. induction l; simpl; eauto using natmap_cons_canon_wf, natmap_wf_inv. Qed. Lemma natmap_lookup_omap_raw {A B} (f : A → option B) l i : mjoin (natmap_omap_raw f l !! i) = mjoin (l !! i) ≫= f. Proof. revert i. induction l; intros [|?]; simpl; autorewrite with natmap; auto. Qed. Hint Rewrite @natmap_lookup_omap_raw : natmap. Global Instance natmap_omap: OMap natmap := λ A B f m, let (l,Hl) := m in NatMap _ (natmap_omap_raw_wf f _ Hl). Definition natmap_merge_raw {A B C} (f : option A → option B → option C) : natmap_raw A → natmap_raw B → natmap_raw C := fix go l1 l2 := match l1, l2 with | [], l2 => natmap_omap_raw (f None ∘ Some) l2 | l1, [] => natmap_omap_raw (flip f None ∘ Some) l1 | o1 :: l1, o2 :: l2 => natmap_cons_canon (f o1 o2) (go l1 l2) end. Lemma natmap_merge_wf {A B C} (f : option A → option B → option C) l1 l2 : natmap_wf l1 → natmap_wf l2 → natmap_wf (natmap_merge_raw f l1 l2). Proof. revert l2. induction l1; intros [|??]; simpl; eauto using natmap_omap_raw_wf, natmap_cons_canon_wf, natmap_wf_inv. Qed. Lemma natmap_lookup_merge_raw {A B C} (f : option A → option B → option C) l1 l2 i : f None None = None → mjoin (natmap_merge_raw f l1 l2 !! i) = f (mjoin (l1 !! i)) (mjoin (l2 !! i)). Proof. intros. revert i l2. induction l1; intros [|?] [|??]; simpl; autorewrite with natmap; auto; match goal with |- context [?o ≫= _] => by destruct o end. Qed. Instance natmap_merge: Merge natmap := λ A B C f m1 m2, let (l1, Hl1) := m1 in let (l2, Hl2) := m2 in NatMap (natmap_merge_raw f l1 l2) (natmap_merge_wf _ _ _ Hl1 Hl2). Fixpoint natmap_to_list_raw {A} (i : nat) (l : natmap_raw A) : list (nat * A) := match l with | [] => [] | None :: l => natmap_to_list_raw (S i) l | Some x :: l => (i,x) :: natmap_to_list_raw (S i) l end. Lemma natmap_elem_of_to_list_raw_aux {A} j (l : natmap_raw A) i x : (i,x) ∈ natmap_to_list_raw j l ↔ ∃ i', i = i' + j ∧ mjoin (l !! i') = Some x. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 163 `````` - revert j. induction l as [|[y|] l IH]; intros j; simpl. `````` Robbert Krebbers committed Nov 11, 2015 164 `````` + by rewrite elem_of_nil. `````` Robbert Krebbers committed Feb 17, 2016 165 `````` + rewrite elem_of_cons. intros [?|?]; simplify_eq. `````` Robbert Krebbers committed Feb 17, 2016 166 167 `````` * by exists 0. * destruct (IH (S j)) as (i'&?&?); auto. `````` Robbert Krebbers committed Nov 11, 2015 168 169 170 `````` exists (S i'); simpl; auto with lia. + intros. destruct (IH (S j)) as (i'&?&?); auto. exists (S i'); simpl; auto with lia. `````` Robbert Krebbers committed Feb 17, 2016 171 `````` - intros (i'&?&Hi'). subst. revert i' j Hi'. `````` Robbert Krebbers committed Nov 11, 2015 172 173 `````` induction l as [|[y|] l IH]; intros i j ?; simpl. + done. `````` Robbert Krebbers committed Feb 17, 2016 174 `````` + destruct i as [|i]; simplify_eq/=; [left|]. `````` Robbert Krebbers committed Nov 11, 2015 175 `````` right. rewrite <-Nat.add_succ_r. by apply (IH i (S j)). `````` Robbert Krebbers committed Feb 17, 2016 176 `````` + destruct i as [|i]; simplify_eq/=. `````` Robbert Krebbers committed Nov 11, 2015 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 `````` rewrite <-Nat.add_succ_r. by apply (IH i (S j)). Qed. Lemma natmap_elem_of_to_list_raw {A} (l : natmap_raw A) i x : (i,x) ∈ natmap_to_list_raw 0 l ↔ mjoin (l !! i) = Some x. Proof. rewrite natmap_elem_of_to_list_raw_aux. setoid_rewrite Nat.add_0_r. naive_solver. Qed. Lemma natmap_to_list_raw_nodup {A} i (l : natmap_raw A) : NoDup (natmap_to_list_raw i l). Proof. revert i. induction l as [|[?|] ? IH]; simpl; try constructor; auto. rewrite natmap_elem_of_to_list_raw_aux. intros (?&?&?). lia. Qed. Instance natmap_to_list {A} : FinMapToList nat A (natmap A) := λ m, let (l,_) := m in natmap_to_list_raw 0 l. Definition natmap_map_raw {A B} (f : A → B) : natmap_raw A → natmap_raw B := fmap (fmap f). Lemma natmap_map_wf {A B} (f : A → B) l : natmap_wf l → natmap_wf (natmap_map_raw f l). Proof. unfold natmap_map_raw, natmap_wf. rewrite fmap_last. destruct (last l). by apply fmap_is_Some. done. Qed. Lemma natmap_lookup_map_raw {A B} (f : A → B) i l : mjoin (natmap_map_raw f l !! i) = f <\$> mjoin (l !! i). Proof. unfold natmap_map_raw. rewrite list_lookup_fmap. by destruct (l !! i). Qed. Instance natmap_map: FMap natmap := λ A B f m, let (l,Hl) := m in NatMap (natmap_map_raw f l) (natmap_map_wf _ _ Hl). Instance: FinMap nat natmap. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 213 `````` - unfold lookup, natmap_lookup. intros A [l1 Hl1] [l2 Hl2] E. `````` Robbert Krebbers committed Nov 11, 2015 214 215 216 217 218 219 220 221 222 223 224 225 `````` apply natmap_eq. revert l2 Hl1 Hl2 E. simpl. induction l1 as [|[x|] l1 IH]; intros [|[y|] l2] Hl1 Hl2 E; simpl in *. + done. + by specialize (E 0). + destruct (natmap_wf_lookup (None :: l2)) as (i&?&?); auto with congruence. + by specialize (E 0). + f_equal. apply (E 0). apply IH; eauto using natmap_wf_inv. intros i. apply (E (S i)). + by specialize (E 0). + destruct (natmap_wf_lookup (None :: l1)) as (i&?&?); auto with congruence. + by specialize (E 0). + f_equal. apply IH; eauto using natmap_wf_inv. intros i. apply (E (S i)). `````` Robbert Krebbers committed Feb 17, 2016 226 227 228 229 230 231 232 233 `````` - done. - intros ?? [??] ?. apply natmap_lookup_alter_raw. - intros ?? [??] ??. apply natmap_lookup_alter_raw_ne. - intros ??? [??] ?. apply natmap_lookup_map_raw. - intros ? [??]. by apply natmap_to_list_raw_nodup. - intros ? [??] ??. by apply natmap_elem_of_to_list_raw. - intros ??? [??] ?. by apply natmap_lookup_omap_raw. - intros ????? [??] [??] ?. by apply natmap_lookup_merge_raw. `````` Robbert Krebbers committed Nov 11, 2015 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 ``````Qed. Fixpoint strip_Nones {A} (l : list (option A)) : list (option A) := match l with None :: l => strip_Nones l | _ => l end. Lemma list_to_natmap_wf {A} (l : list (option A)) : natmap_wf (reverse (strip_Nones (reverse l))). Proof. unfold natmap_wf. rewrite last_reverse. induction (reverse l) as [|[]]; simpl; eauto. Qed. Definition list_to_natmap {A} (l : list (option A)) : natmap A := NatMap (reverse (strip_Nones (reverse l))) (list_to_natmap_wf l). Lemma list_to_natmap_spec {A} (l : list (option A)) i : list_to_natmap l !! i = mjoin (l !! i). Proof. unfold lookup at 1, natmap_lookup, list_to_natmap; simpl. rewrite <-(reverse_involutive l) at 2. revert i. induction (reverse l) as [|[x|] l' IH]; intros i; simpl; auto. rewrite reverse_cons, IH. clear IH. revert i. induction (reverse l'); intros [|?]; simpl; auto. Qed. (** Finally, we can construct sets of [nat]s satisfying extensional equality. *) Notation natset := (mapset natmap). Instance natmap_dom {A} : Dom (natmap A) natset := mapset_dom. Instance: FinMapDom nat natmap natset := mapset_dom_spec. (* Fixpoint avoids this definition from being unfolded *) Fixpoint of_bools (βs : list bool) : natset := let f (β : bool) := if β then Some () else None in Mapset \$ list_to_natmap \$ f <\$> βs. Definition to_bools (sz : nat) (X : natset) : list bool := let f (mu : option ()) := match mu with Some _ => true | None => false end in resize sz false \$ f <\$> natmap_car (mapset_car X). Lemma of_bools_unfold βs : let f (β : bool) := if β then Some () else None in of_bools βs = Mapset \$ list_to_natmap \$ f <\$> βs. Proof. by destruct βs. Qed. Lemma elem_of_of_bools βs i : i ∈ of_bools βs ↔ βs !! i = Some true. Proof. rewrite of_bools_unfold; unfold elem_of, mapset_elem_of; simpl. rewrite list_to_natmap_spec, list_lookup_fmap. destruct (βs !! i) as [[]|]; compute; intuition congruence. Qed. Lemma of_bools_union βs1 βs2 : length βs1 = length βs2 → of_bools (βs1 ||* βs2) = of_bools βs1 ∪ of_bools βs2. Proof. rewrite <-Forall2_same_length; intros Hβs. apply elem_of_equiv_L. intros i. rewrite elem_of_union, !elem_of_of_bools. revert i. induction Hβs as [|[] []]; intros [|?]; naive_solver. Qed. Lemma to_bools_length (X : natset) sz : length (to_bools sz X) = sz. Proof. apply resize_length. Qed. Lemma lookup_to_bools_ge sz X i : sz ≤ i → to_bools sz X !! i = None. Proof. by apply lookup_resize_old. Qed. Lemma lookup_to_bools sz X i β : i < sz → to_bools sz X !! i = Some β ↔ (i ∈ X ↔ β = true). Proof. unfold to_bools, elem_of, mapset_elem_of, lookup at 2, natmap_lookup; simpl. intros. destruct (mapset_car X) as [l ?]; simpl. destruct (l !! i) as [mu|] eqn:Hmu; simpl. { rewrite lookup_resize, list_lookup_fmap, Hmu by (rewrite ?fmap_length; eauto using lookup_lt_Some). destruct mu as [[]|], β; simpl; intuition congruence. } rewrite lookup_resize_new by (rewrite ?fmap_length; eauto using lookup_ge_None_1); destruct β; intuition congruence. Qed. Lemma lookup_to_bools_true sz X i : i < sz → to_bools sz X !! i = Some true ↔ i ∈ X. Proof. intros. rewrite lookup_to_bools by done. intuition. Qed. Lemma lookup_to_bools_false sz X i : i < sz → to_bools sz X !! i = Some false ↔ i ∉ X. Proof. intros. rewrite lookup_to_bools by done. naive_solver. Qed. Lemma to_bools_union sz X1 X2 : to_bools sz (X1 ∪ X2) = to_bools sz X1 ||* to_bools sz X2. Proof. apply list_eq; intros i; rewrite lookup_zip_with. destruct (decide (i < sz)); [|by rewrite !lookup_to_bools_ge by lia]. apply option_eq; intros β. rewrite lookup_to_bools, elem_of_union by done; intros. destruct (decide (i ∈ X1)), (decide (i ∈ X2)); repeat first [ rewrite (λ X H, proj2 (lookup_to_bools_true sz X i H)) by done | rewrite (λ X H, proj2 (lookup_to_bools_false sz X i H)) by done]; destruct β; naive_solver. Qed. Lemma to_of_bools βs sz : to_bools sz (of_bools βs) = resize sz false βs. Proof. apply list_eq; intros i. destruct (decide (i < sz)); [|by rewrite lookup_to_bools_ge, lookup_resize_old by lia]. apply option_eq; intros β. rewrite lookup_to_bools, elem_of_of_bools by done. destruct (decide (i < length βs)). { rewrite lookup_resize by done. destruct (lookup_lt_is_Some_2 βs i) as [[]]; destruct β; naive_solver. } rewrite lookup_resize_new, lookup_ge_None_2 by lia. destruct β; naive_solver. Qed. (** A [natmap A] forms a stack with elements of type [A] and possible holes *) Definition natmap_push {A} (o : option A) (m : natmap A) : natmap A := let (l,Hl) := m in NatMap _ (natmap_cons_canon_wf o l Hl). Definition natmap_pop_raw {A} (l : natmap_raw A) : natmap_raw A := tail l. Lemma natmap_pop_wf {A} (l : natmap_raw A) : natmap_wf l → natmap_wf (natmap_pop_raw l). Proof. destruct l; simpl; eauto using natmap_wf_inv. Qed. Definition natmap_pop {A} (m : natmap A) : natmap A := let (l,Hl) := m in NatMap _ (natmap_pop_wf _ Hl). Lemma lookup_natmap_push_O {A} o (m : natmap A) : natmap_push o m !! 0 = o. Proof. by destruct o, m as [[|??]]. Qed. Lemma lookup_natmap_push_S {A} o (m : natmap A) i : natmap_push o m !! S i = m !! i. Proof. by destruct o, m as [[|??]]. Qed. Lemma lookup_natmap_pop {A} (m : natmap A) i : natmap_pop m !! i = m !! S i. Proof. by destruct m as [[|??]]. Qed. Lemma natmap_push_pop {A} (m : natmap A) : natmap_push (m !! 0) (natmap_pop m) = m. Proof. apply map_eq. intros i. destruct i. `````` Robbert Krebbers committed Feb 17, 2016 356 357 `````` - by rewrite lookup_natmap_push_O. - by rewrite lookup_natmap_push_S, lookup_natmap_pop. `````` Robbert Krebbers committed Nov 11, 2015 358 359 360 ``````Qed. Lemma natmap_pop_push {A} o (m : natmap A) : natmap_pop (natmap_push o m) = m. Proof. apply natmap_eq. by destruct o, m as [[|??]]. Qed.``````