natmap.v 10.5 KB
 Robbert Krebbers committed Mar 25, 2013 1 2 ``````(* Copyright (c) 2012-2013, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) `````` Robbert Krebbers committed May 07, 2013 3 4 5 6 ``````(** 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. *) Require Import fin_maps mapset. `````` Robbert Krebbers committed Mar 25, 2013 7 8 9 `````` Notation natmap_raw A := (list (option A)). Definition natmap_wf {A} (l : natmap_raw A) := `````` Robbert Krebbers committed May 07, 2013 10 `````` match last l with None => True | Some x => is_Some x end. `````` Robbert Krebbers committed Mar 25, 2013 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 ``````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. * destruct IH as (i&x&?); eauto using natmap_wf_inv. { intro. subst. inversion Hwf. } by exists (S i) x. Qed. Definition natmap (A : Type) : Type := sig (@natmap_wf A). Instance natmap_empty {A} : Empty (natmap A) := [] ↾ I. Instance natmap_lookup {A} : Lookup nat A (natmap A) := `````` Robbert Krebbers committed May 07, 2013 32 `````` λ i m, match m with exist l _ => mjoin (l !! i) end. `````` Robbert Krebbers committed Mar 25, 2013 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 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 `````` 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). Proof. unfold natmap_wf, last. induction i as [|i]; simpl; repeat case_match; simplify_equality; eauto. by destruct i. Qed. 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, `````` Robbert Krebbers committed May 07, 2013 92 `````` match m with exist l Hl => _↾natmap_alter_wf f i l Hl end. `````` Robbert Krebbers committed Mar 25, 2013 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 ``````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_merge_aux {A B} (f : option A → option B) : natmap_raw A → natmap_raw B := fix go l := match l with | [] => [] | o :: l => natmap_cons_canon (f o) (go l) end. Lemma natmap_merge_aux_wf {A B} (f : option A → option B) l : natmap_wf l → natmap_wf (natmap_merge_aux f l). Proof. induction l; simpl; eauto using natmap_cons_canon_wf, natmap_wf_inv. Qed. Lemma natmap_lookup_merge_aux {A B} (f : option A → option B) l i : f None = None → mjoin (natmap_merge_aux f l !! i) = f (mjoin (l !! i)). Proof. revert i. induction l; intros [|?]; simpl; autorewrite with natmap; auto. Qed. Hint Rewrite @natmap_lookup_merge_aux : natmap. 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_merge_aux (f None) l2 | l1, [] => natmap_merge_aux (flip f None) 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_merge_aux_wf, natmap_cons_canon_wf, natmap_wf_inv. Qed. `````` Robbert Krebbers committed May 07, 2013 139 140 ``````Lemma natmap_lookup_merge_raw {A B C} (f : option A → option B → option C) l1 l2 i : f None None = None → `````` Robbert Krebbers committed Mar 25, 2013 141 142 143 144 145 146 `````` 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. Qed. Instance natmap_merge: Merge natmap := λ A B C f m1 m2, `````` Robbert Krebbers committed May 07, 2013 147 148 149 150 `````` match m1, m2 with | exist l1 Hl1, exist l2 Hl2 => natmap_merge_raw f _ _ ↾ natmap_merge_wf _ _ _ Hl1 Hl2 end. `````` Robbert Krebbers committed Mar 25, 2013 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 `````` 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. * revert j. induction l as [|[y|] l IH]; intros j; simpl. + by rewrite elem_of_nil. + rewrite elem_of_cons. intros [?|?]; simplify_equality. - by exists 0. - destruct (IH (S j)) as (i'&?&?); auto. exists (S i'); simpl; auto with lia. + intros. destruct (IH (S j)) as (i'&?&?); auto. exists (S i'); simpl; auto with lia. * intros (i'&?&Hi'). subst. revert i' j Hi'. induction l as [|[y|] l IH]; intros i j ?; simpl. + done. + destruct i as [|i]; simplify_equality; [left|]. right. rewrite NPeano.Nat.add_succ_comm. by apply (IH i (S j)). + destruct i as [|i]; simplify_equality. rewrite NPeano.Nat.add_succ_comm. 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 plus_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, `````` Robbert Krebbers committed May 07, 2013 190 `````` match m with exist l _ => natmap_to_list_raw 0 l end. `````` Robbert Krebbers committed Mar 25, 2013 191 192 193 194 195 196 197 198 199 200 201 202 `````` 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_wf, last. induction l; simpl; repeat case_match; simplify_equality; eauto. simpl. by rewrite fmap_is_Some. Qed. Lemma natmap_lookup_map_raw {A B} (f : A → B) i l : mjoin (natmap_map_raw f l !! i) = f <\$> mjoin (l !! i). `````` Robbert Krebbers committed May 07, 2013 203 204 205 ``````Proof. unfold natmap_map_raw. rewrite list_lookup_fmap. by destruct (l !! i). Qed. `````` Robbert Krebbers committed Mar 25, 2013 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 ``````Instance natmap_map: FMap natmap := λ A B f m, natmap_map_raw f _ ↾ natmap_map_wf _ _ (proj2_sig m). Instance: FinMap nat natmap. Proof. split. * unfold lookup, natmap_lookup. intros A [l1 Hl1] [l2 Hl2] E. apply (sig_eq_pi _). 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)). * 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_merge_raw. Qed. `````` Robbert Krebbers committed May 07, 2013 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 `````` (** 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. (** 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 := match m with exist l Hl => _↾natmap_cons_canon_wf o l Hl end. 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 := match m with exist l Hl => _↾natmap_pop_wf _ Hl end. 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. * by rewrite lookup_natmap_push_O. * by rewrite lookup_natmap_push_S, lookup_natmap_pop. Qed. Lemma natmap_pop_push {A} o (m : natmap A) : natmap_pop (natmap_push o m) = m. Proof. apply (sig_eq_pi _). by destruct o, m as [[|??]]. Qed.``````