list.v 153 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1 2 3 4
(* Copyright (c) 2012-2015, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects general purpose definitions and theorems on lists that
are not in the Coq standard library. *)
5 6
From Coq Require Export Permutation.
From prelude Require Export numbers base decidable option.
Robbert Krebbers's avatar
Robbert Krebbers committed
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Arguments length {_} _.
Arguments cons {_} _ _.
Arguments app {_} _ _.
Arguments Permutation {_} _ _.
Arguments Forall_cons {_} _ _ _ _ _.

Notation tail := tl.
Notation take := firstn.
Notation drop := skipn.

Arguments take {_} !_ !_ /.
Arguments drop {_} !_ !_ /.

Notation "(::)" := cons (only parsing) : C_scope.
Notation "( x ::)" := (cons x) (only parsing) : C_scope.
Notation "(:: l )" := (λ x, cons x l) (only parsing) : C_scope.
Notation "(++)" := app (only parsing) : C_scope.
Notation "( l ++)" := (app l) (only parsing) : C_scope.
Notation "(++ k )" := (λ l, app l k) (only parsing) : C_scope.

Infix "≡ₚ" := Permutation (at level 70, no associativity) : C_scope.
Notation "(≡ₚ)" := Permutation (only parsing) : C_scope.
Notation "( x ≡ₚ)" := (Permutation x) (only parsing) : C_scope.
Notation "(≡ₚ x )" := (λ y, y ≡ₚ x) (only parsing) : C_scope.
Notation "(≢ₚ)" := (λ x y, ¬x ≡ₚ y) (only parsing) : C_scope.
Notation "x ≢ₚ y":= (¬x ≡ₚ y) (at level 70, no associativity) : C_scope.
Notation "( x ≢ₚ)" := (λ y, x ≢ₚ y) (only parsing) : C_scope.
Notation "(≢ₚ x )" := (λ y, y ≢ₚ x) (only parsing) : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
37 38 39
Instance maybe_cons {A} : Maybe2 (@cons A) := λ l,
  match l with x :: l => Some (x,l) | _ => None end.

Robbert Krebbers's avatar
Robbert Krebbers committed
40
(** * Definitions *)
41 42 43 44 45 46
(** Setoid equality lifted to lists *)
Inductive list_equiv `{Equiv A} : Equiv (list A) :=
  | nil_equiv : [] ≡ []
  | cons_equiv x y l k : x ≡ y → l ≡ k → x :: l ≡ y :: k.
Existing Instance list_equiv.

Robbert Krebbers's avatar
Robbert Krebbers committed
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 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222
(** The operation [l !! i] gives the [i]th element of the list [l], or [None]
in case [i] is out of bounds. *)
Instance list_lookup {A} : Lookup nat A (list A) :=
  fix go i l {struct l} : option A := let _ : Lookup _ _ _ := @go in
  match l with
  | [] => None | x :: l => match i with 0 => Some x | S i => l !! i end
  end.

(** The operation [alter f i l] applies the function [f] to the [i]th element
of [l]. In case [i] is out of bounds, the list is returned unchanged. *)
Instance list_alter {A} : Alter nat A (list A) := λ f,
  fix go i l {struct l} :=
  match l with
  | [] => []
  | x :: l => match i with 0 => f x :: l | S i => x :: go i l end
  end.

(** The operation [<[i:=x]> l] overwrites the element at position [i] with the
value [x]. In case [i] is out of bounds, the list is returned unchanged. *)
Instance list_insert {A} : Insert nat A (list A) :=
  fix go i y l {struct l} := let _ : Insert _ _ _ := @go in
  match l with
  | [] => []
  | x :: l => match i with 0 => y :: l | S i => x :: <[i:=y]>l end
  end.
Fixpoint list_inserts {A} (i : nat) (k l : list A) : list A :=
  match k with
  | [] => l
  | y :: k => <[i:=y]>(list_inserts (S i) k l)
  end.

(** The operation [delete i l] removes the [i]th element of [l] and moves
all consecutive elements one position ahead. In case [i] is out of bounds,
the list is returned unchanged. *)
Instance list_delete {A} : Delete nat (list A) :=
  fix go (i : nat) (l : list A) {struct l} : list A :=
  match l with
  | [] => []
  | x :: l => match i with 0 => l | S i => x :: @delete _ _ go i l end
  end.

(** The function [option_list o] converts an element [Some x] into the
singleton list [[x]], and [None] into the empty list [[]]. *)
Definition option_list {A} : option A → list A := option_rect _ (λ x, [x]) [].
Definition list_singleton {A} (l : list A) : option A :=
  match l with [x] => Some x | _ => None end.

(** The function [filter P l] returns the list of elements of [l] that
satisfies [P]. The order remains unchanged. *)
Instance list_filter {A} : Filter A (list A) :=
  fix go P _ l := let _ : Filter _ _ := @go in
  match l with
  | [] => []
  | x :: l => if decide (P x) then x :: filter P l else filter P l
  end.

(** The function [list_find P l] returns the first index [i] whose element
satisfies the predicate [P]. *)
Definition list_find {A} P `{∀ x, Decision (P x)} : list A → option (nat * A) :=
  fix go l :=
  match l with
  | [] => None
  | x :: l => if decide (P x) then Some (0,x) else prod_map S id <$> go l
  end.

(** The function [replicate n x] generates a list with length [n] of elements
with value [x]. *)
Fixpoint replicate {A} (n : nat) (x : A) : list A :=
  match n with 0 => [] | S n => x :: replicate n x end.

(** The function [reverse l] returns the elements of [l] in reverse order. *)
Definition reverse {A} (l : list A) : list A := rev_append l [].

(** The function [last l] returns the last element of the list [l], or [None]
if the list [l] is empty. *)
Fixpoint last {A} (l : list A) : option A :=
  match l with [] => None | [x] => Some x | _ :: l => last l end.

(** The function [resize n y l] takes the first [n] elements of [l] in case
[length l ≤ n], and otherwise appends elements with value [x] to [l] to obtain
a list of length [n]. *)
Fixpoint resize {A} (n : nat) (y : A) (l : list A) : list A :=
  match l with
  | [] => replicate n y
  | x :: l => match n with 0 => [] | S n => x :: resize n y l end
  end.
Arguments resize {_} !_ _ !_.

(** The function [reshape k l] transforms [l] into a list of lists whose sizes
are specified by [k]. In case [l] is too short, the resulting list will be
padded with empty lists. In case [l] is too long, it will be truncated. *)
Fixpoint reshape {A} (szs : list nat) (l : list A) : list (list A) :=
  match szs with
  | [] => [] | sz :: szs => take sz l :: reshape szs (drop sz l)
  end.

Definition sublist_lookup {A} (i n : nat) (l : list A) : option (list A) :=
  guard (i + n ≤ length l); Some (take n (drop i l)).
Definition sublist_alter {A} (f : list A → list A)
    (i n : nat) (l : list A) : list A :=
  take i l ++ f (take n (drop i l)) ++ drop (i + n) l.

(** Functions to fold over a list. We redefine [foldl] with the arguments in
the same order as in Haskell. *)
Notation foldr := fold_right.
Definition foldl {A B} (f : A → B → A) : A → list B → A :=
  fix go a l := match l with [] => a | x :: l => go (f a x) l end.

(** The monadic operations. *)
Instance list_ret: MRet list := λ A x, x :: @nil A.
Instance list_fmap : FMap list := λ A B f,
  fix go (l : list A) := match l with [] => [] | x :: l => f x :: go l end.
Instance list_omap : OMap list := λ A B f,
  fix go (l : list A) :=
  match l with
  | [] => []
  | x :: l => match f x with Some y => y :: go l | None => go l end
  end.
Instance list_bind : MBind list := λ A B f,
  fix go (l : list A) := match l with [] => [] | x :: l => f x ++ go l end.
Instance list_join: MJoin list :=
  fix go A (ls : list (list A)) : list A :=
  match ls with [] => [] | l :: ls => l ++ @mjoin _ go _ ls end.
Definition mapM `{MBind M, MRet M} {A B} (f : A → M B) : list A → M (list B) :=
  fix go l :=
  match l with [] => mret [] | x :: l => y ← f x; k ← go l; mret (y :: k) end.

(** We define stronger variants of map and fold that allow the mapped
function to use the index of the elements. *)
Definition imap_go {A B} (f : nat → A → B) : nat → list A → list B :=
  fix go (n : nat) (l : list A) :=
  match l with [] => [] | x :: l => f n x :: go (S n) l end.
Definition imap {A B} (f : nat → A → B) : list A → list B := imap_go f 0.
Definition zipped_map {A B} (f : list A → list A → A → B) :
  list A → list A → list B := fix go l k :=
  match k with [] => [] | x :: k => f l k x :: go (x :: l) k end.

Definition imap2_go {A B C} (f : nat → A → B → C) :
    nat → list A → list B → list C:=
  fix go (n : nat) (l : list A) (k : list B) :=
  match l, k with
  | [], _ |_, [] => [] | x :: l, y :: k => f n x y :: go (S n) l k
  end.
Definition imap2 {A B C} (f : nat → A → B → C) :
  list A → list B → list C := imap2_go f 0.

Inductive zipped_Forall {A} (P : list A → list A → A → Prop) :
    list A → list A → Prop :=
  | zipped_Forall_nil l : zipped_Forall P l []
  | zipped_Forall_cons l k x :
     P l k x → zipped_Forall P (x :: l) k → zipped_Forall P l (x :: k).
Arguments zipped_Forall_nil {_ _} _.
Arguments zipped_Forall_cons {_ _} _ _ _ _ _.

(** The function [mask f βs l] applies the function [f] to elements in [l] at
positions that are [true] in [βs]. *)
Fixpoint mask {A} (f : A → A) (βs : list bool) (l : list A) : list A :=
  match βs, l with
  | β :: βs, x :: l => (if β then f x else x) :: mask f βs l
  | _, _ => l
  end.

(** The function [permutations l] yields all permutations of [l]. *)
Fixpoint interleave {A} (x : A) (l : list A) : list (list A) :=
  match l with
  | [] => [[x]]| y :: l => (x :: y :: l) :: ((y ::) <$> interleave x l)
  end.
Fixpoint permutations {A} (l : list A) : list (list A) :=
  match l with [] => [[]] | x :: l => permutations l ≫= interleave x end.

(** The predicate [suffix_of] holds if the first list is a suffix of the second.
The predicate [prefix_of] holds if the first list is a prefix of the second. *)
Definition suffix_of {A} : relation (list A) := λ l1 l2, ∃ k, l2 = k ++ l1.
Definition prefix_of {A} : relation (list A) := λ l1 l2, ∃ k, l2 = l1 ++ k.
Infix "`suffix_of`" := suffix_of (at level 70) : C_scope.
Infix "`prefix_of`" := prefix_of (at level 70) : C_scope.
223 224
Hint Extern 0 (_ `prefix_of` _) => reflexivity.
Hint Extern 0 (_ `suffix_of` _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251

Section prefix_suffix_ops.
  Context `{∀ x y : A, Decision (x = y)}.
  Definition max_prefix_of : list A → list A → list A * list A * list A :=
    fix go l1 l2 :=
    match l1, l2 with
    | [], l2 => ([], l2, [])
    | l1, [] => (l1, [], [])
    | x1 :: l1, x2 :: l2 =>
      if decide_rel (=) x1 x2
      then prod_map id (x1 ::) (go l1 l2) else (x1 :: l1, x2 :: l2, [])
    end.
  Definition max_suffix_of (l1 l2 : list A) : list A * list A * list A :=
    match max_prefix_of (reverse l1) (reverse l2) with
    | (k1, k2, k3) => (reverse k1, reverse k2, reverse k3)
    end.
  Definition strip_prefix (l1 l2 : list A) := (max_prefix_of l1 l2).1.2.
  Definition strip_suffix (l1 l2 : list A) := (max_suffix_of l1 l2).1.2.
End prefix_suffix_ops.

(** A list [l1] is a sublist of [l2] if [l2] is obtained by removing elements
from [l1] without changing the order. *)
Inductive sublist {A} : relation (list A) :=
  | sublist_nil : sublist [] []
  | sublist_skip x l1 l2 : sublist l1 l2 → sublist (x :: l1) (x :: l2)
  | sublist_cons x l1 l2 : sublist l1 l2 → sublist l1 (x :: l2).
Infix "`sublist`" := sublist (at level 70) : C_scope.
252
Hint Extern 0 (_ `sublist` _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
253 254 255 256 257 258 259 260 261 262

(** A list [l2] contains a list [l1] if [l2] is obtained by removing elements
from [l1] while possiblity changing the order. *)
Inductive contains {A} : relation (list A) :=
  | contains_nil : contains [] []
  | contains_skip x l1 l2 : contains l1 l2 → contains (x :: l1) (x :: l2)
  | contains_swap x y l : contains (y :: x :: l) (x :: y :: l)
  | contains_cons x l1 l2 : contains l1 l2 → contains l1 (x :: l2)
  | contains_trans l1 l2 l3 : contains l1 l2 → contains l2 l3 → contains l1 l3.
Infix "`contains`" := contains (at level 70) : C_scope.
263
Hint Extern 0 (_ `contains` _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
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

Section contains_dec_help.
  Context {A} {dec : ∀ x y : A, Decision (x = y)}.
  Fixpoint list_remove (x : A) (l : list A) : option (list A) :=
    match l with
    | [] => None
    | y :: l => if decide (x = y) then Some l else (y ::) <$> list_remove x l
    end.
  Fixpoint list_remove_list (k : list A) (l : list A) : option (list A) :=
    match k with
    | [] => Some l | x :: k => list_remove x l ≫= list_remove_list k
    end.
End contains_dec_help.

Inductive Forall3 {A B C} (P : A → B → C → Prop) :
     list A → list B → list C → Prop :=
  | Forall3_nil : Forall3 P [] [] []
  | Forall3_cons x y z l k k' :
     P x y z → Forall3 P l k k' → Forall3 P (x :: l) (y :: k) (z :: k').

(** Set operations on lists *)
Section list_set.
  Context {A} {dec : ∀ x y : A, Decision (x = y)}.
  Global Instance elem_of_list_dec {dec : ∀ x y : A, Decision (x = y)}
    (x : A) : ∀ l, Decision (x ∈ l).
  Proof.
   refine (
    fix go l :=
    match l return Decision (x ∈ l) with
    | [] => right _
    | y :: l => cast_if_or (decide (x = y)) (go l)
    end); clear go dec; subst; try (by constructor); abstract by inversion 1.
  Defined.
  Fixpoint remove_dups (l : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel (∈) x l then remove_dups l else x :: remove_dups l
    end.
  Fixpoint list_difference (l k : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel (∈) x k
      then list_difference l k else x :: list_difference l k
    end.
  Definition list_union (l k : list A) : list A := list_difference l k ++ k.
  Fixpoint list_intersection (l k : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel (∈) x k
      then x :: list_intersection l k else list_intersection l k
    end.
  Definition list_intersection_with (f : A → A → option A) :
    list A → list A → list A := fix go l k :=
    match l with
    | [] => []
    | x :: l => foldr (λ y,
        match f x y with None => id | Some z => (z ::) end) (go l k) k
    end.
End list_set.

(** * Basic tactics on lists *)
328
(** The tactic [discriminate_list] discharges a goal if it contains
Robbert Krebbers's avatar
Robbert Krebbers committed
329 330
a list equality involving [(::)] and [(++)] of two lists that have a different
length as one of its hypotheses. *)
331
Tactic Notation "discriminate_list" hyp(H) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
332 333
  apply (f_equal length) in H;
  repeat (csimpl in H || rewrite app_length in H); exfalso; lia.
334 335
Tactic Notation "discriminate_list" :=
  match goal with H : @eq (list _) _ _ |- _ => discriminate_list H end.
Robbert Krebbers's avatar
Robbert Krebbers committed
336

337
(** The tactic [simplify_list_eq] simplifies hypotheses involving
Robbert Krebbers's avatar
Robbert Krebbers committed
338 339
equalities on lists using injectivity of [(::)] and [(++)]. Also, it simplifies
lookups in singleton lists. *)
340
Lemma app_inj_1 {A} (l1 k1 l2 k2 : list A) :
Robbert Krebbers's avatar
Robbert Krebbers committed
341 342
  length l1 = length k1 → l1 ++ l2 = k1 ++ k2 → l1 = k1 ∧ l2 = k2.
Proof. revert k1. induction l1; intros [|??]; naive_solver. Qed.
343
Lemma app_inj_2 {A} (l1 k1 l2 k2 : list A) :
Robbert Krebbers's avatar
Robbert Krebbers committed
344 345
  length l2 = length k2 → l1 ++ l2 = k1 ++ k2 → l1 = k1 ∧ l2 = k2.
Proof.
346
  intros ? Hl. apply app_inj_1; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
347 348
  apply (f_equal length) in Hl. rewrite !app_length in Hl. lia.
Qed.
349
Ltac simplify_list_eq :=
Robbert Krebbers's avatar
Robbert Krebbers committed
350
  repeat match goal with
351
  | _ => progress simplify_eq/=
Robbert Krebbers's avatar
Robbert Krebbers committed
352 353
  | H : _ ++ _ = _ ++ _ |- _ => first
    [ apply app_inv_head in H | apply app_inv_tail in H
354 355
    | apply app_inj_1 in H; [destruct H|done]
    | apply app_inj_2 in H; [destruct H|done] ]
Robbert Krebbers's avatar
Robbert Krebbers committed
356 357 358 359 360 361 362 363 364 365
  | H : [?x] !! ?i = Some ?y |- _ =>
    destruct i; [change (Some x = Some y) in H | discriminate]
  end.

(** * General theorems *)
Section general_properties.
Context {A : Type}.
Implicit Types x y z : A.
Implicit Types l k : list A.

366 367 368 369 370
Section setoid.
  Context `{Equiv A} `{!Equivalence ((≡) : relation A)}.
  Global Instance map_equivalence : Equivalence ((≡) : relation (list A)).
  Proof.
    split.
371 372 373
    - intros l; induction l; constructor; auto.
    - induction 1; constructor; auto.
    - intros l1 l2 l3 Hl; revert l3.
374
      induction Hl; inversion_clear 1; constructor; try etrans; eauto.
375 376 377 378 379 380 381 382 383
  Qed.
  Global Instance cons_proper : Proper ((≡) ==> (≡) ==> (≡)) (@cons A).
  Proof. by constructor. Qed.
  Global Instance app_proper : Proper ((≡) ==> (≡) ==> (≡)) (@app A).
  Proof.
    induction 1 as [|x y l k ?? IH]; intros ?? Htl; simpl; auto.
    by apply cons_equiv, IH.
  Qed.
  Global Instance list_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (list A).
384
  Proof. induction 1; f_equal; fold_leibniz; auto. Qed.
385 386
End setoid.

387
Global Instance: Inj2 (=) (=) (=) (@cons A).
Robbert Krebbers's avatar
Robbert Krebbers committed
388
Proof. by injection 1. Qed.
389
Global Instance: ∀ k, Inj (=) (=) (k ++).
Robbert Krebbers's avatar
Robbert Krebbers committed
390
Proof. intros ???. apply app_inv_head. Qed.
391
Global Instance: ∀ k, Inj (=) (=) (++ k).
Robbert Krebbers's avatar
Robbert Krebbers committed
392
Proof. intros ???. apply app_inv_tail. Qed.
393
Global Instance: Assoc (=) (@app A).
Robbert Krebbers's avatar
Robbert Krebbers committed
394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409
Proof. intros ???. apply app_assoc. Qed.
Global Instance: LeftId (=) [] (@app A).
Proof. done. Qed.
Global Instance: RightId (=) [] (@app A).
Proof. intro. apply app_nil_r. Qed.

Lemma app_nil l1 l2 : l1 ++ l2 = [] ↔ l1 = [] ∧ l2 = [].
Proof. split. apply app_eq_nil. by intros [-> ->]. Qed.
Lemma app_singleton l1 l2 x :
  l1 ++ l2 = [x] ↔ l1 = [] ∧ l2 = [x] ∨ l1 = [x] ∧ l2 = [].
Proof. split. apply app_eq_unit. by intros [[-> ->]|[-> ->]]. Qed.
Lemma cons_middle x l1 l2 : l1 ++ x :: l2 = l1 ++ [x] ++ l2.
Proof. done. Qed.
Lemma list_eq l1 l2 : (∀ i, l1 !! i = l2 !! i) → l1 = l2.
Proof.
  revert l2. induction l1; intros [|??] H.
410 411 412 413
  - done.
  - discriminate (H 0).
  - discriminate (H 0).
  - f_equal; [by injection (H 0)|]. apply (IHl1 _ $ λ i, H (S i)).
Robbert Krebbers's avatar
Robbert Krebbers committed
414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434
Qed.
Global Instance list_eq_dec {dec : ∀ x y, Decision (x = y)} : ∀ l k,
  Decision (l = k) := list_eq_dec dec.
Global Instance list_eq_nil_dec l : Decision (l = []).
Proof. by refine match l with [] => left _ | _ => right _ end. Defined.
Lemma list_singleton_reflect l :
  option_reflect (λ x, l = [x]) (length l ≠ 1) (list_singleton l).
Proof. by destruct l as [|? []]; constructor. Defined.

Definition nil_length : length (@nil A) = 0 := eq_refl.
Definition cons_length x l : length (x :: l) = S (length l) := eq_refl.
Lemma nil_or_length_pos l : l = [] ∨ length l ≠ 0.
Proof. destruct l; simpl; auto with lia. Qed.
Lemma nil_length_inv l : length l = 0 → l = [].
Proof. by destruct l. Qed.
Lemma lookup_nil i : @nil A !! i = None.
Proof. by destruct i. Qed.
Lemma lookup_tail l i : tail l !! i = l !! S i.
Proof. by destruct l. Qed.
Lemma lookup_lt_Some l i x : l !! i = Some x → i < length l.
Proof.
435
  revert i. induction l; intros [|?] ?; simplify_eq/=; auto with arith.
Robbert Krebbers's avatar
Robbert Krebbers committed
436 437 438 439 440
Qed.
Lemma lookup_lt_is_Some_1 l i : is_Some (l !! i) → i < length l.
Proof. intros [??]; eauto using lookup_lt_Some. Qed.
Lemma lookup_lt_is_Some_2 l i : i < length l → is_Some (l !! i).
Proof.
441
  revert i. induction l; intros [|?] ?; simplify_eq/=; eauto with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
442 443 444 445 446 447 448 449 450 451 452 453 454 455
Qed.
Lemma lookup_lt_is_Some l i : is_Some (l !! i) ↔ i < length l.
Proof. split; auto using lookup_lt_is_Some_1, lookup_lt_is_Some_2. Qed.
Lemma lookup_ge_None l i : l !! i = None ↔ length l ≤ i.
Proof. rewrite eq_None_not_Some, lookup_lt_is_Some. lia. Qed.
Lemma lookup_ge_None_1 l i : l !! i = None → length l ≤ i.
Proof. by rewrite lookup_ge_None. Qed.
Lemma lookup_ge_None_2 l i : length l ≤ i → l !! i = None.
Proof. by rewrite lookup_ge_None. Qed.
Lemma list_eq_same_length l1 l2 n :
  length l2 = n → length l1 = n →
  (∀ i x y, i < n → l1 !! i = Some x → l2 !! i = Some y → x = y) → l1 = l2.
Proof.
  intros <- Hlen Hl; apply list_eq; intros i. destruct (l2 !! i) as [x|] eqn:Hx.
456
  - destruct (lookup_lt_is_Some_2 l1 i) as [y Hy].
Robbert Krebbers's avatar
Robbert Krebbers committed
457 458
    { rewrite Hlen; eauto using lookup_lt_Some. }
    rewrite Hy; f_equal; apply (Hl i); eauto using lookup_lt_Some.
459
  - by rewrite lookup_ge_None, Hlen, <-lookup_ge_None.
Robbert Krebbers's avatar
Robbert Krebbers committed
460 461 462 463 464 465 466 467 468 469 470 471 472
Qed.
Lemma lookup_app_l l1 l2 i : i < length l1 → (l1 ++ l2) !! i = l1 !! i.
Proof. revert i. induction l1; intros [|?]; simpl; auto with lia. Qed.
Lemma lookup_app_l_Some l1 l2 i x : l1 !! i = Some x → (l1 ++ l2) !! i = Some x.
Proof. intros. rewrite lookup_app_l; eauto using lookup_lt_Some. Qed.
Lemma lookup_app_r l1 l2 i :
  length l1 ≤ i → (l1 ++ l2) !! i = l2 !! (i - length l1).
Proof. revert i. induction l1; intros [|?]; simpl; auto with lia. Qed.
Lemma lookup_app_Some l1 l2 i x :
  (l1 ++ l2) !! i = Some x ↔
    l1 !! i = Some x ∨ length l1 ≤ i ∧ l2 !! (i - length l1) = Some x.
Proof.
  split.
473
  - revert i. induction l1 as [|y l1 IH]; intros [|i] ?;
474
      simplify_eq/=; auto with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
475
    destruct (IH i) as [?|[??]]; auto with lia.
476
  - intros [?|[??]]; auto using lookup_app_l_Some. by rewrite lookup_app_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
477 478 479 480 481 482
Qed.
Lemma list_lookup_middle l1 l2 x n :
  n = length l1 → (l1 ++ x :: l2) !! n = Some x.
Proof. intros ->. by induction l1. Qed.

Lemma list_insert_alter l i x : <[i:=x]>l = alter (λ _, x) i l.
483
Proof. by revert i; induction l; intros []; intros; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
484
Lemma alter_length f l i : length (alter f i l) = length l.
485
Proof. revert i. by induction l; intros [|?]; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
486
Lemma insert_length l i x : length (<[i:=x]>l) = length l.
487
Proof. revert i. by induction l; intros [|?]; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
488 489 490 491 492 493 494
Lemma list_lookup_alter f l i : alter f i l !! i = f <$> l !! i.
Proof. revert i. induction l. done. intros [|i]. done. apply (IHl i). Qed.
Lemma list_lookup_alter_ne f l i j : i ≠ j → alter f i l !! j = l !! j.
Proof.
  revert i j. induction l; [done|]. intros [][] ?; csimpl; auto with congruence.
Qed.
Lemma list_lookup_insert l i x : i < length l → <[i:=x]>l !! i = Some x.
495
Proof. revert i. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
496 497 498 499 500 501 502 503 504 505
Lemma list_lookup_insert_ne l i j x : i ≠ j → <[i:=x]>l !! j = l !! j.
Proof.
  revert i j. induction l; [done|]. intros [] [] ?; simpl; auto with congruence.
Qed.
Lemma list_lookup_insert_Some l i x j y :
  <[i:=x]>l !! j = Some y ↔
    i = j ∧ x = y ∧ j < length l ∨ i ≠ j ∧ l !! j = Some y.
Proof.
  destruct (decide (i = j)) as [->|];
    [split|rewrite list_lookup_insert_ne by done; tauto].
506
  - intros Hy. assert (j < length l).
Robbert Krebbers's avatar
Robbert Krebbers committed
507 508
    { rewrite <-(insert_length l j x); eauto using lookup_lt_Some. }
    rewrite list_lookup_insert in Hy by done; naive_solver.
509
  - intros [(?&?&?)|[??]]; rewrite ?list_lookup_insert; naive_solver.
Robbert Krebbers's avatar
Robbert Krebbers committed
510 511 512
Qed.
Lemma list_insert_commute l i j x y :
  i ≠ j → <[i:=x]>(<[j:=y]>l) = <[j:=y]>(<[i:=x]>l).
513
Proof. revert i j. by induction l; intros [|?] [|?] ?; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
514 515 516
Lemma list_lookup_other l i x :
  length l ≠ 1 → l !! i = Some x → ∃ j y, j ≠ i ∧ l !! j = Some y.
Proof.
517
  intros. destruct i, l as [|x0 [|x1 l]]; simplify_eq/=.
518 519
  - by exists 1, x1.
  - by exists 0, x0.
Robbert Krebbers's avatar
Robbert Krebbers committed
520 521 522
Qed.
Lemma alter_app_l f l1 l2 i :
  i < length l1 → alter f i (l1 ++ l2) = alter f i l1 ++ l2.
523
Proof. revert i. induction l1; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
524 525
Lemma alter_app_r f l1 l2 i :
  alter f (length l1 + i) (l1 ++ l2) = l1 ++ alter f i l2.
526
Proof. revert i. induction l1; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
527 528 529 530 531 532 533
Lemma alter_app_r_alt f l1 l2 i :
  length l1 ≤ i → alter f i (l1 ++ l2) = l1 ++ alter f (i - length l1) l2.
Proof.
  intros. assert (i = length l1 + (i - length l1)) as Hi by lia.
  rewrite Hi at 1. by apply alter_app_r.
Qed.
Lemma list_alter_id f l i : (∀ x, f x = x) → alter f i l = l.
534
Proof. intros ?. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
535 536
Lemma list_alter_ext f g l k i :
  (∀ x, l !! i = Some x → f x = g x) → l = k → alter f i l = alter g i k.
537
Proof. intros H ->. revert i H. induction k; intros [|?] ?; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
538 539
Lemma list_alter_compose f g l i :
  alter (f ∘ g) i l = alter f i (alter g i l).
540
Proof. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
541 542
Lemma list_alter_commute f g l i j :
  i ≠ j → alter f i (alter g j l) = alter g j (alter f i l).
543
Proof. revert i j. induction l; intros [|?][|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
544 545
Lemma insert_app_l l1 l2 i x :
  i < length l1 → <[i:=x]>(l1 ++ l2) = <[i:=x]>l1 ++ l2.
546
Proof. revert i. induction l1; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
547
Lemma insert_app_r l1 l2 i x : <[length l1+i:=x]>(l1 ++ l2) = l1 ++ <[i:=x]>l2.
548
Proof. revert i. induction l1; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
549 550 551 552 553 554 555
Lemma insert_app_r_alt l1 l2 i x :
  length l1 ≤ i → <[i:=x]>(l1 ++ l2) = l1 ++ <[i - length l1:=x]>l2.
Proof.
  intros. assert (i = length l1 + (i - length l1)) as Hi by lia.
  rewrite Hi at 1. by apply insert_app_r.
Qed.
Lemma delete_middle l1 l2 x : delete (length l1) (l1 ++ x :: l2) = l1 ++ l2.
556
Proof. induction l1; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594

Lemma inserts_length l i k : length (list_inserts i k l) = length l.
Proof.
  revert i. induction k; intros ?; csimpl; rewrite ?insert_length; auto.
Qed.
Lemma list_lookup_inserts l i k j :
  i ≤ j < i + length k → j < length l →
  list_inserts i k l !! j = k !! (j - i).
Proof.
  revert i j. induction k as [|y k IH]; csimpl; intros i j ??; [lia|].
  destruct (decide (i = j)) as [->|].
  { by rewrite list_lookup_insert, Nat.sub_diag
      by (rewrite inserts_length; lia). }
  rewrite list_lookup_insert_ne, IH by lia.
  by replace (j - i) with (S (j - S i)) by lia.
Qed.
Lemma list_lookup_inserts_lt l i k j :
  j < i → list_inserts i k l !! j = l !! j.
Proof.
  revert i j. induction k; intros i j ?; csimpl;
    rewrite ?list_lookup_insert_ne by lia; auto with lia.
Qed.
Lemma list_lookup_inserts_ge l i k j :
  i + length k ≤ j → list_inserts i k l !! j = l !! j.
Proof.
  revert i j. induction k; csimpl; intros i j ?;
    rewrite ?list_lookup_insert_ne by lia; auto with lia.
Qed.
Lemma list_lookup_inserts_Some l i k j y :
  list_inserts i k l !! j = Some y ↔
    (j < i ∨ i + length k ≤ j) ∧ l !! j = Some y ∨
    i ≤ j < i + length k ∧ j < length l ∧ k !! (j - i) = Some y.
Proof.
  destruct (decide (j < i)).
  { rewrite list_lookup_inserts_lt by done; intuition lia. }
  destruct (decide (i + length k ≤ j)).
  { rewrite list_lookup_inserts_ge by done; intuition lia. }
  split.
595
  - intros Hy. assert (j < length l).
Robbert Krebbers's avatar
Robbert Krebbers committed
596 597
    { rewrite <-(inserts_length l i k); eauto using lookup_lt_Some. }
    rewrite list_lookup_inserts in Hy by lia. intuition lia.
598
  - intuition. by rewrite list_lookup_inserts by lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622
Qed.
Lemma list_insert_inserts_lt l i j x k :
  i < j → <[i:=x]>(list_inserts j k l) = list_inserts j k (<[i:=x]>l).
Proof.
  revert i j. induction k; intros i j ?; simpl;
    rewrite 1?list_insert_commute by lia; auto with f_equal.
Qed.

(** ** Properties of the [elem_of] predicate *)
Lemma not_elem_of_nil x : x ∉ [].
Proof. by inversion 1. Qed.
Lemma elem_of_nil x : x ∈ [] ↔ False.
Proof. intuition. by destruct (not_elem_of_nil x). Qed.
Lemma elem_of_nil_inv l : (∀ x, x ∉ l) → l = [].
Proof. destruct l. done. by edestruct 1; constructor. Qed.
Lemma elem_of_not_nil x l : x ∈ l → l ≠ [].
Proof. intros ? ->. by apply (elem_of_nil x). Qed.
Lemma elem_of_cons l x y : x ∈ y :: l ↔ x = y ∨ x ∈ l.
Proof. by split; [inversion 1; subst|intros [->|?]]; constructor. Qed.
Lemma not_elem_of_cons l x y : x ∉ y :: l ↔ x ≠ y ∧ x ∉ l.
Proof. rewrite elem_of_cons. tauto. Qed.
Lemma elem_of_app l1 l2 x : x ∈ l1 ++ l2 ↔ x ∈ l1 ∨ x ∈ l2.
Proof.
  induction l1.
623 624
  - split; [by right|]. intros [Hx|]; [|done]. by destruct (elem_of_nil x).
  - simpl. rewrite !elem_of_cons, IHl1. tauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
625 626 627 628 629 630 631 632 633 634
Qed.
Lemma not_elem_of_app l1 l2 x : x ∉ l1 ++ l2 ↔ x ∉ l1 ∧ x ∉ l2.
Proof. rewrite elem_of_app. tauto. Qed.
Lemma elem_of_list_singleton x y : x ∈ [y] ↔ x = y.
Proof. rewrite elem_of_cons, elem_of_nil. tauto. Qed.
Global Instance elem_of_list_permutation_proper x : Proper ((≡ₚ) ==> iff) (x ∈).
Proof. induction 1; rewrite ?elem_of_nil, ?elem_of_cons; intuition. Qed.
Lemma elem_of_list_split l x : x ∈ l → ∃ l1 l2, l = l1 ++ x :: l2.
Proof.
  induction 1 as [x l|x y l ? [l1 [l2 ->]]]; [by eexists [], l|].
635
  by exists (y :: l1), l2.
Robbert Krebbers's avatar
Robbert Krebbers committed
636 637 638 639 640 641 642 643
Qed.
Lemma elem_of_list_lookup_1 l x : x ∈ l → ∃ i, l !! i = Some x.
Proof.
  induction 1 as [|???? IH]; [by exists 0 |].
  destruct IH as [i ?]; auto. by exists (S i).
Qed.
Lemma elem_of_list_lookup_2 l i x : l !! i = Some x → x ∈ l.
Proof.
644
  revert i. induction l; intros [|i] ?; simplify_eq/=; constructor; eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
645 646 647 648 649 650 651
Qed.
Lemma elem_of_list_lookup l x : x ∈ l ↔ ∃ i, l !! i = Some x.
Proof. firstorder eauto using elem_of_list_lookup_1, elem_of_list_lookup_2. Qed.
Lemma elem_of_list_omap {B} (f : A → option B) l (y : B) :
  y ∈ omap f l ↔ ∃ x, x ∈ l ∧ f x = Some y.
Proof.
  split.
652
  - induction l as [|x l]; csimpl; repeat case_match; inversion 1; subst;
Robbert Krebbers's avatar
Robbert Krebbers committed
653
      setoid_rewrite elem_of_cons; naive_solver.
654
  - intros (x&Hx&?). by induction Hx; csimpl; repeat case_match;
655
      simplify_eq; try constructor; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671
Qed.

(** ** Properties of the [NoDup] predicate *)
Lemma NoDup_nil : NoDup (@nil A) ↔ True.
Proof. split; constructor. Qed.
Lemma NoDup_cons x l : NoDup (x :: l) ↔ x ∉ l ∧ NoDup l.
Proof. split. by inversion 1. intros [??]. by constructor. Qed.
Lemma NoDup_cons_11 x l : NoDup (x :: l) → x ∉ l.
Proof. rewrite NoDup_cons. by intros [??]. Qed.
Lemma NoDup_cons_12 x l : NoDup (x :: l) → NoDup l.
Proof. rewrite NoDup_cons. by intros [??]. Qed.
Lemma NoDup_singleton x : NoDup [x].
Proof. constructor. apply not_elem_of_nil. constructor. Qed.
Lemma NoDup_app l k : NoDup (l ++ k) ↔ NoDup l ∧ (∀ x, x ∈ l → x ∉ k) ∧ NoDup k.
Proof.
  induction l; simpl.
672 673
  - rewrite NoDup_nil. setoid_rewrite elem_of_nil. naive_solver.
  - rewrite !NoDup_cons.
Robbert Krebbers's avatar
Robbert Krebbers committed
674 675 676 677 678
    setoid_rewrite elem_of_cons. setoid_rewrite elem_of_app. naive_solver.
Qed.
Global Instance NoDup_proper: Proper ((≡ₚ) ==> iff) (@NoDup A).
Proof.
  induction 1 as [|x l k Hlk IH | |].
679 680 681 682
  - by rewrite !NoDup_nil.
  - by rewrite !NoDup_cons, IH, Hlk.
  - rewrite !NoDup_cons, !elem_of_cons. intuition.
  - intuition.
Robbert Krebbers's avatar
Robbert Krebbers committed
683 684 685 686 687
Qed.
Lemma NoDup_lookup l i j x :
  NoDup l → l !! i = Some x → l !! j = Some x → i = j.
Proof.
  intros Hl. revert i j. induction Hl as [|x' l Hx Hl IH].
688 689
  { intros; simplify_eq. }
  intros [|i] [|j] ??; simplify_eq/=; eauto with f_equal;
Robbert Krebbers's avatar
Robbert Krebbers committed
690 691 692 693 694 695 696
    exfalso; eauto using elem_of_list_lookup_2.
Qed.
Lemma NoDup_alt l :
  NoDup l ↔ ∀ i j x, l !! i = Some x → l !! j = Some x → i = j.
Proof.
  split; eauto using NoDup_lookup.
  induction l as [|x l IH]; intros Hl; constructor.
697
  - rewrite elem_of_list_lookup. intros [i ?].
Robbert Krebbers's avatar
Robbert Krebbers committed
698
    by feed pose proof (Hl (S i) 0 x); auto.
699
  - apply IH. intros i j x' ??. by apply (inj S), (Hl (S i) (S j) x').
Robbert Krebbers's avatar
Robbert Krebbers committed
700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720
Qed.

Section no_dup_dec.
  Context `{!∀ x y, Decision (x = y)}.
  Global Instance NoDup_dec: ∀ l, Decision (NoDup l) :=
    fix NoDup_dec l :=
    match l return Decision (NoDup l) with
    | [] => left NoDup_nil_2
    | x :: l =>
      match decide_rel (∈) x l with
      | left Hin => right (λ H, NoDup_cons_11 _ _ H Hin)
      | right Hin =>
        match NoDup_dec l with
        | left H => left (NoDup_cons_2 _ _ Hin H)
        | right H => right (H ∘ NoDup_cons_12 _ _)
        end
      end
    end.
  Lemma elem_of_remove_dups l x : x ∈ remove_dups l ↔ x ∈ l.
  Proof.
    split; induction l; simpl; repeat case_decide;
721
      rewrite ?elem_of_cons; intuition (simplify_eq; auto).
Robbert Krebbers's avatar
Robbert Krebbers committed
722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740
  Qed.
  Lemma NoDup_remove_dups l : NoDup (remove_dups l).
  Proof.
    induction l; simpl; repeat case_decide; try constructor; auto.
    by rewrite elem_of_remove_dups.
  Qed.
End no_dup_dec.

(** ** Set operations on lists *)
Section list_set.
  Context {dec : ∀ x y, Decision (x = y)}.
  Lemma elem_of_list_difference l k x : x ∈ list_difference l k ↔ x ∈ l ∧ x ∉ k.
  Proof.
    split; induction l; simpl; try case_decide;
      rewrite ?elem_of_nil, ?elem_of_cons; intuition congruence.
  Qed.
  Lemma NoDup_list_difference l k : NoDup l → NoDup (list_difference l k).
  Proof.
    induction 1; simpl; try case_decide.
741 742 743
    - constructor.
    - done.
    - constructor. rewrite elem_of_list_difference; intuition. done.
Robbert Krebbers's avatar
Robbert Krebbers committed
744 745 746 747 748 749 750 751 752
  Qed.
  Lemma elem_of_list_union l k x : x ∈ list_union l k ↔ x ∈ l ∨ x ∈ k.
  Proof.
    unfold list_union. rewrite elem_of_app, elem_of_list_difference.
    intuition. case (decide (x ∈ k)); intuition.
  Qed.
  Lemma NoDup_list_union l k : NoDup l → NoDup k → NoDup (list_union l k).
  Proof.
    intros. apply NoDup_app. repeat split.
753 754 755
    - by apply NoDup_list_difference.
    - intro. rewrite elem_of_list_difference. intuition.
    - done.
Robbert Krebbers's avatar
Robbert Krebbers committed
756 757 758 759 760 761 762 763 764 765
  Qed.
  Lemma elem_of_list_intersection l k x :
    x ∈ list_intersection l k ↔ x ∈ l ∧ x ∈ k.
  Proof.
    split; induction l; simpl; repeat case_decide;
      rewrite ?elem_of_nil, ?elem_of_cons; intuition congruence.
  Qed.
  Lemma NoDup_list_intersection l k : NoDup l → NoDup (list_intersection l k).
  Proof.
    induction 1; simpl; try case_decide.
766 767 768
    - constructor.
    - constructor. rewrite elem_of_list_intersection; intuition. done.
    - done.
Robbert Krebbers's avatar
Robbert Krebbers committed
769 770 771 772 773 774
  Qed.
  Lemma elem_of_list_intersection_with f l k x :
    x ∈ list_intersection_with f l k ↔ ∃ x1 x2,
      x1 ∈ l ∧ x2 ∈ k ∧ f x1 x2 = Some x.
  Proof.
    split.
775
    - induction l as [|x1 l IH]; simpl; [by rewrite elem_of_nil|].
Robbert Krebbers's avatar
Robbert Krebbers committed
776 777 778 779 780 781
      intros Hx. setoid_rewrite elem_of_cons.
      cut ((∃ x2, x2 ∈ k ∧ f x1 x2 = Some x)
        ∨ x ∈ list_intersection_with f l k); [naive_solver|].
      clear IH. revert Hx. generalize (list_intersection_with f l k).
      induction k; simpl; [by auto|].
      case_match; setoid_rewrite elem_of_cons; naive_solver.
782
    - intros (x1&x2&Hx1&Hx2&Hx). induction Hx1 as [x1|x1 ? l ? IH]; simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815
      + generalize (list_intersection_with f l k).
        induction Hx2; simpl; [by rewrite Hx; left |].
        case_match; simpl; try setoid_rewrite elem_of_cons; auto.
      + generalize (IH Hx). clear Hx IH Hx2.
        generalize (list_intersection_with f l k).
        induction k; simpl; intros; [done|].
        case_match; simpl; rewrite ?elem_of_cons; auto.
  Qed.
End list_set.

(** ** Properties of the [filter] function *)
Section filter.
  Context (P : A → Prop) `{∀ x, Decision (P x)}.
  Lemma elem_of_list_filter l x : x ∈ filter P l ↔ P x ∧ x ∈ l.
  Proof.
    unfold filter. induction l; simpl; repeat case_decide;
       rewrite ?elem_of_nil, ?elem_of_cons; naive_solver.
  Qed.
  Lemma NoDup_filter l : NoDup l → NoDup (filter P l).
  Proof.
    unfold filter. induction 1; simpl; repeat case_decide;
      rewrite ?NoDup_nil, ?NoDup_cons, ?elem_of_list_filter; tauto.
  Qed.
End filter.

(** ** Properties of the [find] function *)
Section find.
  Context (P : A → Prop) `{∀ x, Decision (P x)}.
  Lemma list_find_Some l i x :
    list_find P l = Some (i,x) → l !! i = Some x ∧ P x.
  Proof.
    revert i; induction l; intros [] ?;
      repeat (match goal with x : prod _ _ |- _ => destruct x end
816
              || simplify_option_eq); eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
817 818 819
  Qed.
  Lemma list_find_elem_of l x : x ∈ l → P x → is_Some (list_find P l).
  Proof.
820
    induction 1 as [|x y l ? IH]; intros; simplify_option_eq; eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849
    by destruct IH as [[i x'] ->]; [|exists (S i, x')].
  Qed.
End find.

(** ** Properties of the [reverse] function *)
Lemma reverse_nil : reverse [] = @nil A.
Proof. done. Qed.
Lemma reverse_singleton x : reverse [x] = [x].
Proof. done. Qed.
Lemma reverse_cons l x : reverse (x :: l) = reverse l ++ [x].
Proof. unfold reverse. by rewrite <-!rev_alt. Qed.
Lemma reverse_snoc l x : reverse (l ++ [x]) = x :: reverse l.
Proof. unfold reverse. by rewrite <-!rev_alt, rev_unit. Qed.
Lemma reverse_app l1 l2 : reverse (l1 ++ l2) = reverse l2 ++ reverse l1.
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_app_distr. Qed.
Lemma reverse_length l : length (reverse l) = length l.
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_length. Qed.
Lemma reverse_involutive l : reverse (reverse l) = l.
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_involutive. Qed.
Lemma elem_of_reverse_2 x l : x ∈ l → x ∈ reverse l.
Proof.
  induction 1; rewrite reverse_cons, elem_of_app,
    ?elem_of_list_singleton; intuition.
Qed.
Lemma elem_of_reverse x l : x ∈ reverse l ↔ x ∈ l.
Proof.
  split; auto using elem_of_reverse_2.
  intros. rewrite <-(reverse_involutive l). by apply elem_of_reverse_2.
Qed.
850
Global Instance: Inj (=) (=) (@reverse A).
Robbert Krebbers's avatar
Robbert Krebbers committed
851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876
Proof.
  intros l1 l2 Hl.
  by rewrite <-(reverse_involutive l1), <-(reverse_involutive l2), Hl.
Qed.
Lemma sum_list_with_app (f : A → nat) l k :
  sum_list_with f (l ++ k) = sum_list_with f l + sum_list_with f k.
Proof. induction l; simpl; lia. Qed.
Lemma sum_list_with_reverse (f : A → nat) l :
  sum_list_with f (reverse l) = sum_list_with f l.
Proof.
  induction l; simpl; rewrite ?reverse_cons, ?sum_list_with_app; simpl; lia.
Qed.

(** ** Properties of the [last] function *)
Lemma last_snoc x l : last (l ++ [x]) = Some x.
Proof. induction l as [|? []]; simpl; auto. Qed.
Lemma last_reverse l : last (reverse l) = head l.
Proof. by destruct l as [|x l]; rewrite ?reverse_cons, ?last_snoc. Qed.
Lemma head_reverse l : head (reverse l) = last l.
Proof. by rewrite <-last_reverse, reverse_involutive. Qed.

(** ** Properties of the [take] function *)
Definition take_drop i l : take i l ++ drop i l = l := firstn_skipn i l.
Lemma take_drop_middle l i x :
  l !! i = Some x → take i l ++ x :: drop (S i) l = l.
Proof.
877
  revert i x. induction l; intros [|?] ??; simplify_eq/=; f_equal; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
878 879 880 881
Qed.
Lemma take_nil n : take n (@nil A) = [].
Proof. by destruct n. Qed.
Lemma take_app l k : take (length l) (l ++ k) = l.
882
Proof. induction l; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
883 884 885
Lemma take_app_alt l k n : n = length l → take n (l ++ k) = l.
Proof. intros ->. by apply take_app. Qed.
Lemma take_app3_alt l1 l2 l3 n : n = length l1 → take n ((l1 ++ l2) ++ l3) = l1.
886
Proof. intros ->. by rewrite <-(assoc_L (++)), take_app. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
887
Lemma take_app_le l k n : n ≤ length l → take n (l ++ k) = take n l.
888
Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
889 890
Lemma take_plus_app l k n m :
  length l = n → take (n + m) (l ++ k) = l ++ take m k.
891
Proof. intros <-. induction l; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
892 893
Lemma take_app_ge l k n :
  length l ≤ n → take n (l ++ k) = l ++ take (n - length l) k.
894
Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
895
Lemma take_ge l n : length l ≤ n → take n l = l.
896
Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
897
Lemma take_take l n m : take n (take m l) = take (min n m) l.
898
Proof. revert n m. induction l; intros [|?] [|?]; f_equal/=; auto. Qed.
899
Lemma take_idemp l n : take n (take n l) = take n l.
Robbert Krebbers's avatar
Robbert Krebbers committed
900 901
Proof. by rewrite take_take, Min.min_idempotent. Qed.
Lemma take_length l n : length (take n l) = min n (length l).
902
Proof. revert n. induction l; intros [|?]; f_equal/=; done. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
903 904 905 906 907 908 909 910 911 912 913 914 915 916 917
Lemma take_length_le l n : n ≤ length l → length (take n l) = n.
Proof. rewrite take_length. apply Min.min_l. Qed.
Lemma take_length_ge l n : length l ≤ n → length (take n l) = length l.
Proof. rewrite take_length. apply Min.min_r. Qed.
Lemma take_drop_commute l n m : take n (drop m l) = drop m (take (m + n) l).
Proof.
  revert n m. induction l; intros [|?][|?]; simpl; auto using take_nil with lia.
Qed.
Lemma lookup_take l n i : i < n → take n l !! i = l !! i.
Proof. revert n i. induction l; intros [|n] [|i] ?; simpl; auto with lia. Qed.
Lemma lookup_take_ge l n i : n ≤ i → take n l !! i = None.
Proof. revert n i. induction l; intros [|?] [|?] ?; simpl; auto with lia. Qed.
Lemma take_alter f l n i : n ≤ i → take n (alter f i l) = take n l.
Proof.
  intros. apply list_eq. intros j. destruct (le_lt_dec n j).
918 919
  - by rewrite !lookup_take_ge.
  - by rewrite !lookup_take, !list_lookup_alter_ne by lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
920 921 922 923
Qed.
Lemma take_insert l n i x : n ≤ i → take n (<[i:=x]>l) = take n l.
Proof.
  intros. apply list_eq. intros j. destruct (le_lt_dec n j).
924 925
  - by rewrite !lookup_take_ge.
  - by rewrite !lookup_take, !list_lookup_insert_ne by lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
926 927 928 929 930 931 932 933
Qed.

(** ** Properties of the [drop] function *)
Lemma drop_0 l : drop 0 l = l.
Proof. done. Qed.
Lemma drop_nil n : drop n (@nil A) = [].
Proof. by destruct n. Qed.
Lemma drop_length l n : length (drop n l) = length l - n.
934
Proof. revert n. by induction l; intros [|i]; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
935 936 937 938 939 940 941 942 943 944 945 946 947 948 949
Lemma drop_ge l n : length l ≤ n → drop n l = [].
Proof. revert n. induction l; intros [|??]; simpl in *; auto with lia. Qed.
Lemma drop_all l : drop (length l) l = [].
Proof. by apply drop_ge. Qed.
Lemma drop_drop l n1 n2 : drop n1 (drop n2 l) = drop (n2 + n1) l.
Proof. revert n2. induction l; intros [|?]; simpl; rewrite ?drop_nil; auto. Qed.
Lemma drop_app_le l k n :
  n ≤ length l → drop n (l ++ k) = drop n l ++ k.
Proof. revert n. induction l; intros [|?]; simpl; auto with lia. Qed.
Lemma drop_app l k : drop (length l) (l ++ k) = k.
Proof. by rewrite drop_app_le, drop_all. Qed.
Lemma drop_app_alt l k n : n = length l → drop n (l ++ k) = k.
Proof. intros ->. by apply drop_app. Qed.
Lemma drop_app3_alt l1 l2 l3 n :
  n = length l1 → drop n ((l1 ++ l2) ++ l3) = l2 ++ l3.
950
Proof. intros ->. by rewrite <-(assoc_L (++)), drop_app. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972
Lemma drop_app_ge l k n :
  length l ≤ n → drop n (l ++ k) = drop (n - length l) k.
Proof.
  intros. rewrite <-(Nat.sub_add (length l) n) at 1 by done.
  by rewrite Nat.add_comm, <-drop_drop, drop_app.
Qed.
Lemma drop_plus_app l k n m :
  length l = n → drop (n + m) (l ++ k) = drop m k.
Proof. intros <-. by rewrite <-drop_drop, drop_app. Qed.
Lemma lookup_drop l n i : drop n l !! i = l !! (n + i).
Proof. revert n i. induction l; intros [|i] ?; simpl; auto. Qed.
Lemma drop_alter f l n i : i < n → drop n (alter f i l) = drop n l.
Proof.
  intros. apply list_eq. intros j.
  by rewrite !lookup_drop, !list_lookup_alter_ne by lia.
Qed.
Lemma drop_insert l n i x : i < n → drop n (<[i:=x]>l) = drop n l.
Proof.
  intros. apply list_eq. intros j.
  by rewrite !lookup_drop, !list_lookup_insert_ne by lia.
Qed.
Lemma delete_take_drop l i : delete i l = take i l ++ drop (S i) l.
973
Proof. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
974
Lemma take_take_drop l n m : take n l ++ take m (drop n l) = take (n + m) l.
975
Proof. revert n m. induction l; intros [|?] [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
976 977 978
Lemma drop_take_drop l n m : n ≤ m → drop n (take m l) ++ drop m l = drop n l.
Proof.
  revert n m. induction l; intros [|?] [|?] ?;
979
    f_equal/=; auto using take_drop with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
980 981 982 983 984 985 986 987 988
Qed.

(** ** Properties of the [replicate] function *)
Lemma replicate_length n x : length (replicate n x) = n.
Proof. induction n; simpl; auto. Qed.
Lemma lookup_replicate n x y i :
  replicate n x !! i = Some y ↔ y = x ∧ i < n.
Proof.
  split.
989 990
  - revert i. induction n; intros [|?]; naive_solver auto with lia.
  - intros [-> Hi]. revert i Hi.
Robbert Krebbers's avatar
Robbert Krebbers committed
991 992 993 994 995 996 997 998 999 1000
    induction n; intros [|?]; naive_solver auto with lia.
Qed.
Lemma lookup_replicate_1 n x y i :
  replicate n x !! i = Some y → y = x ∧ i < n.
Proof. by rewrite lookup_replicate. Qed.
Lemma lookup_replicate_2 n x i : i < n → replicate n x !! i = Some x.
Proof. by rewrite lookup_replicate. Qed.
Lemma lookup_replicate_None n x i : n ≤ i ↔ replicate n x !! i = None.
Proof.
  rewrite eq_None_not_Some, Nat.le_ngt. split.
1001 1002
  - intros Hin [x' Hx']; destruct Hin. rewrite lookup_replicate in Hx'; tauto.
  - intros Hx ?. destruct Hx. exists x; auto using lookup_replicate_2.
Robbert Krebbers's avatar
Robbert Krebbers committed
1003 1004
Qed.
Lemma insert_replicate x n i : <[i:=x]>(replicate n x) = replicate n x.
1005
Proof. revert i. induction n; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1006 1007 1008 1009 1010 1011
Lemma elem_of_replicate_inv x n y : x ∈ replicate n y → x = y.
Proof. induction n; simpl; rewrite ?elem_of_nil, ?elem_of_cons; intuition. Qed.
Lemma replicate_S n x : replicate (S n) x = x :: replicate  n x.
Proof. done. Qed.
Lemma replicate_plus n m x :
  replicate (n + m) x = replicate n x ++ replicate m x.
1012
Proof. induction n; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1013
Lemma take_replicate n m x : take n (replicate m x) = replicate (min n m) x.
1014
Proof. revert m. by induction n; intros [|?]; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1015 1016 1017
Lemma take_replicate_plus n m x : take n (replicate (n + m) x) = replicate n x.
Proof. by rewrite take_replicate, min_l by lia. Qed.
Lemma drop_replicate n m x : drop n (replicate m x) = replicate (m - n) x.
1018
Proof. revert m. by induction n; intros [|?]; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1019 1020 1021 1022 1023 1024
Lemma drop_replicate_plus n m x : drop n (replicate (n + m) x) = replicate m x.
Proof. rewrite drop_replicate. f_equal. lia. Qed.
Lemma replicate_as_elem_of x n l :
  replicate n x = l ↔ length l = n ∧ ∀ y, y ∈ l → y = x.
Proof.
  split; [intros <-; eauto using elem_of_replicate_inv, replicate_length|].
1025
  intros [<- Hl]. symmetry. induction l as [|y l IH]; f_equal/=.
1026 1027
  - apply Hl. by left.
  - apply IH. intros ??. apply Hl. by right.
Robbert Krebbers's avatar
Robbert Krebbers committed
1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039
Qed.
Lemma reverse_replicate n x : reverse (replicate n x) = replicate n x.
Proof.
  symmetry. apply replicate_as_elem_of.
  rewrite reverse_length, replicate_length. split; auto.
  intros y. rewrite elem_of_reverse. by apply elem_of_replicate_inv.
Qed.
Lemma replicate_false βs n : length βs = n → replicate n false =.>* βs.
Proof. intros <-. by induction βs; simpl; constructor. Qed.

(** ** Properties of the [resize] function *)
Lemma resize_spec l n x : resize n x l = take n l ++ replicate (n - length l) x.
1040
Proof. revert n. induction l; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1041 1042 1043
Lemma resize_0 l x : resize 0 x l = [].
Proof. by destruct l. Qed.
Lemma resize_nil n x : resize n x [] = replicate n x.
1044
Proof. rewrite resize_spec. rewrite take_nil. f_equal/=. lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059
Lemma resize_ge l n x :
  length l ≤ n → resize n x l = l ++ replicate (n - length l) x.
Proof. intros. by rewrite resize_spec, take_ge. Qed.
Lemma resize_le l n x : n ≤ length l → resize n x l = take n l.
Proof.
  intros. rewrite resize_spec, (proj2 (Nat.sub_0_le _ _)) by done.
  simpl. by rewrite (right_id_L [] (++)).
Qed.
Lemma resize_all l x : resize (length l) x l = l.
Proof. intros. by rewrite resize_le, take_ge. Qed.
Lemma resize_all_alt l n x : n = length l → resize n x l = l.
Proof. intros ->. by rewrite resize_all. Qed.
Lemma resize_plus l n m x :
  resize (n + m) x l = resize n x l ++ resize m x (drop n l).
Proof.
1060
  revert n m. induction l; intros [|?] [|?]; f_equal/=; auto.
1061 1062
  - by rewrite Nat.add_0_r, (right_id_L [] (++)).
  - by rewrite replicate_plus.
Robbert Krebbers's avatar
Robbert Krebbers committed
1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076
Qed.
Lemma resize_plus_eq l n m x :
  length l = n → resize (n + m) x l = l ++ replicate m x.
Proof. intros <-. by rewrite resize_plus, resize_all, drop_all, resize_nil. Qed.
Lemma resize_app_le l1 l2 n x :
  n ≤ length l1 → resize n x (l1 ++ l2) = resize n x l1.
Proof.
  intros. by rewrite !resize_le, take_app_le by (rewrite ?app_length; lia).
Qed.
Lemma resize_app l1 l2 n x : n = length l1 → resize n x (l1 ++ l2) = l1.
Proof. intros ->. by rewrite resize_app_le, resize_all. Qed.
Lemma resize_app_ge l1 l2 n x :
  length l1 ≤ n → resize n x (l1 ++ l2) = l1 ++ resize (n - length l1) x l2.
Proof.
1077
  intros. rewrite !resize_spec, take_app_ge, (assoc_L (++)) by done.
Robbert Krebbers's avatar
Robbert Krebbers committed
1078 1079 1080 1081 1082
  do 2 f_equal. rewrite app_length. lia.
Qed.
Lemma resize_length l n x : length (resize n x l) = n.
Proof. rewrite resize_spec, app_length, replicate_length, take_length. lia. Qed.
Lemma resize_replicate x n m : resize n x (replicate m x) = replicate n x.
1083
Proof. revert m. induction n; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1084 1085 1086
Lemma resize_resize l n m x : n ≤ m → resize n x (resize m x l) = resize n x l.
Proof.
  revert n m. induction l; simpl.
1087
  - intros. by rewrite !resize_nil, resize_replicate.
1088
  - intros [|?] [|?] ?; f_equal/=; auto with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
1089
Qed.
1090
Lemma resize_idemp l n x : resize n x (resize n x l) = resize n x l.
Robbert Krebbers's avatar
Robbert Krebbers committed
1091 1092
Proof. by rewrite resize_resize. Qed.
Lemma resize_take_le l n m x : n ≤ m → resize n x (take m l) = resize n x l.
1093
Proof. revert n m. induction l; intros [|?][|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1094 1095 1096 1097
Lemma resize_take_eq l n x : resize n x (take n l) = resize n x l.
Proof. by rewrite resize_take_le. Qed.
Lemma take_resize l n m x : take n (resize m x l) = resize (min n m) x l.
Proof.
1098
  revert n m. induction l; intros [|?][|?]; f_equal/=; auto using take_replicate.
Robbert Krebbers's avatar
Robbert Krebbers committed
1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109
Qed.
Lemma take_resize_le l n m x : n ≤ m → take n (resize m x l) = resize n x l.
Proof. intros. by rewrite take_resize, Min.min_l. Qed.
Lemma take_resize_eq l n x : take n (resize n x l) = resize n x l.
Proof. intros. by rewrite take_resize, Min.min_l. Qed.
Lemma take_resize_plus l n m x : take n (resize (n + m) x l) = resize n x l.
Proof. by rewrite take_resize, min_l by lia. Qed.
Lemma drop_resize_le l n m x :
  n ≤ m → drop n (resize m x l) = resize (m - n) x (drop n l).
Proof.
  revert n m. induction l; simpl.
1110 1111
  - intros. by rewrite drop_nil, !resize_nil, drop_replicate.
  - intros [|?] [|?] ?; simpl; try case_match; auto with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
1112 1113 1114 1115 1116 1117 1118
Qed.
Lemma drop_resize_plus l n m x :
  drop n (resize (n + m) x l) = resize m x (drop n l).
Proof. rewrite drop_resize_le by lia. f_equal. lia. Qed.
Lemma lookup_resize l n x i : i < n → i < length l → resize n x l !! i = l !! i.
Proof.
  intros ??. destruct (decide (n < length l)).
1119 1120
  - by rewrite resize_le, lookup_take by lia.
  - by rewrite resize_ge, lookup_app_l by lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139
Qed.
Lemma lookup_resize_new l n x i :
  length l ≤ i → i < n → resize n x l !! i = Some x.
Proof.
  intros ??. rewrite resize_ge by lia.
  replace i with (length l + (i - length l)) by lia.
  by rewrite lookup_app_r, lookup_replicate_2 by lia.
Qed.
Lemma lookup_resize_old l n x i : n ≤ i → resize n x l !! i = None.
Proof. intros ?. apply lookup_ge_None_2. by rewrite resize_length. Qed.
End general_properties.

Section more_general_properties.
Context {A : Type}.
Implicit Types x y z : A.
Implicit Types l k : list A.

(** ** Properties of the [reshape] function *)
Lemma reshape_length szs l : length (reshape szs l) = length szs.
1140
Proof. revert l. by induction szs; intros; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153
Lemma join_reshape szs l :
  sum_list szs = length l → mjoin (reshape szs l) = l.
Proof.
  revert l. induction szs as [|sz szs IH]; simpl; intros l Hl; [by destruct l|].
  by rewrite IH, take_drop by (rewrite drop_length; lia).
Qed.
Lemma sum_list_replicate n m : sum_list (replicate m n) = m * n.
Proof. induction m; simpl; auto. Qed.

(** ** Properties of [sublist_lookup] and [sublist_alter] *)
Lemma sublist_lookup_length l i n k :
  sublist_lookup i n l = Some k → length k = n.
Proof.
1154
  unfold sublist_lookup; intros; simplify_option_eq.
Robbert Krebbers's avatar
Robbert Krebbers committed
1155 1156 1157 1158 1159 1160 1161 1162 1163
  rewrite take_length, drop_length; lia.
Qed.
Lemma sublist_lookup_all l n : length l = n → sublist_lookup 0 n l = Some l.
Proof.
  intros. unfold sublist_lookup; case_option_guard; [|lia].
  by rewrite take_ge by (rewrite drop_length; lia).
Qed.
Lemma sublist_lookup_Some l i n :
  i + n ≤ length l → sublist_lookup i n l = Some (take n (drop i l)).
1164
Proof. by unfold sublist_lookup; intros; simplify_option_eq. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1165 1166
Lemma sublist_lookup_None l i n :
  length l < i + n → sublist_lookup i n l = None.
1167
Proof. by unfold sublist_lookup; intros; simplify_option_eq by lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181
Lemma sublist_eq l k n :
  (n | length l) → (n | length k) →
  (∀ i, sublist_lookup (i * n) n l = sublist_lookup (i * n) n k) → l = k.
Proof.
  revert l k. assert (∀ l i,
    n ≠ 0 → (n | length l) → ¬n * i `div` n + n ≤ length l → length l ≤ i).
  { intros l i ? [j ->] Hjn. apply Nat.nlt_ge; contradict Hjn.
    rewrite <-Nat.mul_succ_r, (Nat.mul_comm n).
    apply Nat.mul_le_mono_r, Nat.le_succ_l, Nat.div_lt_upper_bound; lia. }
  intros l k Hl Hk Hlookup. destruct (decide (n = 0)) as [->|].
  { by rewrite (nil_length_inv l),
      (nil_length_inv k) by eauto using Nat.divide_0_l. }
  apply list_eq; intros i. specialize (Hlookup (i `div` n)).
  rewrite (Nat.mul_comm _ n) in Hlookup.
1182
  unfold sublist_lookup in *; simplify_option_eq;
Robbert Krebbers's avatar
Robbert Krebbers committed
1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205
    [|by rewrite !lookup_ge_None_2 by auto].
  apply (f_equal (!! i `mod` n)) in Hlookup.
  by rewrite !lookup_take, !lookup_drop, <-!Nat.div_mod in Hlookup
    by (auto using Nat.mod_upper_bound with lia).
Qed.
Lemma sublist_eq_same_length l k j n :
  length l = j * n → length k = j * n →
  (∀ i,i < j → sublist_lookup (i * n) n l = sublist_lookup (i * n) n k) → l = k.
Proof.
  intros Hl Hk ?. destruct (decide (n = 0)) as [->|].
  { by rewrite (nil_length_inv l), (nil_length_inv k) by lia. }
  apply sublist_eq with n; [by exists j|by exists j|].
  intros i. destruct (decide (i < j)); [by auto|].
  assert (∀ m, m = j * n → m < i * n + n).
  { intros ? ->. replace (i * n + n) with (S i * n) by lia.
    apply Nat.mul_lt_mono_pos_r; lia. }
  by rewrite !sublist_lookup_None by auto.
Qed.
Lemma sublist_lookup_reshape l i n m :
  0 < n → length l = m * n →
  reshape (replicate m n) l !! i = sublist_lookup (i * n) n l.
Proof.
  intros Hn Hl. unfold sublist_lookup.  apply option_eq; intros x; split.
1206
  - intros Hx. case_option_guard as Hi.
Robbert Krebbers's avatar
Robbert Krebbers committed
1207
    { f_equal. clear Hi. revert i l Hl Hx.
1208
      induction m as [|m IH]; intros [|i] l ??; simplify_eq/=; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
1209 1210 1211 1212
      rewrite <-drop_drop. apply IH; rewrite ?drop_length; auto with lia. }
    destruct Hi. rewrite Hl, <-Nat.mul_succ_l.
    apply Nat.mul_le_mono_r, Nat.le_succ_l. apply lookup_lt_Some in Hx.
    by rewrite reshape_length, replicate_length in Hx.
1213
  - intros Hx. case_option_guard as Hi; simplify_eq/=.
Robbert Krebbers's avatar
Robbert Krebbers committed
1214 1215 1216 1217 1218 1219 1220 1221
    revert i l Hl Hi. induction m as [|m IH]; [auto with lia|].
    intros [|i] l ??; simpl; [done|]. rewrite <-drop_drop.
    rewrite IH; rewrite ?drop_length; auto with lia.
Qed.
Lemma sublist_lookup_compose l1 l2 l3 i n j m :
  sublist_lookup i n l1 = Some l2 → sublist_lookup j m l2 = Some l3 →
  sublist_lookup (i + j) m l1 = Some l3.
Proof.
1222
  unfold sublist_lookup; intros; simplify_option_eq;
Robbert Krebbers's avatar
Robbert Krebbers committed
1223 1224 1225 1226 1227 1228 1229 1230 1231
    repeat match goal with
    | H : _ ≤ length _ |- _ => rewrite take_length, drop_length in H
    end; rewrite ?take_drop_commute, ?drop_drop, ?take_take,
      ?Min.min_l, Nat.add_assoc by lia; auto with lia.
Qed.
Lemma sublist_alter_length f l i n k :
  sublist_lookup i n l = Some k → length (f k) = n →
  length (sublist_alter f i n l) = length l.
Proof.
1232
  unfold sublist_alter, sublist_lookup. intros Hk ?; simplify_option_eq.
Robbert Krebbers's avatar
Robbert Krebbers committed
1233 1234 1235 1236 1237 1238 1239
  rewrite !app_length, Hk, !take_length, !drop_length; lia.
Qed.
Lemma sublist_lookup_alter f l i n k :
  sublist_lookup i n l = Some k → length (f k) = n →
  sublist_lookup i n (sublist_alter f i n l) = f <$> sublist_lookup i n l.
Proof.
  unfold sublist_lookup. intros Hk ?. erewrite sublist_alter_length by eauto.
1240
  unfold sublist_alter; simplify_option_eq.
Robbert Krebbers's avatar
Robbert Krebbers committed
1241 1242 1243 1244 1245 1246 1247
  by rewrite Hk, drop_app_alt, take_app_alt by (rewrite ?take_length; lia).
Qed.
Lemma sublist_lookup_alter_ne f l i j n k :
  sublist_lookup j n l = Some k → length (f k) = n → i + n ≤ j ∨ j + n ≤ i →
  sublist_lookup i n (sublist_alter f j n l) = sublist_lookup i n l.
Proof.
  unfold sublist_lookup. intros Hk Hi ?. erewrite sublist_alter_length by eauto.
1248
  unfold sublist_alter; simplify_option_eq; f_equal; rewrite Hk.
Robbert Krebbers's avatar
Robbert Krebbers committed
1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264
  apply list_eq; intros ii.
  destruct (decide (ii < length (f k))); [|by rewrite !lookup_take_ge by lia].
  rewrite !lookup_take, !lookup_drop by done. destruct (decide (i + ii < j)).
  { by rewrite lookup_app_l, lookup_take by (rewrite ?take_length; lia). }
  rewrite lookup_app_r by (rewrite take_length; lia).
  rewrite take_length_le, lookup_app_r, lookup_drop by lia. f_equal; lia.
Qed.
Lemma sublist_alter_all f l n : length l = n → sublist_alter f 0 n l = f l.
Proof.
  intros <-. unfold sublist_alter; simpl.
  by rewrite drop_all, (right_id_L [] (++)), take_ge.
Qed.
Lemma sublist_alter_compose f g l i n k :
  sublist_lookup i n l = Some k → length (f k) = n → length (g k) = n →
  sublist_alter (f ∘ g) i n l = sublist_alter f i n (sublist_alter g i n l).
Proof.
1265
  unfold sublist_alter, sublist_lookup. intros Hk ??; simplify_option_eq.
1266
  by rewrite !take_app_alt, drop_app_alt, !(assoc_L (++)), drop_app_alt,
Robbert Krebbers's avatar
Robbert Krebbers committed
1267 1268 1269 1270 1271 1272 1273
    take_app_alt by (rewrite ?app_length, ?take_length, ?Hk; lia).
Qed.

(** ** Properties of the [mask] function *)
Lemma mask_nil f βs : mask f βs (@nil A) = [].
Proof. by destruct βs. Qed.
Lemma mask_length f βs l : length (mask f βs l) = length l.
1274
Proof. revert βs. induction l; intros [|??]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1275
Lemma mask_true f l n : length l ≤ n → mask f (replicate n true) l = f <$> l.
1276
Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1277
Lemma mask_false f l n : mask f (replicate n false) l = l.
1278
Proof. revert l. induction n; intros [|??]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1279 1280 1281
Lemma mask_app f βs1 βs2 l :
  mask f (βs1 ++ βs2) l
  = mask f βs1 (take (length βs1) l) ++ mask f βs2 (drop (length βs1) l).
1282
Proof. revert l. induction βs1;intros [|??]; f_equal/=; auto using mask_nil. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1283 1284 1285
Lemma mask_app_2 f βs l1 l2 :
  mask f βs (l1 ++ l2)
  = mask f (take (length l1) βs) l1 ++ mask f (drop (length l1) βs) l2.
1286
Proof. revert βs. induction l1; intros [|??]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1287
Lemma take_mask f βs l n : take n (mask f βs l) = mask f (take n βs) (take n l).
1288
Proof. revert n βs. induction l; intros [|?] [|[] ?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1289 1290
Lemma drop_mask f βs l n : drop n (mask f βs l) = mask f (drop n βs) (drop n l).
Proof.
1291
  revert n βs. induction l; intros [|?] [|[] ?]; f_equal/=; auto using mask_nil.
Robbert Krebbers's avatar
Robbert Krebbers committed
1292 1293 1294 1295 1296
Qed.
Lemma sublist_lookup_mask f βs l i n :
  sublist_lookup i n (mask f βs l)
  = mask f (take n (drop i βs)) <$> sublist_lookup i n l.
Proof.
1297
  unfold sublist_lookup; rewrite mask_length; simplify_option_eq; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
1298 1299 1300 1301 1302 1303
  by rewrite drop_mask, take_mask.
Qed.
Lemma mask_mask f g βs1 βs2 l :
  (∀ x, f (g x) = f x) → βs1 =.>* βs2 →
  mask f βs2 (mask g βs1 l) = mask f βs2 l.
Proof.
1304
  intros ? Hβs. revert l. by induction Hβs as [|[] []]; intros [|??]; f_equal/=.
Robbert Krebbers's avatar
Robbert Krebbers committed
1305 1306 1307 1308
Qed.
Lemma lookup_mask f βs l i :
  βs !! i = Some true → mask f βs l !! i = f <$> l !! i.
Proof.
1309
  revert i βs. induction l; intros [] [] ?; simplify_eq/=; f_equal; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
1310 1311 1312 1313
Qed.
Lemma lookup_mask_notin f βs l i :
  βs !! i ≠ Some true → mask f βs l !! i = l !! i.
Proof.
1314
  revert i βs. induction l; intros [] [|[]] ?; simplify_eq/=; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
1315 1316 1317 1318
Qed.

(** ** Properties of the [seq] function *)
Lemma fmap_seq j n : S <$> seq j n = seq (S j) n.
1319
Proof. revert j. induction n; intros; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344
Lemma lookup_seq j n i : i < n → seq j n !! i = Some (j + i).
Proof.
  revert j i. induction n as [|n IH]; intros j [|i] ?; simpl; auto with lia.
  rewrite IH; auto with lia.
Qed.
Lemma lookup_seq_ge j n i : n ≤ i → seq j n !! i = None.
Proof. revert j i. induction n; intros j [|i] ?; simpl; auto with lia. Qed.
Lemma lookup_seq_inv j n i j' : seq j n !! i = Some j' → j' = j + i ∧ i < n.
Proof.
  destruct (le_lt_dec n i); [by rewrite lookup_seq_ge|].
  rewrite lookup_seq by done. intuition congruence.
Qed.

(** ** Properties of the [Permutation] predicate *)
Lemma Permutation_nil l : l ≡ₚ [] ↔ l = [].
Proof. split. by intro; apply Permutation_nil. by intros ->. Qed.
Lemma Permutation_singleton l x : l ≡ₚ [x] ↔ l = [x].
Proof. split. by intro; apply Permutation_length_1_inv. by intros ->. Qed.
Definition Permutation_skip := @perm_skip A.
Definition Permutation_swap := @perm_swap A.
Definition Permutation_singleton_inj := @Permutation_length_1 A.

Global Existing Instance Permutation_app'.
Global Instance: Proper ((≡ₚ) ==> (=)) (@length A).
Proof. induction 1; simpl; auto with lia. Qed.
1345
Global Instance: Comm (≡ₚ) (@app A).
Robbert Krebbers's avatar
Robbert Krebbers committed
1346 1347
Proof.
  intros l1. induction l1 as [|x l1 IH]; intros l2; simpl.
1348 1349
  - by rewrite (right_id_L [] (++)).
  - rewrite Permutation_middle, IH. simpl. by rewrite Permutation_middle.
Robbert Krebbers's avatar
Robbert Krebbers committed
1350
Qed.
1351
Global Instance: ∀ x : A, Inj (≡ₚ) (≡ₚ) (x ::).
Robbert Krebbers's avatar
Robbert Krebbers committed
1352
Proof. red. eauto using Permutation_cons_inv. Qed.
1353
Global Instance: ∀ k : list A, Inj (≡ₚ) (≡ₚ) (k ++).
Robbert Krebbers's avatar
Robbert Krebbers committed
1354 1355
Proof.
  red. induction k as [|x k IH]; intros l1 l2; simpl; auto.
1356
  intros. by apply IH, (inj (x ::)).
Robbert Krebbers's avatar
Robbert Krebbers committed
1357
Qed.
1358
Global Instance: ∀ k : list A, Inj (≡ₚ) (≡ₚ) (++ k).
Robbert Krebbers's avatar
Robbert Krebbers committed
1359
Proof.
1360
  intros k l1 l2. rewrite !(comm (++) _ k). by apply (inj (k ++)).
Robbert Krebbers's avatar
Robbert Krebbers committed
1361 1362 1363 1364
Qed.
Lemma replicate_Permutation n x l : replicate n x ≡ₚ l → replicate n x = l.
Proof.
  intros Hl. apply replicate_as_elem_of. split.
1365 1366
  - by rewrite <-Hl, replicate_length.
  - intros y. rewrite <-Hl. by apply elem_of_replicate_inv.
Robbert Krebbers's avatar
Robbert Krebbers committed
1367 1368 1369 1370
Qed.
Lemma reverse_Permutation l : reverse l ≡ₚ l.
Proof.
  induction l as [|x l IH]; [done|].
1371
  by rewrite reverse_cons, (comm (++)), IH.
Robbert Krebbers's avatar
Robbert Krebbers committed
1372
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1373 1374
Lemma delete_Permutation l i x : l !! i = Some x → l ≡ₚ x :: delete i l.
Proof.
1375
  revert i; induction l as [|y l IH]; intros [|i] ?; simplify_eq/=; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
1376 1377
  by rewrite Permutation_swap, <-(IH i).
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1378 1379 1380 1381 1382

(** ** Properties of the [prefix_of] and [suffix_of] predicates *)
Global Instance: PreOrder (@prefix_of A).
Proof.
  split.
1383 1384
  - intros ?. eexists []. by rewrite (right_id_L [] (++)).
  - intros ???[k1->] [k2->]. exists (k1 ++ k2). by rewrite (assoc_L (++)).
Robbert Krebbers's avatar
Robbert Krebbers committed
1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395
Qed.
Lemma prefix_of_nil l : [] `prefix_of` l.
Proof. by exists l. Qed.
Lemma prefix_of_nil_not x l : ¬x :: l `prefix_of` [].
Proof. by intros [k ?]. Qed.
Lemma prefix_of_cons x l1 l2 : l1 `prefix_of` l2 → x :: l1 `prefix_of` x :: l2.
Proof. intros [k ->]. by exists k. Qed.
Lemma prefix_of_cons_alt x y l1 l2 :
  x = y → l1 `prefix_of` l2 → x :: l1 `prefix_of` y :: l2.
Proof. intros ->. apply prefix_of_cons. Qed.
Lemma prefix_of_cons_inv_1 x y l1 l2 : x :: l1 `prefix_of` y :: l2 → x = y.
1396
Proof. by intros [k ?]; simplify_eq/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1397 1398
Lemma prefix_of_cons_inv_2 x y l1 l2 :
  x :: l1 `prefix_of` y :: l2 → l1 `prefix_of` l2.
1399
Proof. intros [k ?]; simplify_eq/=. by exists k. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1400
Lemma prefix_of_app k l1 l2 : l1 `prefix_of` l2 → k ++ l1 `prefix_of` k ++ l2.
1401
Proof. intros [k' ->]. exists k'. by rewrite (assoc_L (++)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1402 1403 1404 1405
Lemma prefix_of_app_alt k1 k2 l1 l2 :
  k1 = k2 → l1 `prefix_of` l2 → k1 ++ l1 `prefix_of` k2 ++ l2.
Proof. intros ->. apply prefix_of_app. Qed.
Lemma prefix_of_app_l l1 l2 l3 : l1 ++ l3 `prefix_of` l2 → l1 `prefix_of` l2.
1406
Proof. intros [k ->]. exists (l3 ++ k). by rewrite (assoc_L (++)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1407
Lemma prefix_of_app_r l1 l2 l3 : l1 `prefix_of` l2 → l1 `prefix_of` l2 ++ l3.
1408
Proof. intros [k ->]. exists (k ++ l3). by rewrite (assoc_L (++)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1409 1410 1411
Lemma prefix_of_length l1 l2 : l1 `prefix_of` l2 → length l1 ≤ length l2.
Proof. intros [? ->]. rewrite app_length. lia. Qed.
Lemma prefix_of_snoc_not l x : ¬l ++ [x] `prefix_of` l.
1412
Proof. intros [??]. discriminate_list. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1413 1414 1415
Global Instance: PreOrder (@suffix_of A).
Proof.
  split.
1416 1417
  - intros ?. by eexists [].
  - intros ???[k1->] [k2->]. exists (k2 ++ k1). by rewrite (assoc_L (++)).
Robbert Krebbers's avatar
Robbert Krebbers committed
1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440
Qed.
Global Instance prefix_of_dec `{∀ x y, Decision (x = y)} : ∀ l1 l2,
    Decision (l1 `prefix_of` l2) := fix go l1 l2 :=
  match l1, l2 return { l1 `prefix_of` l2 } + { ¬l1 `prefix_of` l2 } with
  | [], _ => left (prefix_of_nil _)
  | _, [] => right (prefix_of_nil_not _ _)
  | x :: l1, y :: l2 =>
    match decide_rel (=) x y with
    | left Hxy =>
      match go l1 l2 with
      | left Hl1l2 => left (prefix_of_cons_alt _ _ _ _ Hxy Hl1l2)
      | right Hl1l2 => right (Hl1l2 ∘ prefix_of_cons_inv_2 _ _ _ _)
      end
    | right Hxy => right (Hxy ∘ prefix_of_cons_inv_1 _ _ _ _)
    end
  end.

Section prefix_ops.
  Context `{∀ x y, Decision (x = y)}.
  Lemma max_prefix_of_fst l1 l2 :
    l1 = (max_prefix_of l1 l2).2 ++ (max_prefix_of l1 l2).1.1.
  Proof.
    revert l2. induction l1; intros [|??]; simpl;
1441
      repeat case_decide; f_equal/=; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
1442 1443 1444 1445 1446
  Qed.
  Lemma max_prefix_of_fst_alt l1 l2 k1 k2 k3 :
    max_prefix_of l1 l2 = (k1, k2, k3) → l1 = k3 ++ k1.
  Proof.
    intros. pose proof (max_prefix_of_fst l1 l2).
1447
    by destruct (max_prefix_of l1 l2) as [[]?]; simplify_eq.
Robbert Krebbers's avatar
Robbert Krebbers committed
1448 1449 1450 1451 1452 1453 1454 1455 1456 1457
  Qed.
  Lemma max_prefix_of_fst_prefix l1 l2 : (max_prefix_of l1 l2).2 `prefix_of` l1.
  Proof. eexists. apply max_prefix_of_fst. Qed.
  Lemma max_prefix_of_fst_prefix_alt l1 l2 k1 k2 k3 :
    max_prefix_of l1 l2 = (k1, k2, k3) → k3 `prefix_of` l1.
  Proof. eexists. eauto using max_prefix_of_fst_alt. Qed.
  Lemma max_prefix_of_snd l1 l2 :
    l2 = (max_prefix_of l1 l2).2 ++ (max_prefix_of l1 l2).1.2.
  Proof.
    revert l2. induction l1; intros [|??]; simpl;
1458
      repeat case_decide; f_equal/=; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
1459 1460 1461 1462 1463
  Qed.
  Lemma max_prefix_of_snd_alt l1 l2 k1 k2 k3 :
    max_prefix_of l1 l2 = (k1, k2, k3) → l2 = k3 ++ k2.
  Proof.
    intro. pose proof (max_prefix_of_snd l1 l2).
1464
    by destruct (max_prefix_of l1 l2) as [[]?]; simplify_eq.
Robbert Krebbers's avatar
Robbert Krebbers committed
1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481
  Qed.
  Lemma max_prefix_of_snd_prefix l1 l2 : (max_prefix_of l1 l2).2 `prefix_of` l2.
  Proof. eexists. apply max_prefix_of_snd. Qed.
  Lemma max_prefix_of_snd_prefix_alt l1 l2 k1 k2 k3 :
    max_prefix_of l1 l2 = (k1,k2,k3) → k3 `prefix_of` l2.
  Proof. eexists. eauto using max_prefix_of_snd_alt. Qed.
  Lemma max_prefix_of_max l1 l2 k :
    k `prefix_of` l1 → k `prefix_of` l2 → k `prefix_of` (max_prefix_of l1 l2).2.
  Proof.
    intros [l1' ->] [l2' ->]. by induction k; simpl; repeat case_decide;
      simpl; auto using prefix_of_nil, prefix_of_cons.
  Qed.
  Lemma max_prefix_of_max_alt l1 l2 k1 k2 k3 k :
    max_prefix_of l1 l2 = (k1,k2,k3) →
    k `prefix_of` l1 → k `prefix_of` l2 → k `prefix_of` k3.
  Proof.
    intro. pose proof (max_prefix_of_max l1 l2 k).
1482
    by destruct (max_prefix_of l1 l2) as [[]?]; simplify_eq.
Robbert Krebbers's avatar
Robbert Krebbers committed
1483 1484 1485 1486 1487 1488
  Qed.
  Lemma max_prefix_of_max_snoc l1 l2 k1 k2 k3 x1 x2 :
    max_prefix_of l1 l2 = (x1 :: k1, x2 :: k2, k3) → x1 ≠ x2.
  Proof.
    intros Hl ->. destruct (prefix_of_snoc_not k3 x2).
    eapply max_prefix_of_max_alt; eauto.
1489
    - rewrite (max_prefix_of_fst_alt _ _ _ _ _ Hl).
Robbert Krebbers's avatar
Robbert Krebbers committed
1490
      apply prefix_of_app, prefix_of_cons, prefix_of_nil.
1491
    - rewrite (max_prefix_of_snd_alt _ _ _ _ _ Hl).
Robbert Krebbers's avatar
Robbert Krebbers committed
1492 1493 1494 1495 1496 1497 1498 1499
      apply prefix_of_app, prefix_of_cons, prefix_of_nil.
  Qed.
End prefix_ops.

Lemma prefix_suffix_reverse l1 l2 :
  l1 `prefix_of` l2 ↔ reverse l1 `suffix_of` reverse l2.
Proof.
  split; intros [k E]; exists (reverse k).
1500 1501
  - by rewrite E, reverse_app.
  - by rewrite <-(reverse_involutive l2), E, reverse_app, reverse_involutive.
Robbert Krebbers's avatar
Robbert Krebbers committed
1502 1503 1504 1505 1506 1507 1508
Qed.
Lemma suffix_prefix_reverse l1 l2 :
  l1 `suffix_of` l2 ↔ reverse l1 `prefix_of` reverse l2.
Proof. by rewrite prefix_suffix_reverse, !reverse_involutive. Qed.
Lemma suffix_of_nil l : [] `suffix_of` l.
Proof. exists l. by rewrite (right_id_L [] (++)). Qed.
Lemma suffix_of_nil_inv l : l `suffix_of` [] → l = [].
1509
Proof. by intros [[|?] ?]; simplify_list_eq. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1510 1511 1512 1513
Lemma suffix_of_cons_nil_inv x l : ¬x :: l `suffix_of` [].
Proof. by intros [[] ?]. Qed.
Lemma suffix_of_snoc l1 l2 x :
  l1 `suffix_of` l2 → l1 ++ [x] `suffix_of` l2 ++ [x].
1514
Proof. intros [k ->]. exists k. by rewrite (assoc_L (++)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1515 1516 1517 1518
Lemma suffix_of_snoc_alt x y l1 l2 :
  x = y → l1 `suffix_of` l2 → l1 ++ [x] `suffix_of` l2 ++ [y].
Proof. intros ->. apply suffix_of_snoc. Qed.
Lemma suffix_of_app l1 l2 k : l1 `suffix_of` l2 → l1 ++ k `suffix_of` l2 ++ k.
1519
Proof. intros [k' ->]. exists k'. by rewrite (assoc_L (++)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1520 1521 1522 1523 1524
Lemma suffix_of_app_alt l1 l2 k1 k2 :
  k1 = k2 → l1 `suffix_of` l2 → l1 ++ k1 `suffix_of` l2 ++ k2.
Proof. intros ->. apply suffix_of_app. Qed.
Lemma suffix_of_snoc_inv_1 x y l1 l2 :
  l1 ++ [x] `suffix_of` l2 ++ [y] → x = y.
1525
Proof. intros [k' E]. rewrite (assoc_L (++)) in E. by simplify_list_eq. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1526 1527 1528
Lemma suffix_of_snoc_inv_2 x y l1 l2 :
  l1 ++ [x] `suffix_of` l2 ++ [y] → l1 `suffix_of` l2.
Proof.
1529
  intros [k' E]. exists k'. rewrite (assoc_L (++)) in E. by simplify_list_eq.
Robbert Krebbers's avatar
Robbert Krebbers committed
1530 1531 1532 1533
Qed.
Lemma suffix_of_app_inv l1 l2 k :
  l1 ++ k `suffix_of` l2 ++ k → l1 `suffix_of` l2.
Proof.