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(* 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. *)
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From Coq Require Export Permutation.
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From iris.prelude Require Export numbers base option.
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Arguments length {_} _.
Arguments cons {_} _ _.
Arguments app {_} _ _.
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Instance: Params (@length) 1.
Instance: Params (@cons) 1.
Instance: Params (@app) 1.
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Notation tail := tl.
Notation take := firstn.
Notation drop := skipn.

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

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Instance: Params (@tail) 1.
Instance: Params (@take) 1.
Instance: Params (@drop) 1.

Arguments Permutation {_} _ _.
Arguments Forall_cons {_} _ _ _ _ _.
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Remove Hints Permutation_cons : typeclass_instances.
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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.

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Instance maybe_cons {A} : Maybe2 (@cons A) := λ l,
  match l with x :: l => Some (x,l) | _ => None end.

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(** * Definitions *)
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(** 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.

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(** 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.
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Instance: Params (@list_inserts) 1.
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(** 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]) [].
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Instance: Params (@option_list) 1.
Instance maybe_list_singleton {A} : Maybe (λ x : A, [x]) := λ l,
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  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.
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Instance: Params (@list_find) 3.
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(** 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.
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Instance: Params (@replicate) 2.
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(** The function [reverse l] returns the elements of [l] in reverse order. *)
Definition reverse {A} (l : list A) : list A := rev_append l [].
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Instance: Params (@reverse) 1.
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(** 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.
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Instance: Params (@last) 1.
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(** 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 {_} !_ _ !_.
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Instance: Params (@resize) 2.
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(** 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.
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Instance: Params (@reshape) 2.
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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.
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Arguments imap : simpl never.

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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.
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Hint Extern 0 (_ `prefix_of` _) => reflexivity.
Hint Extern 0 (_ `suffix_of` _) => reflexivity.
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Section prefix_suffix_ops.
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  Context `{EqDecision A}.

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  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.
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Hint Extern 0 (_ `sublist` _) => reflexivity.
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(** 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.
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Hint Extern 0 (_ `contains` _) => reflexivity.
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Section contains_dec_help.
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  Context `{EqDecision A}.
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  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').

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(** Set operations on lists *)
Instance list_subseteq {A} : SubsetEq (list A) := λ l1 l2, ∀ x, x ∈ l1 → x ∈ l2.
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Section list_set.
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  Context `{dec : EqDecision A}.
  Global Instance elem_of_list_dec (x : A) : ∀ l, Decision (x ∈ l).
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  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 *)
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(** The tactic [discriminate_list] discharges a goal if it contains
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a list equality involving [(::)] and [(++)] of two lists that have a different
length as one of its hypotheses. *)
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Tactic Notation "discriminate_list" hyp(H) :=
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  apply (f_equal length) in H;
  repeat (csimpl in H || rewrite app_length in H); exfalso; lia.
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Tactic Notation "discriminate_list" :=
  match goal with H : @eq (list _) _ _ |- _ => discriminate_list H end.
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(** The tactic [simplify_list_eq] simplifies hypotheses involving
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equalities on lists using injectivity of [(::)] and [(++)]. Also, it simplifies
lookups in singleton lists. *)
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Lemma app_inj_1 {A} (l1 k1 l2 k2 : list A) :
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  length l1 = length k1 → l1 ++ l2 = k1 ++ k2 → l1 = k1 ∧ l2 = k2.
Proof. revert k1. induction l1; intros [|??]; naive_solver. Qed.
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Lemma app_inj_2 {A} (l1 k1 l2 k2 : list A) :
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  length l2 = length k2 → l1 ++ l2 = k1 ++ k2 → l1 = k1 ∧ l2 = k2.
Proof.
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  intros ? Hl. apply app_inj_1; auto.
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  apply (f_equal length) in Hl. rewrite !app_length in Hl. lia.
Qed.
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Ltac simplify_list_eq :=
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  repeat match goal with
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  | _ => progress simplify_eq/=
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  | H : _ ++ _ = _ ++ _ |- _ => first
    [ apply app_inv_head in H | apply app_inv_tail in H
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    | apply app_inj_1 in H; [destruct H|done]
    | apply app_inj_2 in H; [destruct H|done] ]
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  | 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.

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Global Instance: Inj2 (=) (=) (=) (@cons A).
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Proof. by injection 1. Qed.
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Global Instance: ∀ k, Inj (=) (=) (k ++).
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Proof. intros ???. apply app_inv_head. Qed.
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Global Instance: ∀ k, Inj (=) (=) (++ k).
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Proof. intros ???. apply app_inv_tail. Qed.
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Global Instance: Assoc (=) (@app A).
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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.
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  revert l2. induction l1 as [|x l1 IH]; intros [|y l2] H.
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  - done.
  - discriminate (H 0).
  - discriminate (H 0).
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  - f_equal; [by injection (H 0)|]. apply (IH _ $ λ i, H (S i)).
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Qed.
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Global Instance list_eq_dec {dec : EqDecision A} : EqDecision (list A) :=
  list_eq_dec dec.
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Global Instance list_eq_nil_dec l : Decision (l = []).
Proof. by refine match l with [] => left _ | _ => right _ end. Defined.
Lemma list_singleton_reflect l :
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  option_reflect (λ x, l = [x]) (length l ≠ 1) (maybe (λ x, [x]) l).
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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.
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Proof. revert i. induction l; intros [|?] ?; naive_solver auto with arith. Qed.
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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).
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Proof. revert i. induction l; intros [|?] ?; naive_solver eauto with lia. Qed.
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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.
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  - destruct (lookup_lt_is_Some_2 l1 i) as [y Hy].
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    { rewrite Hlen; eauto using lookup_lt_Some. }
    rewrite Hy; f_equal; apply (Hl i); eauto using lookup_lt_Some.
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  - by rewrite lookup_ge_None, Hlen, <-lookup_ge_None.
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Qed.
Lemma lookup_app_l l1 l2 i : i < length l1 → (l1 ++ l2) !! i = l1 !! i.
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Proof. revert i. induction l1; intros [|?]; naive_solver auto with lia. Qed.
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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.
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  - revert i. induction l1 as [|y l1 IH]; intros [|i] ?;
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      simplify_eq/=; auto with lia.
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    destruct (IH i) as [?|[??]]; auto with lia.
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  - intros [?|[??]]; auto using lookup_app_l_Some. by rewrite lookup_app_r.
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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.
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Proof. by revert i; induction l; intros []; intros; f_equal/=. Qed.
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Lemma alter_length f l i : length (alter f i l) = length l.
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Proof. revert i. by induction l; intros [|?]; f_equal/=. Qed.
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Lemma insert_length l i x : length (<[i:=x]>l) = length l.
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Proof. revert i. by induction l; intros [|?]; f_equal/=. Qed.
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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.
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Proof. revert i j. induction l; [done|]. intros [] []; naive_solver. Qed.
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Lemma list_lookup_insert l i x : i < length l → <[i:=x]>l !! i = Some x.
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Proof. revert i. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
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Lemma list_lookup_insert_ne l i j x : i ≠ j → <[i:=x]>l !! j = l !! j.
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Proof. revert i j. induction l; [done|]. intros [] []; naive_solver. Qed.
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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].
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  - intros Hy. assert (j < length l).
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    { rewrite <-(insert_length l j x); eauto using lookup_lt_Some. }
    rewrite list_lookup_insert in Hy by done; naive_solver.
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  - intros [(?&?&?)|[??]]; rewrite ?list_lookup_insert; naive_solver.
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Qed.
Lemma list_insert_commute l i j x y :
  i ≠ j → <[i:=x]>(<[j:=y]>l) = <[j:=y]>(<[i:=x]>l).
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Proof. revert i j. by induction l; intros [|?] [|?] ?; f_equal/=; auto. Qed.
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Lemma list_lookup_other l i x :
  length l ≠ 1 → l !! i = Some x → ∃ j y, j ≠ i ∧ l !! j = Some y.
Proof.
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  intros. destruct i, l as [|x0 [|x1 l]]; simplify_eq/=.
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  - by exists 1, x1.
  - by exists 0, x0.
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Qed.
Lemma alter_app_l f l1 l2 i :
  i < length l1 → alter f i (l1 ++ l2) = alter f i l1 ++ l2.
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Proof. revert i. induction l1; intros [|?] ?; f_equal/=; auto with lia. Qed.
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Lemma alter_app_r f l1 l2 i :
  alter f (length l1 + i) (l1 ++ l2) = l1 ++ alter f i l2.
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Proof. revert i. induction l1; intros [|?]; f_equal/=; auto. Qed.
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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.
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Proof. intros ?. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
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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.
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Proof. intros H ->. revert i H. induction k; intros [|?] ?; f_equal/=; auto. Qed.
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Lemma list_alter_compose f g l i :
  alter (f ∘ g) i l = alter f i (alter g i l).
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Proof. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
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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).
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Proof. revert i j. induction l; intros [|?][|?] ?; f_equal/=; auto with lia. Qed.
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Lemma insert_app_l l1 l2 i x :
  i < length l1 → <[i:=x]>(l1 ++ l2) = <[i:=x]>l1 ++ l2.
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Proof. revert i. induction l1; intros [|?] ?; f_equal/=; auto with lia. Qed.
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Lemma insert_app_r l1 l2 i x : <[length l1+i:=x]>(l1 ++ l2) = l1 ++ <[i:=x]>l2.
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Proof. revert i. induction l1; intros [|?]; f_equal/=; auto. Qed.
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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.
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Proof. induction l1; f_equal/=; auto. Qed.
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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.
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  - intros Hy. assert (j < length l).
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    { rewrite <-(inserts_length l i k); eauto using lookup_lt_Some. }
    rewrite list_lookup_inserts in Hy by lia. intuition lia.
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  - intuition. by rewrite list_lookup_inserts by lia.
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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.
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  - split; [by right|]. intros [Hx|]; [|done]. by destruct (elem_of_nil x).
  - simpl. rewrite !elem_of_cons, IHl1. tauto.
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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|].
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  by exists (y :: l1), l2.
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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.
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  revert i. induction l; intros [|i] ?; simplify_eq/=; constructor; eauto.
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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.
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  - induction l as [|x l]; csimpl; repeat case_match; inversion 1; subst;
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      setoid_rewrite elem_of_cons; naive_solver.
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  - intros (x&Hx&?). by induction Hx; csimpl; repeat case_match;
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      simplify_eq; try constructor; auto.
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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.
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  - rewrite NoDup_nil. setoid_rewrite elem_of_nil. naive_solver.
  - rewrite !NoDup_cons.
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    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 | |].
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  - by rewrite !NoDup_nil.
  - by rewrite !NoDup_cons, IH, Hlk.
  - rewrite !NoDup_cons, !elem_of_cons. intuition.
  - intuition.
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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].
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  { intros; simplify_eq. }
  intros [|i] [|j] ??; simplify_eq/=; eauto with f_equal;
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    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.
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  - rewrite elem_of_list_lookup. intros [i ?].
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    by feed pose proof (Hl (S i) 0 x); auto.
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  - apply IH. intros i j x' ??. by apply (inj S), (Hl (S i) (S j) x').
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Qed.

Section no_dup_dec.
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  Context `{!EqDecision A}.
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  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;
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      rewrite ?elem_of_cons; intuition (simplify_eq; auto).
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  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.
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  Context `{!EqDecision A}.
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  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.
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    - constructor.
    - done.
    - constructor. rewrite elem_of_list_difference; intuition. done.
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  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.
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    - by apply NoDup_list_difference.
    - intro. rewrite elem_of_list_difference. intuition.
    - done.
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  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.
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    - constructor.
    - constructor. rewrite elem_of_list_intersection; intuition. done.
    - done.
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  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.
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    - induction l as [|x1 l IH]; simpl; [by rewrite elem_of_nil|].
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      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.
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    - intros (x1&x2&Hx1&Hx2&Hx). induction Hx1 as [x1|x1 ? l ? IH]; simpl.
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      + 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.
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    revert i; induction l; intros [] ?; repeat first
      [ match goal with x : prod _ _ |- _ => destruct x end
      | simplify_option_eq ]; eauto.
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  Qed.
  Lemma list_find_elem_of l x : x ∈ l → P x → is_Some (list_find P l).
  Proof.
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    induction 1 as [|x y l ? IH]; intros; simplify_option_eq; eauto.
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    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.
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Global Instance: Inj (=) (=) (@reverse A).
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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.
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  revert i x. induction l; intros [|?] ??; simplify_eq/=; f_equal; auto.
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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.
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Proof. induction l; f_equal/=; auto. Qed.
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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.
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Proof. intros ->. by rewrite <-(assoc_L (++)), take_app. Qed.
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Lemma take_app_le l k n : n ≤ length l → take n (l ++ k) = take n l.
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Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
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Lemma take_plus_app l k n m :
  length l = n → take (n + m) (l ++ k) = l ++ take m k.
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Proof. intros <-. induction l; f_equal/=; auto. Qed.
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Lemma take_app_ge l k n :
  length l ≤ n → take n (l ++ k) = l ++ take (n - length l) k.
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Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
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Lemma take_ge l n : length l ≤ n → take n l = l.
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Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
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Lemma take_take l n m : take n (take m l) = take (min n m) l.
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Proof. revert n m. induction l; intros [|?] [|?]; f_equal/=; auto. Qed.
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Lemma take_idemp l n : take n (take n l) = take n l.
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Proof. by rewrite take_take, Min.min_idempotent. Qed.
Lemma take_length l n : length (take n l) = min n (length l).
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Proof. revert n. induction l; intros [|?]; f_equal/=; done. Qed.
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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).
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  - by rewrite !lookup_take_ge.
  - by rewrite !lookup_take, !list_lookup_alter_ne by lia.
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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).
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  - by rewrite !lookup_take_ge.
  - by rewrite !lookup_take, !list_lookup_insert_ne by lia.
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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.
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Proof. revert n. by induction l; intros [|i]; f_equal/=. Qed.
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Lemma drop_ge l n : length l ≤ n → drop n l = [].
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Proof. revert n. induction l; intros [|?]; simpl in *; auto with lia. Qed.
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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.
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Proof. intros ->. by rewrite <-(assoc_L (++)), drop_app. Qed.
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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.
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Proof. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
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Lemma take_take_drop l n m : take n l ++ take m (drop n l) = take (n + m) l.
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Proof. revert n m. induction l; intros [|?] [|?]; f_equal/=; auto. Qed.
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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 [|?] [|?] ?;
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    f_equal/=; auto using take_drop with lia.
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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.
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  - revert i. induction n; intros [|?]; naive_solver auto with lia.
  - intros [-> Hi]. revert i Hi.
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    induction n; intros [|?]; naive_solver auto with lia.
Qed.
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Lemma elem_of_replicate n x y : y ∈ replicate n x ↔ y = x ∧ n ≠ 0.
Proof.
  rewrite elem_of_list_lookup, Nat.neq_0_lt_0.
  setoid_rewrite lookup_replicate; naive_solver eauto with lia.
Qed.
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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.
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  - intros Hin [x' Hx']; destruct Hin. rewrite lookup_replicate in Hx'; tauto.
  - intros Hx ?. destruct Hx. exists x; auto using lookup_replicate_2.
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Qed.
Lemma insert_replicate x n i : <[i:=x]>(replicate n x) = replicate n x.
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Proof. revert i. induction n; intros [|?]; f_equal/=; auto. Qed.
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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.
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Proof. induction n; f_equal/=; auto. Qed.
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Lemma take_replicate n m x : take n (replicate m x) = replicate (min n m) x.
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Proof. revert m. by induction n; intros [|?]; f_equal/=. Qed.
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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.
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Proof. revert m. by induction n; intros [|?]; f_equal/=. Qed.
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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|].
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  intros [<- Hl]. symmetry. induction l as [|y l IH]; f_equal/=.
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  - apply Hl. by left.
  - apply IH. intros ??. apply Hl. by right.
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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.
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Proof. revert n. induction l; intros [|?]; f_equal/=; auto. Qed.
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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.
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Proof. rewrite resize_spec. rewrite take_nil. f_equal/=. lia. Qed.
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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.
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  revert n m. induction l; intros [|?] [|?]; f_equal/=; auto.
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  - by rewrite Nat.add_0_r, (right_id_L [] (++)).
  - by rewrite replicate_plus.
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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.
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  intros. rewrite !resize_spec, take_app_ge, (assoc_L (++)) by done.
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  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.
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Proof. revert m. induction n; intros [|?]; f_equal/=; auto. Qed.
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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.
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  - intros. by rewrite !resize_nil, resize_replicate.
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  - intros [|?] [|?] ?; f_equal/=; auto with lia.
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Qed.
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Lemma resize_idemp l n x : resize n x (resize n x l) = resize n x l.
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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.
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Proof. revert n m. induction l; intros [|?][|?] ?; f_equal/=; auto with lia. Qed.
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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.
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  revert n m. induction l; intros [|?][|?]; f_equal/=; auto using take_replicate.
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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.
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  - intros. by rewrite drop_nil, !resize_nil, drop_replicate.
  - intros [|?] [|?] ?; simpl; try case_match; auto with lia.
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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)).
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  - by rewrite resize_le, lookup_take by lia.
  - by rewrite resize_ge, lookup_app_l by lia.
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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.

(** ** Properties of the [reshape] function *)
Lemma reshape_length szs l : length (reshape szs l) = length szs.
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Proof. revert l. by induction szs; intros; f_equal/=. Qed.
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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.
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End general_properties.

Section more_general_properties.
Context {A : Type}.
Implicit Types x y z : A.
Implicit Types l k : list A.
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(** ** 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.
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  unfold sublist_lookup; intros; simplify_option_eq.
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  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)).
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Proof. by unfold sublist_lookup; intros; simplify_option_eq. Qed.
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Lemma sublist_lookup_None l i n :
  length l < i + n → sublist_lookup i n l = None.
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Proof. by unfold sublist_lookup; intros; simplify_option_eq by lia. Qed.
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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.
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  unfold sublist_lookup in *; simplify_option_eq;
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    [|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.
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  - intros Hx. case_option_guard as Hi.
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    { f_equal. clear Hi. revert i l Hl Hx.
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      induction m as [|m IH]; intros [|i] l ??; simplify_eq/=; auto.
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      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.
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  - intros Hx. case_option_guard as Hi; simplify_eq/=.
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    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.
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  unfold sublist_lookup; intros; simplify_option_eq;
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    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.
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  unfold sublist_alter, sublist_lookup. intros Hk ?; simplify_option_eq.
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  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.
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  unfold sublist_alter; simplify_option_eq.
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  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.
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  unfold sublist_alter; simplify_option_eq; f_equal; rewrite Hk.
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  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.
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  unfold sublist_alter, sublist_lookup. intros Hk ??; simplify_option_eq.
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  by rewrite !take_app_alt, drop_app_alt, !(assoc_L (++)), drop_app_alt,
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    take_app_alt by (rewrite ?app_length, ?take_length, ?Hk; lia).
Qed.

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(** ** Properties of the [imap] function *)
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Lemma imap_nil {B} (f : nat → A → B) : imap f [] = [].
Proof. done. Qed.
Lemma imap_app {B} (f : nat → A → B) l1 l2 :
  imap f (l1 ++ l2) = imap f l1 ++ imap (λ n, f (length l1 + n)) l2.
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Proof.
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  unfold imap. generalize 0. revert l2.
  induction l1 as [|x l1 IH]; intros l2 n; f_equal/=; auto.
  rewrite IH. f_equal. clear. revert n.
  induction l2; simpl; auto with f_equal lia.
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Qed.
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Lemma imap_cons {B} (f : nat → A → B) x l :
  imap f (x :: l) = f 0 x :: imap (f ∘ S) l.
Proof. apply (imap_app _ [_]). Qed.

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Lemma imap_ext {B} (f g : nat → A → B) l :
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  (∀ i x, l !! i = Some x → f i x = g i x) → imap f l = imap g l.
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Proof.
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  revert f g; induction l as [|x l IH]; intros f g Hfg; auto.
  rewrite !imap_cons; f_equal; eauto.
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Qed.

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Lemma imap_fmap {B C} (f : nat → B → C) (g : A → B) l :
  imap f (g <$> l) = imap (λ n, f n ∘ g) l.
Proof. unfold imap. generalize 0. induction l; csimpl; auto with f_equal. Qed.

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(** ** 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.
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Proof. revert βs. induction l; intros [|??]; f_equal/=; auto. Qed.
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Lemma mask_true f l n : length l ≤ n → mask f (replicate n true) l = f <$> l.
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Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
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Lemma mask_false f l n : mask f (replicate n false) l = l.
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Proof. revert l. induction n; intros [|??]; f_equal/=; auto. Qed.
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Lemma mask_app f βs1 βs2 l :
  mask f (βs1 ++ βs2) l
  = mask f βs1 (take (length βs1) l) ++ mask f βs2 (drop (length βs1) l).
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Proof. revert l. induction βs1;intros [|??]; f_equal/=; auto using mask_nil. Qed.
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Lemma mask_app_2 f βs l1 l2 :
  mask f βs (l1 ++ l2)
  = mask f (take (length l1) βs) l1 ++ mask f (drop (length l1) βs) l2.
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Proof. revert βs. induction l1; intros [|??]; f_equal/=; auto. Qed.
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Lemma take_mask f βs l n : take n (mask f βs l) = mask f (take n βs) (take n l).
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Proof. revert n βs. induction l; intros [|?] [|[] ?]; f_equal/=; auto. Qed.
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Lemma drop_mask f βs l n : drop n (mask f βs l) = mask f (drop n βs) (drop n l).
Proof.
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  revert n βs. induction l; intros [|?] [|[] ?]; f_equal/=; auto using mask_nil.
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Qed.
Lemma sublist_lookup_mask f βs l i n :
  sublist_lookup i n (mask f βs l)
  = mask f (take n (drop i βs)) <$> sublist_lookup i n l.
Proof.
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  unfold sublist_lookup; rewrite mask_length; simplify_option_eq; auto.
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  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.
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  intros ? Hβs. revert l. by induction Hβs as [|[] []]; intros [|??]; f_equal/=.
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Qed.
Lemma lookup_mask f βs l i :
  βs !! i = Some true → mask f βs l !! i = f <$> l !! i.
Proof.
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  revert i βs. induction l; intros [] [] ?; simplify_eq/=; f_equal; auto.
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Qed.
Lemma lookup_mask_notin f βs l i :
  βs !! i ≠ Some true → mask f βs l !! i = l !! i.
Proof.
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  revert i βs. induction l; intros [] [|[]] ?; simplify_eq/=; auto.
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Qed.

(** ** Properties of the [seq] function *)
Lemma fmap_seq j n : S <$> seq j n = seq (S j) n.
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Proof. revert j. induction n; intros; f_equal/=; auto. Qed.
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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.

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Global Instance Permutation_cons : Proper ((≡ₚ) ==> (≡ₚ)) (@cons A x).
Proof. by constructor. Qed.
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Global Existing Instance Permutation_app'.
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Global Instance: Proper ((≡ₚ) ==> (=)) (@length A).
Proof. induction 1; simpl; auto with lia. Qed.
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Global Instance: Comm (≡ₚ) (@app A).
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Proof.
  intros l1. induction l1 as [|x l1 IH]; intros l2; simpl.
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  - by rewrite (right_id_L [] (++)).
  - rewrite Permutation_middle, IH. simpl. by rewrite Permutation_middle.
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Qed.
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Global Instance: ∀ x : A, Inj (≡ₚ) (≡ₚ) (x ::).
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Proof. red. eauto using Permutation_cons_inv. Qed.
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Global Instance: ∀ k : list A, Inj (≡ₚ) (≡ₚ) (k ++).
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Proof.
  red. induction k as [|x k IH]; intros l1 l2; simpl; auto.
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  intros. by apply IH, (inj (x ::)).
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Qed.
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Global Instance: ∀ k : list A, Inj (≡ₚ) (≡ₚ) (++ k).
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Proof. intros k l1 l2. rewrite !(comm (++) _ k). by apply (inj (k ++)). Qed.
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Lemma replicate_Permutation n x l : replicate n x ≡ₚ l → replicate n x = l.
Proof.
  intros Hl. apply replicate_as_elem_of. split.
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  - by rewrite <-Hl, replicate_length.
  - intros y. rewrite <-Hl. by apply elem_of_replicate_inv.
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Qed.
Lemma reverse_Permutation l : reverse l ≡ₚ l.
Proof.
  induction l as [|x l IH]; [done|].
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  by rewrite reverse_cons, (comm (++)), IH.
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Qed.
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Lemma delete_Permutation l i x : l !! i = Some x → l ≡ₚ x :: delete i l.
Proof.
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  revert i; induction l as [|y l IH]; intros [|i] ?; simplify_eq/=; auto.
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  by rewrite Permutation_swap, <-(IH i).
Qed.
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(** ** Properties of the [prefix_of] and [suffix_of] predicates *)
Global Instance: PreOrder (@prefix_of A).
Proof.
  split.
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  - intros ?. eexists []. by rewrite (right_id_L [] (++)).
  - intros ???[k1->] [k2->]. exists (k1 ++ k2). by rewrite (assoc_L (++)).
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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.
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Proof. by intros [k ?]; simplify_eq/=. Qed.
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Lemma prefix_of_cons_inv_2 x y l1 l2 :
  x :: l1 `prefix_of` y :: l2 → l1 `prefix_of` l2.
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Proof. intros [k ?]; simplify_eq/=. by exists k. Qed.
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