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(* Copyright (c) 2012-2013, Robbert Krebbers. *)
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(* 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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Require Import Permutation.
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Require Export numbers base decidable option.
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Arguments length {_} _.
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Arguments cons {_} _ _.
Arguments app {_} _ _.
Arguments Permutation {_} _ _.
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Arguments Forall_cons {_} _ _ _ _ _.
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Notation Forall_nil_2 := Forall_nil.
Notation Forall_cons_2 := Forall_cons.

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Notation tail := tl.
Notation take := firstn.
Notation drop := skipn.
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Notation take_drop := firstn_skipn.
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Arguments take {_} !_ !_ /.
Arguments drop {_} !_ !_ /.

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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.

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(** * Definitions *)
(** The operation [l !! i] gives the [i]th element of the list [l], or [None]
in case [i] is out of bounds. *)
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Instance list_lookup {A} : Lookup nat A (list A) :=
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  fix go (i : nat) (l : list A) {struct l} : option A :=
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  match l with
  | [] => None
  | x :: l =>
    match i with
    | 0 => Some x
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    | S i => @lookup _ _ _ go i l
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    end
  end.
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(** 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. *)
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Instance list_alter {A} (f : A  A) : AlterD nat A (list A) f :=
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  fix go (i : nat) (l : list A) {struct l} :=
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  match l with
  | [] => []
  | x :: l =>
    match i with
    | 0 => f x :: l
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    | S i => x :: @alter _ _ _ f go i l
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    end
  end.
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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. *)
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Instance list_delete {A} : Delete nat (list A) :=
  fix go (i : nat) (l : list A) {struct l} : list A :=
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  match l with
  | [] => []
  | x :: l =>
    match i with
    | 0 => l
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    | S i => x :: @delete _ _ go i l
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    end
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  end.
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(** 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. *)
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Instance list_insert {A} : Insert nat A (list A) := λ i x,
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  alter (λ _, x) i.
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(** The function [option_list o] converts an element [Some x] into the
singleton list [[x]], and [None] into the empty list [[]]. *)
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Definition option_list {A} : option A  list A := option_rect _ (λ x, [x]) [].

(** 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 :=
  match l with
  | [] => []
  | x :: l =>
     if decide (P x)
     then x :: @filter _ _ (@go) _ _ l
     else @filter _ _ (@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 [].

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Fixpoint last' {A} (x : A) (l : list A) : A :=
  match l with
  | [] => x
  | x :: l => last' x l
  end.
Definition last {A} (l : list A) : option A :=
  match l with
  | [] => None
  | x :: l => Some (last' x l)
  end.

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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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(** 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 {A B} (f : A  B) : FMapD list f :=
  fix go (l : list A) :=
  match l with
  | [] => []
  | x :: l => f x :: @fmap _ _ _ f go l
  end.
Instance list_bind {A B} (f : A  list B) : MBindD list f :=
  fix go (l : list A) :=
  match l with
  | [] => []
  | x :: l => f x ++ @mbind _ _ _ f 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.

(** 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 ifoldr {A B} (f : nat  B  A  A)
    (a : nat  A) : nat  list B  A :=
  fix go (n : nat) (l : list B) : A :=
  match l with
  | nil => a n
  | b :: l => f n b (go (S n) l)
  end.

(** Zipping lists. *)
Definition zip_with {A B C} (f : A  B  C) : list A  list B  list C :=
  fix go l1 l2 :=
  match l1, l2 with
  | x1 :: l1, x2 :: l2 => f x1 x2 :: go l1 l2
  | _ , _ => []
  end.
Notation zip := (zip_with pair).

(** 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.

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(** 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. *)
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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.
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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 snd_map (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) := snd $ fst $ max_prefix_of l1 l2.
  Definition strip_suffix (l1 l2 : list A) := snd $ fst $ max_suffix_of l1 l2.
End prefix_suffix_ops.
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(** A list [l1] is a sub list 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_cons x l1 l2 : sublist l1 l2  sublist (x :: l1) (x :: l2)
  | sublist_cons_skip x l1 l2 : sublist l1 l2  sublist l1 (x :: l2).

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(** The [same_length] view allows convenient induction over two lists with the
same length. *)
Inductive same_length {A B} : list A  list B  Prop :=
  | same_length_nil : same_length [] []
  | same_length_cons x y l k :
     same_length l k  same_length (x :: l) (y :: k).

(** * Basic tactics on lists *)
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(** The tactic [discriminate_list_equality] discharges a goal if it contains
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_equality" hyp(H) :=
  apply (f_equal length) in H;
  repeat (simpl in H || rewrite app_length in H);
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  exfalso; lia.
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Tactic Notation "discriminate_list_equality" :=
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  solve [repeat_on_hyps (fun H => discriminate_list_equality H)].
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(** The tactic [simplify_list_equality] simplifies hypotheses involving
equalities on lists using injectivity of [(::)] and [(++)]. Also, it simplifies
lookups in singleton lists. *)
Lemma cons_inv {A} (l1 l2 : list A) x1 x2 :
  x1 :: l1 = x2 :: l2  x1 = x2  l1 = l2.
Proof. by injection 1. Qed.

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Ltac simplify_list_equality := repeat
  match goal with
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  | H : _ :: _ = _ :: _ |- _ =>
     apply cons_inv in H; destruct H
     (* to circumvent bug #2939 in some situations *)
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  | H : _ ++ _ = _ ++ _ |- _ => first
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     [ apply app_inj_tail in H; destruct H
     | apply app_inv_head in H
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     | apply app_inv_tail in H ]
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  | H : [?x] !! ?i = Some ?y |- _ =>
     destruct i; [change (Some x = Some y) in H|discriminate]
  | _ => progress simplify_equality
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  | H : _ |- _ => discriminate_list_equality H
  end.
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(** * General theorems *)
Section general_properties.
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Context {A : Type}.

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Global Instance:  x : A, Injective (=) (=) (x ::).
Proof. by injection 1. Qed.
Global Instance:  l : list A, Injective (=) (=) (:: l).
Proof. by injection 1. Qed.
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Global Instance:  k : list A, Injective (=) (=) (k ++).
Proof. intros ???. apply app_inv_head. Qed.
Global Instance:  k : list A, Injective (=) (=) (++ k).
Proof. intros ???. apply app_inv_tail. Qed.
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Global Instance: Associative (=) (@app A).
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.
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Lemma app_inj (l1 k1 l2 k2 : list A) :
  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 list_eq (l1 l2 : list A) : ( i, l1 !! i = l2 !! i)%C  l1 = l2.
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Proof.
  revert l2. induction l1; intros [|??] H.
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  * done.
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  * discriminate (H 0).
  * discriminate (H 0).
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  * f_equal; [by injection (H 0) |].
    apply IHl1. intro. apply (H (S _)).
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Qed.
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Lemma list_eq_nil (l : list A) : ( i, l !! i = None)  l = nil.
Proof. intros. by apply list_eq. Qed.
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Global Instance list_eq_dec {dec :  x y : A, Decision (x = y)} :  l k,
  Decision (l = k) := list_eq_dec dec.
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Definition list_singleton_dec (l : list A) :
  { x | l = [x] } + { length l  1 }.
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Proof.
 by refine (
  match l with
  | [x] => inleft (x  _)
  | _ => inright _
  end).
Defined.
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Global Instance: Proper (Permutation ==> (=)) (@length A).
Proof. induction 1; simpl; auto with lia. Qed.

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Lemma nil_or_length_pos (l : list A) : l = []  length l  0.
Proof. destruct l; simpl; auto with lia. Qed.
Lemma nil_length (l : list A) : length l = 0  l = [].
Proof. by destruct l. Qed.
Lemma lookup_nil i : @nil A !! i = None.
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Proof. by destruct i. Qed.
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Lemma lookup_tail (l : list A) i : tail l !! i = l !! S i.
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Proof. by destruct l. Qed.
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Lemma lookup_lt_length (l : list A) i :
  is_Some (l !! i)  i < length l.
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Proof.
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  revert i. induction l.
  * split; by inversion 1.
  * intros [|?]; simpl.
    + split; eauto with arith.
    + by rewrite <-NPeano.Nat.succ_lt_mono.
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Qed.
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Lemma lookup_lt_length_1 (l : list A) i :
  is_Some (l !! i)  i < length l.
Proof. apply lookup_lt_length. Qed.
Lemma lookup_lt_length_alt (l : list A) i x :
  l !! i = Some x  i < length l.
Proof. intros Hl. by rewrite <-lookup_lt_length, Hl. Qed.
Lemma lookup_lt_length_2 (l : list A) i :
  i < length l  is_Some (l !! i).
Proof. apply lookup_lt_length. Qed.

Lemma lookup_ge_length (l : list A) i :
  l !! i = None  length l  i.
Proof. rewrite eq_None_not_Some, lookup_lt_length. lia. Qed.
Lemma lookup_ge_length_1 (l : list A) i :
  l !! i = None  length l  i.
Proof. by rewrite lookup_ge_length. Qed.
Lemma lookup_ge_length_2 (l : list A) i :
  length l  i  l !! i = None.
Proof. by rewrite lookup_ge_length. Qed.

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Lemma list_eq_length_eq (l1 l2 : list A) :
  length l2 = length l1 
  ( i x y, l1 !! i = Some x  l2 !! i = Some y  x = y) 
  l1 = l2.
Proof.
  intros Hlength Hlookup. apply list_eq. intros i.
  destruct (l2 !! i) as [x|] eqn:E.
  * feed inversion (lookup_lt_length_2 l1 i) as [y].
    { pose proof (lookup_lt_length_alt l2 i x E). lia. }
    f_equal. eauto.
  * rewrite lookup_ge_length in E |- *. lia.
Qed.

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Lemma lookup_app_l (l1 l2 : list A) 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 : list A) i x :
  l1 !! i = Some x 
  (l1 ++ l2) !! i = Some x.
Proof. intros. rewrite lookup_app_l; eauto using lookup_lt_length_alt. Qed.

Lemma lookup_app_r (l1 l2 : list A) i :
  (l1 ++ l2) !! (length l1 + i) = l2 !! i.
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Proof.
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  revert i.
  induction l1; intros [|i]; simpl in *; simplify_equality; auto.
Qed.
Lemma lookup_app_r_alt (l1 l2 : list A) i :
  length l1  i 
  (l1 ++ l2) !! i = l2 !! (i - length l1).
Proof.
  intros. assert (i = length l1 + (i - length l1)) as Hi by lia.
  rewrite Hi at 1. by apply lookup_app_r.
Qed.
Lemma lookup_app_r_Some (l1 l2 : list A) i x :
  l2 !! i = Some x 
  (l1 ++ l2) !! (length l1 + i) = Some x.
Proof. by rewrite lookup_app_r. Qed.
Lemma lookup_app_r_Some_alt (l1 l2 : list A) i x :
  length l1  i 
  l2 !! (i - length l1) = Some x 
  (l1 ++ l2) !! i = Some x.
Proof. intro. by rewrite lookup_app_r_alt. Qed.

Lemma lookup_app_inv (l1 l2 : list A) i x :
  (l1 ++ l2) !! i = Some x 
  l1 !! i = Some x  l2 !! (i - length l1) = Some x.
Proof.
  revert i.
  induction l1; intros [|i] ?; simpl in *; simplify_equality; auto.
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Qed.

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Lemma list_lookup_middle (l1 l2 : list A) (x : A) :
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  (l1 ++ x :: l2) !! length l1 = Some x.
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Proof. by induction l1; simpl. Qed.
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Lemma alter_length (f : A  A) l i :
  length (alter f i l) = length l.
Proof. revert i. induction l; intros [|?]; simpl; auto with lia. Qed.
Lemma insert_length (l : list A) i x :
  length (<[i:=x]>l) = length l.
Proof. apply alter_length. Qed.

Lemma list_lookup_alter (f : A  A) l i :
  alter f i l !! i = f <$> l !! i.
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Proof. revert i. induction l. done. intros [|i]. done. apply (IHl i). Qed.
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Lemma list_lookup_alter_ne (f : A  A) l i j :
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  i  j  alter f i l !! j = l !! j.
Proof.
  revert i j. induction l; [done|].
  intros [|i] [|j] ?; try done. apply (IHl i). congruence.
Qed.
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Lemma list_lookup_insert (l : list A) i x :
  i < length l 
  <[i:=x]>l !! i = Some x.
Proof.
  intros Hi. unfold insert, list_insert.
  rewrite list_lookup_alter.
  by feed inversion (lookup_lt_length_2 l i).
Qed.
Lemma list_lookup_insert_ne (l : list A) i j x :
  i  j  <[i:=x]>l !! j = l !! j.
Proof. apply list_lookup_alter_ne. Qed.

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Lemma list_lookup_other (l : list A) i x :
  length l  1 
  l !! i = Some x 
   j y, j  i  l !! j = Some y.
Proof.
  intros Hl Hi.
  destruct i; destruct l as [|x0 [|x1 l]]; simpl in *; simplify_equality.
  * by exists 1 x1.
  * by exists 0 x0.
Qed.

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Lemma alter_app_l (f : A  A) (l1 l2 : list A) i :
  i < length l1 
  alter f i (l1 ++ l2) = alter f i l1 ++ l2.
Proof.
  revert i.
  induction l1; intros [|?] ?; simpl in *; f_equal; auto with lia.
Qed.
Lemma alter_app_r (f : A  A) (l1 l2 : list A) i :
  alter f (length l1 + i) (l1 ++ l2) = l1 ++ alter f i l2.
Proof.
  revert i.
  induction l1; intros [|?]; simpl in *; f_equal; auto.
Qed.
Lemma alter_app_r_alt (f : A  A) (l1 l2 : list A) 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.
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Lemma insert_app_l (l1 l2 : list A) i x :
  i < length l1 
  <[i:=x]>(l1 ++ l2) = <[i:=x]>l1 ++ l2.
Proof. apply alter_app_l. Qed.
Lemma insert_app_r (l1 l2 : list A) i x :
  <[length l1 + i:=x]>(l1 ++ l2) = l1 ++ <[i:=x]>l2.
Proof. apply alter_app_r. Qed.
Lemma insert_app_r_alt (l1 l2 : list A) i x :
  length l1  i 
  <[i:=x]>(l1 ++ l2) = l1 ++ <[i - length l1:=x]>l2.
Proof. apply alter_app_r_alt. Qed.

Lemma insert_consecutive_length (l : list A) i k :
  length (insert_consecutive i k l) = length l.
Proof. revert i. by induction k; intros; simpl; rewrite ?insert_length. Qed.
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Lemma delete_middle (l1 l2 : list A) x :
  delete (length l1) (l1 ++ x :: l2) = l1 ++ l2.
Proof. induction l1; simpl; f_equal; auto. Qed.

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(** ** Properties of the [elem_of] predicate *)
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Lemma not_elem_of_nil (x : A) : x  [].
Proof. by inversion 1. Qed.
Lemma elem_of_nil (x : A) : x  []  False.
Proof. intuition. by destruct (not_elem_of_nil x). Qed.
Lemma elem_of_nil_inv (l : list A) : ( x, x  l)  l = [].
Proof. destruct l. done. by edestruct 1; constructor. Qed.
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Lemma elem_of_cons (l : list A) x y :
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  x  y :: l  x = y  x  l.
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Proof.
  split.
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  * inversion 1; subst. by left. by right.
  * intros [?|?]; subst. by left. by right.
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Qed.
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Lemma not_elem_of_cons (l : list A) x y :
  x  y :: l  x  y  x  l.
Proof. rewrite elem_of_cons. tauto. Qed.
Lemma elem_of_app (l1 l2 : list A) x :
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  x  l1 ++ l2  x  l1  x  l2.
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Proof.
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  induction l1.
  * split; [by right|]. intros [Hx|]; [|done].
    by destruct (elem_of_nil x).
  * simpl. rewrite !elem_of_cons, IHl1. tauto.
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Qed.
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Lemma not_elem_of_app (l1 l2 : list A) x :
  x  l1 ++ l2  x  l1  x  l2.
Proof. rewrite elem_of_app. tauto. Qed.

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Lemma elem_of_list_singleton (x y : A) : x  [y]  x = y.
Proof. rewrite elem_of_cons, elem_of_nil. tauto. Qed.
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Global Instance elem_of_list_permutation_proper (x : A) :
  Proper (Permutation ==> iff) (x ).
Proof. induction 1; rewrite ?elem_of_nil, ?elem_of_cons; intuition. Qed.
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Lemma elem_of_list_split (l : list A) x :
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  x  l   l1 l2, l = l1 ++ x :: l2.
Proof.
  induction 1 as [x l|x y l ? [l1 [l2 ?]]].
  * by eexists [], l.
  * subst. by exists (y :: l1) l2.
Qed.
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Global Instance elem_of_list_dec {dec :  x y : A, Decision (x = y)} :
   (x : A) l, Decision (x  l).
Proof.
 intros x. refine (
  fix go l :=
  match l return Decision (x  l) with
  | [] => right (not_elem_of_nil _)
  | y :: l => cast_if_or (decide_rel (=) x y) (go l)
  end); clear go dec; subst; try (by constructor); by inversion 1.
Defined.

Lemma elem_of_list_lookup_1 (l : list A) 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 : list A) i x :
  l !! i = Some x  x  l.
Proof.
  revert i. induction l; intros [|i] ?;
    simpl; simplify_equality; constructor; eauto.
Qed.
Lemma elem_of_list_lookup (l : list A) x :
  x  l   i, l !! i = Some x.
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Proof.
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  firstorder eauto using
    elem_of_list_lookup_1, elem_of_list_lookup_2.
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Qed.
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(** ** Properties of the [NoDup] predicate *)
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Lemma NoDup_nil : NoDup (@nil A)  True.
Proof. split; constructor. Qed.
Lemma NoDup_cons (x : A) l : NoDup (x :: l)  x  l  NoDup l.
Proof. split. by inversion 1. intros [??]. by constructor. Qed.
Lemma NoDup_cons_11 (x : A) l : NoDup (x :: l)  x  l.
Proof. rewrite NoDup_cons. by intros [??]. Qed.
Lemma NoDup_cons_12 (x : A) l : NoDup (x :: l)  NoDup l.
Proof. rewrite NoDup_cons. by intros [??]. Qed.
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Lemma NoDup_singleton (x : A) : NoDup [x].
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Proof. constructor. apply not_elem_of_nil. constructor. Qed.

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Lemma NoDup_app (l k : list A) :
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  NoDup (l ++ k)  NoDup l  ( x, x  l  x  k)  NoDup k.
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Proof.
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  induction l; simpl.
  * 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.
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Qed.

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Global Instance NoDup_proper:
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  Proper (Permutation ==> iff) (@NoDup A).
Proof.
  induction 1 as [|x l k Hlk IH | |].
  * by rewrite !NoDup_nil.
  * by rewrite !NoDup_cons, IH, Hlk.
  * rewrite !NoDup_cons, !elem_of_cons. intuition.
  * intuition.
Qed.
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Lemma NoDup_Permutation (l k : list A) :
  NoDup l  NoDup k  ( x, x  l  x  k)  Permutation l k.
Proof.
  intros Hl. revert k. induction Hl as [|x l Hin ? IH].
  * intros k _ Hk.
    rewrite (elem_of_nil_inv k); [done |].
    intros x. rewrite <-Hk, elem_of_nil. intros [].
  * intros k Hk Hlk.
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    destruct (elem_of_list_split k x) as [l1 [l2 ?]]; subst.
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    { rewrite <-Hlk. by constructor. }
    rewrite <-Permutation_middle, NoDup_cons in Hk.
    destruct Hk as [??].
    apply Permutation_cons_app, IH; [done |].
    intros y. specialize (Hlk y).
    rewrite <-Permutation_middle, !elem_of_cons in Hlk.
    naive_solver.
Qed.
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Global Instance NoDup_dec {dec :  x y : A, Decision (x = y)} :
     (l : list A), Decision (NoDup l) :=
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  fix NoDup_dec l :=
  match l return Decision (NoDup l) with
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  | [] => left NoDup_nil_2
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  | x :: l =>
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    match decide_rel () x l with
    | left Hin => right (λ H, NoDup_cons_11 _ _ H Hin)
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    | right Hin =>
      match NoDup_dec l with
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      | left H => left (NoDup_cons_2 _ _ Hin H)
      | right H => right (H  NoDup_cons_12 _ _)
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      end
    end
  end.

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Section remove_dups.
  Context `{! x y : A, Decision (x = y)}.
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  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.
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  Lemma elem_of_remove_dups l x :
    x  remove_dups l  x  l.
  Proof.
    split; induction l; simpl; repeat case_decide;
      rewrite ?elem_of_cons; intuition (simplify_equality; auto).
  Qed.
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  Lemma remove_dups_nodup l : NoDup (remove_dups l).
  Proof.
    induction l; simpl; repeat case_decide; try constructor; auto.
    by rewrite elem_of_remove_dups.
  Qed.
End remove_dups.
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(** ** Properties of the [filter] function *)
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Lemma elem_of_list_filter `{ x : A, Decision (P x)} 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 filter_nodup P `{ x : A, Decision (P x)} 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.

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(** ** Properties of the [reverse] function *)
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Lemma reverse_nil : reverse [] = @nil A.
Proof. done. Qed.
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Lemma reverse_singleton (x : A) : reverse [x] = [x].
Proof. done. Qed.
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Lemma reverse_cons (l : list A) x : reverse (x :: l) = reverse l ++ [x].
Proof. unfold reverse. by rewrite <-!rev_alt. Qed.
Lemma reverse_snoc (l : list A) x : reverse (l ++ [x]) = x :: reverse l.
Proof. unfold reverse. by rewrite <-!rev_alt, rev_unit. Qed.
Lemma reverse_app (l1 l2 : list A) :
  reverse (l1 ++ l2) = reverse l2 ++ reverse l1.
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_app_distr. Qed.
Lemma reverse_length (l : list A) : length (reverse l) = length l.
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_length. Qed.
Lemma reverse_involutive (l : list A) : reverse (reverse l) = l.
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Proof. unfold reverse. rewrite <-!rev_alt. apply rev_involutive. Qed.
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(** ** Properties of the [take] function *)
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Lemma take_nil n :
  take n (@nil A) = [].
Proof. by destruct n. Qed.
Lemma take_app (l k : list A) :
  take (length l) (l ++ k) = l.
Proof. induction l; simpl; f_equal; auto. Qed.
Lemma take_app_alt (l k : list A) n :
  n = length l 
  take n (l ++ k) = l.
Proof. intros Hn. by rewrite Hn, take_app. Qed.
Lemma take_app_le (l k : list A) n :
  n  length l 
  take n (l ++ k) = take n l.
Proof.
  revert n;
  induction l; intros [|?] ?; simpl in *; f_equal; auto with lia.
Qed.
Lemma take_app_ge (l k : list A) n :
  length l  n 
  take n (l ++ k) = l ++ take (n - length l) k.
Proof.
  revert n;
  induction l; intros [|?] ?; simpl in *; f_equal; auto with lia.
Qed.
Lemma take_ge (l : list A) n :
  length l  n 
  take n l = l.
Proof.
  revert n.
  induction l; intros [|?] ?; simpl in *; f_equal; auto with lia.
Qed.

Lemma take_take (l : list A) n m :
  take n (take m l) = take (min n m) l.
Proof. revert n m. induction l; intros [|?] [|?]; simpl; f_equal; auto. Qed.
Lemma take_idempotent (l : list A) n :
  take n (take n l) = take n l.
Proof. by rewrite take_take, Min.min_idempotent. Qed.

Lemma take_length (l : list A) n :
  length (take n l) = min n (length l).
Proof. revert n. induction l; intros [|?]; simpl; f_equal; done. Qed.
Lemma take_length_alt (l : list A) n :
  n  length l 
  length (take n l) = n.
Proof. rewrite take_length. apply Min.min_l. Qed.

Lemma lookup_take (l : list A) n i :
  i < n  take n l !! i = l !! i.
Proof.
  revert n i. induction l; intros [|n] i ?; trivial.
  * auto with lia.
  * destruct i; simpl; auto with arith.
Qed.
Lemma lookup_take_ge (l : list A) 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 : A  A) 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).
  * by rewrite !lookup_take_ge.
  * by rewrite !lookup_take, !list_lookup_alter_ne by lia.
Qed.
Lemma take_insert (l : list A) n i x :
  n  i  take n (<[i:=x]>l) = take n l.
Proof take_alter _ _ _ _.

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(** ** Properties of the [drop] function *)
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Lemma drop_nil n :
  drop n (@nil A) = [].
Proof. by destruct n. Qed.
Lemma drop_app (l k : list A) :
  drop (length l) (l ++ k) = k.
Proof. induction l; simpl; f_equal; auto. Qed.
Lemma drop_app_alt (l k : list A) n :
  n = length l 
  drop n (l ++ k) = k.
Proof. intros Hn. by rewrite Hn, drop_app. Qed.
Lemma drop_length (l : list A) n :
  length (drop n l) = length l - n.
Proof.
  revert n. by induction l; intros [|i]; simpl; f_equal.
Qed.
Lemma drop_all (l : list A) :
  drop (length l) l = [].
Proof. induction l; simpl; auto. Qed.
Lemma drop_all_alt (l : list A) n :
  n = length l 
  drop n l = [].
Proof. intros. subst. by rewrite drop_all. Qed.

Lemma lookup_drop (l : list A) n i :
  drop n l !! i = l !! (n + i).
Proof. revert n i. induction l; intros [|i] ?; simpl; auto. Qed.
Lemma drop_alter (f : A  A) 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 : list A) n i x :
  i < n  drop n (<[i:=x]>l) = drop n l.
Proof drop_alter _ _ _ _.

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Lemma delete_take_drop (l : list A) i :
  delete i l = take i l ++ drop (S i) l.
Proof. revert i. induction l; intros [|?]; simpl; auto using f_equal. Qed.

(** ** Properties of the [replicate] function *)
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Lemma replicate_length n (x : A) : length (replicate n x) = n.
Proof. induction n; simpl; auto. Qed.
Lemma lookup_replicate n (x : A) i :
  i < n 
  replicate n x !! i = Some x.
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Proof.
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  revert i.
  induction n; intros [|?]; naive_solver auto with lia.
Qed.
Lemma lookup_replicate_inv n (x y : A) i :
  replicate n x !! i = Some y  y = x  i < n.
Proof.
  revert i.
  induction n; intros [|?]; naive_solver auto with lia.
Qed.
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Lemma replicate_S n (x : A) :
  replicate (S n) x = x :: replicate  n x.
Proof. done. Qed.
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Lemma replicate_plus n m (x : A) :
  replicate (n + m) x = replicate n x ++ replicate m x.
Proof. induction n; simpl; f_equal; auto. Qed.

Lemma take_replicate n m (x : A) :
  take n (replicate m x) = replicate (min n m) x.
Proof. revert m. by induction n; intros [|?]; simpl; f_equal. Qed.
Lemma take_replicate_plus n m (x : A) :
  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 : A) :
  drop n (replicate m x) = replicate (m - n) x.
Proof. revert m. by induction n; intros [|?]; simpl; f_equal. Qed.
Lemma drop_replicate_plus n m (x : A) :
  drop n (replicate (n + m) x) = replicate m x.
Proof. rewrite drop_replicate. f_equal. lia. Qed.

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Lemma reverse_replicate n (x : A) :
  reverse (replicate n x) = replicate n x.
Proof.
  induction n as [|n IH]; [done|].
  simpl. rewrite reverse_cons, IH. change [x] with (replicate 1 x).
  by rewrite <-replicate_plus, plus_comm.
Qed.

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(** ** Properties of the [resize] function *)
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Lemma resize_spec (l : list A) n x :
  resize n x l = take n l ++ replicate (n - length l) x.
Proof.
  revert n.
  induction l; intros [|?]; simpl; f_equal; auto.
Qed.
Lemma resize_0 (l : list A) x :
  resize 0 x l = [].
Proof. by destruct l. Qed.
Lemma resize_nil n (x : A) :
  resize n x [] = replicate n x.
Proof. rewrite resize_spec. rewrite take_nil. simpl. f_equal. lia. Qed.
Lemma resize_ge (l : list A) 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 : list A) n x :
  n  length l 
  resize n x l = take n l.
Proof.
  intros. rewrite resize_spec, (proj2 (NPeano.Nat.sub_0_le _ _)) by done.
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  simpl. by rewrite (right_id [] (++)).
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Qed.

Lemma resize_all (l : list A) x :
  resize (length l) x l = l.
Proof. intros. by rewrite resize_le, take_ge. Qed.
Lemma resize_all_alt (l : list A) n x :
  n = length l 
  resize n x l = l.
Proof. intros. subst. by rewrite resize_all. Qed.

Lemma resize_plus (l : list A) n m x :
  resize (n + m) x l = resize n x l ++ resize m x (drop n l).
Proof.
  revert n m.
  induction l; intros [|?] [|?]; simpl; f_equal; auto.
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  * by rewrite plus_0_r, (right_id [] (++)).
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  * by rewrite replicate_plus.
Qed.
Lemma resize_plus_eq (l : list A) n m x :
  length l = n 
  resize (n + m) x l = l ++ replicate m x.
Proof.
  intros. subst.
  by rewrite resize_plus, resize_all, drop_all, resize_nil.
Qed.

Lemma resize_app_le (l1 l2 : list A) 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_ge (l1 l2 : list A) n x :
  length l1  n 
  resize n x (l1 ++ l2) = l1 ++ resize (n - length l1) x l2.
Proof.
  intros.
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  rewrite !resize_spec, take_app_ge, (associative (++)) by done.
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  do 2 f_equal. rewrite app_length. lia.
Qed.

Lemma resize_length (l : list A) n x : length (resize n x l) = n.
Proof.
  rewrite resize_spec, app_length, replicate_length, take_length. lia.
Qed.
Lemma resize_replicate (x : A) n m :
  resize n x (replicate m x) = replicate n x.
Proof. revert m. induction n; intros [|?]; simpl; f_equal; auto. Qed.

Lemma resize_resize (l : list A) n m x :
  n  m 
  resize n x (resize m x l) = resize n x l.
Proof.
  revert n m. induction l; simpl.
  * intros. by rewrite !resize_nil, resize_replicate.
  * intros [|?] [|?] ?; simpl; f_equal; auto with lia.
Qed.
Lemma resize_idempotent (l : list A) n x :
  resize n x (resize n x l) = resize n x l.
Proof. by rewrite resize_resize. Qed.

Lemma resize_take_le (l : list A) n m x :
  n  m 
  resize n x (take m l) = resize n x l.
Proof.
  revert n m.
  induction l; intros [|?] [|?] ?; simpl; f_equal; auto with lia.
Qed.
Lemma resize_take_eq (l : list A) n x :
  resize n x (take n l) = resize n x l.
Proof. by rewrite resize_take_le. Qed.

Lemma take_resize (l : list A) n m x :
  take n (resize m x l) = resize (min n m) x l.
Proof.
  revert n m.
  induction l; intros [|?] [|?]; simpl; f_equal; auto using take_replicate.
Qed.
Lemma take_resize_le (l : list A) 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 : list A) n x :
  take n (resize n x l) = resize n x l.
Proof. intros. by rewrite take_resize, Min.min_l. Qed.
Lemma take_length_resize (l : list A) n x :
  length l  n 
  take (length l) (resize n x l) = l.
Proof. intros. by rewrite take_resize_le, resize_all. Qed.
Lemma take_length_resize_alt (l : list A) n m x :
  m = length l 
  m  n 
  take m (resize n x l) = l.
Proof. intros. subst. by apply take_length_resize. Qed.
Lemma take_resize_plus (l : list A) 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 : list A) 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.
  * intros. by rewrite drop_nil, !resize_nil, drop_replicate.
  * intros [|?] [|?] ?; simpl; try case_match; auto with lia.
Qed.
Lemma drop_resize_plus (l : list A) 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.
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(** ** Properties of the [sublist] predicate *)
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Lemma sublist_nil_l (l : list A) :
  sublist [] l.
Proof. induction l; try constructor; auto. Qed.
Lemma sublist_nil_r (l : list A) :
  sublist l []  l = [].
Proof. split. by inversion 1. intros. subst. constructor. Qed.

Lemma sublist_app_skip_l (k : list A) l1 l2 :
  sublist l1 l2 
  sublist l1 (k ++ l2).
Proof. induction k; try constructor; auto. Qed.
Lemma sublist_app_skip_r (k : list A) l1 l2 :
  sublist l1 l2 
  sublist l1 (l2 ++ k).
Proof. induction 1; simpl; try constructor; auto using sublist_nil_l. Qed.

Lemma sublist_cons_r (x : A) l k :
  sublist l (x :: k)  sublist l k   l', l = x :: l'  sublist l' k.
Proof.
  split.
  * inversion 1; eauto.
  * intros [?|(?&?&?)]; subst; constructor; auto.
Qed.
Lemma sublist_cons_l (x : A) l k :
  sublist (x :: l) k   k1 k2, k = k1 ++ x :: k2  sublist l k2.
Proof.
  split.
  * intros Hlk. induction k as [|y k IH]; inversion Hlk.
    + eexists [], k. by repeat constructor.
    + destruct IH as (k1 & k2 & ? & ?); subst; auto.
      by exists (y :: k1) k2.
  * intros (k1 & k2 & ? & ?). subst.
    by apply sublist_app_skip_l, sublist_cons.
Qed.

Lemma sublist_app_compat (l1 l2 k1 k2 : list A) :
  sublist l1 l2  sublist k1 k2 
  sublist (l1 ++ k1) (l2 ++ k2).
Proof. induction 1; simpl; try constructor; auto. Qed.

Lemma sublist_app_r (l k1 k2 : list A) :
  sublist l (k1 ++ k2)   l1 l2,
    l = l1 ++ l2  sublist l1 k1  sublist l2 k2.
Proof.
  split.
  * revert l k2. induction k1 as [|y k1 IH]; intros l k2; simpl.
    { eexists [], l. by repeat constructor. }
    rewrite sublist_cons_r. intros [?|(l' & ? &?)]; subst.
    + destruct (IH l k2) as (l1&l2&?&?&?); trivial; subst.
      exists l1 l2. auto using sublist_cons_skip.
    + destruct (IH l' k2) as (l1&l2&?&?&?); trivial; subst.
      exists (y :: l1) l2. auto using sublist_cons.
  * intros (?&?&?&?&?); subst. auto using sublist_app_compat.
Qed.
Lemma sublist_app_l (l1 l2 k : list A) :
  sublist (l1 ++ l2) k   k1 k2,
    k = k1 ++ k2  sublist l1 k1  sublist l2 k2.
Proof.
  split.
  * revert l2 k. induction l1 as [|x l1 IH]; intros l2 k; simpl.
    { eexists [], k. by repeat constructor. }
    rewrite sublist_cons_l. intros (k1 & k2 &?&?); subst.
    destruct (IH l2 k2) as (h1 & h2 &?&?&?); trivial; subst.
    exists (k1 ++ x :: h1) h2. rewrite <-(associative (++)).
    auto using sublist_app_skip_l, sublist_cons.
  * intros (?&?&?&?&?); subst. auto using sublist_app_compat.
Qed.

Global Instance: PreOrder (@sublist A).
Proof.
  split.
  * intros l. induction l; constructor; auto.
  * intros l1 l2 l3 Hl12. revert l3. induction Hl12.
    + auto using sublist_nil_l.
    + intros ?. rewrite sublist_cons_l. intros (?&?&?&?); subst.
      eauto using sublist_app_skip_l, sublist_cons.
    + intros ?. rewrite sublist_cons_l. intros (?&?&?&?); subst.
      eauto using sublist_app_skip_l, sublist_cons_skip.
Qed.

Lemma sublist_length (l1 l2 : list A) :
  sublist l1 l2  length l1  length l2.
Proof. induction 1; simpl; auto with arith. Qed.

Lemma sublist_take (l : list A) i :
  sublist (take i l) l.
Proof. rewrite <-(take_drop i l) at 2. by apply sublist_app_skip_r. Qed.
Lemma sublist_drop (l : list A) i :
  sublist (drop i l) l.
Proof. rewrite <-(take_drop i l) at 2. by apply sublist_app_skip_l. Qed.
Lemma sublist_delete (l : list A) i :
  sublist (delete i l) l.
Proof. revert i. by induction l; intros [|?]; simpl; constructor. Qed.
Lemma sublist_delete_list (l : list A) is :
  sublist (delete_list is l) l.
Proof.
  induction is as [|i is IH]; simpl; [done |].
  transitivity (delete_list is l); auto using sublist_delete.
Qed.

Lemma sublist_alt (l1 l2 : list A) :
  sublist l1 l2   is, l1 = delete_list is l2.
Proof.
  split.
  * intros Hl12.
    cut ( k,  is, k ++ l1 = delete_list is (k ++ l2)).
    { intros help. apply (help []). }
    induction Hl12 as [|x l1 l2 _ IH|x l1 l2 _ IH]; intros k.
    + by eexists [].
    + destruct (IH (k ++ [x])) as [is His]. exists is.
      by rewrite <-!(associative (++)) in His.
    + destruct (IH k) as [is His]. exists (is ++ [length k]).
      unfold delete_list. rewrite fold_right_app. simpl.
      by rewrite delete_middle.
  * intros [is ?]. subst. apply sublist_delete_list.
Qed.

Global Instance: AntiSymmetric (@sublist A).
Proof.
  intros l1 l2 Hl12 Hl21. apply sublist_length in Hl21.
  induction Hl12; simpl in *.
  * done.
  * f_equal. auto with arith.
  * apply sublist_length in Hl12. lia.
Qed.
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End general_properties.

(** ** Properties of the [same_length] predicate *)
Instance:  A, Reflexive (@same_length A A).
Proof. intros A l. induction l; constructor; auto. Qed.
Instance:  A, Symmetric (@same_length A A).
Proof. induction 1; constructor; auto. Qed.

Section same_length.
  Context {A B : Type}.

  Lemma same_length_length_1 (l : list A) (k : list B) :
    same_length l k  length l = length k.
  Proof. induction 1; simpl; auto. Qed.
  Lemma same_length_length_2 (l : list A) (k : list B) :
    length l = length k  same_length l k.
  Proof.
    revert k. induction l; intros [|??]; try discriminate;
      constructor; auto with arith.
  Qed.
  Lemma same_length_length (l : list A) (k : list B) :
    same_length l k  length l = length k.
  Proof. split; auto using same_length_length_1, same_length_length_2. Qed.

  Lemma same_length_lookup (l : list A) (k : list B) i :
    same_length l k  is_Some (l !! i)  is_Some (k !! i).
  Proof.
    rewrite same_length_length.
    setoid_rewrite lookup_lt_length.
    intros E. by rewrite E.
  Qed.

  Lemma same_length_take (l1 : list A) (l2 : list B) n :
    same_length l1 l2 
    same_length (take n l1) (take n l2).
  Proof.
    intros Hl. revert n; induction Hl; intros [|n]; constructor; auto.
  Qed.
  Lemma same_length_drop (l1 : list A) (l2 : list B) n :
    same_length l1 l2 
    same_length (drop n l1) (drop n l2).
  Proof.
    intros Hl.
    revert n; induction Hl; intros [|n]; simpl; try constructor; auto.
  Qed.
  Lemma same_length_resize (l1 : list A) (l2 : list B) x1 x2 n :
    same_length (resize n x1 l1) (resize n x2 l2).
  Proof. apply same_length_length. by rewrite !resize_length. Qed.
End same_length.
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(** ** Properties of the [Forall] and [Exists] predicate *)
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Section Forall_Exists.
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  Context {A} (P : A  Prop).
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  Lemma Forall_forall l :
    Forall P l   x, x  l  P x.
  Proof.
    split.
    * induction 1; inversion 1; subst; auto.
    * intros Hin. induction l; constructor.
      + apply Hin. constructor.
      + apply IHl. intros ??. apply Hin. by constructor.
  Qed.

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  Lemma Forall_nil : Forall P []  True.
  Proof. done. Qed.
  Lemma Forall_cons_1 x l : Forall P (x :: l)  P x  Forall P l.
  Proof. by inversion 1. Qed.
  Lemma Forall_cons x l : Forall P (x :: l)  P x  Forall P l.
  Proof. split. by inversion 1. intros [??]. by constructor. Qed.
  Lemma Forall_singleton x : Forall P [x]  P x.
  Proof. rewrite Forall_cons, Forall_nil; tauto. Qed.
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  Lemma Forall_app l1 l2 : Forall P (l1 ++ l2)  Forall P l1  Forall P l2.
  Proof.
    split.
    * induction l1; inversion 1; intuition.
    * intros [H ?]. induction H; simpl; intuition.
  Qed.
  Lemma Forall_true l : ( x, P x)  Forall P l.
  Proof. induction l; auto. Qed.
  Lemma Forall_impl l (Q : A  Prop) :
    Forall P l  ( x, P x  Q x)  Forall Q l.
  Proof. intros H ?. induction H; auto. Defined.
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