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(* Copyright (c) 2012-2019, Coq-std++ developers. *)
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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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From Coq Require Export Permutation.
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From stdpp Require Export numbers base option.
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Set Default Proof Using "Type*".
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Arguments length {_} _ : assert.
Arguments cons {_} _ _ : assert.
Arguments app {_} _ _ : assert.
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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 head {_} _ : assert.
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Arguments tail {_} _ : assert.
Arguments take {_} !_ !_ / : assert.
Arguments drop {_} !_ !_ / : assert.
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Instance: Params (@head) 1 := {}.
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Instance: Params (@tail) 1 := {}.
Instance: Params (@take) 1 := {}.
Instance: Params (@drop) 1 := {}.
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Arguments Permutation {_} _ _ : assert.
Arguments Forall_cons {_} _ _ _ _ _ : assert.
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Remove Hints Permutation_cons : typeclass_instances.
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Notation "(::)" := cons (only parsing) : list_scope.
Notation "( x ::)" := (cons x) (only parsing) : list_scope.
Notation "(:: l )" := (λ x, cons x l) (only parsing) : list_scope.
Notation "(++)" := app (only parsing) : list_scope.
Notation "( l ++)" := (app l) (only parsing) : list_scope.
Notation "(++ k )" := (λ l, app l k) (only parsing) : list_scope.
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Infix "≡ₚ" := Permutation (at level 70, no associativity) : stdpp_scope.
Notation "(≡ₚ)" := Permutation (only parsing) : stdpp_scope.
Notation "( x ≡ₚ)" := (Permutation x) (only parsing) : stdpp_scope.
Notation "(≡ₚ x )" := (λ y, y  x) (only parsing) : stdpp_scope.
Notation "(≢ₚ)" := (λ x y, ¬x  y) (only parsing) : stdpp_scope.
Notation "x ≢ₚ y":= (¬x  y) (at level 70, no associativity) : stdpp_scope.
Notation "( x ≢ₚ)" := (λ y, x ≢ₚ y) (only parsing) : stdpp_scope.
Notation "(≢ₚ x )" := (λ y, y ≢ₚ x) (only parsing) : stdpp_scope.
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Infix "≡ₚ@{ A }" :=
  (@Permutation A) (at level 70, no associativity, only parsing) : stdpp_scope.
Notation "(≡ₚ@{ A } )" := (@Permutation A) (only parsing) : stdpp_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. *)
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Instance list_lookup {A} : Lookup nat A (list A) :=
  fix go i l {struct l} : option A := let _ : Lookup _ _ _ := @go in
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  match l with
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  | [] => None | x :: l => match i with 0 => Some x | S i => l !! i end
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  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} : Alter nat A (list A) := λ f,
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  fix go i l {struct l} :=
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  match l with
  | [] => []
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  | x :: l => match i with 0 => f x :: l | S i => x :: go i l 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) :=
  fix go i y l {struct l} := let _ : Insert _ _ _ := @go in
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  match l with
  | [] => []
  | x :: l => match i with 0 => y :: l | S i => x :: <[i:=y]>l end
  end.
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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. *)
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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
  | [] => []
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  | x :: l => match i with 0 => l | S i => x :: @delete _ _ go i l end
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  end.
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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]) [].
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Instance: Params (@option_list) 1 := {}.
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Instance maybe_list_singleton {A} : Maybe (λ x : A, [x]) := λ l,
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  match l with [x] => Some x | _ => None end.
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(** 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) :=
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  fix go P _ l := let _ : Filter _ _ := @go in
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  match l with
  | [] => []
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  | x :: l => if decide (P x) then x :: filter P l else filter P l
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  end.

(** The function [list_find P l] returns the first index [i] whose element
satisfies the predicate [P]. *)
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Definition list_find {A} P `{ x, Decision (P x)} : list A  option (nat * A) :=
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  fix go l :=
  match l with
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  | [] => None
  | x :: l => if decide (P x) then Some (0,x) else prod_map S id <$> go l
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  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 :=
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  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
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  | x :: l => match n with 0 => [] | S n => x :: resize n y l end
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  end.
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Arguments resize {_} !_ _ !_ : assert.
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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. *)
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Fixpoint reshape {A} (szs : list nat) (l : list A) : list (list A) :=
  match szs with
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  | [] => [] | sz :: szs => take sz l :: reshape szs (drop sz l)
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  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) :=
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  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.
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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 :=
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  fix go a l := match l with [] => a | x :: l => go (f a x) l end.
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(** The monadic operations. *)
Instance list_ret: MRet list := λ A x, x :: @nil A.
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Instance list_fmap : FMap list := λ A B f,
  fix go (l : list A) := match l with [] => [] | x :: l => f x :: go l end.
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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.
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Instance list_bind : MBind list := λ A B f,
  fix go (l : list A) := match l with [] => [] | x :: l => f x ++ go l end.
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Instance list_join: MJoin list :=
  fix go A (ls : list (list A)) : list A :=
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  match ls with [] => [] | l :: ls => l ++ @mjoin _ go _ ls end.
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Definition mapM `{MBind M, MRet M} {A B} (f : A  M B) : list A  M (list B) :=
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  fix go l :=
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  match l with [] => mret [] | x :: l => y  f x; k  go l; mret (y :: k) end.
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(** We define stronger variants of map and fold that allow the mapped
function to use the index of the elements. *)
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Fixpoint imap {A B} (f : nat  A  B) (l : list A) : list B :=
  match l with
  | [] => []
  | x :: l => f 0 x :: imap (f  S) l
  end.
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Definition zipped_map {A B} (f : list A  list A  A  B) :
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    list A  list A  list B := fix go l k :=
  match k with
  | [] => []
  | x :: k => f l k x :: go (x :: l) k
  end.
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Fixpoint imap2 {A B C} (f : nat  A  B  C) (l : list A) (k : list B) : list C :=
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  match l, k with
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  | [], _ | _, [] => []
  | x :: l, y :: k => f 0 x y :: imap2 (f  S) l k
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  end.

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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).
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Arguments zipped_Forall_nil {_ _} _ : assert.
Arguments zipped_Forall_cons {_ _} _ _ _ _ _ : assert.
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(** 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.
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(** The function [permutations l] yields all permutations of [l]. *)
Fixpoint interleave {A} (x : A) (l : list A) : list (list A) :=
  match l with
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  | [] => [[x]]| y :: l => (x :: y :: l) :: ((y ::) <$> interleave x l)
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  end.
Fixpoint permutations {A} (l : list A) : list (list A) :=
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  match l with [] => [[]] | x :: l => permutations l = interleave x end.
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(** The predicate [suffix] holds if the first list is a suffix of the second.
The predicate [prefix] holds if the first list is a prefix of the second. *)
Definition suffix {A} : relation (list A) := λ l1 l2,  k, l2 = k ++ l1.
Definition prefix {A} : relation (list A) := λ l1 l2,  k, l2 = l1 ++ k.
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Infix "`suffix_of`" := suffix (at level 70) : stdpp_scope.
Infix "`prefix_of`" := prefix (at level 70) : stdpp_scope.
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Hint Extern 0 (_ `prefix_of` _) => reflexivity : core.
Hint Extern 0 (_ `suffix_of` _) => reflexivity : core.
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Section prefix_suffix_ops.
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  Context `{EqDecision A}.

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  Definition max_prefix : list A  list A  list A * list A * list A :=
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    fix go l1 l2 :=
    match l1, l2 with
    | [], l2 => ([], l2, [])
    | l1, [] => (l1, [], [])
    | x1 :: l1, x2 :: l2 =>
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      if decide_rel (=) x1 x2
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      then prod_map id (x1 ::) (go l1 l2) else (x1 :: l1, x2 :: l2, [])
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    end.
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  Definition max_suffix (l1 l2 : list A) : list A * list A * list A :=
    match max_prefix (reverse l1) (reverse l2) with
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    | (k1, k2, k3) => (reverse k1, reverse k2, reverse k3)
    end.
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  Definition strip_prefix (l1 l2 : list A) := (max_prefix l1 l2).1.2.
  Definition strip_suffix (l1 l2 : list A) := (max_suffix l1 l2).1.2.
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End prefix_suffix_ops.
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(** A list [l1] is a sublist of [l2] if [l2] is obtained by removing elements
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from [l1] without changing the order. *)
Inductive sublist {A} : relation (list A) :=
  | sublist_nil : sublist [] []
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  | sublist_skip x l1 l2 : sublist l1 l2  sublist (x :: l1) (x :: l2)
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  | sublist_cons x l1 l2 : sublist l1 l2  sublist l1 (x :: l2).
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Infix "`sublist_of`" := sublist (at level 70) : stdpp_scope.
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Hint Extern 0 (_ `sublist_of` _) => reflexivity : core.
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(** A list [l2] submseteq a list [l1] if [l2] is obtained by removing elements
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from [l1] while possiblity changing the order. *)
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Inductive submseteq {A} : relation (list A) :=
  | submseteq_nil : submseteq [] []
  | submseteq_skip x l1 l2 : submseteq l1 l2  submseteq (x :: l1) (x :: l2)
  | submseteq_swap x y l : submseteq (y :: x :: l) (x :: y :: l)
  | submseteq_cons x l1 l2 : submseteq l1 l2  submseteq l1 (x :: l2)
  | submseteq_trans l1 l2 l3 : submseteq l1 l2  submseteq l2 l3  submseteq l1 l3.
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Infix "⊆+" := submseteq (at level 70) : stdpp_scope.
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Hint Extern 0 (_ + _) => reflexivity : core.
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(** Removes [x] from the list [l]. The function returns a [Some] when the
+removal succeeds and [None] when [x] is not in [l]. *)
Fixpoint list_remove `{EqDecision A} (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.

(** Removes all elements in the list [k] from the list [l]. The function returns
a [Some] when the removal succeeds and [None] some element of [k] is not in [l]. *)
Fixpoint list_remove_list `{EqDecision A} (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.
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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}.
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  Global Instance elem_of_list_dec : RelDecision (@{list A}).
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  Proof.
   refine (
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    fix go x l :=
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    match l return Decision (x  l) with
    | [] => right _
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    | y :: l => cast_if_or (decide (x = y)) (go x l)
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    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
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      then list_difference l k else x :: list_difference l k
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    end.
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  Definition list_union (l k : list A) : list A := list_difference l k ++ k.
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  Fixpoint list_intersection (l k : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel () x k
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      then x :: list_intersection l k else list_intersection l k
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    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.
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(** These next functions allow to efficiently encode lists of positives (bit
strings) into a single positive and go in the other direction as well. This is
for example used for the countable instance of lists and in namespaces.
 The main functions are [positives_flatten] and [positives_unflatten]. *)
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Fixpoint positives_flatten_go (xs : list positive) (acc : positive) : positive :=
  match xs with
  | [] => acc
  | x :: xs => positives_flatten_go xs (acc~1~0 ++ Preverse (Pdup x))
  end.

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(** Flatten a list of positives into a single positive by duplicating the bits
of each element, so that:

- [0 -> 00]
- [1 -> 11]

and then separating each element with [10]. *)
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Definition positives_flatten (xs : list positive) : positive :=
  positives_flatten_go xs 1.

Fixpoint positives_unflatten_go
        (p : positive)
        (acc_xs : list positive)
        (acc_elm : positive)
  : option (list positive) :=
  match p with
  | 1 => Some acc_xs
  | p'~0~0 => positives_unflatten_go p' acc_xs (acc_elm~0)
  | p'~1~1 => positives_unflatten_go p' acc_xs (acc_elm~1)
  | p'~1~0 => positives_unflatten_go p' (acc_elm :: acc_xs) 1
  | _ => None
  end%positive.

(** Unflatten a positive into a list of positives, assuming the encoding
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used by [positives_flatten]. *)
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Definition positives_unflatten (p : positive) : option (list positive) :=
  positives_unflatten_go p [] 1.

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(** [seqZ m n] generates the sequence [m], [m + 1], ..., [m + n - 1] 
over integers, provided [n >= 0]. If n < 0, then the range is empty. **)
Definition seqZ (m len: Z) : list Z :=
  (λ i: nat, Z.add i m) <$> (seq 0 (Z.to_nat len)).
Arguments seqZ : simpl never.

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(** * Basic tactics on lists *)
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(** The tactic [discriminate_list] discharges a goal if it submseteq
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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;
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  repeat (csimpl in H || rewrite app_length in H); exfalso; lia.
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Tactic Notation "discriminate_list" :=
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  match goal with H : _ =@{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
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    [ 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 |- _ =>
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    destruct i; [change (Some x = Some y) in H | discriminate]
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  end.
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(** * General theorems *)
Section general_properties.
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Context {A : Type}.
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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.
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Lemma app_nil l1 l2 : l1 ++ l2 = []  l1 = []  l2 = [].
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Proof. split. apply app_eq_nil. by intros [-> ->]. Qed.
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Lemma app_singleton l1 l2 x :
  l1 ++ l2 = [x]  l1 = []  l2 = [x]  l1 = [x]  l2 = [].
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Proof. split. apply app_eq_unit. by intros [[-> ->]|[-> ->]]. Qed.
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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.
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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.
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Lemma nil_or_length_pos l : l = []  length l  0.
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Proof. destruct l; simpl; auto with lia. Qed.
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Lemma nil_length_inv l : length l = 0  l = [].
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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 i : tail l !! i = l !! S i.
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Proof. by destruct l. Qed.
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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.
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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.
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Proof.
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  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.
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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.
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Lemma lookup_app_r l1 l2 i :
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  length l1  i  (l1 ++ l2) !! i = l2 !! (i - length l1).
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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.
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Lemma list_lookup_middle l1 l2 x n :
  n = length l1  (l1 ++ x :: l2) !! n = Some x.
Proof. intros ->. by induction l1. Qed.
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Lemma nth_lookup l i d : nth i l d = default d (l !! i).
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Proof. revert i. induction l as [|x l IH]; intros [|i]; simpl; auto. Qed.
Lemma nth_lookup_Some l i d x : l !! i = Some x  nth i l d = x.
Proof. rewrite nth_lookup. by intros ->. Qed.
Lemma nth_lookup_or_length l i d : {l !! i = Some (nth i l d)} + {length l  i}.
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Proof.
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  rewrite nth_lookup. destruct (l !! i) eqn:?; eauto using lookup_ge_None_1.
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Qed.

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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.
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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 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_insert_id l i x : l !! i = Some x  <[i:=x]>l = l.
Proof. revert i. induction l; intros [|i] [=]; f_equal/=; auto. Qed.
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Lemma list_insert_ge l i x : length l  i  <[i:=x]>l = l.
Proof. revert i. induction l; intros [|i] ?; f_equal/=; auto with lia. Qed.
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Lemma list_insert_insert l i x y :
  <[i:=x]> (<[i:=y]> l) = <[i:=x]> l.
Proof. revert i. induction l; intros [|i]; 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.
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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.
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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 :
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  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.
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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 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.
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Proof.
  intros. assert (i = length l1 + (i - length l1)) as Hi by lia.
  rewrite Hi at 1. by apply insert_app_r.
Qed.
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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.

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(** ** Properties of the [elem_of] predicate *)
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Lemma not_elem_of_nil x : x  [].
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Proof. by inversion 1. Qed.
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Lemma elem_of_nil x : x  []  False.
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Proof. intuition. by destruct (not_elem_of_nil x). Qed.
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Lemma elem_of_nil_inv l : ( x, x  l)  l = [].
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Proof. destruct l. done. by edestruct 1; constructor. Qed.
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Lemma elem_of_not_nil x l : x  l  l  [].
Proof. intros ? ->. by apply (elem_of_nil x). Qed.
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Lemma elem_of_cons l x y : x  y :: l  x = y  x  l.
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Proof. by split; [inversion 1; subst|intros [->|?]]; constructor. Qed.
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Lemma not_elem_of_cons l x y : x  y :: l  x  y  x  l.
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Proof. rewrite elem_of_cons. tauto. Qed.
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Lemma elem_of_app l1 l2 x : x  l1 ++ l2  x  l1  x  l2.
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Proof.
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  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.
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Lemma not_elem_of_app l1 l2 x : x  l1 ++ l2  x  l1  x  l2.
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Proof. rewrite elem_of_app. tauto. Qed.
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Lemma elem_of_list_singleton x y : x  [y]  x = y.
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Proof. rewrite elem_of_cons, elem_of_nil. tauto. Qed.
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Global Instance elem_of_list_permutation_proper x : Proper (() ==> iff) (x ).
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Proof. induction 1; rewrite ?elem_of_nil, ?elem_of_cons; intuition. Qed.
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Lemma elem_of_list_split l x : x  l   l1 l2, l = l1 ++ x :: l2.
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Proof.
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  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.
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Lemma elem_of_list_split_l `{EqDecision A} l x :
  x  l   l1 l2, l = l1 ++ x :: l2  x  l1.
Proof.
  induction 1 as [x l|x y l ? IH].
  { exists [], l. rewrite elem_of_nil. naive_solver. }
  destruct (decide (x = y)) as [->|?].
  - exists [], l. rewrite elem_of_nil. naive_solver.
  - destruct IH as (l1 & l2 & -> & ?).
    exists (y :: l1), l2. rewrite elem_of_cons. naive_solver.
Qed.
Lemma elem_of_list_split_r `{EqDecision A} l x :
  x  l   l1 l2, l = l1 ++ x :: l2  x  l2.
Proof.
  induction l as [|y l IH] using rev_ind.
  { by rewrite elem_of_nil. }
  destruct (decide (x = y)) as [->|].
  - exists l, []. rewrite elem_of_nil. naive_solver.
  - rewrite elem_of_app, elem_of_list_singleton. intros [?| ->]; try done.
    destruct IH as (l1 & l2 & -> & ?); auto.
    exists l1, (l2 ++ [y]).