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(* Copyright (c) 2012-2015, 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 Export 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 tail := tl.
Notation take := firstn.
Notation drop := 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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Infix "≡ₚ" := Permutation (at level 70, no associativity) : C_scope.
Notation "(≡ₚ)" := Permutation (only parsing) : C_scope.
Notation "( x ≡ₚ)" := (Permutation x) (only parsing) : C_scope.
Notation "(≡ₚ x )" := (λ y, y  x) (only parsing) : C_scope.
Notation "(≢ₚ)" := (λ x y, ¬x  y) (only parsing) : C_scope.
Notation "x ≢ₚ y":= (¬x  y) (at level 70, no associativity) : C_scope.
Notation "( x ≢ₚ)" := (λ y, x ≢ₚ y) (only parsing) : C_scope.
Notation "(≢ₚ x )" := (λ y, y ≢ₚ x) (only parsing) : C_scope.

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(** * 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 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,
  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
  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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(** 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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Definition list_singleton {A} (l : list A) : option A :=
  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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(** 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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(** 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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(** 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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(** 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.
Arguments resize {_} !_ _ !_.

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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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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. *)
Definition imap_go {A B} (f : nat  A  B) : nat  list A  list B :=
  fix go (n : nat) (l : list A) :=
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  match l with [] => [] | x :: l => f n x :: go (S n) l end.
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Definition imap {A B} (f : nat  A  B) : list A  list B := imap_go f 0.
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Definition zipped_map {A B} (f : list A  list A  A  B) :
  list A  list A  list B := fix go l k :=
  match k with [] => [] | x :: k => f l k x :: go (x :: l) k end.

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

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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).
Arguments zipped_Forall_nil {_ _} _.
Arguments zipped_Forall_cons {_ _} _ _ _ _ _.
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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_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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Infix "`suffix_of`" := suffix_of (at level 70) : C_scope.
Infix "`prefix_of`" := prefix_of (at level 70) : C_scope.
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Hint Extern 0 (?x `prefix_of` ?y) => reflexivity.
Hint Extern 0 (?x `suffix_of` ?y) => reflexivity.
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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 =>
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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.
  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.
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  Definition strip_prefix (l1 l2 : list A) := (max_prefix_of l1 l2).1.2.
  Definition strip_suffix (l1 l2 : list A) := (max_suffix_of l1 l2).1.2.
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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`" := sublist (at level 70) : C_scope.
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Hint Extern 0 (?x `sublist` ?y) => reflexivity.
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(** A list [l2] contains 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 contains {A} : relation (list A) :=
  | contains_nil : contains [] []
  | contains_skip x l1 l2 : contains l1 l2  contains (x :: l1) (x :: l2)
  | contains_swap x y l : contains (y :: x :: l) (x :: y :: l)
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  | contains_cons x l1 l2 : contains l1 l2  contains l1 (x :: l2)
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  | contains_trans l1 l2 l3 : contains l1 l2  contains l2 l3  contains l1 l3.
Infix "`contains`" := contains (at level 70) : C_scope.
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Hint Extern 0 (?x `contains` ?y) => reflexivity.
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Section contains_dec_help.
  Context {A} {dec :  x y : A, Decision (x = y)}.
  Fixpoint list_remove (x : A) (l : list A) : option (list A) :=
    match l with
    | [] => None
    | y :: l => if decide (x = y) then Some l else (y ::) <$> list_remove x l
    end.
  Fixpoint list_remove_list (k : list A) (l : list A) : option (list A) :=
    match k with
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    | [] => Some l | x :: k => list_remove x l = list_remove_list k
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    end.
End contains_dec_help.
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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 *)
Section list_set.
  Context {A} {dec :  x y : A, Decision (x = y)}.
  Global Instance elem_of_list_dec {dec :  x y : A, Decision (x = y)}
    (x : A) :  l, Decision (x  l).
  Proof.
   refine (
    fix go l :=
    match l return Decision (x  l) with
    | [] => right _
    | y :: l => cast_if_or (decide (x = y)) (go l)
    end); clear go dec; subst; try (by constructor); abstract by inversion 1.
  Defined.
  Fixpoint remove_dups (l : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel () x l then remove_dups l else x :: remove_dups l
    end.
  Fixpoint list_difference (l k : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel () x k
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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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(** * 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;
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  repeat (csimpl in H || rewrite app_length in H); exfalso; lia.
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Tactic Notation "discriminate_list_equality" :=
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  match goal with
  | H : @eq (list _) _ _ |- _ => discriminate_list_equality H
  end.
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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. *)
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Lemma app_injective_1 {A} (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.
Lemma app_injective_2 {A} (l1 k1 l2 k2 : list A) :
  length l2 = length k2  l1 ++ l2 = k1 ++ k2  l1 = k1  l2 = k2.
Proof.
  intros ? Hl. apply app_injective_1; auto.
  apply (f_equal length) in Hl. rewrite !app_length in Hl. lia.
Qed.
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Ltac simplify_list_equality :=
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  repeat match goal with
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  | _ => progress simplify_equality'
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  | H : _ ++ _ = _ ++ _ |- _ => first
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    [ apply app_inv_head in H | apply app_inv_tail in H
    | apply app_injective_1 in H; [destruct H|done]
    | apply app_injective_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: Injective2 (=) (=) (=) (@cons A).
Proof. by injection 1. Qed.
Global Instance:  k, Injective (=) (=) (k ++).
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Proof. intros ???. apply app_inv_head. Qed.
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Global Instance:  k, Injective (=) (=) (++ k).
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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_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.
  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 _ $ λ i, H (S i)).
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Qed.
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Global Instance list_eq_dec {dec :  x y, Decision (x = y)} :  l k,
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  Decision (l = k) := 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 :
  option_reflect (λ x, l = [x]) (length l  1) (list_singleton l).
Proof. by destruct l as [|? []]; constructor. Defined.

Definition nil_length : length (@nil A) = 0 := eq_refl.
Definition cons_length x l : length (x :: l) = S (length l) := eq_refl.
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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.
Proof.
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  revert i. induction l; intros [|?] ?; simplify_equality'; auto with arith.
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Qed.
Lemma lookup_lt_is_Some_1 l i : is_Some (l !! i)  i < length l.
Proof. intros [??]; eauto using lookup_lt_Some. Qed.
Lemma lookup_lt_is_Some_2 l i : i < length l  is_Some (l !! i).
Proof.
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  revert i. induction l; intros [|?] ?; simplify_equality'; eauto with lia.
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Qed.
Lemma lookup_lt_is_Some l i : is_Some (l !! i)  i < length l.
Proof. split; auto using lookup_lt_is_Some_1, lookup_lt_is_Some_2. Qed.
Lemma lookup_ge_None l i : l !! i = None  length l  i.
Proof. rewrite eq_None_not_Some, lookup_lt_is_Some. lia. Qed.
Lemma lookup_ge_None_1 l i : l !! i = None  length l  i.
Proof. by rewrite lookup_ge_None. Qed.
Lemma lookup_ge_None_2 l i : length l  i  l !! i = None.
Proof. by rewrite lookup_ge_None. Qed.
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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.
  * destruct (lookup_lt_is_Some_2 l1 i) as [y Hy].
    { rewrite Hlen; eauto using lookup_lt_Some. }
    rewrite Hy; f_equal; apply (Hl i); eauto using lookup_lt_Some.
  * 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 [|?]; simpl; 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.
  * revert i. induction l1 as [|y l1 IH]; intros [|i] ?;
      simplify_equality'; auto with lia.
    destruct (IH i) as [?|[??]]; auto with lia.
  * intros [?|[??]]; auto using lookup_app_l_Some. by rewrite lookup_app_r.
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 list_insert_alter l i x : <[i:=x]>l = alter (λ _, x) i l.
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.
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  revert i j. induction l; [done|]. intros [][] ?; csimpl; auto with congruence.
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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.
Lemma list_lookup_insert_ne l i j x : i  j  <[i:=x]>l !! j = l !! j.
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Proof.
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  revert i j. induction l; [done|]. intros [] [] ?; simpl; auto with congruence.
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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].
  * intros Hy. assert (j < length l).
    { rewrite <-(insert_length l j x); eauto using lookup_lt_Some. }
    rewrite list_lookup_insert in Hy by done; naive_solver.
  * intros [(?&?&?)|[??]]; rewrite ?list_lookup_insert; naive_solver.
Qed.
Lemma list_insert_commute l i j x y :
  i  j  <[i:=x]>(<[j:=y]>l) = <[j:=y]>(<[i:=x]>l).
Proof. revert i j. by induction l; intros [|?] [|?] ?; f_equal'; auto. Qed.
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Lemma list_lookup_other l i x :
  length l  1  l !! i = Some x   j y, j  i  l !! j = Some y.
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Proof.
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  intros. destruct i, l as [|x0 [|x1 l]]; simplify_equality'.
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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.
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.
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.
  * intros Hy. assert (j < length l).
    { rewrite <-(inserts_length l i k); eauto using lookup_lt_Some. }
    rewrite list_lookup_inserts in Hy by lia. intuition lia.
  * intuition. by rewrite list_lookup_inserts by lia.
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).
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  * 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_lookup_1 l x : x  l   i, l !! i = Some x.
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Proof.
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  induction 1 as [|???? IH]; [by exists 0 |].
  destruct IH as [i ?]; auto. by exists (S i).
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Qed.
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Lemma elem_of_list_lookup_2 l i x : l !! i = Some x  x  l.
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Proof.
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  revert i. induction l; intros [|i] ?; simplify_equality'; constructor; eauto.
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Qed.
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Lemma elem_of_list_lookup l x : x  l   i, l !! i = Some x.
Proof. firstorder eauto using elem_of_list_lookup_1, elem_of_list_lookup_2. Qed.
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Lemma elem_of_list_omap {B} (f : A  option B) l (y : B) :
  y  omap f l   x, x  l  f x = Some y.
Proof.
  split.
  * induction l as [|x l]; csimpl; repeat case_match; inversion 1; subst;
      setoid_rewrite elem_of_cons; naive_solver.
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  * intros (x&Hx&?). by induction Hx; csimpl; repeat case_match;
      simplify_equality; try constructor; auto.
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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.
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Lemma NoDup_cons x l : NoDup (x :: l)  x  l  NoDup l.
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Proof. split. by inversion 1. intros [??]. by constructor. Qed.
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Lemma NoDup_cons_11 x l : NoDup (x :: l)  x  l.
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Proof. rewrite NoDup_cons. by intros [??]. Qed.
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Lemma NoDup_cons_12 x l : NoDup (x :: l)  NoDup l.
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Proof. rewrite NoDup_cons. by intros [??]. Qed.
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Lemma NoDup_singleton x : NoDup [x].
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Proof. constructor. apply not_elem_of_nil. constructor. Qed.
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Lemma NoDup_app l k : NoDup (l ++ k)  NoDup l  ( x, x  l  x  k)  NoDup k.
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Proof.
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  induction l; simpl.
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  * rewrite NoDup_nil. setoid_rewrite elem_of_nil. naive_solver.
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  * 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: Proper (() ==> iff) (@NoDup A).
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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_lookup l i j x :
  NoDup l  l !! i = Some x  l !! j = Some x  i = j.
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Proof.
  intros Hl. revert i j. induction Hl as [|x' l Hx Hl IH].
  { intros; simplify_equality. }
  intros [|i] [|j] ??; simplify_equality'; eauto with f_equal;
    exfalso; eauto using elem_of_list_lookup_2.
Qed.
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Lemma NoDup_alt l :
  NoDup l   i j x, l !! i = Some x  l !! j = Some x  i = j.
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Proof.
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  split; eauto using NoDup_lookup.
  induction l as [|x l IH]; intros Hl; constructor.
  * rewrite elem_of_list_lookup. intros [i ?].
    by feed pose proof (Hl (S i) 0 x); auto.
  * apply IH. intros i j x' ??. by apply (injective S), (Hl (S i) (S j) x').
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Qed.
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Section no_dup_dec.
  Context `{! x y, Decision (x = y)}.
  Global Instance NoDup_dec:  l, Decision (NoDup l) :=
    fix NoDup_dec l :=
    match l return Decision (NoDup l) with
    | [] => left NoDup_nil_2
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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)
      | right Hin =>
        match NoDup_dec l with
        | left H => left (NoDup_cons_2 _ _ Hin H)
        | right H => right (H  NoDup_cons_12 _ _)
        end
      end
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    end.
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  Lemma elem_of_remove_dups l x : x  remove_dups l  x  l.
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  Proof.
    split; induction l; simpl; repeat case_decide;
      rewrite ?elem_of_cons; intuition (simplify_equality; auto).
  Qed.
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  Lemma NoDup_remove_dups l : NoDup (remove_dups l).
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  Proof.
    induction l; simpl; repeat case_decide; try constructor; auto.
    by rewrite elem_of_remove_dups.
  Qed.
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End no_dup_dec.
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(** ** Set operations on lists *)
Section list_set.
  Context {dec :  x y, Decision (x = y)}.
  Lemma elem_of_list_difference l k x : x  list_difference l k  x  l  x  k.
  Proof.
    split; induction l; simpl; try case_decide;
      rewrite ?elem_of_nil, ?elem_of_cons; intuition congruence.
  Qed.
  Lemma NoDup_list_difference l k : NoDup l  NoDup (list_difference l k).
  Proof.
    induction 1; simpl; try case_decide.
    * constructor.
    * done.
    * constructor. rewrite elem_of_list_difference; intuition. done.
  Qed.
  Lemma elem_of_list_union l k x : x  list_union l k  x  l  x  k.
  Proof.
    unfold list_union. rewrite elem_of_app, elem_of_list_difference.
    intuition. case (decide (x  k)); intuition.
  Qed.
  Lemma NoDup_list_union l k : NoDup l  NoDup k  NoDup (list_union l k).
  Proof.
    intros. apply NoDup_app. repeat split.
    * by apply NoDup_list_difference.
    * intro. rewrite elem_of_list_difference. intuition.
    * done.
  Qed.
  Lemma elem_of_list_intersection l k x :
    x  list_intersection l k  x  l  x  k.
  Proof.
    split; induction l; simpl; repeat case_decide;
      rewrite ?elem_of_nil, ?elem_of_cons; intuition congruence.
  Qed.
  Lemma NoDup_list_intersection l k : NoDup l  NoDup (list_intersection l k).
  Proof.
    induction 1; simpl; try case_decide.
    * constructor.
    * constructor. rewrite elem_of_list_intersection; intuition. done.
    * done.
  Qed.
  Lemma elem_of_list_intersection_with f l k x :
    x  list_intersection_with f l k   x1 x2,
      x1  l  x2  k  f x1 x2 = Some x.
  Proof.
    split.
    * induction l as [|x1 l IH]; simpl; [by rewrite elem_of_nil|].
      intros Hx. setoid_rewrite elem_of_cons.
      cut (( x2, x2  k  f x1 x2 = Some x)
         x  list_intersection_with f l k); [naive_solver|].
      clear IH. revert Hx. generalize (list_intersection_with f l k).
      induction k; simpl; [by auto|].
      case_match; setoid_rewrite elem_of_cons; naive_solver.
    * intros (x1&x2&Hx1&Hx2&Hx). induction Hx1 as [x1|x1 ? l ? IH]; simpl.
      + generalize (list_intersection_with f l k).
        induction Hx2; simpl; [by rewrite Hx; left |].
        case_match; simpl; try setoid_rewrite elem_of_cons; auto.
      + generalize (IH Hx). clear Hx IH Hx2.
        generalize (list_intersection_with f l k).
        induction k; simpl; intros; [done|].
        case_match; simpl; rewrite ?elem_of_cons; auto.
  Qed.
End list_set.

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(** ** Properties of the [filter] function *)
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Section filter.
  Context (P : A  Prop) `{ x, Decision (P x)}.
  Lemma elem_of_list_filter l x : x  filter P l  P x  x  l.
  Proof.
    unfold filter. induction l; simpl; repeat case_decide;
       rewrite ?elem_of_nil, ?elem_of_cons; naive_solver.
  Qed.
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  Lemma NoDup_filter l : NoDup l  NoDup (filter P l).
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  Proof.
    unfold filter. induction 1; simpl; repeat case_decide;
      rewrite ?NoDup_nil, ?NoDup_cons, ?elem_of_list_filter; tauto.
  Qed.
End filter.
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(** ** Properties of the [find] function *)
Section find.
  Context (P : A  Prop) `{ x, Decision (P x)}.
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  Lemma list_find_Some l i x :
    list_find P l = Some (i,x)  l !! i = Some x  P x.
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  Proof.
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    revert i; induction l; intros [] ?;
      repeat (match goal with x : prod _ _ |- _ => destruct x end
              || simplify_option_equality); eauto.
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  Qed.
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  Lemma list_find_elem_of l x : x  l  P x  is_Some (list_find P l).
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  Proof.