Library Coq.Lists.List


Require Import PeanoNat.

Set Implicit Arguments.

Basics: definition of polymorphic lists and some operations

The definition of list is now in Init/Datatypes, as well as the definitions of length and app

#[local] Open Scope bool_scope.
Open Scope list_scope.

Standard notations for lists. In a special module to avoid conflicts.
Module ListNotations.
Notation "[ ]" := nil (format "[ ]") : list_scope.
Notation "[ x ]" := (cons x nil) : list_scope.
Notation "[ x ; y ; .. ; z ]" := (cons x (cons y .. (cons z nil) ..))
  (format "[ '[' x ; '/' y ; '/' .. ; '/' z ']' ]") : list_scope.
End ListNotations.

Import ListNotations.

Section Lists.

  Variable A : Type.

Head and tail

  Definition hd (default:A) (l:list A) :=
    match l with
      | [] => default
      | x :: _ => x
    end.

  Definition hd_error (l:list A) :=
    match l with
      | [] => None
      | x :: _ => Some x
    end.

  Definition tl (l:list A) :=
    match l with
      | [] => nil
      | a :: m => m
    end.

The In predicate
  Fixpoint In (a:A) (l:list A) : Prop :=
    match l with
      | [] => False
      | b :: m => b = a \/ In a m
    end.

End Lists.

Section Facts.

  Variable A : Type.

Generic facts

Discrimination
  Theorem nil_cons (x:A) (l:list A) : [] <> x :: l.

Destruction

  Theorem destruct_list (l : list A) : {x:A & {tl:list A | l = x::tl}}+{l = []}.

  Lemma hd_error_tl_repr l (a:A) r :
    hd_error l = Some a /\ tl l = r <-> l = a :: r.

  Lemma hd_error_some_nil l (a:A) : hd_error l = Some a -> l <> nil.

  Theorem length_zero_iff_nil (l : list A):
    length l = 0 <-> l = [].

Head and tail


  Theorem hd_error_nil : hd_error (@nil A) = None.

  Theorem hd_error_cons (l : list A) (x : A) : hd_error (x::l) = Some x.

Facts about app

Discrimination
  Theorem app_cons_not_nil (x y:list A) (a:A) : [] <> x ++ a :: y.

Concat with nil
  Theorem app_nil_l (l:list A) : [] ++ l = l.

  Theorem app_nil_r (l:list A) : l ++ [] = l.


app is associative
  Theorem app_assoc (l m n:list A) : l ++ m ++ n = (l ++ m) ++ n.


app commutes with cons
  Theorem app_comm_cons (x y:list A) (a:A) : a :: (x ++ y) = (a :: x) ++ y.

Facts deduced from the result of a concatenation

  Theorem app_eq_nil (l l':list A) : l ++ l' = [] -> l = [] /\ l' = [].

  Lemma app_eq_cons x y z (a : A):
    x ++ y = a :: z -> (x = nil /\ y = a :: z) \/ exists x', x = a :: x' /\ z = x' ++ y.

  Theorem app_eq_unit (x y:list A) (a:A) :
      x ++ y = [a] -> x = [] /\ y = [a] \/ x = [a] /\ y = [].

  Lemma elt_eq_unit l1 l2 (a b : A) :
    l1 ++ a :: l2 = [b] -> a = b /\ l1 = [] /\ l2 = [].

  Theorem app_eq_app X (x1 x2 y1 y2: list X) : x1++x2 = y1++y2 ->
    exists l, (x1 = y1++l /\ y2 = l++x2) \/ (y1 = x1++l /\ x2 = l++y2).

  Lemma app_inj_tail :
    forall (x y:list A) (a b:A), x ++ [a] = y ++ [b] -> x = y /\ a = b.

  Lemma app_inj_tail_iff :
    forall (x y:list A) (a b:A), x ++ [a] = y ++ [b] <-> x = y /\ a = b.

Compatibility with other operations

  Lemma app_length : forall l l' : list A, length (l++l') = length l + length l'.

  Lemma last_length : forall (l : list A) a, length (l ++ a :: nil) = S (length l).

  Lemma app_inv_head_iff:
   forall l l1 l2 : list A, l ++ l1 = l ++ l2 <-> l1 = l2.

  Lemma app_inv_head:
   forall l l1 l2 : list A, l ++ l1 = l ++ l2 -> l1 = l2.

  Lemma app_inv_tail:
    forall l l1 l2 : list A, l1 ++ l = l2 ++ l -> l1 = l2.

  Lemma app_inv_tail_iff:
    forall l l1 l2 : list A, l1 ++ l = l2 ++ l <-> l1 = l2.

Facts about In

Characterization of In

  Theorem in_eq : forall (a:A) (l:list A), In a (a :: l).

  Theorem in_cons : forall (a b:A) (l:list A), In b l -> In b (a :: l).

  Theorem not_in_cons (x a : A) (l : list A):
    ~ In x (a::l) <-> x<>a /\ ~ In x l.

  Theorem in_nil : forall a:A, ~ In a [].

  Lemma in_app_or : forall (l m:list A) (a:A), In a (l ++ m) -> In a l \/ In a m.

  Lemma in_or_app : forall (l m:list A) (a:A), In a l \/ In a m -> In a (l ++ m).

  Lemma in_app_iff : forall l l' (a:A), In a (l++l') <-> In a l \/ In a l'.

  Theorem in_split : forall x (l:list A), In x l -> exists l1 l2, l = l1++x::l2.

  Lemma in_elt : forall (x:A) l1 l2, In x (l1 ++ x :: l2).

  Lemma in_elt_inv : forall (x y : A) l1 l2,
    In x (l1 ++ y :: l2) -> x = y \/ In x (l1 ++ l2).

  Lemma app_inj_pivot x1 x2 y1 y2 (a : A): x1 ++ a :: x2 = y1 ++ a :: y2 ->
    ((In a x1 /\ In a y2) \/ (In a x2 /\ In a y1)) \/ (x1 = y1 /\ x2 = y2).

Inversion
  Lemma in_inv : forall (a b:A) (l:list A), In b (a :: l) -> a = b \/ In b l.

Decidability of In
  Theorem in_dec :
    (forall x y:A, {x = y} + {x <> y}) ->
    forall (a:A) (l:list A), {In a l} + {~ In a l}.

End Facts.

#[global]
Hint Resolve app_assoc app_assoc_reverse_deprecated: datatypes.
#[global]
Hint Resolve app_comm_cons app_cons_not_nil: datatypes.
#[global]
Hint Immediate app_eq_nil: datatypes.
#[global]
Hint Resolve app_eq_unit app_inj_tail: datatypes.
#[global]
Hint Resolve in_eq in_cons in_inv in_nil in_app_or in_or_app: datatypes.

Local Ltac Tauto.intuition_solver ::= auto with datatypes.

Operations on the elements of a list


Section Elts.

  Variable A : Type.

Nth element of a list


  Fixpoint nth (n:nat) (l:list A) (default:A) {struct l} : A :=
    match n, l with
      | O, x :: l' => x
      | O, [] => default
      | S m, [] => default
      | S m, x :: t => nth m t default
    end.

  Fixpoint nth_ok (n:nat) (l:list A) (default:A) {struct l} : bool :=
    match n, l with
      | O, x :: l' => true
      | O, [] => false
      | S m, [] => false
      | S m, x :: t => nth_ok m t default
    end.

  Lemma nth_in_or_default :
    forall (n:nat) (l:list A) (d:A), {In (nth n l d) l} + {nth n l d = d}.

  Lemma nth_S_cons :
    forall (n:nat) (l:list A) (d a:A),
      In (nth n l d) l -> In (nth (S n) (a :: l) d) (a :: l).

  Fixpoint nth_error (l:list A) (n:nat) {struct n} : option A :=
    match n, l with
      | O, x :: _ => Some x
      | S n, _ :: l => nth_error l n
      | _, _ => None
    end.

  Definition nth_default (default:A) (l:list A) (n:nat) : A :=
    match nth_error l n with
      | Some x => x
      | None => default
    end.

  Lemma nth_default_eq :
    forall n l (d:A), nth_default d l n = nth n l d.

Results about nth

  Lemma nth_In :
    forall (n:nat) (l:list A) (d:A), n < length l -> In (nth n l d) l.

  Lemma In_nth l x d : In x l ->
    exists n, n < length l /\ nth n l d = x.

  Lemma nth_overflow : forall l n d, length l <= n -> nth n l d = d.

  Lemma nth_indep :
    forall l n d d', n < length l -> nth n l d = nth n l d'.

  Lemma app_nth1 :
    forall l l' d n, n < length l -> nth n (l++l') d = nth n l d.

  Lemma app_nth2 :
    forall l l' d n, n >= length l -> nth n (l++l') d = nth (n-length l) l' d.

  Lemma app_nth2_plus : forall l l' d n,
    nth (length l + n) (l ++ l') d = nth n l' d.

  Lemma nth_middle : forall l l' a d,
    nth (length l) (l ++ a :: l') d = a.

  Lemma nth_split n l d : n < length l ->
    exists l1, exists l2, l = l1 ++ nth n l d :: l2 /\ length l1 = n.

  Lemma nth_ext : forall l l' d d', length l = length l' ->
    (forall n, n < length l -> nth n l d = nth n l' d') -> l = l'.

Results about nth_error

  Lemma nth_error_In l n x : nth_error l n = Some x -> In x l.

  Lemma In_nth_error l x : In x l -> exists n, nth_error l n = Some x.

  Lemma nth_error_None l n : nth_error l n = None <-> length l <= n.

  Lemma nth_error_Some l n : nth_error l n <> None <-> n < length l.

  Lemma nth_error_split l n a : nth_error l n = Some a ->
    exists l1, exists l2, l = l1 ++ a :: l2 /\ length l1 = n.

  Lemma nth_error_app1 l l' n : n < length l ->
    nth_error (l++l') n = nth_error l n.

  Lemma nth_error_app2 l l' n : length l <= n ->
    nth_error (l++l') n = nth_error l' (n-length l).

  Lemma nth_error_ext l l':
    (forall n, nth_error l n = nth_error l' n) -> l = l'.

  Lemma unfold_nth_error l n
    : nth_error l n
      = match n, l with
        | O, x :: _ => Some x
        | S n, _ :: l => nth_error l n
        | _, _ => None
        end.

  Lemma nth_error_nil n : nth_error nil n = None.

  Lemma nth_error_cons x xs n
    : nth_error (x :: xs) n
      = match n with
        | O => Some x
        | S n => nth_error xs n
        end.

  Lemma nth_error_O l
    : nth_error l O = hd_error l.

  Lemma nth_error_S l n
    : nth_error l (S n) = nth_error (tl l) n.

Results directly relating nth and nth_error

  Lemma nth_error_nth : forall (l : list A) (n : nat) (x d : A),
    nth_error l n = Some x -> nth n l d = x.

  Lemma nth_error_nth' : forall (l : list A) (n : nat) (d : A),
    n < length l -> nth_error l n = Some (nth n l d).

Last element of a list

last l d returns the last element of the list l, or the default value d if l is empty.

  Fixpoint last (l:list A) (d:A) : A :=
  match l with
    | [] => d
    | [a] => a
    | a :: l => last l d
  end.

  Lemma last_last : forall l a d, last (l ++ [a]) d = a.

removelast l remove the last element of l

  Fixpoint removelast (l:list A) : list A :=
    match l with
      | [] => []
      | [a] => []
      | a :: l => a :: removelast l
    end.

  Lemma app_removelast_last :
    forall l d, l <> [] -> l = removelast l ++ [last l d].

  Lemma exists_last :
    forall l, l <> [] -> { l' : (list A) & { a : A | l = l' ++ [a]}}.

  Lemma removelast_app :
    forall l l', l' <> [] -> removelast (l++l') = l ++ removelast l'.

  Lemma removelast_last : forall l a, removelast (l ++ [a]) = l.

Remove


  Hypothesis eq_dec : forall x y : A, {x = y}+{x <> y}.

  Fixpoint remove (x : A) (l : list A) : list A :=
    match l with
      | [] => []
      | y::tl => if (eq_dec x y) then remove x tl else y::(remove x tl)
    end.

  Lemma remove_cons : forall x l, remove x (x :: l) = remove x l.

  Lemma remove_app : forall x l1 l2,
    remove x (l1 ++ l2) = remove x l1 ++ remove x l2.

  Theorem remove_In : forall (l : list A) (x : A), ~ In x (remove x l).

  Lemma notin_remove: forall l x, ~ In x l -> remove x l = l.

  Lemma in_remove: forall l x y, In x (remove y l) -> In x l /\ x <> y.

  Lemma in_in_remove : forall l x y, x <> y -> In x l -> In x (remove y l).

  Lemma remove_remove_comm : forall l x y,
    remove x (remove y l) = remove y (remove x l).

  Lemma remove_remove_eq : forall l x, remove x (remove x l) = remove x l.

  Lemma remove_length_le : forall l x, length (remove x l) <= length l.

  Lemma remove_length_lt : forall l x, In x l -> length (remove x l) < length l.

Counting occurrences of an element


  Fixpoint count_occ (l : list A) (x : A) : nat :=
    match l with
      | [] => 0
      | y :: tl =>
        let n := count_occ tl x in
        if eq_dec y x then S n else n
    end.

Compatibility of count_occ with operations on list
  Theorem count_occ_In l x : In x l <-> count_occ l x > 0.

  Theorem count_occ_not_In l x : ~ In x l <-> count_occ l x = 0.

  Lemma count_occ_nil x : count_occ [] x = 0.

  Theorem count_occ_inv_nil l :
    (forall x:A, count_occ l x = 0) <-> l = [].

  Lemma count_occ_cons_eq l x y :
    x = y -> count_occ (x::l) y = S (count_occ l y).

  Lemma count_occ_cons_neq l x y :
    x <> y -> count_occ (x::l) y = count_occ l y.

  Lemma count_occ_app l1 l2 x :
    count_occ (l1 ++ l2) x = count_occ l1 x + count_occ l2 x.

  Lemma count_occ_elt_eq l1 l2 x y : x = y ->
    count_occ (l1 ++ x :: l2) y = S (count_occ (l1 ++ l2) y).

  Lemma count_occ_elt_neq l1 l2 x y : x <> y ->
    count_occ (l1 ++ x :: l2) y = count_occ (l1 ++ l2) y.

  Lemma count_occ_bound x l : count_occ l x <= length l.

End Elts.

Manipulating whole lists


Section ListOps.

  Variable A : Type.

Reverse


  Fixpoint rev (l:list A) : list A :=
    match l with
      | [] => []
      | x :: l' => rev l' ++ [x]
    end.

  Lemma rev_app_distr : forall x y:list A, rev (x ++ y) = rev y ++ rev x.

  Remark rev_unit : forall (l:list A) (a:A), rev (l ++ [a]) = a :: rev l.

  Lemma rev_involutive : forall l:list A, rev (rev l) = l.

  Lemma rev_inj (l1 l2: list A):
    rev l1 = rev l2 -> l1 = l2.

  Lemma rev_eq_app : forall l l1 l2, rev l = l1 ++ l2 -> l = rev l2 ++ rev l1.

Reverse Induction Principle on Lists

  Lemma rev_list_ind : forall P:list A-> Prop,
    P [] ->
    (forall (a:A) (l:list A), P (rev l) -> P (rev (a :: l))) ->
    forall l:list A, P (rev l).

  Theorem rev_ind : forall P:list A -> Prop,
    P [] ->
    (forall (x:A) (l:list A), P l -> P (l ++ [x])) -> forall l:list A, P l.

Compatibility with other operations

  Lemma in_rev : forall l x, In x l <-> In x (rev l).

  Lemma rev_length : forall l, length (rev l) = length l.

  Lemma rev_nth : forall l d n, n < length l ->
    nth n (rev l) d = nth (length l - S n) l d.

An alternative tail-recursive definition for reverse

  Fixpoint rev_append (l l': list A) : list A :=
    match l with
      | [] => l'
      | a::l => rev_append l (a::l')
    end.

  Definition rev' l : list A := rev_append l [].

  Lemma rev_append_rev : forall l l', rev_append l l' = rev l ++ l'.

  Lemma rev_alt : forall l, rev l = rev_append l [].

Concatenation


  Fixpoint concat (l : list (list A)) : list A :=
  match l with
  | nil => nil
  | cons x l => x ++ concat l
  end.

  Lemma concat_nil : concat nil = nil.

  Lemma concat_cons : forall x l, concat (cons x l) = x ++ concat l.

  Lemma concat_app : forall l1 l2, concat (l1 ++ l2) = concat l1 ++ concat l2.

  Lemma in_concat : forall l y,
    In y (concat l) <-> exists x, In x l /\ In y x.

Decidable equality on lists


  Hypothesis eq_dec : forall (x y : A), {x = y}+{x <> y}.

  Lemma list_eq_dec : forall l l':list A, {l = l'} + {l <> l'}.

  Lemma count_occ_rev l x : count_occ eq_dec (rev l) x = count_occ eq_dec l x.

End ListOps.

Applying functions to the elements of a list

Map


Section Map.
  Variables (A : Type) (B : Type).
  Variable f : A -> B.

  Fixpoint map (l:list A) : list B :=
    match l with
      | [] => []
      | a :: t => (f a) :: (map t)
    end.

  Lemma map_cons (x:A)(l:list A) : map (x::l) = (f x) :: (map l).

  Lemma in_map :
    forall (l:list A) (x:A), In x l -> In (f x) (map l).

  Lemma in_map_iff : forall l y, In y (map l) <-> exists x, f x = y /\ In x l.

  Lemma map_length : forall l, length (map l) = length l.

  Lemma map_nth : forall l d n,
    nth n (map l) (f d) = f (nth n l d).

  Lemma nth_error_map : forall n l,
    nth_error (map l) n = option_map f (nth_error l n).

  Lemma map_nth_error : forall n l d,
    nth_error l n = Some d -> nth_error (map l) n = Some (f d).

  Lemma map_app : forall l l',
    map (l++l') = (map l)++(map l').

  Lemma map_last : forall l a,
    map (l ++ [a]) = (map l) ++ [f a].

  Lemma map_rev : forall l, map (rev l) = rev (map l).

  Lemma map_eq_nil : forall l, map l = [] -> l = [].

  Lemma map_eq_cons : forall l l' b,
    map l = b :: l' -> exists a tl, l = a :: tl /\ f a = b /\ map tl = l'.

  Lemma map_eq_app : forall l l1 l2,
    map l = l1 ++ l2 -> exists l1' l2', l = l1' ++ l2' /\ map l1' = l1 /\ map l2' = l2.

map and count of occurrences

  Hypothesis decA: forall x1 x2 : A, {x1 = x2} + {x1 <> x2}.
  Hypothesis decB: forall y1 y2 : B, {y1 = y2} + {y1 <> y2}.
  Hypothesis Hfinjective: forall x1 x2: A, (f x1) = (f x2) -> x1 = x2.

  Theorem count_occ_map x l:
    count_occ decA l x = count_occ decB (map l) (f x).

End Map.

Flat Map


Section FlatMap.
  Variables (A : Type) (B : Type).
  Variable f : A -> list B.

flat_map

    Definition flat_map :=
      fix flat_map (l:list A) : list B :=
      match l with
        | nil => nil
        | cons x t => (f x)++(flat_map t)
      end.

    Lemma flat_map_concat_map l :
      flat_map l = concat (map f l).

    Lemma flat_map_app l1 l2 :
      flat_map (l1 ++ l2) = flat_map l1 ++ flat_map l2.

    Lemma in_flat_map l y :
      In y (flat_map l) <-> exists x, In x l /\ In y (f x).

End FlatMap.

Lemma concat_map : forall A B (f : A -> B) l, map f (concat l) = concat (map (map f) l).

Lemma remove_concat A (eq_dec : forall x y : A, {x = y}+{x <> y}) : forall l x,
  remove eq_dec x (concat l) = flat_map (remove eq_dec x) l.

Lemma map_id : forall (A :Type) (l : list A),
  map (fun x => x) l = l.

Lemma map_map : forall (A B C:Type)(f:A->B)(g:B->C) l,
  map g (map f l) = map (fun x => g (f x)) l.

Lemma map_ext_in :
  forall (A B : Type)(f g:A->B) l, (forall a, In a l -> f a = g a) -> map f l = map g l.

Lemma ext_in_map :
  forall (A B : Type)(f g:A->B) l, map f l = map g l -> forall a, In a l -> f a = g a.

Arguments ext_in_map [A B f g l].

Lemma map_ext_in_iff :
   forall (A B : Type)(f g:A->B) l, map f l = map g l <-> forall a, In a l -> f a = g a.

Arguments map_ext_in_iff {A B f g l}.

Lemma map_ext :
  forall (A B : Type)(f g:A->B), (forall a, f a = g a) -> forall l, map f l = map g l.

Lemma flat_map_ext : forall (A B : Type)(f g : A -> list B),
  (forall a, f a = g a) -> forall l, flat_map f l = flat_map g l.

Lemma nth_nth_nth_map A : forall (l : list A) n d ln dn, n < length ln \/ length l <= dn ->
  nth (nth n ln dn) l d = nth n (map (fun x => nth x l d) ln) d.

Left-to-right iterator on lists

Section Fold_Left_Recursor.
  Variables (A : Type) (B : Type).
  Variable f : A -> B -> A.

  Fixpoint fold_left (l:list B) (a0:A) : A :=
    match l with
      | nil => a0
      | cons b t => fold_left t (f a0 b)
    end.

  Lemma fold_left_app : forall (l l':list B)(i:A),
    fold_left (l++l') i = fold_left l' (fold_left l i).

End Fold_Left_Recursor.

Lemma fold_left_length :
  forall (A:Type)(l:list A), fold_left (fun x _ => S x) l 0 = length l.

Right-to-left iterator on lists

Section Fold_Right_Recursor.
  Variables (A : Type) (B : Type).
  Variable f : B -> A -> A.
  Variable a0 : A.

  Fixpoint fold_right (l:list B) : A :=
    match l with
      | nil => a0
      | cons b t => f b (fold_right t)
    end.

End Fold_Right_Recursor.

  Lemma fold_right_app : forall (A B:Type)(f:A->B->B) l l' i,
    fold_right f i (l++l') = fold_right f (fold_right f i l') l.

  Lemma fold_left_rev_right : forall (A B:Type)(f:A->B->B) l i,
    fold_right f i (rev l) = fold_left (fun x y => f y x) l i.

  Theorem fold_symmetric :
    forall (A : Type) (f : A -> A -> A),
    (forall x y z : A, f x (f y z) = f (f x y) z) ->
    forall (a0 : A), (forall y : A, f a0 y = f y a0) ->
    forall (l : list A), fold_left f l a0 = fold_right f a0 l.

(list_power x y) is y^x, or the set of sequences of elts of y indexed by elts of x, sorted in lexicographic order.

  Fixpoint list_power (A B:Type)(l:list A) (l':list B) :
    list (list (A * B)) :=
    match l with
      | nil => cons nil nil
      | cons x t =>
        flat_map (fun f:list (A * B) => map (fun y:B => cons (x, y) f) l')
        (list_power t l')
    end.

Boolean operations over lists


  Section Bool.
    Variable A : Type.
    Variable f : A -> bool.

find whether a boolean function can be satisfied by an elements of the list.

    Fixpoint existsb (l:list A) : bool :=
      match l with
      | nil => false
      | a::l => f a || existsb l
      end.

    Lemma existsb_exists :
      forall l, existsb l = true <-> exists x, In x l /\ f x = true.

    Lemma existsb_nth : forall l n d, n < length l ->
      existsb l = false -> f (nth n l d) = false.

    Lemma existsb_app : forall l1 l2,
      existsb (l1++l2) = existsb l1 || existsb l2.

find whether a boolean function is satisfied by all the elements of a list.

    Fixpoint forallb (l:list A) : bool :=
      match l with
      | nil => true
      | a::l => f a && forallb l
      end.

    Lemma forallb_forall :
      forall l, forallb l = true <-> (forall x, In x l -> f x = true).

    Lemma forallb_app :
      forall l1 l2, forallb (l1++l2) = forallb l1 && forallb l2.

filter

    Fixpoint filter (l:list A) : list A :=
      match l with
      | nil => nil
      | x :: l => if f x then x::(filter l) else filter l
      end.

    Lemma filter_In : forall x l, In x (filter l) <-> In x l /\ f x = true.

    Lemma filter_app (l l':list A) :
      filter (l ++ l') = filter l ++ filter l'.

    Lemma concat_filter_map : forall (l : list (list A)),
      concat (map filter l) = filter (concat l).

    Lemma forallb_filter l: forallb (filter l) = true.

    Lemma forallb_filter_id l: forallb l = true -> filter l = l.

find

    Fixpoint find (l:list A) : option A :=
      match l with
      | nil => None
      | x :: tl => if f x then Some x else find tl
      end.

    Lemma find_some l x : find l = Some x -> In x l /\ f x = true.

    Lemma find_none l : find l = None -> forall x, In x l -> f x = false.

partition

    Fixpoint partition (l:list A) : list A * list A :=
      match l with
      | nil => (nil, nil)
      | x :: tl => let (g,d) := partition tl in
                   if f x then (x::g,d) else (g,x::d)
      end.

  Theorem partition_cons1 a l l1 l2:
    partition l = (l1, l2) ->
    f a = true ->
    partition (a::l) = (a::l1, l2).

  Theorem partition_cons2 a l l1 l2:
    partition l = (l1, l2) ->
    f a=false ->
    partition (a::l) = (l1, a::l2).

  Theorem partition_length l l1 l2:
    partition l = (l1, l2) ->
    length l = length l1 + length l2.

  Theorem partition_inv_nil (l : list A):
    partition l = ([], []) <-> l = [].

  Theorem elements_in_partition l l1 l2:
    partition l = (l1, l2) ->
    forall x:A, In x l <-> In x l1 \/ In x l2.

  End Bool.

Further filtering facts


  Section Filtering.
    Variables (A : Type).

    Lemma filter_ext_in : forall (f g : A -> bool) (l : list A),
      (forall a, In a l -> f a = g a) -> filter f l = filter g l.

    Lemma ext_in_filter : forall (f g : A -> bool) (l : list A),
      filter f l = filter g l -> (forall a, In a l -> f a = g a).

    Lemma filter_ext_in_iff : forall (f g : A -> bool) (l : list A),
      filter f l = filter g l <-> (forall a, In a l -> f a = g a).

    Lemma filter_map : forall (f g : A -> bool) (l : list A),
      filter f l = filter g l <-> map f l = map g l.

    Lemma filter_ext : forall (f g : A -> bool),
      (forall a, f a = g a) -> forall l, filter f l = filter g l.

    Lemma partition_as_filter f (l : list A) : partition f l = (filter f l, filter (fun x => negb (f x)) l).

    Corollary filter_length f (l : list A) : length (filter f l) + length (filter (fun x => negb (f x)) l) = length l.

    Corollary filter_length_le f (l : list A): length (filter f l) <= length l.

    Lemma filter_length_forallb f (l : list A): length (filter f l) = length l -> forallb f l = true.

Remove by filtering

    Hypothesis eq_dec : forall x y : A, {x = y}+{x <> y}.

    Definition remove' (x : A) : list A -> list A :=
      filter (fun y => if eq_dec x y then false else true).

    Lemma remove_alt (x : A) (l : list A) : remove' x l = remove eq_dec x l.

Counting occurrences by filtering

    Definition count_occ' (l : list A) (x : A) : nat :=
      length (filter (fun y => if eq_dec y x then true else false) l).

    Lemma count_occ_alt (l : list A) (x : A) :
      count_occ' l x = count_occ eq_dec l x.

  End Filtering.

Operations on lists of pairs or lists of lists


  Section ListPairs.
    Variables (A : Type) (B : Type).

split derives two lists from a list of pairs

    Fixpoint split (l:list (A*B)) : list A * list B :=
      match l with
      | [] => ([], [])
      | (x,y) :: tl => let (left,right) := split tl in (x::left, y::right)
      end.

    Lemma in_split_l : forall (l:list (A*B))(p:A*B),
      In p l -> In (fst p) (fst (split l)).

    Lemma in_split_r : forall (l:list (A*B))(p:A*B),
      In p l -> In (snd p) (snd (split l)).

    Lemma split_nth : forall (l:list (A*B))(n:nat)(d:A*B),
      nth n l d = (nth n (fst (split l)) (fst d), nth n (snd (split l)) (snd d)).

    Lemma split_length_l : forall (l:list (A*B)),
      length (fst (split l)) = length l.

    Lemma split_length_r : forall (l:list (A*B)),
      length (snd (split l)) = length l.

combine is the opposite of split. Lists given to combine are meant to be of same length. If not, combine stops on the shorter list

    Fixpoint combine (l : list A) (l' : list B) : list (A*B) :=
      match l,l' with
      | x::tl, y::tl' => (x,y)::(combine tl tl')
      | _, _ => nil
      end.

    Lemma split_combine : forall (l: list (A*B)),
      forall l1 l2, split l = (l1, l2) -> combine l1 l2 = l.

    Lemma combine_split : forall (l:list A)(l':list B), length l = length l' ->
      split (combine l l') = (l,l').

    Lemma in_combine_l : forall (l:list A)(l':list B)(x:A)(y:B),
      In (x,y) (combine l l') -> In x l.

    Lemma in_combine_r : forall (l:list A)(l':list B)(x:A)(y:B),
      In (x,y) (combine l l') -> In y l'.

    Lemma combine_length : forall (l:list A)(l':list B),
      length (combine l l') = min (length l) (length l').

    Lemma combine_nth : forall (l:list A)(l':list B)(n:nat)(x:A)(y:B),
      length l = length l' ->
      nth n (combine l l') (x,y) = (nth n l x, nth n l' y).

list_prod has the same signature as combine, but unlike combine, it adds every possible pairs, not only those at the same position.

    Fixpoint list_prod (l:list A) (l':list B) :
      list (A * B) :=
      match l with
      | nil => nil
      | cons x t => (map (fun y:B => (x, y)) l')++(list_prod t l')
      end.

    Lemma in_prod_aux :
      forall (x:A) (y:B) (l:list B),
        In y l -> In (x, y) (map (fun y0:B => (x, y0)) l).

    Lemma in_prod :
      forall (l:list A) (l':list B) (x:A) (y:B),
        In x l -> In y l' -> In (x, y) (list_prod l l').

    Lemma in_prod_iff :
      forall (l:list A)(l':list B)(x:A)(y:B),
        In (x,y) (list_prod l l') <-> In x l /\ In y l'.

    Lemma prod_length : forall (l:list A)(l':list B),
      length (list_prod l l') = (length l) * (length l').

  End ListPairs.

Miscellaneous operations on lists

Length order of lists


Section length_order.
  Variable A : Type.

  Definition lel (l m:list A) := length l <= length m.

  Variables a b : A.
  Variables l m n : list A.

  Lemma lel_refl : lel l l.

  Lemma lel_trans : lel l m -> lel m n -> lel l n.

  Lemma lel_cons_cons : lel l m -> lel (a :: l) (b :: m).

  Lemma lel_cons : lel l m -> lel l (b :: m).

  Lemma lel_tail : lel (a :: l) (b :: m) -> lel l m.

  Lemma lel_nil : forall l':list A, lel l' nil -> nil = l'.
End length_order.

#[global]
Hint Resolve lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons:
  datatypes.

Set inclusion on list


Section SetIncl.

  Variable A : Type.

  Definition incl (l m:list A) := forall a:A, In a l -> In a m.
  #[local]
  Hint Unfold incl : core.

  Lemma incl_nil_l : forall l, incl nil l.

  Lemma incl_l_nil : forall l, incl l nil -> l = nil.

  Lemma incl_refl : forall l:list A, incl l l.
  #[local]
  Hint Resolve incl_refl : core.

  Lemma incl_tl : forall (a:A) (l m:list A), incl l m -> incl l (a :: m).
  #[local]
  Hint Immediate incl_tl : core.

  Lemma incl_tran : forall l m n:list A, incl l m -> incl m n -> incl l n.

  Lemma incl_appl : forall l m n:list A, incl l n -> incl l (n ++ m).
  #[local]
  Hint Immediate incl_appl : core.

  Lemma incl_appr : forall l m n:list A, incl l n -> incl l (m ++ n).
  #[local]
  Hint Immediate incl_appr : core.

  Lemma incl_cons :
    forall (a:A) (l m:list A), In a m -> incl l m -> incl (a :: l) m.
  #[local]
  Hint Resolve incl_cons : core.

  Lemma incl_cons_inv : forall (a:A) (l m:list A),
    incl (a :: l) m -> In a m /\ incl l m.

  Lemma incl_app : forall l m n:list A, incl l n -> incl m n -> incl (l ++ m) n.
  #[local]
  Hint Resolve incl_app : core.

  Lemma incl_app_app : forall l1 l2 m1 m2:list A,
    incl l1 m1 -> incl l2 m2 -> incl (l1 ++ l2) (m1 ++ m2).

  Lemma incl_app_inv : forall l1 l2 m : list A,
    incl (l1 ++ l2) m -> incl l1 m /\ incl l2 m.

  Lemma incl_filter f l : incl (filter f l) l.

  Lemma remove_incl (eq_dec : forall x y : A, {x = y} + {x <> y}) : forall l1 l2 x,
    incl l1 l2 -> incl (remove eq_dec x l1) (remove eq_dec x l2).

End SetIncl.

Lemma incl_map A B (f : A -> B) l1 l2 : incl l1 l2 -> incl (map f l1) (map f l2).

#[global]
Hint Resolve incl_refl incl_tl incl_tran incl_appl incl_appr incl_cons
  incl_app incl_map: datatypes.

Cutting a list at some position


Section Cutting.

  Variable A : Type.

  Fixpoint firstn (n:nat)(l:list A) : list A :=
    match n with
      | 0 => nil
      | S n => match l with
                 | nil => nil
                 | a::l => a::(firstn n l)
               end
    end.

  Lemma firstn_nil n: firstn n [] = [].

  Lemma firstn_cons n a l: firstn (S n) (a::l) = a :: (firstn n l).

  Lemma firstn_all l: firstn (length l) l = l.

  Lemma firstn_all2 n: forall (l:list A), (length l) <= n -> firstn n l = l.

  Lemma firstn_O l: firstn 0 l = [].

  Lemma firstn_le_length n: forall l:list A, length (firstn n l) <= n.

  Lemma firstn_length_le: forall l:list A, forall n:nat,
    n <= length l -> length (firstn n l) = n.

  Lemma firstn_app n:
    forall l1 l2,
    firstn n (l1 ++ l2) = (firstn n l1) ++ (firstn (n - length l1) l2).

  Lemma firstn_app_2 n:
    forall l1 l2,
    firstn ((length l1) + n) (l1 ++ l2) = l1 ++ firstn n l2.

  Lemma firstn_firstn:
    forall l:list A,
    forall i j : nat,
    firstn i (firstn j l) = firstn (min i j) l.

  Fixpoint skipn (n:nat)(l:list A) : list A :=
    match n with
      | 0 => l
      | S n => match l with
                 | nil => nil
                 | a::l => skipn n l
               end
    end.

  Lemma firstn_skipn_comm : forall m n l,
  firstn m (skipn n l) = skipn n (firstn (n + m) l).

  Lemma skipn_firstn_comm : forall m n l,
  skipn m (firstn n l) = firstn (n - m) (skipn m l).

  Lemma skipn_O : forall l, skipn 0 l = l.

  Lemma skipn_nil : forall n, skipn n ([] : list A) = [].

  Lemma skipn_cons n a l: skipn (S n) (a::l) = skipn n l.

  Lemma skipn_all : forall l, skipn (length l) l = nil.

  Lemma skipn_all2 n: forall l, length l <= n -> skipn n l = [].

  Lemma skipn_skipn : forall x y l, skipn x (skipn y l) = skipn (x + y) l.

  Lemma firstn_skipn : forall n l, firstn n l ++ skipn n l = l.

  Lemma firstn_length : forall n l, length (firstn n l) = min n (length l).

  Lemma skipn_length n :
    forall l, length (skipn n l) = length l - n.

  Lemma skipn_app n : forall l1 l2,
    skipn n (l1 ++ l2) = (skipn n l1) ++ (skipn (n - length l1) l2).

  Lemma firstn_skipn_rev: forall x l,
      firstn x l = rev (skipn (length l - x) (rev l)).

  Lemma firstn_rev: forall x l,
    firstn x (rev l) = rev (skipn (length l - x) l).

  Lemma skipn_rev: forall x l,
      skipn x (rev l) = rev (firstn (length l - x) l).

  Lemma removelast_firstn : forall n l, n < length l ->
    removelast (firstn (S n) l) = firstn n l.

  Lemma removelast_firstn_len : forall l,
    removelast l = firstn (pred (length l)) l.

  Lemma firstn_removelast : forall n l, n < length l ->
    firstn n (removelast l) = firstn n l.

End Cutting.

Section CuttingMap.
  Variables A B : Type.
  Variable f : A -> B.

  Lemma firstn_map : forall n l,
      firstn n (map f l) = map f (firstn n l).

  Lemma skipn_map : forall n l,
      skipn n (map f l) = map f (skipn n l).
End CuttingMap.

Combining pairs of lists of possibly-different lengths


Section Combining.
    Variables (A B : Type).

    Lemma combine_nil : forall (l : list A),
      combine l (@nil B) = @nil (A*B).

    Lemma combine_firstn_l : forall (l : list A) (l' : list B),
      combine l l' = combine l (firstn (length l) l').

    Lemma combine_firstn_r : forall (l : list A) (l' : list B),
      combine l l' = combine (firstn (length l') l) l'.

    Lemma combine_firstn : forall (l : list A) (l' : list B) (n : nat),
      firstn n (combine l l') = combine (firstn n l) (firstn n l').

End Combining.

Predicate for List addition/removal (no need for decidability)


Section Add.

  Variable A : Type.

  Inductive Add (a:A) : list A -> list A -> Prop :=
    | Add_head l : Add a l (a::l)
    | Add_cons x l l' : Add a l l' -> Add a (x::l) (x::l').

  Lemma Add_app a l1 l2 : Add a (l1++l2) (l1++a::l2).

  Lemma Add_split a l l' :
    Add a l l' -> exists l1 l2, l = l1++l2 /\ l' = l1++a::l2.

  Lemma Add_in a l l' : Add a l l' ->
   forall x, In x l' <-> In x (a::l).

  Lemma Add_length a l l' : Add a l l' -> length l' = S (length l).

  Lemma Add_inv a l : In a l -> exists l', Add a l' l.

  Lemma incl_Add_inv a l u v :
    ~In a l -> incl (a::l) v -> Add a u v -> incl l u.

End Add.

Lists without redundancy


Section ReDun.

  Variable A : Type.

  Inductive NoDup : list A -> Prop :=
    | NoDup_nil : NoDup nil
    | NoDup_cons : forall x l, ~ In x l -> NoDup l -> NoDup (x::l).

  Lemma NoDup_Add a l l' : Add a l l' -> (NoDup l' <-> NoDup l /\ ~In a l).

  Lemma NoDup_remove l l' a :
    NoDup (l++a::l') -> NoDup (l++l') /\ ~In a (l++l').

  Lemma NoDup_remove_1 l l' a : NoDup (l++a::l') -> NoDup (l++l').

  Lemma NoDup_remove_2 l l' a : NoDup (l++a::l') -> ~In a (l++l').

  Theorem NoDup_cons_iff a l:
    NoDup (a::l) <-> ~ In a l /\ NoDup l.

  Lemma NoDup_app_remove_l l l' : NoDup (l++l') -> NoDup l'.

  Lemma NoDup_app_remove_r l l' : NoDup (l++l') -> NoDup l.

  Lemma NoDup_rev l : NoDup l -> NoDup (rev l).

  Lemma NoDup_filter f l : NoDup l -> NoDup (filter f l).

Effective computation of a list without duplicates

  Hypothesis decA: forall x y : A, {x = y} + {x <> y}.

  Fixpoint nodup (l : list A) : list A :=
    match l with
      | [] => []
      | x::xs => if in_dec decA x xs then nodup xs else x::(nodup xs)
    end.

  Lemma nodup_fixed_point (l : list A) :
    NoDup l -> nodup l = l.

  Lemma nodup_In l x : In x (nodup l) <-> In x l.

  Lemma nodup_incl l1 l2 : incl l1 (nodup l2) <-> incl l1 l2.

  Lemma NoDup_nodup l: NoDup (nodup l).

  Lemma nodup_inv k l a : nodup k = a :: l -> ~ In a l.

  Theorem NoDup_count_occ l:
    NoDup l <-> (forall x:A, count_occ decA l x <= 1).

  Theorem NoDup_count_occ' l:
    NoDup l <-> (forall x:A, In x l -> count_occ decA l x = 1).

Alternative characterisations of being without duplicates, thanks to nth_error and nth

  Lemma NoDup_nth_error l :
    NoDup l <->
    (forall i j, i<length l -> nth_error l i = nth_error l j -> i = j).
  Lemma NoDup_nth l d :
    NoDup l <->
    (forall i j, i<length l -> j<length l ->
       nth i l d = nth j l d -> i = j).

Having NoDup hypotheses bring more precise facts about incl.

  Lemma NoDup_incl_length l l' :
    NoDup l -> incl l l' -> length l <= length l'.

  Lemma NoDup_length_incl l l' :
    NoDup l -> length l' <= length l -> incl l l' -> incl l' l.

  Lemma NoDup_incl_NoDup (l l' : list A) : NoDup l ->
    length l' <= length l -> incl l l' -> NoDup l'.

End ReDun.

NoDup and map
NB: the reciprocal result holds only for injective functions, see FinFun.v

Lemma NoDup_map_inv A B (f:A->B) l : NoDup (map f l) -> NoDup l.

Sequence of natural numbers


Section NatSeq.

seq computes the sequence of len contiguous integers that starts at start. For instance, seq 2 3 is 2::3::4::nil.

  Fixpoint seq (start len:nat) : list nat :=
    match len with
      | 0 => nil
      | S len => start :: seq (S start) len
    end.

  Lemma cons_seq : forall len start, start :: seq (S start) len = seq start (S len).

  Lemma seq_length : forall len start, length (seq start len) = len.

  Lemma seq_nth : forall len start n d,
    n < len -> nth n (seq start len) d = start+n.

  Lemma seq_shift : forall len start,
    map S (seq start len) = seq (S start) len.

  Lemma in_seq len start n :
    In n (seq start len) <-> start <= n < start+len.

  Lemma seq_NoDup len start : NoDup (seq start len).

  Lemma seq_app : forall len1 len2 start,
    seq start (len1 + len2) = seq start len1 ++ seq (start + len1) len2.

  Lemma seq_S : forall len start, seq start (S len) = seq start len ++ [start + len].

End NatSeq.

Section Exists_Forall.

Existential and universal predicates over lists


  Variable A:Type.

  Section One_predicate.

    Variable P:A->Prop.

    Inductive Exists : list A -> Prop :=
      | Exists_cons_hd : forall x l, P x -> Exists (x::l)
      | Exists_cons_tl : forall x l, Exists l -> Exists (x::l).

    #[local]
    Hint Constructors Exists : core.

    Lemma Exists_exists (l:list A) :
      Exists l <-> (exists x, In x l /\ P x).

    Lemma Exists_nth l :
      Exists l <-> exists i d, i < length l /\ P (nth i l d).

    Lemma Exists_nil : Exists nil <-> False.

    Lemma Exists_cons x l:
      Exists (x::l) <-> P x \/ Exists l.

    Lemma Exists_app l1 l2 :
      Exists (l1 ++ l2) <-> Exists l1 \/ Exists l2.

    Lemma Exists_rev l : Exists l -> Exists (rev l).

    Lemma Exists_dec l:
      (forall x:A, {P x} + { ~ P x }) ->
      {Exists l} + {~ Exists l}.

    Lemma Exists_fold_right l :
      Exists l <-> fold_right (fun x => or (P x)) False l.

    Lemma incl_Exists l1 l2 : incl l1 l2 -> Exists l1 -> Exists l2.

    Inductive Forall : list A -> Prop :=
      | Forall_nil : Forall nil
      | Forall_cons : forall x l, P x -> Forall l -> Forall (x::l).

    #[local]
    Hint Constructors Forall : core.

    Lemma Forall_inv : forall (a:A) l, Forall (a :: l) -> P a.

    Theorem Forall_inv_tail : forall (a:A) l, Forall (a :: l) -> Forall l.

    Lemma Forall_nil_iff : Forall [] <-> True.

    Lemma Forall_cons_iff : forall (a:A) l, Forall (a :: l) <-> P a /\ Forall l.

    Lemma Forall_forall (l:list A):
      Forall l <-> (forall x, In x l -> P x).

    Lemma Forall_nth l :
      Forall l <-> forall i d, i < length l -> P (nth i l d).

    Lemma Forall_app l1 l2 :
      Forall (l1 ++ l2) <-> Forall l1 /\ Forall l2.

    Lemma Forall_elt a l1 l2 : Forall (l1 ++ a :: l2) -> P a.

    Lemma Forall_rev l : Forall l -> Forall (rev l).

    Lemma Forall_rect : forall (Q : list A -> Type),
      Q [] -> (forall b l, P b -> Q (b :: l)) -> forall l, Forall l -> Q l.

    Lemma Forall_dec :
      (forall x:A, {P x} + { ~ P x }) ->
      forall l:list A, {Forall l} + {~ Forall l}.

    Lemma Forall_fold_right l :
      Forall l <-> fold_right (fun x => and (P x)) True l.

    Lemma incl_Forall l1 l2 : incl l2 l1 -> Forall l1 -> Forall l2.

  End One_predicate.

  Lemma map_ext_Forall B : forall (f g : A -> B) l,
    Forall (fun x => f x = g x) l -> map f l = map g l.

  Theorem Exists_impl : forall (P Q : A -> Prop), (forall a : A, P a -> Q a) ->
    forall l, Exists P l -> Exists Q l.

  Lemma Exists_or : forall (P Q : A -> Prop) l,
    Exists P l \/ Exists Q l -> Exists (fun x => P x \/ Q x) l.

  Lemma Exists_or_inv : forall (P Q : A -> Prop) l,
    Exists (fun x => P x \/ Q x) l -> Exists P l \/ Exists Q l.

  Lemma Forall_impl : forall (P Q : A -> Prop), (forall a, P a -> Q a) ->
    forall l, Forall P l -> Forall Q l.

  Lemma Forall_and : forall (P Q : A -> Prop) l,
    Forall P l -> Forall Q l -> Forall (fun x => P x /\ Q x) l.

  Lemma Forall_and_inv : forall (P Q : A -> Prop) l,
    Forall (fun x => P x /\ Q x) l -> Forall P l /\ Forall Q l.

  Lemma Forall_Exists_neg (P:A->Prop)(l:list A) :
    Forall (fun x => ~ P x) l <-> ~(Exists P l).

  Lemma Exists_Forall_neg (P:A->Prop)(l:list A) :
    (forall x, P x \/ ~P x) ->
    Exists (fun x => ~ P x) l <-> ~(Forall P l).

  Lemma neg_Forall_Exists_neg (P:A->Prop) (l:list A) :
    (forall x:A, {P x} + { ~ P x }) ->
    ~ Forall P l ->
    Exists (fun x => ~ P x) l.

  Lemma Forall_Exists_dec (P:A->Prop) :
    (forall x:A, {P x} + { ~ P x }) ->
    forall l:list A,
    {Forall P l} + {Exists (fun x => ~ P x) l}.

  Lemma incl_Forall_in_iff l l' :
    incl l l' <-> Forall (fun x => In x l') l.

End Exists_Forall.

#[global]
Hint Constructors Exists : core.
#[global]
Hint Constructors Forall : core.

Lemma Exists_map A B (f : A -> B) P l :
  Exists P (map f l) <-> Exists (fun x => P (f x)) l.

Lemma Exists_concat A P (ls : list (list A)) :
  Exists P (concat ls) <-> Exists (Exists P) ls.

Lemma Exists_flat_map A B P ls (f : A -> list B) :
  Exists P (flat_map f ls) <-> Exists (fun d => Exists P (f d)) ls.

Lemma Forall_map A B (f : A -> B) P l :
  Forall P (map f l) <-> Forall (fun x => P (f x)) l.

Lemma Forall_concat A P (ls : list (list A)) :
  Forall P (concat ls) <-> Forall (Forall P) ls.

Lemma Forall_flat_map A B P ls (f : A -> list B) :
  Forall P (flat_map f ls) <-> Forall (fun d => Forall P (f d)) ls.

Lemma exists_Forall A B : forall (P : A -> B -> Prop) l,
  (exists k, Forall (P k) l) -> Forall (fun x => exists k, P k x) l.

Lemma Forall_image A B : forall (f : A -> B) l,
  Forall (fun y => exists x, y = f x) l <-> exists l', l = map f l'.

Lemma concat_nil_Forall A : forall (l : list (list A)),
  concat l = nil <-> Forall (fun x => x = nil) l.

Lemma in_flat_map_Exists A B : forall (f : A -> list B) x l,
  In x (flat_map f l) <-> Exists (fun y => In x (f y)) l.

Lemma notin_flat_map_Forall A B : forall (f : A -> list B) x l,
  ~ In x (flat_map f l) <-> Forall (fun y => ~ In x (f y)) l.

Section Forall2.

Forall2: stating that elements of two lists are pairwise related.

  Variables A B : Type.
  Variable R : A -> B -> Prop.

  Inductive Forall2 : list A -> list B -> Prop :=
    | Forall2_nil : Forall2 [] []
    | Forall2_cons : forall x y l l',
      R x y -> Forall2 l l' -> Forall2 (x::l) (y::l').

  #[local]
  Hint Constructors Forall2 : core.

  #[deprecated(since = "8.18", note = "Use Forall2_nil instead.")]
  Theorem Forall2_refl : Forall2 [] [].

  Theorem Forall2_cons_iff : forall x y l l',
    Forall2 (x :: l) (y :: l') <-> R x y /\ Forall2 l l'.

  Theorem Forall2_length : forall l l',
    Forall2 l l' -> length l = length l'.

  Theorem Forall2_app_inv_l : forall l1 l2 l',
    Forall2 (l1 ++ l2) l' ->
    exists l1' l2', Forall2 l1 l1' /\ Forall2 l2 l2' /\ l' = l1' ++ l2'.

  Theorem Forall2_app_inv_r : forall l1' l2' l,
    Forall2 l (l1' ++ l2') ->
    exists l1 l2, Forall2 l1 l1' /\ Forall2 l2 l2' /\ l = l1 ++ l2.

  Theorem Forall2_app : forall l1 l2 l1' l2',
    Forall2 l1 l1' -> Forall2 l2 l2' -> Forall2 (l1 ++ l2) (l1' ++ l2').

  Theorem Forall_Exists_exists_Forall2 l1 l2 :
    Forall (fun a => Exists (R a) l2) l1 ->
    exists l2', Forall2 l1 l2' /\ incl l2' l2.
End Forall2.

Lemma Forall2_impl (A B : Type) (R1 R2 : A -> B -> Prop) : (forall a b, R1 a b -> R2 a b) ->
  forall l1 l2, Forall2 R1 l1 l2 -> Forall2 R2 l1 l2.

Lemma Forall2_flip (A B : Type) (R : A -> B -> Prop) l1 l2 :
  Forall2 R l1 l2 -> Forall2 (fun b a => R a b) l2 l1.

#[global]
Hint Constructors Forall2 : core.

Section ForallPairs.

ForallPairs : specifies that a certain relation should always hold when inspecting all possible pairs of elements of a list.

  Variable A : Type.
  Variable R : A -> A -> Prop.

  Definition ForallPairs l :=
    forall a b, In a l -> In b l -> R a b.

ForallOrdPairs : we still check a relation over all pairs of elements of a list, but now the order of elements matters.

  Inductive ForallOrdPairs : list A -> Prop :=
    | FOP_nil : ForallOrdPairs nil
    | FOP_cons : forall a l,
      Forall (R a) l -> ForallOrdPairs l -> ForallOrdPairs (a::l).

  #[local]
  Hint Constructors ForallOrdPairs : core.

  Lemma ForallOrdPairs_In : forall l,
    ForallOrdPairs l ->
    forall x y, In x l -> In y l -> x=y \/ R x y \/ R y x.

ForallPairs implies ForallOrdPairs. The reverse implication is true only when R is symmetric and reflexive.

  Lemma ForallPairs_ForallOrdPairs l: ForallPairs l -> ForallOrdPairs l.

  Lemma ForallOrdPairs_ForallPairs :
    (forall x, R x x) ->
    (forall x y, R x y -> R y x) ->
    forall l, ForallOrdPairs l -> ForallPairs l.
End ForallPairs.

Section Repeat.

  Variable A : Type.
  Fixpoint repeat (x : A) (n: nat ) :=
    match n with
      | O => []
      | S k => x::(repeat x k)
    end.

  Theorem repeat_length x n:
    length (repeat x n) = n.

  Theorem repeat_spec n x y:
    In y (repeat x n) -> y=x.

  Lemma repeat_cons n a :
    a :: repeat a n = repeat a n ++ (a :: nil).

  Lemma repeat_app x n m :
    repeat x (n + m) = repeat x n ++ repeat x m.

  Lemma repeat_eq_app x n l1 l2 :
    repeat x n = l1 ++ l2 -> repeat x (length l1) = l1 /\ repeat x (length l2) = l2.

  Lemma repeat_eq_cons x y n l :
    repeat x n = y :: l -> x = y /\ repeat x (pred n) = l.

  Lemma repeat_eq_elt x y n l1 l2 :
    repeat x n = l1 ++ y :: l2 -> x = y /\ repeat x (length l1) = l1 /\ repeat x (length l2) = l2.

  Lemma Forall_eq_repeat x l :
    Forall (eq x) l -> l = repeat x (length l).

  Hypothesis decA : forall x y : A, {x = y}+{x <> y}.

  Lemma count_occ_repeat_eq x y n : x = y -> count_occ decA (repeat y n) x = n.

  Lemma count_occ_repeat_neq x y n : x <> y -> count_occ decA (repeat y n) x = 0.

  Lemma count_occ_unique x l : count_occ decA l x = length l -> l = repeat x (length l).

  Lemma count_occ_repeat_excl x l :
    (forall y, y <> x -> count_occ decA l y = 0) -> l = repeat x (length l).

  Lemma count_occ_sgt l x : l = x :: nil <->
    count_occ decA l x = 1 /\ forall y, y <> x -> count_occ decA l y = 0.

  Lemma nth_repeat a m n :
    nth n (repeat a m) a = a.

  Lemma nth_error_repeat a m n :
    n < m -> nth_error (repeat a m) n = Some a.

End Repeat.

Lemma repeat_to_concat A n (a:A) :
  repeat a n = concat (repeat [a] n).

Lemma map_repeat A B (a:A) n (f : A -> B):
  map f (repeat a n) = repeat (f a) n.

Lemma rev_repeat A n (a:A):
  rev (repeat a n) = repeat a n.

Sum of elements of a list of nat: list_sum

Definition list_sum l := fold_right plus 0 l.

Lemma list_sum_app : forall l1 l2,
   list_sum (l1 ++ l2) = list_sum l1 + list_sum l2.

Lemma concat_length A l:
  length (concat l) = list_sum (map (@length A) l).

Lemma flat_map_length A B (f: A -> list B) l:
  length (flat_map f l) = list_sum (map (fun x => length (f x)) l).

Corollary flat_map_constant_length A B c (f: A -> list B) l:
  (forall x, In x l -> length (f x) = c) -> length (flat_map f l) = (length l) * c.

Lemma list_power_length (A B:Type)(l:list A) (l':list B):
    length (list_power l l') = (length l')^(length l).

Max of elements of a list of nat: list_max

Definition list_max l := fold_right max 0 l.

Lemma list_max_app : forall l1 l2,
   list_max (l1 ++ l2) = max (list_max l1) (list_max l2).

Lemma list_max_le : forall l n,
  list_max l <= n <-> Forall (fun k => k <= n) l.

Lemma list_max_lt : forall l n, l <> nil ->
  list_max l < n <-> Forall (fun k => k < n) l.

Inversion of predicates over lists based on head symbol


Ltac is_list_constr c :=
 match c with
  | nil => idtac
  | (_::_) => idtac
  | _ => fail
 end.

Ltac invlist f :=
 match goal with
  | H:f ?l |- _ => is_list_constr l; inversion_clear H; invlist f
  | H:f _ ?l |- _ => is_list_constr l; inversion_clear H; invlist f
  | H:f _ _ ?l |- _ => is_list_constr l; inversion_clear H; invlist f
  | H:f _ _ _ ?l |- _ => is_list_constr l; inversion_clear H; invlist f
  | H:f _ _ _ _ ?l |- _ => is_list_constr l; inversion_clear H; invlist f
  | _ => idtac
 end.

Exporting hints and tactics


Global Hint Rewrite
  rev_involutive
  rev_unit
  map_nth
  map_length
  seq_length
  app_length
  rev_length
  app_nil_r
  : list.

Ltac simpl_list := autorewrite with list.
Ltac ssimpl_list := autorewrite with list using simpl.