Library Stdlib.Reals.RList

Require Import List.
Require Import Rbase.
Require Import Rfunctions.
Local Open Scope R_scope.

Fixpoint MaxRlist (l:list R) : R :=
  match l with
    | nil => 0
    | a :: l1 =>
      match l1 with
        | nil => a
        | a' :: l2 => Rmax a (MaxRlist l1)
      end
  end.

Fixpoint MinRlist (l:list R) : R :=
  match l with
    | nil => 1
    | a :: l1 =>
      match l1 with
        | nil => a
        | a' :: l2 => Rmin a (MinRlist l1)
      end
  end.

Lemma MaxRlist_P1 : forall (l:list R) (x:R), In x l -> x <= MaxRlist l.

Fixpoint AbsList (l:list R) (x:R) : list R :=
  match l with
    | nil => nil
    | a :: l' => (Rabs (a - x) / 2) :: (AbsList l' x)
  end.

Lemma MinRlist_P1 : forall (l:list R) (x:R), In x l -> MinRlist l <= x.

Lemma AbsList_P1 :
  forall (l:list R) (x y:R), In y l -> In (Rabs (y - x) / 2) (AbsList l x).

Lemma MinRlist_P2 :
  forall l:list R, (forall y:R, In y l -> 0 < y) -> 0 < MinRlist l.

Lemma AbsList_P2 :
  forall (l:list R) (x y:R),
    In y (AbsList l x) -> exists z : R, In z l /\ y = Rabs (z - x) / 2.

Lemma MaxRlist_P2 :
  forall l:list R, (exists y : R, In y l) -> In (MaxRlist l) l.

Fixpoint pos_Rl (l:list R) (i:nat) : R :=
  match l with
    | nil => 0
    | a :: l' => match i with
                     | O => a
                     | S i' => pos_Rl l' i'
                   end
  end.

Lemma pos_Rl_P1 :
  forall (l:list R) (a:R),
    (0 < length l)%nat ->
    pos_Rl (a :: l) (length l) = pos_Rl l (pred (length l)).

Lemma pos_Rl_P2 :
  forall (l:list R) (x:R),
    In x l <-> (exists i : nat, (i < length l)%nat /\ x = pos_Rl l i).

Lemma Rlist_P1 :
  forall (l:list R) (P:R -> R -> Prop),
    (forall x:R, In x l -> exists y : R, P x y) ->
    exists l' : list R,
      length l = length l' /\
      (forall i:nat, (i < length l)%nat -> P (pos_Rl l i) (pos_Rl l' i)).

Definition ordered_Rlist (l:list R) : Prop :=
  forall i:nat, (i < pred (length l))%nat -> pos_Rl l i <= pos_Rl l (S i).

Fixpoint insert (l:list R) (x:R) : list R :=
  match l with
    | nil => x :: nil
    | a :: l' =>
      match Rle_dec a x with
        | left _ => a :: (insert l' x)
        | right _ => x :: l
      end
  end.

Fixpoint cons_ORlist (k l:list R) : list R :=
  match k with
    | nil => l
    | a :: k' => cons_ORlist k' (insert l a)
  end.

Fixpoint mid_Rlist (l:list R) (x:R) : list R :=
  match l with
    | nil => nil
    | a :: l' => ((x + a) / 2) :: (mid_Rlist l' a)
  end.

Definition Rtail (l:list R) : list R :=
  match l with
    | nil => nil
    | a :: l' => l'
  end.

Definition FF (l:list R) (f:R -> R) : list R :=
  match l with
    | nil => nil
    | a :: l' => map f (mid_Rlist l' a)
  end.

Lemma RList_P0 :
  forall (l:list R) (a:R),
    pos_Rl (insert l a) 0 = a \/ pos_Rl (insert l a) 0 = pos_Rl l 0.

Lemma RList_P1 :
  forall (l:list R) (a:R), ordered_Rlist l -> ordered_Rlist (insert l a).

Lemma RList_P2 :
  forall l1 l2:list R, ordered_Rlist l2 -> ordered_Rlist (cons_ORlist l1 l2).

Lemma RList_P3 :
  forall (l:list R) (x:R),
    In x l <-> (exists i : nat, x = pos_Rl l i /\ (i < length l)%nat).

Lemma RList_P4 :
  forall (l1:list R) (a:R), ordered_Rlist (a :: l1) -> ordered_Rlist l1.

Lemma RList_P5 :
  forall (l:list R) (x:R), ordered_Rlist l -> In x l -> pos_Rl l 0 <= x.

Lemma RList_P6 :
  forall l:list R,
    ordered_Rlist l <->
    (forall i j:nat,
      (i <= j)%nat -> (j < length l)%nat -> pos_Rl l i <= pos_Rl l j).

Lemma RList_P7 :
  forall (l:list R) (x:R),
    ordered_Rlist l -> In x l -> x <= pos_Rl l (pred (length l)).

Lemma RList_P8 :
  forall (l:list R) (a x:R), In x (insert l a) <-> x = a \/ In x l.

Lemma RList_P9 :
  forall (l1 l2:list R) (x:R), In x (cons_ORlist l1 l2) <-> In x l1 \/ In x l2.

Lemma RList_P10 :
  forall (l:list R) (a:R), length (insert l a) = S (length l).

Lemma RList_P11 :
  forall l1 l2:list R,
    length (cons_ORlist l1 l2) = (length l1 + length l2)%nat.

Lemma RList_P12 :
  forall (l:list R) (i:nat) (f:R -> R),
    (i < length l)%nat -> pos_Rl (map f l) i = f (pos_Rl l i).

Lemma RList_P13 :
  forall (l:list R) (i:nat) (a:R),
    (i < pred (length l))%nat ->
    pos_Rl (mid_Rlist l a) (S i) = (pos_Rl l i + pos_Rl l (S i)) / 2.

Lemma RList_P14 : forall (l:list R) (a:R), length (mid_Rlist l a) = length l.

Lemma RList_P15 :
  forall l1 l2:list R,
    ordered_Rlist l1 ->
    ordered_Rlist l2 ->
    pos_Rl l1 0 = pos_Rl l2 0 -> pos_Rl (cons_ORlist l1 l2) 0 = pos_Rl l1 0.

Lemma RList_P16 :
  forall l1 l2:list R,
    ordered_Rlist l1 ->
    ordered_Rlist l2 ->
    pos_Rl l1 (pred (length l1)) = pos_Rl l2 (pred (length l2)) ->
    pos_Rl (cons_ORlist l1 l2) (pred (length (cons_ORlist l1 l2))) =
    pos_Rl l1 (pred (length l1)).

Lemma RList_P17 :
  forall (l1:list R) (x:R) (i:nat),
    ordered_Rlist l1 ->
    In x l1 ->
    pos_Rl l1 i < x -> (i < pred (length l1))%nat -> pos_Rl l1 (S i) <= x.

Lemma RList_P18 :
  forall (l:list R) (f:R -> R), length (map f l) = length l.

Lemma RList_P19 :
  forall l:list R,
    l <> nil -> exists r : R, (exists r0 : list R, l = r :: r0).

Lemma RList_P20 :
  forall l:list R,
    (2 <= length l)%nat ->
    exists r : R,
      (exists r1 : R, (exists l' : list R, l = r :: r1 :: l')).

Lemma RList_P21 : forall l l':list R, l = l' -> Rtail l = Rtail l'.

Lemma RList_P22 :
  forall l1 l2:list R, l1 <> nil -> pos_Rl (app l1 l2) 0 = pos_Rl l1 0.

Lemma RList_P24 :
  forall l1 l2:list R,
    l2 <> nil ->
    pos_Rl (app l1 l2) (pred (length (app l1 l2))) =
    pos_Rl l2 (pred (length l2)).

Lemma RList_P25 :
  forall l1 l2:list R,
    ordered_Rlist l1 ->
    ordered_Rlist l2 ->
    pos_Rl l1 (pred (length l1)) <= pos_Rl l2 0 ->
    ordered_Rlist (app l1 l2).

Lemma RList_P26 :
  forall (l1 l2:list R) (i:nat),
    (i < length l1)%nat -> pos_Rl (app l1 l2) i = pos_Rl l1 i.

Lemma RList_P29 :
  forall (l2 l1:list R) (i:nat),
    (length l1 <= i)%nat ->
    (i < length (app l1 l2))%nat ->
    pos_Rl (app l1 l2) i = pos_Rl l2 (i - length l1).