(* --------------------------------------------------------------------
 * Copyright (c) - 2012--2015 - IMDEA Software Institute
 * Copyright (c) - 2012--2015 - Inria
 * 
 * Distributed under the terms of the CeCILL-B-V1 license
 * -------------------------------------------------------------------- *)

require import FSet. import Interval.
require import Int.
require import Real.
require import Monoid.

(* temporary workaround *)
op intval : int -> int -> int set = interval.

lemma intval_def i i1 i2:
  mem i (intval i1 i2)  <=> (i1<= i /\ i <= i2)
by [].

lemma intval_card_0 i1 (i2 : int):
  i1 <= i2 =>
  card (intval i1 i2) = i2 - i1 + 1
by [].

lemma intval_card_1 (i1 i2:int):
  i2 < i1 => card (intval i1 i2) = 0
by [].

op int_sum: (int -> real) -> int set -> real = Mrplus.sum.

pred is_cnst (f : int -> 'a) = forall i1 i2, f i1 = f i2.

lemma int_sum_const (f : int -> real) (s: int set):
  is_cnst f => int_sum f s = (card s)%r * f 0.
proof strict.
by intros=> is_cnst;
   rewrite /int_sum (Mrplus.NatMul.sum_const (f 0)) // => x hh;
   apply is_cnst.
qed.