(* --------------------------------------------------------------------
 * Copyright (c) - 2012--2016 - IMDEA Software Institute
 * Copyright (c) - 2012--2021 - Inria
 * Copyright (c) - 2012--2021 - Ecole Polytechnique
 *
 * Distributed under the terms of the CeCILL-B-V1 license
 * -------------------------------------------------------------------- *)

require import AllCore Distr.

type input.
type output.

module type Adv = {
  proc main(_:input) : output 
}.

lemma equiv_not_pr_half (A<:Adv) (Pre: input -> glob A -> bool) (E: glob A -> output -> bool): 
  equiv [A.main ~ A.main : ={arg, glob A} /\ Pre arg{1} (glob A){1} ==> E (glob A){1} res{1} = !E (glob A){2} res{2}] =>
  forall a &m, 
  Pre a (glob A){m} =>
  Pr[A.main(a) @ &m : true] = 1%r =>
  Pr[A.main(a) @ &m : E (glob A) res] = 1%r/2%r.
proof.
  move => he a &m hpre hll.
  have : Pr[A.main(a) @ &m : E (glob A) res] = Pr[A.main(a) @ &m : !E (glob A) res].
  + by byequiv he => // /#.
  by rewrite Pr [mu_not] hll /#. 
qed.