require import Distr Real StdOrder.
(*---*) import RealOrder.

op max (x y : real) = if x <= y then y else x.

type t.

(* We want the bad event to be defined on both sides, *
 * so we assume that all the variables that are used  *
 * to define victory conditions and bad events are    *
 * stored in a separate module. (Note: the empty      *
 * signature could be instantiated with anything,     *
 * including the concrete experiment themselves       *
 * if their glob types match.)                        *)
module type Mem = { }.

module type Exp = {
  proc main(): t
}.

lemma Pr_split (G <: Exp) (Mem <: Mem) (A: (glob Mem) -> t -> bool) (F: (glob Mem) -> t -> bool) &m:
  Pr[G.main() @ &m: A (glob Mem) res /\ F (glob Mem) res]
  + Pr[G.main() @ &m: A (glob Mem) res /\ !F (glob Mem) res]
  = Pr[G.main() @ &m: A (glob Mem) res].
proof.
have <-: Pr[G.main() @ &m: (A (glob Mem) res /\ F (glob Mem) res) \/
                           (A (glob Mem) res /\ !F (glob Mem) res)]
         = Pr[G.main() @ &m: A (glob Mem) res].
+ by rewrite Pr [mu_eq]=> /#.
by rewrite Pr [mu_disjoint]=> /#.
qed.

lemma FundamentalLemma (G1 <: Exp) (G2 <: Exp) (Mem <: Mem)
                       (A: (glob Mem) -> t -> bool) (B: (glob Mem) -> t -> bool)
                       (F: (glob Mem) -> t -> bool) &m:
  Pr[G1.main() @ &m: A (glob Mem) res /\ !F (glob Mem) res]
  = Pr[G2.main() @ &m: B (glob Mem) res /\ !F (glob Mem) res] =>
  `|Pr[G1.main() @ &m: A (glob Mem) res] - Pr[G2.main() @ &m: B (glob Mem) res]|
  <= max Pr[G1.main() @ &m: F (glob Mem) res] Pr[G2.main() @ &m: F (glob Mem) res].
proof.
rewrite -(Pr_split G1 Mem A F &m) -(Pr_split G2 Mem B F &m)=> ->.
have ->: forall (x y z:real), x + y - (z + y) = x - z by smt().
apply (ler_trans (max Pr[G1.main() @ &m: A (glob Mem) res /\ F (glob Mem) res]
                      Pr[G2.main() @ &m: B (glob Mem) res /\ F (glob Mem) res])).
+ smt(ge0_mu).
have -> //: forall (x y x' y':real), x <= x' => y <= y' => max x y <= max x' y' by smt().
+ by rewrite -(Pr_split G1 Mem F A &m) andbC; smt(ge0_mu).
by rewrite -(Pr_split G2 Mem F B &m) andbC; smt(ge0_mu).
qed.