https://github.com/EasyCrypt/easycrypt
Tip revision: acfd4ea7d779487e774eb6aa8c3deeae783aafd8 authored by Pierre-Yves Strub on 16 April 2020, 14:56:46 UTC
Definition of quotient types w.r.t. a equivalence relation
Definition of quotient types w.r.t. a equivalence relation
Tip revision: acfd4ea
FundamentalLemma.ec
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.