swh:1:snp:04e159a4411e97cbe416dcf21d082639f654120b
Tip revision: f7b8664dcf5237042389e655a2e37b09177167f5 authored by Alley Stoughton on 30 June 2021, 15:32:30 UTC
Added Above Threshold and Report Noisy Max examples, which check
Added Above Threshold and Report Noisy Max examples, which check
Tip revision: f7b8664
OldDistr.ec
(* --------------------------------------------------------------------
* Copyright (c) - 2012--2016 - IMDEA Software Institute
* Copyright (c) - 2012--2018 - Inria
* Copyright (c) - 2012--2018 - Ecole Polytechnique
*
* Distributed under the terms of the CeCILL-B-V1 license
* -------------------------------------------------------------------- *)
(* -------------------------------------------------------------------- *)
require import AllCore StdRing StdOrder.
(*---*) import RField RealOrder.
op charfun (p:'a -> bool) x: real = if p x then 1%r else 0%r.
op mu1 (d:'a distr) x: real = mu d (pred1 x).
op weight (d:'a distr): real = mu d predT.
op in_supp x (d:'a distr) : bool = 0%r < mu1 d x.
op support (d:'a distr) x = in_supp x d.
pred is_lossless (d : 'a distr) = mu d predT = 1%r.
pred is_full (d : 'a distr) = forall x, support d x.
pred is_subuniform (d : 'a distr) = forall (x y:'a),
support d x =>
support d y =>
mu d (pred1 x) = mu d (pred1 y).
pred is_uniform (d : 'a distr) =
is_lossless d
/\ is_subuniform d.
pred is_subuniform_over (d : 'a distr) (p : 'a -> bool) =
(forall x, support d x <=> p x)
/\ is_subuniform d.
pred is_uniform_over (d : 'a distr) (p : 'a -> bool) =
(forall x, support d x <=> p x)
/\ is_uniform d.
(** Point-wise equality *)
pred (==)(d d':'a distr) =
(forall x, mu1 d x = mu1 d' x).
(** Event-wise equality *)
pred (===)(d d':'a distr) =
forall p, mu d p = mu d' p.
(** Axioms *)
axiom mu_bounded (d:'a distr) (p:'a -> bool):
0%r <= mu d p <= 1%r.
(* now mu0 *)
axiom mu_false (d:'a distr): mu d pred0 = 0%r.
axiom mu_sub (d:'a distr) (p q:('a -> bool)):
p <= q => mu d p <= mu d q.
axiom mu_supp_in (d:'a distr) p:
mu d p = mu d predT <=>
support d <= p.
axiom mu_or (d:'a distr) (p q:('a -> bool)):
mu d (predU p q) = mu d p + mu d q - mu d (predI p q).
axiom pw_eq (d d':'a distr):
d == d' <=> d = d'.
axiom uniform_unique (d d':'a distr):
support d = support d' =>
is_uniform d =>
is_uniform d' =>
d = d'.
(** Lemmas *)
lemma witness_nzero P (d:'a distr):
0%r < mu d P => (exists x, P x ).
proof.
have: P <> pred0 => (exists x, P x).
apply absurd=> /=.
have -> h: (!exists (x:'a), P x) = forall (x:'a), !P x by smt.
by apply fun_ext=> x; rewrite h.
smt.
qed.
lemma ew_eq (d d':'a distr):
d === d' => d = d'.
proof.
move=> ew_eq; rewrite -pw_eq=> x.
by rewrite /mu1 ew_eq.
qed.
lemma nosmt mu_or_le (d:'a distr) (p q:'a -> bool) r1 r2:
mu d p <= r1 => mu d q <= r2 =>
mu d (predU p q) <= r1 + r2 by [].
lemma nosmt mu_and (d:'a distr) (p q:'a -> bool):
mu d (predI p q) = mu d p + mu d q - mu d (predU p q)
by [].
lemma nosmt mu_and_le_l (d:'a distr) (p q:'a -> bool) r:
mu d p <= r =>
mu d (predI p q) <= r.
proof.
apply (ler_trans (mu d p)).
by apply mu_sub; rewrite /predI=> x.
qed.
lemma nosmt mu_and_le_r (d:'a distr) (p q:'a -> bool) r :
mu d q <= r =>
mu d (predI p q) <= r.
proof.
apply (ler_trans (mu d q)).
by apply mu_sub; rewrite /predI=> x.
qed.
lemma mu_supp (d:'a distr):
mu d (support d) = mu d predT.
proof. by rewrite mu_supp_in. qed.
lemma mu_eq (d:'a distr) (p q:'a -> bool):
p == q => mu d p = mu d q.
proof.
by move=> ext_p_q; congr=> //; apply fun_ext.
qed.
lemma mu_disjoint (d:'a distr) (p q:('a -> bool)):
(predI p q) <= pred0 =>
mu d (predU p q) = mu d p + mu d q.
proof.
move=> and_p_q_false; rewrite mu_or.
have ->: (predI p q) = pred0 by apply subpred_asym.
by rewrite mu_false.
qed.
lemma mu_not (d:'a distr) (p:('a -> bool)):
mu d (predC p) = mu d predT - mu d p.
proof.
have: mu d (predC p) + mu d p = mu d predT; [rewrite -mu_disjoint | smt].
(* rewrite seems to unroll too much *)
+ by rewrite predCI; apply/(subpred_refl<:'a> pred0).
+ by rewrite predCU.
qed.
lemma mu_split (d:'a distr) (p q:('a -> bool)):
mu d p = mu d (predI p q) + mu d (predI p (predC q)).
proof.
rewrite -mu_disjoint; first smt.
by apply mu_eq=> x; rewrite /predI /predC /predU !(andbC (p x)) orDandN.
qed.
lemma mu_support (p:('a -> bool)) (d:'a distr):
mu d p = mu d (predI p (support d)).
proof.
apply/ler_anti; split => [|_]; last by apply/mu_sub/predIsubpredl.
have ->: forall (p q:'a -> bool), (predI p q) = predC (predU (predC p) (predC q)).
by (move=> p1 p2; apply fun_ext; delta; smt). (* delta *)
by rewrite mu_not mu_or !mu_not mu_supp; smt.
qed.
lemma witness_support P (d:'a distr):
0%r < mu d P <=> (exists x, P x /\ in_supp x d).
proof.
split=> [|[] x [x_in_P x_in_d]].
rewrite mu_support=> nzero.
apply witness_nzero in nzero; case nzero=> x.
rewrite /predI //= => p_supp.
by exists x.
have: mu d (pred1 x) <= mu d P /\ 0%r < mu d (pred1 x); last smt.
split=> [|//=].
by rewrite mu_sub // /Core.(<=) /pred1 => x0 <<-.
qed.
lemma mu_sub_support (d:'a distr) (p q:('a -> bool)):
(predI p (support d)) <= (predI q (support d)) =>
mu d p <= mu d q.
proof.
by move=> ple_p_q; rewrite (mu_support p) (mu_support q);
apply mu_sub.
qed.
lemma mu_eq_support (d:'a distr) (p q:('a -> bool)):
(predI p (support d)) = (predI q (support d)) =>
mu d p = mu d q.
proof.
by move=> eq_supp;
rewrite (mu_support p) (mu_support q);
apply mu_eq; rewrite eq_supp.
qed.
lemma weight_0_mu (d:'a distr):
weight d = 0%r => forall p, mu d p = 0%r
by [].
lemma mu_one (P:'a -> bool) (d:'a distr):
P == predT =>
weight d = 1%r =>
mu d P = 1%r.
proof.
move=> heq <-.
rewrite /weight.
congr=> //.
by apply fun_ext.
qed.