(* Copyright 2012-2015 by Adam Petcher.				* * Use of this source code is governed by the license described	* * in the LICENSE file at the root of the source tree.		*)
(* RndNat -- a computation that returns a random nat uniformly distributed in a specified range. *)
Set Implicit Arguments.
Definition RndNat_unchecked(n : nat) :=
v <-$ {0,1} ^ (lognat n);
let n := (bvToNat v) in
ret n.
Definition ltNatBool := fun x1 x2 => if (lt_dec x1 x2) then true else false.
(* RndNat uses Repeat to get a number strictly less than n *)
Definition RndNat(n : nat) :=
(Repeat (RndNat_unchecked n) (fun x => (ltNatBool x n))).
Notation "[ 0 '..' n )" := (RndNat n)
(right associativity, at level 77) : comp_scope.
intuition.
unfold RndNat.
eapply (well_formed_Repeat).
unfold eq_dec.
intuition.
eapply eq_nat_dec.
unfold RndNat_unchecked.
eapply well_formed_Bind.
eapply well_formed_Rnd.
intuition.
eapply well_formed_Ret.
eapply filter_In.
intuition.
2:{
unfold ltNatBool.
assert ((if (lt_dec O n) then true else false) = true).
destruct (lt_dec O n).
trivial.
omega.
eapply H0.
}
simpl.
eapply Fold.in_getUnique.
eapply Fold.in_flatten.
econstructor.
split.
eapply in_map_iff.
econstructor.
split.
eauto.
eapply in_getAllBvectors.
specialize (bvNat_zero (lognat n)); intuition.
simpl in *.
left.
eapply H0.
Qed.
intuition.
unfold RndNat in *.
simpl in *.
eapply filter_In in H.
intuition.
unfold ltNatBool in *.
destruct (lt_dec x n); congruence.
Qed.
Hint Resolve well_formed_RndNat : wftac.
intuition.
unfold RndNat_unchecked.
eapply getSupport_In_Seq.
eapply in_getAllBvectors.
simpl.
left.
eapply bvToNat_natToBv_inverse.
eapply lognat_monotonic.
omega.
Qed.
Local Open Scope rat_scope.
intuition.
eapply evalDist_Repeat_eq; intuition.
unfold RndNat_unchecked.
remember (lognat n) as length.
eapply (evalDist_iso
(BVxor length (BVxor length (natToBv length v1) (natToBv length v2)))
(BVxor length (BVxor length (natToBv length v1) (natToBv length v2)))); intuition.
(* eapply getSupport_In_evalDist. intuition. eapply getSupport_In_evalDist. intuition. *)
remember (BVxor length (natToBv length v1) (natToBv length v2)) as z.
rewrite <- BVxor_assoc.
rewrite BVxor_same_id.
rewrite BVxor_id_l.
trivial.
rewrite <- BVxor_assoc.
rewrite BVxor_same_id.
rewrite BVxor_id_l.
trivial.
simpl.
eapply in_getAllBvectors.
eapply uniformity.
simpl.
destruct (EqDec_dec nat_EqDec (bvToNat x) v2); subst.
destruct (EqDec_dec nat_EqDec
(bvToNat
(BVxor (lognat n)
(BVxor (lognat n) (natToBv (lognat n) v1)
(natToBv (lognat n) (bvToNat x))) x)) v1).
intuition.
exfalso.
eapply n0.
rewrite natToBv_bvToNat_inverse.
rewrite BVxor_assoc.
rewrite BVxor_same_id.
rewrite BVxor_id_r.
eapply bvToNat_natToBv_inverse.
eapply lognat_monotonic.
omega.
destruct (EqDec_dec nat_EqDec
(bvToNat
(BVxor (lognat n)
(BVxor (lognat n) (natToBv (lognat n) v1)
(natToBv (lognat n) v2)) x)) v1).
exfalso.
eapply n0.
rewrite BVxor_assoc in e.
assert ((BVxor (lognat n) (natToBv (lognat n) v2) x) = (Bvect_false (lognat n))).
eapply BVxor_id_r_inv.
eapply bvToNat_natToBv_eq.
eauto.
apply BVxor_id_inv in H2.
rewrite <- H2.
eapply bvToNat_natToBv_inverse.
eapply lognat_monotonic.
omega.
intuition.
unfold ltNatBool.
destruct (lt_dec v1 n); destruct (lt_dec v2 n); congruence.
eapply filter_In; intuition.
eapply RndNat_unchecked_lt_support.
trivial.
unfold ltNatBool.
destruct (lt_dec v1 n); congruence.
Qed.
intuition.
simpl.
eapply filter_In.
split.
eapply in_getUnique.
eapply in_flatten.
econstructor.
split.
eapply in_map_iff.
econstructor.
split.
eapply eq_refl.
eapply in_getAllBvectors.
simpl.
left.
eapply bvToNat_natToBv_inverse.
eapply lognat_monotonic.
trivial.
unfold ltNatBool.
destruct (lt_dec x k); intuition.
Qed.
intuition.
rewrite (@Permutation_length _ _ (forNats n)).
eapply forNats_length.
eapply NoDup_Permutation.
eapply getSupport_NoDup.
eapply forNats_NoDup.
intuition.
eapply forNats_In.
eapply RndNat_support_lt; intuition.
eapply in_getSupport_RndNat.
eapply forNats_In; intuition.
Qed.
intuition.
eapply (@ratMult_same_r_inv _ (n / 1)).
rewrite ratMult_eq_rat1.
eapply eqRat_trans.
2:{
eapply evalDist_lossless.
eapply well_formed_RndNat.
destruct nzn.
apply agz.
}
symmetry.
eapply eqRat_trans.
eapply sumList_body_eq; intuition.
eapply (@RndNat_uniform _ i).
eapply RndNat_support_lt; intuition.
trivial.
rewrite sumList_body_const.
rewrite RndNat_support_length.
intuition.
intuition.
eapply rat_num_nz.
destruct nzn.
eapply agz.
eauto.
Qed.

Theorem RndNat_seq :
forall (n : posnat)(A : Set)(c : nat -> Comp A) a,
evalDist (x <-$ RndNat n; c x) a ==
(1 / n) * sumList (allNatsLt n) (fun z => evalDist (c z) a).