(* 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.		*)
(* Rules used to prove facts based on the distribution semantics *)
Set Implicit Arguments.
Local Open Scope rat_scope.
Local Open Scope comp_scope.
Local Open Scope list_scope.
intuition.
unfold evalDist.
eapply eqRat_refl.
Qed.
intros.
specialize (getSupport_correct c1).
specialize (getSupport_correct c2).
intros.
inversion H0; clear H0.
inversion H1; clear H1.
intuition.
apply H3.
intuition.
eapply H4; eauto.
eapply eqRat_trans.
eapply H.
eauto.
eapply H4.
intuition.
eapply H3; eauto.
eapply eqRat_trans.
eapply eqRat_symm.
eapply H.
eauto.
Qed.
intuition.
eapply NoDup_Permutation; intuition.
apply in_map_iff in H3.
destruct H3.
intuition.
subst.
eapply H0.
trivial.
eapply in_map_iff.
econstructor.
split.
2:{
eapply H0.
assert (x = f (f_inv x)).
symmetry.
eapply H.
trivial.
rewrite H4 in H3.
eapply H3.
}
eapply H.
trivial.
Qed.
induction ls; intuition.
simpl in *.
intuition.
eapply H1.
rewrite <- (H a).
rewrite <- (H a0).
f_equal.
trivial.
eapply IHls; eauto.
Qed.
induction ls; intuition; simpl in *.
econstructor.
inversion H0; clear H0; subst.
econstructor; eauto.
intuition.
apply in_map_iff in H0.
destruct H0.
intuition.
assert (x = a).
rewrite <- (H x); intuition.
rewrite <- (H a); intuition.
f_equal.
trivial.
subst.
intuition.
Qed.
intuition.
specialize (getSupport_correct c); intuition.
inversion H.
eauto.
Qed.
induction ls; simpl in *; intuition.
inversion H; clear H; subst.
destruct (P a).
econstructor; eauto.
intuition.
eapply filter_In in H.
intuition.
eapply IHls; eauto.
Qed.
intuition.
eapply NoDup_Permutation.
eapply map_NoDup.
intuition.
eapply getSupport_NoDup.
eapply getSupport_NoDup.
intuition.
apply in_map_iff in H3.
destruct H3.
intuition.
subst.
eapply getSupport_In_evalDist.
intuition.
eapply getSupport_In_evalDist.
eauto.
eapply eqRat_trans.
eapply H1.
trivial.
trivial.
eapply in_map_iff.
econstructor.
split.
eapply H.
trivial.
eapply getSupport_In_evalDist.
intuition.
eapply getSupport_In_evalDist.
eapply H3.
rewrite H2.
trivial.
trivial.
Qed.
intuition.
eapply NoDup_Permutation.
eapply filter_NoDup.
eapply support_NoDup.
eapply filter_NoDup.
eapply support_NoDup.
intuition.
eapply filter_In in H0.
eapply filter_In.
intuition.
eapply getSupport_In_evalDist.
apply getSupport_In_evalDist in H1.
intuition.
apply H1.
eapply eqRat_trans.
eapply H.
trivial.
trivial.
intuition.
eapply filter_In in H0.
eapply filter_In.
intuition.
eapply getSupport_In_evalDist.
apply getSupport_In_evalDist in H1.
intuition.
apply H1.
eapply eqRat_trans.
eapply eqRat_symm.
eapply H.
trivial.
trivial.
Qed.
intuition.
rewrite <- map_id.
eapply Permutation_sym.
eapply evalDist_getSupport_perm; intuition.
symmetry.
eauto.
Qed.
intuition.
unfold sumList.
eapply fold_add_f_inverse;
intuition.
Qed.
intuition.
simpl.
eapply eqRat_trans.
eapply fold_add_rat_perm.
eapply Permutation_sym.
eapply evalDist_getSupport_perm; eauto.
intuition.
symmetry.
eauto.
intuition.
rewrite <- (H a0) at 1.
eapply H2.
eauto.
trivial.
eapply eqRat_refl.
intuition.
eapply ratMult_eqRat_compat.
rewrite <- H at 1.
eauto.
trivial.
rewrite <- H at 1.
eapply H3.
eauto.
trivial.
eapply eqRat_trans.
2:{
eapply fold_add_f_inverse; eauto.
eapply eqRat_refl.
}
simpl.
eapply eqRat_refl.
Qed.
intuition.
eapply (distro_iso_eq (fun b => b)(fun b => b)); intuition.
eapply getSupport_In_evalDist.
intuition.
eapply getSupport_In_evalDist.
eauto.
rewrite H.
trivial.
eapply H0.
eapply getSupport_In_evalDist.
intuition.
eapply getSupport_In_evalDist.
eauto.
rewrite <- H.
trivial.
Qed.
intuition.
simpl.
rewrite sumList_body_eq.
2:{
intuition.
rewrite H0 at 1.
eapply eqRat_refl.
trivial.
}
rewrite sumList_factor_constant_r.
rewrite evalDist_lossless.
eapply ratMult_1_l.
trivial.
Qed.

Lemma sumList_filter : forall (A : Set)(ls : list A)(f : A -> Rat)(P : A -> bool) init,
fold_left (fun r a => r + if (P a) then (f a) else 0) ls init  ==
fold_left (fun r a => r + (f a)) (filter P ls) init.