intuition.
(* special case: the events have probability 0 *)
destruct (eq_Rat_dec (evalDist c1 x) 0).
exists (a <-$ c1;
b <-$ c2;
ret (a, b)).
intuition.
unfold marginal_l.
dist_inline_first.
rewrite <- evalDist_right_ident.
dist_skip.
dist_inline_first.
dist_irr_r.
dist_compute.
unfold marginal_r.
dist_inline_first.
rewrite <- evalDist_right_ident.
dist_irr_r.
dist_inline_first.
dist_skip.
dist_compute.
repeat simp_in_support.
simpl in *.
exfalso.
eapply getSupport_not_In_evalDist.
eapply e.
trivial.
repeat simp_in_support.
simpl in *.
exfalso.
eapply getSupport_not_In_evalDist.
rewrite <- H1.
trivial.
trivial.
(* special case: the events have probability 1 *)
destruct (eq_Rat_dec (evalDist c1 x) 1).
exists (a <-$ c1;
b <-$ c2;
ret (a, b)).
split.
intuition.
unfold marginal_l.
dist_inline_first.
rewrite <- evalDist_right_ident.
dist_skip.
dist_inline_first.
dist_irr_r.
dist_compute.
split.
intuition.
unfold marginal_r.
dist_inline_first.
rewrite <- evalDist_right_ident.
dist_irr_r.
dist_inline_first.
dist_skip.
dist_compute.
intros.
repeat simp_in_support; simpl in *.
erewrite (evalDist_1) in H5; intuition.
2:{
rewrite <- H1.
trivial.
}
simpl in *; intuition.
erewrite (evalDist_1) in H3; eauto.
simpl in *; intuition.
(* general case *)
exists (a <-$ c1;
b <-$ if (eqb a x) then (ret y) else (Repeat c2 (fun b => negb (eqb b y)));
ret (a, b)).
split.
intuition.
unfold marginal_l.
dist_inline_first.
rewrite <- evalDist_right_ident.
dist_skip.
dist_inline_first.
dist_irr_r.
case_eq (eqb x0 x); intuition; wftac.
remember (getSupport c2) as s.
destruct s.
specialize (getSupport_length_nz H0); intuition.
rewrite <- Heqs in H4.
simpl in *.
omega.
destruct (EqDec_dec _ b y); subst.
destruct s.
exfalso.
eapply n0.
rewrite H1.
rewrite <- (evalDist_lossless H0).
rewrite <- Heqs.
rewrite sumList_cons.
unfold sumList; simpl.
eapply ratAdd_0_r.
eapply (@well_formed_Repeat _ _ _ _ b).
trivial.
eapply filter_In; intuition.
rewrite <- Heqs.
simpl.
intuition.
case_eq (eqb b y); intuition.
rewrite eqb_leibniz in H4.
subst.
specialize (getSupport_NoDup c2); intuition.
rewrite <- Heqs in H4.
inversion H4; clear H4; subst.
simpl in *.
intuition.
eapply (@well_formed_Repeat _ _ _ _ b).
trivial.
eapply filter_In; intuition.
rewrite <- Heqs; simpl; intuition.
case_eq (eqb b y); intuition.
rewrite eqb_leibniz in H4; subst.
intuition.
dist_compute.
split.
intuition.
unfold marginal_r.
dist_inline_first.
rewrite evalDist_seq_step.
rewrite (sumList_filter_partition (eqb x)).
symmetry.
eapply eqRat_trans.
eapply ratAdd_eqRat_compat.
eapply sumList_body_eq.
intros.
eapply ratMult_eqRat_compat.
eapply eqRat_refl.
apply filter_In in H2.
intuition.
rewrite eqb_leibniz in H4.
subst.
rewrite eqb_refl.
dist_inline_l.
simpl snd.
eapply eqRat_refl.
eapply sumList_body_eq; intuition.
apply filter_In in H2.
intuition.
case_eq (eqb x a); intuition.
rewrite eqb_leibniz in H2.
subst.
rewrite eqb_refl in H4.
simpl in *.
discriminate.
rewrite evalDist_assoc at 1.
case_eq (eqb a x); intuition.
rewrite eqb_leibniz in H5.
subst.
rewrite eqb_refl in H2.
discriminate.
eapply ratMult_eqRat_compat.
eapply eqRat_refl.
eapply evalDist_seq_eq.
intuition.
eapply eqRat_refl.
intuition.
dist_simp.
unfold snd.
eapply eqRat_refl.
intuition.
rewrite (@sumList_exactly_one _ x).
case_eq (eqb y r2); intuition.
rewrite eqb_leibniz in H2.
subst.
assert (Pr[ret true] == 1).
dist_compute.
rewrite H2.
rewrite ratMult_1_r.
clear H2.
rewrite H1.
symmetry.
rewrite ratAdd_0_r at 1.
eapply ratAdd_eqRat_compat; intuition.
symmetry.
eapply sumList_0.
intuition.
apply filter_In in H2.
intuition.
assert (Pr  [x0 <-$ Repeat c2 (fun b : B => negb (eqb b r2)); ret eqb x0 r2 ] ==
Pr  [x0 <-$ Repeat c2 (fun b : B => negb (eqb b r2)); ret eqb r2 x0 ] ).
dist_skip; dist_compute.
rewrite H2.
clear H2.
rewrite <- evalDist_event_equiv.
eapply ratMult_0.
right.
eapply evalDist_Repeat_0; intuition.
subst.
rewrite eqb_refl in H5.
simpl in *; discriminate.
assert (Pr [ret false] == 0).
dist_compute.
rewrite H3.
clear H3.
rewrite ratMult_0_r.
rewrite <- ratAdd_0_l.
rewrite sumList_factor_constant_r.
rewrite sumList_filter_complement.
rewrite H1.
rewrite evalDist_seq_step.
destruct (in_dec (EqDec_dec _) r2 (getSupport c2)).
rewrite (@sumList_exactly_one _ r2).
assert (Pr  [ret eqb r2 r2 ] == 1).
dist_compute.
rewrite H3.
clear H3.
rewrite ratMult_1_r.
simpl.
unfold indicator.
case_eq (eqb r2 y); intuition.
rewrite eqb_leibniz in H3; subst.
rewrite eqb_refl in H2.
discriminate.
simpl.
rewrite ratMult_1_l.
symmetry.
rewrite <- ratMult_1_l at 1.
rewrite <- ratMult_assoc.
eapply ratMult_eqRat_compat; intuition.
symmetry.
eapply eqRat_trans.
eapply ratMult_eqRat_compat.
eapply eqRat_refl.
eapply ratInverse_eqRat_compat.
intuition.
assert ( (filter (fun b : B => negb (eqb b y)) (getSupport c2)) =
(filter (fun b : B => negb (eqb y b)) (getSupport c2))).
eapply filter_ext; intuition.
rewrite eqb_symm.
trivial.
rewrite H5 in H4.
clear H5.
eapply n0.
eapply leRat_impl_eqRat.
eapply evalDist_le_1.
eapply ratSubtract_0_inv.
rewrite sumList_filter_complement in H4.
rewrite H1.
intuition.
trivial.
assert ((filter (fun b : B => negb (eqb b y)) (getSupport c2)) =
(filter (fun b : B => negb (eqb y b)) (getSupport c2))).
eapply filter_ext.
intuition.
rewrite eqb_symm.
intuition.
rewrite H4.
clear H4.
eapply sumList_filter_complement.
trivial.
rewrite ratMult_comm.
eapply ratInverse_prod_1.
intuition.
eapply n0.
eapply leRat_impl_eqRat.
eapply evalDist_le_1.
eapply ratSubtract_0_inv.
rewrite H1.
intuition.
eapply getSupport_NoDup.
simpl.
eapply filter_In; intuition.
rewrite eqb_symm.
rewrite H2.
trivial.
intuition.
dist_compute.
eapply eqRat_trans.
eapply ratMult_eqRat_compat.
eapply eqRat_refl.
eapply sumList_0.
intuition.
simpl in H3.
apply filter_In in H3.
intuition.
case_eq (eqb a r2); intuition.
rewrite eqb_leibniz in H3; subst.
intuition.
dist_compute.
rewrite ratMult_0_r.
symmetry.
eapply getSupport_not_In_evalDist.
trivial.
trivial.
eapply filter_NoDup.
eapply getSupport_NoDup.
eapply filter_In; intuition.
eapply getSupport_In_evalDist; intuition.
eapply eqb_refl.
intuition.
apply filter_In in H2.
intuition.
rewrite eqb_leibniz in H5.
intuition.
intros.
repeat simp_in_support;
simpl in *; intuition.
rewrite eqb_refl in Heqx2.
discriminate.
symmetry in Heqx2.
rewrite eqb_leibniz in Heqx2.
trivial.
simpl in *.
eapply filter_In in H5.
intuition.
rewrite eqb_refl in H4.
simpl in *.
discriminate.
Grab Existential Variables.
unfold eq_dec; intuition.
eapply (EqDec_dec _).
unfold eq_dec; intuition.
eapply (EqDec_dec _).
intuition.
intuition.
intuition.
intuition.
intuition.
Qed.

Theorem le_impl_comp_spec :
forall (A B : Set){eqda : EqDec A}{eqdb : EqDec B}(c1 : Comp A)(c2 : Comp B),
well_formed_comp c1 ->
well_formed_comp c2 ->
forall x y,
evalDist c1 x <= evalDist c2 y ->
comp_spec (fun a b => a = x -> b = y) c1 c2.