intuition.
simpl.
case_eq (P x); intuition.
unfold indicator.
rewrite H.
rewrite ratMult_1_l.
rewrite <- sumList_factor_constant_l.
symmetry.
rewrite (sumList_filter_partition (fun p : A * B => P (fst p))).
symmetry.
rewrite ratAdd_0_r.
eapply ratAdd_eqRat_compat.
eapply sumList_body_eq; intuition.
apply filter_In in H0; intuition.
rewrite H2.
destruct (EqDec_dec eqd (fst a) x).
rewrite ratMult_1_l.
repeat rewrite ratMult_1_r.
eapply ratMult_eqRat_compat; intuition.
rewrite flatten_map_eq.
eapply ratInverse_eqRat_compat.
intuition.
destruct a.
simpl in *.
subst.
eapply sumList_nz; eauto.
exists (x, b).
intuition.
eapply filter_In; intuition.
eapply getSupport_In_evalDist; eauto.
rewrite sumList_comm.
symmetry.
rewrite (sumList_filter_partition (fun p : A * B => P (fst p))).
symmetry.
rewrite ratAdd_0_r.
eapply ratAdd_eqRat_compat.
eapply sumList_body_eq; intuition.
apply filter_In in H0; intuition.
rewrite sumList_factor_constant_l.
rewrite <- ratMult_1_r at 1.
eapply ratMult_eqRat_compat; intuition.
symmetry.
destruct a0.
simpl in *.
rewrite (@sumList_exactly_one _ a0).
destruct (EqDec_dec eqd a0 a0); intuition.
eapply filter_NoDup.
eapply getUnique_NoDup.
eapply filter_In; intuition.
eapply in_getUnique.
eapply in_map_iff.
econstructor. split; eauto.
simpl.
trivial.
intuition.
destruct (EqDec_dec eqd a0 b0); subst; intuition.
symmetry.
eapply sumList_0.
intuition.
eapply sumList_0; intuition.
apply filter_In in H3; intuition.
apply filter_In in H0; intuition.
destruct a0.
simpl in *.
destruct (EqDec_dec eqd a0 a1); subst.
destruct (P a1); simpl in *; discriminate.
eapply ratMult_0_r.
repeat rewrite ratMult_0_r.
intuition.
symmetry.
eapply sumList_0; intuition.
eapply filter_In in H0; intuition.
destruct (EqDec_dec eqd (fst a) x); subst.
destruct (P (fst a)); simpl in *; discriminate.
repeat rewrite ratMult_0_r.
intuition.
unfold indicator.
rewrite H.
repeat rewrite ratMult_0_l.
eapply sumList_0; intuition.
apply filter_In in H0.
intuition.
rewrite H2.
rewrite ratMult_1_l.
destruct a.
simpl in *.
destruct (EqDec_dec eqd a x); subst; try congruence.
rewrite ratMult_0_r.
intuition.
Qed.

Theorem repeat_fission' :
forall (A B : Set)(eqda : EqDec A)(eqdb : EqDec B)(c : Comp A)(P1 : A -> bool)(P2 : B -> bool)(f : A -> Comp B) x,
well_formed_comp c ->
(exists a, In a (filter P1 (getSupport c))) ->
(forall a, In a (getSupport c) -> well_formed_comp (f a)) ->
(forall a b, In a (getSupport c) -> In b (getSupport (f a)) -> (P1 a = P2 b)) ->
evalDist (a <-$ Repeat c P1; f a) x ==
evalDist
(Repeat (a <-$ c; f a) P2) x.