induction n; intuition.
unfold forNats.
unfold compMap.
unfold expRat.
eapply ratDistance_le.
eapply evalDist_le_1.
eapply evalDist_le_1.
destruct H0; intuition.
unfold forNats.
fold forNats.
unfold compMap.
fold compMap.
unfold expRat.
fold expRat.
eapply leRat_trans.
eapply eqRat_impl_leRat.
eapply ratDistance_eqRat_compat.
eapply eqRat_refl.
eapply evalDist_seq.
intuition.
intuition.
eapply repeat_unroll_eq; intuition.
econstructor; eauto.
intuition.
eapply eqRat_refl.
assert (forall x,
evalDist
(a <-$
(b <-$ c;
lsb' <-$ compMap eqda (fun _ : nat => c) (forNats n); ret b :: lsb');
ret hd_error (map f (filter P a))) x ==
evalDist
(a <-$ c;
a' <-$
(
lsa <-$ compMap eqda (fun _ : nat => c) (forNats n);
ret hd_error (map f (filter P lsa)));
ret (if (P a) then Some (f a) else a')) x).
intuition.
inline_first.
comp_skip.
inline_first.
comp_skip.
comp_simp.
simpl.
case_eq (P x2); intuition.
simpl.
unfold value.
intuition.
rewrite H1.
clear H1.
eapply leRat_trans.
eapply eqRat_impl_leRat.
eapply ratDistance_eqRat_compat.
eapply eqRat_refl.
comp_inline_l.
eapply eqRat_refl.
assert
((|
evalDist
(a <-$ c;
a' <-$
(lsa <-$ compMap eqda (fun _ : nat => c) (forNats n);
ret hd_error (map f (filter P lsa))); ret (if P a then Some (f a) else a')) x -
evalDist
(a <-$ c; r2 <-$ (if P a then ret a else Repeat c P); ret Some (f r2)) x |) ==
(|
sumList (getSupport c) (fun a => evalDist c a *
(evalDist
(a' <-$
(lsa <-$ compMap eqda (fun _ : nat => c) (forNats n);
ret hd_error (map f (filter P lsa))); ret (if P a then Some (f a) else a'))) x) -
sumList (getSupport c) (fun a => evalDist c a *
(evalDist
(r2 <-$ (if P a then ret a else Repeat c P); ret Some (f r2)) x)) | )).
simpl.
intuition.
rewrite H1.
clear H1.
eapply leRat_trans.
eapply sumList_distance_prod.
cbv beta.
rewrite (sumList_filter_partition P).
eapply leRat_trans.
eapply ratAdd_leRat_compat.
eapply eqRat_impl_leRat.
eapply sumList_0; intuition.
apply filter_In in H1; intuition.
rewrite H3.
eapply eqRat_trans.
eapply ratMult_eqRat_compat.
eapply eqRat_refl.
eapply eqRat_trans.
eapply ratDistance_eqRat_compat.
comp_irr_l.
wftac.
eapply compMap_wf; intuition.
eapply eqRat_refl.
comp_ret_l.
eapply eqRat_refl.
rewrite <- ratIdentityIndiscernables.
intuition.
eapply ratMult_0_r.
eapply sumList_le; intuition.
apply filter_In in H1; intuition.
destruct (P a).
simpl in *.
discriminate.
eapply ratMult_leRat_compat.
eapply leRat_refl.
rewrite evalDist_right_ident.
eapply IHn; intuition.
econstructor; eauto.
rewrite <- ratAdd_0_l.
rewrite sumList_factor_constant_r.
eapply ratMult_leRat_compat; intuition.
simpl.
eapply eqRat_impl_leRat.
symmetry.
rewrite (sumList_filter_partition (P )).
symmetry.
rewrite ratAdd_0_l.
eapply ratAdd_eqRat_compat; intuition.
symmetry.
eapply sumList_0; intuition.
apply filter_In in H2.
intuition.
destruct (EqDec_dec bool_EqDec (negb (P a)) true).
destruct (P a).
simpl in *.
discriminate.
discriminate.
eapply ratMult_0_r.
eapply sumList_body_eq; intuition.
apply filter_In in H2.
intuition.
destruct (EqDec_dec bool_EqDec (negb (P a)) true).
rewrite ratMult_1_r.
intuition.
congruence.
Qed.

(* Specifications: *)
Theorem compMap_head :
forall (A B : Set)(eqd : EqDec B)(f : A -> Comp B)(ls : list A) x,
(forall a, In a ls -> well_formed_comp (f a)) ->
evalDist (ls' <-$ compMap _ f ls; ret (head ls')) x ==
evalDist (match (head ls) with
| None => ret None
| Some a => b <-$ f a; ret (Some b)
end) x.