Section SumList.
Variable A : Set.
Definition sumList(ls : list A)(f : A -> Rat) := fold_left (fun a b => a + (f b)) ls 0.
(* Facts about sumList *)
Abort.
Abort.
induction ls; intuition; simpl in *.
eapply eqRat_refl.
eapply eqRat_trans.
eapply IHls.
eapply eqRat_trans.
eapply ratAdd_assoc.
eapply ratAdd_eqRat_compat.
eapply eqRat_refl.
eapply eqRat_symm.
eapply IHls.
Qed.
induction ls; intuition; simpl in *.
eapply IHls; eauto.
eapply ratAdd_eqRat_compat; eauto.
Qed.
induction 1; intuition; simpl in *.
eapply IHPermutation; eauto.
eapply ratAdd_eqRat_compat; eauto.
eapply fold_add_body_eq.
eapply eqRat_trans.
eapply ratAdd_assoc.
eapply eqRat_trans.
2:{
eapply eqRat_symm.
eapply ratAdd_assoc.
}
eapply ratAdd_eqRat_compat; eauto.
eapply eqRat_trans.
eapply ratAdd_comm.
eapply ratAdd_eqRat_compat; eauto.
intuition.
eapply eqRat_trans; eauto.
eapply eqRat_trans.
eapply fold_add_body_eq.
eapply eqRat_symm; eauto.
intuition.
eapply eqRat_symm.
eapply H2.
eapply Permutation_in.
eapply Permutation_sym.
eauto.
eauto.
simpl.
eapply IHPermutation2.
trivial.
intuition.
eapply H2.
eapply Permutation_in.
eapply Permutation_sym.
eauto.
eauto.
Qed.
induction ls; intuition; simpl in *.
eapply IHls; eauto.
rewrite H0.
eapply ratAdd_eqRat_compat; eauto.
eapply eqRat_refl.
intuition.
Qed.
induction ls; intuition; simpl in *.
intuition.
unfold sumList.
simpl.
eapply eqRat_refl.
intuition; subst.
unfold sumList in *; simpl in *.
eapply eqRat_trans.
eapply ratAdd_0.
eapply eqRat_trans.
eapply ratAdd_comm.
eapply eqRat_trans.
eapply eqRat_symm.
eapply fold_add_init.
eapply eqRat_trans.
eapply fold_add_body_eq.
eapply ratAdd_comm.
intuition.
eapply eqRat_refl.
eapply H.
eapply eqRat_refl.
eapply IHls.
unfold sumList in *; simpl in *.
eapply ratAdd_0.
eapply eqRat_trans.
eapply eqRat_symm.
eapply fold_add_init.
eapply eqRat_trans.
eapply fold_add_body_eq.
eapply ratAdd_comm.
intuition.
eapply eqRat_refl.
eapply H.
trivial.
unfold sumList.
simpl.
eapply eqRat_trans.
eapply fold_add_body_eq.
eapply ratAdd_comm.
intuition.
eapply eqRat_refl.
eapply eqRat_trans.
eapply fold_add_init.
eapply ratAdd_0.
intuition.
unfold sumList in *.
eapply IHls.
intuition.
Qed.

Lemma sumList_nz : forall (ls : list A) f,
(~sumList ls f == 0) <-> exists a : _, In a ls /\ (~f a == 0).