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.

(* Specifications: *)
Lemma fold_add_f_inverse : forall (B : Set)(ls : list A)(f : A -> B)(f_inv : B -> A) fa init1 init2,
(init1 == init2) ->
(forall a, In a ls -> f_inv (f a) = a) ->
fold_left (fun r b => r + fa (f_inv b)) (map f ls) init1 ==
fold_left (fun r a => r + (fa a)) ls init2.