(** Two-argument functions that allow to reorder their arguments. *)
Definition transpose2 (f : A -> B -> B) :=
  forall (x y : A) (z z': B), eqB z z' -> eqB (f x (f y z)) (f y (f x z')).(** A version of transpose with restriction on where it should hold *)
Definition transpose_restr2 (R : A -> A -> Prop)(f : A -> B -> B) :=
  forall (x y : A) (z z': B), R x y -> eqB z z' -> eqB (f x (f y z)) (f y (f x z')).

(* Specifications: *)
Lemma fold_right_commutes_restr2 : forall s1 s2 x (i j:B) (heqij: eqB i j), ForallOrdPairs R (s1++x::s2) -> eqB (fold_right f i (s1++x::s2)) (f x (fold_right f j (s1++s2))).
Lemma fold_right_equivlistA_restr2 : forall s s' i j, NoDupA s -> NoDupA s' -> ForallOrdPairs R s -> equivlistA s s' -> eqB i j -> eqB (fold_right f i s) (fold_right f j s').
Lemma fold_right_add_restr2 : forall s' s i j x, NoDupA s -> NoDupA s' -> eqB i j -> ForallOrdPairs R s' -> ~ InA x s -> equivlistA s' (x::s) -> eqB (fold_right f i s') (f x (fold_right f j s)).
Lemma fold_right_commutes2 : forall s1 s2 i x x', eqA x x' -> eqB (fold_right f i (s1++x::s2)) (f x' (fold_right f i (s1++s2))).
Lemma fold_right_equivlistA2 : forall s s' i j, NoDupA s -> NoDupA s' -> eqB i j -> equivlistA s s' -> eqB (fold_right f i s) (fold_right f j s').
Lemma fold_right_add2 : forall s' s i j x, NoDupA s -> NoDupA s' -> eqB i j -> ~ InA x s -> equivlistA s' (x::s) -> eqB (fold_right f i s') (f x (fold_right f j s)).
Lemma InA_dec : forall x l, { InA x l } + { ~ InA x l }.