(** We can deduce a [sgn] function from a [compare] function *)
Module Type ZDecAxiomsSig := ZAxiomsMiniSig <+ HasCompare.
Module Type ZDecAxiomsSig' := ZAxiomsMiniSig' <+ HasCompare.
Module Type GenericSgn (Import Z : ZDecAxiomsSig')
(Import ZP : ZMulOrderProp Z) <: HasSgn Z.
Definition sgn n :=
match compare 0 n with Eq => 0 | Lt => 1 | Gt => -1 end.

Lemma abs_case_strong :
forall (P:t->Prop) n, Proper (eq==>iff) P ->
(0<=n -> P n) -> (n<=0 -> P (-n)) -> P (abs n).