# Library Coq.Numbers.Integer.Abstract.ZAxioms

Require Export NZAxioms.
Module Type Opp (Import T:Typ).
Parameter Inline opp : t -> t.
End Opp.
Module Type OppNotation (T:Typ)(Import O : Opp T).
Notation "- x" := (opp x) (at level 35, right associativity).
End OppNotation.
Module Type Opp' (T:Typ) := Opp T <+ OppNotation T.
We obtain integers by postulating that every number has a predecessor.
Module Type IsOpp (Import Z : NZAxiomsSig')(Import O : Opp' Z).
Declare Instance opp_wd : Proper (eq ==> eq) opp.
Axiom succ_pred : forall n, S (P n) == n.
Axiom opp_0 : - 0 == 0.
Axiom opp_succ : forall n, - ( S n ) == P (- n).
End IsOpp.
Module Type ZAxiomsSig := NZOrdAxiomsSig <+ Opp <+ IsOpp.
Module Type ZAxiomsSig' := NZOrdAxiomsSig' <+ Opp' <+ IsOpp.