Module Type Int.
Parameter t : Set.
Bind Scope Int_scope with t.
Parameter i2z : t -> Z.
Parameter _0 : t.
Parameter _1 : t.
Parameter _2 : t.
Parameter _3 : t.
Parameter add : t -> t -> t.
Parameter opp : t -> t.
Parameter sub : t -> t -> t.
Parameter mul : t -> t -> t.
Parameter max : t -> t -> t.
Notation "0" := _0 : Int_scope.
Notation "1" := _1 : Int_scope.
Notation "2" := _2 : Int_scope.
Notation "3" := _3 : Int_scope.
Infix "+" := add : Int_scope.
Infix "-" := sub : Int_scope.
Infix "*" := mul : Int_scope.
Notation "- x" := (opp x) : Int_scope.
(** For logical relations, we can rely on their counterparts in Z, since they don't appear after extraction. Moreover, using tactics like omega is easier this way. *)
Notation "x == y" := (i2z x = i2z y)
(at level 70, y at next level, no associativity) : Int_scope.
Notation "x <= y" := (i2z x <= i2z y)%Z : Int_scope.
Notation "x < y" := (i2z x < i2z y)%Z : Int_scope.
Notation "x >= y" := (i2z x >= i2z y)%Z : Int_scope.
Notation "x > y" := (i2z x > i2z y)%Z : Int_scope.
Notation "x <= y <= z" := (x <= y /\ y <= z) : Int_scope.
Notation "x <= y < z" := (x <= y /\ y < z) : Int_scope.
Notation "x < y < z" := (x < y /\ y < z) : Int_scope.
Notation "x < y <= z" := (x < y /\ y <= z) : Int_scope.
(** Informative comparisons. *)
Infix "=?" := eqb.
Infix "<?" := ltb.
Infix "<=?" := leb.
(** For compatibility, some decidability fonctions (informative). *)

Axiom gt_le_dec : forall x y : t, {x > y} + {x <= y}.
Axiom ge_lt_dec :  forall x y : t, {x >= y} + {x < y}.
Axiom eq_dec : forall x y : t, { x == y } + {~ x==y }.