Require Import Sumbool.

Require Import BinInt.
Require Import Zorder.
Require Import Zcompare.
Local Open Scope Z_scope.


Lemma Zcompare_rect (P:Type) (n m:Z) :
  ((n ?= m) = Eq -> P) -> ((n ?= m) = Lt -> P) -> ((n ?= m) = Gt -> P) -> P.

Lemma Zcompare_rec (P:Set) (n m:Z) :
  ((n ?= m) = Eq -> P) -> ((n ?= m) = Lt -> P) -> ((n ?= m) = Gt -> P) -> P.

Notation Z_eq_dec := Z.eq_dec (compat "8.3").

Section decidability.

  Variables x y : Z.

  Definition Z_lt_dec : {x < y} + {~ x < y}.

  Definition Z_le_dec : {x <= y} + {~ x <= y}.

  Definition Z_gt_dec : {x > y} + {~ x > y}.

  Definition Z_ge_dec : {x >= y} + {~ x >= y}.

  Definition Z_lt_ge_dec : {x < y} + {x >= y}.

  Lemma Z_lt_le_dec : {x < y} + {y <= x}.

  Definition Z_le_gt_dec : {x <= y} + {x > y}.

  Definition Z_gt_le_dec : {x > y} + {x <= y}.

  Definition Z_ge_lt_dec : {x >= y} + {x < y}.

  Definition Z_le_lt_eq_dec : x <= y -> {x < y} + {x = y}.

End decidability.

Lemma Zlt_cotrans : forall n m:Z, n < m -> forall p:Z, {n < p} + {p < m}.

Lemma Zlt_cotrans_pos : forall n m:Z, 0 < n + m -> {0 < n} + {0 < m}.

Lemma Zlt_cotrans_neg : forall n m:Z, n + m < 0 -> {n < 0} + {m < 0}.

Lemma not_Zeq_inf : forall n m:Z, n <> m -> {n < m} + {m < n}.

Lemma Z_dec : forall n m:Z, {n < m} + {n > m} + {n = m}.

Lemma Z_dec´ : forall n m:Z, {n < m} + {m < n} + {n = m}.