Library Coq.Arith.Bool_nat


Require Export Compare_dec.

Require Export Peano_dec.

Require Import Sumbool.




Open Local Scope nat_scope.




Implicit Types m n x y : nat.

The decidability of equality and order relations over type nat give some boolean functions with the adequate specification.





Definition notzerop n := sumbool_not _ _ (zerop n).

Definition lt_ge_dec : forall x y, {x < y} + {x >= y} :=

  fun n m => sumbool_not _ _ (le_lt_dec m n).




Definition nat_lt_ge_bool x y := bool_of_sumbool (lt_ge_dec x y).

Definition nat_ge_lt_bool x y :=

  bool_of_sumbool (sumbool_not _ _ (lt_ge_dec x y)).




Definition nat_le_gt_bool x y := bool_of_sumbool (le_gt_dec x y).

Definition nat_gt_le_bool x y :=

  bool_of_sumbool (sumbool_not _ _ (le_gt_dec x y)).




Definition nat_eq_bool x y := bool_of_sumbool (eq_nat_dec x y).

Definition nat_noteq_bool x y :=

  bool_of_sumbool (sumbool_not _ _ (eq_nat_dec x y)).




Definition zerop_bool x := bool_of_sumbool (zerop x).

Definition notzerop_bool x := bool_of_sumbool (notzerop x).