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Diffstat (limited to 'theories7/Arith/Bool_nat.v')
-rw-r--r-- | theories7/Arith/Bool_nat.v | 43 |
1 files changed, 43 insertions, 0 deletions
diff --git a/theories7/Arith/Bool_nat.v b/theories7/Arith/Bool_nat.v new file mode 100644 index 00000000..c36f8f15 --- /dev/null +++ b/theories7/Arith/Bool_nat.v @@ -0,0 +1,43 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(* $Id: Bool_nat.v,v 1.1.2.1 2004/07/16 19:31:23 herbelin Exp $ *) + +Require Export Compare_dec. +Require Export Peano_dec. +Require Sumbool. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type 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:nat] (sumbool_not ? ? (zerop n)). +Definition lt_ge_dec : (x,y:nat){(lt x y)}+{(ge x y)} := + [n,m:nat] (sumbool_not ? ? (le_lt_dec m n)). + +Definition nat_lt_ge_bool := + [x,y:nat](bool_of_sumbool (lt_ge_dec x y)). +Definition nat_ge_lt_bool := + [x,y:nat](bool_of_sumbool (sumbool_not ? ? (lt_ge_dec x y))). + +Definition nat_le_gt_bool := + [x,y:nat](bool_of_sumbool (le_gt_dec x y)). +Definition nat_gt_le_bool := + [x,y:nat](bool_of_sumbool (sumbool_not ? ? (le_gt_dec x y))). + +Definition nat_eq_bool := + [x,y:nat](bool_of_sumbool (eq_nat_dec x y)). +Definition nat_noteq_bool := + [x,y:nat](bool_of_sumbool (sumbool_not ? ? (eq_nat_dec x y))). + +Definition zerop_bool := [x:nat](bool_of_sumbool (zerop x)). +Definition notzerop_bool := [x:nat](bool_of_sumbool (notzerop x)). |