diff options
Diffstat (limited to 'theories/ZArith/Zabs.v')
| -rw-r--r-- | theories/ZArith/Zabs.v | 56 |
1 files changed, 28 insertions, 28 deletions
diff --git a/theories/ZArith/Zabs.v b/theories/ZArith/Zabs.v index 23473e93..08d1a931 100644 --- a/theories/ZArith/Zabs.v +++ b/theories/ZArith/Zabs.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -27,17 +27,17 @@ Local Open Scope Z_scope. (**********************************************************************) (** * Properties of absolute value *) -Notation Zabs_eq := Z.abs_eq (only parsing). -Notation Zabs_non_eq := Z.abs_neq (only parsing). -Notation Zabs_Zopp := Z.abs_opp (only parsing). -Notation Zabs_pos := Z.abs_nonneg (only parsing). -Notation Zabs_involutive := Z.abs_involutive (only parsing). -Notation Zabs_eq_case := Z.abs_eq_cases (only parsing). -Notation Zabs_triangle := Z.abs_triangle (only parsing). -Notation Zsgn_Zabs := Z.sgn_abs (only parsing). -Notation Zabs_Zsgn := Z.abs_sgn (only parsing). -Notation Zabs_Zmult := Z.abs_mul (only parsing). -Notation Zabs_square := Z.abs_square (only parsing). +Notation Zabs_eq := Z.abs_eq (compat "8.3"). +Notation Zabs_non_eq := Z.abs_neq (compat "8.3"). +Notation Zabs_Zopp := Z.abs_opp (compat "8.3"). +Notation Zabs_pos := Z.abs_nonneg (compat "8.3"). +Notation Zabs_involutive := Z.abs_involutive (compat "8.3"). +Notation Zabs_eq_case := Z.abs_eq_cases (compat "8.3"). +Notation Zabs_triangle := Z.abs_triangle (compat "8.3"). +Notation Zsgn_Zabs := Z.sgn_abs (compat "8.3"). +Notation Zabs_Zsgn := Z.abs_sgn (compat "8.3"). +Notation Zabs_Zmult := Z.abs_mul (compat "8.3"). +Notation Zabs_square := Z.abs_square (compat "8.3"). (** * Proving a property of the absolute value by cases *) @@ -68,38 +68,38 @@ Qed. (** * Some results about the sign function. *) -Notation Zsgn_Zmult := Z.sgn_mul (only parsing). -Notation Zsgn_Zopp := Z.sgn_opp (only parsing). -Notation Zsgn_pos := Z.sgn_pos_iff (only parsing). -Notation Zsgn_neg := Z.sgn_neg_iff (only parsing). -Notation Zsgn_null := Z.sgn_null_iff (only parsing). +Notation Zsgn_Zmult := Z.sgn_mul (compat "8.3"). +Notation Zsgn_Zopp := Z.sgn_opp (compat "8.3"). +Notation Zsgn_pos := Z.sgn_pos_iff (compat "8.3"). +Notation Zsgn_neg := Z.sgn_neg_iff (compat "8.3"). +Notation Zsgn_null := Z.sgn_null_iff (compat "8.3"). (** A characterization of the sign function: *) Lemma Zsgn_spec x : - 0 < x /\ Zsgn x = 1 \/ - 0 = x /\ Zsgn x = 0 \/ - 0 > x /\ Zsgn x = -1. + 0 < x /\ Z.sgn x = 1 \/ + 0 = x /\ Z.sgn x = 0 \/ + 0 > x /\ Z.sgn x = -1. Proof. intros. Z.swap_greater. apply Z.sgn_spec. Qed. (** Compatibility *) -Notation inj_Zabs_nat := Zabs2Nat.id_abs (only parsing). -Notation Zabs_nat_Z_of_nat := Zabs2Nat.id (only parsing). -Notation Zabs_nat_mult := Zabs2Nat.inj_mul (only parsing). -Notation Zabs_nat_Zsucc := Zabs2Nat.inj_succ (only parsing). -Notation Zabs_nat_Zplus := Zabs2Nat.inj_add (only parsing). -Notation Zabs_nat_Zminus := (fun n m => Zabs2Nat.inj_sub m n) (only parsing). -Notation Zabs_nat_compare := Zabs2Nat.inj_compare (only parsing). +Notation inj_Zabs_nat := Zabs2Nat.id_abs (compat "8.3"). +Notation Zabs_nat_Z_of_nat := Zabs2Nat.id (compat "8.3"). +Notation Zabs_nat_mult := Zabs2Nat.inj_mul (compat "8.3"). +Notation Zabs_nat_Zsucc := Zabs2Nat.inj_succ (compat "8.3"). +Notation Zabs_nat_Zplus := Zabs2Nat.inj_add (compat "8.3"). +Notation Zabs_nat_Zminus := (fun n m => Zabs2Nat.inj_sub m n) (compat "8.3"). +Notation Zabs_nat_compare := Zabs2Nat.inj_compare (compat "8.3"). Lemma Zabs_nat_le n m : 0 <= n <= m -> (Z.abs_nat n <= Z.abs_nat m)%nat. Proof. intros (H,H'). apply Zabs2Nat.inj_le; trivial. now transitivity n. Qed. -Lemma Zabs_nat_lt n m : 0 <= n < m -> (Zabs_nat n < Zabs_nat m)%nat. +Lemma Zabs_nat_lt n m : 0 <= n < m -> (Z.abs_nat n < Z.abs_nat m)%nat. Proof. intros (H,H'). apply Zabs2Nat.inj_lt; trivial. transitivity n; trivial. now apply Z.lt_le_incl. |