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-rw-r--r--theories/Numbers/Integer/Abstract/ZAdd.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZAddOrder.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZAxioms.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZBase.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZBits.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZDivEucl.v18
-rw-r--r--theories/Numbers/Integer/Abstract/ZDivFloor.v18
-rw-r--r--theories/Numbers/Integer/Abstract/ZDivTrunc.v18
-rw-r--r--theories/Numbers/Integer/Abstract/ZGcd.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZLcm.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZLt.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZMaxMin.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZMul.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZMulOrder.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZParity.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZPow.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZProperties.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZSgnAbs.v10
18 files changed, 132 insertions, 72 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZAdd.v b/theories/Numbers/Integer/Abstract/ZAdd.v
index f7fdc179..c4c5174d 100644
--- a/theories/Numbers/Integer/Abstract/ZAdd.v
+++ b/theories/Numbers/Integer/Abstract/ZAdd.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
diff --git a/theories/Numbers/Integer/Abstract/ZAddOrder.v b/theories/Numbers/Integer/Abstract/ZAddOrder.v
index 6bf5e9d4..7f5b0df6 100644
--- a/theories/Numbers/Integer/Abstract/ZAddOrder.v
+++ b/theories/Numbers/Integer/Abstract/ZAddOrder.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
diff --git a/theories/Numbers/Integer/Abstract/ZAxioms.v b/theories/Numbers/Integer/Abstract/ZAxioms.v
index ad10e65f..4f1ab775 100644
--- a/theories/Numbers/Integer/Abstract/ZAxioms.v
+++ b/theories/Numbers/Integer/Abstract/ZAxioms.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
diff --git a/theories/Numbers/Integer/Abstract/ZBase.v b/theories/Numbers/Integer/Abstract/ZBase.v
index 9b1b30f8..7fdd018d 100644
--- a/theories/Numbers/Integer/Abstract/ZBase.v
+++ b/theories/Numbers/Integer/Abstract/ZBase.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
diff --git a/theories/Numbers/Integer/Abstract/ZBits.v b/theories/Numbers/Integer/Abstract/ZBits.v
index c919e121..2da44528 100644
--- a/theories/Numbers/Integer/Abstract/ZBits.v
+++ b/theories/Numbers/Integer/Abstract/ZBits.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import
diff --git a/theories/Numbers/Integer/Abstract/ZDivEucl.v b/theories/Numbers/Integer/Abstract/ZDivEucl.v
index c2fa69e5..d7f25a66 100644
--- a/theories/Numbers/Integer/Abstract/ZDivEucl.v
+++ b/theories/Numbers/Integer/Abstract/ZDivEucl.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import ZAxioms ZMulOrder ZSgnAbs NZDiv.
@@ -602,6 +604,14 @@ Proof.
apply div_mod; order.
Qed.
+Lemma mod_div: forall a b, b~=0 ->
+ a mod b / b == 0.
+Proof.
+ intros a b Hb.
+ rewrite div_small_iff by assumption.
+ auto using mod_always_pos.
+Qed.
+
(** A last inequality: *)
Theorem div_mul_le:
diff --git a/theories/Numbers/Integer/Abstract/ZDivFloor.v b/theories/Numbers/Integer/Abstract/ZDivFloor.v
index 310748dd..a0d1821b 100644
--- a/theories/Numbers/Integer/Abstract/ZDivFloor.v
+++ b/theories/Numbers/Integer/Abstract/ZDivFloor.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import ZAxioms ZMulOrder ZSgnAbs NZDiv.
@@ -650,6 +652,14 @@ Proof.
apply div_mod; order.
Qed.
+Lemma mod_div: forall a b, b~=0 ->
+ a mod b / b == 0.
+Proof.
+ intros a b Hb.
+ rewrite div_small_iff by assumption.
+ auto using mod_bound_or.
+Qed.
+
(** A last inequality: *)
Theorem div_mul_le:
diff --git a/theories/Numbers/Integer/Abstract/ZDivTrunc.v b/theories/Numbers/Integer/Abstract/ZDivTrunc.v
index 04301077..31e42738 100644
--- a/theories/Numbers/Integer/Abstract/ZDivTrunc.v
+++ b/theories/Numbers/Integer/Abstract/ZDivTrunc.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import ZAxioms ZMulOrder ZSgnAbs NZDiv.
@@ -622,6 +624,14 @@ Proof.
apply quot_rem; order.
Qed.
+Lemma rem_quot: forall a b, b~=0 ->
+ a rem b รท b == 0.
+Proof.
+ intros a b Hb.
+ rewrite quot_small_iff by assumption.
+ auto using rem_bound_abs.
+Qed.
+
(** A last inequality: *)
Theorem quot_mul_le:
diff --git a/theories/Numbers/Integer/Abstract/ZGcd.v b/theories/Numbers/Integer/Abstract/ZGcd.v
index 30adaeb4..f0b7bf9d 100644
--- a/theories/Numbers/Integer/Abstract/ZGcd.v
+++ b/theories/Numbers/Integer/Abstract/ZGcd.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Properties of the greatest common divisor *)
diff --git a/theories/Numbers/Integer/Abstract/ZLcm.v b/theories/Numbers/Integer/Abstract/ZLcm.v
index ef33308c..0ab528de 100644
--- a/theories/Numbers/Integer/Abstract/ZLcm.v
+++ b/theories/Numbers/Integer/Abstract/ZLcm.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import ZAxioms ZMulOrder ZSgnAbs ZGcd ZDivTrunc ZDivFloor.
diff --git a/theories/Numbers/Integer/Abstract/ZLt.v b/theories/Numbers/Integer/Abstract/ZLt.v
index 0c92918d..726b041c 100644
--- a/theories/Numbers/Integer/Abstract/ZLt.v
+++ b/theories/Numbers/Integer/Abstract/ZLt.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
diff --git a/theories/Numbers/Integer/Abstract/ZMaxMin.v b/theories/Numbers/Integer/Abstract/ZMaxMin.v
index 24a47f00..f3f3a861 100644
--- a/theories/Numbers/Integer/Abstract/ZMaxMin.v
+++ b/theories/Numbers/Integer/Abstract/ZMaxMin.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import ZAxioms ZMulOrder GenericMinMax.
diff --git a/theories/Numbers/Integer/Abstract/ZMul.v b/theories/Numbers/Integer/Abstract/ZMul.v
index 830c0a7d..120647dc 100644
--- a/theories/Numbers/Integer/Abstract/ZMul.v
+++ b/theories/Numbers/Integer/Abstract/ZMul.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
diff --git a/theories/Numbers/Integer/Abstract/ZMulOrder.v b/theories/Numbers/Integer/Abstract/ZMulOrder.v
index 320c8f35..cd9523d3 100644
--- a/theories/Numbers/Integer/Abstract/ZMulOrder.v
+++ b/theories/Numbers/Integer/Abstract/ZMulOrder.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
diff --git a/theories/Numbers/Integer/Abstract/ZParity.v b/theories/Numbers/Integer/Abstract/ZParity.v
index 952d8e9c..a5e53b36 100644
--- a/theories/Numbers/Integer/Abstract/ZParity.v
+++ b/theories/Numbers/Integer/Abstract/ZParity.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import Bool ZMulOrder NZParity.
diff --git a/theories/Numbers/Integer/Abstract/ZPow.v b/theories/Numbers/Integer/Abstract/ZPow.v
index 02b8501c..a4b964e5 100644
--- a/theories/Numbers/Integer/Abstract/ZPow.v
+++ b/theories/Numbers/Integer/Abstract/ZPow.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Properties of the power function *)
diff --git a/theories/Numbers/Integer/Abstract/ZProperties.v b/theories/Numbers/Integer/Abstract/ZProperties.v
index 1dec3c58..e4b997cf 100644
--- a/theories/Numbers/Integer/Abstract/ZProperties.v
+++ b/theories/Numbers/Integer/Abstract/ZProperties.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Export ZAxioms ZMaxMin ZSgnAbs ZParity ZPow ZDivTrunc ZDivFloor
diff --git a/theories/Numbers/Integer/Abstract/ZSgnAbs.v b/theories/Numbers/Integer/Abstract/ZSgnAbs.v
index a10552ab..dda12872 100644
--- a/theories/Numbers/Integer/Abstract/ZSgnAbs.v
+++ b/theories/Numbers/Integer/Abstract/ZSgnAbs.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Properties of [abs] and [sgn] *)
generated by cgit v1.2.3 (git 2.39.1) at 2024-05-19 16:32:44 +0000