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-rw-r--r--theories/Numbers/Integer/Abstract/ZAdd.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZAddOrder.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZAxioms.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZBase.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZBits.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZDivEucl.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZDivFloor.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZDivTrunc.v10
-rw-r--r--theories/Numbers/Integer/Abstract/ZGcd.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZLcm.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZLt.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZMaxMin.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZMul.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZMulOrder.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZParity.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZPow.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZProperties.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZSgnAbs.v2
18 files changed, 42 insertions, 18 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZAdd.v b/theories/Numbers/Integer/Abstract/ZAdd.v
index f7fdc17950..9d94ff65d5 100644
--- a/theories/Numbers/Integer/Abstract/ZAdd.v
+++ b/theories/Numbers/Integer/Abstract/ZAdd.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZAddOrder.v b/theories/Numbers/Integer/Abstract/ZAddOrder.v
index 6bf5e9d4c5..84bb1a24a5 100644
--- a/theories/Numbers/Integer/Abstract/ZAddOrder.v
+++ b/theories/Numbers/Integer/Abstract/ZAddOrder.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZAxioms.v b/theories/Numbers/Integer/Abstract/ZAxioms.v
index ad10e65f00..6efb7ea235 100644
--- a/theories/Numbers/Integer/Abstract/ZAxioms.v
+++ b/theories/Numbers/Integer/Abstract/ZAxioms.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZBase.v b/theories/Numbers/Integer/Abstract/ZBase.v
index 9b1b30f878..b2014135f1 100644
--- a/theories/Numbers/Integer/Abstract/ZBase.v
+++ b/theories/Numbers/Integer/Abstract/ZBase.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZBits.v b/theories/Numbers/Integer/Abstract/ZBits.v
index c919e121db..68d27fec93 100644
--- a/theories/Numbers/Integer/Abstract/ZBits.v
+++ b/theories/Numbers/Integer/Abstract/ZBits.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZDivEucl.v b/theories/Numbers/Integer/Abstract/ZDivEucl.v
index c2fa69e535..967b68d365 100644
--- a/theories/Numbers/Integer/Abstract/ZDivEucl.v
+++ b/theories/Numbers/Integer/Abstract/ZDivEucl.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
@@ -602,6 +602,14 @@ 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 310748ddb3..a9077127ed 100644
--- a/theories/Numbers/Integer/Abstract/ZDivFloor.v
+++ b/theories/Numbers/Integer/Abstract/ZDivFloor.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
@@ -650,6 +650,14 @@ 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 0430107792..bbb8ad5ae9 100644
--- a/theories/Numbers/Integer/Abstract/ZDivTrunc.v
+++ b/theories/Numbers/Integer/Abstract/ZDivTrunc.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
@@ -622,6 +622,14 @@ 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 30adaeb4b1..1144bd2bfa 100644
--- a/theories/Numbers/Integer/Abstract/ZGcd.v
+++ b/theories/Numbers/Integer/Abstract/ZGcd.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZLcm.v b/theories/Numbers/Integer/Abstract/ZLcm.v
index ef33308c28..4b0f9f9789 100644
--- a/theories/Numbers/Integer/Abstract/ZLcm.v
+++ b/theories/Numbers/Integer/Abstract/ZLcm.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZLt.v b/theories/Numbers/Integer/Abstract/ZLt.v
index 0c92918d13..f43d74d706 100644
--- a/theories/Numbers/Integer/Abstract/ZLt.v
+++ b/theories/Numbers/Integer/Abstract/ZLt.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZMaxMin.v b/theories/Numbers/Integer/Abstract/ZMaxMin.v
index 24a47f002f..40c3980a5b 100644
--- a/theories/Numbers/Integer/Abstract/ZMaxMin.v
+++ b/theories/Numbers/Integer/Abstract/ZMaxMin.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZMul.v b/theories/Numbers/Integer/Abstract/ZMul.v
index 830c0a7da0..2550ae1c5e 100644
--- a/theories/Numbers/Integer/Abstract/ZMul.v
+++ b/theories/Numbers/Integer/Abstract/ZMul.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZMulOrder.v b/theories/Numbers/Integer/Abstract/ZMulOrder.v
index 320c8f355a..adea36f0e8 100644
--- a/theories/Numbers/Integer/Abstract/ZMulOrder.v
+++ b/theories/Numbers/Integer/Abstract/ZMulOrder.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZParity.v b/theories/Numbers/Integer/Abstract/ZParity.v
index 952d8e9c12..db379474bf 100644
--- a/theories/Numbers/Integer/Abstract/ZParity.v
+++ b/theories/Numbers/Integer/Abstract/ZParity.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZPow.v b/theories/Numbers/Integer/Abstract/ZPow.v
index 02b8501c79..478724e2ec 100644
--- a/theories/Numbers/Integer/Abstract/ZPow.v
+++ b/theories/Numbers/Integer/Abstract/ZPow.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZProperties.v b/theories/Numbers/Integer/Abstract/ZProperties.v
index 1dec3c5869..4a0ce5e5bf 100644
--- a/theories/Numbers/Integer/Abstract/ZProperties.v
+++ b/theories/Numbers/Integer/Abstract/ZProperties.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Integer/Abstract/ZSgnAbs.v b/theories/Numbers/Integer/Abstract/ZSgnAbs.v
index a10552ab3b..b2f77c3f59 100644
--- a/theories/Numbers/Integer/Abstract/ZSgnAbs.v
+++ b/theories/Numbers/Integer/Abstract/ZSgnAbs.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
generated by cgit v1.2.3 (git 2.25.1) at 2026-05-02 20:58:11 +0000