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-rw-r--r--theories/Numbers/Natural/Abstract/NAdd.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NAddOrder.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NAxioms.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NBase.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NBits.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NDefOps.v3
-rw-r--r--theories/Numbers/Natural/Abstract/NDiv.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NGcd.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NIso.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NLcm.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NLog.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NMaxMin.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NMulOrder.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NOrder.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NParity.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NPow.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NProperties.v23
-rw-r--r--theories/Numbers/Natural/Abstract/NSqrt.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NStrongRec.v7
-rw-r--r--theories/Numbers/Natural/Abstract/NSub.v2
20 files changed, 38 insertions, 29 deletions
diff --git a/theories/Numbers/Natural/Abstract/NAdd.v b/theories/Numbers/Natural/Abstract/NAdd.v
index e2dabf0e..638cfc7e 100644
--- a/theories/Numbers/Natural/Abstract/NAdd.v
+++ b/theories/Numbers/Natural/Abstract/NAdd.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NAddOrder.v b/theories/Numbers/Natural/Abstract/NAddOrder.v
index 94dc9e79..144bce72 100644
--- a/theories/Numbers/Natural/Abstract/NAddOrder.v
+++ b/theories/Numbers/Natural/Abstract/NAddOrder.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NAxioms.v b/theories/Numbers/Natural/Abstract/NAxioms.v
index c783478d..d300f857 100644
--- a/theories/Numbers/Natural/Abstract/NAxioms.v
+++ b/theories/Numbers/Natural/Abstract/NAxioms.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NBase.v b/theories/Numbers/Natural/Abstract/NBase.v
index a2bf4109..40453214 100644
--- a/theories/Numbers/Natural/Abstract/NBase.v
+++ b/theories/Numbers/Natural/Abstract/NBase.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NBits.v b/theories/Numbers/Natural/Abstract/NBits.v
index d368cc4e..6f8a8fe5 100644
--- a/theories/Numbers/Natural/Abstract/NBits.v
+++ b/theories/Numbers/Natural/Abstract/NBits.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NDefOps.v b/theories/Numbers/Natural/Abstract/NDefOps.v
index 882bb850..892b9266 100644
--- a/theories/Numbers/Natural/Abstract/NDefOps.v
+++ b/theories/Numbers/Natural/Abstract/NDefOps.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
@@ -133,7 +133,6 @@ Proof.
intros m n; unfold ltb at 1.
f_equiv.
rewrite recursion_succ; f_equiv'.
-reflexivity.
Qed.
(* Above, we rewrite applications of function. Is it possible to rewrite
diff --git a/theories/Numbers/Natural/Abstract/NDiv.v b/theories/Numbers/Natural/Abstract/NDiv.v
index 90a4e9e8..fb68c139 100644
--- a/theories/Numbers/Natural/Abstract/NDiv.v
+++ b/theories/Numbers/Natural/Abstract/NDiv.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NGcd.v b/theories/Numbers/Natural/Abstract/NGcd.v
index ff38b364..a1f4ddf8 100644
--- a/theories/Numbers/Natural/Abstract/NGcd.v
+++ b/theories/Numbers/Natural/Abstract/NGcd.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NIso.v b/theories/Numbers/Natural/Abstract/NIso.v
index 00f91311..c296315d 100644
--- a/theories/Numbers/Natural/Abstract/NIso.v
+++ b/theories/Numbers/Natural/Abstract/NIso.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NLcm.v b/theories/Numbers/Natural/Abstract/NLcm.v
index 6236360b..0fe8e105 100644
--- a/theories/Numbers/Natural/Abstract/NLcm.v
+++ b/theories/Numbers/Natural/Abstract/NLcm.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NLog.v b/theories/Numbers/Natural/Abstract/NLog.v
index f3418ef8..605c0aad 100644
--- a/theories/Numbers/Natural/Abstract/NLog.v
+++ b/theories/Numbers/Natural/Abstract/NLog.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NMaxMin.v b/theories/Numbers/Natural/Abstract/NMaxMin.v
index 9fa4114b..e0710561 100644
--- a/theories/Numbers/Natural/Abstract/NMaxMin.v
+++ b/theories/Numbers/Natural/Abstract/NMaxMin.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NMulOrder.v b/theories/Numbers/Natural/Abstract/NMulOrder.v
index 2aaf20ef..c41275d2 100644
--- a/theories/Numbers/Natural/Abstract/NMulOrder.v
+++ b/theories/Numbers/Natural/Abstract/NMulOrder.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NOrder.v b/theories/Numbers/Natural/Abstract/NOrder.v
index a46207ec..90053a73 100644
--- a/theories/Numbers/Natural/Abstract/NOrder.v
+++ b/theories/Numbers/Natural/Abstract/NOrder.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NParity.v b/theories/Numbers/Natural/Abstract/NParity.v
index 62d71eae..b3526c9a 100644
--- a/theories/Numbers/Natural/Abstract/NParity.v
+++ b/theories/Numbers/Natural/Abstract/NParity.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NPow.v b/theories/Numbers/Natural/Abstract/NPow.v
index c35757af..9cc23004 100644
--- a/theories/Numbers/Natural/Abstract/NPow.v
+++ b/theories/Numbers/Natural/Abstract/NPow.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NProperties.v b/theories/Numbers/Natural/Abstract/NProperties.v
index 575ede4f..cb3501d4 100644
--- a/theories/Numbers/Natural/Abstract/NProperties.v
+++ b/theories/Numbers/Natural/Abstract/NProperties.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
@@ -9,9 +9,20 @@
Require Export NAxioms.
Require Import NMaxMin NParity NPow NSqrt NLog NDiv NGcd NLcm NBits.
-(** This functor summarizes all known facts about N. *)
+(** The two following functors summarize all known facts about N.
-Module Type NProp (N:NAxiomsSig) :=
- NMaxMinProp N <+ NParityProp N <+ NPowProp N <+ NSqrtProp N
- <+ NLog2Prop N <+ NDivProp N <+ NGcdProp N <+ NLcmProp N
- <+ NBitsProp N.
+ - [NBasicProp] provides properties of basic functions:
+ + - * min max <= <
+
+ - [NExtraProp] provides properties of advanced functions:
+ pow, sqrt, log2, div, gcd, and bitwise functions.
+
+ If necessary, the earlier all-in-one functor [NProp]
+ could be re-obtained via [NBasicProp <+ NExtraProp] *)
+
+Module Type NBasicProp (N:NAxiomsMiniSig) := NMaxMinProp N.
+
+Module Type NExtraProp (N:NAxiomsSig)(P:NBasicProp N) :=
+ NParityProp N P <+ NPowProp N P <+ NSqrtProp N P
+ <+ NLog2Prop N P <+ NDivProp N P <+ NGcdProp N P <+ NLcmProp N P
+ <+ NBitsProp N P.
diff --git a/theories/Numbers/Natural/Abstract/NSqrt.v b/theories/Numbers/Natural/Abstract/NSqrt.v
index bc989a81..8dc66884 100644
--- a/theories/Numbers/Natural/Abstract/NSqrt.v
+++ b/theories/Numbers/Natural/Abstract/NSqrt.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Numbers/Natural/Abstract/NStrongRec.v b/theories/Numbers/Natural/Abstract/NStrongRec.v
index 7ec44dec..896ffc18 100644
--- a/theories/Numbers/Natural/Abstract/NStrongRec.v
+++ b/theories/Numbers/Natural/Abstract/NStrongRec.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
@@ -13,7 +13,7 @@ and proves its properties *)
Require Export NSub.
-Ltac f_equiv' := repeat progress (f_equiv; try intros ? ? ?; auto).
+Ltac f_equiv' := repeat (repeat f_equiv; try intros ? ? ?; auto).
Module NStrongRecProp (Import N : NAxiomsRecSig').
Include NSubProp N.
@@ -24,7 +24,7 @@ Variable A : Type.
Variable Aeq : relation A.
Variable Aeq_equiv : Equivalence Aeq.
-(** [strong_rec] allows to define a recursive function [phi] given by
+(** [strong_rec] allows defining a recursive function [phi] given by
an equation [phi(n) = F(phi)(n)] where recursive calls to [phi]
in [F] are made on strictly lower numbers than [n].
@@ -82,7 +82,6 @@ Proof.
intros. unfold strong_rec0.
f_equiv.
rewrite recursion_succ; f_equiv'.
-reflexivity.
Qed.
Lemma strong_rec_0 : forall a,
diff --git a/theories/Numbers/Natural/Abstract/NSub.v b/theories/Numbers/Natural/Abstract/NSub.v
index db61dd9b..18ebe77b 100644
--- a/theories/Numbers/Natural/Abstract/NSub.v
+++ b/theories/Numbers/Natural/Abstract/NSub.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \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 2.39.1) at 2024-06-06 15:41:45 +0000