Imported Upstream version 8.0pl3+8.1alphaupstream/8.0pl3+8.1alpha

1 files changed, 0 insertions, 65 deletions

diff --git a/theories7/Arith/Euclid.v b/theories7/Arith/Euclid.v
deleted file mode 100644
index adeaf713..00000000
--- a/theories7/Arith/Euclid.v
+++ /dev/null
@@ -1,65 +0,0 @@
-(************************************************************************)
-(*  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        *)
-(************************************************************************)
-
-(*i $Id: Euclid.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*)
-
-Require Mult.
-Require Compare_dec.
-Require Wf_nat.
-
-V7only [Import nat_scope.].
-Open Local Scope nat_scope.
-
-Implicit Variables Type a,b,n,q,r:nat.
-
-Inductive diveucl [a,b:nat] : Set 
-      := divex : (q,r:nat)(gt b r)->(a=(plus (mult q b) r))->(diveucl a b).
-
-
-Lemma eucl_dev : (b:nat)(gt b O)->(a:nat)(diveucl a b).
-Intros b H a; Pattern a; Apply gt_wf_rec; Intros n H0.
-Elim (le_gt_dec b n).
-Intro lebn.
-Elim (H0 (minus n b)); Auto with arith.
-Intros q r g e.
-Apply divex with (S q) r; Simpl; Auto with arith.
-Elim plus_assoc_l.
-Elim e; Auto with arith.
-Intros gtbn.
-Apply divex with O n; Simpl; Auto with arith.
-Qed.
-
-Lemma quotient : (b:nat)(gt b O)->
-     (a:nat){q:nat|(EX r:nat | (a=(plus (mult q b) r))/\(gt b r))}.
-Intros b H a; Pattern a; Apply gt_wf_rec; Intros n H0.
-Elim (le_gt_dec b n).
-Intro lebn.
-Elim (H0 (minus n b)); Auto with arith.
-Intros q Hq; Exists (S q).
-Elim Hq; Intros r Hr.
-Exists r; Simpl; Elim Hr; Intros.
-Elim plus_assoc_l.
-Elim H1; Auto with arith.
-Intros gtbn.
-Exists O; Exists n; Simpl; Auto with arith.
-Qed.
-
-Lemma modulo : (b:nat)(gt b O)->
-     (a:nat){r:nat|(EX q:nat | (a=(plus (mult q b) r))/\(gt b r))}.
-Intros b H a; Pattern a; Apply gt_wf_rec; Intros n H0.
-Elim (le_gt_dec b n).
-Intro lebn.
-Elim (H0 (minus n b)); Auto with arith.
-Intros r Hr; Exists r.
-Elim Hr; Intros q Hq.
-Elim Hq; Intros; Exists (S q); Simpl.
-Elim plus_assoc_l.
-Elim H1; Auto with arith.
-Intros gtbn.
-Exists n; Exists O; Simpl; Auto with arith.
-Qed.