1 files changed, 0 insertions, 109 deletions
diff --git a/theories7/Arith/Compare_dec.v b/theories7/Arith/Compare_dec.v
deleted file mode 100755
index 504c0562..00000000
--- a/theories7/Arith/Compare_dec.v
+++ /dev/null
@@ -1,109 +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: Compare_dec.v,v 1.1.2.1 2004/07/16 19:31:23 herbelin Exp $ i*)
-
-Require Le.
-Require Lt.
-Require Gt.
-Require Decidable.
-
-V7only [Import nat_scope.].
-Open Local Scope nat_scope.
-
-Implicit Variables Type m,n,x,y:nat.
-
-Definition zerop : (n:nat){n=O}+{lt O n}.
-NewDestruct n; Auto with arith.
-Defined.
-
-Definition lt_eq_lt_dec : (n,m:nat){(lt n m)}+{n=m}+{(lt m n)}.
-Proof.
-NewInduction n; Destruct m; Auto with arith.
-Intros m0; Elim (IHn m0); Auto with arith.
-NewInduction 1; Auto with arith.
-Defined.
-
-Lemma gt_eq_gt_dec : (n,m:nat)({(gt m n)}+{n=m})+{(gt n m)}.
-Proof lt_eq_lt_dec.
-
-Lemma le_lt_dec : (n,m:nat) {le n m} + {lt m n}.
-Proof.
-NewInduction n.
-Auto with arith.
-NewInduction m.
-Auto with arith.
-Elim (IHn m); Auto with arith.
-Defined.
-
-Definition le_le_S_dec : (n,m:nat) {le n m} + {le (S m) n}.
-Proof.
-Exact le_lt_dec.
-Defined.
-
-Definition le_ge_dec : (n,m:nat) {le n m} + {ge n m}.
-Proof.
-Intros; Elim (le_lt_dec n m); Auto with arith.
-Defined.
-
-Definition le_gt_dec : (n,m:nat){(le n m)}+{(gt n m)}.
-Proof.
-Exact le_lt_dec.
-Defined.
-
-Definition le_lt_eq_dec : (n,m:nat)(le n m)->({(lt n m)}+{n=m}).
-Proof.
-Intros; Elim (lt_eq_lt_dec n m); Auto with arith.
-Intros; Absurd (lt m n); Auto with arith.
-Defined.
-
-(** Proofs of decidability *)
-
-Theorem dec_le:(x,y:nat)(decidable (le x y)).
-Intros x y; Unfold decidable ; Elim (le_gt_dec x y); [
-  Auto with arith
-| Intro; Right; Apply gt_not_le; Assumption].
-Qed.
-
-Theorem dec_lt:(x,y:nat)(decidable (lt x y)).
-Intros x y; Unfold lt; Apply dec_le.
-Qed.
-
-Theorem dec_gt:(x,y:nat)(decidable (gt x y)).
-Intros x y; Unfold gt; Apply dec_lt.
-Qed.
-
-Theorem dec_ge:(x,y:nat)(decidable (ge x y)).
-Intros x y; Unfold ge; Apply dec_le.
-Qed.
-
-Theorem not_eq : (x,y:nat) ~ x=y -> (lt x y) \/ (lt y x).
-Intros x y H; Elim (lt_eq_lt_dec x y); [
-  Intros H1; Elim H1; [ Auto with arith | Intros H2; Absurd x=y; Assumption]
-| Auto with arith].
-Qed.
-
-
-Theorem not_le : (x,y:nat) ~(le x y) -> (gt x y).
-Intros x y H; Elim (le_gt_dec x y);
-  [ Intros H1; Absurd (le x y); Assumption | Trivial with arith ].
-Qed.
-
-Theorem not_gt : (x,y:nat) ~(gt x y) -> (le x y).
-Intros x y H; Elim (le_gt_dec x y);
-  [ Trivial with arith | Intros H1; Absurd (gt x y); Assumption].
-Qed.
-
-Theorem not_ge : (x,y:nat) ~(ge x y) -> (lt x y).
-Intros x y H; Exact (not_le y x H).
-Qed.
-
-Theorem not_lt : (x,y:nat) ~(lt x y) -> (ge x y).
-Intros x y H; Exact (not_gt y x H). 
-Qed.
-