diff options
Diffstat (limited to 'theories7/Arith/Compare_dec.v')
-rwxr-xr-x | theories7/Arith/Compare_dec.v | 109 |
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. - |