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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(*i $Id$ i*)
Require Le.
Require Lt.
Require Gt.
Require Decidable.
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; NewInduction m; Auto with arith.
Elim (IHn m).
NewInduction 1; Auto with arith.
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.
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