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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 Mult.
Require Compare_dec.
Require Wf_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.
Save.
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.
Save.
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.
Save.
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