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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Mult.
Require Import Compare_dec.
Require Import Wf_nat.
Local Open Scope nat_scope.
Implicit Types a b n q r : nat.
Inductive diveucl a b : Set :=
divex : forall q r, b > r -> a = q * b + r -> diveucl a b.
Lemma eucl_dev : forall n, n > 0 -> forall m:nat, diveucl m n.
Proof.
intros b H a; pattern a; apply gt_wf_rec; intros n H0.
elim (le_gt_dec b n).
intro lebn.
elim (H0 (n - b)); auto with arith.
intros q r g e.
apply divex with (S q) r; simpl; auto with arith.
elim plus_assoc.
elim e; auto with arith.
intros gtbn.
apply divex with 0 n; simpl; auto with arith.
Defined.
Lemma quotient :
forall n,
n > 0 ->
forall m:nat, {q : nat | exists r : nat, m = q * n + r /\ n > r}.
Proof.
intros b H a; pattern a; apply gt_wf_rec; intros n H0.
elim (le_gt_dec b n).
intro lebn.
elim (H0 (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.
elim H1; auto with arith.
intros gtbn.
exists 0; exists n; simpl; auto with arith.
Defined.
Lemma modulo :
forall n,
n > 0 ->
forall m:nat, {r : nat | exists q : nat, m = q * n + r /\ n > r}.
Proof.
intros b H a; pattern a; apply gt_wf_rec; intros n H0.
elim (le_gt_dec b n).
intro lebn.
elim (H0 (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.
elim H1; auto with arith.
intros gtbn.
exists n; exists 0; simpl; auto with arith.
Defined.
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