(* Copyright (C) 2005-2008 Sebastien Briais *)
(* http://lamp.epfl.ch/~sbriais/ *)
(* This library is free software; you can redistribute it and/or modify *)
(* it under the terms of the GNU Lesser General Public License as *)
(* published by the Free Software Foundation; either version 2.1 of the *)
(* License, or (at your option) any later version. *)
(* This library is distributed in the hope that it will be useful, but *)
(* WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU *)
(* Lesser General Public License for more details. *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this library; if not, write to the Free Software *)
(* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
(** b | a if there is q such that a = b * q*)
Definition divides (a b:nat) := exists q:nat,a = (b*q).
intros.
red.
exists n.
auto with arith.
Qed.
intros.
red.
exists O.
auto with arith.
Qed.
intros.
red.
exists 1.
auto with arith.
Qed.
unfold divides.
intros.
elim H;intro q;intro.
elim H0;intro q';intro.
rewrite H2 in H1.
exists (q' * q).
rewrite H1.
auto with arith.
Qed.
unfold divides.
intros.
elim H;intro q;intro.
elim H0;intro q';intro.
rewrite H2 in H1.
assert ((a = 0) \/ (q' * q)=1).
apply mult_lemma4.
replace (a*(q'*q)) with (a*q'*q);try (auto with arith).
case H3;intro.
rewrite H4 in H2;simpl in H2;rewrite H2;trivial.
elim (mult_lemma5 q' q H4);intros.
rewrite H5 in H2;rewrite mult_comm in H2;simpl in H2;rewrite plus_comm in H2;simpl in H2;symmetry;trivial.
Qed.
intros.
red.
intro.
apply H.
apply divides_antisym;trivial.
apply one_min_div.
Qed.
unfold divides.
intros.
elim H;intro q;intro.
elim H0;intro q';intro.
exists (q+q').
rewrite H1;rewrite H2.
ring.
Qed.
unfold divides.
intros.
elim H;intro q;intro.
exists (b * q).
rewrite H0.
ring.
Qed.
unfold divides.
intros.
elim H;intro q;intro.
elim H0;intro q';intro.
rewrite H1;rewrite H2.
exists (q'-q).
rewrite (mult_comm d q');rewrite (mult_comm d q);rewrite (mult_comm d (q'-q));auto with arith.
Qed.
intros.
apply (lt_wf_rec a (fun x:nat => (divides x b)->{q:nat | x = b*q}));trivial.
intro.
case n;intros.
exists 0;auto with arith.
elim (H0 ((S n0)-b)).
intro q;intro.
exists (S q).
replace (S n0) with (b+(S n0-b)).
rewrite p;rewrite plus_comm;auto with arith.
symmetry.
apply le_plus_minus.
elim H1;intros.
rewrite H2.
replace (b <= b*x) with (1*b <= b*x);rewrite (mult_comm b x).
apply mult_le_compat_r.
destruct x;[rewrite mult_comm in H2;discriminate | auto with arith].
simpl;auto with arith.
destruct b.
elim H1;simpl;intros;discriminate.
omega.
apply (divides_minus b b (S n0));[apply divides_refl | trivial].
Qed.
(** we can now define the quotient of a by b in case of b | a *)
Definition quo (a b:nat) (H:(divides a b)) := let (q,_):=(quo_dec a b H) in q.

(** the quotient is the quotient! *)
Lemma quo_is_quo : forall (a b:nat)(H:divides a b),a=(mult b (quo a b H)).