(* 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 *)
Unset Standard Proposition Elimination Names.
intros.
elim H0;intro q;intro.
replace b with (b*1);try ring.
rewrite H1.
apply mult_le_compat;try omega.
destruct q;omega.
Qed.
intros.
apply (lt_wf_rec a (fun a:nat =>{q : nat &  {r : nat | a = b * q + r /\ r < b}})).
intros.
case (le_lt_dec b n);intro.
elim (H0 (n-b)).
intro q;intro.
elim p;intro r;intro.
exists (q+1);exists r.
split;try tauto.
rewrite (le_plus_minus b n);trivial.
elim p0;intros.
rewrite H1;ring.
omega.
exists 0;exists n.
split;try tauto.
ring.
Qed.
Definition quotient_euclide (a b:nat)(H:(b<>O)) := let (q,_) := (euclide a b H) in q.
Definition remainder_euclide (a b:nat)(H:(b<>O)) := let (_,e0) := (euclide a b H) in let (r,_) := e0 in r.
unfold quotient_euclide;unfold remainder_euclide;intros.
generalize (euclide a b H);intros.
elim s;intro q;intro.
elim p;intro r;intro.
tauto.
Qed.
unfold remainder_euclide;intros.
generalize (euclide a b H);intros.
elim s;intro q;intro.
elim p;intro r;intro.
tauto.
Qed.
intros.
rewrite H1 in H0.
case (lt_eq_lt_dec q q');intro.
case s;intro.
rewrite (le_plus_minus q q') in H0;try (auto with arith).
rewrite mult_plus_distr_l in H0.
assert (b*(q'-q)+r' = r).
apply plus_reg_l with (b*q).
rewrite plus_assoc;trivial.
assert (0<(q'-q));try omega.
assert (b<=b*(q'-q));try omega.
case (mult_O_le b (q'-q));intro;try omega.
rewrite mult_comm;trivial.
split;try tauto.
rewrite <- e in H0.
symmetry;apply plus_reg_l with (b*q);trivial.
rewrite (le_plus_minus q' q) in H0;try (auto with arith).
rewrite mult_plus_distr_l in H0.
assert (r'=(b*(q-q')+r)).
apply plus_reg_l with (b*q').
rewrite plus_assoc;trivial.
assert (0<(q-q'));try omega.
assert (b<=b*(q-q'));try omega.
case (mult_O_le b (q-q'));intro;try omega.
rewrite mult_comm;trivial.
Qed.
intros.
red.
split;intro.
generalize (quo_rem_euclide a b H);intro.
generalize (rem_euclide a b H);intro.
elim H0;intro q;intro.
assert (a=b*q+0).
rewrite plus_comm;simpl;trivial.
assert (0<b);try omega.
generalize (euclide_unique a b (quotient_euclide a b H) (remainder_euclide a b H) q 0 H H1 H4 H2 H5).
intros;tauto.
generalize (quo_rem_euclide a b H).
rewrite H0;rewrite plus_comm;simpl.
intro;exists (quotient_euclide a b H);trivial.
Qed.
intros.
case (eq_nat_dec b 0).
case (eq_nat_dec a 0);intros.
rewrite e;left;apply zero_max_div.
right;rewrite e;intro.
elim H;intro q;intro.
simpl in H0;apply n;trivial.
intro.
case (eq_nat_dec (remainder_euclide a b n) 0);[left | right];intros;elim (divides_euclide a b n);auto.
Qed.

(** if a property about integer is decidable then it is decidable if there is an integer less than n that satisfies this property *)
Lemma dec_impl_lt_dec : forall (P:nat->Prop),(forall (n:nat),{(P n)}+{~(P n)})->(forall (m:nat),{n:nat | (n<m)/\(P(n))}+{(forall (n:nat),(n<m)->~(P n))}).