(* 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.
(** d is a common divisor of a and b if d | a and d | b *)
Definition is_cd (d a b : nat) := (divides a d)/\(divides b d).
(** d is a greatest common divisor of a and b if it is a maximal common divisor *)
Definition is_gcd (d a b:nat) := (is_cd d a b)/\(forall (d':nat),(is_cd d' a b)->(divides d d')).
unfold is_gcd.
intros.
elim H;elim H0;intros.
apply divides_antisym;auto.
Qed.
unfold is_gcd.
intros.
elim H;intros.
split.
red;red in H0;tauto.
intros.
apply H1.
red;red in H2;tauto.
Qed.
unfold is_gcd.
intro.
split.
red;split;[apply zero_max_div | apply divides_refl].
unfold is_cd;tauto.
Qed.
unfold is_gcd.
intros.
split.
red;split;[apply divides_refl | apply one_min_div].
unfold is_cd;tauto.
Qed.
intros.
unfold is_gcd.
split;intro.
elim H0;intros.
split.
red in H1;red.
elim H1;intros.
split;try tauto.
apply divides_minus;trivial.
unfold is_cd;intros.
apply H2;red;elim H3;intros.
split;[tauto | rewrite (le_plus_minus a b H);apply divides_plus;trivial].
elim H0;unfold is_cd;intros.
split.
split;[tauto | elim H1;intros;rewrite (le_plus_minus a b H);apply divides_plus;trivial].
intros.
elim H3;intros;apply H2.
split;try (apply divides_minus);trivial.
Qed.
unfold is_gcd.
intros.
unfold is_cd.
split;try tauto.
split;apply divides_refl.
Qed.
unfold is_gcd;unfold is_cd;intros;tauto.
Qed.
unfold is_gcd;unfold is_cd;intros;tauto.
Qed.
(** we now show that gcd(a,b) exists for all a and b (we even have an algorithm) *)
Definition f (x:nat*nat) := (fst x)+(snd x).
Definition R (x y:nat*nat) := (f x)<(f y).
unfold R.
apply (well_founded_ltof (nat*nat) f).
Qed.
apply (induction_ltof2 (nat*nat) f (fun x:nat*nat => {d:nat | (is_gcd d (fst x) (snd x))})).
unfold ltof.
unfold f.
intros.
case (lt_eq_lt_dec (fst x) (snd x));intro.
case s;intro.
destruct (fst x).
exists (snd x);apply gcd_zero.
elim (H (S n,snd x-S n)).
simpl;intro d;intro.
exists d.
elim (gcd_minus d (S n) (snd x));try (auto with arith).
simpl.
omega.
rewrite e;exists (snd x);apply gcd_refl.
destruct (snd x).
exists (fst x);apply gcd_sym;apply gcd_zero.
elim (H (S n,fst x-S n)).
simpl;intro d;intro.
exists d.
apply gcd_sym.
elim (gcd_minus d (S n) (fst x));try (auto with arith).
simpl.
omega.
Qed.
intros.
elim (gcd_exists_prod (a,b)).
simpl;intro d;intro;exists d;trivial.
Qed.
(** take the first projection of the proof *)
Definition gcd (a b:nat) := let (d,_):=(gcd_exists a b) in d.
intros.
unfold gcd.
generalize (gcd_exists a b).
intro;elim s;intro d;trivial.
Qed.
(** a and b are relatively prime if gcd(a,b)=1 *)
Definition rel_prime (a b:nat) := (is_gcd 1 a b).
unfold rel_prime.
intros;apply gcd_sym;trivial.
Qed.
unfold rel_prime.
intros;apply gcd_sym;apply gcd_one.
Qed.
unfold rel_prime.
intros.
generalize (quo_is_quo a d (gcd_div_l d a b H));intro.
generalize (quo_is_quo b d (gcd_div_r d a b H));intro.
unfold is_gcd;split;unfold is_cd.
split;apply one_min_div.
intros.
elim H3;intros.
elim H4;intro q;intro.
elim H5;intro q';intro.
rewrite H6 in H1.
rewrite H7 in H2.
assert (divides d (d*d')).
red in H;elim H;intros.
apply H9;red;split;[exists q;rewrite H1;ring | exists q';rewrite H2;ring].
elim H8;intros.
exists x.
apply mult_lemma6 with d;trivial.
rewrite mult_assoc;rewrite <- H9;auto with arith.
Qed.
unfold is_gcd.
intros.
elim H0;intros.
intro.
elim H1;intros.
elim H5;intros.
rewrite H3 in H6;simpl in H6;auto.
Qed.
intros.
elim (gcd_minus ((u+v)*x-v*y) x y H);intros.
apply H2.
rewrite mult_plus_distr_r;rewrite <- minus_minus_lemma1;try (auto with arith);rewrite <- mult_minus_lemma2;trivial.
Qed.
intros.
elim (gcd_minus (v*y-(u+v)*x) x y H);intros.
apply H2.
rewrite mult_plus_distr_r;rewrite plus_comm;rewrite <- minus_minus_lemma2;rewrite <- mult_minus_lemma2;trivial.
Qed.
apply (induction_ltof2 (nat*nat) f (fun x:nat*nat => ({y:nat*nat | (is_gcd ((fst y)*(fst x)-(snd y)*(snd x)) (fst x) (snd x))}+{y:nat*nat | (is_gcd ((snd y)*(snd x)-(fst y)*(fst x)) (fst x) (snd x))})%type)).
unfold ltof.
unfold f.
intros.
case (lt_eq_lt_dec (fst x) (snd x));intro.
case s;intro.
destruct (fst x).
right;exists (0,1);simpl;rewrite <- minus_n_O;rewrite plus_comm;simpl;apply gcd_zero.
elim (H (S n,snd x-S n));try (intro;simpl).
elim a;intro y;intro.
left;exists ((fst y)+(snd y),(snd y)).
simpl;apply bezout_aux1;try (auto with arith).
elim b;intro y;intro.
right;exists ((fst y)+(snd y),(snd y)).
simpl;apply bezout_aux2;try (auto with arith).
simpl;omega.
rewrite e;left;exists (1,0);simpl;rewrite <- minus_n_O;rewrite plus_comm;simpl;apply gcd_refl.
destruct (snd x).
left;exists (1,0);simpl;rewrite <- minus_n_O;rewrite plus_comm;simpl;apply gcd_sym;apply gcd_zero.
elim (H (S n,fst x-S n));try (intro;simpl).
elim a;intro y;intro.
right;exists ((snd y),(fst y)+(snd y));apply gcd_sym.
simpl;apply bezout_aux1;try (auto with arith).
elim b;intro y;intro.
left;exists ((snd y),(fst y)+(snd y));apply gcd_sym.
simpl;apply bezout_aux2;try (auto with arith).
simpl;omega.
Qed.
intros.
elim (bezout_exists_prod (a,b));intro.
elim a0;destruct x;simpl;intros.
left;exists n;exists n0;rewrite mult_comm;rewrite (mult_comm b);trivial.
elim b0;destruct x;simpl;intros.
right;exists n;exists n0;rewrite mult_comm;rewrite (mult_comm a);trivial.
Qed.
intros.
elim (bezout_exists a b);intro.
elim a0;intro u;intro;elim p;intro v;intro;exists u;exists v;left;apply (gcd_unique d (a*u-b*v) a b);trivial.
elim b0;intro u;intro;elim p;intro v;intro;exists u;exists v;right;apply (gcd_unique d (b*v-a*u) a b);trivial.
Qed.
intros.
unfold rel_prime.
split;intro.
apply bezout;trivial.
elim H;intro u;intro H0.
elim H0;intro v;intro.
unfold is_gcd;unfold is_cd.
split.
split;apply one_min_div.
intros.
elim H2;intros.
elim H3;intro q;intro.
elim H4;intro q';intro.
rewrite H5 in H1;rewrite H6 in H1.
case H1;intro.
exists (q*u-q'*v);rewrite mult_minus_lemma2;rewrite mult_assoc;rewrite mult_assoc;trivial.
exists (q'*v-q*u);rewrite mult_minus_lemma2;rewrite mult_assoc;rewrite mult_assoc;trivial.
Qed.
unfold is_gcd;unfold is_cd.
intros.
elim H;intros.
elim H0;intros.
split.
elim H2;intro q;intro.
elim H3;intro q';intro.
rewrite H4;rewrite mult_assoc.
rewrite H5;rewrite mult_assoc.
split;[exists q;trivial | exists q';trivial].
intros.
elim H4;intros.
elim (bezout d a b);try (unfold is_gcd;unfold is_cd;trivial).
intro u;intro.
elim H7;intro v;intro.
elim H5;intro q;intro.
elim H6;intro q';intro.
case H8;intro;[exists (q*u-q'*v) | exists (q'*v-q*u)];rewrite mult_minus_lemma2;rewrite mult_assoc;rewrite mult_assoc;rewrite <- H9;rewrite <- H10;rewrite H11;rewrite mult_minus_lemma2;rewrite mult_assoc;rewrite mult_assoc;trivial.
Qed.
unfold rel_prime.
intros.
elim (bezout 1 a d H);intro u;intro.
elim H1;intro v;intro.
elim H0;intro q;intro.
case H2;intro;[exists (q*u-b*v) | exists (b*v-q*u)];rewrite mult_minus_lemma2;rewrite mult_assoc;rewrite mult_assoc;rewrite <- H3;rewrite (mult_comm a b);rewrite (mult_comm d b);rewrite <- mult_assoc;rewrite <- mult_assoc;rewrite <- mult_minus_lemma2;rewrite <- H4;auto with arith.
Qed.
intros.
generalize (quo_rem_euclide a b H);intro.
red;split;intro.
rewrite H0 in H1.
elim H1;intros.
unfold is_gcd;unfold is_cd.
elim H2;intros.
split.
split;try tauto.
elim H4;intro q;intro.
elim H5;intro q';intro.
replace (b*(quotient_euclide a b H)) with (d*q'*(quotient_euclide a b H)) in H6.
assert ((remainder_euclide a b H)=(d*q-d*q'*(quotient_euclide a b H))).
rewrite <- H6;rewrite minus_plus;trivial.
rewrite <- mult_assoc in H8;rewrite <- mult_minus_lemma2 in H8.
exists (q-q'*(quotient_euclide a b H));trivial.
rewrite <- H7;trivial.
intros.
elim H6;intros.
apply H3.
unfold is_cd;split;try tauto.
elim H7;intro q;intro.
elim H8;intro q';intro.
rewrite H10.
replace (b*(quotient_euclide a b H)) with (d'*q*(quotient_euclide a b H)).
rewrite <- mult_assoc;rewrite <- mult_plus_distr_l.
exists (q*(quotient_euclide a b H)+q');trivial.
rewrite <- H9;trivial.
unfold is_gcd;unfold is_cd.
unfold is_gcd in H1;unfold is_cd in H1.
elim H1;intros.
elim H2;intros.
rewrite H0.
split.
split;try tauto.
elim H4;intro q;intro.
elim H5;intro q';intro.
rewrite H7.
replace (b*(quotient_euclide a b H)) with (d*q*(quotient_euclide a b H)).
rewrite <- mult_assoc;rewrite <- mult_plus_distr_l.
exists (q*(quotient_euclide a b H)+q');trivial.
rewrite <- H6;trivial.
intros.
apply H3.
split;try tauto.
elim H6;intros.
elim H7;intro q;intro.
elim H8;intro q';intro.
assert ((remainder_euclide a b H)=b*(quotient_euclide a b H)+(remainder_euclide a b H)-b*(quotient_euclide a b H)).
rewrite minus_plus;trivial.
rewrite H9 in H11.
exists (q-q'*(quotient_euclide a b H)).
rewrite mult_minus_lemma2;rewrite mult_assoc.
rewrite <- H10;trivial.
Qed.
apply (induction_ltof2 (nat*nat) f (fun x:nat*nat => {d:nat | (is_gcd d (fst x) (snd x))})).
unfold ltof;unfold f;intros.
case (lt_eq_lt_dec (fst x) (snd x));intro.
case s;intro.
case (eq_nat_dec (fst x) 0);intro.
rewrite e;exists (snd x);apply gcd_zero.
elim (H ((fst x),(remainder_euclide (snd x) (fst x) n)));simpl.
intro d;intro.
exists d.
apply gcd_sym.
elim (gcd_euclide d (snd x) (fst x) n);auto.
generalize (rem_euclide (snd x) (fst x) n);try omega.
rewrite e;exists (snd x);apply gcd_refl.
case (eq_nat_dec (snd x) 0);intro.
rewrite e;exists (fst x);apply gcd_sym;apply gcd_zero.
elim (H ((snd x),(remainder_euclide (fst x) (snd x) n)));simpl.
intro d;intro.
exists d.
elim (gcd_euclide d (fst x) (snd x) n);auto.
generalize (rem_euclide (fst x) (snd x) n);try omega.
Qed.
intros.
elim (gcd_exists_prod_bis (a,b));intro d;simpl;intros.
exists d;trivial.
Qed.
intros.
unfold rel_prime.
generalize (gcd_is_gcd a b);intro.
case (eq_nat_dec (gcd a b) 1);intro.
left;rewrite e in H;trivial.
right;intro;apply n;apply (gcd_unique (gcd a b) 1 a b);trivial.
Qed.
intros.
split.
split;try (apply one_min_div).
intros.
elim H1;intros.
case (rel_prime_dec b d');intro.
assert (divides c d').
apply gauss with b;trivial.
elim H0;intros.
apply H6;unfold is_cd;tauto.
generalize (gcd_is_gcd b d');intro.
assert ((gcd b d')<>1).
intro;apply n.
unfold rel_prime;rewrite <- H5;trivial.
generalize (gcd_div_l (gcd b d') b d' H4);intro.
generalize (gcd_div_r (gcd b d') b d' H4);intro.
assert (divides a (gcd b d')).
apply divides_trans with d';[apply H2 | apply H7].
elim H5.
apply divides_antisym.
apply one_min_div.
elim H;intros;apply H10;unfold is_cd;tauto.
Qed.
intros.
split;split;[split | intros | split | intros];try (apply one_min_div);elim H0;intros;elim H;intros;apply H4;split;trivial;elim H2;intro q;intro;rewrite H5;[exists (q*c) | exists (q*b)];ring.
Qed.
induction n;simpl;intros.
unfold rel_prime;apply gcd_sym;apply gcd_one.
generalize (IHn H);intro.
apply rel_prime_mult;trivial.
Qed.
destruct n;simpl;intros.
inversion H.
elim (mult_rel_prime a d (power d n));auto.
Qed.

(** if n>0 and m>0 then gcd(a^n,b^m)=1 iff gcd(a,b)=1 *)
Lemma power_power_rel_prime : forall (a n b m:nat),(n>0)->(m>0)->((rel_prime (power a n) (power b m))<->(rel_prime a b)).