--- a/src/HOL/Library/GCD.thy Sat Jan 06 20:47:09 2007 +0100
+++ b/src/HOL/Library/GCD.thy Sat Jan 06 20:56:44 2007 +0100
@@ -195,4 +195,111 @@
lemma gcd_add_mult: "gcd (m, k * m + n) = gcd (m, n)"
by (induct k) (simp_all add: add_assoc)
+ (* Division by gcd yields rrelatively primes *)
+
+
+lemma div_gcd_relprime:
+ assumes nz:"a\<noteq>0 \<or> b\<noteq>0"
+ shows "gcd (a div gcd(a,b), b div gcd(a,b)) = 1"
+proof-
+ let ?g = "gcd (a,b)"
+ let ?a' = "a div ?g"
+ let ?b' = "b div ?g"
+ let ?g' = "gcd (?a', ?b')"
+ have dvdg: "?g dvd a" "?g dvd b" by simp_all
+ have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
+ from dvdg dvdg' obtain ka kb ka' kb' where
+ kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
+ unfolding dvd_def by blast
+ hence "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all
+ hence dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
+ by (auto simp add: dvd_mult_div_cancel[OF dvdg(1)]
+ dvd_mult_div_cancel[OF dvdg(2)] dvd_def)
+ have "?g \<noteq> 0" using nz by (simp add: gcd_zero)
+ hence gp: "?g > 0" by simp
+ from gcd_greatest[OF dvdgg'] have "?g * ?g' dvd ?g" .
+ with dvd_mult_cancel1[OF gp] show "?g' = 1" by simp
+qed
+
+ (* gcd on integers *)
+constdefs igcd:: "int \<Rightarrow> int \<Rightarrow> int"
+ "igcd i j \<equiv> int (gcd (nat (abs i),nat (abs j)))"
+lemma igcd_dvd1[simp]:"igcd i j dvd i"
+ by (simp add: igcd_def int_dvd_iff)
+
+lemma igcd_dvd2[simp]:"igcd i j dvd j"
+by (simp add: igcd_def int_dvd_iff)
+
+lemma igcd_pos: "igcd i j \<ge> 0"
+by (simp add: igcd_def)
+lemma igcd0[simp]: "(igcd i j = 0) = (i = 0 \<and> j = 0)"
+by (simp add: igcd_def gcd_zero) arith
+
+lemma igcd_commute: "igcd i j = igcd j i"
+ unfolding igcd_def by (simp add: gcd_commute)
+lemma igcd_neg1[simp]: "igcd (- i) j = igcd i j"
+ unfolding igcd_def by simp
+lemma igcd_neg2[simp]: "igcd i (- j) = igcd i j"
+ unfolding igcd_def by simp
+lemma zrelprime_dvd_mult: "igcd i j = 1 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k"
+ unfolding igcd_def
+proof-
+ assume H: "int (gcd (nat \<bar>i\<bar>, nat \<bar>j\<bar>)) = 1" "i dvd k * j"
+ hence g: "gcd (nat \<bar>i\<bar>, nat \<bar>j\<bar>) = 1" by simp
+ from H(2) obtain h where h:"k*j = i*h" unfolding dvd_def by blast
+ have th: "nat \<bar>i\<bar> dvd nat \<bar>k\<bar> * nat \<bar>j\<bar>"
+ unfolding dvd_def
+ by (rule_tac x= "nat \<bar>h\<bar>" in exI,simp add: h nat_abs_mult_distrib[symmetric])
+ from relprime_dvd_mult[OF g th] obtain h' where h': "nat \<bar>k\<bar> = nat \<bar>i\<bar> * h'"
+ unfolding dvd_def by blast
+ from h' have "int (nat \<bar>k\<bar>) = int (nat \<bar>i\<bar> * h')" by simp
+ hence "\<bar>k\<bar> = \<bar>i\<bar> * int h'" by (simp add: int_mult)
+ then show ?thesis
+ apply (subst zdvd_abs1[symmetric])
+ apply (subst zdvd_abs2[symmetric])
+ apply (unfold dvd_def)
+ apply (rule_tac x="int h'" in exI, simp)
+ done
+qed
+
+lemma int_nat_abs: "int (nat (abs x)) = abs x" by arith
+lemma igcd_greatest: assumes km:"k dvd m" and kn:"k dvd n" shows "k dvd igcd m n"
+proof-
+ let ?k' = "nat \<bar>k\<bar>"
+ let ?m' = "nat \<bar>m\<bar>"
+ let ?n' = "nat \<bar>n\<bar>"
+ from km kn have dvd': "?k' dvd ?m'" "?k' dvd ?n'"
+ unfolding zdvd_int by (simp_all only: int_nat_abs zdvd_abs1 zdvd_abs2)
+ from gcd_greatest[OF dvd'] have "int (nat \<bar>k\<bar>) dvd igcd m n"
+ unfolding igcd_def by (simp only: zdvd_int)
+ hence "\<bar>k\<bar> dvd igcd m n" by (simp only: int_nat_abs)
+ thus "k dvd igcd m n" by (simp add: zdvd_abs1)
+qed
+
+lemma div_igcd_relprime:
+ assumes nz:"a\<noteq>0 \<or> b\<noteq>0"
+ shows "igcd (a div (igcd a b)) (b div (igcd a b)) = 1"
+proof-
+ from nz have nz': "nat \<bar>a\<bar> \<noteq> 0 \<or> nat \<bar>b\<bar> \<noteq> 0" by simp
+ have th1: "(1::int) = int 1" by simp
+ let ?g = "igcd a b"
+ let ?a' = "a div ?g"
+ let ?b' = "b div ?g"
+ let ?g' = "igcd ?a' ?b'"
+ have dvdg: "?g dvd a" "?g dvd b" by (simp_all add: igcd_dvd1 igcd_dvd2)
+ have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: igcd_dvd1 igcd_dvd2)
+ from dvdg dvdg' obtain ka kb ka' kb' where
+ kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
+ unfolding dvd_def by blast
+ hence "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all
+ hence dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
+ by (auto simp add: zdvd_mult_div_cancel[OF dvdg(1)]
+ zdvd_mult_div_cancel[OF dvdg(2)] dvd_def)
+ have "?g \<noteq> 0" using nz by simp
+ hence gp: "?g \<noteq> 0" using igcd_pos[where i="a" and j="b"] by arith
+ from igcd_greatest[OF dvdgg'] have "?g * ?g' dvd ?g" .
+ with zdvd_mult_cancel1[OF gp] have "\<bar>?g'\<bar> = 1" by simp
+ with igcd_pos show "?g' = 1" by simp
+qed
+
end