--- a/src/HOL/Library/Primes.thy Mon Jul 21 13:37:05 2008 +0200
+++ b/src/HOL/Library/Primes.thy Mon Jul 21 13:37:10 2008 +0200
@@ -117,7 +117,6 @@
lemma divides_mul_r: "(a::nat) dvd b ==> (a * c) dvd (b * c)" by presburger
lemma divides_cases: "(n::nat) dvd m ==> m = 0 \<or> m = n \<or> 2 * n <= m"
by (auto simp add: dvd_def)
-lemma divides_le: "m dvd n ==> m <= n \<or> n = (0::nat)" by (auto simp add: dvd_def)
lemma divides_div_not: "(x::nat) = (q * n) + r \<Longrightarrow> 0 < r \<Longrightarrow> r < n ==> ~(n dvd x)"
proof(auto simp add: dvd_def)
@@ -178,236 +177,6 @@
lemma divides_rexp:
"x dvd y \<Longrightarrow> (x::nat) dvd (y^(Suc n))" by (simp add: dvd_mult2[of x y])
-text {* The Bezout theorem is a bit ugly for N; it'd be easier for Z *}
-lemma ind_euclid:
- assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
- and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
- shows "P a b"
-proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
- fix n a b
- assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
- have "a = b \<or> a < b \<or> b < a" by arith
- moreover {assume eq: "a= b"
- from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq by simp}
- moreover
- {assume lt: "a < b"
- hence "a + b - a < n \<or> a = 0" using H(2) by arith
- moreover
- {assume "a =0" with z c have "P a b" by blast }
- moreover
- {assume ab: "a + b - a < n"
- have th0: "a + b - a = a + (b - a)" using lt by arith
- from add[rule_format, OF H(1)[rule_format, OF ab th0]]
- have "P a b" by (simp add: th0[symmetric])}
- ultimately have "P a b" by blast}
- moreover
- {assume lt: "a > b"
- hence "b + a - b < n \<or> b = 0" using H(2) by arith
- moreover
- {assume "b =0" with z c have "P a b" by blast }
- moreover
- {assume ab: "b + a - b < n"
- have th0: "b + a - b = b + (a - b)" using lt by arith
- from add[rule_format, OF H(1)[rule_format, OF ab th0]]
- have "P b a" by (simp add: th0[symmetric])
- hence "P a b" using c by blast }
- ultimately have "P a b" by blast}
-ultimately show "P a b" by blast
-qed
-
-lemma bezout_lemma:
- assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
- shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and> (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
-using ex
-apply clarsimp
-apply (rule_tac x="d" in exI, simp add: dvd_add)
-apply (case_tac "a * x = b * y + d" , simp_all)
-apply (rule_tac x="x + y" in exI)
-apply (rule_tac x="y" in exI)
-apply algebra
-apply (rule_tac x="x" in exI)
-apply (rule_tac x="x + y" in exI)
-apply algebra
-done
-
-lemma bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
-apply(induct a b rule: ind_euclid)
-apply blast
-apply clarify
-apply (rule_tac x="a" in exI, simp add: dvd_add)
-apply clarsimp
-apply (rule_tac x="d" in exI)
-apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
-apply (rule_tac x="x+y" in exI)
-apply (rule_tac x="y" in exI)
-apply algebra
-apply (rule_tac x="x" in exI)
-apply (rule_tac x="x+y" in exI)
-apply algebra
-done
-
-lemma bezout: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x - b * y = d \<or> b * x - a * y = d)"
-using bezout_add[of a b]
-apply clarsimp
-apply (rule_tac x="d" in exI, simp)
-apply (rule_tac x="x" in exI)
-apply (rule_tac x="y" in exI)
-apply auto
-done
-
-text {* We can get a stronger version with a nonzeroness assumption. *}
-
-lemma bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
- shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
-proof-
- from nz have ap: "a > 0" by simp
- from bezout_add[of a b]
- have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or> (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
- moreover
- {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
- from H have ?thesis by blast }
- moreover
- {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
- {assume b0: "b = 0" with H have ?thesis by simp}
- moreover
- {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
- from divides_le[OF H(2)] b have "d < b \<or> d = b" using le_less by blast
- moreover
- {assume db: "d=b"
- from prems have ?thesis apply simp
- apply (rule exI[where x = b], simp)
- apply (rule exI[where x = b])
- by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
- moreover
- {assume db: "d < b"
- {assume "x=0" hence ?thesis using prems by simp }
- moreover
- {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
-
- from db have "d \<le> b - 1" by simp
- hence "d*b \<le> b*(b - 1)" by simp
- with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
- have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
- from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)" by simp
- hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
- by (simp only: mult_assoc right_distrib)
- hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)" by algebra
- hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
- hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
- by (simp only: diff_add_assoc[OF dble, of d, symmetric])
- hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
- by (simp only: diff_mult_distrib2 add_commute mult_ac)
- hence ?thesis using H(1,2)
- apply -
- apply (rule exI[where x=d], simp)
- apply (rule exI[where x="(b - 1) * y"])
- by (rule exI[where x="x*(b - 1) - d"], simp)}
- ultimately have ?thesis by blast}
- ultimately have ?thesis by blast}
- ultimately have ?thesis by blast}
- ultimately show ?thesis by blast
-qed
-
-text {* Greatest common divisor. *}
-lemma gcd_unique: "d dvd a\<and>d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
-proof(auto)
- assume H: "d dvd a" "d dvd b" "\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d"
- from H(3)[rule_format] gcd_dvd1[of a b] gcd_dvd2[of a b]
- have th: "gcd a b dvd d" by blast
- from dvd_anti_sym[OF th gcd_greatest[OF H(1,2)]] show "d = gcd a b" by blast
-qed
-
-lemma gcd_eq: assumes H: "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd u \<and> d dvd v"
- shows "gcd x y = gcd u v"
-proof-
- from H have "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd gcd u v" by simp
- with gcd_unique[of "gcd u v" x y] show ?thesis by auto
-qed
-
-lemma bezout_gcd: "\<exists>x y. a * x - b * y = gcd a b \<or> b * x - a * y = gcd a b"
-proof-
- let ?g = "gcd a b"
- from bezout[of a b] obtain d x y where d: "d dvd a" "d dvd b" "a * x - b * y = d \<or> b * x - a * y = d" by blast
- from d(1,2) have "d dvd ?g" by simp
- then obtain k where k: "?g = d*k" unfolding dvd_def by blast
- from d(3) have "(a * x - b * y)*k = d*k \<or> (b * x - a * y)*k = d*k" by blast
- hence "a * x * k - b * y*k = d*k \<or> b * x * k - a * y*k = d*k"
- by (simp only: diff_mult_distrib)
- hence "a * (x * k) - b * (y*k) = ?g \<or> b * (x * k) - a * (y*k) = ?g"
- by (simp add: k mult_assoc)
- thus ?thesis by blast
-qed
-
-lemma bezout_gcd_strong: assumes a: "a \<noteq> 0"
- shows "\<exists>x y. a * x = b * y + gcd a b"
-proof-
- let ?g = "gcd a b"
- from bezout_add_strong[OF a, of b]
- obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
- from d(1,2) have "d dvd ?g" by simp
- then obtain k where k: "?g = d*k" unfolding dvd_def by blast
- from d(3) have "a * x * k = (b * y + d) *k " by auto
- hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
- thus ?thesis by blast
-qed
-
-lemma gcd_mult_distrib: "gcd(a * c) (b * c) = c * gcd a b"
-by(simp add: gcd_mult_distrib2 mult_commute)
-
-lemma gcd_bezout: "(\<exists>x y. a * x - b * y = d \<or> b * x - a * y = d) \<longleftrightarrow> gcd a b dvd d"
- (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
- let ?g = "gcd a b"
- {assume H: ?rhs then obtain k where k: "d = ?g*k" unfolding dvd_def by blast
- from bezout_gcd[of a b] obtain x y where xy: "a * x - b * y = ?g \<or> b * x - a * y = ?g"
- by blast
- hence "(a * x - b * y)*k = ?g*k \<or> (b * x - a * y)*k = ?g*k" by auto
- hence "a * x*k - b * y*k = ?g*k \<or> b * x * k - a * y*k = ?g*k"
- by (simp only: diff_mult_distrib)
- hence "a * (x*k) - b * (y*k) = d \<or> b * (x * k) - a * (y*k) = d"
- by (simp add: k[symmetric] mult_assoc)
- hence ?lhs by blast}
- moreover
- {fix x y assume H: "a * x - b * y = d \<or> b * x - a * y = d"
- have dv: "?g dvd a*x" "?g dvd b * y" "?g dvd b*x" "?g dvd a * y"
- using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
- from dvd_diff[OF dv(1,2)] dvd_diff[OF dv(3,4)] H
- have ?rhs by auto}
- ultimately show ?thesis by blast
-qed
-
-lemma gcd_bezout_sum: assumes H:"a * x + b * y = d" shows "gcd a b dvd d"
-proof-
- let ?g = "gcd a b"
- have dv: "?g dvd a*x" "?g dvd b * y"
- using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
- from dvd_add[OF dv] H
- show ?thesis by auto
-qed
-
-lemma gcd_mult': "gcd b (a * b) = b"
-by (simp add: gcd_mult mult_commute[of a b])
-
-lemma gcd_add: "gcd(a + b) b = gcd a b"
- "gcd(b + a) b = gcd a b" "gcd a (a + b) = gcd a b" "gcd a (b + a) = gcd a b"
-apply (simp_all add: gcd_add1)
-by (simp add: gcd_commute gcd_add1)
-
-lemma gcd_sub: "b <= a ==> gcd(a - b) b = gcd a b" "a <= b ==> gcd a (b - a) = gcd a b"
-proof-
- {fix a b assume H: "b \<le> (a::nat)"
- hence th: "a - b + b = a" by arith
- from gcd_add(1)[of "a - b" b] th have "gcd(a - b) b = gcd a b" by simp}
- note th = this
-{
- assume ab: "b \<le> a"
- from th[OF ab] show "gcd (a - b) b = gcd a b" by blast
-next
- assume ab: "a \<le> b"
- from th[OF ab] show "gcd a (b - a) = gcd a b"
- by (simp add: gcd_commute)}
-qed
-
text {* Coprimality *}
lemma coprime: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"