--- a/src/HOL/Number_Theory/Euclidean_Algorithm.thy Wed Jul 08 14:01:39 2015 +0200
+++ b/src/HOL/Number_Theory/Euclidean_Algorithm.thy Wed Jul 08 14:01:40 2015 +0200
@@ -753,84 +753,19 @@
"lcm a b = normalize (a * b) div gcd a b"
by (simp add: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
+subclass semiring_gcd
+ apply standard
+ using gcd_right_idem
+ apply (simp_all add: lcm_gcd gcd_greatest_iff gcd_proj1_if_dvd)
+ done
+
lemma lcm_gcd_prod:
"lcm a b * gcd a b = normalize (a * b)"
by (simp add: lcm_gcd)
-lemma lcm_dvd1 [iff]:
- "a dvd lcm a b"
-proof (cases "a*b = 0")
- assume "a * b \<noteq> 0"
- hence "gcd a b \<noteq> 0" by simp
- let ?c = "1 div unit_factor (a * b)"
- from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (unit_factor (a * b))" by simp
- from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
- by (simp add: div_mult_swap unit_div_commute)
- hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
- with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b"
- by (subst (asm) div_mult_self2_is_id, simp_all)
- also have "... = a * (?c * b div gcd a b)"
- by (metis div_mult_swap gcd_dvd2 mult_assoc)
- finally show ?thesis by (rule dvdI)
-qed (auto simp add: lcm_gcd)
-
-lemma lcm_least:
- "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
-proof (cases "k = 0")
- let ?nf = unit_factor
- assume "k \<noteq> 0"
- hence "is_unit (?nf k)" by simp
- hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
- assume A: "a dvd k" "b dvd k"
- hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto
- from A obtain r s where ar: "k = a * r" and bs: "k = b * s"
- unfolding dvd_def by blast
- with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0"
- by auto (drule sym [of 0], simp)
- hence "is_unit (?nf (r * s))" by simp
- let ?c = "?nf k div ?nf (r*s)"
- from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div)
- hence "?c \<noteq> 0" using not_is_unit_0 by fast
- from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
- by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
- also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
- by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps)
- also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close>
- by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
- finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
- by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
- hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
- by (simp add: algebra_simps)
- hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close>
- by (metis div_mult_self2_is_id)
- also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
- by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')
- also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
- by (simp add: algebra_simps)
- finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close>
- by (metis mult.commute div_mult_self2_is_id)
- hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close>
- by (metis div_mult_self2_is_id mult_assoc)
- also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close>
- by (simp add: unit_simps)
- finally show ?thesis by (rule dvdI)
-qed simp
-
lemma lcm_zero:
"lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
-proof -
- let ?nf = unit_factor
- {
- assume "a \<noteq> 0" "b \<noteq> 0"
- hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
- moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp
- ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
- } moreover {
- assume "a = 0 \<or> b = 0"
- hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
- }
- ultimately show ?thesis by blast
-qed
+ by (fact lcm_eq_0_iff)
lemmas lcm_0_iff = lcm_zero
@@ -844,46 +779,15 @@
by (metis nonzero_mult_divide_cancel_left)
qed
-lemma unit_factor_lcm [simp]:
- "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
- by (simp add: dvd_unit_factor_div lcm_gcd)
-
-lemma lcm_dvd2 [iff]: "b dvd lcm a b"
- using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)
+declare unit_factor_lcm [simp]
lemma lcmI:
assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d"
and "unit_factor c = (if c = 0 then 0 else 1)"
shows "c = lcm a b"
- by (rule associated_eqI)
- (auto simp: assms associated_def intro: lcm_least, simp_all add: lcm_gcd)
+ by (rule associated_eqI) (auto simp: assms intro: associatedI lcm_least)
-sublocale lcm!: abel_semigroup lcm
-proof
- fix a b c
- show "lcm (lcm a b) c = lcm a (lcm b c)"
- proof (rule lcmI)
- have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
- then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)
-
- have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
- hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)
- moreover have "c dvd lcm (lcm a b) c" by simp
- ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)
-
- fix l assume "a dvd l" and "lcm b c dvd l"
- have "b dvd lcm b c" by simp
- from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans)
- have "c dvd lcm b c" by simp
- from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans)
- from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least)
- from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least)
- qed (simp add: lcm_zero)
-next
- fix a b
- show "lcm a b = lcm b a"
- by (simp add: lcm_gcd ac_simps)
-qed
+sublocale lcm!: abel_semigroup lcm ..
lemma dvd_lcm_D1:
"lcm m n dvd k \<Longrightarrow> m dvd k"
@@ -913,13 +817,9 @@
finally show "lcm a b = 1" .
qed
-lemma lcm_0_left [simp]:
- "lcm 0 a = 0"
- by (rule sym, rule lcmI, simp_all)
-
-lemma lcm_0 [simp]:
+lemma lcm_0:
"lcm a 0 = 0"
- by (rule sym, rule lcmI, simp_all)
+ by (fact lcm_0_right)
lemma lcm_unique:
"a dvd d \<and> b dvd d \<and>
@@ -935,14 +835,6 @@
"k dvd n \<Longrightarrow> k dvd lcm m n"
by (metis lcm_dvd2 dvd_trans)
-lemma lcm_1_left [simp]:
- "lcm 1 a = normalize a"
- by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
-
-lemma lcm_1_right [simp]:
- "lcm a 1 = normalize a"
- using lcm_1_left [of a] by (simp add: ac_simps)
-
lemma lcm_coprime:
"gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)"
by (subst lcm_gcd) simp
@@ -1119,7 +1011,8 @@
qed
ultimately have "euclidean_size l = euclidean_size (gcd l l')"
by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
- with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
+ with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'"
+ using dvd_euclidean_size_eq_imp_dvd by auto
hence "l dvd l'" by (blast dest: dvd_gcd_D2)
}
@@ -1246,7 +1139,7 @@
"Lcm (insert a A) = lcm a (Lcm A)"
proof (rule lcmI)
fix l assume "a dvd l" and "Lcm A dvd l"
- hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)
+ then have "\<forall>a\<in>A. a dvd l" by (auto intro: dvd_trans [of _ "Lcm A" l])
with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_least)
qed (auto intro: Lcm_least dvd_Lcm)
@@ -1313,6 +1206,9 @@
"normalize (Gcd A) = Gcd A"
by (cases "Gcd A = 0") (auto intro: associated_eqI)
+subclass semiring_Gcd
+ by standard (simp_all add: Gcd_greatest)
+
lemma GcdI:
assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b"
and "unit_factor b = (if b = 0 then 0 else 1)"
@@ -1323,28 +1219,12 @@
"Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_greatest)
-lemma Gcd_0_iff:
- "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
- apply (rule iffI)
- apply (rule subsetI, drule Gcd_dvd, simp)
- apply (auto intro: GcdI[symmetric])
- done
-
-lemma Gcd_empty [simp]:
- "Gcd {} = 0"
- by (simp add: Gcd_0_iff)
+subclass semiring_Lcm
+ by standard (simp add: Lcm_Gcd)
lemma Gcd_1:
"1 \<in> A \<Longrightarrow> Gcd A = 1"
- by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
-
-lemma Gcd_insert [simp]:
- "Gcd (insert a A) = gcd a (Gcd A)"
-proof (rule gcdI)
- fix l assume "l dvd a" and "l dvd Gcd A"
- hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)
- with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd Gcd_greatest)
-qed (auto intro: Gcd_greatest)
+ by (auto intro!: Gcd_eq_1_I)
lemma Gcd_finite:
assumes "finite A"
@@ -1357,22 +1237,10 @@
using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"
- by (simp add: gcd_0)
+ by simp
lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
- by (simp add: gcd_0)
-
-subclass semiring_gcd
- apply standard
- using gcd_right_idem
- apply (simp_all add: lcm_gcd gcd_greatest_iff gcd_proj1_if_dvd)
- done
-
-subclass semiring_Gcd
- by standard (simp_all add: Gcd_greatest)
-
-subclass semiring_Lcm
- by standard (simp add: Lcm_Gcd)
+ by simp
end
@@ -1515,4 +1383,19 @@
end
+(*instance nat :: euclidean_semiring_gcd
+proof (standard, auto intro!: ext)
+ fix m n :: nat
+ show *: "gcd m n = gcd_eucl m n"
+ proof (induct m n rule: gcd_eucl_induct)
+ case zero then show ?case by (simp add: gcd_eucl_0)
+ next
+ case (mod m n)
+ with gcd_eucl_non_0 [of n m, symmetric]
+ show ?case by (simp add: gcd_non_0_nat)
+ qed
+ show "lcm m n = lcm_eucl m n"
+ by (simp add: lcm_eucl_def lcm_gcd * [symmetric] lcm_nat_def)
+qed*)
+
end