author | haftmann |
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(* Author: Manuel Eberl *) |
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section \<open>Abstract euclidean algorithm\<close> |
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theory Euclidean_Algorithm |
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imports Main "~~/src/HOL/GCD" "~~/src/HOL/Library/Polynomial" |
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begin |
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text \<open> |
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A Euclidean semiring is a semiring upon which the Euclidean algorithm can be |
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implemented. It must provide: |
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\begin{itemize} |
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\item division with remainder |
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\item a size function such that @{term "size (a mod b) < size b"} |
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for any @{term "b \<noteq> 0"} |
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\end{itemize} |
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The existence of these functions makes it possible to derive gcd and lcm functions |
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for any Euclidean semiring. |
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\<close> |
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class euclidean_semiring = semiring_div + normalization_semidom + |
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fixes euclidean_size :: "'a \<Rightarrow> nat" |
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assumes mod_size_less: |
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"b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b" |
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assumes size_mult_mono: |
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"b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)" |
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begin |
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lemma euclidean_division: |
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fixes a :: 'a and b :: 'a |
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assumes "b \<noteq> 0" |
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obtains s and t where "a = s * b + t" |
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and "euclidean_size t < euclidean_size b" |
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proof - |
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from div_mod_equality [of a b 0] |
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have "a = a div b * b + a mod b" by simp |
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with that and assms show ?thesis by (auto simp add: mod_size_less) |
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qed |
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lemma dvd_euclidean_size_eq_imp_dvd: |
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assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b" |
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shows "a dvd b" |
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proof (rule ccontr) |
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assume "\<not> a dvd b" |
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then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd) |
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from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff) |
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from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast |
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with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto |
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with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b" |
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using size_mult_mono by force |
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moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close> |
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have "euclidean_size (b mod a) < euclidean_size a" |
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using mod_size_less by blast |
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ultimately show False using size_eq by simp |
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qed |
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function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" |
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where |
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"gcd_eucl a b = (if b = 0 then normalize a else gcd_eucl b (a mod b))" |
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by pat_completeness simp |
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termination |
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by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less) |
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declare gcd_eucl.simps [simp del] |
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lemma gcd_eucl_induct [case_names zero mod]: |
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assumes H1: "\<And>b. P b 0" |
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and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b" |
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shows "P a b" |
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proof (induct a b rule: gcd_eucl.induct) |
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case ("1" a b) |
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show ?case |
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proof (cases "b = 0") |
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case True then show "P a b" by simp (rule H1) |
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next |
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case False |
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then have "P b (a mod b)" |
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by (rule "1.hyps") |
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with \<open>b \<noteq> 0\<close> show "P a b" |
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by (blast intro: H2) |
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qed |
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qed |
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definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" |
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where |
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"lcm_eucl a b = normalize (a * b) div gcd_eucl a b" |
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definition Lcm_eucl :: "'a set \<Rightarrow> 'a" -- \<open> |
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Somewhat complicated definition of Lcm that has the advantage of working |
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for infinite sets as well\<close> |
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where |
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"Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then |
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let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = |
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(LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n) |
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in normalize l |
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else 0)" |
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definition Gcd_eucl :: "'a set \<Rightarrow> 'a" |
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where |
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"Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}" |
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lemma gcd_eucl_0: |
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"gcd_eucl a 0 = normalize a" |
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by (simp add: gcd_eucl.simps [of a 0]) |
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lemma gcd_eucl_0_left: |
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"gcd_eucl 0 a = normalize a" |
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by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a]) |
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lemma gcd_eucl_non_0: |
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"b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)" |
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by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0]) |
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end |
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class euclidean_ring = euclidean_semiring + idom |
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begin |
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function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where |
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"euclid_ext a b = |
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(if b = 0 then |
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(1 div unit_factor a, 0, normalize a) |
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else |
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case euclid_ext b (a mod b) of |
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(s, t, c) \<Rightarrow> (t, s - t * (a div b), c))" |
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by pat_completeness simp |
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termination |
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by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less) |
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declare euclid_ext.simps [simp del] |
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lemma euclid_ext_0: |
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"euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)" |
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by (simp add: euclid_ext.simps [of a 0]) |
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lemma euclid_ext_left_0: |
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"euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)" |
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by (simp add: euclid_ext_0 euclid_ext.simps [of 0 a]) |
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lemma euclid_ext_non_0: |
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"b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of |
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(s, t, c) \<Rightarrow> (t, s - t * (a div b), c))" |
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by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0]) |
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lemma euclid_ext_code [code]: |
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"euclid_ext a b = (if b = 0 then (1 div unit_factor a, 0, normalize a) |
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else let (s, t, c) = euclid_ext b (a mod b) in (t, s - t * (a div b), c))" |
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by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0]) |
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lemma euclid_ext_correct: |
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"case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c" |
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proof (induct a b rule: gcd_eucl_induct) |
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case (zero a) then show ?case |
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by (simp add: euclid_ext_0 ac_simps) |
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next |
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case (mod a b) |
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obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)" |
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by (cases "euclid_ext b (a mod b)") blast |
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with mod have "c = s * b + t * (a mod b)" by simp |
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also have "... = t * ((a div b) * b + a mod b) + (s - t * (a div b)) * b" |
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by (simp add: algebra_simps) |
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also have "(a div b) * b + a mod b = a" using mod_div_equality . |
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finally show ?case |
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by (subst euclid_ext.simps) (simp add: stc mod ac_simps) |
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qed |
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definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a" |
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where |
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"euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))" |
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lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)" |
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by (simp add: euclid_ext'_def euclid_ext_0) |
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|
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lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)" |
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174 |
by (simp add: euclid_ext'_def euclid_ext_left_0) |
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|
175 |
|
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|
176 |
lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)), |
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|
177 |
fst (euclid_ext' b (a mod b)) - snd (euclid_ext' b (a mod b)) * (a div b))" |
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|
178 |
by (simp add: euclid_ext'_def euclid_ext_non_0 split_def) |
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|
179 |
|
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180 |
end |
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181 |
|
58023 | 182 |
class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd + |
183 |
assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl" |
|
184 |
assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl" |
|
185 |
begin |
|
186 |
||
187 |
lemma gcd_0_left: |
|
60634 | 188 |
"gcd 0 a = normalize a" |
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189 |
unfolding gcd_gcd_eucl by (fact gcd_eucl_0_left) |
58023 | 190 |
|
191 |
lemma gcd_0: |
|
60634 | 192 |
"gcd a 0 = normalize a" |
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|
193 |
unfolding gcd_gcd_eucl by (fact gcd_eucl_0) |
58023 | 194 |
|
195 |
lemma gcd_non_0: |
|
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|
196 |
"b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)" |
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|
197 |
unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0) |
58023 | 198 |
|
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|
199 |
lemma gcd_dvd1 [iff]: "gcd a b dvd a" |
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200 |
and gcd_dvd2 [iff]: "gcd a b dvd b" |
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|
201 |
by (induct a b rule: gcd_eucl_induct) |
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|
202 |
(simp_all add: gcd_0 gcd_non_0 dvd_mod_iff) |
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|
203 |
|
58023 | 204 |
lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m" |
205 |
by (rule dvd_trans, assumption, rule gcd_dvd1) |
|
206 |
||
207 |
lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n" |
|
208 |
by (rule dvd_trans, assumption, rule gcd_dvd2) |
|
209 |
||
210 |
lemma gcd_greatest: |
|
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|
211 |
fixes k a b :: 'a |
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|
212 |
shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b" |
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|
213 |
proof (induct a b rule: gcd_eucl_induct) |
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|
214 |
case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0) |
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|
215 |
next |
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|
216 |
case (mod a b) |
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|
217 |
then show ?case |
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|
218 |
by (simp add: gcd_non_0 dvd_mod_iff) |
58023 | 219 |
qed |
220 |
||
221 |
lemma dvd_gcd_iff: |
|
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|
222 |
"k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b" |
58023 | 223 |
by (blast intro!: gcd_greatest intro: dvd_trans) |
224 |
||
225 |
lemmas gcd_greatest_iff = dvd_gcd_iff |
|
226 |
||
227 |
lemma gcd_zero [simp]: |
|
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|
228 |
"gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" |
58023 | 229 |
by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+ |
230 |
||
60634 | 231 |
lemma unit_factor_gcd [simp]: |
232 |
"unit_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b") |
|
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233 |
by (induct a b rule: gcd_eucl_induct) |
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|
234 |
(auto simp add: gcd_0 gcd_non_0) |
58023 | 235 |
|
236 |
lemma gcdI: |
|
60634 | 237 |
assumes "c dvd a" and "c dvd b" and greatest: "\<And>d. d dvd a \<Longrightarrow> d dvd b \<Longrightarrow> d dvd c" |
238 |
and "unit_factor c = (if c = 0 then 0 else 1)" |
|
239 |
shows "c = gcd a b" |
|
240 |
by (rule associated_eqI) (auto simp: assms associated_def intro: gcd_greatest) |
|
58023 | 241 |
|
242 |
sublocale gcd!: abel_semigroup gcd |
|
243 |
proof |
|
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|
244 |
fix a b c |
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|
245 |
show "gcd (gcd a b) c = gcd a (gcd b c)" |
58023 | 246 |
proof (rule gcdI) |
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|
247 |
have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all |
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|
248 |
then show "gcd (gcd a b) c dvd a" by (rule dvd_trans) |
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|
249 |
have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all |
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|
250 |
hence "gcd (gcd a b) c dvd b" by (rule dvd_trans) |
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|
251 |
moreover have "gcd (gcd a b) c dvd c" by simp |
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|
252 |
ultimately show "gcd (gcd a b) c dvd gcd b c" |
58023 | 253 |
by (rule gcd_greatest) |
60634 | 254 |
show "unit_factor (gcd (gcd a b) c) = (if gcd (gcd a b) c = 0 then 0 else 1)" |
58023 | 255 |
by auto |
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|
256 |
fix l assume "l dvd a" and "l dvd gcd b c" |
58023 | 257 |
with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2] |
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|
258 |
have "l dvd b" and "l dvd c" by blast+ |
60526 | 259 |
with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c" |
58023 | 260 |
by (intro gcd_greatest) |
261 |
qed |
|
262 |
next |
|
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|
263 |
fix a b |
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|
264 |
show "gcd a b = gcd b a" |
58023 | 265 |
by (rule gcdI) (simp_all add: gcd_greatest) |
266 |
qed |
|
267 |
||
268 |
lemma gcd_unique: "d dvd a \<and> d dvd b \<and> |
|
60634 | 269 |
unit_factor d = (if d = 0 then 0 else 1) \<and> |
58023 | 270 |
(\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b" |
271 |
by (rule, auto intro: gcdI simp: gcd_greatest) |
|
272 |
||
273 |
lemma gcd_dvd_prod: "gcd a b dvd k * b" |
|
274 |
using mult_dvd_mono [of 1] by auto |
|
275 |
||
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|
276 |
lemma gcd_1_left [simp]: "gcd 1 a = 1" |
58023 | 277 |
by (rule sym, rule gcdI, simp_all) |
278 |
||
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279 |
lemma gcd_1 [simp]: "gcd a 1 = 1" |
58023 | 280 |
by (rule sym, rule gcdI, simp_all) |
281 |
||
282 |
lemma gcd_proj2_if_dvd: |
|
60634 | 283 |
"b dvd a \<Longrightarrow> gcd a b = normalize b" |
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|
284 |
by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0) |
58023 | 285 |
|
286 |
lemma gcd_proj1_if_dvd: |
|
60634 | 287 |
"a dvd b \<Longrightarrow> gcd a b = normalize a" |
58023 | 288 |
by (subst gcd.commute, simp add: gcd_proj2_if_dvd) |
289 |
||
60634 | 290 |
lemma gcd_proj1_iff: "gcd m n = normalize m \<longleftrightarrow> m dvd n" |
58023 | 291 |
proof |
60634 | 292 |
assume A: "gcd m n = normalize m" |
58023 | 293 |
show "m dvd n" |
294 |
proof (cases "m = 0") |
|
295 |
assume [simp]: "m \<noteq> 0" |
|
60634 | 296 |
from A have B: "m = gcd m n * unit_factor m" |
58023 | 297 |
by (simp add: unit_eq_div2) |
298 |
show ?thesis by (subst B, simp add: mult_unit_dvd_iff) |
|
299 |
qed (insert A, simp) |
|
300 |
next |
|
301 |
assume "m dvd n" |
|
60634 | 302 |
then show "gcd m n = normalize m" by (rule gcd_proj1_if_dvd) |
58023 | 303 |
qed |
304 |
||
60634 | 305 |
lemma gcd_proj2_iff: "gcd m n = normalize n \<longleftrightarrow> n dvd m" |
306 |
using gcd_proj1_iff [of n m] by (simp add: ac_simps) |
|
58023 | 307 |
|
308 |
lemma gcd_mod1 [simp]: |
|
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|
309 |
"gcd (a mod b) b = gcd a b" |
58023 | 310 |
by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff) |
311 |
||
312 |
lemma gcd_mod2 [simp]: |
|
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|
313 |
"gcd a (b mod a) = gcd a b" |
58023 | 314 |
by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff) |
315 |
||
316 |
lemma gcd_mult_distrib': |
|
60634 | 317 |
"normalize c * gcd a b = gcd (c * a) (c * b)" |
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|
318 |
proof (cases "c = 0") |
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|
319 |
case True then show ?thesis by (simp_all add: gcd_0) |
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|
320 |
next |
60634 | 321 |
case False then have [simp]: "is_unit (unit_factor c)" by simp |
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|
322 |
show ?thesis |
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|
323 |
proof (induct a b rule: gcd_eucl_induct) |
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|
324 |
case (zero a) show ?case |
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|
325 |
proof (cases "a = 0") |
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|
326 |
case True then show ?thesis by (simp add: gcd_0) |
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|
327 |
next |
60634 | 328 |
case False |
329 |
then show ?thesis by (simp add: gcd_0 normalize_mult) |
|
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|
330 |
qed |
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|
331 |
case (mod a b) |
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|
332 |
then show ?case by (simp add: mult_mod_right gcd.commute) |
58023 | 333 |
qed |
334 |
qed |
|
335 |
||
336 |
lemma gcd_mult_distrib: |
|
60634 | 337 |
"k * gcd a b = gcd (k * a) (k * b) * unit_factor k" |
58023 | 338 |
proof- |
60634 | 339 |
have "normalize k * gcd a b = gcd (k * a) (k * b)" |
340 |
by (simp add: gcd_mult_distrib') |
|
341 |
then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k" |
|
342 |
by simp |
|
343 |
then have "normalize k * unit_factor k * gcd a b = gcd (k * a) (k * b) * unit_factor k" |
|
344 |
by (simp only: ac_simps) |
|
345 |
then show ?thesis |
|
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|
346 |
by simp |
58023 | 347 |
qed |
348 |
||
349 |
lemma euclidean_size_gcd_le1 [simp]: |
|
350 |
assumes "a \<noteq> 0" |
|
351 |
shows "euclidean_size (gcd a b) \<le> euclidean_size a" |
|
352 |
proof - |
|
353 |
have "gcd a b dvd a" by (rule gcd_dvd1) |
|
354 |
then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast |
|
60526 | 355 |
with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto |
58023 | 356 |
qed |
357 |
||
358 |
lemma euclidean_size_gcd_le2 [simp]: |
|
359 |
"b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b" |
|
360 |
by (subst gcd.commute, rule euclidean_size_gcd_le1) |
|
361 |
||
362 |
lemma euclidean_size_gcd_less1: |
|
363 |
assumes "a \<noteq> 0" and "\<not>a dvd b" |
|
364 |
shows "euclidean_size (gcd a b) < euclidean_size a" |
|
365 |
proof (rule ccontr) |
|
366 |
assume "\<not>euclidean_size (gcd a b) < euclidean_size a" |
|
60526 | 367 |
with \<open>a \<noteq> 0\<close> have "euclidean_size (gcd a b) = euclidean_size a" |
58023 | 368 |
by (intro le_antisym, simp_all) |
369 |
with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd) |
|
370 |
hence "a dvd b" using dvd_gcd_D2 by blast |
|
60526 | 371 |
with \<open>\<not>a dvd b\<close> show False by contradiction |
58023 | 372 |
qed |
373 |
||
374 |
lemma euclidean_size_gcd_less2: |
|
375 |
assumes "b \<noteq> 0" and "\<not>b dvd a" |
|
376 |
shows "euclidean_size (gcd a b) < euclidean_size b" |
|
377 |
using assms by (subst gcd.commute, rule euclidean_size_gcd_less1) |
|
378 |
||
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|
379 |
lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c" |
58023 | 380 |
apply (rule gcdI) |
381 |
apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps) |
|
382 |
apply (rule gcd_dvd2) |
|
383 |
apply (rule gcd_greatest, simp add: unit_simps, assumption) |
|
60634 | 384 |
apply (subst unit_factor_gcd, simp add: gcd_0) |
58023 | 385 |
done |
386 |
||
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|
387 |
lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c" |
58023 | 388 |
by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute) |
389 |
||
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|
390 |
lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c" |
60433 | 391 |
by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1) |
58023 | 392 |
|
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|
393 |
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c" |
60433 | 394 |
by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2) |
58023 | 395 |
|
60634 | 396 |
lemma normalize_gcd_left [simp]: |
397 |
"gcd (normalize a) b = gcd a b" |
|
398 |
proof (cases "a = 0") |
|
399 |
case True then show ?thesis |
|
400 |
by simp |
|
401 |
next |
|
402 |
case False then have "is_unit (unit_factor a)" |
|
403 |
by simp |
|
404 |
moreover have "normalize a = a div unit_factor a" |
|
405 |
by simp |
|
406 |
ultimately show ?thesis |
|
407 |
by (simp only: gcd_div_unit1) |
|
408 |
qed |
|
409 |
||
410 |
lemma normalize_gcd_right [simp]: |
|
411 |
"gcd a (normalize b) = gcd a b" |
|
412 |
using normalize_gcd_left [of b a] by (simp add: ac_simps) |
|
413 |
||
414 |
lemma gcd_idem: "gcd a a = normalize a" |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
415 |
by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all) |
58023 | 416 |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
417 |
lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b" |
58023 | 418 |
apply (rule gcdI) |
419 |
apply (simp add: ac_simps) |
|
420 |
apply (rule gcd_dvd2) |
|
421 |
apply (rule gcd_greatest, erule (1) gcd_greatest, assumption) |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
422 |
apply simp |
58023 | 423 |
done |
424 |
||
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
425 |
lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b" |
58023 | 426 |
apply (rule gcdI) |
427 |
apply simp |
|
428 |
apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2) |
|
429 |
apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption) |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
430 |
apply simp |
58023 | 431 |
done |
432 |
||
433 |
lemma comp_fun_idem_gcd: "comp_fun_idem gcd" |
|
434 |
proof |
|
435 |
fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a" |
|
436 |
by (simp add: fun_eq_iff ac_simps) |
|
437 |
next |
|
438 |
fix a show "gcd a \<circ> gcd a = gcd a" |
|
439 |
by (simp add: fun_eq_iff gcd_left_idem) |
|
440 |
qed |
|
441 |
||
442 |
lemma coprime_dvd_mult: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
443 |
assumes "gcd c b = 1" and "c dvd a * b" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
444 |
shows "c dvd a" |
58023 | 445 |
proof - |
60634 | 446 |
let ?nf = "unit_factor" |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
447 |
from assms gcd_mult_distrib [of a c b] |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
448 |
have A: "a = gcd (a * c) (a * b) * ?nf a" by simp |
60526 | 449 |
from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest) |
58023 | 450 |
qed |
451 |
||
452 |
lemma coprime_dvd_mult_iff: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
453 |
"gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)" |
58023 | 454 |
by (rule, rule coprime_dvd_mult, simp_all) |
455 |
||
456 |
lemma gcd_dvd_antisym: |
|
457 |
"gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d" |
|
458 |
proof (rule gcdI) |
|
459 |
assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b" |
|
460 |
have "gcd c d dvd c" by simp |
|
461 |
with A show "gcd a b dvd c" by (rule dvd_trans) |
|
462 |
have "gcd c d dvd d" by simp |
|
463 |
with A show "gcd a b dvd d" by (rule dvd_trans) |
|
60634 | 464 |
show "unit_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
465 |
by simp |
58023 | 466 |
fix l assume "l dvd c" and "l dvd d" |
467 |
hence "l dvd gcd c d" by (rule gcd_greatest) |
|
468 |
from this and B show "l dvd gcd a b" by (rule dvd_trans) |
|
469 |
qed |
|
470 |
||
471 |
lemma gcd_mult_cancel: |
|
472 |
assumes "gcd k n = 1" |
|
473 |
shows "gcd (k * m) n = gcd m n" |
|
474 |
proof (rule gcd_dvd_antisym) |
|
475 |
have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps) |
|
60526 | 476 |
also note \<open>gcd k n = 1\<close> |
58023 | 477 |
finally have "gcd (gcd (k * m) n) k = 1" by simp |
478 |
hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps) |
|
479 |
moreover have "gcd (k * m) n dvd n" by simp |
|
480 |
ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest) |
|
481 |
have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all |
|
482 |
then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest) |
|
483 |
qed |
|
484 |
||
485 |
lemma coprime_crossproduct: |
|
486 |
assumes [simp]: "gcd a d = 1" "gcd b c = 1" |
|
487 |
shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs") |
|
488 |
proof |
|
489 |
assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono) |
|
490 |
next |
|
491 |
assume ?lhs |
|
60526 | 492 |
from \<open>?lhs\<close> have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left) |
58023 | 493 |
hence "a dvd b" by (simp add: coprime_dvd_mult_iff) |
60526 | 494 |
moreover from \<open>?lhs\<close> have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left) |
58023 | 495 |
hence "b dvd a" by (simp add: coprime_dvd_mult_iff) |
60526 | 496 |
moreover from \<open>?lhs\<close> have "c dvd d * b" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
497 |
unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps) |
58023 | 498 |
hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute) |
60526 | 499 |
moreover from \<open>?lhs\<close> have "d dvd c * a" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
500 |
unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps) |
58023 | 501 |
hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute) |
502 |
ultimately show ?rhs unfolding associated_def by simp |
|
503 |
qed |
|
504 |
||
505 |
lemma gcd_add1 [simp]: |
|
506 |
"gcd (m + n) n = gcd m n" |
|
507 |
by (cases "n = 0", simp_all add: gcd_non_0) |
|
508 |
||
509 |
lemma gcd_add2 [simp]: |
|
510 |
"gcd m (m + n) = gcd m n" |
|
511 |
using gcd_add1 [of n m] by (simp add: ac_simps) |
|
512 |
||
60572
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
513 |
lemma gcd_add_mult: |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
514 |
"gcd m (k * m + n) = gcd m n" |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
515 |
proof - |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
516 |
have "gcd m ((k * m + n) mod m) = gcd m (k * m + n)" |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
517 |
by (fact gcd_mod2) |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
518 |
then show ?thesis by simp |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
519 |
qed |
58023 | 520 |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
521 |
lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1" |
58023 | 522 |
by (rule sym, rule gcdI, simp_all) |
523 |
||
524 |
lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)" |
|
59061 | 525 |
by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2) |
58023 | 526 |
|
527 |
lemma div_gcd_coprime: |
|
528 |
assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0" |
|
529 |
defines [simp]: "d \<equiv> gcd a b" |
|
530 |
defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d" |
|
531 |
shows "gcd a' b' = 1" |
|
532 |
proof (rule coprimeI) |
|
533 |
fix l assume "l dvd a'" "l dvd b'" |
|
534 |
then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
535 |
moreover have "a = a' * d" "b = b' * d" by simp_all |
58023 | 536 |
ultimately have "a = (l * d) * s" "b = (l * d) * t" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
537 |
by (simp_all only: ac_simps) |
58023 | 538 |
hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left) |
539 |
hence "l*d dvd d" by (simp add: gcd_greatest) |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
540 |
then obtain u where "d = l * d * u" .. |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
541 |
then have "d * (l * u) = d" by (simp add: ac_simps) |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
542 |
moreover from nz have "d \<noteq> 0" by simp |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
543 |
with div_mult_self1_is_id have "d * (l * u) div d = l * u" . |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
544 |
ultimately have "1 = l * u" |
60526 | 545 |
using \<open>d \<noteq> 0\<close> by simp |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
546 |
then show "l dvd 1" .. |
58023 | 547 |
qed |
548 |
||
549 |
lemma coprime_mult: |
|
550 |
assumes da: "gcd d a = 1" and db: "gcd d b = 1" |
|
551 |
shows "gcd d (a * b) = 1" |
|
552 |
apply (subst gcd.commute) |
|
553 |
using da apply (subst gcd_mult_cancel) |
|
554 |
apply (subst gcd.commute, assumption) |
|
555 |
apply (subst gcd.commute, rule db) |
|
556 |
done |
|
557 |
||
558 |
lemma coprime_lmult: |
|
559 |
assumes dab: "gcd d (a * b) = 1" |
|
560 |
shows "gcd d a = 1" |
|
561 |
proof (rule coprimeI) |
|
562 |
fix l assume "l dvd d" and "l dvd a" |
|
563 |
hence "l dvd a * b" by simp |
|
60526 | 564 |
with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest) |
58023 | 565 |
qed |
566 |
||
567 |
lemma coprime_rmult: |
|
568 |
assumes dab: "gcd d (a * b) = 1" |
|
569 |
shows "gcd d b = 1" |
|
570 |
proof (rule coprimeI) |
|
571 |
fix l assume "l dvd d" and "l dvd b" |
|
572 |
hence "l dvd a * b" by simp |
|
60526 | 573 |
with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest) |
58023 | 574 |
qed |
575 |
||
576 |
lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1" |
|
577 |
using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast |
|
578 |
||
579 |
lemma gcd_coprime: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
580 |
assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b" |
58023 | 581 |
shows "gcd a' b' = 1" |
582 |
proof - |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
583 |
from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp |
58023 | 584 |
with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" . |
585 |
also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+ |
|
586 |
also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+ |
|
587 |
finally show ?thesis . |
|
588 |
qed |
|
589 |
||
590 |
lemma coprime_power: |
|
591 |
assumes "0 < n" |
|
592 |
shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1" |
|
593 |
using assms proof (induct n) |
|
594 |
case (Suc n) then show ?case |
|
595 |
by (cases n) (simp_all add: coprime_mul_eq) |
|
596 |
qed simp |
|
597 |
||
598 |
lemma gcd_coprime_exists: |
|
599 |
assumes nz: "gcd a b \<noteq> 0" |
|
600 |
shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1" |
|
601 |
apply (rule_tac x = "a div gcd a b" in exI) |
|
602 |
apply (rule_tac x = "b div gcd a b" in exI) |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
603 |
apply (insert nz, auto intro: div_gcd_coprime) |
58023 | 604 |
done |
605 |
||
606 |
lemma coprime_exp: |
|
607 |
"gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1" |
|
608 |
by (induct n, simp_all add: coprime_mult) |
|
609 |
||
610 |
lemma coprime_exp2 [intro]: |
|
611 |
"gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1" |
|
612 |
apply (rule coprime_exp) |
|
613 |
apply (subst gcd.commute) |
|
614 |
apply (rule coprime_exp) |
|
615 |
apply (subst gcd.commute) |
|
616 |
apply assumption |
|
617 |
done |
|
618 |
||
619 |
lemma gcd_exp: |
|
620 |
"gcd (a^n) (b^n) = (gcd a b) ^ n" |
|
621 |
proof (cases "a = 0 \<and> b = 0") |
|
622 |
assume "a = 0 \<and> b = 0" |
|
623 |
then show ?thesis by (cases n, simp_all add: gcd_0_left) |
|
624 |
next |
|
625 |
assume A: "\<not>(a = 0 \<and> b = 0)" |
|
626 |
hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)" |
|
627 |
using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime) |
|
628 |
hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp |
|
629 |
also note gcd_mult_distrib |
|
60634 | 630 |
also have "unit_factor ((gcd a b)^n) = 1" |
631 |
by (simp add: unit_factor_power A) |
|
58023 | 632 |
also have "(gcd a b)^n * (a div gcd a b)^n = a^n" |
633 |
by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp) |
|
634 |
also have "(gcd a b)^n * (b div gcd a b)^n = b^n" |
|
635 |
by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp) |
|
636 |
finally show ?thesis by simp |
|
637 |
qed |
|
638 |
||
639 |
lemma coprime_common_divisor: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
640 |
"gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
641 |
apply (subgoal_tac "a dvd gcd a b") |
59061 | 642 |
apply simp |
58023 | 643 |
apply (erule (1) gcd_greatest) |
644 |
done |
|
645 |
||
646 |
lemma division_decomp: |
|
647 |
assumes dc: "a dvd b * c" |
|
648 |
shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c" |
|
649 |
proof (cases "gcd a b = 0") |
|
650 |
assume "gcd a b = 0" |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
651 |
hence "a = 0 \<and> b = 0" by simp |
58023 | 652 |
hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp |
653 |
then show ?thesis by blast |
|
654 |
next |
|
655 |
let ?d = "gcd a b" |
|
656 |
assume "?d \<noteq> 0" |
|
657 |
from gcd_coprime_exists[OF this] |
|
658 |
obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1" |
|
659 |
by blast |
|
660 |
from ab'(1) have "a' dvd a" unfolding dvd_def by blast |
|
661 |
with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp |
|
662 |
from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp |
|
663 |
hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac) |
|
60526 | 664 |
with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp |
58023 | 665 |
with coprime_dvd_mult[OF ab'(3)] |
666 |
have "a' dvd c" by (subst (asm) ac_simps, blast) |
|
667 |
with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac) |
|
668 |
then show ?thesis by blast |
|
669 |
qed |
|
670 |
||
60433 | 671 |
lemma pow_divs_pow: |
58023 | 672 |
assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0" |
673 |
shows "a dvd b" |
|
674 |
proof (cases "gcd a b = 0") |
|
675 |
assume "gcd a b = 0" |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
676 |
then show ?thesis by simp |
58023 | 677 |
next |
678 |
let ?d = "gcd a b" |
|
679 |
assume "?d \<noteq> 0" |
|
680 |
from n obtain m where m: "n = Suc m" by (cases n, simp_all) |
|
60526 | 681 |
from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero) |
682 |
from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>] |
|
58023 | 683 |
obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1" |
684 |
by blast |
|
685 |
from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n" |
|
686 |
by (simp add: ab'(1,2)[symmetric]) |
|
687 |
hence "?d^n * a'^n dvd ?d^n * b'^n" |
|
688 |
by (simp only: power_mult_distrib ac_simps) |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
689 |
with zn have "a'^n dvd b'^n" by simp |
58023 | 690 |
hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m) |
691 |
hence "a' dvd b'^m * b'" by (simp add: m ac_simps) |
|
692 |
with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]] |
|
693 |
have "a' dvd b'" by (subst (asm) ac_simps, blast) |
|
694 |
hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp) |
|
695 |
with ab'(1,2) show ?thesis by simp |
|
696 |
qed |
|
697 |
||
60433 | 698 |
lemma pow_divs_eq [simp]: |
58023 | 699 |
"n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b" |
60433 | 700 |
by (auto intro: pow_divs_pow dvd_power_same) |
58023 | 701 |
|
60433 | 702 |
lemma divs_mult: |
58023 | 703 |
assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1" |
704 |
shows "m * n dvd r" |
|
705 |
proof - |
|
706 |
from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'" |
|
707 |
unfolding dvd_def by blast |
|
708 |
from mr n' have "m dvd n'*n" by (simp add: ac_simps) |
|
709 |
hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp |
|
710 |
then obtain k where k: "n' = m*k" unfolding dvd_def by blast |
|
711 |
with n' have "r = m * n * k" by (simp add: mult_ac) |
|
712 |
then show ?thesis unfolding dvd_def by blast |
|
713 |
qed |
|
714 |
||
715 |
lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1" |
|
716 |
by (subst add_commute, simp) |
|
717 |
||
718 |
lemma setprod_coprime [rule_format]: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
719 |
"(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1" |
58023 | 720 |
apply (cases "finite A") |
721 |
apply (induct set: finite) |
|
722 |
apply (auto simp add: gcd_mult_cancel) |
|
723 |
done |
|
724 |
||
725 |
lemma coprime_divisors: |
|
726 |
assumes "d dvd a" "e dvd b" "gcd a b = 1" |
|
727 |
shows "gcd d e = 1" |
|
728 |
proof - |
|
729 |
from assms obtain k l where "a = d * k" "b = e * l" |
|
730 |
unfolding dvd_def by blast |
|
731 |
with assms have "gcd (d * k) (e * l) = 1" by simp |
|
732 |
hence "gcd (d * k) e = 1" by (rule coprime_lmult) |
|
733 |
also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps) |
|
734 |
finally have "gcd e d = 1" by (rule coprime_lmult) |
|
735 |
then show ?thesis by (simp add: ac_simps) |
|
736 |
qed |
|
737 |
||
738 |
lemma invertible_coprime: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
739 |
assumes "a * b mod m = 1" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
740 |
shows "coprime a m" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
741 |
proof - |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
742 |
from assms have "coprime m (a * b mod m)" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
743 |
by simp |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
744 |
then have "coprime m (a * b)" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
745 |
by simp |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
746 |
then have "coprime m a" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
747 |
by (rule coprime_lmult) |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
748 |
then show ?thesis |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
749 |
by (simp add: ac_simps) |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
750 |
qed |
58023 | 751 |
|
752 |
lemma lcm_gcd: |
|
60634 | 753 |
"lcm a b = normalize (a * b) div gcd a b" |
754 |
by (simp add: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def) |
|
58023 | 755 |
|
756 |
lemma lcm_gcd_prod: |
|
60634 | 757 |
"lcm a b * gcd a b = normalize (a * b)" |
758 |
by (simp add: lcm_gcd) |
|
58023 | 759 |
|
760 |
lemma lcm_dvd1 [iff]: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
761 |
"a dvd lcm a b" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
762 |
proof (cases "a*b = 0") |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
763 |
assume "a * b \<noteq> 0" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
764 |
hence "gcd a b \<noteq> 0" by simp |
60634 | 765 |
let ?c = "1 div unit_factor (a * b)" |
766 |
from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (unit_factor (a * b))" by simp |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
767 |
from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b" |
60432 | 768 |
by (simp add: div_mult_swap unit_div_commute) |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
769 |
hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp |
60526 | 770 |
with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b" |
58023 | 771 |
by (subst (asm) div_mult_self2_is_id, simp_all) |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
772 |
also have "... = a * (?c * b div gcd a b)" |
58023 | 773 |
by (metis div_mult_swap gcd_dvd2 mult_assoc) |
774 |
finally show ?thesis by (rule dvdI) |
|
58953 | 775 |
qed (auto simp add: lcm_gcd) |
58023 | 776 |
|
777 |
lemma lcm_least: |
|
778 |
"\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k" |
|
779 |
proof (cases "k = 0") |
|
60634 | 780 |
let ?nf = unit_factor |
58023 | 781 |
assume "k \<noteq> 0" |
782 |
hence "is_unit (?nf k)" by simp |
|
783 |
hence "?nf k \<noteq> 0" by (metis not_is_unit_0) |
|
784 |
assume A: "a dvd k" "b dvd k" |
|
60526 | 785 |
hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto |
58023 | 786 |
from A obtain r s where ar: "k = a * r" and bs: "k = b * s" |
787 |
unfolding dvd_def by blast |
|
60526 | 788 |
with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0" |
58953 | 789 |
by auto (drule sym [of 0], simp) |
58023 | 790 |
hence "is_unit (?nf (r * s))" by simp |
791 |
let ?c = "?nf k div ?nf (r*s)" |
|
60526 | 792 |
from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div) |
58023 | 793 |
hence "?c \<noteq> 0" using not_is_unit_0 by fast |
794 |
from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)" |
|
58953 | 795 |
by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps) |
58023 | 796 |
also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)" |
60526 | 797 |
by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps) |
798 |
also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close> |
|
58023 | 799 |
by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps) |
800 |
finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b" |
|
801 |
by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac) |
|
802 |
hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)" |
|
803 |
by (simp add: algebra_simps) |
|
60526 | 804 |
hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close> |
58023 | 805 |
by (metis div_mult_self2_is_id) |
806 |
also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)" |
|
807 |
by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib') |
|
808 |
also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b" |
|
809 |
by (simp add: algebra_simps) |
|
60526 | 810 |
finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close> |
58023 | 811 |
by (metis mult.commute div_mult_self2_is_id) |
60526 | 812 |
hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close> |
58023 | 813 |
by (metis div_mult_self2_is_id mult_assoc) |
60526 | 814 |
also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close> |
58023 | 815 |
by (simp add: unit_simps) |
816 |
finally show ?thesis by (rule dvdI) |
|
817 |
qed simp |
|
818 |
||
819 |
lemma lcm_zero: |
|
820 |
"lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" |
|
821 |
proof - |
|
60634 | 822 |
let ?nf = unit_factor |
58023 | 823 |
{ |
824 |
assume "a \<noteq> 0" "b \<noteq> 0" |
|
825 |
hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors) |
|
60526 | 826 |
moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp |
58023 | 827 |
ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp) |
828 |
} moreover { |
|
829 |
assume "a = 0 \<or> b = 0" |
|
830 |
hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd) |
|
831 |
} |
|
832 |
ultimately show ?thesis by blast |
|
833 |
qed |
|
834 |
||
835 |
lemmas lcm_0_iff = lcm_zero |
|
836 |
||
837 |
lemma gcd_lcm: |
|
838 |
assumes "lcm a b \<noteq> 0" |
|
60634 | 839 |
shows "gcd a b = normalize (a * b) div lcm a b" |
840 |
proof - |
|
841 |
have "lcm a b * gcd a b = normalize (a * b)" |
|
842 |
by (fact lcm_gcd_prod) |
|
843 |
with assms show ?thesis |
|
844 |
by (metis nonzero_mult_divide_cancel_left) |
|
58023 | 845 |
qed |
846 |
||
60634 | 847 |
lemma unit_factor_lcm [simp]: |
848 |
"unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)" |
|
849 |
by (simp add: dvd_unit_factor_div lcm_gcd) |
|
58023 | 850 |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
851 |
lemma lcm_dvd2 [iff]: "b dvd lcm a b" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
852 |
using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps) |
58023 | 853 |
|
854 |
lemma lcmI: |
|
60634 | 855 |
assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d" |
856 |
and "unit_factor c = (if c = 0 then 0 else 1)" |
|
857 |
shows "c = lcm a b" |
|
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60634
diff
changeset
|
858 |
by (rule associated_eqI) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60634
diff
changeset
|
859 |
(auto simp: assms associated_def intro: lcm_least, simp_all add: lcm_gcd) |
58023 | 860 |
|
861 |
sublocale lcm!: abel_semigroup lcm |
|
862 |
proof |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
863 |
fix a b c |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
864 |
show "lcm (lcm a b) c = lcm a (lcm b c)" |
58023 | 865 |
proof (rule lcmI) |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
866 |
have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
867 |
then show "a dvd lcm (lcm a b) c" by (rule dvd_trans) |
58023 | 868 |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
869 |
have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
870 |
hence "b dvd lcm (lcm a b) c" by (rule dvd_trans) |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
871 |
moreover have "c dvd lcm (lcm a b) c" by simp |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
872 |
ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least) |
58023 | 873 |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
874 |
fix l assume "a dvd l" and "lcm b c dvd l" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
875 |
have "b dvd lcm b c" by simp |
60526 | 876 |
from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans) |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
877 |
have "c dvd lcm b c" by simp |
60526 | 878 |
from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans) |
879 |
from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least) |
|
880 |
from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least) |
|
58023 | 881 |
qed (simp add: lcm_zero) |
882 |
next |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
883 |
fix a b |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
884 |
show "lcm a b = lcm b a" |
58023 | 885 |
by (simp add: lcm_gcd ac_simps) |
886 |
qed |
|
887 |
||
888 |
lemma dvd_lcm_D1: |
|
889 |
"lcm m n dvd k \<Longrightarrow> m dvd k" |
|
890 |
by (rule dvd_trans, rule lcm_dvd1, assumption) |
|
891 |
||
892 |
lemma dvd_lcm_D2: |
|
893 |
"lcm m n dvd k \<Longrightarrow> n dvd k" |
|
894 |
by (rule dvd_trans, rule lcm_dvd2, assumption) |
|
895 |
||
896 |
lemma gcd_dvd_lcm [simp]: |
|
897 |
"gcd a b dvd lcm a b" |
|
898 |
by (metis dvd_trans gcd_dvd2 lcm_dvd2) |
|
899 |
||
900 |
lemma lcm_1_iff: |
|
901 |
"lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b" |
|
902 |
proof |
|
903 |
assume "lcm a b = 1" |
|
59061 | 904 |
then show "is_unit a \<and> is_unit b" by auto |
58023 | 905 |
next |
906 |
assume "is_unit a \<and> is_unit b" |
|
59061 | 907 |
hence "a dvd 1" and "b dvd 1" by simp_all |
908 |
hence "is_unit (lcm a b)" by (rule lcm_least) |
|
60634 | 909 |
hence "lcm a b = unit_factor (lcm a b)" |
910 |
by (blast intro: sym is_unit_unit_factor) |
|
60526 | 911 |
also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close> |
59061 | 912 |
by auto |
58023 | 913 |
finally show "lcm a b = 1" . |
914 |
qed |
|
915 |
||
916 |
lemma lcm_0_left [simp]: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
917 |
"lcm 0 a = 0" |
58023 | 918 |
by (rule sym, rule lcmI, simp_all) |
919 |
||
920 |
lemma lcm_0 [simp]: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
921 |
"lcm a 0 = 0" |
58023 | 922 |
by (rule sym, rule lcmI, simp_all) |
923 |
||
924 |
lemma lcm_unique: |
|
925 |
"a dvd d \<and> b dvd d \<and> |
|
60634 | 926 |
unit_factor d = (if d = 0 then 0 else 1) \<and> |
58023 | 927 |
(\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b" |
928 |
by (rule, auto intro: lcmI simp: lcm_least lcm_zero) |
|
929 |
||
930 |
lemma dvd_lcm_I1 [simp]: |
|
931 |
"k dvd m \<Longrightarrow> k dvd lcm m n" |
|
932 |
by (metis lcm_dvd1 dvd_trans) |
|
933 |
||
934 |
lemma dvd_lcm_I2 [simp]: |
|
935 |
"k dvd n \<Longrightarrow> k dvd lcm m n" |
|
936 |
by (metis lcm_dvd2 dvd_trans) |
|
937 |
||
938 |
lemma lcm_1_left [simp]: |
|
60634 | 939 |
"lcm 1 a = normalize a" |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
940 |
by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all) |
58023 | 941 |
|
942 |
lemma lcm_1_right [simp]: |
|
60634 | 943 |
"lcm a 1 = normalize a" |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
944 |
using lcm_1_left [of a] by (simp add: ac_simps) |
58023 | 945 |
|
946 |
lemma lcm_coprime: |
|
60634 | 947 |
"gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)" |
58023 | 948 |
by (subst lcm_gcd) simp |
949 |
||
950 |
lemma lcm_proj1_if_dvd: |
|
60634 | 951 |
"b dvd a \<Longrightarrow> lcm a b = normalize a" |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
952 |
by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all) |
58023 | 953 |
|
954 |
lemma lcm_proj2_if_dvd: |
|
60634 | 955 |
"a dvd b \<Longrightarrow> lcm a b = normalize b" |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
956 |
using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps) |
58023 | 957 |
|
958 |
lemma lcm_proj1_iff: |
|
60634 | 959 |
"lcm m n = normalize m \<longleftrightarrow> n dvd m" |
58023 | 960 |
proof |
60634 | 961 |
assume A: "lcm m n = normalize m" |
58023 | 962 |
show "n dvd m" |
963 |
proof (cases "m = 0") |
|
964 |
assume [simp]: "m \<noteq> 0" |
|
60634 | 965 |
from A have B: "m = lcm m n * unit_factor m" |
58023 | 966 |
by (simp add: unit_eq_div2) |
967 |
show ?thesis by (subst B, simp) |
|
968 |
qed simp |
|
969 |
next |
|
970 |
assume "n dvd m" |
|
60634 | 971 |
then show "lcm m n = normalize m" by (rule lcm_proj1_if_dvd) |
58023 | 972 |
qed |
973 |
||
974 |
lemma lcm_proj2_iff: |
|
60634 | 975 |
"lcm m n = normalize n \<longleftrightarrow> m dvd n" |
58023 | 976 |
using lcm_proj1_iff [of n m] by (simp add: ac_simps) |
977 |
||
978 |
lemma euclidean_size_lcm_le1: |
|
979 |
assumes "a \<noteq> 0" and "b \<noteq> 0" |
|
980 |
shows "euclidean_size a \<le> euclidean_size (lcm a b)" |
|
981 |
proof - |
|
982 |
have "a dvd lcm a b" by (rule lcm_dvd1) |
|
983 |
then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast |
|
60526 | 984 |
with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero) |
58023 | 985 |
then show ?thesis by (subst A, intro size_mult_mono) |
986 |
qed |
|
987 |
||
988 |
lemma euclidean_size_lcm_le2: |
|
989 |
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)" |
|
990 |
using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps) |
|
991 |
||
992 |
lemma euclidean_size_lcm_less1: |
|
993 |
assumes "b \<noteq> 0" and "\<not>b dvd a" |
|
994 |
shows "euclidean_size a < euclidean_size (lcm a b)" |
|
995 |
proof (rule ccontr) |
|
996 |
from assms have "a \<noteq> 0" by auto |
|
997 |
assume "\<not>euclidean_size a < euclidean_size (lcm a b)" |
|
60526 | 998 |
with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a" |
58023 | 999 |
by (intro le_antisym, simp, intro euclidean_size_lcm_le1) |
1000 |
with assms have "lcm a b dvd a" |
|
1001 |
by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero) |
|
1002 |
hence "b dvd a" by (rule dvd_lcm_D2) |
|
60526 | 1003 |
with \<open>\<not>b dvd a\<close> show False by contradiction |
58023 | 1004 |
qed |
1005 |
||
1006 |
lemma euclidean_size_lcm_less2: |
|
1007 |
assumes "a \<noteq> 0" and "\<not>a dvd b" |
|
1008 |
shows "euclidean_size b < euclidean_size (lcm a b)" |
|
1009 |
using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps) |
|
1010 |
||
1011 |
lemma lcm_mult_unit1: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1012 |
"is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c" |
58023 | 1013 |
apply (rule lcmI) |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1014 |
apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1) |
58023 | 1015 |
apply (rule lcm_dvd2) |
1016 |
apply (rule lcm_least, simp add: unit_simps, assumption) |
|
60634 | 1017 |
apply (subst unit_factor_lcm, simp add: lcm_zero) |
58023 | 1018 |
done |
1019 |
||
1020 |
lemma lcm_mult_unit2: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1021 |
"is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1022 |
using lcm_mult_unit1 [of a c b] by (simp add: ac_simps) |
58023 | 1023 |
|
1024 |
lemma lcm_div_unit1: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1025 |
"is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c" |
60433 | 1026 |
by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1) |
58023 | 1027 |
|
1028 |
lemma lcm_div_unit2: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1029 |
"is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c" |
60433 | 1030 |
by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2) |
58023 | 1031 |
|
60634 | 1032 |
lemma normalize_lcm_left [simp]: |
1033 |
"lcm (normalize a) b = lcm a b" |
|
1034 |
proof (cases "a = 0") |
|
1035 |
case True then show ?thesis |
|
1036 |
by simp |
|
1037 |
next |
|
1038 |
case False then have "is_unit (unit_factor a)" |
|
1039 |
by simp |
|
1040 |
moreover have "normalize a = a div unit_factor a" |
|
1041 |
by simp |
|
1042 |
ultimately show ?thesis |
|
1043 |
by (simp only: lcm_div_unit1) |
|
1044 |
qed |
|
1045 |
||
1046 |
lemma normalize_lcm_right [simp]: |
|
1047 |
"lcm a (normalize b) = lcm a b" |
|
1048 |
using normalize_lcm_left [of b a] by (simp add: ac_simps) |
|
1049 |
||
58023 | 1050 |
lemma lcm_left_idem: |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1051 |
"lcm a (lcm a b) = lcm a b" |
58023 | 1052 |
apply (rule lcmI) |
1053 |
apply simp |
|
1054 |
apply (subst lcm.assoc [symmetric], rule lcm_dvd2) |
|
1055 |
apply (rule lcm_least, assumption) |
|
1056 |
apply (erule (1) lcm_least) |
|
1057 |
apply (auto simp: lcm_zero) |
|
1058 |
done |
|
1059 |
||
1060 |
lemma lcm_right_idem: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1061 |
"lcm (lcm a b) b = lcm a b" |
58023 | 1062 |
apply (rule lcmI) |
1063 |
apply (subst lcm.assoc, rule lcm_dvd1) |
|
1064 |
apply (rule lcm_dvd2) |
|
1065 |
apply (rule lcm_least, erule (1) lcm_least, assumption) |
|
1066 |
apply (auto simp: lcm_zero) |
|
1067 |
done |
|
1068 |
||
1069 |
lemma comp_fun_idem_lcm: "comp_fun_idem lcm" |
|
1070 |
proof |
|
1071 |
fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a" |
|
1072 |
by (simp add: fun_eq_iff ac_simps) |
|
1073 |
next |
|
1074 |
fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def |
|
1075 |
by (intro ext, simp add: lcm_left_idem) |
|
1076 |
qed |
|
1077 |
||
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1078 |
lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A" |
60634 | 1079 |
and Lcm_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm A dvd b" |
1080 |
and unit_factor_Lcm [simp]: |
|
1081 |
"unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" |
|
58023 | 1082 |
proof - |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1083 |
have "(\<forall>a\<in>A. a dvd Lcm A) \<and> (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> Lcm A dvd l') \<and> |
60634 | 1084 |
unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis) |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1085 |
proof (cases "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)") |
58023 | 1086 |
case False |
1087 |
hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def) |
|
1088 |
with False show ?thesis by auto |
|
1089 |
next |
|
1090 |
case True |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1091 |
then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l\<^sub>0)" by blast |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1092 |
def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1093 |
def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1094 |
have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" |
58023 | 1095 |
apply (subst n_def) |
1096 |
apply (rule LeastI[of _ "euclidean_size l\<^sub>0"]) |
|
1097 |
apply (rule exI[of _ l\<^sub>0]) |
|
1098 |
apply (simp add: l\<^sub>0_props) |
|
1099 |
done |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1100 |
from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n" |
58023 | 1101 |
unfolding l_def by simp_all |
1102 |
{ |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1103 |
fix l' assume "\<forall>a\<in>A. a dvd l'" |
60526 | 1104 |
with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd l l'" by (auto intro: gcd_greatest) |
1105 |
moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1106 |
ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')" |
58023 | 1107 |
by (intro exI[of _ "gcd l l'"], auto) |
1108 |
hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le) |
|
1109 |
moreover have "euclidean_size (gcd l l') \<le> n" |
|
1110 |
proof - |
|
1111 |
have "gcd l l' dvd l" by simp |
|
1112 |
then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast |
|
60526 | 1113 |
with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto |
58023 | 1114 |
hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)" |
1115 |
by (rule size_mult_mono) |
|
60526 | 1116 |
also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> .. |
1117 |
also note \<open>euclidean_size l = n\<close> |
|
58023 | 1118 |
finally show "euclidean_size (gcd l l') \<le> n" . |
1119 |
qed |
|
1120 |
ultimately have "euclidean_size l = euclidean_size (gcd l l')" |
|
60526 | 1121 |
by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>) |
1122 |
with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd) |
|
58023 | 1123 |
hence "l dvd l'" by (blast dest: dvd_gcd_D2) |
1124 |
} |
|
1125 |
||
60634 | 1126 |
with \<open>(\<forall>a\<in>A. a dvd l)\<close> and unit_factor_is_unit[OF \<open>l \<noteq> 0\<close>] and \<open>l \<noteq> 0\<close> |
1127 |
have "(\<forall>a\<in>A. a dvd normalize l) \<and> |
|
1128 |
(\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and> |
|
1129 |
unit_factor (normalize l) = |
|
1130 |
(if normalize l = 0 then 0 else 1)" |
|
58023 | 1131 |
by (auto simp: unit_simps) |
60634 | 1132 |
also from True have "normalize l = Lcm A" |
58023 | 1133 |
by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def) |
1134 |
finally show ?thesis . |
|
1135 |
qed |
|
1136 |
note A = this |
|
1137 |
||
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1138 |
{fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast} |
60634 | 1139 |
{fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm A dvd b" using A by blast} |
1140 |
from A show "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast |
|
58023 | 1141 |
qed |
60634 | 1142 |
|
1143 |
lemma normalize_Lcm [simp]: |
|
1144 |
"normalize (Lcm A) = Lcm A" |
|
1145 |
by (cases "Lcm A = 0") (auto intro: associated_eqI) |
|
1146 |
||
58023 | 1147 |
lemma LcmI: |
60634 | 1148 |
assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c" |
1149 |
and "unit_factor b = (if b = 0 then 0 else 1)" shows "b = Lcm A" |
|
1150 |
by (rule associated_eqI) (auto simp: assms associated_def intro: Lcm_least) |
|
58023 | 1151 |
|
1152 |
lemma Lcm_subset: |
|
1153 |
"A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B" |
|
60634 | 1154 |
by (blast intro: Lcm_least dvd_Lcm) |
58023 | 1155 |
|
1156 |
lemma Lcm_Un: |
|
1157 |
"Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)" |
|
1158 |
apply (rule lcmI) |
|
1159 |
apply (blast intro: Lcm_subset) |
|
1160 |
apply (blast intro: Lcm_subset) |
|
60634 | 1161 |
apply (intro Lcm_least ballI, elim UnE) |
58023 | 1162 |
apply (rule dvd_trans, erule dvd_Lcm, assumption) |
1163 |
apply (rule dvd_trans, erule dvd_Lcm, assumption) |
|
1164 |
apply simp |
|
1165 |
done |
|
1166 |
||
1167 |
lemma Lcm_1_iff: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1168 |
"Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)" |
58023 | 1169 |
proof |
1170 |
assume "Lcm A = 1" |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1171 |
then show "\<forall>a\<in>A. is_unit a" by auto |
58023 | 1172 |
qed (rule LcmI [symmetric], auto) |
1173 |
||
1174 |
lemma Lcm_no_units: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1175 |
"Lcm A = Lcm (A - {a. is_unit a})" |
58023 | 1176 |
proof - |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1177 |
have "(A - {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1178 |
hence "Lcm A = lcm (Lcm (A - {a. is_unit a})) (Lcm {a\<in>A. is_unit a})" |
60634 | 1179 |
by (simp add: Lcm_Un [symmetric]) |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1180 |
also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff) |
58023 | 1181 |
finally show ?thesis by simp |
1182 |
qed |
|
1183 |
||
1184 |
lemma Lcm_empty [simp]: |
|
1185 |
"Lcm {} = 1" |
|
1186 |
by (simp add: Lcm_1_iff) |
|
1187 |
||
1188 |
lemma Lcm_eq_0 [simp]: |
|
1189 |
"0 \<in> A \<Longrightarrow> Lcm A = 0" |
|
1190 |
by (drule dvd_Lcm) simp |
|
1191 |
||
1192 |
lemma Lcm0_iff': |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1193 |
"Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" |
58023 | 1194 |
proof |
1195 |
assume "Lcm A = 0" |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1196 |
show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" |
58023 | 1197 |
proof |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1198 |
assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1199 |
then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l\<^sub>0)" by blast |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1200 |
def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1201 |
def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1202 |
have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" |
58023 | 1203 |
apply (subst n_def) |
1204 |
apply (rule LeastI[of _ "euclidean_size l\<^sub>0"]) |
|
1205 |
apply (rule exI[of _ l\<^sub>0]) |
|
1206 |
apply (simp add: l\<^sub>0_props) |
|
1207 |
done |
|
1208 |
from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all |
|
60634 | 1209 |
hence "normalize l \<noteq> 0" by simp |
1210 |
also from ex have "normalize l = Lcm A" |
|
58023 | 1211 |
by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def) |
60526 | 1212 |
finally show False using \<open>Lcm A = 0\<close> by contradiction |
58023 | 1213 |
qed |
1214 |
qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False) |
|
1215 |
||
1216 |
lemma Lcm0_iff [simp]: |
|
1217 |
"finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A" |
|
1218 |
proof - |
|
1219 |
assume "finite A" |
|
1220 |
have "0 \<in> A \<Longrightarrow> Lcm A = 0" by (intro dvd_0_left dvd_Lcm) |
|
1221 |
moreover { |
|
1222 |
assume "0 \<notin> A" |
|
1223 |
hence "\<Prod>A \<noteq> 0" |
|
60526 | 1224 |
apply (induct rule: finite_induct[OF \<open>finite A\<close>]) |
58023 | 1225 |
apply simp |
1226 |
apply (subst setprod.insert, assumption, assumption) |
|
1227 |
apply (rule no_zero_divisors) |
|
1228 |
apply blast+ |
|
1229 |
done |
|
60526 | 1230 |
moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1231 |
ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast |
58023 | 1232 |
with Lcm0_iff' have "Lcm A \<noteq> 0" by simp |
1233 |
} |
|
1234 |
ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast |
|
1235 |
qed |
|
1236 |
||
1237 |
lemma Lcm_no_multiple: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1238 |
"(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0" |
58023 | 1239 |
proof - |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1240 |
assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)" |
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1241 |
hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast |
58023 | 1242 |
then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False) |
1243 |
qed |
|
1244 |
||
1245 |
lemma Lcm_insert [simp]: |
|
1246 |
"Lcm (insert a A) = lcm a (Lcm A)" |
|
1247 |
proof (rule lcmI) |
|
1248 |
fix l assume "a dvd l" and "Lcm A dvd l" |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1249 |
hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm) |
60634 | 1250 |
with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_least) |
1251 |
qed (auto intro: Lcm_least dvd_Lcm) |
|
58023 | 1252 |
|
1253 |
lemma Lcm_finite: |
|
1254 |
assumes "finite A" |
|
1255 |
shows "Lcm A = Finite_Set.fold lcm 1 A" |
|
60526 | 1256 |
by (induct rule: finite.induct[OF \<open>finite A\<close>]) |
58023 | 1257 |
(simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm]) |
1258 |
||
60431
db9c67b760f1
dropped warnings by dropping ineffective code declarations
haftmann
parents:
60430
diff
changeset
|
1259 |
lemma Lcm_set [code_unfold]: |
58023 | 1260 |
"Lcm (set xs) = fold lcm xs 1" |
1261 |
using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps) |
|
1262 |
||
1263 |
lemma Lcm_singleton [simp]: |
|
60634 | 1264 |
"Lcm {a} = normalize a" |
58023 | 1265 |
by simp |
1266 |
||
1267 |
lemma Lcm_2 [simp]: |
|
1268 |
"Lcm {a,b} = lcm a b" |
|
60634 | 1269 |
by simp |
58023 | 1270 |
|
1271 |
lemma Lcm_coprime: |
|
1272 |
assumes "finite A" and "A \<noteq> {}" |
|
1273 |
assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1" |
|
60634 | 1274 |
shows "Lcm A = normalize (\<Prod>A)" |
58023 | 1275 |
using assms proof (induct rule: finite_ne_induct) |
1276 |
case (insert a A) |
|
1277 |
have "Lcm (insert a A) = lcm a (Lcm A)" by simp |
|
60634 | 1278 |
also from insert have "Lcm A = normalize (\<Prod>A)" by blast |
58023 | 1279 |
also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2) |
1280 |
also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto |
|
60634 | 1281 |
with insert have "lcm a (\<Prod>A) = normalize (\<Prod>(insert a A))" |
58023 | 1282 |
by (simp add: lcm_coprime) |
1283 |
finally show ?case . |
|
1284 |
qed simp |
|
1285 |
||
1286 |
lemma Lcm_coprime': |
|
1287 |
"card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1) |
|
60634 | 1288 |
\<Longrightarrow> Lcm A = normalize (\<Prod>A)" |
58023 | 1289 |
by (rule Lcm_coprime) (simp_all add: card_eq_0_iff) |
1290 |
||
1291 |
lemma Gcd_Lcm: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1292 |
"Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}" |
58023 | 1293 |
by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def) |
1294 |
||
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1295 |
lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a" |
60634 | 1296 |
and Gcd_greatest: "(\<And>a. a \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> b dvd Gcd A" |
1297 |
and unit_factor_Gcd [simp]: |
|
1298 |
"unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)" |
|
58023 | 1299 |
proof - |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1300 |
fix a assume "a \<in> A" |
60634 | 1301 |
hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_least) blast |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1302 |
then show "Gcd A dvd a" by (simp add: Gcd_Lcm) |
58023 | 1303 |
next |
60634 | 1304 |
fix g' assume "\<And>a. a \<in> A \<Longrightarrow> g' dvd a" |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1305 |
hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast |
58023 | 1306 |
then show "g' dvd Gcd A" by (simp add: Gcd_Lcm) |
1307 |
next |
|
60634 | 1308 |
show "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
1309 |
by (simp add: Gcd_Lcm) |
58023 | 1310 |
qed |
1311 |
||
60634 | 1312 |
lemma normalize_Gcd [simp]: |
1313 |
"normalize (Gcd A) = Gcd A" |
|
1314 |
by (cases "Gcd A = 0") (auto intro: associated_eqI) |
|
1315 |
||
58023 | 1316 |
lemma GcdI: |
60634 | 1317 |
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" |
1318 |
and "unit_factor b = (if b = 0 then 0 else 1)" |
|
1319 |
shows "b = Gcd A" |
|
1320 |
by (rule associated_eqI) (auto simp: assms associated_def intro: Gcd_greatest) |
|
58023 | 1321 |
|
1322 |
lemma Lcm_Gcd: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1323 |
"Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}" |
60634 | 1324 |
by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_greatest) |
58023 | 1325 |
|
1326 |
lemma Gcd_0_iff: |
|
1327 |
"Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}" |
|
1328 |
apply (rule iffI) |
|
1329 |
apply (rule subsetI, drule Gcd_dvd, simp) |
|
1330 |
apply (auto intro: GcdI[symmetric]) |
|
1331 |
done |
|
1332 |
||
1333 |
lemma Gcd_empty [simp]: |
|
1334 |
"Gcd {} = 0" |
|
1335 |
by (simp add: Gcd_0_iff) |
|
1336 |
||
1337 |
lemma Gcd_1: |
|
1338 |
"1 \<in> A \<Longrightarrow> Gcd A = 1" |
|
1339 |
by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd) |
|
1340 |
||
1341 |
lemma Gcd_insert [simp]: |
|
1342 |
"Gcd (insert a A) = gcd a (Gcd A)" |
|
1343 |
proof (rule gcdI) |
|
1344 |
fix l assume "l dvd a" and "l dvd Gcd A" |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1345 |
hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd) |
60634 | 1346 |
with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd Gcd_greatest) |
1347 |
qed (auto intro: Gcd_greatest) |
|
58023 | 1348 |
|
1349 |
lemma Gcd_finite: |
|
1350 |
assumes "finite A" |
|
1351 |
shows "Gcd A = Finite_Set.fold gcd 0 A" |
|
60526 | 1352 |
by (induct rule: finite.induct[OF \<open>finite A\<close>]) |
58023 | 1353 |
(simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd]) |
1354 |
||
60431
db9c67b760f1
dropped warnings by dropping ineffective code declarations
haftmann
parents:
60430
diff
changeset
|
1355 |
lemma Gcd_set [code_unfold]: |
58023 | 1356 |
"Gcd (set xs) = fold gcd xs 0" |
1357 |
using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps) |
|
1358 |
||
60634 | 1359 |
lemma Gcd_singleton [simp]: "Gcd {a} = normalize a" |
58023 | 1360 |
by (simp add: gcd_0) |
1361 |
||
1362 |
lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b" |
|
60634 | 1363 |
by (simp add: gcd_0) |
58023 | 1364 |
|
60439
b765e08f8bc0
proper subclass instances for existing gcd (semi)rings
haftmann
parents:
60438
diff
changeset
|
1365 |
subclass semiring_gcd |
b765e08f8bc0
proper subclass instances for existing gcd (semi)rings
haftmann
parents:
60438
diff
changeset
|
1366 |
by unfold_locales (simp_all add: gcd_greatest_iff) |
b765e08f8bc0
proper subclass instances for existing gcd (semi)rings
haftmann
parents:
60438
diff
changeset
|
1367 |
|
58023 | 1368 |
end |
1369 |
||
60526 | 1370 |
text \<open> |
58023 | 1371 |
A Euclidean ring is a Euclidean semiring with additive inverses. It provides a |
1372 |
few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring. |
|
60526 | 1373 |
\<close> |
58023 | 1374 |
|
1375 |
class euclidean_ring_gcd = euclidean_semiring_gcd + idom |
|
1376 |
begin |
|
1377 |
||
1378 |
subclass euclidean_ring .. |
|
1379 |
||
60439
b765e08f8bc0
proper subclass instances for existing gcd (semi)rings
haftmann
parents:
60438
diff
changeset
|
1380 |
subclass ring_gcd .. |
b765e08f8bc0
proper subclass instances for existing gcd (semi)rings
haftmann
parents:
60438
diff
changeset
|
1381 |
|
60572
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1382 |
lemma euclid_ext_gcd [simp]: |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1383 |
"(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b" |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1384 |
by (induct a b rule: gcd_eucl_induct) |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1385 |
(simp_all add: euclid_ext_0 gcd_0 euclid_ext_non_0 ac_simps split: prod.split prod.split_asm) |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1386 |
|
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1387 |
lemma euclid_ext_gcd' [simp]: |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1388 |
"euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b" |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1389 |
by (insert euclid_ext_gcd[of a b], drule (1) subst, simp) |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1390 |
|
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1391 |
lemma euclid_ext'_correct: |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1392 |
"fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b" |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1393 |
proof- |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1394 |
obtain s t c where "euclid_ext a b = (s,t,c)" |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1395 |
by (cases "euclid_ext a b", blast) |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1396 |
with euclid_ext_correct[of a b] euclid_ext_gcd[of a b] |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1397 |
show ?thesis unfolding euclid_ext'_def by simp |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1398 |
qed |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1399 |
|
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1400 |
lemma bezout: "\<exists>s t. s * a + t * b = gcd a b" |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1401 |
using euclid_ext'_correct by blast |
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1402 |
|
58023 | 1403 |
lemma gcd_neg1 [simp]: |
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1404 |
"gcd (-a) b = gcd a b" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
1405 |
by (rule sym, rule gcdI, simp_all add: gcd_greatest) |
58023 | 1406 |
|
1407 |
lemma gcd_neg2 [simp]: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1408 |
"gcd a (-b) = gcd a b" |
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
1409 |
by (rule sym, rule gcdI, simp_all add: gcd_greatest) |
58023 | 1410 |
|
1411 |
lemma gcd_neg_numeral_1 [simp]: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1412 |
"gcd (- numeral n) a = gcd (numeral n) a" |
58023 | 1413 |
by (fact gcd_neg1) |
1414 |
||
1415 |
lemma gcd_neg_numeral_2 [simp]: |
|
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset
|
1416 |
"gcd a (- numeral n) = gcd a (numeral n)" |
58023 | 1417 |
by (fact gcd_neg2) |
1418 |
||
1419 |
lemma gcd_diff1: "gcd (m - n) n = gcd m n" |
|
1420 |
by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric], subst gcd_add1, simp) |
|
1421 |
||
1422 |
lemma gcd_diff2: "gcd (n - m) n = gcd m n" |
|
1423 |
by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1) |
|
1424 |
||
1425 |
lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1" |
|
1426 |
proof - |
|
1427 |
have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute) |
|
1428 |
also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp |
|
1429 |
also have "\<dots> = 1" by (rule coprime_plus_one) |
|
1430 |
finally show ?thesis . |
|
1431 |
qed |
|
1432 |
||
60430
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standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
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diff
changeset
|
1433 |
lemma lcm_neg1 [simp]: "lcm (-a) b = lcm a b" |
58023 | 1434 |
by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero) |
1435 |
||
60430
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standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
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diff
changeset
|
1436 |
lemma lcm_neg2 [simp]: "lcm a (-b) = lcm a b" |
58023 | 1437 |
by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero) |
1438 |
||
60430
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standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
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diff
changeset
|
1439 |
lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) a = lcm (numeral n) a" |
58023 | 1440 |
by (fact lcm_neg1) |
1441 |
||
60430
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standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
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diff
changeset
|
1442 |
lemma lcm_neg_numeral_2 [simp]: "lcm a (- numeral n) = lcm a (numeral n)" |
58023 | 1443 |
by (fact lcm_neg2) |
1444 |
||
60572
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset
|
1445 |
end |
58023 | 1446 |
|
1447 |
||
60572
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streamlined definitions and primitive lemma of euclidean algorithm, including code generation
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diff
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|
1448 |
subsection \<open>Typical instances\<close> |
58023 | 1449 |
|
1450 |
instantiation nat :: euclidean_semiring |
|
1451 |
begin |
|
1452 |
||
1453 |
definition [simp]: |
|
1454 |
"euclidean_size_nat = (id :: nat \<Rightarrow> nat)" |
|
1455 |
||
1456 |
definition [simp]: |
|
60634 | 1457 |
"unit_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)" |
58023 | 1458 |
|
1459 |
instance proof |
|
59061 | 1460 |
qed simp_all |
58023 | 1461 |
|
1462 |
end |
|
1463 |
||
1464 |
instantiation int :: euclidean_ring |
|
1465 |
begin |
|
1466 |
||
1467 |
definition [simp]: |
|
1468 |
"euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)" |
|
1469 |
||
1470 |
definition [simp]: |
|
60634 | 1471 |
"unit_factor_int = (sgn :: int \<Rightarrow> int)" |
58023 | 1472 |
|
60580 | 1473 |
instance |
60634 | 1474 |
by standard (auto simp add: abs_mult nat_mult_distrib sgn_times split: abs_split) |
58023 | 1475 |
|
1476 |
end |
|
1477 |
||
60572
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haftmann
parents:
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diff
changeset
|
1478 |
instantiation poly :: (field) euclidean_ring |
60571 | 1479 |
begin |
1480 |
||
1481 |
definition euclidean_size_poly :: "'a poly \<Rightarrow> nat" |
|
60600
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haftmann
parents:
60599
diff
changeset
|
1482 |
where "euclidean_size p = (if p = 0 then 0 else Suc (degree p))" |
60571 | 1483 |
|
60634 | 1484 |
lemma euclidenan_size_poly_minus_one_degree [simp]: |
1485 |
"euclidean_size p - 1 = degree p" |
|
1486 |
by (simp add: euclidean_size_poly_def) |
|
60571 | 1487 |
|
60600
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haftmann
parents:
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diff
changeset
|
1488 |
lemma euclidean_size_poly_0 [simp]: |
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1489 |
"euclidean_size (0::'a poly) = 0" |
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1490 |
by (simp add: euclidean_size_poly_def) |
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1491 |
|
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
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diff
changeset
|
1492 |
lemma euclidean_size_poly_not_0 [simp]: |
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1493 |
"p \<noteq> 0 \<Longrightarrow> euclidean_size p = Suc (degree p)" |
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1494 |
by (simp add: euclidean_size_poly_def) |
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1495 |
|
60571 | 1496 |
instance |
60600
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1497 |
proof |
60571 | 1498 |
fix p q :: "'a poly" |
60600
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1499 |
assume "q \<noteq> 0" |
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1500 |
then have "p mod q = 0 \<or> degree (p mod q) < degree q" |
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1501 |
by (rule degree_mod_less [of q p]) |
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1502 |
with \<open>q \<noteq> 0\<close> show "euclidean_size (p mod q) < euclidean_size q" |
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1503 |
by (cases "p mod q = 0") simp_all |
60571 | 1504 |
next |
1505 |
fix p q :: "'a poly" |
|
1506 |
assume "q \<noteq> 0" |
|
60600
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1507 |
from \<open>q \<noteq> 0\<close> have "degree p \<le> degree (p * q)" |
60571 | 1508 |
by (rule degree_mult_right_le) |
60600
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1509 |
with \<open>q \<noteq> 0\<close> show "euclidean_size p \<le> euclidean_size (p * q)" |
87fbfea0bd0a
simplified termination criterion for euclidean algorithm (again)
haftmann
parents:
60599
diff
changeset
|
1510 |
by (cases "p = 0") simp_all |
60571 | 1511 |
qed |
1512 |
||
58023 | 1513 |
end |
60571 | 1514 |
|
1515 |
end |