author | wenzelm |
Fri, 17 Apr 2015 22:15:35 +0200 | |
changeset 60125 | 2944cc4f4f56 |
parent 59061 | 67771d267ff2 |
child 60430 | ce559c850a27 |
permissions | -rw-r--r-- |
58023 | 1 |
(* Author: Manuel Eberl *) |
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section {* Abstract euclidean algorithm *} |
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theory Euclidean_Algorithm |
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imports Complex_Main |
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begin |
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context semiring_div |
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begin |
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||
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abbreviation is_unit :: "'a \<Rightarrow> bool" |
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where |
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"is_unit x \<equiv> x dvd 1" |
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|
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definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool" |
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where |
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"associated x y \<longleftrightarrow> x dvd y \<and> y dvd x" |
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||
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definition ring_inv :: "'a \<Rightarrow> 'a" |
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where |
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"ring_inv x = 1 div x" |
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||
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lemma unit_prod [intro]: |
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"is_unit x \<Longrightarrow> is_unit y \<Longrightarrow> is_unit (x * y)" |
|
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by (subst mult_1_left [of 1, symmetric], rule mult_dvd_mono) |
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|
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lemma unit_ring_inv: |
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"is_unit y \<Longrightarrow> x div y = x * ring_inv y" |
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by (simp add: div_mult_swap ring_inv_def) |
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lemma unit_ring_inv_ring_inv [simp]: |
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"is_unit x \<Longrightarrow> ring_inv (ring_inv x) = x" |
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unfolding ring_inv_def |
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by (metis div_mult_mult1_if div_mult_self1_is_id dvd_mult_div_cancel mult_1_right) |
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||
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lemma inv_imp_eq_ring_inv: |
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"a * b = 1 \<Longrightarrow> ring_inv a = b" |
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by (metis dvd_mult_div_cancel dvd_mult_right mult_1_right mult.left_commute one_dvd ring_inv_def) |
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lemma ring_inv_is_inv1 [simp]: |
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"is_unit a \<Longrightarrow> a * ring_inv a = 1" |
|
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unfolding ring_inv_def by simp |
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|
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lemma ring_inv_is_inv2 [simp]: |
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"is_unit a \<Longrightarrow> ring_inv a * a = 1" |
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by (simp add: ac_simps) |
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||
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lemma unit_ring_inv_unit [simp, intro]: |
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assumes "is_unit x" |
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shows "is_unit (ring_inv x)" |
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proof - |
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from assms have "1 = ring_inv x * x" by simp |
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then show "is_unit (ring_inv x)" by (rule dvdI) |
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qed |
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lemma mult_unit_dvd_iff: |
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"is_unit y \<Longrightarrow> x * y dvd z \<longleftrightarrow> x dvd z" |
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proof |
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assume "is_unit y" "x * y dvd z" |
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then show "x dvd z" by (simp add: dvd_mult_left) |
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next |
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assume "is_unit y" "x dvd z" |
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then obtain k where "z = x * k" unfolding dvd_def by blast |
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with `is_unit y` have "z = (x * y) * (ring_inv y * k)" |
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by (simp add: mult_ac) |
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then show "x * y dvd z" by (rule dvdI) |
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qed |
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lemma div_unit_dvd_iff: |
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"is_unit y \<Longrightarrow> x div y dvd z \<longleftrightarrow> x dvd z" |
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by (subst unit_ring_inv) (assumption, simp add: mult_unit_dvd_iff) |
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lemma dvd_mult_unit_iff: |
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"is_unit y \<Longrightarrow> x dvd z * y \<longleftrightarrow> x dvd z" |
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proof |
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assume "is_unit y" and "x dvd z * y" |
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have "z * y dvd z * (y * ring_inv y)" by (subst mult_assoc [symmetric]) simp |
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also from `is_unit y` have "y * ring_inv y = 1" by simp |
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finally have "z * y dvd z" by simp |
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with `x dvd z * y` show "x dvd z" by (rule dvd_trans) |
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next |
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assume "x dvd z" |
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then show "x dvd z * y" by simp |
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qed |
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lemma dvd_div_unit_iff: |
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"is_unit y \<Longrightarrow> x dvd z div y \<longleftrightarrow> x dvd z" |
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by (subst unit_ring_inv) (assumption, simp add: dvd_mult_unit_iff) |
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lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff dvd_div_unit_iff |
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lemma unit_div [intro]: |
|
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"is_unit x \<Longrightarrow> is_unit y \<Longrightarrow> is_unit (x div y)" |
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by (subst unit_ring_inv) (assumption, rule unit_prod, simp_all) |
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lemma unit_div_mult_swap: |
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"is_unit z \<Longrightarrow> x * (y div z) = x * y div z" |
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by (simp only: unit_ring_inv [of _ y] unit_ring_inv [of _ "x*y"] ac_simps) |
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lemma unit_div_commute: |
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"is_unit y \<Longrightarrow> x div y * z = x * z div y" |
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by (simp only: unit_ring_inv [of _ x] unit_ring_inv [of _ "x*z"] ac_simps) |
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lemma unit_imp_dvd [dest]: |
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"is_unit y \<Longrightarrow> y dvd x" |
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by (rule dvd_trans [of _ 1]) simp_all |
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lemma dvd_unit_imp_unit: |
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"is_unit y \<Longrightarrow> x dvd y \<Longrightarrow> is_unit x" |
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by (rule dvd_trans) |
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lemma ring_inv_0 [simp]: |
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"ring_inv 0 = 0" |
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unfolding ring_inv_def by simp |
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lemma unit_ring_inv'1: |
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assumes "is_unit y" |
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shows "x div (y * z) = x * ring_inv y div z" |
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proof - |
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from assms have "x div (y * z) = x * (ring_inv y * y) div (y * z)" |
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by simp |
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also have "... = y * (x * ring_inv y) div (y * z)" |
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by (simp only: mult_ac) |
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also have "... = x * ring_inv y div z" |
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by (cases "y = 0", simp, rule div_mult_mult1) |
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finally show ?thesis . |
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qed |
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lemma associated_comm: |
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"associated x y \<Longrightarrow> associated y x" |
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by (simp add: associated_def) |
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lemma associated_0 [simp]: |
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"associated 0 b \<longleftrightarrow> b = 0" |
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"associated a 0 \<longleftrightarrow> a = 0" |
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unfolding associated_def by simp_all |
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lemma associated_unit: |
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"is_unit x \<Longrightarrow> associated x y \<Longrightarrow> is_unit y" |
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unfolding associated_def using dvd_unit_imp_unit by auto |
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lemma is_unit_1 [simp]: |
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"is_unit 1" |
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59061 | 145 |
by simp |
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lemma not_is_unit_0 [simp]: |
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"\<not> is_unit 0" |
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by auto |
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lemma unit_mult_left_cancel: |
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assumes "is_unit x" |
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shows "(x * y) = (x * z) \<longleftrightarrow> y = z" |
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proof - |
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from assms have "x \<noteq> 0" by auto |
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then show ?thesis by (metis div_mult_self1_is_id) |
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qed |
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lemma unit_mult_right_cancel: |
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"is_unit x \<Longrightarrow> (y * x) = (z * x) \<longleftrightarrow> y = z" |
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by (simp add: ac_simps unit_mult_left_cancel) |
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lemma unit_div_cancel: |
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"is_unit x \<Longrightarrow> (y div x) = (z div x) \<longleftrightarrow> y = z" |
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apply (subst unit_ring_inv[of _ y], assumption) |
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apply (subst unit_ring_inv[of _ z], assumption) |
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apply (rule unit_mult_right_cancel, erule unit_ring_inv_unit) |
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done |
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lemma unit_eq_div1: |
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"is_unit y \<Longrightarrow> x div y = z \<longleftrightarrow> x = z * y" |
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apply (subst unit_ring_inv, assumption) |
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apply (subst unit_mult_right_cancel[symmetric], assumption) |
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apply (subst mult_assoc, subst ring_inv_is_inv2, assumption, simp) |
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done |
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lemma unit_eq_div2: |
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"is_unit y \<Longrightarrow> x = z div y \<longleftrightarrow> x * y = z" |
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by (subst (1 2) eq_commute, simp add: unit_eq_div1, subst eq_commute, rule refl) |
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lemma associated_iff_div_unit: |
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"associated x y \<longleftrightarrow> (\<exists>z. is_unit z \<and> x = z * y)" |
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proof |
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assume "associated x y" |
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show "\<exists>z. is_unit z \<and> x = z * y" |
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proof (cases "x = 0") |
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assume "x = 0" |
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then show "\<exists>z. is_unit z \<and> x = z * y" using `associated x y` |
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by (intro exI[of _ 1], simp add: associated_def) |
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next |
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assume [simp]: "x \<noteq> 0" |
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hence [simp]: "x dvd y" "y dvd x" using `associated x y` |
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unfolding associated_def by simp_all |
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hence "1 = x div y * (y div x)" |
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by (simp add: div_mult_swap) |
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hence "is_unit (x div y)" .. |
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moreover have "x = (x div y) * y" by simp |
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ultimately show ?thesis by blast |
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qed |
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next |
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assume "\<exists>z. is_unit z \<and> x = z * y" |
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then obtain z where "is_unit z" and "x = z * y" by blast |
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hence "y = x * ring_inv z" by (simp add: algebra_simps) |
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hence "x dvd y" by simp |
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moreover from `x = z * y` have "y dvd x" by simp |
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ultimately show "associated x y" unfolding associated_def by simp |
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qed |
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lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff |
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dvd_div_unit_iff unit_div_mult_swap unit_div_commute |
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unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel |
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unit_eq_div1 unit_eq_div2 |
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end |
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context ring_div |
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begin |
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lemma is_unit_neg [simp]: |
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"is_unit (- x) \<Longrightarrow> is_unit x" |
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by simp |
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lemma is_unit_neg_1 [simp]: |
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"is_unit (-1)" |
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by simp |
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end |
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lemma is_unit_nat [simp]: |
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"is_unit (x::nat) \<longleftrightarrow> x = 1" |
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59061 | 231 |
by simp |
58023 | 232 |
|
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lemma is_unit_int: |
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"is_unit (x::int) \<longleftrightarrow> x = 1 \<or> x = -1" |
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by auto |
58023 | 236 |
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text {* |
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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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\item a normalisation factor such that two associated numbers are equal iff |
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they are the same when divided by their normalisation factors. |
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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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*} |
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class euclidean_semiring = semiring_div + |
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fixes euclidean_size :: "'a \<Rightarrow> nat" |
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fixes normalisation_factor :: "'a \<Rightarrow> 'a" |
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assumes mod_size_less [simp]: |
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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 * b) \<ge> euclidean_size a" |
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assumes normalisation_factor_is_unit [intro,simp]: |
|
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"a \<noteq> 0 \<Longrightarrow> is_unit (normalisation_factor a)" |
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assumes normalisation_factor_mult: "normalisation_factor (a * b) = |
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normalisation_factor a * normalisation_factor b" |
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assumes normalisation_factor_unit: "is_unit x \<Longrightarrow> normalisation_factor x = x" |
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assumes normalisation_factor_0 [simp]: "normalisation_factor 0 = 0" |
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begin |
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lemma normalisation_factor_dvd [simp]: |
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"a \<noteq> 0 \<Longrightarrow> normalisation_factor a dvd b" |
|
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by (rule unit_imp_dvd, simp) |
|
268 |
||
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lemma normalisation_factor_1 [simp]: |
|
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"normalisation_factor 1 = 1" |
|
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by (simp add: normalisation_factor_unit) |
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||
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lemma normalisation_factor_0_iff [simp]: |
|
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"normalisation_factor x = 0 \<longleftrightarrow> x = 0" |
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proof |
|
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assume "normalisation_factor x = 0" |
|
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hence "\<not> is_unit (normalisation_factor x)" |
|
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by (metis not_is_unit_0) |
|
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then show "x = 0" by force |
|
280 |
next |
|
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assume "x = 0" |
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then show "normalisation_factor x = 0" by simp |
|
283 |
qed |
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284 |
||
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lemma normalisation_factor_pow: |
|
286 |
"normalisation_factor (x ^ n) = normalisation_factor x ^ n" |
|
287 |
by (induct n) (simp_all add: normalisation_factor_mult power_Suc2) |
|
288 |
||
289 |
lemma normalisation_correct [simp]: |
|
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"normalisation_factor (x div normalisation_factor x) = (if x = 0 then 0 else 1)" |
|
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proof (cases "x = 0", simp) |
|
292 |
assume "x \<noteq> 0" |
|
293 |
let ?nf = "normalisation_factor" |
|
294 |
from normalisation_factor_is_unit[OF `x \<noteq> 0`] have "?nf x \<noteq> 0" |
|
295 |
by (metis not_is_unit_0) |
|
296 |
have "?nf (x div ?nf x) * ?nf (?nf x) = ?nf (x div ?nf x * ?nf x)" |
|
297 |
by (simp add: normalisation_factor_mult) |
|
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also have "x div ?nf x * ?nf x = x" using `x \<noteq> 0` |
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by simp |
58023 | 300 |
also have "?nf (?nf x) = ?nf x" using `x \<noteq> 0` |
301 |
normalisation_factor_is_unit normalisation_factor_unit by simp |
|
302 |
finally show ?thesis using `x \<noteq> 0` and `?nf x \<noteq> 0` |
|
303 |
by (metis div_mult_self2_is_id div_self) |
|
304 |
qed |
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305 |
||
306 |
lemma normalisation_0_iff [simp]: |
|
307 |
"x div normalisation_factor x = 0 \<longleftrightarrow> x = 0" |
|
308 |
by (cases "x = 0", simp, subst unit_eq_div1, blast, simp) |
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309 |
||
310 |
lemma associated_iff_normed_eq: |
|
311 |
"associated a b \<longleftrightarrow> a div normalisation_factor a = b div normalisation_factor b" |
|
312 |
proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalisation_0_iff, rule iffI) |
|
313 |
let ?nf = normalisation_factor |
|
314 |
assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b" |
|
315 |
hence "a = b * (?nf a div ?nf b)" |
|
316 |
apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast) |
|
317 |
apply (subst div_mult_swap, simp, simp) |
|
318 |
done |
|
319 |
with `a \<noteq> 0` `b \<noteq> 0` have "\<exists>z. is_unit z \<and> a = z * b" |
|
320 |
by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac) |
|
321 |
with associated_iff_div_unit show "associated a b" by simp |
|
322 |
next |
|
323 |
let ?nf = normalisation_factor |
|
324 |
assume "a \<noteq> 0" "b \<noteq> 0" "associated a b" |
|
325 |
with associated_iff_div_unit obtain z where "is_unit z" and "a = z * b" by blast |
|
326 |
then show "a div ?nf a = b div ?nf b" |
|
327 |
apply (simp only: `a = z * b` normalisation_factor_mult normalisation_factor_unit) |
|
328 |
apply (rule div_mult_mult1, force) |
|
329 |
done |
|
330 |
qed |
|
331 |
||
332 |
lemma normed_associated_imp_eq: |
|
333 |
"associated a b \<Longrightarrow> normalisation_factor a \<in> {0, 1} \<Longrightarrow> normalisation_factor b \<in> {0, 1} \<Longrightarrow> a = b" |
|
334 |
by (simp add: associated_iff_normed_eq, elim disjE, simp_all) |
|
335 |
||
336 |
lemmas normalisation_factor_dvd_iff [simp] = |
|
337 |
unit_dvd_iff [OF normalisation_factor_is_unit] |
|
338 |
||
339 |
lemma euclidean_division: |
|
340 |
fixes a :: 'a and b :: 'a |
|
341 |
assumes "b \<noteq> 0" |
|
342 |
obtains s and t where "a = s * b + t" |
|
343 |
and "euclidean_size t < euclidean_size b" |
|
344 |
proof - |
|
345 |
from div_mod_equality[of a b 0] |
|
346 |
have "a = a div b * b + a mod b" by simp |
|
347 |
with that and assms show ?thesis by force |
|
348 |
qed |
|
349 |
||
350 |
lemma dvd_euclidean_size_eq_imp_dvd: |
|
351 |
assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b" |
|
352 |
shows "a dvd b" |
|
353 |
proof (subst dvd_eq_mod_eq_0, rule ccontr) |
|
354 |
assume "b mod a \<noteq> 0" |
|
355 |
from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff) |
|
356 |
from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast |
|
357 |
with `b mod a \<noteq> 0` have "c \<noteq> 0" by auto |
|
358 |
with `b mod a = b * c` have "euclidean_size (b mod a) \<ge> euclidean_size b" |
|
359 |
using size_mult_mono by force |
|
360 |
moreover from `a \<noteq> 0` have "euclidean_size (b mod a) < euclidean_size a" |
|
361 |
using mod_size_less by blast |
|
362 |
ultimately show False using size_eq by simp |
|
363 |
qed |
|
364 |
||
365 |
function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" |
|
366 |
where |
|
367 |
"gcd_eucl a b = (if b = 0 then a div normalisation_factor a else gcd_eucl b (a mod b))" |
|
368 |
by (pat_completeness, simp) |
|
369 |
termination by (relation "measure (euclidean_size \<circ> snd)", simp_all) |
|
370 |
||
371 |
declare gcd_eucl.simps [simp del] |
|
372 |
||
373 |
lemma gcd_induct: "\<lbrakk>\<And>b. P b 0; \<And>a b. 0 \<noteq> b \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b\<rbrakk> \<Longrightarrow> P a b" |
|
374 |
proof (induct a b rule: gcd_eucl.induct) |
|
375 |
case ("1" m n) |
|
376 |
then show ?case by (cases "n = 0") auto |
|
377 |
qed |
|
378 |
||
379 |
definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" |
|
380 |
where |
|
381 |
"lcm_eucl a b = a * b div (gcd_eucl a b * normalisation_factor (a * b))" |
|
382 |
||
383 |
(* Somewhat complicated definition of Lcm that has the advantage of working |
|
384 |
for infinite sets as well *) |
|
385 |
||
386 |
definition Lcm_eucl :: "'a set \<Rightarrow> 'a" |
|
387 |
where |
|
388 |
"Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) then |
|
389 |
let l = SOME l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = |
|
390 |
(LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n) |
|
391 |
in l div normalisation_factor l |
|
392 |
else 0)" |
|
393 |
||
394 |
definition Gcd_eucl :: "'a set \<Rightarrow> 'a" |
|
395 |
where |
|
396 |
"Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}" |
|
397 |
||
398 |
end |
|
399 |
||
400 |
class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd + |
|
401 |
assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl" |
|
402 |
assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl" |
|
403 |
begin |
|
404 |
||
405 |
lemma gcd_red: |
|
406 |
"gcd x y = gcd y (x mod y)" |
|
407 |
by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl) |
|
408 |
||
409 |
lemma gcd_non_0: |
|
410 |
"y \<noteq> 0 \<Longrightarrow> gcd x y = gcd y (x mod y)" |
|
411 |
by (rule gcd_red) |
|
412 |
||
413 |
lemma gcd_0_left: |
|
414 |
"gcd 0 x = x div normalisation_factor x" |
|
415 |
by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def) |
|
416 |
||
417 |
lemma gcd_0: |
|
418 |
"gcd x 0 = x div normalisation_factor x" |
|
419 |
by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def) |
|
420 |
||
421 |
lemma gcd_dvd1 [iff]: "gcd x y dvd x" |
|
422 |
and gcd_dvd2 [iff]: "gcd x y dvd y" |
|
423 |
proof (induct x y rule: gcd_eucl.induct) |
|
424 |
fix x y :: 'a |
|
425 |
assume IH1: "y \<noteq> 0 \<Longrightarrow> gcd y (x mod y) dvd y" |
|
426 |
assume IH2: "y \<noteq> 0 \<Longrightarrow> gcd y (x mod y) dvd (x mod y)" |
|
427 |
||
428 |
have "gcd x y dvd x \<and> gcd x y dvd y" |
|
429 |
proof (cases "y = 0") |
|
430 |
case True |
|
431 |
then show ?thesis by (cases "x = 0", simp_all add: gcd_0) |
|
432 |
next |
|
433 |
case False |
|
434 |
with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff) |
|
435 |
qed |
|
436 |
then show "gcd x y dvd x" "gcd x y dvd y" by simp_all |
|
437 |
qed |
|
438 |
||
439 |
lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m" |
|
440 |
by (rule dvd_trans, assumption, rule gcd_dvd1) |
|
441 |
||
442 |
lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n" |
|
443 |
by (rule dvd_trans, assumption, rule gcd_dvd2) |
|
444 |
||
445 |
lemma gcd_greatest: |
|
446 |
fixes k x y :: 'a |
|
447 |
shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y" |
|
448 |
proof (induct x y rule: gcd_eucl.induct) |
|
449 |
case (1 x y) |
|
450 |
show ?case |
|
451 |
proof (cases "y = 0") |
|
452 |
assume "y = 0" |
|
453 |
with 1 show ?thesis by (cases "x = 0", simp_all add: gcd_0) |
|
454 |
next |
|
455 |
assume "y \<noteq> 0" |
|
456 |
with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff) |
|
457 |
qed |
|
458 |
qed |
|
459 |
||
460 |
lemma dvd_gcd_iff: |
|
461 |
"k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y" |
|
462 |
by (blast intro!: gcd_greatest intro: dvd_trans) |
|
463 |
||
464 |
lemmas gcd_greatest_iff = dvd_gcd_iff |
|
465 |
||
466 |
lemma gcd_zero [simp]: |
|
467 |
"gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0" |
|
468 |
by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+ |
|
469 |
||
470 |
lemma normalisation_factor_gcd [simp]: |
|
471 |
"normalisation_factor (gcd x y) = (if x = 0 \<and> y = 0 then 0 else 1)" (is "?f x y = ?g x y") |
|
472 |
proof (induct x y rule: gcd_eucl.induct) |
|
473 |
fix x y :: 'a |
|
474 |
assume IH: "y \<noteq> 0 \<Longrightarrow> ?f y (x mod y) = ?g y (x mod y)" |
|
475 |
then show "?f x y = ?g x y" by (cases "y = 0", auto simp: gcd_non_0 gcd_0) |
|
476 |
qed |
|
477 |
||
478 |
lemma gcdI: |
|
479 |
"k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> (\<And>l. l dvd x \<Longrightarrow> l dvd y \<Longrightarrow> l dvd k) |
|
480 |
\<Longrightarrow> normalisation_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd x y" |
|
481 |
by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest) |
|
482 |
||
483 |
sublocale gcd!: abel_semigroup gcd |
|
484 |
proof |
|
485 |
fix x y z |
|
486 |
show "gcd (gcd x y) z = gcd x (gcd y z)" |
|
487 |
proof (rule gcdI) |
|
488 |
have "gcd (gcd x y) z dvd gcd x y" "gcd x y dvd x" by simp_all |
|
489 |
then show "gcd (gcd x y) z dvd x" by (rule dvd_trans) |
|
490 |
have "gcd (gcd x y) z dvd gcd x y" "gcd x y dvd y" by simp_all |
|
491 |
hence "gcd (gcd x y) z dvd y" by (rule dvd_trans) |
|
492 |
moreover have "gcd (gcd x y) z dvd z" by simp |
|
493 |
ultimately show "gcd (gcd x y) z dvd gcd y z" |
|
494 |
by (rule gcd_greatest) |
|
495 |
show "normalisation_factor (gcd (gcd x y) z) = (if gcd (gcd x y) z = 0 then 0 else 1)" |
|
496 |
by auto |
|
497 |
fix l assume "l dvd x" and "l dvd gcd y z" |
|
498 |
with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2] |
|
499 |
have "l dvd y" and "l dvd z" by blast+ |
|
500 |
with `l dvd x` show "l dvd gcd (gcd x y) z" |
|
501 |
by (intro gcd_greatest) |
|
502 |
qed |
|
503 |
next |
|
504 |
fix x y |
|
505 |
show "gcd x y = gcd y x" |
|
506 |
by (rule gcdI) (simp_all add: gcd_greatest) |
|
507 |
qed |
|
508 |
||
509 |
lemma gcd_unique: "d dvd a \<and> d dvd b \<and> |
|
510 |
normalisation_factor d = (if d = 0 then 0 else 1) \<and> |
|
511 |
(\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b" |
|
512 |
by (rule, auto intro: gcdI simp: gcd_greatest) |
|
513 |
||
514 |
lemma gcd_dvd_prod: "gcd a b dvd k * b" |
|
515 |
using mult_dvd_mono [of 1] by auto |
|
516 |
||
517 |
lemma gcd_1_left [simp]: "gcd 1 x = 1" |
|
518 |
by (rule sym, rule gcdI, simp_all) |
|
519 |
||
520 |
lemma gcd_1 [simp]: "gcd x 1 = 1" |
|
521 |
by (rule sym, rule gcdI, simp_all) |
|
522 |
||
523 |
lemma gcd_proj2_if_dvd: |
|
524 |
"y dvd x \<Longrightarrow> gcd x y = y div normalisation_factor y" |
|
525 |
by (cases "y = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0) |
|
526 |
||
527 |
lemma gcd_proj1_if_dvd: |
|
528 |
"x dvd y \<Longrightarrow> gcd x y = x div normalisation_factor x" |
|
529 |
by (subst gcd.commute, simp add: gcd_proj2_if_dvd) |
|
530 |
||
531 |
lemma gcd_proj1_iff: "gcd m n = m div normalisation_factor m \<longleftrightarrow> m dvd n" |
|
532 |
proof |
|
533 |
assume A: "gcd m n = m div normalisation_factor m" |
|
534 |
show "m dvd n" |
|
535 |
proof (cases "m = 0") |
|
536 |
assume [simp]: "m \<noteq> 0" |
|
537 |
from A have B: "m = gcd m n * normalisation_factor m" |
|
538 |
by (simp add: unit_eq_div2) |
|
539 |
show ?thesis by (subst B, simp add: mult_unit_dvd_iff) |
|
540 |
qed (insert A, simp) |
|
541 |
next |
|
542 |
assume "m dvd n" |
|
543 |
then show "gcd m n = m div normalisation_factor m" by (rule gcd_proj1_if_dvd) |
|
544 |
qed |
|
545 |
||
546 |
lemma gcd_proj2_iff: "gcd m n = n div normalisation_factor n \<longleftrightarrow> n dvd m" |
|
547 |
by (subst gcd.commute, simp add: gcd_proj1_iff) |
|
548 |
||
549 |
lemma gcd_mod1 [simp]: |
|
550 |
"gcd (x mod y) y = gcd x y" |
|
551 |
by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff) |
|
552 |
||
553 |
lemma gcd_mod2 [simp]: |
|
554 |
"gcd x (y mod x) = gcd x y" |
|
555 |
by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff) |
|
556 |
||
557 |
lemma normalisation_factor_dvd' [simp]: |
|
558 |
"normalisation_factor x dvd x" |
|
559 |
by (cases "x = 0", simp_all) |
|
560 |
||
561 |
lemma gcd_mult_distrib': |
|
562 |
"k div normalisation_factor k * gcd x y = gcd (k*x) (k*y)" |
|
563 |
proof (induct x y rule: gcd_eucl.induct) |
|
564 |
case (1 x y) |
|
565 |
show ?case |
|
566 |
proof (cases "y = 0") |
|
567 |
case True |
|
568 |
then show ?thesis by (simp add: normalisation_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd) |
|
569 |
next |
|
570 |
case False |
|
571 |
hence "k div normalisation_factor k * gcd x y = gcd (k * y) (k * (x mod y))" |
|
572 |
using 1 by (subst gcd_red, simp) |
|
573 |
also have "... = gcd (k * x) (k * y)" |
|
574 |
by (simp add: mult_mod_right gcd.commute) |
|
575 |
finally show ?thesis . |
|
576 |
qed |
|
577 |
qed |
|
578 |
||
579 |
lemma gcd_mult_distrib: |
|
580 |
"k * gcd x y = gcd (k*x) (k*y) * normalisation_factor k" |
|
581 |
proof- |
|
582 |
let ?nf = "normalisation_factor" |
|
583 |
from gcd_mult_distrib' |
|
584 |
have "gcd (k*x) (k*y) = k div ?nf k * gcd x y" .. |
|
585 |
also have "... = k * gcd x y div ?nf k" |
|
586 |
by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalisation_factor_dvd) |
|
587 |
finally show ?thesis |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
588 |
by simp |
58023 | 589 |
qed |
590 |
||
591 |
lemma euclidean_size_gcd_le1 [simp]: |
|
592 |
assumes "a \<noteq> 0" |
|
593 |
shows "euclidean_size (gcd a b) \<le> euclidean_size a" |
|
594 |
proof - |
|
595 |
have "gcd a b dvd a" by (rule gcd_dvd1) |
|
596 |
then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast |
|
597 |
with `a \<noteq> 0` show ?thesis by (subst (2) A, intro size_mult_mono) auto |
|
598 |
qed |
|
599 |
||
600 |
lemma euclidean_size_gcd_le2 [simp]: |
|
601 |
"b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b" |
|
602 |
by (subst gcd.commute, rule euclidean_size_gcd_le1) |
|
603 |
||
604 |
lemma euclidean_size_gcd_less1: |
|
605 |
assumes "a \<noteq> 0" and "\<not>a dvd b" |
|
606 |
shows "euclidean_size (gcd a b) < euclidean_size a" |
|
607 |
proof (rule ccontr) |
|
608 |
assume "\<not>euclidean_size (gcd a b) < euclidean_size a" |
|
609 |
with `a \<noteq> 0` have "euclidean_size (gcd a b) = euclidean_size a" |
|
610 |
by (intro le_antisym, simp_all) |
|
611 |
with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd) |
|
612 |
hence "a dvd b" using dvd_gcd_D2 by blast |
|
613 |
with `\<not>a dvd b` show False by contradiction |
|
614 |
qed |
|
615 |
||
616 |
lemma euclidean_size_gcd_less2: |
|
617 |
assumes "b \<noteq> 0" and "\<not>b dvd a" |
|
618 |
shows "euclidean_size (gcd a b) < euclidean_size b" |
|
619 |
using assms by (subst gcd.commute, rule euclidean_size_gcd_less1) |
|
620 |
||
621 |
lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (x*a) y = gcd x y" |
|
622 |
apply (rule gcdI) |
|
623 |
apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps) |
|
624 |
apply (rule gcd_dvd2) |
|
625 |
apply (rule gcd_greatest, simp add: unit_simps, assumption) |
|
626 |
apply (subst normalisation_factor_gcd, simp add: gcd_0) |
|
627 |
done |
|
628 |
||
629 |
lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd x (y*a) = gcd x y" |
|
630 |
by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute) |
|
631 |
||
632 |
lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (x div a) y = gcd x y" |
|
633 |
by (simp add: unit_ring_inv gcd_mult_unit1) |
|
634 |
||
635 |
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd x (y div a) = gcd x y" |
|
636 |
by (simp add: unit_ring_inv gcd_mult_unit2) |
|
637 |
||
638 |
lemma gcd_idem: "gcd x x = x div normalisation_factor x" |
|
639 |
by (cases "x = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all) |
|
640 |
||
641 |
lemma gcd_right_idem: "gcd (gcd p q) q = gcd p q" |
|
642 |
apply (rule gcdI) |
|
643 |
apply (simp add: ac_simps) |
|
644 |
apply (rule gcd_dvd2) |
|
645 |
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
|
646 |
apply simp |
58023 | 647 |
done |
648 |
||
649 |
lemma gcd_left_idem: "gcd p (gcd p q) = gcd p q" |
|
650 |
apply (rule gcdI) |
|
651 |
apply simp |
|
652 |
apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2) |
|
653 |
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
|
654 |
apply simp |
58023 | 655 |
done |
656 |
||
657 |
lemma comp_fun_idem_gcd: "comp_fun_idem gcd" |
|
658 |
proof |
|
659 |
fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a" |
|
660 |
by (simp add: fun_eq_iff ac_simps) |
|
661 |
next |
|
662 |
fix a show "gcd a \<circ> gcd a = gcd a" |
|
663 |
by (simp add: fun_eq_iff gcd_left_idem) |
|
664 |
qed |
|
665 |
||
666 |
lemma coprime_dvd_mult: |
|
667 |
assumes "gcd k n = 1" and "k dvd m * n" |
|
668 |
shows "k dvd m" |
|
669 |
proof - |
|
670 |
let ?nf = "normalisation_factor" |
|
671 |
from assms gcd_mult_distrib [of m k n] |
|
672 |
have A: "m = gcd (m * k) (m * n) * ?nf m" by simp |
|
673 |
from `k dvd m * n` show ?thesis by (subst A, simp_all add: gcd_greatest) |
|
674 |
qed |
|
675 |
||
676 |
lemma coprime_dvd_mult_iff: |
|
677 |
"gcd k n = 1 \<Longrightarrow> (k dvd m * n) = (k dvd m)" |
|
678 |
by (rule, rule coprime_dvd_mult, simp_all) |
|
679 |
||
680 |
lemma gcd_dvd_antisym: |
|
681 |
"gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d" |
|
682 |
proof (rule gcdI) |
|
683 |
assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b" |
|
684 |
have "gcd c d dvd c" by simp |
|
685 |
with A show "gcd a b dvd c" by (rule dvd_trans) |
|
686 |
have "gcd c d dvd d" by simp |
|
687 |
with A show "gcd a b dvd d" by (rule dvd_trans) |
|
688 |
show "normalisation_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
|
689 |
by simp |
58023 | 690 |
fix l assume "l dvd c" and "l dvd d" |
691 |
hence "l dvd gcd c d" by (rule gcd_greatest) |
|
692 |
from this and B show "l dvd gcd a b" by (rule dvd_trans) |
|
693 |
qed |
|
694 |
||
695 |
lemma gcd_mult_cancel: |
|
696 |
assumes "gcd k n = 1" |
|
697 |
shows "gcd (k * m) n = gcd m n" |
|
698 |
proof (rule gcd_dvd_antisym) |
|
699 |
have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps) |
|
700 |
also note `gcd k n = 1` |
|
701 |
finally have "gcd (gcd (k * m) n) k = 1" by simp |
|
702 |
hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps) |
|
703 |
moreover have "gcd (k * m) n dvd n" by simp |
|
704 |
ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest) |
|
705 |
have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all |
|
706 |
then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest) |
|
707 |
qed |
|
708 |
||
709 |
lemma coprime_crossproduct: |
|
710 |
assumes [simp]: "gcd a d = 1" "gcd b c = 1" |
|
711 |
shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs") |
|
712 |
proof |
|
713 |
assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono) |
|
714 |
next |
|
715 |
assume ?lhs |
|
716 |
from `?lhs` have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left) |
|
717 |
hence "a dvd b" by (simp add: coprime_dvd_mult_iff) |
|
718 |
moreover from `?lhs` have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left) |
|
719 |
hence "b dvd a" by (simp add: coprime_dvd_mult_iff) |
|
720 |
moreover from `?lhs` have "c dvd d * b" |
|
59009
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generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
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58953
diff
changeset
|
721 |
unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps) |
58023 | 722 |
hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute) |
723 |
moreover from `?lhs` have "d dvd c * a" |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
724 |
unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps) |
58023 | 725 |
hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute) |
726 |
ultimately show ?rhs unfolding associated_def by simp |
|
727 |
qed |
|
728 |
||
729 |
lemma gcd_add1 [simp]: |
|
730 |
"gcd (m + n) n = gcd m n" |
|
731 |
by (cases "n = 0", simp_all add: gcd_non_0) |
|
732 |
||
733 |
lemma gcd_add2 [simp]: |
|
734 |
"gcd m (m + n) = gcd m n" |
|
735 |
using gcd_add1 [of n m] by (simp add: ac_simps) |
|
736 |
||
737 |
lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n" |
|
738 |
by (subst gcd.commute, subst gcd_red, simp) |
|
739 |
||
740 |
lemma coprimeI: "(\<And>l. \<lbrakk>l dvd x; l dvd y\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd x y = 1" |
|
741 |
by (rule sym, rule gcdI, simp_all) |
|
742 |
||
743 |
lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)" |
|
59061 | 744 |
by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2) |
58023 | 745 |
|
746 |
lemma div_gcd_coprime: |
|
747 |
assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0" |
|
748 |
defines [simp]: "d \<equiv> gcd a b" |
|
749 |
defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d" |
|
750 |
shows "gcd a' b' = 1" |
|
751 |
proof (rule coprimeI) |
|
752 |
fix l assume "l dvd a'" "l dvd b'" |
|
753 |
then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast |
|
59009
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parents:
58953
diff
changeset
|
754 |
moreover have "a = a' * d" "b = b' * d" by simp_all |
58023 | 755 |
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
|
756 |
by (simp_all only: ac_simps) |
58023 | 757 |
hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left) |
758 |
hence "l*d dvd d" by (simp add: gcd_greatest) |
|
59009
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generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
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58953
diff
changeset
|
759 |
then obtain u where "d = l * d * u" .. |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
760 |
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
|
761 |
moreover from nz have "d \<noteq> 0" by simp |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
762 |
with div_mult_self1_is_id have "d * (l * u) div d = l * u" . |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
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parents:
58953
diff
changeset
|
763 |
ultimately have "1 = l * u" |
348561aa3869
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parents:
58953
diff
changeset
|
764 |
using `d \<noteq> 0` by simp |
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haftmann
parents:
58953
diff
changeset
|
765 |
then show "l dvd 1" .. |
58023 | 766 |
qed |
767 |
||
768 |
lemma coprime_mult: |
|
769 |
assumes da: "gcd d a = 1" and db: "gcd d b = 1" |
|
770 |
shows "gcd d (a * b) = 1" |
|
771 |
apply (subst gcd.commute) |
|
772 |
using da apply (subst gcd_mult_cancel) |
|
773 |
apply (subst gcd.commute, assumption) |
|
774 |
apply (subst gcd.commute, rule db) |
|
775 |
done |
|
776 |
||
777 |
lemma coprime_lmult: |
|
778 |
assumes dab: "gcd d (a * b) = 1" |
|
779 |
shows "gcd d a = 1" |
|
780 |
proof (rule coprimeI) |
|
781 |
fix l assume "l dvd d" and "l dvd a" |
|
782 |
hence "l dvd a * b" by simp |
|
783 |
with `l dvd d` and dab show "l dvd 1" by (auto intro: gcd_greatest) |
|
784 |
qed |
|
785 |
||
786 |
lemma coprime_rmult: |
|
787 |
assumes dab: "gcd d (a * b) = 1" |
|
788 |
shows "gcd d b = 1" |
|
789 |
proof (rule coprimeI) |
|
790 |
fix l assume "l dvd d" and "l dvd b" |
|
791 |
hence "l dvd a * b" by simp |
|
792 |
with `l dvd d` and dab show "l dvd 1" by (auto intro: gcd_greatest) |
|
793 |
qed |
|
794 |
||
795 |
lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1" |
|
796 |
using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast |
|
797 |
||
798 |
lemma gcd_coprime: |
|
799 |
assumes z: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b" |
|
800 |
shows "gcd a' b' = 1" |
|
801 |
proof - |
|
59009
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haftmann
parents:
58953
diff
changeset
|
802 |
from z have "a \<noteq> 0 \<or> b \<noteq> 0" by simp |
58023 | 803 |
with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" . |
804 |
also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+ |
|
805 |
also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+ |
|
806 |
finally show ?thesis . |
|
807 |
qed |
|
808 |
||
809 |
lemma coprime_power: |
|
810 |
assumes "0 < n" |
|
811 |
shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1" |
|
812 |
using assms proof (induct n) |
|
813 |
case (Suc n) then show ?case |
|
814 |
by (cases n) (simp_all add: coprime_mul_eq) |
|
815 |
qed simp |
|
816 |
||
817 |
lemma gcd_coprime_exists: |
|
818 |
assumes nz: "gcd a b \<noteq> 0" |
|
819 |
shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1" |
|
820 |
apply (rule_tac x = "a div gcd a b" in exI) |
|
821 |
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
|
822 |
apply (insert nz, auto intro: div_gcd_coprime) |
58023 | 823 |
done |
824 |
||
825 |
lemma coprime_exp: |
|
826 |
"gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1" |
|
827 |
by (induct n, simp_all add: coprime_mult) |
|
828 |
||
829 |
lemma coprime_exp2 [intro]: |
|
830 |
"gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1" |
|
831 |
apply (rule coprime_exp) |
|
832 |
apply (subst gcd.commute) |
|
833 |
apply (rule coprime_exp) |
|
834 |
apply (subst gcd.commute) |
|
835 |
apply assumption |
|
836 |
done |
|
837 |
||
838 |
lemma gcd_exp: |
|
839 |
"gcd (a^n) (b^n) = (gcd a b) ^ n" |
|
840 |
proof (cases "a = 0 \<and> b = 0") |
|
841 |
assume "a = 0 \<and> b = 0" |
|
842 |
then show ?thesis by (cases n, simp_all add: gcd_0_left) |
|
843 |
next |
|
844 |
assume A: "\<not>(a = 0 \<and> b = 0)" |
|
845 |
hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)" |
|
846 |
using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime) |
|
847 |
hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp |
|
848 |
also note gcd_mult_distrib |
|
849 |
also have "normalisation_factor ((gcd a b)^n) = 1" |
|
850 |
by (simp add: normalisation_factor_pow A) |
|
851 |
also have "(gcd a b)^n * (a div gcd a b)^n = a^n" |
|
852 |
by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp) |
|
853 |
also have "(gcd a b)^n * (b div gcd a b)^n = b^n" |
|
854 |
by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp) |
|
855 |
finally show ?thesis by simp |
|
856 |
qed |
|
857 |
||
858 |
lemma coprime_common_divisor: |
|
859 |
"gcd a b = 1 \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> is_unit x" |
|
860 |
apply (subgoal_tac "x dvd gcd a b") |
|
59061 | 861 |
apply simp |
58023 | 862 |
apply (erule (1) gcd_greatest) |
863 |
done |
|
864 |
||
865 |
lemma division_decomp: |
|
866 |
assumes dc: "a dvd b * c" |
|
867 |
shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c" |
|
868 |
proof (cases "gcd a b = 0") |
|
869 |
assume "gcd a b = 0" |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
870 |
hence "a = 0 \<and> b = 0" by simp |
58023 | 871 |
hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp |
872 |
then show ?thesis by blast |
|
873 |
next |
|
874 |
let ?d = "gcd a b" |
|
875 |
assume "?d \<noteq> 0" |
|
876 |
from gcd_coprime_exists[OF this] |
|
877 |
obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1" |
|
878 |
by blast |
|
879 |
from ab'(1) have "a' dvd a" unfolding dvd_def by blast |
|
880 |
with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp |
|
881 |
from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp |
|
882 |
hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac) |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
883 |
with `?d \<noteq> 0` have "a' dvd b' * c" by simp |
58023 | 884 |
with coprime_dvd_mult[OF ab'(3)] |
885 |
have "a' dvd c" by (subst (asm) ac_simps, blast) |
|
886 |
with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac) |
|
887 |
then show ?thesis by blast |
|
888 |
qed |
|
889 |
||
890 |
lemma pow_divides_pow: |
|
891 |
assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0" |
|
892 |
shows "a dvd b" |
|
893 |
proof (cases "gcd a b = 0") |
|
894 |
assume "gcd a b = 0" |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
895 |
then show ?thesis by simp |
58023 | 896 |
next |
897 |
let ?d = "gcd a b" |
|
898 |
assume "?d \<noteq> 0" |
|
899 |
from n obtain m where m: "n = Suc m" by (cases n, simp_all) |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
900 |
from `?d \<noteq> 0` have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero) |
58023 | 901 |
from gcd_coprime_exists[OF `?d \<noteq> 0`] |
902 |
obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1" |
|
903 |
by blast |
|
904 |
from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n" |
|
905 |
by (simp add: ab'(1,2)[symmetric]) |
|
906 |
hence "?d^n * a'^n dvd ?d^n * b'^n" |
|
907 |
by (simp only: power_mult_distrib ac_simps) |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
908 |
with zn have "a'^n dvd b'^n" by simp |
58023 | 909 |
hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m) |
910 |
hence "a' dvd b'^m * b'" by (simp add: m ac_simps) |
|
911 |
with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]] |
|
912 |
have "a' dvd b'" by (subst (asm) ac_simps, blast) |
|
913 |
hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp) |
|
914 |
with ab'(1,2) show ?thesis by simp |
|
915 |
qed |
|
916 |
||
917 |
lemma pow_divides_eq [simp]: |
|
918 |
"n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b" |
|
919 |
by (auto intro: pow_divides_pow dvd_power_same) |
|
920 |
||
921 |
lemma divides_mult: |
|
922 |
assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1" |
|
923 |
shows "m * n dvd r" |
|
924 |
proof - |
|
925 |
from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'" |
|
926 |
unfolding dvd_def by blast |
|
927 |
from mr n' have "m dvd n'*n" by (simp add: ac_simps) |
|
928 |
hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp |
|
929 |
then obtain k where k: "n' = m*k" unfolding dvd_def by blast |
|
930 |
with n' have "r = m * n * k" by (simp add: mult_ac) |
|
931 |
then show ?thesis unfolding dvd_def by blast |
|
932 |
qed |
|
933 |
||
934 |
lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1" |
|
935 |
by (subst add_commute, simp) |
|
936 |
||
937 |
lemma setprod_coprime [rule_format]: |
|
938 |
"(\<forall>i\<in>A. gcd (f i) x = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) x = 1" |
|
939 |
apply (cases "finite A") |
|
940 |
apply (induct set: finite) |
|
941 |
apply (auto simp add: gcd_mult_cancel) |
|
942 |
done |
|
943 |
||
944 |
lemma coprime_divisors: |
|
945 |
assumes "d dvd a" "e dvd b" "gcd a b = 1" |
|
946 |
shows "gcd d e = 1" |
|
947 |
proof - |
|
948 |
from assms obtain k l where "a = d * k" "b = e * l" |
|
949 |
unfolding dvd_def by blast |
|
950 |
with assms have "gcd (d * k) (e * l) = 1" by simp |
|
951 |
hence "gcd (d * k) e = 1" by (rule coprime_lmult) |
|
952 |
also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps) |
|
953 |
finally have "gcd e d = 1" by (rule coprime_lmult) |
|
954 |
then show ?thesis by (simp add: ac_simps) |
|
955 |
qed |
|
956 |
||
957 |
lemma invertible_coprime: |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
958 |
assumes "x * y mod m = 1" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
959 |
shows "coprime x m" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
960 |
proof - |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
961 |
from assms have "coprime m (x * y mod m)" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
962 |
by simp |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
963 |
then have "coprime m (x * y)" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
964 |
by simp |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
965 |
then have "coprime m x" |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
966 |
by (rule coprime_lmult) |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
967 |
then show ?thesis |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
968 |
by (simp add: ac_simps) |
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
969 |
qed |
58023 | 970 |
|
971 |
lemma lcm_gcd: |
|
972 |
"lcm a b = a * b div (gcd a b * normalisation_factor (a*b))" |
|
973 |
by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def) |
|
974 |
||
975 |
lemma lcm_gcd_prod: |
|
976 |
"lcm a b * gcd a b = a * b div normalisation_factor (a*b)" |
|
977 |
proof (cases "a * b = 0") |
|
978 |
let ?nf = normalisation_factor |
|
979 |
assume "a * b \<noteq> 0" |
|
58953 | 980 |
hence "gcd a b \<noteq> 0" by simp |
58023 | 981 |
from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))" |
982 |
by (simp add: mult_ac) |
|
983 |
also from `a * b \<noteq> 0` have "... = a * b div ?nf (a*b)" |
|
58953 | 984 |
by (simp_all add: unit_ring_inv'1 unit_ring_inv) |
58023 | 985 |
finally show ?thesis . |
58953 | 986 |
qed (auto simp add: lcm_gcd) |
58023 | 987 |
|
988 |
lemma lcm_dvd1 [iff]: |
|
989 |
"x dvd lcm x y" |
|
990 |
proof (cases "x*y = 0") |
|
991 |
assume "x * y \<noteq> 0" |
|
58953 | 992 |
hence "gcd x y \<noteq> 0" by simp |
58023 | 993 |
let ?c = "ring_inv (normalisation_factor (x*y))" |
994 |
from `x * y \<noteq> 0` have [simp]: "is_unit (normalisation_factor (x*y))" by simp |
|
995 |
from lcm_gcd_prod[of x y] have "lcm x y * gcd x y = x * ?c * y" |
|
996 |
by (simp add: mult_ac unit_ring_inv) |
|
997 |
hence "lcm x y * gcd x y div gcd x y = x * ?c * y div gcd x y" by simp |
|
998 |
with `gcd x y \<noteq> 0` have "lcm x y = x * ?c * y div gcd x y" |
|
999 |
by (subst (asm) div_mult_self2_is_id, simp_all) |
|
1000 |
also have "... = x * (?c * y div gcd x y)" |
|
1001 |
by (metis div_mult_swap gcd_dvd2 mult_assoc) |
|
1002 |
finally show ?thesis by (rule dvdI) |
|
58953 | 1003 |
qed (auto simp add: lcm_gcd) |
58023 | 1004 |
|
1005 |
lemma lcm_least: |
|
1006 |
"\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k" |
|
1007 |
proof (cases "k = 0") |
|
1008 |
let ?nf = normalisation_factor |
|
1009 |
assume "k \<noteq> 0" |
|
1010 |
hence "is_unit (?nf k)" by simp |
|
1011 |
hence "?nf k \<noteq> 0" by (metis not_is_unit_0) |
|
1012 |
assume A: "a dvd k" "b dvd k" |
|
58953 | 1013 |
hence "gcd a b \<noteq> 0" using `k \<noteq> 0` by auto |
58023 | 1014 |
from A obtain r s where ar: "k = a * r" and bs: "k = b * s" |
1015 |
unfolding dvd_def by blast |
|
58953 | 1016 |
with `k \<noteq> 0` have "r * s \<noteq> 0" |
1017 |
by auto (drule sym [of 0], simp) |
|
58023 | 1018 |
hence "is_unit (?nf (r * s))" by simp |
1019 |
let ?c = "?nf k div ?nf (r*s)" |
|
1020 |
from `is_unit (?nf k)` and `is_unit (?nf (r * s))` have "is_unit ?c" by (rule unit_div) |
|
1021 |
hence "?c \<noteq> 0" using not_is_unit_0 by fast |
|
1022 |
from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)" |
|
58953 | 1023 |
by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps) |
58023 | 1024 |
also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)" |
1025 |
by (subst (3) `k = a * r`, subst (3) `k = b * s`, simp add: algebra_simps) |
|
1026 |
also have "... = ?c * r*s * k * gcd a b" using `r * s \<noteq> 0` |
|
1027 |
by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps) |
|
1028 |
finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b" |
|
1029 |
by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac) |
|
1030 |
hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)" |
|
1031 |
by (simp add: algebra_simps) |
|
1032 |
hence "?c * k * gcd a b = a * b * gcd s r" using `r * s \<noteq> 0` |
|
1033 |
by (metis div_mult_self2_is_id) |
|
1034 |
also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)" |
|
1035 |
by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib') |
|
1036 |
also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b" |
|
1037 |
by (simp add: algebra_simps) |
|
1038 |
finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using `gcd a b \<noteq> 0` |
|
1039 |
by (metis mult.commute div_mult_self2_is_id) |
|
1040 |
hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using `?c \<noteq> 0` |
|
1041 |
by (metis div_mult_self2_is_id mult_assoc) |
|
1042 |
also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using `is_unit ?c` |
|
1043 |
by (simp add: unit_simps) |
|
1044 |
finally show ?thesis by (rule dvdI) |
|
1045 |
qed simp |
|
1046 |
||
1047 |
lemma lcm_zero: |
|
1048 |
"lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" |
|
1049 |
proof - |
|
1050 |
let ?nf = normalisation_factor |
|
1051 |
{ |
|
1052 |
assume "a \<noteq> 0" "b \<noteq> 0" |
|
1053 |
hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors) |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
1054 |
moreover from `a \<noteq> 0` and `b \<noteq> 0` have "gcd a b \<noteq> 0" by simp |
58023 | 1055 |
ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp) |
1056 |
} moreover { |
|
1057 |
assume "a = 0 \<or> b = 0" |
|
1058 |
hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd) |
|
1059 |
} |
|
1060 |
ultimately show ?thesis by blast |
|
1061 |
qed |
|
1062 |
||
1063 |
lemmas lcm_0_iff = lcm_zero |
|
1064 |
||
1065 |
lemma gcd_lcm: |
|
1066 |
assumes "lcm a b \<noteq> 0" |
|
1067 |
shows "gcd a b = a * b div (lcm a b * normalisation_factor (a * b))" |
|
1068 |
proof- |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
1069 |
from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero) |
58023 | 1070 |
let ?c = "normalisation_factor (a*b)" |
1071 |
from `lcm a b \<noteq> 0` have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors) |
|
1072 |
hence "is_unit ?c" by simp |
|
1073 |
from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b" |
|
1074 |
by (subst (2) div_mult_self2_is_id[OF `lcm a b \<noteq> 0`, symmetric], simp add: mult_ac) |
|
1075 |
also from `is_unit ?c` have "... = a * b div (?c * lcm a b)" |
|
1076 |
by (simp only: unit_ring_inv'1 unit_ring_inv) |
|
1077 |
finally show ?thesis by (simp only: ac_simps) |
|
1078 |
qed |
|
1079 |
||
1080 |
lemma normalisation_factor_lcm [simp]: |
|
1081 |
"normalisation_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)" |
|
1082 |
proof (cases "a = 0 \<or> b = 0") |
|
1083 |
case True then show ?thesis |
|
58953 | 1084 |
by (auto simp add: lcm_gcd) |
58023 | 1085 |
next |
1086 |
case False |
|
1087 |
let ?nf = normalisation_factor |
|
1088 |
from lcm_gcd_prod[of a b] |
|
1089 |
have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)" |
|
1090 |
by (metis div_by_0 div_self normalisation_correct normalisation_factor_0 normalisation_factor_mult) |
|
1091 |
also have "... = (if a*b = 0 then 0 else 1)" |
|
58953 | 1092 |
by simp |
1093 |
finally show ?thesis using False by simp |
|
58023 | 1094 |
qed |
1095 |
||
1096 |
lemma lcm_dvd2 [iff]: "y dvd lcm x y" |
|
1097 |
using lcm_dvd1 [of y x] by (simp add: lcm_gcd ac_simps) |
|
1098 |
||
1099 |
lemma lcmI: |
|
1100 |
"\<lbrakk>x dvd k; y dvd k; \<And>l. x dvd l \<Longrightarrow> y dvd l \<Longrightarrow> k dvd l; |
|
1101 |
normalisation_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm x y" |
|
1102 |
by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least) |
|
1103 |
||
1104 |
sublocale lcm!: abel_semigroup lcm |
|
1105 |
proof |
|
1106 |
fix x y z |
|
1107 |
show "lcm (lcm x y) z = lcm x (lcm y z)" |
|
1108 |
proof (rule lcmI) |
|
1109 |
have "x dvd lcm x y" and "lcm x y dvd lcm (lcm x y) z" by simp_all |
|
1110 |
then show "x dvd lcm (lcm x y) z" by (rule dvd_trans) |
|
1111 |
||
1112 |
have "y dvd lcm x y" and "lcm x y dvd lcm (lcm x y) z" by simp_all |
|
1113 |
hence "y dvd lcm (lcm x y) z" by (rule dvd_trans) |
|
1114 |
moreover have "z dvd lcm (lcm x y) z" by simp |
|
1115 |
ultimately show "lcm y z dvd lcm (lcm x y) z" by (rule lcm_least) |
|
1116 |
||
1117 |
fix l assume "x dvd l" and "lcm y z dvd l" |
|
1118 |
have "y dvd lcm y z" by simp |
|
1119 |
from this and `lcm y z dvd l` have "y dvd l" by (rule dvd_trans) |
|
1120 |
have "z dvd lcm y z" by simp |
|
1121 |
from this and `lcm y z dvd l` have "z dvd l" by (rule dvd_trans) |
|
1122 |
from `x dvd l` and `y dvd l` have "lcm x y dvd l" by (rule lcm_least) |
|
1123 |
from this and `z dvd l` show "lcm (lcm x y) z dvd l" by (rule lcm_least) |
|
1124 |
qed (simp add: lcm_zero) |
|
1125 |
next |
|
1126 |
fix x y |
|
1127 |
show "lcm x y = lcm y x" |
|
1128 |
by (simp add: lcm_gcd ac_simps) |
|
1129 |
qed |
|
1130 |
||
1131 |
lemma dvd_lcm_D1: |
|
1132 |
"lcm m n dvd k \<Longrightarrow> m dvd k" |
|
1133 |
by (rule dvd_trans, rule lcm_dvd1, assumption) |
|
1134 |
||
1135 |
lemma dvd_lcm_D2: |
|
1136 |
"lcm m n dvd k \<Longrightarrow> n dvd k" |
|
1137 |
by (rule dvd_trans, rule lcm_dvd2, assumption) |
|
1138 |
||
1139 |
lemma gcd_dvd_lcm [simp]: |
|
1140 |
"gcd a b dvd lcm a b" |
|
1141 |
by (metis dvd_trans gcd_dvd2 lcm_dvd2) |
|
1142 |
||
1143 |
lemma lcm_1_iff: |
|
1144 |
"lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b" |
|
1145 |
proof |
|
1146 |
assume "lcm a b = 1" |
|
59061 | 1147 |
then show "is_unit a \<and> is_unit b" by auto |
58023 | 1148 |
next |
1149 |
assume "is_unit a \<and> is_unit b" |
|
59061 | 1150 |
hence "a dvd 1" and "b dvd 1" by simp_all |
1151 |
hence "is_unit (lcm a b)" by (rule lcm_least) |
|
58023 | 1152 |
hence "lcm a b = normalisation_factor (lcm a b)" |
1153 |
by (subst normalisation_factor_unit, simp_all) |
|
59061 | 1154 |
also have "\<dots> = 1" using `is_unit a \<and> is_unit b` |
1155 |
by auto |
|
58023 | 1156 |
finally show "lcm a b = 1" . |
1157 |
qed |
|
1158 |
||
1159 |
lemma lcm_0_left [simp]: |
|
1160 |
"lcm 0 x = 0" |
|
1161 |
by (rule sym, rule lcmI, simp_all) |
|
1162 |
||
1163 |
lemma lcm_0 [simp]: |
|
1164 |
"lcm x 0 = 0" |
|
1165 |
by (rule sym, rule lcmI, simp_all) |
|
1166 |
||
1167 |
lemma lcm_unique: |
|
1168 |
"a dvd d \<and> b dvd d \<and> |
|
1169 |
normalisation_factor d = (if d = 0 then 0 else 1) \<and> |
|
1170 |
(\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b" |
|
1171 |
by (rule, auto intro: lcmI simp: lcm_least lcm_zero) |
|
1172 |
||
1173 |
lemma dvd_lcm_I1 [simp]: |
|
1174 |
"k dvd m \<Longrightarrow> k dvd lcm m n" |
|
1175 |
by (metis lcm_dvd1 dvd_trans) |
|
1176 |
||
1177 |
lemma dvd_lcm_I2 [simp]: |
|
1178 |
"k dvd n \<Longrightarrow> k dvd lcm m n" |
|
1179 |
by (metis lcm_dvd2 dvd_trans) |
|
1180 |
||
1181 |
lemma lcm_1_left [simp]: |
|
1182 |
"lcm 1 x = x div normalisation_factor x" |
|
1183 |
by (cases "x = 0") (simp, rule sym, rule lcmI, simp_all) |
|
1184 |
||
1185 |
lemma lcm_1_right [simp]: |
|
1186 |
"lcm x 1 = x div normalisation_factor x" |
|
1187 |
by (simp add: ac_simps) |
|
1188 |
||
1189 |
lemma lcm_coprime: |
|
1190 |
"gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalisation_factor (a*b)" |
|
1191 |
by (subst lcm_gcd) simp |
|
1192 |
||
1193 |
lemma lcm_proj1_if_dvd: |
|
1194 |
"y dvd x \<Longrightarrow> lcm x y = x div normalisation_factor x" |
|
1195 |
by (cases "x = 0") (simp, rule sym, rule lcmI, simp_all) |
|
1196 |
||
1197 |
lemma lcm_proj2_if_dvd: |
|
1198 |
"x dvd y \<Longrightarrow> lcm x y = y div normalisation_factor y" |
|
1199 |
using lcm_proj1_if_dvd [of x y] by (simp add: ac_simps) |
|
1200 |
||
1201 |
lemma lcm_proj1_iff: |
|
1202 |
"lcm m n = m div normalisation_factor m \<longleftrightarrow> n dvd m" |
|
1203 |
proof |
|
1204 |
assume A: "lcm m n = m div normalisation_factor m" |
|
1205 |
show "n dvd m" |
|
1206 |
proof (cases "m = 0") |
|
1207 |
assume [simp]: "m \<noteq> 0" |
|
1208 |
from A have B: "m = lcm m n * normalisation_factor m" |
|
1209 |
by (simp add: unit_eq_div2) |
|
1210 |
show ?thesis by (subst B, simp) |
|
1211 |
qed simp |
|
1212 |
next |
|
1213 |
assume "n dvd m" |
|
1214 |
then show "lcm m n = m div normalisation_factor m" by (rule lcm_proj1_if_dvd) |
|
1215 |
qed |
|
1216 |
||
1217 |
lemma lcm_proj2_iff: |
|
1218 |
"lcm m n = n div normalisation_factor n \<longleftrightarrow> m dvd n" |
|
1219 |
using lcm_proj1_iff [of n m] by (simp add: ac_simps) |
|
1220 |
||
1221 |
lemma euclidean_size_lcm_le1: |
|
1222 |
assumes "a \<noteq> 0" and "b \<noteq> 0" |
|
1223 |
shows "euclidean_size a \<le> euclidean_size (lcm a b)" |
|
1224 |
proof - |
|
1225 |
have "a dvd lcm a b" by (rule lcm_dvd1) |
|
1226 |
then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast |
|
1227 |
with `a \<noteq> 0` and `b \<noteq> 0` have "c \<noteq> 0" by (auto simp: lcm_zero) |
|
1228 |
then show ?thesis by (subst A, intro size_mult_mono) |
|
1229 |
qed |
|
1230 |
||
1231 |
lemma euclidean_size_lcm_le2: |
|
1232 |
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)" |
|
1233 |
using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps) |
|
1234 |
||
1235 |
lemma euclidean_size_lcm_less1: |
|
1236 |
assumes "b \<noteq> 0" and "\<not>b dvd a" |
|
1237 |
shows "euclidean_size a < euclidean_size (lcm a b)" |
|
1238 |
proof (rule ccontr) |
|
1239 |
from assms have "a \<noteq> 0" by auto |
|
1240 |
assume "\<not>euclidean_size a < euclidean_size (lcm a b)" |
|
1241 |
with `a \<noteq> 0` and `b \<noteq> 0` have "euclidean_size (lcm a b) = euclidean_size a" |
|
1242 |
by (intro le_antisym, simp, intro euclidean_size_lcm_le1) |
|
1243 |
with assms have "lcm a b dvd a" |
|
1244 |
by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero) |
|
1245 |
hence "b dvd a" by (rule dvd_lcm_D2) |
|
1246 |
with `\<not>b dvd a` show False by contradiction |
|
1247 |
qed |
|
1248 |
||
1249 |
lemma euclidean_size_lcm_less2: |
|
1250 |
assumes "a \<noteq> 0" and "\<not>a dvd b" |
|
1251 |
shows "euclidean_size b < euclidean_size (lcm a b)" |
|
1252 |
using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps) |
|
1253 |
||
1254 |
lemma lcm_mult_unit1: |
|
1255 |
"is_unit a \<Longrightarrow> lcm (x*a) y = lcm x y" |
|
1256 |
apply (rule lcmI) |
|
1257 |
apply (rule dvd_trans[of _ "x*a"], simp, rule lcm_dvd1) |
|
1258 |
apply (rule lcm_dvd2) |
|
1259 |
apply (rule lcm_least, simp add: unit_simps, assumption) |
|
1260 |
apply (subst normalisation_factor_lcm, simp add: lcm_zero) |
|
1261 |
done |
|
1262 |
||
1263 |
lemma lcm_mult_unit2: |
|
1264 |
"is_unit a \<Longrightarrow> lcm x (y*a) = lcm x y" |
|
1265 |
using lcm_mult_unit1 [of a y x] by (simp add: ac_simps) |
|
1266 |
||
1267 |
lemma lcm_div_unit1: |
|
1268 |
"is_unit a \<Longrightarrow> lcm (x div a) y = lcm x y" |
|
1269 |
by (simp add: unit_ring_inv lcm_mult_unit1) |
|
1270 |
||
1271 |
lemma lcm_div_unit2: |
|
1272 |
"is_unit a \<Longrightarrow> lcm x (y div a) = lcm x y" |
|
1273 |
by (simp add: unit_ring_inv lcm_mult_unit2) |
|
1274 |
||
1275 |
lemma lcm_left_idem: |
|
1276 |
"lcm p (lcm p q) = lcm p q" |
|
1277 |
apply (rule lcmI) |
|
1278 |
apply simp |
|
1279 |
apply (subst lcm.assoc [symmetric], rule lcm_dvd2) |
|
1280 |
apply (rule lcm_least, assumption) |
|
1281 |
apply (erule (1) lcm_least) |
|
1282 |
apply (auto simp: lcm_zero) |
|
1283 |
done |
|
1284 |
||
1285 |
lemma lcm_right_idem: |
|
1286 |
"lcm (lcm p q) q = lcm p q" |
|
1287 |
apply (rule lcmI) |
|
1288 |
apply (subst lcm.assoc, rule lcm_dvd1) |
|
1289 |
apply (rule lcm_dvd2) |
|
1290 |
apply (rule lcm_least, erule (1) lcm_least, assumption) |
|
1291 |
apply (auto simp: lcm_zero) |
|
1292 |
done |
|
1293 |
||
1294 |
lemma comp_fun_idem_lcm: "comp_fun_idem lcm" |
|
1295 |
proof |
|
1296 |
fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a" |
|
1297 |
by (simp add: fun_eq_iff ac_simps) |
|
1298 |
next |
|
1299 |
fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def |
|
1300 |
by (intro ext, simp add: lcm_left_idem) |
|
1301 |
qed |
|
1302 |
||
1303 |
lemma dvd_Lcm [simp]: "x \<in> A \<Longrightarrow> x dvd Lcm A" |
|
1304 |
and Lcm_dvd [simp]: "(\<forall>x\<in>A. x dvd l') \<Longrightarrow> Lcm A dvd l'" |
|
1305 |
and normalisation_factor_Lcm [simp]: |
|
1306 |
"normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" |
|
1307 |
proof - |
|
1308 |
have "(\<forall>x\<in>A. x dvd Lcm A) \<and> (\<forall>l'. (\<forall>x\<in>A. x dvd l') \<longrightarrow> Lcm A dvd l') \<and> |
|
1309 |
normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis) |
|
1310 |
proof (cases "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l)") |
|
1311 |
case False |
|
1312 |
hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def) |
|
1313 |
with False show ?thesis by auto |
|
1314 |
next |
|
1315 |
case True |
|
1316 |
then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l\<^sub>0)" by blast |
|
1317 |
def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n" |
|
1318 |
def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n" |
|
1319 |
have "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n" |
|
1320 |
apply (subst n_def) |
|
1321 |
apply (rule LeastI[of _ "euclidean_size l\<^sub>0"]) |
|
1322 |
apply (rule exI[of _ l\<^sub>0]) |
|
1323 |
apply (simp add: l\<^sub>0_props) |
|
1324 |
done |
|
1325 |
from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>x\<in>A. x dvd l" and "euclidean_size l = n" |
|
1326 |
unfolding l_def by simp_all |
|
1327 |
{ |
|
1328 |
fix l' assume "\<forall>x\<in>A. x dvd l'" |
|
1329 |
with `\<forall>x\<in>A. x dvd l` have "\<forall>x\<in>A. x dvd gcd l l'" by (auto intro: gcd_greatest) |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
1330 |
moreover from `l \<noteq> 0` have "gcd l l' \<noteq> 0" by simp |
58023 | 1331 |
ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')" |
1332 |
by (intro exI[of _ "gcd l l'"], auto) |
|
1333 |
hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le) |
|
1334 |
moreover have "euclidean_size (gcd l l') \<le> n" |
|
1335 |
proof - |
|
1336 |
have "gcd l l' dvd l" by simp |
|
1337 |
then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast |
|
1338 |
with `l \<noteq> 0` have "a \<noteq> 0" by auto |
|
1339 |
hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)" |
|
1340 |
by (rule size_mult_mono) |
|
1341 |
also have "gcd l l' * a = l" using `l = gcd l l' * a` .. |
|
1342 |
also note `euclidean_size l = n` |
|
1343 |
finally show "euclidean_size (gcd l l') \<le> n" . |
|
1344 |
qed |
|
1345 |
ultimately have "euclidean_size l = euclidean_size (gcd l l')" |
|
1346 |
by (intro le_antisym, simp_all add: `euclidean_size l = n`) |
|
1347 |
with `l \<noteq> 0` have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd) |
|
1348 |
hence "l dvd l'" by (blast dest: dvd_gcd_D2) |
|
1349 |
} |
|
1350 |
||
1351 |
with `(\<forall>x\<in>A. x dvd l)` and normalisation_factor_is_unit[OF `l \<noteq> 0`] and `l \<noteq> 0` |
|
1352 |
have "(\<forall>x\<in>A. x dvd l div normalisation_factor l) \<and> |
|
1353 |
(\<forall>l'. (\<forall>x\<in>A. x dvd l') \<longrightarrow> l div normalisation_factor l dvd l') \<and> |
|
1354 |
normalisation_factor (l div normalisation_factor l) = |
|
1355 |
(if l div normalisation_factor l = 0 then 0 else 1)" |
|
1356 |
by (auto simp: unit_simps) |
|
1357 |
also from True have "l div normalisation_factor l = Lcm A" |
|
1358 |
by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def) |
|
1359 |
finally show ?thesis . |
|
1360 |
qed |
|
1361 |
note A = this |
|
1362 |
||
1363 |
{fix x assume "x \<in> A" then show "x dvd Lcm A" using A by blast} |
|
1364 |
{fix l' assume "\<forall>x\<in>A. x dvd l'" then show "Lcm A dvd l'" using A by blast} |
|
1365 |
from A show "normalisation_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast |
|
1366 |
qed |
|
1367 |
||
1368 |
lemma LcmI: |
|
1369 |
"(\<And>x. x\<in>A \<Longrightarrow> x dvd l) \<Longrightarrow> (\<And>l'. (\<forall>x\<in>A. x dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow> |
|
1370 |
normalisation_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A" |
|
1371 |
by (intro normed_associated_imp_eq) |
|
1372 |
(auto intro: Lcm_dvd dvd_Lcm simp: associated_def) |
|
1373 |
||
1374 |
lemma Lcm_subset: |
|
1375 |
"A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B" |
|
1376 |
by (blast intro: Lcm_dvd dvd_Lcm) |
|
1377 |
||
1378 |
lemma Lcm_Un: |
|
1379 |
"Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)" |
|
1380 |
apply (rule lcmI) |
|
1381 |
apply (blast intro: Lcm_subset) |
|
1382 |
apply (blast intro: Lcm_subset) |
|
1383 |
apply (intro Lcm_dvd ballI, elim UnE) |
|
1384 |
apply (rule dvd_trans, erule dvd_Lcm, assumption) |
|
1385 |
apply (rule dvd_trans, erule dvd_Lcm, assumption) |
|
1386 |
apply simp |
|
1387 |
done |
|
1388 |
||
1389 |
lemma Lcm_1_iff: |
|
1390 |
"Lcm A = 1 \<longleftrightarrow> (\<forall>x\<in>A. is_unit x)" |
|
1391 |
proof |
|
1392 |
assume "Lcm A = 1" |
|
59061 | 1393 |
then show "\<forall>x\<in>A. is_unit x" by auto |
58023 | 1394 |
qed (rule LcmI [symmetric], auto) |
1395 |
||
1396 |
lemma Lcm_no_units: |
|
1397 |
"Lcm A = Lcm (A - {x. is_unit x})" |
|
1398 |
proof - |
|
1399 |
have "(A - {x. is_unit x}) \<union> {x\<in>A. is_unit x} = A" by blast |
|
1400 |
hence "Lcm A = lcm (Lcm (A - {x. is_unit x})) (Lcm {x\<in>A. is_unit x})" |
|
1401 |
by (simp add: Lcm_Un[symmetric]) |
|
1402 |
also have "Lcm {x\<in>A. is_unit x} = 1" by (simp add: Lcm_1_iff) |
|
1403 |
finally show ?thesis by simp |
|
1404 |
qed |
|
1405 |
||
1406 |
lemma Lcm_empty [simp]: |
|
1407 |
"Lcm {} = 1" |
|
1408 |
by (simp add: Lcm_1_iff) |
|
1409 |
||
1410 |
lemma Lcm_eq_0 [simp]: |
|
1411 |
"0 \<in> A \<Longrightarrow> Lcm A = 0" |
|
1412 |
by (drule dvd_Lcm) simp |
|
1413 |
||
1414 |
lemma Lcm0_iff': |
|
1415 |
"Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l))" |
|
1416 |
proof |
|
1417 |
assume "Lcm A = 0" |
|
1418 |
show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l))" |
|
1419 |
proof |
|
1420 |
assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l)" |
|
1421 |
then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l\<^sub>0)" by blast |
|
1422 |
def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n" |
|
1423 |
def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n" |
|
1424 |
have "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l) \<and> euclidean_size l = n" |
|
1425 |
apply (subst n_def) |
|
1426 |
apply (rule LeastI[of _ "euclidean_size l\<^sub>0"]) |
|
1427 |
apply (rule exI[of _ l\<^sub>0]) |
|
1428 |
apply (simp add: l\<^sub>0_props) |
|
1429 |
done |
|
1430 |
from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all |
|
1431 |
hence "l div normalisation_factor l \<noteq> 0" by simp |
|
1432 |
also from ex have "l div normalisation_factor l = Lcm A" |
|
1433 |
by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def) |
|
1434 |
finally show False using `Lcm A = 0` by contradiction |
|
1435 |
qed |
|
1436 |
qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False) |
|
1437 |
||
1438 |
lemma Lcm0_iff [simp]: |
|
1439 |
"finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A" |
|
1440 |
proof - |
|
1441 |
assume "finite A" |
|
1442 |
have "0 \<in> A \<Longrightarrow> Lcm A = 0" by (intro dvd_0_left dvd_Lcm) |
|
1443 |
moreover { |
|
1444 |
assume "0 \<notin> A" |
|
1445 |
hence "\<Prod>A \<noteq> 0" |
|
1446 |
apply (induct rule: finite_induct[OF `finite A`]) |
|
1447 |
apply simp |
|
1448 |
apply (subst setprod.insert, assumption, assumption) |
|
1449 |
apply (rule no_zero_divisors) |
|
1450 |
apply blast+ |
|
1451 |
done |
|
59010 | 1452 |
moreover from `finite A` have "\<forall>x\<in>A. x dvd \<Prod>A" by blast |
58023 | 1453 |
ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l)" by blast |
1454 |
with Lcm0_iff' have "Lcm A \<noteq> 0" by simp |
|
1455 |
} |
|
1456 |
ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast |
|
1457 |
qed |
|
1458 |
||
1459 |
lemma Lcm_no_multiple: |
|
1460 |
"(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>x\<in>A. \<not>x dvd m)) \<Longrightarrow> Lcm A = 0" |
|
1461 |
proof - |
|
1462 |
assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>x\<in>A. \<not>x dvd m)" |
|
1463 |
hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>x\<in>A. x dvd l))" by blast |
|
1464 |
then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False) |
|
1465 |
qed |
|
1466 |
||
1467 |
lemma Lcm_insert [simp]: |
|
1468 |
"Lcm (insert a A) = lcm a (Lcm A)" |
|
1469 |
proof (rule lcmI) |
|
1470 |
fix l assume "a dvd l" and "Lcm A dvd l" |
|
1471 |
hence "\<forall>x\<in>A. x dvd l" by (blast intro: dvd_trans dvd_Lcm) |
|
1472 |
with `a dvd l` show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd) |
|
1473 |
qed (auto intro: Lcm_dvd dvd_Lcm) |
|
1474 |
||
1475 |
lemma Lcm_finite: |
|
1476 |
assumes "finite A" |
|
1477 |
shows "Lcm A = Finite_Set.fold lcm 1 A" |
|
1478 |
by (induct rule: finite.induct[OF `finite A`]) |
|
1479 |
(simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm]) |
|
1480 |
||
1481 |
lemma Lcm_set [code, code_unfold]: |
|
1482 |
"Lcm (set xs) = fold lcm xs 1" |
|
1483 |
using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps) |
|
1484 |
||
1485 |
lemma Lcm_singleton [simp]: |
|
1486 |
"Lcm {a} = a div normalisation_factor a" |
|
1487 |
by simp |
|
1488 |
||
1489 |
lemma Lcm_2 [simp]: |
|
1490 |
"Lcm {a,b} = lcm a b" |
|
1491 |
by (simp only: Lcm_insert Lcm_empty lcm_1_right) |
|
1492 |
(cases "b = 0", simp, rule lcm_div_unit2, simp) |
|
1493 |
||
1494 |
lemma Lcm_coprime: |
|
1495 |
assumes "finite A" and "A \<noteq> {}" |
|
1496 |
assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1" |
|
1497 |
shows "Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)" |
|
1498 |
using assms proof (induct rule: finite_ne_induct) |
|
1499 |
case (insert a A) |
|
1500 |
have "Lcm (insert a A) = lcm a (Lcm A)" by simp |
|
1501 |
also from insert have "Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)" by blast |
|
1502 |
also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2) |
|
1503 |
also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto |
|
1504 |
with insert have "lcm a (\<Prod>A) = \<Prod>(insert a A) div normalisation_factor (\<Prod>(insert a A))" |
|
1505 |
by (simp add: lcm_coprime) |
|
1506 |
finally show ?case . |
|
1507 |
qed simp |
|
1508 |
||
1509 |
lemma Lcm_coprime': |
|
1510 |
"card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1) |
|
1511 |
\<Longrightarrow> Lcm A = \<Prod>A div normalisation_factor (\<Prod>A)" |
|
1512 |
by (rule Lcm_coprime) (simp_all add: card_eq_0_iff) |
|
1513 |
||
1514 |
lemma Gcd_Lcm: |
|
1515 |
"Gcd A = Lcm {d. \<forall>x\<in>A. d dvd x}" |
|
1516 |
by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def) |
|
1517 |
||
1518 |
lemma Gcd_dvd [simp]: "x \<in> A \<Longrightarrow> Gcd A dvd x" |
|
1519 |
and dvd_Gcd [simp]: "(\<forall>x\<in>A. g' dvd x) \<Longrightarrow> g' dvd Gcd A" |
|
1520 |
and normalisation_factor_Gcd [simp]: |
|
1521 |
"normalisation_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)" |
|
1522 |
proof - |
|
1523 |
fix x assume "x \<in> A" |
|
1524 |
hence "Lcm {d. \<forall>x\<in>A. d dvd x} dvd x" by (intro Lcm_dvd) blast |
|
1525 |
then show "Gcd A dvd x" by (simp add: Gcd_Lcm) |
|
1526 |
next |
|
1527 |
fix g' assume "\<forall>x\<in>A. g' dvd x" |
|
1528 |
hence "g' dvd Lcm {d. \<forall>x\<in>A. d dvd x}" by (intro dvd_Lcm) blast |
|
1529 |
then show "g' dvd Gcd A" by (simp add: Gcd_Lcm) |
|
1530 |
next |
|
1531 |
show "normalisation_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
|
1532 |
by (simp add: Gcd_Lcm) |
58023 | 1533 |
qed |
1534 |
||
1535 |
lemma GcdI: |
|
1536 |
"(\<And>x. x\<in>A \<Longrightarrow> l dvd x) \<Longrightarrow> (\<And>l'. (\<forall>x\<in>A. l' dvd x) \<Longrightarrow> l' dvd l) \<Longrightarrow> |
|
1537 |
normalisation_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Gcd A" |
|
1538 |
by (intro normed_associated_imp_eq) |
|
1539 |
(auto intro: Gcd_dvd dvd_Gcd simp: associated_def) |
|
1540 |
||
1541 |
lemma Lcm_Gcd: |
|
1542 |
"Lcm A = Gcd {m. \<forall>x\<in>A. x dvd m}" |
|
1543 |
by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_dvd) |
|
1544 |
||
1545 |
lemma Gcd_0_iff: |
|
1546 |
"Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}" |
|
1547 |
apply (rule iffI) |
|
1548 |
apply (rule subsetI, drule Gcd_dvd, simp) |
|
1549 |
apply (auto intro: GcdI[symmetric]) |
|
1550 |
done |
|
1551 |
||
1552 |
lemma Gcd_empty [simp]: |
|
1553 |
"Gcd {} = 0" |
|
1554 |
by (simp add: Gcd_0_iff) |
|
1555 |
||
1556 |
lemma Gcd_1: |
|
1557 |
"1 \<in> A \<Longrightarrow> Gcd A = 1" |
|
1558 |
by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd) |
|
1559 |
||
1560 |
lemma Gcd_insert [simp]: |
|
1561 |
"Gcd (insert a A) = gcd a (Gcd A)" |
|
1562 |
proof (rule gcdI) |
|
1563 |
fix l assume "l dvd a" and "l dvd Gcd A" |
|
1564 |
hence "\<forall>x\<in>A. l dvd x" by (blast intro: dvd_trans Gcd_dvd) |
|
1565 |
with `l dvd a` show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd) |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
1566 |
qed auto |
58023 | 1567 |
|
1568 |
lemma Gcd_finite: |
|
1569 |
assumes "finite A" |
|
1570 |
shows "Gcd A = Finite_Set.fold gcd 0 A" |
|
1571 |
by (induct rule: finite.induct[OF `finite A`]) |
|
1572 |
(simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd]) |
|
1573 |
||
1574 |
lemma Gcd_set [code, code_unfold]: |
|
1575 |
"Gcd (set xs) = fold gcd xs 0" |
|
1576 |
using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps) |
|
1577 |
||
1578 |
lemma Gcd_singleton [simp]: "Gcd {a} = a div normalisation_factor a" |
|
1579 |
by (simp add: gcd_0) |
|
1580 |
||
1581 |
lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b" |
|
1582 |
by (simp only: Gcd_insert Gcd_empty gcd_0) (cases "b = 0", simp, rule gcd_div_unit2, simp) |
|
1583 |
||
1584 |
end |
|
1585 |
||
1586 |
text {* |
|
1587 |
A Euclidean ring is a Euclidean semiring with additive inverses. It provides a |
|
1588 |
few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring. |
|
1589 |
*} |
|
1590 |
||
1591 |
class euclidean_ring = euclidean_semiring + idom |
|
1592 |
||
1593 |
class euclidean_ring_gcd = euclidean_semiring_gcd + idom |
|
1594 |
begin |
|
1595 |
||
1596 |
subclass euclidean_ring .. |
|
1597 |
||
1598 |
lemma gcd_neg1 [simp]: |
|
1599 |
"gcd (-x) y = gcd x y" |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
1600 |
by (rule sym, rule gcdI, simp_all add: gcd_greatest) |
58023 | 1601 |
|
1602 |
lemma gcd_neg2 [simp]: |
|
1603 |
"gcd x (-y) = gcd x y" |
|
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset
|
1604 |
by (rule sym, rule gcdI, simp_all add: gcd_greatest) |
58023 | 1605 |
|
1606 |
lemma gcd_neg_numeral_1 [simp]: |
|
1607 |
"gcd (- numeral n) x = gcd (numeral n) x" |
|
1608 |
by (fact gcd_neg1) |
|
1609 |
||
1610 |
lemma gcd_neg_numeral_2 [simp]: |
|
1611 |
"gcd x (- numeral n) = gcd x (numeral n)" |
|
1612 |
by (fact gcd_neg2) |
|
1613 |
||
1614 |
lemma gcd_diff1: "gcd (m - n) n = gcd m n" |
|
1615 |
by (subst diff_conv_add_uminus, subst gcd_neg2[symmetric], subst gcd_add1, simp) |
|
1616 |
||
1617 |
lemma gcd_diff2: "gcd (n - m) n = gcd m n" |
|
1618 |
by (subst gcd_neg1[symmetric], simp only: minus_diff_eq gcd_diff1) |
|
1619 |
||
1620 |
lemma coprime_minus_one [simp]: "gcd (n - 1) n = 1" |
|
1621 |
proof - |
|
1622 |
have "gcd (n - 1) n = gcd n (n - 1)" by (fact gcd.commute) |
|
1623 |
also have "\<dots> = gcd ((n - 1) + 1) (n - 1)" by simp |
|
1624 |
also have "\<dots> = 1" by (rule coprime_plus_one) |
|
1625 |
finally show ?thesis . |
|
1626 |
qed |
|
1627 |
||
1628 |
lemma lcm_neg1 [simp]: "lcm (-x) y = lcm x y" |
|
1629 |
by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero) |
|
1630 |
||
1631 |
lemma lcm_neg2 [simp]: "lcm x (-y) = lcm x y" |
|
1632 |
by (rule sym, rule lcmI, simp_all add: lcm_least lcm_zero) |
|
1633 |
||
1634 |
lemma lcm_neg_numeral_1 [simp]: "lcm (- numeral n) x = lcm (numeral n) x" |
|
1635 |
by (fact lcm_neg1) |
|
1636 |
||
1637 |
lemma lcm_neg_numeral_2 [simp]: "lcm x (- numeral n) = lcm x (numeral n)" |
|
1638 |
by (fact lcm_neg2) |
|
1639 |
||
1640 |
function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where |
|
1641 |
"euclid_ext a b = |
|
1642 |
(if b = 0 then |
|
1643 |
let x = ring_inv (normalisation_factor a) in (x, 0, a * x) |
|
1644 |
else |
|
1645 |
case euclid_ext b (a mod b) of |
|
1646 |
(s,t,c) \<Rightarrow> (t, s - t * (a div b), c))" |
|
1647 |
by (pat_completeness, simp) |
|
1648 |
termination by (relation "measure (euclidean_size \<circ> snd)", simp_all) |
|
1649 |
||
1650 |
declare euclid_ext.simps [simp del] |
|
1651 |
||
1652 |
lemma euclid_ext_0: |
|
1653 |
"euclid_ext a 0 = (ring_inv (normalisation_factor a), 0, a * ring_inv (normalisation_factor a))" |
|
1654 |
by (subst euclid_ext.simps, simp add: Let_def) |
|
1655 |
||
1656 |
lemma euclid_ext_non_0: |
|
1657 |
"b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of |
|
1658 |
(s,t,c) \<Rightarrow> (t, s - t * (a div b), c))" |
|
1659 |
by (subst euclid_ext.simps, simp) |
|
1660 |
||
1661 |
definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a" |
|
1662 |
where |
|
1663 |
"euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))" |
|
1664 |
||
1665 |
lemma euclid_ext_gcd [simp]: |
|
1666 |
"(case euclid_ext a b of (_,_,t) \<Rightarrow> t) = gcd a b" |
|
1667 |
proof (induct a b rule: euclid_ext.induct) |
|
1668 |
case (1 a b) |
|
1669 |
then show ?case |
|
1670 |
proof (cases "b = 0") |
|
1671 |
case True |
|
1672 |
then show ?thesis by (cases "a = 0") |
|
1673 |
(simp_all add: euclid_ext_0 unit_div mult_ac unit_simps gcd_0) |
|
1674 |
next |
|
1675 |
case False with 1 show ?thesis |
|
1676 |
by (simp add: euclid_ext_non_0 ac_simps split: prod.split prod.split_asm) |
|
1677 |
qed |
|
1678 |
qed |
|
1679 |
||
1680 |
lemma euclid_ext_gcd' [simp]: |
|
1681 |
"euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b" |
|
1682 |
by (insert euclid_ext_gcd[of a b], drule (1) subst, simp) |
|
1683 |
||
1684 |
lemma euclid_ext_correct: |
|
1685 |
"case euclid_ext x y of (s,t,c) \<Rightarrow> s*x + t*y = c" |
|
1686 |
proof (induct x y rule: euclid_ext.induct) |
|
1687 |
case (1 x y) |
|
1688 |
show ?case |
|
1689 |
proof (cases "y = 0") |
|
1690 |
case True |
|
1691 |
then show ?thesis by (simp add: euclid_ext_0 mult_ac) |
|
1692 |
next |
|
1693 |
case False |
|
1694 |
obtain s t c where stc: "euclid_ext y (x mod y) = (s,t,c)" |
|
1695 |
by (cases "euclid_ext y (x mod y)", blast) |
|
1696 |
from 1 have "c = s * y + t * (x mod y)" by (simp add: stc False) |
|
1697 |
also have "... = t*((x div y)*y + x mod y) + (s - t * (x div y))*y" |
|
1698 |
by (simp add: algebra_simps) |
|
1699 |
also have "(x div y)*y + x mod y = x" using mod_div_equality . |
|
1700 |
finally show ?thesis |
|
1701 |
by (subst euclid_ext.simps, simp add: False stc) |
|
1702 |
qed |
|
1703 |
qed |
|
1704 |
||
1705 |
lemma euclid_ext'_correct: |
|
1706 |
"fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b" |
|
1707 |
proof- |
|
1708 |
obtain s t c where "euclid_ext a b = (s,t,c)" |
|
1709 |
by (cases "euclid_ext a b", blast) |
|
1710 |
with euclid_ext_correct[of a b] euclid_ext_gcd[of a b] |
|
1711 |
show ?thesis unfolding euclid_ext'_def by simp |
|
1712 |
qed |
|
1713 |
||
1714 |
lemma bezout: "\<exists>s t. s * x + t * y = gcd x y" |
|
1715 |
using euclid_ext'_correct by blast |
|
1716 |
||
1717 |
lemma euclid_ext'_0 [simp]: "euclid_ext' x 0 = (ring_inv (normalisation_factor x), 0)" |
|
1718 |
by (simp add: bezw_def euclid_ext'_def euclid_ext_0) |
|
1719 |
||
1720 |
lemma euclid_ext'_non_0: "y \<noteq> 0 \<Longrightarrow> euclid_ext' x y = (snd (euclid_ext' y (x mod y)), |
|
1721 |
fst (euclid_ext' y (x mod y)) - snd (euclid_ext' y (x mod y)) * (x div y))" |
|
1722 |
by (cases "euclid_ext y (x mod y)") |
|
1723 |
(simp add: euclid_ext'_def euclid_ext_non_0) |
|
1724 |
||
1725 |
end |
|
1726 |
||
1727 |
instantiation nat :: euclidean_semiring |
|
1728 |
begin |
|
1729 |
||
1730 |
definition [simp]: |
|
1731 |
"euclidean_size_nat = (id :: nat \<Rightarrow> nat)" |
|
1732 |
||
1733 |
definition [simp]: |
|
1734 |
"normalisation_factor_nat (n::nat) = (if n = 0 then 0 else 1 :: nat)" |
|
1735 |
||
1736 |
instance proof |
|
59061 | 1737 |
qed simp_all |
58023 | 1738 |
|
1739 |
end |
|
1740 |
||
1741 |
instantiation int :: euclidean_ring |
|
1742 |
begin |
|
1743 |
||
1744 |
definition [simp]: |
|
1745 |
"euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)" |
|
1746 |
||
1747 |
definition [simp]: |
|
1748 |
"normalisation_factor_int = (sgn :: int \<Rightarrow> int)" |
|
1749 |
||
1750 |
instance proof |
|
1751 |
case goal2 then show ?case by (auto simp add: abs_mult nat_mult_distrib) |
|
1752 |
next |
|
59061 | 1753 |
case goal3 then show ?case by (simp add: zsgn_def) |
58023 | 1754 |
next |
59061 | 1755 |
case goal5 then show ?case by (auto simp: zsgn_def) |
58023 | 1756 |
next |
59061 | 1757 |
case goal6 then show ?case by (auto split: abs_split simp: zsgn_def) |
58023 | 1758 |
qed (auto simp: sgn_times split: abs_split) |
1759 |
||
1760 |
end |
|
1761 |
||
1762 |
end |
|
1763 |