author | haftmann |
Thu, 23 Nov 2017 17:03:27 +0000 | |
changeset 67087 | 733017b19de9 |
parent 66579 | 2db3fe23fdaf |
child 67343 | f0f13aa282f4 |
permissions | -rw-r--r-- |
41959 | 1 |
(* Title: HOL/Algebra/Divisibility.thy |
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Author: Clemens Ballarin |
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Author: Stephan Hohe |
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*) |
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section \<open>Divisibility in monoids and rings\<close> |
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theory Divisibility |
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imports "HOL-Library.Permutation" Coset Group |
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begin |
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section \<open>Factorial Monoids\<close> |
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subsection \<open>Monoids with Cancellation Law\<close> |
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locale monoid_cancel = monoid + |
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assumes l_cancel: "\<lbrakk>c \<otimes> a = c \<otimes> b; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b" |
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and r_cancel: "\<lbrakk>a \<otimes> c = b \<otimes> c; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b" |
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lemma (in monoid) monoid_cancelI: |
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assumes l_cancel: "\<And>a b c. \<lbrakk>c \<otimes> a = c \<otimes> b; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b" |
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and r_cancel: "\<And>a b c. \<lbrakk>a \<otimes> c = b \<otimes> c; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b" |
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shows "monoid_cancel G" |
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by standard fact+ |
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lemma (in monoid_cancel) is_monoid_cancel: "monoid_cancel G" .. |
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sublocale group \<subseteq> monoid_cancel |
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by standard simp_all |
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||
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locale comm_monoid_cancel = monoid_cancel + comm_monoid |
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lemma comm_monoid_cancelI: |
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fixes G (structure) |
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assumes "comm_monoid G" |
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assumes cancel: "\<And>a b c. \<lbrakk>a \<otimes> c = b \<otimes> c; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b" |
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shows "comm_monoid_cancel G" |
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proof - |
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interpret comm_monoid G by fact |
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show "comm_monoid_cancel G" |
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by unfold_locales (metis assms(2) m_ac(2))+ |
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qed |
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lemma (in comm_monoid_cancel) is_comm_monoid_cancel: "comm_monoid_cancel G" |
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by intro_locales |
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sublocale comm_group \<subseteq> comm_monoid_cancel .. |
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subsection \<open>Products of Units in Monoids\<close> |
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lemma (in monoid) Units_m_closed[simp, intro]: |
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assumes h1unit: "h1 \<in> Units G" |
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and h2unit: "h2 \<in> Units G" |
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shows "h1 \<otimes> h2 \<in> Units G" |
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unfolding Units_def |
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using assms |
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by auto (metis Units_inv_closed Units_l_inv Units_m_closed Units_r_inv) |
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lemma (in monoid) prod_unit_l: |
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assumes abunit[simp]: "a \<otimes> b \<in> Units G" |
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and aunit[simp]: "a \<in> Units G" |
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and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" |
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shows "b \<in> Units G" |
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proof - |
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have c: "inv (a \<otimes> b) \<otimes> a \<in> carrier G" by simp |
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have "(inv (a \<otimes> b) \<otimes> a) \<otimes> b = inv (a \<otimes> b) \<otimes> (a \<otimes> b)" |
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by (simp add: m_assoc) |
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also have "\<dots> = \<one>" by simp |
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finally have li: "(inv (a \<otimes> b) \<otimes> a) \<otimes> b = \<one>" . |
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have "\<one> = inv a \<otimes> a" by (simp add: Units_l_inv[symmetric]) |
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also have "\<dots> = inv a \<otimes> \<one> \<otimes> a" by simp |
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also have "\<dots> = inv a \<otimes> ((a \<otimes> b) \<otimes> inv (a \<otimes> b)) \<otimes> a" |
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by (simp add: Units_r_inv[OF abunit, symmetric] del: Units_r_inv) |
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also have "\<dots> = ((inv a \<otimes> a) \<otimes> b) \<otimes> inv (a \<otimes> b) \<otimes> a" |
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by (simp add: m_assoc del: Units_l_inv) |
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also have "\<dots> = b \<otimes> inv (a \<otimes> b) \<otimes> a" by simp |
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also have "\<dots> = b \<otimes> (inv (a \<otimes> b) \<otimes> a)" by (simp add: m_assoc) |
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finally have ri: "b \<otimes> (inv (a \<otimes> b) \<otimes> a) = \<one> " by simp |
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from c li ri show "b \<in> Units G" by (auto simp: Units_def) |
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qed |
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lemma (in monoid) prod_unit_r: |
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assumes abunit[simp]: "a \<otimes> b \<in> Units G" |
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and bunit[simp]: "b \<in> Units G" |
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and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" |
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shows "a \<in> Units G" |
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proof - |
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have c: "b \<otimes> inv (a \<otimes> b) \<in> carrier G" by simp |
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have "a \<otimes> (b \<otimes> inv (a \<otimes> b)) = (a \<otimes> b) \<otimes> inv (a \<otimes> b)" |
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by (simp add: m_assoc del: Units_r_inv) |
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also have "\<dots> = \<one>" by simp |
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finally have li: "a \<otimes> (b \<otimes> inv (a \<otimes> b)) = \<one>" . |
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have "\<one> = b \<otimes> inv b" by (simp add: Units_r_inv[symmetric]) |
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also have "\<dots> = b \<otimes> \<one> \<otimes> inv b" by simp |
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also have "\<dots> = b \<otimes> (inv (a \<otimes> b) \<otimes> (a \<otimes> b)) \<otimes> inv b" |
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by (simp add: Units_l_inv[OF abunit, symmetric] del: Units_l_inv) |
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also have "\<dots> = (b \<otimes> inv (a \<otimes> b) \<otimes> a) \<otimes> (b \<otimes> inv b)" |
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by (simp add: m_assoc del: Units_l_inv) |
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also have "\<dots> = b \<otimes> inv (a \<otimes> b) \<otimes> a" by simp |
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finally have ri: "(b \<otimes> inv (a \<otimes> b)) \<otimes> a = \<one> " by simp |
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from c li ri show "a \<in> Units G" by (auto simp: Units_def) |
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qed |
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lemma (in comm_monoid) unit_factor: |
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assumes abunit: "a \<otimes> b \<in> Units G" |
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and [simp]: "a \<in> carrier G" "b \<in> carrier G" |
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shows "a \<in> Units G" |
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using abunit[simplified Units_def] |
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proof clarsimp |
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fix i |
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assume [simp]: "i \<in> carrier G" |
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have carr': "b \<otimes> i \<in> carrier G" by simp |
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have "(b \<otimes> i) \<otimes> a = (i \<otimes> b) \<otimes> a" by (simp add: m_comm) |
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also have "\<dots> = i \<otimes> (b \<otimes> a)" by (simp add: m_assoc) |
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also have "\<dots> = i \<otimes> (a \<otimes> b)" by (simp add: m_comm) |
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also assume "i \<otimes> (a \<otimes> b) = \<one>" |
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finally have li': "(b \<otimes> i) \<otimes> a = \<one>" . |
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have "a \<otimes> (b \<otimes> i) = a \<otimes> b \<otimes> i" by (simp add: m_assoc) |
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also assume "a \<otimes> b \<otimes> i = \<one>" |
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finally have ri': "a \<otimes> (b \<otimes> i) = \<one>" . |
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from carr' li' ri' |
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show "a \<in> Units G" by (simp add: Units_def, fast) |
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qed |
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subsection \<open>Divisibility and Association\<close> |
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subsubsection \<open>Function definitions\<close> |
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definition factor :: "[_, 'a, 'a] \<Rightarrow> bool" (infix "divides\<index>" 65) |
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where "a divides\<^bsub>G\<^esub> b \<longleftrightarrow> (\<exists>c\<in>carrier G. b = a \<otimes>\<^bsub>G\<^esub> c)" |
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definition associated :: "[_, 'a, 'a] \<Rightarrow> bool" (infix "\<sim>\<index>" 55) |
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where "a \<sim>\<^bsub>G\<^esub> b \<longleftrightarrow> a divides\<^bsub>G\<^esub> b \<and> b divides\<^bsub>G\<^esub> a" |
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abbreviation "division_rel G \<equiv> \<lparr>carrier = carrier G, eq = op \<sim>\<^bsub>G\<^esub>, le = op divides\<^bsub>G\<^esub>\<rparr>" |
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definition properfactor :: "[_, 'a, 'a] \<Rightarrow> bool" |
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where "properfactor G a b \<longleftrightarrow> a divides\<^bsub>G\<^esub> b \<and> \<not>(b divides\<^bsub>G\<^esub> a)" |
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definition irreducible :: "[_, 'a] \<Rightarrow> bool" |
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where "irreducible G a \<longleftrightarrow> a \<notin> Units G \<and> (\<forall>b\<in>carrier G. properfactor G b a \<longrightarrow> b \<in> Units G)" |
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definition prime :: "[_, 'a] \<Rightarrow> bool" |
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where "prime G p \<longleftrightarrow> |
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p \<notin> Units G \<and> |
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(\<forall>a\<in>carrier G. \<forall>b\<in>carrier G. p divides\<^bsub>G\<^esub> (a \<otimes>\<^bsub>G\<^esub> b) \<longrightarrow> p divides\<^bsub>G\<^esub> a \<or> p divides\<^bsub>G\<^esub> b)" |
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subsubsection \<open>Divisibility\<close> |
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lemma dividesI: |
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fixes G (structure) |
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assumes carr: "c \<in> carrier G" |
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and p: "b = a \<otimes> c" |
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shows "a divides b" |
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unfolding factor_def using assms by fast |
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lemma dividesI' [intro]: |
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fixes G (structure) |
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assumes p: "b = a \<otimes> c" |
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and carr: "c \<in> carrier G" |
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shows "a divides b" |
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using assms by (fast intro: dividesI) |
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lemma dividesD: |
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fixes G (structure) |
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assumes "a divides b" |
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shows "\<exists>c\<in>carrier G. b = a \<otimes> c" |
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using assms unfolding factor_def by fast |
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lemma dividesE [elim]: |
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fixes G (structure) |
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assumes d: "a divides b" |
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and elim: "\<And>c. \<lbrakk>b = a \<otimes> c; c \<in> carrier G\<rbrakk> \<Longrightarrow> P" |
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shows "P" |
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proof - |
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from dividesD[OF d] obtain c where "c \<in> carrier G" and "b = a \<otimes> c" by auto |
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then show P by (elim elim) |
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qed |
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lemma (in monoid) divides_refl[simp, intro!]: |
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assumes carr: "a \<in> carrier G" |
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shows "a divides a" |
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by (intro dividesI[of "\<one>"]) (simp_all add: carr) |
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lemma (in monoid) divides_trans [trans]: |
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assumes dvds: "a divides b" "b divides c" |
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and acarr: "a \<in> carrier G" |
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shows "a divides c" |
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using dvds[THEN dividesD] by (blast intro: dividesI m_assoc acarr) |
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lemma (in monoid) divides_mult_lI [intro]: |
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assumes ab: "a divides b" |
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and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
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shows "(c \<otimes> a) divides (c \<otimes> b)" |
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using ab |
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apply (elim dividesE) |
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apply (simp add: m_assoc[symmetric] carr) |
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apply (fast intro: dividesI) |
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done |
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lemma (in monoid_cancel) divides_mult_l [simp]: |
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assumes carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
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shows "(c \<otimes> a) divides (c \<otimes> b) = a divides b" |
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apply safe |
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apply (elim dividesE, intro dividesI, assumption) |
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apply (rule l_cancel[of c]) |
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apply (simp add: m_assoc carr)+ |
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apply (fast intro: carr) |
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done |
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lemma (in comm_monoid) divides_mult_rI [intro]: |
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assumes ab: "a divides b" |
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and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
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shows "(a \<otimes> c) divides (b \<otimes> c)" |
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using carr ab |
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apply (simp add: m_comm[of a c] m_comm[of b c]) |
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apply (rule divides_mult_lI, assumption+) |
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done |
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lemma (in comm_monoid_cancel) divides_mult_r [simp]: |
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assumes carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
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shows "(a \<otimes> c) divides (b \<otimes> c) = a divides b" |
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using carr by (simp add: m_comm[of a c] m_comm[of b c]) |
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lemma (in monoid) divides_prod_r: |
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assumes ab: "a divides b" |
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and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
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shows "a divides (b \<otimes> c)" |
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using ab carr by (fast intro: m_assoc) |
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lemma (in comm_monoid) divides_prod_l: |
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assumes carr[intro]: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
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and ab: "a divides b" |
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shows "a divides (c \<otimes> b)" |
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using ab carr |
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apply (simp add: m_comm[of c b]) |
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apply (fast intro: divides_prod_r) |
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done |
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lemma (in monoid) unit_divides: |
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assumes uunit: "u \<in> Units G" |
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and acarr: "a \<in> carrier G" |
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shows "u divides a" |
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proof (intro dividesI[of "(inv u) \<otimes> a"], fast intro: uunit acarr) |
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from uunit acarr have xcarr: "inv u \<otimes> a \<in> carrier G" by fast |
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from uunit acarr have "u \<otimes> (inv u \<otimes> a) = (u \<otimes> inv u) \<otimes> a" |
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by (fast intro: m_assoc[symmetric]) |
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also have "\<dots> = \<one> \<otimes> a" by (simp add: Units_r_inv[OF uunit]) |
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also from acarr have "\<dots> = a" by simp |
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finally show "a = u \<otimes> (inv u \<otimes> a)" .. |
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qed |
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lemma (in comm_monoid) divides_unit: |
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assumes udvd: "a divides u" |
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and carr: "a \<in> carrier G" "u \<in> Units G" |
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shows "a \<in> Units G" |
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using udvd carr by (blast intro: unit_factor) |
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lemma (in comm_monoid) Unit_eq_dividesone: |
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assumes ucarr: "u \<in> carrier G" |
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shows "u \<in> Units G = u divides \<one>" |
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using ucarr by (fast dest: divides_unit intro: unit_divides) |
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subsubsection \<open>Association\<close> |
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lemma associatedI: |
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fixes G (structure) |
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assumes "a divides b" "b divides a" |
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shows "a \<sim> b" |
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using assms by (simp add: associated_def) |
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lemma (in monoid) associatedI2: |
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assumes uunit[simp]: "u \<in> Units G" |
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and a: "a = b \<otimes> u" |
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and bcarr[simp]: "b \<in> carrier G" |
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shows "a \<sim> b" |
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using uunit bcarr |
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unfolding a |
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apply (intro associatedI) |
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apply (rule dividesI[of "inv u"], simp) |
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apply (simp add: m_assoc Units_closed) |
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apply fast |
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done |
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lemma (in monoid) associatedI2': |
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assumes "a = b \<otimes> u" |
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and "u \<in> Units G" |
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and "b \<in> carrier G" |
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shows "a \<sim> b" |
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using assms by (intro associatedI2) |
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lemma associatedD: |
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fixes G (structure) |
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assumes "a \<sim> b" |
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shows "a divides b" |
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using assms by (simp add: associated_def) |
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lemma (in monoid_cancel) associatedD2: |
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assumes assoc: "a \<sim> b" |
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and carr: "a \<in> carrier G" "b \<in> carrier G" |
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shows "\<exists>u\<in>Units G. a = b \<otimes> u" |
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using assoc |
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unfolding associated_def |
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proof clarify |
320 |
assume "b divides a" |
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then obtain u where ucarr: "u \<in> carrier G" and a: "a = b \<otimes> u" |
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by (rule dividesE) |
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assume "a divides b" |
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then obtain u' where u'carr: "u' \<in> carrier G" and b: "b = a \<otimes> u'" |
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by (rule dividesE) |
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note carr = carr ucarr u'carr |
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||
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from carr have "a \<otimes> \<one> = a" by simp |
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also have "\<dots> = b \<otimes> u" by (simp add: a) |
331 |
also have "\<dots> = a \<otimes> u' \<otimes> u" by (simp add: b) |
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also from carr have "\<dots> = a \<otimes> (u' \<otimes> u)" by (simp add: m_assoc) |
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finally have "a \<otimes> \<one> = a \<otimes> (u' \<otimes> u)" . |
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with carr have u1: "\<one> = u' \<otimes> u" by (fast dest: l_cancel) |
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336 |
from carr have "b \<otimes> \<one> = b" by simp |
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also have "\<dots> = a \<otimes> u'" by (simp add: b) |
338 |
also have "\<dots> = b \<otimes> u \<otimes> u'" by (simp add: a) |
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also from carr have "\<dots> = b \<otimes> (u \<otimes> u')" by (simp add: m_assoc) |
340 |
finally have "b \<otimes> \<one> = b \<otimes> (u \<otimes> u')" . |
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with carr have u2: "\<one> = u \<otimes> u'" by (fast dest: l_cancel) |
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||
343 |
from u'carr u1[symmetric] u2[symmetric] have "\<exists>u'\<in>carrier G. u' \<otimes> u = \<one> \<and> u \<otimes> u' = \<one>" |
|
344 |
by fast |
|
345 |
then have "u \<in> Units G" |
|
346 |
by (simp add: Units_def ucarr) |
|
347 |
with ucarr a show "\<exists>u\<in>Units G. a = b \<otimes> u" by fast |
|
27701 | 348 |
qed |
349 |
||
350 |
lemma associatedE: |
|
351 |
fixes G (structure) |
|
352 |
assumes assoc: "a \<sim> b" |
|
353 |
and e: "\<lbrakk>a divides b; b divides a\<rbrakk> \<Longrightarrow> P" |
|
354 |
shows "P" |
|
355 |
proof - |
|
63832 | 356 |
from assoc have "a divides b" "b divides a" |
357 |
by (simp_all add: associated_def) |
|
358 |
then show P by (elim e) |
|
27701 | 359 |
qed |
360 |
||
361 |
lemma (in monoid_cancel) associatedE2: |
|
362 |
assumes assoc: "a \<sim> b" |
|
363 |
and e: "\<And>u. \<lbrakk>a = b \<otimes> u; u \<in> Units G\<rbrakk> \<Longrightarrow> P" |
|
364 |
and carr: "a \<in> carrier G" "b \<in> carrier G" |
|
365 |
shows "P" |
|
366 |
proof - |
|
63832 | 367 |
from assoc and carr have "\<exists>u\<in>Units G. a = b \<otimes> u" |
368 |
by (rule associatedD2) |
|
369 |
then obtain u where "u \<in> Units G" "a = b \<otimes> u" |
|
370 |
by auto |
|
371 |
then show P by (elim e) |
|
27701 | 372 |
qed |
373 |
||
374 |
lemma (in monoid) associated_refl [simp, intro!]: |
|
375 |
assumes "a \<in> carrier G" |
|
376 |
shows "a \<sim> a" |
|
63832 | 377 |
using assms by (fast intro: associatedI) |
27701 | 378 |
|
379 |
lemma (in monoid) associated_sym [sym]: |
|
380 |
assumes "a \<sim> b" |
|
381 |
and "a \<in> carrier G" "b \<in> carrier G" |
|
382 |
shows "b \<sim> a" |
|
63832 | 383 |
using assms by (iprover intro: associatedI elim: associatedE) |
27701 | 384 |
|
385 |
lemma (in monoid) associated_trans [trans]: |
|
386 |
assumes "a \<sim> b" "b \<sim> c" |
|
387 |
and "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
|
388 |
shows "a \<sim> c" |
|
63832 | 389 |
using assms by (iprover intro: associatedI divides_trans elim: associatedE) |
390 |
||
391 |
lemma (in monoid) division_equiv [intro, simp]: "equivalence (division_rel G)" |
|
27701 | 392 |
apply unfold_locales |
63832 | 393 |
apply simp_all |
394 |
apply (metis associated_def) |
|
27701 | 395 |
apply (iprover intro: associated_trans) |
396 |
done |
|
397 |
||
398 |
||
61382 | 399 |
subsubsection \<open>Division and associativity\<close> |
27701 | 400 |
|
401 |
lemma divides_antisym: |
|
402 |
fixes G (structure) |
|
403 |
assumes "a divides b" "b divides a" |
|
404 |
and "a \<in> carrier G" "b \<in> carrier G" |
|
405 |
shows "a \<sim> b" |
|
63832 | 406 |
using assms by (fast intro: associatedI) |
27701 | 407 |
|
408 |
lemma (in monoid) divides_cong_l [trans]: |
|
63832 | 409 |
assumes "x \<sim> x'" |
410 |
and "x' divides y" |
|
411 |
and [simp]: "x \<in> carrier G" "x' \<in> carrier G" "y \<in> carrier G" |
|
27701 | 412 |
shows "x divides y" |
413 |
proof - |
|
63832 | 414 |
from assms(1) have "x divides x'" by (simp add: associatedD) |
415 |
also note assms(2) |
|
416 |
finally show "x divides y" by simp |
|
27701 | 417 |
qed |
418 |
||
419 |
lemma (in monoid) divides_cong_r [trans]: |
|
63832 | 420 |
assumes "x divides y" |
421 |
and "y \<sim> y'" |
|
422 |
and [simp]: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G" |
|
27701 | 423 |
shows "x divides y'" |
424 |
proof - |
|
63832 | 425 |
note assms(1) |
426 |
also from assms(2) have "y divides y'" by (simp add: associatedD) |
|
427 |
finally show "x divides y'" by simp |
|
27701 | 428 |
qed |
429 |
||
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
430 |
lemma (in monoid) division_weak_partial_order [simp, intro!]: |
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
431 |
"weak_partial_order (division_rel G)" |
27701 | 432 |
apply unfold_locales |
63832 | 433 |
apply simp_all |
434 |
apply (simp add: associated_sym) |
|
435 |
apply (blast intro: associated_trans) |
|
436 |
apply (simp add: divides_antisym) |
|
437 |
apply (blast intro: divides_trans) |
|
27701 | 438 |
apply (blast intro: divides_cong_l divides_cong_r associated_sym) |
439 |
done |
|
440 |
||
63832 | 441 |
|
61382 | 442 |
subsubsection \<open>Multiplication and associativity\<close> |
27701 | 443 |
|
444 |
lemma (in monoid_cancel) mult_cong_r: |
|
445 |
assumes "b \<sim> b'" |
|
446 |
and carr: "a \<in> carrier G" "b \<in> carrier G" "b' \<in> carrier G" |
|
447 |
shows "a \<otimes> b \<sim> a \<otimes> b'" |
|
63832 | 448 |
using assms |
449 |
apply (elim associatedE2, intro associatedI2) |
|
450 |
apply (auto intro: m_assoc[symmetric]) |
|
451 |
done |
|
27701 | 452 |
|
453 |
lemma (in comm_monoid_cancel) mult_cong_l: |
|
454 |
assumes "a \<sim> a'" |
|
455 |
and carr: "a \<in> carrier G" "a' \<in> carrier G" "b \<in> carrier G" |
|
456 |
shows "a \<otimes> b \<sim> a' \<otimes> b" |
|
63832 | 457 |
using assms |
458 |
apply (elim associatedE2, intro associatedI2) |
|
459 |
apply assumption |
|
460 |
apply (simp add: m_assoc Units_closed) |
|
461 |
apply (simp add: m_comm Units_closed) |
|
462 |
apply simp_all |
|
463 |
done |
|
27701 | 464 |
|
465 |
lemma (in monoid_cancel) assoc_l_cancel: |
|
466 |
assumes carr: "a \<in> carrier G" "b \<in> carrier G" "b' \<in> carrier G" |
|
467 |
and "a \<otimes> b \<sim> a \<otimes> b'" |
|
468 |
shows "b \<sim> b'" |
|
63832 | 469 |
using assms |
470 |
apply (elim associatedE2, intro associatedI2) |
|
471 |
apply assumption |
|
472 |
apply (rule l_cancel[of a]) |
|
473 |
apply (simp add: m_assoc Units_closed) |
|
474 |
apply fast+ |
|
475 |
done |
|
27701 | 476 |
|
477 |
lemma (in comm_monoid_cancel) assoc_r_cancel: |
|
478 |
assumes "a \<otimes> b \<sim> a' \<otimes> b" |
|
479 |
and carr: "a \<in> carrier G" "a' \<in> carrier G" "b \<in> carrier G" |
|
480 |
shows "a \<sim> a'" |
|
63832 | 481 |
using assms |
482 |
apply (elim associatedE2, intro associatedI2) |
|
483 |
apply assumption |
|
484 |
apply (rule r_cancel[of a b]) |
|
485 |
apply (metis Units_closed assms(3) assms(4) m_ac) |
|
486 |
apply fast+ |
|
487 |
done |
|
27701 | 488 |
|
489 |
||
61382 | 490 |
subsubsection \<open>Units\<close> |
27701 | 491 |
|
492 |
lemma (in monoid_cancel) assoc_unit_l [trans]: |
|
63832 | 493 |
assumes "a \<sim> b" |
494 |
and "b \<in> Units G" |
|
495 |
and "a \<in> carrier G" |
|
27701 | 496 |
shows "a \<in> Units G" |
63832 | 497 |
using assms by (fast elim: associatedE2) |
27701 | 498 |
|
499 |
lemma (in monoid_cancel) assoc_unit_r [trans]: |
|
63832 | 500 |
assumes aunit: "a \<in> Units G" |
501 |
and asc: "a \<sim> b" |
|
27701 | 502 |
and bcarr: "b \<in> carrier G" |
503 |
shows "b \<in> Units G" |
|
63832 | 504 |
using aunit bcarr associated_sym[OF asc] by (blast intro: assoc_unit_l) |
27701 | 505 |
|
506 |
lemma (in comm_monoid) Units_cong: |
|
507 |
assumes aunit: "a \<in> Units G" and asc: "a \<sim> b" |
|
508 |
and bcarr: "b \<in> carrier G" |
|
509 |
shows "b \<in> Units G" |
|
63832 | 510 |
using assms by (blast intro: divides_unit elim: associatedE) |
27701 | 511 |
|
512 |
lemma (in monoid) Units_assoc: |
|
513 |
assumes units: "a \<in> Units G" "b \<in> Units G" |
|
514 |
shows "a \<sim> b" |
|
63832 | 515 |
using units by (fast intro: associatedI unit_divides) |
516 |
||
517 |
lemma (in monoid) Units_are_ones: "Units G {.=}\<^bsub>(division_rel G)\<^esub> {\<one>}" |
|
518 |
apply (simp add: set_eq_def elem_def, rule, simp_all) |
|
27701 | 519 |
proof clarsimp |
520 |
fix a |
|
521 |
assume aunit: "a \<in> Units G" |
|
522 |
show "a \<sim> \<one>" |
|
63832 | 523 |
apply (rule associatedI) |
524 |
apply (fast intro: dividesI[of "inv a"] aunit Units_r_inv[symmetric]) |
|
525 |
apply (fast intro: dividesI[of "a"] l_one[symmetric] Units_closed[OF aunit]) |
|
526 |
done |
|
27701 | 527 |
next |
528 |
have "\<one> \<in> Units G" by simp |
|
529 |
moreover have "\<one> \<sim> \<one>" by simp |
|
530 |
ultimately show "\<exists>a \<in> Units G. \<one> \<sim> a" by fast |
|
531 |
qed |
|
532 |
||
63832 | 533 |
lemma (in comm_monoid) Units_Lower: "Units G = Lower (division_rel G) (carrier G)" |
534 |
apply (simp add: Units_def Lower_def) |
|
535 |
apply (rule, rule) |
|
536 |
apply clarsimp |
|
537 |
apply (rule unit_divides) |
|
538 |
apply (unfold Units_def, fast) |
|
539 |
apply assumption |
|
540 |
apply clarsimp |
|
541 |
apply (metis Unit_eq_dividesone Units_r_inv_ex m_ac(2) one_closed) |
|
542 |
done |
|
27701 | 543 |
|
544 |
||
61382 | 545 |
subsubsection \<open>Proper factors\<close> |
27701 | 546 |
|
547 |
lemma properfactorI: |
|
548 |
fixes G (structure) |
|
549 |
assumes "a divides b" |
|
550 |
and "\<not>(b divides a)" |
|
551 |
shows "properfactor G a b" |
|
63832 | 552 |
using assms unfolding properfactor_def by simp |
27701 | 553 |
|
554 |
lemma properfactorI2: |
|
555 |
fixes G (structure) |
|
556 |
assumes advdb: "a divides b" |
|
557 |
and neq: "\<not>(a \<sim> b)" |
|
558 |
shows "properfactor G a b" |
|
63846 | 559 |
proof (rule properfactorI, rule advdb, rule notI) |
27701 | 560 |
assume "b divides a" |
561 |
with advdb have "a \<sim> b" by (rule associatedI) |
|
562 |
with neq show "False" by fast |
|
563 |
qed |
|
564 |
||
565 |
lemma (in comm_monoid_cancel) properfactorI3: |
|
566 |
assumes p: "p = a \<otimes> b" |
|
567 |
and nunit: "b \<notin> Units G" |
|
568 |
and carr: "a \<in> carrier G" "b \<in> carrier G" "p \<in> carrier G" |
|
569 |
shows "properfactor G a p" |
|
63832 | 570 |
unfolding p |
571 |
using carr |
|
572 |
apply (intro properfactorI, fast) |
|
27701 | 573 |
proof (clarsimp, elim dividesE) |
574 |
fix c |
|
575 |
assume ccarr: "c \<in> carrier G" |
|
576 |
note [simp] = carr ccarr |
|
577 |
||
578 |
have "a \<otimes> \<one> = a" by simp |
|
579 |
also assume "a = a \<otimes> b \<otimes> c" |
|
580 |
also have "\<dots> = a \<otimes> (b \<otimes> c)" by (simp add: m_assoc) |
|
581 |
finally have "a \<otimes> \<one> = a \<otimes> (b \<otimes> c)" . |
|
582 |
||
63832 | 583 |
then have rinv: "\<one> = b \<otimes> c" by (intro l_cancel[of "a" "\<one>" "b \<otimes> c"], simp+) |
27701 | 584 |
also have "\<dots> = c \<otimes> b" by (simp add: m_comm) |
585 |
finally have linv: "\<one> = c \<otimes> b" . |
|
586 |
||
63832 | 587 |
from ccarr linv[symmetric] rinv[symmetric] have "b \<in> Units G" |
588 |
unfolding Units_def by fastforce |
|
589 |
with nunit show False .. |
|
27701 | 590 |
qed |
591 |
||
592 |
lemma properfactorE: |
|
593 |
fixes G (structure) |
|
594 |
assumes pf: "properfactor G a b" |
|
595 |
and r: "\<lbrakk>a divides b; \<not>(b divides a)\<rbrakk> \<Longrightarrow> P" |
|
596 |
shows "P" |
|
63832 | 597 |
using pf unfolding properfactor_def by (fast intro: r) |
27701 | 598 |
|
599 |
lemma properfactorE2: |
|
600 |
fixes G (structure) |
|
601 |
assumes pf: "properfactor G a b" |
|
602 |
and elim: "\<lbrakk>a divides b; \<not>(a \<sim> b)\<rbrakk> \<Longrightarrow> P" |
|
603 |
shows "P" |
|
63832 | 604 |
using pf unfolding properfactor_def by (fast elim: elim associatedE) |
27701 | 605 |
|
606 |
lemma (in monoid) properfactor_unitE: |
|
607 |
assumes uunit: "u \<in> Units G" |
|
608 |
and pf: "properfactor G a u" |
|
609 |
and acarr: "a \<in> carrier G" |
|
610 |
shows "P" |
|
63832 | 611 |
using pf unit_divides[OF uunit acarr] by (fast elim: properfactorE) |
27701 | 612 |
|
613 |
lemma (in monoid) properfactor_divides: |
|
614 |
assumes pf: "properfactor G a b" |
|
615 |
shows "a divides b" |
|
63832 | 616 |
using pf by (elim properfactorE) |
27701 | 617 |
|
618 |
lemma (in monoid) properfactor_trans1 [trans]: |
|
619 |
assumes dvds: "a divides b" "properfactor G b c" |
|
620 |
and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
|
621 |
shows "properfactor G a c" |
|
63832 | 622 |
using dvds carr |
623 |
apply (elim properfactorE, intro properfactorI) |
|
624 |
apply (iprover intro: divides_trans)+ |
|
625 |
done |
|
27701 | 626 |
|
627 |
lemma (in monoid) properfactor_trans2 [trans]: |
|
628 |
assumes dvds: "properfactor G a b" "b divides c" |
|
629 |
and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
|
630 |
shows "properfactor G a c" |
|
63832 | 631 |
using dvds carr |
632 |
apply (elim properfactorE, intro properfactorI) |
|
633 |
apply (iprover intro: divides_trans)+ |
|
634 |
done |
|
27701 | 635 |
|
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
636 |
lemma properfactor_lless: |
27701 | 637 |
fixes G (structure) |
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
638 |
shows "properfactor G = lless (division_rel G)" |
63832 | 639 |
apply (rule ext) |
640 |
apply (rule ext) |
|
641 |
apply rule |
|
642 |
apply (fastforce elim: properfactorE2 intro: weak_llessI) |
|
643 |
apply (fastforce elim: weak_llessE intro: properfactorI2) |
|
644 |
done |
|
27701 | 645 |
|
646 |
lemma (in monoid) properfactor_cong_l [trans]: |
|
647 |
assumes x'x: "x' \<sim> x" |
|
648 |
and pf: "properfactor G x y" |
|
649 |
and carr: "x \<in> carrier G" "x' \<in> carrier G" "y \<in> carrier G" |
|
650 |
shows "properfactor G x' y" |
|
63832 | 651 |
using pf |
652 |
unfolding properfactor_lless |
|
27701 | 653 |
proof - |
29237 | 654 |
interpret weak_partial_order "division_rel G" .. |
63832 | 655 |
from x'x have "x' .=\<^bsub>division_rel G\<^esub> x" by simp |
27701 | 656 |
also assume "x \<sqsubset>\<^bsub>division_rel G\<^esub> y" |
63832 | 657 |
finally show "x' \<sqsubset>\<^bsub>division_rel G\<^esub> y" by (simp add: carr) |
27701 | 658 |
qed |
659 |
||
660 |
lemma (in monoid) properfactor_cong_r [trans]: |
|
661 |
assumes pf: "properfactor G x y" |
|
662 |
and yy': "y \<sim> y'" |
|
663 |
and carr: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G" |
|
664 |
shows "properfactor G x y'" |
|
63832 | 665 |
using pf |
666 |
unfolding properfactor_lless |
|
27701 | 667 |
proof - |
29237 | 668 |
interpret weak_partial_order "division_rel G" .. |
27701 | 669 |
assume "x \<sqsubset>\<^bsub>division_rel G\<^esub> y" |
670 |
also from yy' |
|
63832 | 671 |
have "y .=\<^bsub>division_rel G\<^esub> y'" by simp |
672 |
finally show "x \<sqsubset>\<^bsub>division_rel G\<^esub> y'" by (simp add: carr) |
|
27701 | 673 |
qed |
674 |
||
675 |
lemma (in monoid_cancel) properfactor_mult_lI [intro]: |
|
676 |
assumes ab: "properfactor G a b" |
|
677 |
and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
|
678 |
shows "properfactor G (c \<otimes> a) (c \<otimes> b)" |
|
63832 | 679 |
using ab carr by (fastforce elim: properfactorE intro: properfactorI) |
27701 | 680 |
|
681 |
lemma (in monoid_cancel) properfactor_mult_l [simp]: |
|
682 |
assumes carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
|
683 |
shows "properfactor G (c \<otimes> a) (c \<otimes> b) = properfactor G a b" |
|
63832 | 684 |
using carr by (fastforce elim: properfactorE intro: properfactorI) |
27701 | 685 |
|
686 |
lemma (in comm_monoid_cancel) properfactor_mult_rI [intro]: |
|
687 |
assumes ab: "properfactor G a b" |
|
688 |
and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
|
689 |
shows "properfactor G (a \<otimes> c) (b \<otimes> c)" |
|
63832 | 690 |
using ab carr by (fastforce elim: properfactorE intro: properfactorI) |
27701 | 691 |
|
692 |
lemma (in comm_monoid_cancel) properfactor_mult_r [simp]: |
|
693 |
assumes carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
|
694 |
shows "properfactor G (a \<otimes> c) (b \<otimes> c) = properfactor G a b" |
|
63832 | 695 |
using carr by (fastforce elim: properfactorE intro: properfactorI) |
27701 | 696 |
|
697 |
lemma (in monoid) properfactor_prod_r: |
|
698 |
assumes ab: "properfactor G a b" |
|
699 |
and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
|
700 |
shows "properfactor G a (b \<otimes> c)" |
|
63832 | 701 |
by (intro properfactor_trans2[OF ab] divides_prod_r) simp_all |
27701 | 702 |
|
703 |
lemma (in comm_monoid) properfactor_prod_l: |
|
704 |
assumes ab: "properfactor G a b" |
|
705 |
and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
|
706 |
shows "properfactor G a (c \<otimes> b)" |
|
63832 | 707 |
by (intro properfactor_trans2[OF ab] divides_prod_l) simp_all |
27701 | 708 |
|
709 |
||
61382 | 710 |
subsection \<open>Irreducible Elements and Primes\<close> |
711 |
||
712 |
subsubsection \<open>Irreducible elements\<close> |
|
27701 | 713 |
|
714 |
lemma irreducibleI: |
|
715 |
fixes G (structure) |
|
716 |
assumes "a \<notin> Units G" |
|
717 |
and "\<And>b. \<lbrakk>b \<in> carrier G; properfactor G b a\<rbrakk> \<Longrightarrow> b \<in> Units G" |
|
718 |
shows "irreducible G a" |
|
63832 | 719 |
using assms unfolding irreducible_def by blast |
27701 | 720 |
|
721 |
lemma irreducibleE: |
|
722 |
fixes G (structure) |
|
723 |
assumes irr: "irreducible G a" |
|
63832 | 724 |
and elim: "\<lbrakk>a \<notin> Units G; \<forall>b. b \<in> carrier G \<and> properfactor G b a \<longrightarrow> b \<in> Units G\<rbrakk> \<Longrightarrow> P" |
27701 | 725 |
shows "P" |
63832 | 726 |
using assms unfolding irreducible_def by blast |
27701 | 727 |
|
728 |
lemma irreducibleD: |
|
729 |
fixes G (structure) |
|
730 |
assumes irr: "irreducible G a" |
|
63832 | 731 |
and pf: "properfactor G b a" |
732 |
and bcarr: "b \<in> carrier G" |
|
27701 | 733 |
shows "b \<in> Units G" |
63832 | 734 |
using assms by (fast elim: irreducibleE) |
27701 | 735 |
|
736 |
lemma (in monoid_cancel) irreducible_cong [trans]: |
|
737 |
assumes irred: "irreducible G a" |
|
738 |
and aa': "a \<sim> a'" |
|
739 |
and carr[simp]: "a \<in> carrier G" "a' \<in> carrier G" |
|
740 |
shows "irreducible G a'" |
|
63832 | 741 |
using assms |
742 |
apply (elim irreducibleE, intro irreducibleI) |
|
743 |
apply simp_all |
|
744 |
apply (metis assms(2) assms(3) assoc_unit_l) |
|
745 |
apply (metis assms(2) assms(3) assms(4) associated_sym properfactor_cong_r) |
|
746 |
done |
|
27701 | 747 |
|
748 |
lemma (in monoid) irreducible_prod_rI: |
|
749 |
assumes airr: "irreducible G a" |
|
750 |
and bunit: "b \<in> Units G" |
|
751 |
and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" |
|
752 |
shows "irreducible G (a \<otimes> b)" |
|
63832 | 753 |
using airr carr bunit |
754 |
apply (elim irreducibleE, intro irreducibleI, clarify) |
|
755 |
apply (subgoal_tac "a \<in> Units G", simp) |
|
756 |
apply (intro prod_unit_r[of a b] carr bunit, assumption) |
|
63847 | 757 |
apply (metis assms(2,3) associatedI2 m_closed properfactor_cong_r) |
63832 | 758 |
done |
27701 | 759 |
|
760 |
lemma (in comm_monoid) irreducible_prod_lI: |
|
761 |
assumes birr: "irreducible G b" |
|
762 |
and aunit: "a \<in> Units G" |
|
763 |
and carr [simp]: "a \<in> carrier G" "b \<in> carrier G" |
|
764 |
shows "irreducible G (a \<otimes> b)" |
|
63832 | 765 |
apply (subst m_comm, simp+) |
766 |
apply (intro irreducible_prod_rI assms) |
|
767 |
done |
|
27701 | 768 |
|
769 |
lemma (in comm_monoid_cancel) irreducible_prodE [elim]: |
|
770 |
assumes irr: "irreducible G (a \<otimes> b)" |
|
771 |
and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" |
|
772 |
and e1: "\<lbrakk>irreducible G a; b \<in> Units G\<rbrakk> \<Longrightarrow> P" |
|
773 |
and e2: "\<lbrakk>a \<in> Units G; irreducible G b\<rbrakk> \<Longrightarrow> P" |
|
63832 | 774 |
shows P |
775 |
using irr |
|
27701 | 776 |
proof (elim irreducibleE) |
777 |
assume abnunit: "a \<otimes> b \<notin> Units G" |
|
778 |
and isunit[rule_format]: "\<forall>ba. ba \<in> carrier G \<and> properfactor G ba (a \<otimes> b) \<longrightarrow> ba \<in> Units G" |
|
63832 | 779 |
show P |
27701 | 780 |
proof (cases "a \<in> Units G") |
63832 | 781 |
case aunit: True |
27701 | 782 |
have "irreducible G b" |
63846 | 783 |
proof (rule irreducibleI, rule notI) |
27701 | 784 |
assume "b \<in> Units G" |
785 |
with aunit have "(a \<otimes> b) \<in> Units G" by fast |
|
786 |
with abnunit show "False" .. |
|
787 |
next |
|
788 |
fix c |
|
789 |
assume ccarr: "c \<in> carrier G" |
|
790 |
and "properfactor G c b" |
|
63832 | 791 |
then have "properfactor G c (a \<otimes> b)" by (simp add: properfactor_prod_l[of c b a]) |
792 |
with ccarr show "c \<in> Units G" by (fast intro: isunit) |
|
27701 | 793 |
qed |
63832 | 794 |
with aunit show "P" by (rule e2) |
27701 | 795 |
next |
63832 | 796 |
case anunit: False |
27701 | 797 |
with carr have "properfactor G b (b \<otimes> a)" by (fast intro: properfactorI3) |
63832 | 798 |
then have bf: "properfactor G b (a \<otimes> b)" by (subst m_comm[of a b], simp+) |
799 |
then have bunit: "b \<in> Units G" by (intro isunit, simp) |
|
27701 | 800 |
|
801 |
have "irreducible G a" |
|
63846 | 802 |
proof (rule irreducibleI, rule notI) |
27701 | 803 |
assume "a \<in> Units G" |
804 |
with bunit have "(a \<otimes> b) \<in> Units G" by fast |
|
805 |
with abnunit show "False" .. |
|
806 |
next |
|
807 |
fix c |
|
808 |
assume ccarr: "c \<in> carrier G" |
|
809 |
and "properfactor G c a" |
|
63832 | 810 |
then have "properfactor G c (a \<otimes> b)" |
811 |
by (simp add: properfactor_prod_r[of c a b]) |
|
812 |
with ccarr show "c \<in> Units G" by (fast intro: isunit) |
|
27701 | 813 |
qed |
814 |
from this bunit show "P" by (rule e1) |
|
815 |
qed |
|
816 |
qed |
|
817 |
||
818 |
||
61382 | 819 |
subsubsection \<open>Prime elements\<close> |
27701 | 820 |
|
821 |
lemma primeI: |
|
822 |
fixes G (structure) |
|
823 |
assumes "p \<notin> Units G" |
|
824 |
and "\<And>a b. \<lbrakk>a \<in> carrier G; b \<in> carrier G; p divides (a \<otimes> b)\<rbrakk> \<Longrightarrow> p divides a \<or> p divides b" |
|
825 |
shows "prime G p" |
|
63832 | 826 |
using assms unfolding prime_def by blast |
27701 | 827 |
|
828 |
lemma primeE: |
|
829 |
fixes G (structure) |
|
830 |
assumes pprime: "prime G p" |
|
831 |
and e: "\<lbrakk>p \<notin> Units G; \<forall>a\<in>carrier G. \<forall>b\<in>carrier G. |
|
63832 | 832 |
p divides a \<otimes> b \<longrightarrow> p divides a \<or> p divides b\<rbrakk> \<Longrightarrow> P" |
27701 | 833 |
shows "P" |
63832 | 834 |
using pprime unfolding prime_def by (blast dest: e) |
27701 | 835 |
|
836 |
lemma (in comm_monoid_cancel) prime_divides: |
|
837 |
assumes carr: "a \<in> carrier G" "b \<in> carrier G" |
|
838 |
and pprime: "prime G p" |
|
839 |
and pdvd: "p divides a \<otimes> b" |
|
840 |
shows "p divides a \<or> p divides b" |
|
63832 | 841 |
using assms by (blast elim: primeE) |
27701 | 842 |
|
843 |
lemma (in monoid_cancel) prime_cong [trans]: |
|
844 |
assumes pprime: "prime G p" |
|
845 |
and pp': "p \<sim> p'" |
|
846 |
and carr[simp]: "p \<in> carrier G" "p' \<in> carrier G" |
|
847 |
shows "prime G p'" |
|
63832 | 848 |
using pprime |
849 |
apply (elim primeE, intro primeI) |
|
850 |
apply (metis assms(2) assms(3) assoc_unit_l) |
|
851 |
apply (metis assms(2) assms(3) assms(4) associated_sym divides_cong_l m_closed) |
|
852 |
done |
|
853 |
||
27701 | 854 |
|
61382 | 855 |
subsection \<open>Factorization and Factorial Monoids\<close> |
856 |
||
857 |
subsubsection \<open>Function definitions\<close> |
|
27701 | 858 |
|
63832 | 859 |
definition factors :: "[_, 'a list, 'a] \<Rightarrow> bool" |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
860 |
where "factors G fs a \<longleftrightarrow> (\<forall>x \<in> (set fs). irreducible G x) \<and> foldr (op \<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> = a" |
35847 | 861 |
|
63832 | 862 |
definition wfactors ::"[_, 'a list, 'a] \<Rightarrow> bool" |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
863 |
where "wfactors G fs a \<longleftrightarrow> (\<forall>x \<in> (set fs). irreducible G x) \<and> foldr (op \<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> \<sim>\<^bsub>G\<^esub> a" |
27701 | 864 |
|
63832 | 865 |
abbreviation list_assoc :: "('a,_) monoid_scheme \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "[\<sim>]\<index>" 44) |
866 |
where "list_assoc G \<equiv> list_all2 (op \<sim>\<^bsub>G\<^esub>)" |
|
867 |
||
868 |
definition essentially_equal :: "[_, 'a list, 'a list] \<Rightarrow> bool" |
|
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
869 |
where "essentially_equal G fs1 fs2 \<longleftrightarrow> (\<exists>fs1'. fs1 <~~> fs1' \<and> fs1' [\<sim>]\<^bsub>G\<^esub> fs2)" |
27701 | 870 |
|
871 |
||
872 |
locale factorial_monoid = comm_monoid_cancel + |
|
63832 | 873 |
assumes factors_exist: "\<lbrakk>a \<in> carrier G; a \<notin> Units G\<rbrakk> \<Longrightarrow> \<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs a" |
874 |
and factors_unique: |
|
875 |
"\<lbrakk>factors G fs a; factors G fs' a; a \<in> carrier G; a \<notin> Units G; |
|
876 |
set fs \<subseteq> carrier G; set fs' \<subseteq> carrier G\<rbrakk> \<Longrightarrow> essentially_equal G fs fs'" |
|
27701 | 877 |
|
878 |
||
61382 | 879 |
subsubsection \<open>Comparing lists of elements\<close> |
880 |
||
881 |
text \<open>Association on lists\<close> |
|
27701 | 882 |
|
883 |
lemma (in monoid) listassoc_refl [simp, intro]: |
|
884 |
assumes "set as \<subseteq> carrier G" |
|
885 |
shows "as [\<sim>] as" |
|
63832 | 886 |
using assms by (induct as) simp_all |
27701 | 887 |
|
888 |
lemma (in monoid) listassoc_sym [sym]: |
|
889 |
assumes "as [\<sim>] bs" |
|
63832 | 890 |
and "set as \<subseteq> carrier G" |
891 |
and "set bs \<subseteq> carrier G" |
|
27701 | 892 |
shows "bs [\<sim>] as" |
63832 | 893 |
using assms |
27701 | 894 |
proof (induct as arbitrary: bs, simp) |
895 |
case Cons |
|
63832 | 896 |
then show ?case |
897 |
apply (induct bs) |
|
898 |
apply simp |
|
27701 | 899 |
apply clarsimp |
900 |
apply (iprover intro: associated_sym) |
|
63832 | 901 |
done |
27701 | 902 |
qed |
903 |
||
904 |
lemma (in monoid) listassoc_trans [trans]: |
|
905 |
assumes "as [\<sim>] bs" and "bs [\<sim>] cs" |
|
906 |
and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" and "set cs \<subseteq> carrier G" |
|
907 |
shows "as [\<sim>] cs" |
|
63832 | 908 |
using assms |
909 |
apply (simp add: list_all2_conv_all_nth set_conv_nth, safe) |
|
910 |
apply (rule associated_trans) |
|
911 |
apply (subgoal_tac "as ! i \<sim> bs ! i", assumption) |
|
912 |
apply (simp, simp) |
|
913 |
apply blast+ |
|
914 |
done |
|
27701 | 915 |
|
916 |
lemma (in monoid_cancel) irrlist_listassoc_cong: |
|
917 |
assumes "\<forall>a\<in>set as. irreducible G a" |
|
918 |
and "as [\<sim>] bs" |
|
919 |
and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" |
|
920 |
shows "\<forall>a\<in>set bs. irreducible G a" |
|
63832 | 921 |
using assms |
922 |
apply (clarsimp simp add: list_all2_conv_all_nth set_conv_nth) |
|
923 |
apply (blast intro: irreducible_cong) |
|
924 |
done |
|
27701 | 925 |
|
926 |
||
61382 | 927 |
text \<open>Permutations\<close> |
27701 | 928 |
|
929 |
lemma perm_map [intro]: |
|
930 |
assumes p: "a <~~> b" |
|
931 |
shows "map f a <~~> map f b" |
|
63832 | 932 |
using p by induct auto |
27701 | 933 |
|
934 |
lemma perm_map_switch: |
|
935 |
assumes m: "map f a = map f b" and p: "b <~~> c" |
|
936 |
shows "\<exists>d. a <~~> d \<and> map f d = map f c" |
|
63832 | 937 |
using p m by (induct arbitrary: a) (simp, force, force, blast) |
27701 | 938 |
|
939 |
lemma (in monoid) perm_assoc_switch: |
|
63832 | 940 |
assumes a:"as [\<sim>] bs" and p: "bs <~~> cs" |
941 |
shows "\<exists>bs'. as <~~> bs' \<and> bs' [\<sim>] cs" |
|
942 |
using p a |
|
943 |
apply (induct bs cs arbitrary: as, simp) |
|
944 |
apply (clarsimp simp add: list_all2_Cons2, blast) |
|
945 |
apply (clarsimp simp add: list_all2_Cons2) |
|
946 |
apply blast |
|
947 |
apply blast |
|
948 |
done |
|
27701 | 949 |
|
950 |
lemma (in monoid) perm_assoc_switch_r: |
|
63832 | 951 |
assumes p: "as <~~> bs" and a:"bs [\<sim>] cs" |
952 |
shows "\<exists>bs'. as [\<sim>] bs' \<and> bs' <~~> cs" |
|
953 |
using p a |
|
954 |
apply (induct as bs arbitrary: cs, simp) |
|
955 |
apply (clarsimp simp add: list_all2_Cons1, blast) |
|
956 |
apply (clarsimp simp add: list_all2_Cons1) |
|
957 |
apply blast |
|
958 |
apply blast |
|
959 |
done |
|
27701 | 960 |
|
961 |
declare perm_sym [sym] |
|
962 |
||
963 |
lemma perm_setP: |
|
964 |
assumes perm: "as <~~> bs" |
|
965 |
and as: "P (set as)" |
|
966 |
shows "P (set bs)" |
|
967 |
proof - |
|
63832 | 968 |
from perm have "mset as = mset bs" |
969 |
by (simp add: mset_eq_perm) |
|
970 |
then have "set as = set bs" |
|
971 |
by (rule mset_eq_setD) |
|
972 |
with as show "P (set bs)" |
|
973 |
by simp |
|
27701 | 974 |
qed |
975 |
||
63832 | 976 |
lemmas (in monoid) perm_closed = perm_setP[of _ _ "\<lambda>as. as \<subseteq> carrier G"] |
977 |
||
978 |
lemmas (in monoid) irrlist_perm_cong = perm_setP[of _ _ "\<lambda>as. \<forall>a\<in>as. irreducible G a"] |
|
27701 | 979 |
|
980 |
||
61382 | 981 |
text \<open>Essentially equal factorizations\<close> |
27701 | 982 |
|
983 |
lemma (in monoid) essentially_equalI: |
|
984 |
assumes ex: "fs1 <~~> fs1'" "fs1' [\<sim>] fs2" |
|
985 |
shows "essentially_equal G fs1 fs2" |
|
63832 | 986 |
using ex unfolding essentially_equal_def by fast |
27701 | 987 |
|
988 |
lemma (in monoid) essentially_equalE: |
|
989 |
assumes ee: "essentially_equal G fs1 fs2" |
|
990 |
and e: "\<And>fs1'. \<lbrakk>fs1 <~~> fs1'; fs1' [\<sim>] fs2\<rbrakk> \<Longrightarrow> P" |
|
991 |
shows "P" |
|
63832 | 992 |
using ee unfolding essentially_equal_def by (fast intro: e) |
27701 | 993 |
|
994 |
lemma (in monoid) ee_refl [simp,intro]: |
|
995 |
assumes carr: "set as \<subseteq> carrier G" |
|
996 |
shows "essentially_equal G as as" |
|
63832 | 997 |
using carr by (fast intro: essentially_equalI) |
27701 | 998 |
|
999 |
lemma (in monoid) ee_sym [sym]: |
|
1000 |
assumes ee: "essentially_equal G as bs" |
|
1001 |
and carr: "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" |
|
1002 |
shows "essentially_equal G bs as" |
|
63832 | 1003 |
using ee |
27701 | 1004 |
proof (elim essentially_equalE) |
1005 |
fix fs |
|
1006 |
assume "as <~~> fs" "fs [\<sim>] bs" |
|
63847 | 1007 |
from perm_assoc_switch_r [OF this] obtain fs' where a: "as [\<sim>] fs'" and p: "fs' <~~> bs" |
1008 |
by blast |
|
27701 | 1009 |
from p have "bs <~~> fs'" by (rule perm_sym) |
63832 | 1010 |
with a[symmetric] carr show ?thesis |
1011 |
by (iprover intro: essentially_equalI perm_closed) |
|
27701 | 1012 |
qed |
1013 |
||
1014 |
lemma (in monoid) ee_trans [trans]: |
|
1015 |
assumes ab: "essentially_equal G as bs" and bc: "essentially_equal G bs cs" |
|
63832 | 1016 |
and ascarr: "set as \<subseteq> carrier G" |
27701 | 1017 |
and bscarr: "set bs \<subseteq> carrier G" |
1018 |
and cscarr: "set cs \<subseteq> carrier G" |
|
1019 |
shows "essentially_equal G as cs" |
|
63832 | 1020 |
using ab bc |
27701 | 1021 |
proof (elim essentially_equalE) |
1022 |
fix abs bcs |
|
63847 | 1023 |
assume "abs [\<sim>] bs" and pb: "bs <~~> bcs" |
1024 |
from perm_assoc_switch [OF this] obtain bs' where p: "abs <~~> bs'" and a: "bs' [\<sim>] bcs" |
|
1025 |
by blast |
|
27701 | 1026 |
|
1027 |
assume "as <~~> abs" |
|
63832 | 1028 |
with p have pp: "as <~~> bs'" by fast |
27701 | 1029 |
|
1030 |
from pp ascarr have c1: "set bs' \<subseteq> carrier G" by (rule perm_closed) |
|
1031 |
from pb bscarr have c2: "set bcs \<subseteq> carrier G" by (rule perm_closed) |
|
1032 |
note a |
|
1033 |
also assume "bcs [\<sim>] cs" |
|
63832 | 1034 |
finally (listassoc_trans) have "bs' [\<sim>] cs" by (simp add: c1 c2 cscarr) |
1035 |
with pp show ?thesis |
|
1036 |
by (rule essentially_equalI) |
|
27701 | 1037 |
qed |
1038 |
||
1039 |
||
61382 | 1040 |
subsubsection \<open>Properties of lists of elements\<close> |
1041 |
||
1042 |
text \<open>Multiplication of factors in a list\<close> |
|
27701 | 1043 |
|
1044 |
lemma (in monoid) multlist_closed [simp, intro]: |
|
1045 |
assumes ascarr: "set fs \<subseteq> carrier G" |
|
1046 |
shows "foldr (op \<otimes>) fs \<one> \<in> carrier G" |
|
63832 | 1047 |
using ascarr by (induct fs) simp_all |
27701 | 1048 |
|
1049 |
lemma (in comm_monoid) multlist_dividesI (*[intro]*): |
|
1050 |
assumes "f \<in> set fs" and "f \<in> carrier G" and "set fs \<subseteq> carrier G" |
|
1051 |
shows "f divides (foldr (op \<otimes>) fs \<one>)" |
|
63832 | 1052 |
using assms |
1053 |
apply (induct fs) |
|
1054 |
apply simp |
|
1055 |
apply (case_tac "f = a") |
|
1056 |
apply simp |
|
1057 |
apply (fast intro: dividesI) |
|
1058 |
apply clarsimp |
|
1059 |
apply (metis assms(2) divides_prod_l multlist_closed) |
|
1060 |
done |
|
27701 | 1061 |
|
1062 |
lemma (in comm_monoid_cancel) multlist_listassoc_cong: |
|
1063 |
assumes "fs [\<sim>] fs'" |
|
1064 |
and "set fs \<subseteq> carrier G" and "set fs' \<subseteq> carrier G" |
|
1065 |
shows "foldr (op \<otimes>) fs \<one> \<sim> foldr (op \<otimes>) fs' \<one>" |
|
63832 | 1066 |
using assms |
27701 | 1067 |
proof (induct fs arbitrary: fs', simp) |
1068 |
case (Cons a as fs') |
|
63832 | 1069 |
then show ?case |
1070 |
apply (induct fs', simp) |
|
27701 | 1071 |
proof clarsimp |
1072 |
fix b bs |
|
63832 | 1073 |
assume "a \<sim> b" |
27701 | 1074 |
and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" |
1075 |
and ascarr: "set as \<subseteq> carrier G" |
|
63832 | 1076 |
then have p: "a \<otimes> foldr op \<otimes> as \<one> \<sim> b \<otimes> foldr op \<otimes> as \<one>" |
1077 |
by (fast intro: mult_cong_l) |
|
27701 | 1078 |
also |
63832 | 1079 |
assume "as [\<sim>] bs" |
1080 |
and bscarr: "set bs \<subseteq> carrier G" |
|
1081 |
and "\<And>fs'. \<lbrakk>as [\<sim>] fs'; set fs' \<subseteq> carrier G\<rbrakk> \<Longrightarrow> foldr op \<otimes> as \<one> \<sim> foldr op \<otimes> fs' \<one>" |
|
1082 |
then have "foldr op \<otimes> as \<one> \<sim> foldr op \<otimes> bs \<one>" by simp |
|
1083 |
with ascarr bscarr bcarr have "b \<otimes> foldr op \<otimes> as \<one> \<sim> b \<otimes> foldr op \<otimes> bs \<one>" |
|
1084 |
by (fast intro: mult_cong_r) |
|
1085 |
finally show "a \<otimes> foldr op \<otimes> as \<one> \<sim> b \<otimes> foldr op \<otimes> bs \<one>" |
|
1086 |
by (simp add: ascarr bscarr acarr bcarr) |
|
27701 | 1087 |
qed |
1088 |
qed |
|
1089 |
||
1090 |
lemma (in comm_monoid) multlist_perm_cong: |
|
1091 |
assumes prm: "as <~~> bs" |
|
1092 |
and ascarr: "set as \<subseteq> carrier G" |
|
1093 |
shows "foldr (op \<otimes>) as \<one> = foldr (op \<otimes>) bs \<one>" |
|
63832 | 1094 |
using prm ascarr |
1095 |
apply (induct, simp, clarsimp simp add: m_ac, clarsimp) |
|
27701 | 1096 |
proof clarsimp |
1097 |
fix xs ys zs |
|
1098 |
assume "xs <~~> ys" "set xs \<subseteq> carrier G" |
|
63832 | 1099 |
then have "set ys \<subseteq> carrier G" by (rule perm_closed) |
27701 | 1100 |
moreover assume "set ys \<subseteq> carrier G \<Longrightarrow> foldr op \<otimes> ys \<one> = foldr op \<otimes> zs \<one>" |
1101 |
ultimately show "foldr op \<otimes> ys \<one> = foldr op \<otimes> zs \<one>" by simp |
|
1102 |
qed |
|
1103 |
||
1104 |
lemma (in comm_monoid_cancel) multlist_ee_cong: |
|
1105 |
assumes "essentially_equal G fs fs'" |
|
1106 |
and "set fs \<subseteq> carrier G" and "set fs' \<subseteq> carrier G" |
|
1107 |
shows "foldr (op \<otimes>) fs \<one> \<sim> foldr (op \<otimes>) fs' \<one>" |
|
63832 | 1108 |
using assms |
1109 |
apply (elim essentially_equalE) |
|
1110 |
apply (simp add: multlist_perm_cong multlist_listassoc_cong perm_closed) |
|
1111 |
done |
|
27701 | 1112 |
|
1113 |
||
61382 | 1114 |
subsubsection \<open>Factorization in irreducible elements\<close> |
27701 | 1115 |
|
1116 |
lemma wfactorsI: |
|
28599 | 1117 |
fixes G (structure) |
27701 | 1118 |
assumes "\<forall>f\<in>set fs. irreducible G f" |
1119 |
and "foldr (op \<otimes>) fs \<one> \<sim> a" |
|
1120 |
shows "wfactors G fs a" |
|
63832 | 1121 |
using assms unfolding wfactors_def by simp |
27701 | 1122 |
|
1123 |
lemma wfactorsE: |
|
28599 | 1124 |
fixes G (structure) |
27701 | 1125 |
assumes wf: "wfactors G fs a" |
1126 |
and e: "\<lbrakk>\<forall>f\<in>set fs. irreducible G f; foldr (op \<otimes>) fs \<one> \<sim> a\<rbrakk> \<Longrightarrow> P" |
|
1127 |
shows "P" |
|
63832 | 1128 |
using wf unfolding wfactors_def by (fast dest: e) |
27701 | 1129 |
|
1130 |
lemma (in monoid) factorsI: |
|
1131 |
assumes "\<forall>f\<in>set fs. irreducible G f" |
|
1132 |
and "foldr (op \<otimes>) fs \<one> = a" |
|
1133 |
shows "factors G fs a" |
|
63832 | 1134 |
using assms unfolding factors_def by simp |
27701 | 1135 |
|
1136 |
lemma factorsE: |
|
28599 | 1137 |
fixes G (structure) |
27701 | 1138 |
assumes f: "factors G fs a" |
1139 |
and e: "\<lbrakk>\<forall>f\<in>set fs. irreducible G f; foldr (op \<otimes>) fs \<one> = a\<rbrakk> \<Longrightarrow> P" |
|
1140 |
shows "P" |
|
63832 | 1141 |
using f unfolding factors_def by (simp add: e) |
27701 | 1142 |
|
1143 |
lemma (in monoid) factors_wfactors: |
|
1144 |
assumes "factors G as a" and "set as \<subseteq> carrier G" |
|
1145 |
shows "wfactors G as a" |
|
63832 | 1146 |
using assms by (blast elim: factorsE intro: wfactorsI) |
27701 | 1147 |
|
1148 |
lemma (in monoid) wfactors_factors: |
|
1149 |
assumes "wfactors G as a" and "set as \<subseteq> carrier G" |
|
1150 |
shows "\<exists>a'. factors G as a' \<and> a' \<sim> a" |
|
63832 | 1151 |
using assms by (blast elim: wfactorsE intro: factorsI) |
27701 | 1152 |
|
1153 |
lemma (in monoid) factors_closed [dest]: |
|
1154 |
assumes "factors G fs a" and "set fs \<subseteq> carrier G" |
|
1155 |
shows "a \<in> carrier G" |
|
63832 | 1156 |
using assms by (elim factorsE, clarsimp) |
27701 | 1157 |
|
1158 |
lemma (in monoid) nunit_factors: |
|
1159 |
assumes anunit: "a \<notin> Units G" |
|
1160 |
and fs: "factors G as a" |
|
1161 |
shows "length as > 0" |
|
46129 | 1162 |
proof - |
1163 |
from anunit Units_one_closed have "a \<noteq> \<one>" by auto |
|
1164 |
with fs show ?thesis by (auto elim: factorsE) |
|
1165 |
qed |
|
27701 | 1166 |
|
1167 |
lemma (in monoid) unit_wfactors [simp]: |
|
1168 |
assumes aunit: "a \<in> Units G" |
|
1169 |
shows "wfactors G [] a" |
|
63832 | 1170 |
using aunit by (intro wfactorsI) (simp, simp add: Units_assoc) |
27701 | 1171 |
|
1172 |
lemma (in comm_monoid_cancel) unit_wfactors_empty: |
|
1173 |
assumes aunit: "a \<in> Units G" |
|
1174 |
and wf: "wfactors G fs a" |
|
1175 |
and carr[simp]: "set fs \<subseteq> carrier G" |
|
1176 |
shows "fs = []" |
|
63846 | 1177 |
proof (cases fs) |
1178 |
case Nil |
|
1179 |
then show ?thesis . |
|
1180 |
next |
|
1181 |
case fs: (Cons f fs') |
|
63832 | 1182 |
from carr have fcarr[simp]: "f \<in> carrier G" and carr'[simp]: "set fs' \<subseteq> carrier G" |
1183 |
by (simp_all add: fs) |
|
1184 |
||
1185 |
from fs wf have "irreducible G f" by (simp add: wfactors_def) |
|
1186 |
then have fnunit: "f \<notin> Units G" by (fast elim: irreducibleE) |
|
1187 |
||
1188 |
from fs wf have a: "f \<otimes> foldr (op \<otimes>) fs' \<one> \<sim> a" by (simp add: wfactors_def) |
|
27701 | 1189 |
|
1190 |
note aunit |
|
1191 |
also from fs wf |
|
63832 | 1192 |
have a: "f \<otimes> foldr (op \<otimes>) fs' \<one> \<sim> a" by (simp add: wfactors_def) |
1193 |
have "a \<sim> f \<otimes> foldr (op \<otimes>) fs' \<one>" |
|
1194 |
by (simp add: Units_closed[OF aunit] a[symmetric]) |
|
1195 |
finally have "f \<otimes> foldr (op \<otimes>) fs' \<one> \<in> Units G" by simp |
|
1196 |
then have "f \<in> Units G" by (intro unit_factor[of f], simp+) |
|
63846 | 1197 |
with fnunit show ?thesis by contradiction |
27701 | 1198 |
qed |
1199 |
||
1200 |
||
61382 | 1201 |
text \<open>Comparing wfactors\<close> |
27701 | 1202 |
|
1203 |
lemma (in comm_monoid_cancel) wfactors_listassoc_cong_l: |
|
1204 |
assumes fact: "wfactors G fs a" |
|
1205 |
and asc: "fs [\<sim>] fs'" |
|
1206 |
and carr: "a \<in> carrier G" "set fs \<subseteq> carrier G" "set fs' \<subseteq> carrier G" |
|
1207 |
shows "wfactors G fs' a" |
|
63832 | 1208 |
using fact |
1209 |
apply (elim wfactorsE, intro wfactorsI) |
|
1210 |
apply (metis assms(2) assms(4) assms(5) irrlist_listassoc_cong) |
|
27701 | 1211 |
proof - |
63832 | 1212 |
from asc[symmetric] have "foldr op \<otimes> fs' \<one> \<sim> foldr op \<otimes> fs \<one>" |
1213 |
by (simp add: multlist_listassoc_cong carr) |
|
27701 | 1214 |
also assume "foldr op \<otimes> fs \<one> \<sim> a" |
63832 | 1215 |
finally show "foldr op \<otimes> fs' \<one> \<sim> a" by (simp add: carr) |
27701 | 1216 |
qed |
1217 |
||
1218 |
lemma (in comm_monoid) wfactors_perm_cong_l: |
|
1219 |
assumes "wfactors G fs a" |
|
1220 |
and "fs <~~> fs'" |
|
1221 |
and "set fs \<subseteq> carrier G" |
|
1222 |
shows "wfactors G fs' a" |
|
63832 | 1223 |
using assms |
1224 |
apply (elim wfactorsE, intro wfactorsI) |
|
1225 |
apply (rule irrlist_perm_cong, assumption+) |
|
1226 |
apply (simp add: multlist_perm_cong[symmetric]) |
|
1227 |
done |
|
27701 | 1228 |
|
1229 |
lemma (in comm_monoid_cancel) wfactors_ee_cong_l [trans]: |
|
1230 |
assumes ee: "essentially_equal G as bs" |
|
1231 |
and bfs: "wfactors G bs b" |
|
1232 |
and carr: "b \<in> carrier G" "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" |
|
1233 |
shows "wfactors G as b" |
|
63832 | 1234 |
using ee |
27701 | 1235 |
proof (elim essentially_equalE) |
1236 |
fix fs |
|
1237 |
assume prm: "as <~~> fs" |
|
63832 | 1238 |
with carr have fscarr: "set fs \<subseteq> carrier G" by (simp add: perm_closed) |
27701 | 1239 |
|
1240 |
note bfs |
|
1241 |
also assume [symmetric]: "fs [\<sim>] bs" |
|
1242 |
also (wfactors_listassoc_cong_l) |
|
63832 | 1243 |
note prm[symmetric] |
27701 | 1244 |
finally (wfactors_perm_cong_l) |
63832 | 1245 |
show "wfactors G as b" by (simp add: carr fscarr) |
27701 | 1246 |
qed |
1247 |
||
1248 |
lemma (in monoid) wfactors_cong_r [trans]: |
|
1249 |
assumes fac: "wfactors G fs a" and aa': "a \<sim> a'" |
|
1250 |
and carr[simp]: "a \<in> carrier G" "a' \<in> carrier G" "set fs \<subseteq> carrier G" |
|
1251 |
shows "wfactors G fs a'" |
|
63832 | 1252 |
using fac |
27701 | 1253 |
proof (elim wfactorsE, intro wfactorsI) |
1254 |
assume "foldr op \<otimes> fs \<one> \<sim> a" also note aa' |
|
1255 |
finally show "foldr op \<otimes> fs \<one> \<sim> a'" by simp |
|
1256 |
qed |
|
1257 |
||
1258 |
||
61382 | 1259 |
subsubsection \<open>Essentially equal factorizations\<close> |
27701 | 1260 |
|
1261 |
lemma (in comm_monoid_cancel) unitfactor_ee: |
|
1262 |
assumes uunit: "u \<in> Units G" |
|
1263 |
and carr: "set as \<subseteq> carrier G" |
|
63832 | 1264 |
shows "essentially_equal G (as[0 := (as!0 \<otimes> u)]) as" |
1265 |
(is "essentially_equal G ?as' as") |
|
1266 |
using assms |
|
1267 |
apply (intro essentially_equalI[of _ ?as'], simp) |
|
1268 |
apply (cases as, simp) |
|
1269 |
apply (clarsimp, fast intro: associatedI2[of u]) |
|
1270 |
done |
|
27701 | 1271 |
|
1272 |
lemma (in comm_monoid_cancel) factors_cong_unit: |
|
63832 | 1273 |
assumes uunit: "u \<in> Units G" |
1274 |
and anunit: "a \<notin> Units G" |
|
27701 | 1275 |
and afs: "factors G as a" |
1276 |
and ascarr: "set as \<subseteq> carrier G" |
|
63832 | 1277 |
shows "factors G (as[0 := (as!0 \<otimes> u)]) (a \<otimes> u)" |
1278 |
(is "factors G ?as' ?a'") |
|
1279 |
using assms |
|
1280 |
apply (elim factorsE, clarify) |
|
1281 |
apply (cases as) |
|
1282 |
apply (simp add: nunit_factors) |
|
1283 |
apply clarsimp |
|
1284 |
apply (elim factorsE, intro factorsI) |
|
1285 |
apply (clarsimp, fast intro: irreducible_prod_rI) |
|
1286 |
apply (simp add: m_ac Units_closed) |
|
1287 |
done |
|
27701 | 1288 |
|
1289 |
lemma (in comm_monoid) perm_wfactorsD: |
|
1290 |
assumes prm: "as <~~> bs" |
|
63832 | 1291 |
and afs: "wfactors G as a" |
1292 |
and bfs: "wfactors G bs b" |
|
27701 | 1293 |
and [simp]: "a \<in> carrier G" "b \<in> carrier G" |
63832 | 1294 |
and ascarr [simp]: "set as \<subseteq> carrier G" |
27701 | 1295 |
shows "a \<sim> b" |
63832 | 1296 |
using afs bfs |
27701 | 1297 |
proof (elim wfactorsE) |
1298 |
from prm have [simp]: "set bs \<subseteq> carrier G" by (simp add: perm_closed) |
|
1299 |
assume "foldr op \<otimes> as \<one> \<sim> a" |
|
63832 | 1300 |
then have "a \<sim> foldr op \<otimes> as \<one>" by (rule associated_sym, simp+) |
27701 | 1301 |
also from prm |
63832 | 1302 |
have "foldr op \<otimes> as \<one> = foldr op \<otimes> bs \<one>" by (rule multlist_perm_cong, simp) |
27701 | 1303 |
also assume "foldr op \<otimes> bs \<one> \<sim> b" |
63832 | 1304 |
finally show "a \<sim> b" by simp |
27701 | 1305 |
qed |
1306 |
||
1307 |
lemma (in comm_monoid_cancel) listassoc_wfactorsD: |
|
1308 |
assumes assoc: "as [\<sim>] bs" |
|
63832 | 1309 |
and afs: "wfactors G as a" |
1310 |
and bfs: "wfactors G bs b" |
|
27701 | 1311 |
and [simp]: "a \<in> carrier G" "b \<in> carrier G" |
1312 |
and [simp]: "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" |
|
1313 |
shows "a \<sim> b" |
|
63832 | 1314 |
using afs bfs |
27701 | 1315 |
proof (elim wfactorsE) |
1316 |
assume "foldr op \<otimes> as \<one> \<sim> a" |
|
63832 | 1317 |
then have "a \<sim> foldr op \<otimes> as \<one>" by (rule associated_sym, simp+) |
27701 | 1318 |
also from assoc |
63832 | 1319 |
have "foldr op \<otimes> as \<one> \<sim> foldr op \<otimes> bs \<one>" by (rule multlist_listassoc_cong, simp+) |
27701 | 1320 |
also assume "foldr op \<otimes> bs \<one> \<sim> b" |
63832 | 1321 |
finally show "a \<sim> b" by simp |
27701 | 1322 |
qed |
1323 |
||
1324 |
lemma (in comm_monoid_cancel) ee_wfactorsD: |
|
1325 |
assumes ee: "essentially_equal G as bs" |
|
1326 |
and afs: "wfactors G as a" and bfs: "wfactors G bs b" |
|
1327 |
and [simp]: "a \<in> carrier G" "b \<in> carrier G" |
|
1328 |
and ascarr[simp]: "set as \<subseteq> carrier G" and bscarr[simp]: "set bs \<subseteq> carrier G" |
|
1329 |
shows "a \<sim> b" |
|
63832 | 1330 |
using ee |
27701 | 1331 |
proof (elim essentially_equalE) |
1332 |
fix fs |
|
1333 |
assume prm: "as <~~> fs" |
|
63832 | 1334 |
then have as'carr[simp]: "set fs \<subseteq> carrier G" |
1335 |
by (simp add: perm_closed) |
|
1336 |
from afs prm have afs': "wfactors G fs a" |
|
1337 |
by (rule wfactors_perm_cong_l) simp |
|
27701 | 1338 |
assume "fs [\<sim>] bs" |
63832 | 1339 |
from this afs' bfs show "a \<sim> b" |
1340 |
by (rule listassoc_wfactorsD) simp_all |
|
27701 | 1341 |
qed |
1342 |
||
1343 |
lemma (in comm_monoid_cancel) ee_factorsD: |
|
1344 |
assumes ee: "essentially_equal G as bs" |
|
1345 |
and afs: "factors G as a" and bfs:"factors G bs b" |
|
1346 |
and "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" |
|
1347 |
shows "a \<sim> b" |
|
63832 | 1348 |
using assms by (blast intro: factors_wfactors dest: ee_wfactorsD) |
27701 | 1349 |
|
1350 |
lemma (in factorial_monoid) ee_factorsI: |
|
1351 |
assumes ab: "a \<sim> b" |
|
1352 |
and afs: "factors G as a" and anunit: "a \<notin> Units G" |
|
1353 |
and bfs: "factors G bs b" and bnunit: "b \<notin> Units G" |
|
1354 |
and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G" |
|
1355 |
shows "essentially_equal G as bs" |
|
1356 |
proof - |
|
1357 |
note carr[simp] = factors_closed[OF afs ascarr] ascarr[THEN subsetD] |
|
63832 | 1358 |
factors_closed[OF bfs bscarr] bscarr[THEN subsetD] |
1359 |
||
63847 | 1360 |
from ab carr obtain u where uunit: "u \<in> Units G" and a: "a = b \<otimes> u" |
1361 |
by (elim associatedE2) |
|
63832 | 1362 |
|
1363 |
from uunit bscarr have ee: "essentially_equal G (bs[0 := (bs!0 \<otimes> u)]) bs" |
|
1364 |
(is "essentially_equal G ?bs' bs") |
|
1365 |
by (rule unitfactor_ee) |
|
1366 |
||
1367 |
from bscarr uunit have bs'carr: "set ?bs' \<subseteq> carrier G" |
|
1368 |
by (cases bs) (simp_all add: Units_closed) |
|
1369 |
||
1370 |
from uunit bnunit bfs bscarr have fac: "factors G ?bs' (b \<otimes> u)" |
|
1371 |
by (rule factors_cong_unit) |
|
27701 | 1372 |
|
1373 |
from afs fac[simplified a[symmetric]] ascarr bs'carr anunit |
|
63832 | 1374 |
have "essentially_equal G as ?bs'" |
1375 |
by (blast intro: factors_unique) |
|
27701 | 1376 |
also note ee |
63832 | 1377 |
finally show "essentially_equal G as bs" |
1378 |
by (simp add: ascarr bscarr bs'carr) |
|
27701 | 1379 |
qed |
1380 |
||
1381 |
lemma (in factorial_monoid) ee_wfactorsI: |
|
1382 |
assumes asc: "a \<sim> b" |
|
1383 |
and asf: "wfactors G as a" and bsf: "wfactors G bs b" |
|
1384 |
and acarr[simp]: "a \<in> carrier G" and bcarr[simp]: "b \<in> carrier G" |
|
1385 |
and ascarr[simp]: "set as \<subseteq> carrier G" and bscarr[simp]: "set bs \<subseteq> carrier G" |
|
1386 |
shows "essentially_equal G as bs" |
|
63832 | 1387 |
using assms |
27701 | 1388 |
proof (cases "a \<in> Units G") |
63832 | 1389 |
case aunit: True |
27701 | 1390 |
also note asc |
1391 |
finally have bunit: "b \<in> Units G" by simp |
|
1392 |
||
63832 | 1393 |
from aunit asf ascarr have e: "as = []" |
1394 |
by (rule unit_wfactors_empty) |
|
1395 |
from bunit bsf bscarr have e': "bs = []" |
|
1396 |
by (rule unit_wfactors_empty) |
|
27701 | 1397 |
|
1398 |
have "essentially_equal G [] []" |
|
63832 | 1399 |
by (fast intro: essentially_equalI) |
1400 |
then show ?thesis |
|
1401 |
by (simp add: e e') |
|
27701 | 1402 |
next |
63832 | 1403 |
case anunit: False |
27701 | 1404 |
have bnunit: "b \<notin> Units G" |
1405 |
proof clarify |
|
1406 |
assume "b \<in> Units G" |
|
1407 |
also note asc[symmetric] |
|
1408 |
finally have "a \<in> Units G" by simp |
|
63832 | 1409 |
with anunit show False .. |
27701 | 1410 |
qed |
1411 |
||
63847 | 1412 |
from wfactors_factors[OF asf ascarr] obtain a' where fa': "factors G as a'" and a': "a' \<sim> a" |
1413 |
by blast |
|
63832 | 1414 |
from fa' ascarr have a'carr[simp]: "a' \<in> carrier G" |
1415 |
by fast |
|
27701 | 1416 |
|
1417 |
have a'nunit: "a' \<notin> Units G" |
|
63832 | 1418 |
proof clarify |
27701 | 1419 |
assume "a' \<in> Units G" |
1420 |
also note a' |
|
1421 |
finally have "a \<in> Units G" by simp |
|
1422 |
with anunit |
|
63832 | 1423 |
show "False" .. |
27701 | 1424 |
qed |
1425 |
||
63847 | 1426 |
from wfactors_factors[OF bsf bscarr] obtain b' where fb': "factors G bs b'" and b': "b' \<sim> b" |
1427 |
by blast |
|
63832 | 1428 |
from fb' bscarr have b'carr[simp]: "b' \<in> carrier G" |
1429 |
by fast |
|
27701 | 1430 |
|
1431 |
have b'nunit: "b' \<notin> Units G" |
|
63832 | 1432 |
proof clarify |
27701 | 1433 |
assume "b' \<in> Units G" |
1434 |
also note b' |
|
1435 |
finally have "b \<in> Units G" by simp |
|
63832 | 1436 |
with bnunit show False .. |
27701 | 1437 |
qed |
1438 |
||
1439 |
note a' |
|
1440 |
also note asc |
|
1441 |
also note b'[symmetric] |
|
63832 | 1442 |
finally have "a' \<sim> b'" by simp |
1443 |
from this fa' a'nunit fb' b'nunit ascarr bscarr show "essentially_equal G as bs" |
|
1444 |
by (rule ee_factorsI) |
|
27701 | 1445 |
qed |
1446 |
||
1447 |
lemma (in factorial_monoid) ee_wfactors: |
|
1448 |
assumes asf: "wfactors G as a" |
|
1449 |
and bsf: "wfactors G bs b" |
|
1450 |
and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" |
|
1451 |
and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G" |
|
1452 |
shows asc: "a \<sim> b = essentially_equal G as bs" |
|
63832 | 1453 |
using assms by (fast intro: ee_wfactorsI ee_wfactorsD) |
27701 | 1454 |
|
1455 |
lemma (in factorial_monoid) wfactors_exist [intro, simp]: |
|
1456 |
assumes acarr[simp]: "a \<in> carrier G" |
|
1457 |
shows "\<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs a" |
|
1458 |
proof (cases "a \<in> Units G") |
|
63832 | 1459 |
case True |
1460 |
then have "wfactors G [] a" by (rule unit_wfactors) |
|
1461 |
then show ?thesis by (intro exI) force |
|
27701 | 1462 |
next |
63832 | 1463 |
case False |
63847 | 1464 |
with factors_exist [OF acarr] obtain fs where fscarr: "set fs \<subseteq> carrier G" and f: "factors G fs a" |
1465 |
by blast |
|
27701 | 1466 |
from f have "wfactors G fs a" by (rule factors_wfactors) fact |
63832 | 1467 |
with fscarr show ?thesis by fast |
27701 | 1468 |
qed |
1469 |
||
1470 |
lemma (in monoid) wfactors_prod_exists [intro, simp]: |
|
1471 |
assumes "\<forall>a \<in> set as. irreducible G a" and "set as \<subseteq> carrier G" |
|
1472 |
shows "\<exists>a. a \<in> carrier G \<and> wfactors G as a" |
|
63832 | 1473 |
unfolding wfactors_def using assms by blast |
27701 | 1474 |
|
1475 |
lemma (in factorial_monoid) wfactors_unique: |
|
63832 | 1476 |
assumes "wfactors G fs a" |
1477 |
and "wfactors G fs' a" |
|
27701 | 1478 |
and "a \<in> carrier G" |
63832 | 1479 |
and "set fs \<subseteq> carrier G" |
1480 |
and "set fs' \<subseteq> carrier G" |
|
27701 | 1481 |
shows "essentially_equal G fs fs'" |
63832 | 1482 |
using assms by (fast intro: ee_wfactorsI[of a a]) |
27701 | 1483 |
|
1484 |
lemma (in monoid) factors_mult_single: |
|
1485 |
assumes "irreducible G a" and "factors G fb b" and "a \<in> carrier G" |
|
1486 |
shows "factors G (a # fb) (a \<otimes> b)" |
|
63832 | 1487 |
using assms unfolding factors_def by simp |
27701 | 1488 |
|
1489 |
lemma (in monoid_cancel) wfactors_mult_single: |
|
1490 |
assumes f: "irreducible G a" "wfactors G fb b" |
|
63832 | 1491 |
"a \<in> carrier G" "b \<in> carrier G" "set fb \<subseteq> carrier G" |
27701 | 1492 |
shows "wfactors G (a # fb) (a \<otimes> b)" |
63832 | 1493 |
using assms unfolding wfactors_def by (simp add: mult_cong_r) |
27701 | 1494 |
|
1495 |
lemma (in monoid) factors_mult: |
|
1496 |
assumes factors: "factors G fa a" "factors G fb b" |
|
63832 | 1497 |
and ascarr: "set fa \<subseteq> carrier G" |
1498 |
and bscarr: "set fb \<subseteq> carrier G" |
|
27701 | 1499 |
shows "factors G (fa @ fb) (a \<otimes> b)" |
63832 | 1500 |
using assms |
1501 |
unfolding factors_def |
|
1502 |
apply safe |
|
1503 |
apply force |
|
1504 |
apply hypsubst_thin |
|
1505 |
apply (induct fa) |
|
1506 |
apply simp |
|
1507 |
apply (simp add: m_assoc) |
|
1508 |
done |
|
27701 | 1509 |
|
1510 |
lemma (in comm_monoid_cancel) wfactors_mult [intro]: |
|
1511 |
assumes asf: "wfactors G as a" and bsf:"wfactors G bs b" |
|
1512 |
and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" |
|
1513 |
and ascarr: "set as \<subseteq> carrier G" and bscarr:"set bs \<subseteq> carrier G" |
|
1514 |
shows "wfactors G (as @ bs) (a \<otimes> b)" |
|
63832 | 1515 |
using wfactors_factors[OF asf ascarr] and wfactors_factors[OF bsf bscarr] |
1516 |
proof clarsimp |
|
27701 | 1517 |
fix a' b' |
1518 |
assume asf': "factors G as a'" and a'a: "a' \<sim> a" |
|
63832 | 1519 |
and bsf': "factors G bs b'" and b'b: "b' \<sim> b" |
27701 | 1520 |
from asf' have a'carr: "a' \<in> carrier G" by (rule factors_closed) fact |
1521 |
from bsf' have b'carr: "b' \<in> carrier G" by (rule factors_closed) fact |
|
1522 |
||
1523 |
note carr = acarr bcarr a'carr b'carr ascarr bscarr |
|
1524 |
||
63832 | 1525 |
from asf' bsf' have "factors G (as @ bs) (a' \<otimes> b')" |
1526 |
by (rule factors_mult) fact+ |
|
1527 |
||
1528 |
with carr have abf': "wfactors G (as @ bs) (a' \<otimes> b')" |
|
1529 |
by (intro factors_wfactors) simp_all |
|
1530 |
also from b'b carr have trb: "a' \<otimes> b' \<sim> a' \<otimes> b" |
|
1531 |
by (intro mult_cong_r) |
|
1532 |
also from a'a carr have tra: "a' \<otimes> b \<sim> a \<otimes> b" |
|
1533 |
by (intro mult_cong_l) |
|
1534 |
finally show "wfactors G (as @ bs) (a \<otimes> b)" |
|
1535 |
by (simp add: carr) |
|
27701 | 1536 |
qed |
1537 |
||
1538 |
lemma (in comm_monoid) factors_dividesI: |
|
63832 | 1539 |
assumes "factors G fs a" |
1540 |
and "f \<in> set fs" |
|
27701 | 1541 |
and "set fs \<subseteq> carrier G" |
1542 |
shows "f divides a" |
|
63832 | 1543 |
using assms by (fast elim: factorsE intro: multlist_dividesI) |
27701 | 1544 |
|
1545 |
lemma (in comm_monoid) wfactors_dividesI: |
|
1546 |
assumes p: "wfactors G fs a" |
|
1547 |
and fscarr: "set fs \<subseteq> carrier G" and acarr: "a \<in> carrier G" |
|
1548 |
and f: "f \<in> set fs" |
|
1549 |
shows "f divides a" |
|
63832 | 1550 |
using wfactors_factors[OF p fscarr] |
1551 |
proof clarsimp |
|
27701 | 1552 |
fix a' |
63832 | 1553 |
assume fsa': "factors G fs a'" and a'a: "a' \<sim> a" |
1554 |
with fscarr have a'carr: "a' \<in> carrier G" |
|
1555 |
by (simp add: factors_closed) |
|
1556 |
||
1557 |
from fsa' fscarr f have "f divides a'" |
|
1558 |
by (fast intro: factors_dividesI) |
|
27701 | 1559 |
also note a'a |
63832 | 1560 |
finally show "f divides a" |
1561 |
by (simp add: f fscarr[THEN subsetD] acarr a'carr) |
|
27701 | 1562 |
qed |
1563 |
||
1564 |
||
61382 | 1565 |
subsubsection \<open>Factorial monoids and wfactors\<close> |
27701 | 1566 |
|
1567 |
lemma (in comm_monoid_cancel) factorial_monoidI: |
|
63832 | 1568 |
assumes wfactors_exists: "\<And>a. a \<in> carrier G \<Longrightarrow> \<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs a" |
1569 |
and wfactors_unique: |
|
1570 |
"\<And>a fs fs'. \<lbrakk>a \<in> carrier G; set fs \<subseteq> carrier G; set fs' \<subseteq> carrier G; |
|
1571 |
wfactors G fs a; wfactors G fs' a\<rbrakk> \<Longrightarrow> essentially_equal G fs fs'" |
|
27701 | 1572 |
shows "factorial_monoid G" |
28823 | 1573 |
proof |
27701 | 1574 |
fix a |
1575 |
assume acarr: "a \<in> carrier G" and anunit: "a \<notin> Units G" |
|
1576 |
||
1577 |
from wfactors_exists[OF acarr] |
|
63832 | 1578 |
obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" |
63847 | 1579 |
by blast |
1580 |
from wfactors_factors [OF afs ascarr] obtain a' where afs': "factors G as a'" and a'a: "a' \<sim> a" |
|
1581 |
by blast |
|
63832 | 1582 |
from afs' ascarr have a'carr: "a' \<in> carrier G" |
1583 |
by fast |
|
27701 | 1584 |
have a'nunit: "a' \<notin> Units G" |
1585 |
proof clarify |
|
1586 |
assume "a' \<in> Units G" |
|
1587 |
also note a'a |
|
1588 |
finally have "a \<in> Units G" by (simp add: acarr) |
|
63832 | 1589 |
with anunit show False .. |
27701 | 1590 |
qed |
1591 |
||
63847 | 1592 |
from a'carr acarr a'a obtain u where uunit: "u \<in> Units G" and a': "a' = a \<otimes> u" |
63832 | 1593 |
by (blast elim: associatedE2) |
27701 | 1594 |
|
1595 |
note [simp] = acarr Units_closed[OF uunit] Units_inv_closed[OF uunit] |
|
1596 |
||
1597 |
have "a = a \<otimes> \<one>" by simp |
|
57865 | 1598 |
also have "\<dots> = a \<otimes> (u \<otimes> inv u)" by (simp add: uunit) |
27701 | 1599 |
also have "\<dots> = a' \<otimes> inv u" by (simp add: m_assoc[symmetric] a'[symmetric]) |
63832 | 1600 |
finally have a: "a = a' \<otimes> inv u" . |
1601 |
||
1602 |
from ascarr uunit have cr: "set (as[0:=(as!0 \<otimes> inv u)]) \<subseteq> carrier G" |
|
1603 |
by (cases as) auto |
|
1604 |
||
1605 |
from afs' uunit a'nunit acarr ascarr have "factors G (as[0:=(as!0 \<otimes> inv u)]) a" |
|
1606 |
by (simp add: a factors_cong_unit) |
|
1607 |
with cr show "\<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs a" |
|
1608 |
by fast |
|
27701 | 1609 |
qed (blast intro: factors_wfactors wfactors_unique) |
1610 |
||
1611 |
||
61382 | 1612 |
subsection \<open>Factorizations as Multisets\<close> |
1613 |
||
1614 |
text \<open>Gives useful operations like intersection\<close> |
|
27701 | 1615 |
|
1616 |
(* FIXME: use class_of x instead of closure_of {x} *) |
|
1617 |
||
63832 | 1618 |
abbreviation "assocs G x \<equiv> eq_closure_of (division_rel G) {x}" |
1619 |
||
1620 |
definition "fmset G as = mset (map (\<lambda>a. assocs G a) as)" |
|
27701 | 1621 |
|
1622 |
||
61382 | 1623 |
text \<open>Helper lemmas\<close> |
27701 | 1624 |
|
1625 |
lemma (in monoid) assocs_repr_independence: |
|
1626 |
assumes "y \<in> assocs G x" |
|
1627 |
and "x \<in> carrier G" |
|
1628 |
shows "assocs G x = assocs G y" |
|
63832 | 1629 |
using assms |
1630 |
apply safe |
|
1631 |
apply (elim closure_ofE2, intro closure_ofI2[of _ _ y]) |
|
1632 |
apply (clarsimp, iprover intro: associated_trans associated_sym, simp+) |
|
1633 |
apply (elim closure_ofE2, intro closure_ofI2[of _ _ x]) |
|
1634 |
apply (clarsimp, iprover intro: associated_trans, simp+) |
|
1635 |
done |
|
27701 | 1636 |
|
1637 |
lemma (in monoid) assocs_self: |
|
1638 |
assumes "x \<in> carrier G" |
|
1639 |
shows "x \<in> assocs G x" |
|
63832 | 1640 |
using assms by (fastforce intro: closure_ofI2) |
27701 | 1641 |
|
1642 |
lemma (in monoid) assocs_repr_independenceD: |
|
1643 |
assumes repr: "assocs G x = assocs G y" |
|
1644 |
and ycarr: "y \<in> carrier G" |
|
1645 |
shows "y \<in> assocs G x" |
|
63832 | 1646 |
unfolding repr using ycarr by (intro assocs_self) |
27701 | 1647 |
|
1648 |
lemma (in comm_monoid) assocs_assoc: |
|
1649 |
assumes "a \<in> assocs G b" |
|
1650 |
and "b \<in> carrier G" |
|
1651 |
shows "a \<sim> b" |
|
63832 | 1652 |
using assms by (elim closure_ofE2) simp |
1653 |
||
1654 |
lemmas (in comm_monoid) assocs_eqD = assocs_repr_independenceD[THEN assocs_assoc] |
|
27701 | 1655 |
|
1656 |
||
61382 | 1657 |
subsubsection \<open>Comparing multisets\<close> |
27701 | 1658 |
|
1659 |
lemma (in monoid) fmset_perm_cong: |
|
1660 |
assumes prm: "as <~~> bs" |
|
1661 |
shows "fmset G as = fmset G bs" |
|
63832 | 1662 |
using perm_map[OF prm] unfolding mset_eq_perm fmset_def by blast |
27701 | 1663 |
|
1664 |
lemma (in comm_monoid_cancel) eqc_listassoc_cong: |
|
1665 |
assumes "as [\<sim>] bs" |
|
1666 |
and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" |
|
1667 |
shows "map (assocs G) as = map (assocs G) bs" |
|
63832 | 1668 |
using assms |
1669 |
apply (induct as arbitrary: bs, simp) |
|
1670 |
apply (clarsimp simp add: Cons_eq_map_conv list_all2_Cons1, safe) |
|
1671 |
apply (clarsimp elim!: closure_ofE2) defer 1 |
|
1672 |
apply (clarsimp elim!: closure_ofE2) defer 1 |
|
27701 | 1673 |
proof - |
1674 |
fix a x z |
|
1675 |
assume carr[simp]: "a \<in> carrier G" "x \<in> carrier G" "z \<in> carrier G" |
|
1676 |
assume "x \<sim> a" |
|
1677 |
also assume "a \<sim> z" |
|
1678 |
finally have "x \<sim> z" by simp |
|
63832 | 1679 |
with carr show "x \<in> assocs G z" |
1680 |
by (intro closure_ofI2) simp_all |
|
27701 | 1681 |
next |
1682 |
fix a x z |
|
1683 |
assume carr[simp]: "a \<in> carrier G" "x \<in> carrier G" "z \<in> carrier G" |
|
1684 |
assume "x \<sim> z" |
|
1685 |
also assume [symmetric]: "a \<sim> z" |
|
1686 |
finally have "x \<sim> a" by simp |
|
63832 | 1687 |
with carr show "x \<in> assocs G a" |
1688 |
by (intro closure_ofI2) simp_all |
|
27701 | 1689 |
qed |
1690 |
||
1691 |
lemma (in comm_monoid_cancel) fmset_listassoc_cong: |
|
63832 | 1692 |
assumes "as [\<sim>] bs" |
27701 | 1693 |
and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" |
1694 |
shows "fmset G as = fmset G bs" |
|
63832 | 1695 |
using assms unfolding fmset_def by (simp add: eqc_listassoc_cong) |
27701 | 1696 |
|
1697 |
lemma (in comm_monoid_cancel) ee_fmset: |
|
63832 | 1698 |
assumes ee: "essentially_equal G as bs" |
27701 | 1699 |
and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G" |
1700 |
shows "fmset G as = fmset G bs" |
|
63832 | 1701 |
using ee |
27701 | 1702 |
proof (elim essentially_equalE) |
1703 |
fix as' |
|
1704 |
assume prm: "as <~~> as'" |
|
63832 | 1705 |
from prm ascarr have as'carr: "set as' \<subseteq> carrier G" |
1706 |
by (rule perm_closed) |
|
1707 |
||
1708 |
from prm have "fmset G as = fmset G as'" |
|
1709 |
by (rule fmset_perm_cong) |
|
27701 | 1710 |
also assume "as' [\<sim>] bs" |
63832 | 1711 |
with as'carr bscarr have "fmset G as' = fmset G bs" |
1712 |
by (simp add: fmset_listassoc_cong) |
|
1713 |
finally show "fmset G as = fmset G bs" . |
|
27701 | 1714 |
qed |
1715 |
||
1716 |
lemma (in monoid_cancel) fmset_ee__hlp_induct: |
|
1717 |
assumes prm: "cas <~~> cbs" |
|
1718 |
and cdef: "cas = map (assocs G) as" "cbs = map (assocs G) bs" |
|
63832 | 1719 |
shows "\<forall>as bs. (cas <~~> cbs \<and> cas = map (assocs G) as \<and> |
1720 |
cbs = map (assocs G) bs) \<longrightarrow> (\<exists>as'. as <~~> as' \<and> map (assocs G) as' = cbs)" |
|
1721 |
apply (rule perm.induct[of cas cbs], rule prm) |
|
1722 |
apply safe |
|
1723 |
apply (simp_all del: mset_map) |
|
1724 |
apply (simp add: map_eq_Cons_conv) |
|
1725 |
apply blast |
|
1726 |
apply force |
|
27701 | 1727 |
proof - |
1728 |
fix ys as bs |
|
1729 |
assume p1: "map (assocs G) as <~~> ys" |
|
1730 |
and r1[rule_format]: |
|
63832 | 1731 |
"\<forall>asa bs. map (assocs G) as = map (assocs G) asa \<and> ys = map (assocs G) bs |
1732 |
\<longrightarrow> (\<exists>as'. asa <~~> as' \<and> map (assocs G) as' = map (assocs G) bs)" |
|
27701 | 1733 |
and p2: "ys <~~> map (assocs G) bs" |
63832 | 1734 |
and r2[rule_format]: "\<forall>as bsa. ys = map (assocs G) as \<and> map (assocs G) bs = map (assocs G) bsa |
1735 |
\<longrightarrow> (\<exists>as'. as <~~> as' \<and> map (assocs G) as' = map (assocs G) bsa)" |
|
27701 | 1736 |
and p3: "map (assocs G) as <~~> map (assocs G) bs" |
1737 |
||
63832 | 1738 |
from p1 have "mset (map (assocs G) as) = mset ys" |
1739 |
by (simp add: mset_eq_perm del: mset_map) |
|
1740 |
then have setys: "set (map (assocs G) as) = set ys" |
|
1741 |
by (rule mset_eq_setD) |
|
1742 |
||
1743 |
have "set (map (assocs G) as) = {assocs G x | x. x \<in> set as}" by auto |
|
27701 | 1744 |
with setys have "set ys \<subseteq> { assocs G x | x. x \<in> set as}" by simp |
63832 | 1745 |
then have "\<exists>yy. ys = map (assocs G) yy" |
63847 | 1746 |
proof (induct ys) |
1747 |
case Nil |
|
1748 |
then show ?case by simp |
|
1749 |
next |
|
1750 |
case Cons |
|
1751 |
then show ?case |
|
1752 |
proof clarsimp |
|
1753 |
fix yy x |
|
1754 |
show "\<exists>yya. assocs G x # map (assocs G) yy = map (assocs G) yya" |
|
1755 |
by (rule exI[of _ "x#yy"]) simp |
|
1756 |
qed |
|
27701 | 1757 |
qed |
63847 | 1758 |
then obtain yy where ys: "ys = map (assocs G) yy" .. |
63832 | 1759 |
|
1760 |
from p1 ys have "\<exists>as'. as <~~> as' \<and> map (assocs G) as' = map (assocs G) yy" |
|
1761 |
by (intro r1) simp |
|
1762 |
then obtain as' where asas': "as <~~> as'" and as'yy: "map (assocs G) as' = map (assocs G) yy" |
|
1763 |
by auto |
|
1764 |
||
1765 |
from p2 ys have "\<exists>as'. yy <~~> as' \<and> map (assocs G) as' = map (assocs G) bs" |
|
1766 |
by (intro r2) simp |
|
1767 |
then obtain as'' where yyas'': "yy <~~> as''" and as''bs: "map (assocs G) as'' = map (assocs G) bs" |
|
1768 |
by auto |
|
1769 |
||
63847 | 1770 |
from perm_map_switch [OF as'yy yyas''] |
1771 |
obtain cs where as'cs: "as' <~~> cs" and csas'': "map (assocs G) cs = map (assocs G) as''" |
|
1772 |
by blast |
|
1773 |
||
1774 |
from asas' and as'cs have ascs: "as <~~> cs" |
|
1775 |
by fast |
|
1776 |
from csas'' and as''bs have "map (assocs G) cs = map (assocs G) bs" |
|
1777 |
by simp |
|
1778 |
with ascs show "\<exists>as'. as <~~> as' \<and> map (assocs G) as' = map (assocs G) bs" |
|
1779 |
by fast |
|
27701 | 1780 |
qed |
1781 |
||
1782 |
lemma (in comm_monoid_cancel) fmset_ee: |
|
1783 |
assumes mset: "fmset G as = fmset G bs" |
|
1784 |
and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G" |
|
1785 |
shows "essentially_equal G as bs" |
|
1786 |
proof - |
|
63832 | 1787 |
from mset have mpp: "map (assocs G) as <~~> map (assocs G) bs" |
1788 |
by (simp add: fmset_def mset_eq_perm del: mset_map) |
|
27701 | 1789 |
|
63847 | 1790 |
define cas where "cas = map (assocs G) as" |
1791 |
define cbs where "cbs = map (assocs G) bs" |
|
1792 |
||
1793 |
from cas_def cbs_def mpp have [rule_format]: |
|
63832 | 1794 |
"\<forall>as bs. (cas <~~> cbs \<and> cas = map (assocs G) as \<and> cbs = map (assocs G) bs) |
1795 |
\<longrightarrow> (\<exists>as'. as <~~> as' \<and> map (assocs G) as' = cbs)" |
|
1796 |
by (intro fmset_ee__hlp_induct, simp+) |
|
63847 | 1797 |
with mpp cas_def cbs_def have "\<exists>as'. as <~~> as' \<and> map (assocs G) as' = map (assocs G) bs" |
63832 | 1798 |
by simp |
1799 |
||
1800 |
then obtain as' where tp: "as <~~> as'" and tm: "map (assocs G) as' = map (assocs G) bs" |
|
1801 |
by auto |
|
1802 |
from tm have lene: "length as' = length bs" |
|
1803 |
by (rule map_eq_imp_length_eq) |
|
1804 |
from tp have "set as = set as'" |
|
1805 |
by (simp add: mset_eq_perm mset_eq_setD) |
|
1806 |
with ascarr have as'carr: "set as' \<subseteq> carrier G" |
|
1807 |
by simp |
|
27701 | 1808 |
|
63847 | 1809 |
from tm as'carr[THEN subsetD] bscarr[THEN subsetD] have "as' [\<sim>] bs" |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44655
diff
changeset
|
1810 |
by (induct as' arbitrary: bs) (simp, fastforce dest: assocs_eqD[THEN associated_sym]) |
63832 | 1811 |
with tp show "essentially_equal G as bs" |
1812 |
by (fast intro: essentially_equalI) |
|
27701 | 1813 |
qed |
1814 |
||
1815 |
lemma (in comm_monoid_cancel) ee_is_fmset: |
|
1816 |
assumes "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" |
|
1817 |
shows "essentially_equal G as bs = (fmset G as = fmset G bs)" |
|
63832 | 1818 |
using assms by (fast intro: ee_fmset fmset_ee) |
27701 | 1819 |
|
1820 |
||
61382 | 1821 |
subsubsection \<open>Interpreting multisets as factorizations\<close> |
27701 | 1822 |
|
1823 |
lemma (in monoid) mset_fmsetEx: |
|
60495 | 1824 |
assumes elems: "\<And>X. X \<in> set_mset Cs \<Longrightarrow> \<exists>x. P x \<and> X = assocs G x" |
27701 | 1825 |
shows "\<exists>cs. (\<forall>c \<in> set cs. P c) \<and> fmset G cs = Cs" |
1826 |
proof - |
|
63847 | 1827 |
from surjE[OF surj_mset] obtain Cs' where Cs: "Cs = mset Cs'" |
1828 |
by blast |
|
60515 | 1829 |
have "\<exists>cs. (\<forall>c \<in> set cs. P c) \<and> mset (map (assocs G) cs) = Cs" |
63832 | 1830 |
using elems |
1831 |
unfolding Cs |
|
27701 | 1832 |
apply (induct Cs', simp) |
63524
4ec755485732
adding mset_map to the simp rules
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63167
diff
changeset
|
1833 |
proof (clarsimp simp del: mset_map) |
63832 | 1834 |
fix a Cs' cs |
27701 | 1835 |
assume ih: "\<And>X. X = a \<or> X \<in> set Cs' \<Longrightarrow> \<exists>x. P x \<and> X = assocs G x" |
1836 |
and csP: "\<forall>x\<in>set cs. P x" |
|
60515 | 1837 |
and mset: "mset (map (assocs G) cs) = mset Cs'" |
63847 | 1838 |
from ih obtain c where cP: "P c" and a: "a = assocs G c" |
1839 |
by auto |
|
1840 |
from cP csP have tP: "\<forall>x\<in>set (c#cs). P x" |
|
1841 |
by simp |
|
1842 |
from mset a have "mset (map (assocs G) (c#cs)) = add_mset a (mset Cs')" |
|
1843 |
by simp |
|
1844 |
with tP show "\<exists>cs. (\<forall>x\<in>set cs. P x) \<and> mset (map (assocs G) cs) = add_mset a (mset Cs')" |
|
1845 |
by fast |
|
60143 | 1846 |
qed |
63832 | 1847 |
then show ?thesis by (simp add: fmset_def) |
27701 | 1848 |
qed |
1849 |
||
1850 |
lemma (in monoid) mset_wfactorsEx: |
|
63832 | 1851 |
assumes elems: "\<And>X. X \<in> set_mset Cs \<Longrightarrow> \<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" |
27701 | 1852 |
shows "\<exists>c cs. c \<in> carrier G \<and> set cs \<subseteq> carrier G \<and> wfactors G cs c \<and> fmset G cs = Cs" |
1853 |
proof - |
|
1854 |
have "\<exists>cs. (\<forall>c\<in>set cs. c \<in> carrier G \<and> irreducible G c) \<and> fmset G cs = Cs" |
|
63832 | 1855 |
by (intro mset_fmsetEx, rule elems) |
1856 |
then obtain cs where p[rule_format]: "\<forall>c\<in>set cs. c \<in> carrier G \<and> irreducible G c" |
|
1857 |
and Cs[symmetric]: "fmset G cs = Cs" by auto |
|
1858 |
from p have cscarr: "set cs \<subseteq> carrier G" by fast |
|
1859 |
from p have "\<exists>c. c \<in> carrier G \<and> wfactors G cs c" |
|
1860 |
by (intro wfactors_prod_exists) auto |
|
1861 |
then obtain c where ccarr: "c \<in> carrier G" and cfs: "wfactors G cs c" by auto |
|
1862 |
with cscarr Cs show ?thesis by fast |
|
27701 | 1863 |
qed |
1864 |
||
1865 |
||
61382 | 1866 |
subsubsection \<open>Multiplication on multisets\<close> |
27701 | 1867 |
|
1868 |
lemma (in factorial_monoid) mult_wfactors_fmset: |
|
63832 | 1869 |
assumes afs: "wfactors G as a" |
1870 |
and bfs: "wfactors G bs b" |
|
1871 |
and cfs: "wfactors G cs (a \<otimes> b)" |
|
27701 | 1872 |
and carr: "a \<in> carrier G" "b \<in> carrier G" |
1873 |
"set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" "set cs \<subseteq> carrier G" |
|
1874 |
shows "fmset G cs = fmset G as + fmset G bs" |
|
1875 |
proof - |
|
63832 | 1876 |
from assms have "wfactors G (as @ bs) (a \<otimes> b)" |
1877 |
by (intro wfactors_mult) |
|
1878 |
with carr cfs have "essentially_equal G cs (as@bs)" |
|
1879 |
by (intro ee_wfactorsI[of "a\<otimes>b" "a\<otimes>b"]) simp_all |
|
1880 |
with carr have "fmset G cs = fmset G (as@bs)" |
|
1881 |
by (intro ee_fmset) simp_all |
|
1882 |
also have "fmset G (as@bs) = fmset G as + fmset G bs" |
|
1883 |
by (simp add: fmset_def) |
|
27701 | 1884 |
finally show "fmset G cs = fmset G as + fmset G bs" . |
1885 |
qed |
|
1886 |
||
1887 |
lemma (in factorial_monoid) mult_factors_fmset: |
|
63832 | 1888 |
assumes afs: "factors G as a" |
1889 |
and bfs: "factors G bs b" |
|
1890 |
and cfs: "factors G cs (a \<otimes> b)" |
|
27701 | 1891 |
and "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" "set cs \<subseteq> carrier G" |
1892 |
shows "fmset G cs = fmset G as + fmset G bs" |
|
63832 | 1893 |
using assms by (blast intro: factors_wfactors mult_wfactors_fmset) |
27701 | 1894 |
|
1895 |
lemma (in comm_monoid_cancel) fmset_wfactors_mult: |
|
1896 |
assumes mset: "fmset G cs = fmset G as + fmset G bs" |
|
1897 |
and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
|
63832 | 1898 |
"set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" "set cs \<subseteq> carrier G" |
27701 | 1899 |
and fs: "wfactors G as a" "wfactors G bs b" "wfactors G cs c" |
1900 |
shows "c \<sim> a \<otimes> b" |
|
1901 |
proof - |
|
63832 | 1902 |
from carr fs have m: "wfactors G (as @ bs) (a \<otimes> b)" |
1903 |
by (intro wfactors_mult) |
|
1904 |
||
1905 |
from mset have "fmset G cs = fmset G (as@bs)" |
|
1906 |
by (simp add: fmset_def) |
|
1907 |
then have "essentially_equal G cs (as@bs)" |
|
1908 |
by (rule fmset_ee) (simp_all add: carr) |
|
1909 |
then show "c \<sim> a \<otimes> b" |
|
1910 |
by (rule ee_wfactorsD[of "cs" "as@bs"]) (simp_all add: assms m) |
|
27701 | 1911 |
qed |
1912 |
||
1913 |
||
61382 | 1914 |
subsubsection \<open>Divisibility on multisets\<close> |
27701 | 1915 |
|
1916 |
lemma (in factorial_monoid) divides_fmsubset: |
|
1917 |
assumes ab: "a divides b" |
|
63832 | 1918 |
and afs: "wfactors G as a" |
1919 |
and bfs: "wfactors G bs b" |
|
27701 | 1920 |
and carr: "a \<in> carrier G" "b \<in> carrier G" "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" |
64587 | 1921 |
shows "fmset G as \<subseteq># fmset G bs" |
63832 | 1922 |
using ab |
27701 | 1923 |
proof (elim dividesE) |
1924 |
fix c |
|
1925 |
assume ccarr: "c \<in> carrier G" |
|
63847 | 1926 |
from wfactors_exist [OF this] |
1927 |
obtain cs where cscarr: "set cs \<subseteq> carrier G" and cfs: "wfactors G cs c" |
|
1928 |
by blast |
|
27701 | 1929 |
note carr = carr ccarr cscarr |
1930 |
||
1931 |
assume "b = a \<otimes> c" |
|
63832 | 1932 |
with afs bfs cfs carr have "fmset G bs = fmset G as + fmset G cs" |
1933 |
by (intro mult_wfactors_fmset[OF afs cfs]) simp_all |
|
1934 |
then show ?thesis by simp |
|
27701 | 1935 |
qed |
1936 |
||
1937 |
lemma (in comm_monoid_cancel) fmsubset_divides: |
|
64587 | 1938 |
assumes msubset: "fmset G as \<subseteq># fmset G bs" |
63832 | 1939 |
and afs: "wfactors G as a" |
1940 |
and bfs: "wfactors G bs b" |
|
1941 |
and acarr: "a \<in> carrier G" |
|
1942 |
and bcarr: "b \<in> carrier G" |
|
1943 |
and ascarr: "set as \<subseteq> carrier G" |
|
1944 |
and bscarr: "set bs \<subseteq> carrier G" |
|
27701 | 1945 |
shows "a divides b" |
1946 |
proof - |
|
1947 |
from afs have airr: "\<forall>a \<in> set as. irreducible G a" by (fast elim: wfactorsE) |
|
1948 |
from bfs have birr: "\<forall>b \<in> set bs. irreducible G b" by (fast elim: wfactorsE) |
|
1949 |
||
1950 |
have "\<exists>c cs. c \<in> carrier G \<and> set cs \<subseteq> carrier G \<and> wfactors G cs c \<and> fmset G cs = fmset G bs - fmset G as" |
|
1951 |
proof (intro mset_wfactorsEx, simp) |
|
1952 |
fix X |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
61382
diff
changeset
|
1953 |
assume "X \<in># fmset G bs - fmset G as" |
63832 | 1954 |
then have "X \<in># fmset G bs" by (rule in_diffD) |
1955 |
then have "X \<in> set (map (assocs G) bs)" by (simp add: fmset_def) |
|
1956 |
then have "\<exists>x. x \<in> set bs \<and> X = assocs G x" by (induct bs) auto |
|
1957 |
then obtain x where xbs: "x \<in> set bs" and X: "X = assocs G x" by auto |
|
27701 | 1958 |
with bscarr have xcarr: "x \<in> carrier G" by fast |
1959 |
from xbs birr have xirr: "irreducible G x" by simp |
|
1960 |
||
63832 | 1961 |
from xcarr and xirr and X show "\<exists>x. x \<in> carrier G \<and> irreducible G x \<and> X = assocs G x" |
1962 |
by fast |
|
27701 | 1963 |
qed |
63832 | 1964 |
then obtain c cs |
1965 |
where ccarr: "c \<in> carrier G" |
|
1966 |
and cscarr: "set cs \<subseteq> carrier G" |
|
27701 | 1967 |
and csf: "wfactors G cs c" |
1968 |
and csmset: "fmset G cs = fmset G bs - fmset G as" by auto |
|
1969 |
||
1970 |
from csmset msubset |
|
63832 | 1971 |
have "fmset G bs = fmset G as + fmset G cs" |
1972 |
by (simp add: multiset_eq_iff subseteq_mset_def) |
|
1973 |
then have basc: "b \<sim> a \<otimes> c" |
|
1974 |
by (rule fmset_wfactors_mult) fact+ |
|
1975 |
then show ?thesis |
|
27701 | 1976 |
proof (elim associatedE2) |
1977 |
fix u |
|
1978 |
assume "u \<in> Units G" "b = a \<otimes> c \<otimes> u" |
|
63832 | 1979 |
with acarr ccarr show "a divides b" |
1980 |
by (fast intro: dividesI[of "c \<otimes> u"] m_assoc) |
|
1981 |
qed (simp_all add: acarr bcarr ccarr) |
|
27701 | 1982 |
qed |
1983 |
||
1984 |
lemma (in factorial_monoid) divides_as_fmsubset: |
|
63832 | 1985 |
assumes "wfactors G as a" |
1986 |
and "wfactors G bs b" |
|
1987 |
and "a \<in> carrier G" |
|
1988 |
and "b \<in> carrier G" |
|
1989 |
and "set as \<subseteq> carrier G" |
|
1990 |
and "set bs \<subseteq> carrier G" |
|
64587 | 1991 |
shows "a divides b = (fmset G as \<subseteq># fmset G bs)" |
63832 | 1992 |
using assms |
1993 |
by (blast intro: divides_fmsubset fmsubset_divides) |
|
27701 | 1994 |
|
1995 |
||
61382 | 1996 |
text \<open>Proper factors on multisets\<close> |
27701 | 1997 |
|
1998 |
lemma (in factorial_monoid) fmset_properfactor: |
|
64587 | 1999 |
assumes asubb: "fmset G as \<subseteq># fmset G bs" |
27701 | 2000 |
and anb: "fmset G as \<noteq> fmset G bs" |
63832 | 2001 |
and "wfactors G as a" |
2002 |
and "wfactors G bs b" |
|
2003 |
and "a \<in> carrier G" |
|
2004 |
and "b \<in> carrier G" |
|
2005 |
and "set as \<subseteq> carrier G" |
|
2006 |
and "set bs \<subseteq> carrier G" |
|
27701 | 2007 |
shows "properfactor G a b" |
63832 | 2008 |
apply (rule properfactorI) |
2009 |
apply (rule fmsubset_divides[of as bs], fact+) |
|
27701 | 2010 |
proof |
2011 |
assume "b divides a" |
|
64587 | 2012 |
then have "fmset G bs \<subseteq># fmset G as" |
63832 | 2013 |
by (rule divides_fmsubset) fact+ |
2014 |
with asubb have "fmset G as = fmset G bs" |
|
2015 |
by (rule subset_mset.antisym) |
|
2016 |
with anb show False .. |
|
27701 | 2017 |
qed |
2018 |
||
2019 |
lemma (in factorial_monoid) properfactor_fmset: |
|
2020 |
assumes pf: "properfactor G a b" |
|
63832 | 2021 |
and "wfactors G as a" |
2022 |
and "wfactors G bs b" |
|
2023 |
and "a \<in> carrier G" |
|
2024 |
and "b \<in> carrier G" |
|
2025 |
and "set as \<subseteq> carrier G" |
|
2026 |
and "set bs \<subseteq> carrier G" |
|
64587 | 2027 |
shows "fmset G as \<subseteq># fmset G bs \<and> fmset G as \<noteq> fmset G bs" |
63832 | 2028 |
using pf |
2029 |
apply (elim properfactorE) |
|
2030 |
apply rule |
|
2031 |
apply (intro divides_fmsubset, assumption) |
|
2032 |
apply (rule assms)+ |
|
2033 |
using assms(2,3,4,6,7) divides_as_fmsubset |
|
2034 |
apply auto |
|
2035 |
done |
|
27701 | 2036 |
|
61382 | 2037 |
subsection \<open>Irreducible Elements are Prime\<close> |
27701 | 2038 |
|
63633 | 2039 |
lemma (in factorial_monoid) irreducible_prime: |
27701 | 2040 |
assumes pirr: "irreducible G p" |
2041 |
and pcarr: "p \<in> carrier G" |
|
2042 |
shows "prime G p" |
|
63832 | 2043 |
using pirr |
27701 | 2044 |
proof (elim irreducibleE, intro primeI) |
2045 |
fix a b |
|
2046 |
assume acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" |
|
2047 |
and pdvdab: "p divides (a \<otimes> b)" |
|
2048 |
and pnunit: "p \<notin> Units G" |
|
2049 |
assume irreduc[rule_format]: |
|
63832 | 2050 |
"\<forall>b. b \<in> carrier G \<and> properfactor G b p \<longrightarrow> b \<in> Units G" |
63847 | 2051 |
from pdvdab obtain c where ccarr: "c \<in> carrier G" and abpc: "a \<otimes> b = p \<otimes> c" |
2052 |
by (rule dividesE) |
|
2053 |
||
2054 |
from wfactors_exist [OF acarr] |
|
2055 |
obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" |
|
2056 |
by blast |
|
2057 |
||
2058 |
from wfactors_exist [OF bcarr] |
|
2059 |
obtain bs where bscarr: "set bs \<subseteq> carrier G" and bfs: "wfactors G bs b" |
|
63832 | 2060 |
by auto |
2061 |
||
63847 | 2062 |
from wfactors_exist [OF ccarr] |
2063 |
obtain cs where cscarr: "set cs \<subseteq> carrier G" and cfs: "wfactors G cs c" |
|
63832 | 2064 |
by auto |
27701 | 2065 |
|
2066 |
note carr[simp] = pcarr acarr bcarr ccarr ascarr bscarr cscarr |
|
2067 |
||
63832 | 2068 |
from afs and bfs have abfs: "wfactors G (as @ bs) (a \<otimes> b)" |
2069 |
by (rule wfactors_mult) fact+ |
|
2070 |
||
2071 |
from pirr cfs have pcfs: "wfactors G (p # cs) (p \<otimes> c)" |
|
2072 |
by (rule wfactors_mult_single) fact+ |
|
2073 |
with abpc have abfs': "wfactors G (p # cs) (a \<otimes> b)" |
|
2074 |
by simp |
|
2075 |
||
2076 |
from abfs' abfs have "essentially_equal G (p # cs) (as @ bs)" |
|
2077 |
by (rule wfactors_unique) simp+ |
|
2078 |
||
63847 | 2079 |
then obtain ds where "p # cs <~~> ds" and dsassoc: "ds [\<sim>] (as @ bs)" |
63832 | 2080 |
by (fast elim: essentially_equalE) |
27701 | 2081 |
then have "p \<in> set ds" |
63832 | 2082 |
by (simp add: perm_set_eq[symmetric]) |
63847 | 2083 |
with dsassoc obtain p' where "p' \<in> set (as@bs)" and pp': "p \<sim> p'" |
63832 | 2084 |
unfolding list_all2_conv_all_nth set_conv_nth by force |
2085 |
then consider "p' \<in> set as" | "p' \<in> set bs" by auto |
|
2086 |
then show "p divides a \<or> p divides b" |
|
2087 |
proof cases |
|
2088 |
case 1 |
|
27701 | 2089 |
with ascarr have [simp]: "p' \<in> carrier G" by fast |
2090 |
||
2091 |
note pp' |
|
2092 |
also from afs |
|
63832 | 2093 |
have "p' divides a" by (rule wfactors_dividesI) fact+ |
2094 |
finally have "p divides a" by simp |
|
2095 |
then show ?thesis .. |
|
2096 |
next |
|
2097 |
case 2 |
|
27701 | 2098 |
with bscarr have [simp]: "p' \<in> carrier G" by fast |
2099 |
||
2100 |
note pp' |
|
2101 |
also from bfs |
|
63832 | 2102 |
have "p' divides b" by (rule wfactors_dividesI) fact+ |
2103 |
finally have "p divides b" by simp |
|
2104 |
then show ?thesis .. |
|
2105 |
qed |
|
27701 | 2106 |
qed |
2107 |
||
2108 |
||
63167 | 2109 |
\<comment>"A version using @{const factors}, more complicated" |
63633 | 2110 |
lemma (in factorial_monoid) factors_irreducible_prime: |
27701 | 2111 |
assumes pirr: "irreducible G p" |
2112 |
and pcarr: "p \<in> carrier G" |
|
2113 |
shows "prime G p" |
|
63832 | 2114 |
using pirr |
2115 |
apply (elim irreducibleE, intro primeI) |
|
2116 |
apply assumption |
|
27701 | 2117 |
proof - |
2118 |
fix a b |
|
63832 | 2119 |
assume acarr: "a \<in> carrier G" |
27701 | 2120 |
and bcarr: "b \<in> carrier G" |
2121 |
and pdvdab: "p divides (a \<otimes> b)" |
|
63832 | 2122 |
assume irreduc[rule_format]: "\<forall>b. b \<in> carrier G \<and> properfactor G b p \<longrightarrow> b \<in> Units G" |
63847 | 2123 |
from pdvdab obtain c where ccarr: "c \<in> carrier G" and abpc: "a \<otimes> b = p \<otimes> c" |
2124 |
by (rule dividesE) |
|
27701 | 2125 |
note [simp] = pcarr acarr bcarr ccarr |
2126 |
||
2127 |
show "p divides a \<or> p divides b" |
|
2128 |
proof (cases "a \<in> Units G") |
|
63832 | 2129 |
case aunit: True |
27701 | 2130 |
|
2131 |
note pdvdab |
|
2132 |
also have "a \<otimes> b = b \<otimes> a" by (simp add: m_comm) |
|
63832 | 2133 |
also from aunit have bab: "b \<otimes> a \<sim> b" |
2134 |
by (intro associatedI2[of "a"], simp+) |
|
2135 |
finally have "p divides b" by simp |
|
2136 |
then show ?thesis .. |
|
27701 | 2137 |
next |
63832 | 2138 |
case anunit: False |
2139 |
show ?thesis |
|
27701 | 2140 |
proof (cases "b \<in> Units G") |
63832 | 2141 |
case bunit: True |
27701 | 2142 |
note pdvdab |
2143 |
also from bunit |
|
63832 | 2144 |
have baa: "a \<otimes> b \<sim> a" |
2145 |
by (intro associatedI2[of "b"], simp+) |
|
2146 |
finally have "p divides a" by simp |
|
2147 |
then show ?thesis .. |
|
27701 | 2148 |
next |
63832 | 2149 |
case bnunit: False |
27701 | 2150 |
have cnunit: "c \<notin> Units G" |
63846 | 2151 |
proof |
27701 | 2152 |
assume cunit: "c \<in> Units G" |
63832 | 2153 |
from bnunit have "properfactor G a (a \<otimes> b)" |
2154 |
by (intro properfactorI3[of _ _ b], simp+) |
|
27701 | 2155 |
also note abpc |
63832 | 2156 |
also from cunit have "p \<otimes> c \<sim> p" |
2157 |
by (intro associatedI2[of c], simp+) |
|
2158 |
finally have "properfactor G a p" by simp |
|
2159 |
with acarr have "a \<in> Units G" by (fast intro: irreduc) |
|
2160 |
with anunit show False .. |
|
27701 | 2161 |
qed |
2162 |
||
2163 |
have abnunit: "a \<otimes> b \<notin> Units G" |
|
2164 |
proof clarsimp |
|
63832 | 2165 |
assume "a \<otimes> b \<in> Units G" |
2166 |
then have "a \<in> Units G" by (rule unit_factor) fact+ |
|
2167 |
with anunit show False .. |
|
27701 | 2168 |
qed |
2169 |
||
63847 | 2170 |
from factors_exist [OF acarr anunit] |
2171 |
obtain as where ascarr: "set as \<subseteq> carrier G" and afac: "factors G as a" |
|
2172 |
by blast |
|
2173 |
||
2174 |
from factors_exist [OF bcarr bnunit] |
|
2175 |
obtain bs where bscarr: "set bs \<subseteq> carrier G" and bfac: "factors G bs b" |
|
2176 |
by blast |
|
2177 |
||
2178 |
from factors_exist [OF ccarr cnunit] |
|
2179 |
obtain cs where cscarr: "set cs \<subseteq> carrier G" and cfac: "factors G cs c" |
|
63832 | 2180 |
by auto |
27701 | 2181 |
|
2182 |
note [simp] = ascarr bscarr cscarr |
|
2183 |
||
63832 | 2184 |
from afac and bfac have abfac: "factors G (as @ bs) (a \<otimes> b)" |
2185 |
by (rule factors_mult) fact+ |
|
2186 |
||
2187 |
from pirr cfac have pcfac: "factors G (p # cs) (p \<otimes> c)" |
|
2188 |
by (rule factors_mult_single) fact+ |
|
2189 |
with abpc have abfac': "factors G (p # cs) (a \<otimes> b)" |
|
2190 |
by simp |
|
2191 |
||
2192 |
from abfac' abfac have "essentially_equal G (p # cs) (as @ bs)" |
|
2193 |
by (rule factors_unique) (fact | simp)+ |
|
63847 | 2194 |
then obtain ds where "p # cs <~~> ds" and dsassoc: "ds [\<sim>] (as @ bs)" |
63832 | 2195 |
by (fast elim: essentially_equalE) |
27701 | 2196 |
then have "p \<in> set ds" |
63832 | 2197 |
by (simp add: perm_set_eq[symmetric]) |
63847 | 2198 |
with dsassoc obtain p' where "p' \<in> set (as@bs)" and pp': "p \<sim> p'" |
63832 | 2199 |
unfolding list_all2_conv_all_nth set_conv_nth by force |
2200 |
then consider "p' \<in> set as" | "p' \<in> set bs" by auto |
|
2201 |
then show "p divides a \<or> p divides b" |
|
2202 |
proof cases |
|
2203 |
case 1 |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
2204 |
with ascarr have [simp]: "p' \<in> carrier G" by fast |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
2205 |
|
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
2206 |
note pp' |
63832 | 2207 |
also from afac 1 have "p' divides a" by (rule factors_dividesI) fact+ |
2208 |
finally have "p divides a" by simp |
|
2209 |
then show ?thesis .. |
|
2210 |
next |
|
2211 |
case 2 |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
2212 |
with bscarr have [simp]: "p' \<in> carrier G" by fast |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
2213 |
|
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
2214 |
note pp' |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
2215 |
also from bfac |
63832 | 2216 |
have "p' divides b" by (rule factors_dividesI) fact+ |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
2217 |
finally have "p divides b" by simp |
63832 | 2218 |
then show ?thesis .. |
2219 |
qed |
|
27701 | 2220 |
qed |
2221 |
qed |
|
2222 |
qed |
|
2223 |
||
2224 |
||
61382 | 2225 |
subsection \<open>Greatest Common Divisors and Lowest Common Multiples\<close> |
2226 |
||
2227 |
subsubsection \<open>Definitions\<close> |
|
27701 | 2228 |
|
63832 | 2229 |
definition isgcd :: "[('a,_) monoid_scheme, 'a, 'a, 'a] \<Rightarrow> bool" ("(_ gcdof\<index> _ _)" [81,81,81] 80) |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
2230 |
where "x gcdof\<^bsub>G\<^esub> a b \<longleftrightarrow> x divides\<^bsub>G\<^esub> a \<and> x divides\<^bsub>G\<^esub> b \<and> |
35847 | 2231 |
(\<forall>y\<in>carrier G. (y divides\<^bsub>G\<^esub> a \<and> y divides\<^bsub>G\<^esub> b \<longrightarrow> y divides\<^bsub>G\<^esub> x))" |
2232 |
||
63832 | 2233 |
definition islcm :: "[_, 'a, 'a, 'a] \<Rightarrow> bool" ("(_ lcmof\<index> _ _)" [81,81,81] 80) |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
2234 |
where "x lcmof\<^bsub>G\<^esub> a b \<longleftrightarrow> a divides\<^bsub>G\<^esub> x \<and> b divides\<^bsub>G\<^esub> x \<and> |
35847 | 2235 |
(\<forall>y\<in>carrier G. (a divides\<^bsub>G\<^esub> y \<and> b divides\<^bsub>G\<^esub> y \<longrightarrow> x divides\<^bsub>G\<^esub> y))" |
2236 |
||
63832 | 2237 |
definition somegcd :: "('a,_) monoid_scheme \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
2238 |
where "somegcd G a b = (SOME x. x \<in> carrier G \<and> x gcdof\<^bsub>G\<^esub> a b)" |
35847 | 2239 |
|
63832 | 2240 |
definition somelcm :: "('a,_) monoid_scheme \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
2241 |
where "somelcm G a b = (SOME x. x \<in> carrier G \<and> x lcmof\<^bsub>G\<^esub> a b)" |
35847 | 2242 |
|
63832 | 2243 |
definition "SomeGcd G A = inf (division_rel G) A" |
27701 | 2244 |
|
2245 |
||
2246 |
locale gcd_condition_monoid = comm_monoid_cancel + |
|
63832 | 2247 |
assumes gcdof_exists: "\<lbrakk>a \<in> carrier G; b \<in> carrier G\<rbrakk> \<Longrightarrow> \<exists>c. c \<in> carrier G \<and> c gcdof a b" |
27701 | 2248 |
|
2249 |
locale primeness_condition_monoid = comm_monoid_cancel + |
|
63832 | 2250 |
assumes irreducible_prime: "\<lbrakk>a \<in> carrier G; irreducible G a\<rbrakk> \<Longrightarrow> prime G a" |
27701 | 2251 |
|
2252 |
locale divisor_chain_condition_monoid = comm_monoid_cancel + |
|
63832 | 2253 |
assumes division_wellfounded: "wf {(x, y). x \<in> carrier G \<and> y \<in> carrier G \<and> properfactor G x y}" |
27701 | 2254 |
|
2255 |
||
61382 | 2256 |
subsubsection \<open>Connections to \texttt{Lattice.thy}\<close> |
27701 | 2257 |
|
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2258 |
lemma gcdof_greatestLower: |
27701 | 2259 |
fixes G (structure) |
2260 |
assumes carr[simp]: "a \<in> carrier G" "b \<in> carrier G" |
|
63832 | 2261 |
shows "(x \<in> carrier G \<and> x gcdof a b) = greatest (division_rel G) x (Lower (division_rel G) {a, b})" |
2262 |
by (auto simp: isgcd_def greatest_def Lower_def elem_def) |
|
27701 | 2263 |
|
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2264 |
lemma lcmof_leastUpper: |
27701 | 2265 |
fixes G (structure) |
2266 |
assumes carr[simp]: "a \<in> carrier G" "b \<in> carrier G" |
|
63832 | 2267 |
shows "(x \<in> carrier G \<and> x lcmof a b) = least (division_rel G) x (Upper (division_rel G) {a, b})" |
2268 |
by (auto simp: islcm_def least_def Upper_def elem_def) |
|
27701 | 2269 |
|
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2270 |
lemma somegcd_meet: |
27701 | 2271 |
fixes G (structure) |
2272 |
assumes carr: "a \<in> carrier G" "b \<in> carrier G" |
|
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2273 |
shows "somegcd G a b = meet (division_rel G) a b" |
63832 | 2274 |
by (simp add: somegcd_def meet_def inf_def gcdof_greatestLower[OF carr]) |
27701 | 2275 |
|
2276 |
lemma (in monoid) isgcd_divides_l: |
|
2277 |
assumes "a divides b" |
|
2278 |
and "a \<in> carrier G" "b \<in> carrier G" |
|
2279 |
shows "a gcdof a b" |
|
63832 | 2280 |
using assms unfolding isgcd_def by fast |
27701 | 2281 |
|
2282 |
lemma (in monoid) isgcd_divides_r: |
|
2283 |
assumes "b divides a" |
|
2284 |
and "a \<in> carrier G" "b \<in> carrier G" |
|
2285 |
shows "b gcdof a b" |
|
63832 | 2286 |
using assms unfolding isgcd_def by fast |
27701 | 2287 |
|
2288 |
||
61382 | 2289 |
subsubsection \<open>Existence of gcd and lcm\<close> |
27701 | 2290 |
|
2291 |
lemma (in factorial_monoid) gcdof_exists: |
|
63832 | 2292 |
assumes acarr: "a \<in> carrier G" |
2293 |
and bcarr: "b \<in> carrier G" |
|
27701 | 2294 |
shows "\<exists>c. c \<in> carrier G \<and> c gcdof a b" |
2295 |
proof - |
|
63847 | 2296 |
from wfactors_exist [OF acarr] |
2297 |
obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" |
|
2298 |
by blast |
|
63832 | 2299 |
from afs have airr: "\<forall>a \<in> set as. irreducible G a" |
2300 |
by (fast elim: wfactorsE) |
|
2301 |
||
63847 | 2302 |
from wfactors_exist [OF bcarr] |
2303 |
obtain bs where bscarr: "set bs \<subseteq> carrier G" and bfs: "wfactors G bs b" |
|
2304 |
by blast |
|
63832 | 2305 |
from bfs have birr: "\<forall>b \<in> set bs. irreducible G b" |
2306 |
by (fast elim: wfactorsE) |
|
2307 |
||
2308 |
have "\<exists>c cs. c \<in> carrier G \<and> set cs \<subseteq> carrier G \<and> wfactors G cs c \<and> |
|
63919
9aed2da07200
# after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63847
diff
changeset
|
2309 |
fmset G cs = fmset G as \<inter># fmset G bs" |
27701 | 2310 |
proof (intro mset_wfactorsEx) |
2311 |
fix X |
|
63919
9aed2da07200
# after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63847
diff
changeset
|
2312 |
assume "X \<in># fmset G as \<inter># fmset G bs" |
63832 | 2313 |
then have "X \<in># fmset G as" by simp |
2314 |
then have "X \<in> set (map (assocs G) as)" |
|
2315 |
by (simp add: fmset_def) |
|
2316 |
then have "\<exists>x. X = assocs G x \<and> x \<in> set as" |
|
2317 |
by (induct as) auto |
|
2318 |
then obtain x where X: "X = assocs G x" and xas: "x \<in> set as" |
|
63847 | 2319 |
by blast |
63832 | 2320 |
with ascarr have xcarr: "x \<in> carrier G" |
63847 | 2321 |
by blast |
63832 | 2322 |
from xas airr have xirr: "irreducible G x" |
2323 |
by simp |
|
2324 |
from xcarr and xirr and X show "\<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" |
|
63847 | 2325 |
by blast |
27701 | 2326 |
qed |
63832 | 2327 |
then obtain c cs |
2328 |
where ccarr: "c \<in> carrier G" |
|
2329 |
and cscarr: "set cs \<subseteq> carrier G" |
|
27701 | 2330 |
and csirr: "wfactors G cs c" |
63919
9aed2da07200
# after multiset intersection and union symbol
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63847
diff
changeset
|
2331 |
and csmset: "fmset G cs = fmset G as \<inter># fmset G bs" |
63832 | 2332 |
by auto |
27701 | 2333 |
|
2334 |
have "c gcdof a b" |
|
2335 |
proof (simp add: isgcd_def, safe) |
|
2336 |
from csmset |
|
64587 | 2337 |
have "fmset G cs \<subseteq># fmset G as" |
63832 | 2338 |
by (simp add: multiset_inter_def subset_mset_def) |
2339 |
then show "c divides a" by (rule fmsubset_divides) fact+ |
|
27701 | 2340 |
next |
64587 | 2341 |
from csmset have "fmset G cs \<subseteq># fmset G bs" |
63832 | 2342 |
by (simp add: multiset_inter_def subseteq_mset_def, force) |
2343 |
then show "c divides b" |
|
2344 |
by (rule fmsubset_divides) fact+ |
|
27701 | 2345 |
next |
2346 |
fix y |
|
63847 | 2347 |
assume "y \<in> carrier G" |
2348 |
from wfactors_exist [OF this] |
|
2349 |
obtain ys where yscarr: "set ys \<subseteq> carrier G" and yfs: "wfactors G ys y" |
|
2350 |
by blast |
|
27701 | 2351 |
|
2352 |
assume "y divides a" |
|
64587 | 2353 |
then have ya: "fmset G ys \<subseteq># fmset G as" |
63832 | 2354 |
by (rule divides_fmsubset) fact+ |
27701 | 2355 |
|
2356 |
assume "y divides b" |
|
64587 | 2357 |
then have yb: "fmset G ys \<subseteq># fmset G bs" |
63832 | 2358 |
by (rule divides_fmsubset) fact+ |
2359 |
||
64587 | 2360 |
from ya yb csmset have "fmset G ys \<subseteq># fmset G cs" |
63832 | 2361 |
by (simp add: subset_mset_def) |
2362 |
then show "y divides c" |
|
2363 |
by (rule fmsubset_divides) fact+ |
|
27701 | 2364 |
qed |
63832 | 2365 |
with ccarr show "\<exists>c. c \<in> carrier G \<and> c gcdof a b" |
2366 |
by fast |
|
27701 | 2367 |
qed |
2368 |
||
2369 |
lemma (in factorial_monoid) lcmof_exists: |
|
63832 | 2370 |
assumes acarr: "a \<in> carrier G" |
2371 |
and bcarr: "b \<in> carrier G" |
|
27701 | 2372 |
shows "\<exists>c. c \<in> carrier G \<and> c lcmof a b" |
2373 |
proof - |
|
63847 | 2374 |
from wfactors_exist [OF acarr] |
2375 |
obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" |
|
2376 |
by blast |
|
63832 | 2377 |
from afs have airr: "\<forall>a \<in> set as. irreducible G a" |
2378 |
by (fast elim: wfactorsE) |
|
2379 |
||
63847 | 2380 |
from wfactors_exist [OF bcarr] |
2381 |
obtain bs where bscarr: "set bs \<subseteq> carrier G" and bfs: "wfactors G bs b" |
|
2382 |
by blast |
|
63832 | 2383 |
from bfs have birr: "\<forall>b \<in> set bs. irreducible G b" |
2384 |
by (fast elim: wfactorsE) |
|
2385 |
||
2386 |
have "\<exists>c cs. c \<in> carrier G \<and> set cs \<subseteq> carrier G \<and> wfactors G cs c \<and> |
|
2387 |
fmset G cs = (fmset G as - fmset G bs) + fmset G bs" |
|
27701 | 2388 |
proof (intro mset_wfactorsEx) |
2389 |
fix X |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
61382
diff
changeset
|
2390 |
assume "X \<in># (fmset G as - fmset G bs) + fmset G bs" |
63832 | 2391 |
then have "X \<in># fmset G as \<or> X \<in># fmset G bs" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
61382
diff
changeset
|
2392 |
by (auto dest: in_diffD) |
63832 | 2393 |
then consider "X \<in> set_mset (fmset G as)" | "X \<in> set_mset (fmset G bs)" |
2394 |
by fast |
|
2395 |
then show "\<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" |
|
2396 |
proof cases |
|
2397 |
case 1 |
|
2398 |
then have "X \<in> set (map (assocs G) as)" by (simp add: fmset_def) |
|
2399 |
then have "\<exists>x. x \<in> set as \<and> X = assocs G x" by (induct as) auto |
|
2400 |
then obtain x where xas: "x \<in> set as" and X: "X = assocs G x" by auto |
|
27701 | 2401 |
with ascarr have xcarr: "x \<in> carrier G" by fast |
2402 |
from xas airr have xirr: "irreducible G x" by simp |
|
63832 | 2403 |
from xcarr and xirr and X show ?thesis by fast |
2404 |
next |
|
2405 |
case 2 |
|
2406 |
then have "X \<in> set (map (assocs G) bs)" by (simp add: fmset_def) |
|
2407 |
then have "\<exists>x. x \<in> set bs \<and> X = assocs G x" by (induct as) auto |
|
2408 |
then obtain x where xbs: "x \<in> set bs" and X: "X = assocs G x" by auto |
|
27701 | 2409 |
with bscarr have xcarr: "x \<in> carrier G" by fast |
2410 |
from xbs birr have xirr: "irreducible G x" by simp |
|
63832 | 2411 |
from xcarr and xirr and X show ?thesis by fast |
2412 |
qed |
|
27701 | 2413 |
qed |
63832 | 2414 |
then obtain c cs |
2415 |
where ccarr: "c \<in> carrier G" |
|
2416 |
and cscarr: "set cs \<subseteq> carrier G" |
|
27701 | 2417 |
and csirr: "wfactors G cs c" |
63832 | 2418 |
and csmset: "fmset G cs = fmset G as - fmset G bs + fmset G bs" |
2419 |
by auto |
|
27701 | 2420 |
|
2421 |
have "c lcmof a b" |
|
2422 |
proof (simp add: islcm_def, safe) |
|
64587 | 2423 |
from csmset have "fmset G as \<subseteq># fmset G cs" |
63832 | 2424 |
by (simp add: subseteq_mset_def, force) |
2425 |
then show "a divides c" |
|
2426 |
by (rule fmsubset_divides) fact+ |
|
27701 | 2427 |
next |
64587 | 2428 |
from csmset have "fmset G bs \<subseteq># fmset G cs" |
63832 | 2429 |
by (simp add: subset_mset_def) |
2430 |
then show "b divides c" |
|
2431 |
by (rule fmsubset_divides) fact+ |
|
27701 | 2432 |
next |
2433 |
fix y |
|
63847 | 2434 |
assume "y \<in> carrier G" |
2435 |
from wfactors_exist [OF this] |
|
2436 |
obtain ys where yscarr: "set ys \<subseteq> carrier G" and yfs: "wfactors G ys y" |
|
2437 |
by blast |
|
27701 | 2438 |
|
2439 |
assume "a divides y" |
|
64587 | 2440 |
then have ya: "fmset G as \<subseteq># fmset G ys" |
63832 | 2441 |
by (rule divides_fmsubset) fact+ |
27701 | 2442 |
|
2443 |
assume "b divides y" |
|
64587 | 2444 |
then have yb: "fmset G bs \<subseteq># fmset G ys" |
63832 | 2445 |
by (rule divides_fmsubset) fact+ |
2446 |
||
64587 | 2447 |
from ya yb csmset have "fmset G cs \<subseteq># fmset G ys" |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
60143
diff
changeset
|
2448 |
apply (simp add: subseteq_mset_def, clarify) |
27701 | 2449 |
apply (case_tac "count (fmset G as) a < count (fmset G bs) a") |
2450 |
apply simp |
|
2451 |
apply simp |
|
63832 | 2452 |
done |
2453 |
then show "c divides y" |
|
2454 |
by (rule fmsubset_divides) fact+ |
|
27701 | 2455 |
qed |
63832 | 2456 |
with ccarr show "\<exists>c. c \<in> carrier G \<and> c lcmof a b" |
2457 |
by fast |
|
27701 | 2458 |
qed |
2459 |
||
2460 |
||
61382 | 2461 |
subsection \<open>Conditions for Factoriality\<close> |
2462 |
||
2463 |
subsubsection \<open>Gcd condition\<close> |
|
27701 | 2464 |
|
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2465 |
lemma (in gcd_condition_monoid) division_weak_lower_semilattice [simp]: |
63832 | 2466 |
"weak_lower_semilattice (division_rel G)" |
27701 | 2467 |
proof - |
29237 | 2468 |
interpret weak_partial_order "division_rel G" .. |
27701 | 2469 |
show ?thesis |
63832 | 2470 |
apply (unfold_locales, simp_all) |
27701 | 2471 |
proof - |
2472 |
fix x y |
|
2473 |
assume carr: "x \<in> carrier G" "y \<in> carrier G" |
|
63847 | 2474 |
from gcdof_exists [OF this] obtain z where zcarr: "z \<in> carrier G" and isgcd: "z gcdof x y" |
2475 |
by blast |
|
63832 | 2476 |
with carr have "greatest (division_rel G) z (Lower (division_rel G) {x, y})" |
2477 |
by (subst gcdof_greatestLower[symmetric], simp+) |
|
2478 |
then show "\<exists>z. greatest (division_rel G) z (Lower (division_rel G) {x, y})" |
|
2479 |
by fast |
|
27701 | 2480 |
qed |
2481 |
qed |
|
2482 |
||
2483 |
lemma (in gcd_condition_monoid) gcdof_cong_l: |
|
2484 |
assumes a'a: "a' \<sim> a" |
|
2485 |
and agcd: "a gcdof b c" |
|
2486 |
and a'carr: "a' \<in> carrier G" and carr': "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
|
2487 |
shows "a' gcdof b c" |
|
2488 |
proof - |
|
2489 |
note carr = a'carr carr' |
|
29237 | 2490 |
interpret weak_lower_semilattice "division_rel G" by simp |
27701 | 2491 |
have "a' \<in> carrier G \<and> a' gcdof b c" |
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2492 |
apply (simp add: gcdof_greatestLower carr') |
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2493 |
apply (subst greatest_Lower_cong_l[of _ a]) |
63832 | 2494 |
apply (simp add: a'a) |
2495 |
apply (simp add: carr) |
|
27701 | 2496 |
apply (simp add: carr) |
2497 |
apply (simp add: carr) |
|
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2498 |
apply (simp add: gcdof_greatestLower[symmetric] agcd carr) |
63832 | 2499 |
done |
2500 |
then show ?thesis .. |
|
27701 | 2501 |
qed |
2502 |
||
2503 |
lemma (in gcd_condition_monoid) gcd_closed [simp]: |
|
2504 |
assumes carr: "a \<in> carrier G" "b \<in> carrier G" |
|
2505 |
shows "somegcd G a b \<in> carrier G" |
|
2506 |
proof - |
|
29237 | 2507 |
interpret weak_lower_semilattice "division_rel G" by simp |
27701 | 2508 |
show ?thesis |
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2509 |
apply (simp add: somegcd_meet[OF carr]) |
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2510 |
apply (rule meet_closed[simplified], fact+) |
63832 | 2511 |
done |
27701 | 2512 |
qed |
2513 |
||
2514 |
lemma (in gcd_condition_monoid) gcd_isgcd: |
|
2515 |
assumes carr: "a \<in> carrier G" "b \<in> carrier G" |
|
2516 |
shows "(somegcd G a b) gcdof a b" |
|
2517 |
proof - |
|
63832 | 2518 |
interpret weak_lower_semilattice "division_rel G" |
2519 |
by simp |
|
2520 |
from carr have "somegcd G a b \<in> carrier G \<and> (somegcd G a b) gcdof a b" |
|
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2521 |
apply (subst gcdof_greatestLower, simp, simp) |
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2522 |
apply (simp add: somegcd_meet[OF carr] meet_def) |
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2523 |
apply (rule inf_of_two_greatest[simplified], assumption+) |
63832 | 2524 |
done |
2525 |
then show "(somegcd G a b) gcdof a b" |
|
2526 |
by simp |
|
27701 | 2527 |
qed |
2528 |
||
2529 |
lemma (in gcd_condition_monoid) gcd_exists: |
|
2530 |
assumes carr: "a \<in> carrier G" "b \<in> carrier G" |
|
2531 |
shows "\<exists>x\<in>carrier G. x = somegcd G a b" |
|
2532 |
proof - |
|
63832 | 2533 |
interpret weak_lower_semilattice "division_rel G" |
2534 |
by simp |
|
27701 | 2535 |
show ?thesis |
55242
413ec965f95d
Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents:
53374
diff
changeset
|
2536 |
by (metis carr(1) carr(2) gcd_closed) |
27701 | 2537 |
qed |
2538 |
||
2539 |
lemma (in gcd_condition_monoid) gcd_divides_l: |
|
2540 |
assumes carr: "a \<in> carrier G" "b \<in> carrier G" |
|
2541 |
shows "(somegcd G a b) divides a" |
|
2542 |
proof - |
|
63832 | 2543 |
interpret weak_lower_semilattice "division_rel G" |
2544 |
by simp |
|
27701 | 2545 |
show ?thesis |
55242
413ec965f95d
Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents:
53374
diff
changeset
|
2546 |
by (metis carr(1) carr(2) gcd_isgcd isgcd_def) |
27701 | 2547 |
qed |
2548 |
||
2549 |
lemma (in gcd_condition_monoid) gcd_divides_r: |
|
2550 |
assumes carr: "a \<in> carrier G" "b \<in> carrier G" |
|
2551 |
shows "(somegcd G a b) divides b" |
|
2552 |
proof - |
|
63832 | 2553 |
interpret weak_lower_semilattice "division_rel G" |
2554 |
by simp |
|
27701 | 2555 |
show ?thesis |
55242
413ec965f95d
Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents:
53374
diff
changeset
|
2556 |
by (metis carr gcd_isgcd isgcd_def) |
27701 | 2557 |
qed |
2558 |
||
2559 |
lemma (in gcd_condition_monoid) gcd_divides: |
|
2560 |
assumes sub: "z divides x" "z divides y" |
|
2561 |
and L: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
|
2562 |
shows "z divides (somegcd G x y)" |
|
2563 |
proof - |
|
63832 | 2564 |
interpret weak_lower_semilattice "division_rel G" |
2565 |
by simp |
|
27701 | 2566 |
show ?thesis |
55242
413ec965f95d
Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents:
53374
diff
changeset
|
2567 |
by (metis gcd_isgcd isgcd_def assms) |
27701 | 2568 |
qed |
2569 |
||
2570 |
lemma (in gcd_condition_monoid) gcd_cong_l: |
|
2571 |
assumes xx': "x \<sim> x'" |
|
2572 |
and carr: "x \<in> carrier G" "x' \<in> carrier G" "y \<in> carrier G" |
|
2573 |
shows "somegcd G x y \<sim> somegcd G x' y" |
|
2574 |
proof - |
|
63832 | 2575 |
interpret weak_lower_semilattice "division_rel G" |
2576 |
by simp |
|
27701 | 2577 |
show ?thesis |
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2578 |
apply (simp add: somegcd_meet carr) |
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2579 |
apply (rule meet_cong_l[simplified], fact+) |
63832 | 2580 |
done |
27701 | 2581 |
qed |
2582 |
||
2583 |
lemma (in gcd_condition_monoid) gcd_cong_r: |
|
2584 |
assumes carr: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G" |
|
2585 |
and yy': "y \<sim> y'" |
|
2586 |
shows "somegcd G x y \<sim> somegcd G x y'" |
|
2587 |
proof - |
|
29237 | 2588 |
interpret weak_lower_semilattice "division_rel G" by simp |
27701 | 2589 |
show ?thesis |
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2590 |
apply (simp add: somegcd_meet carr) |
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2591 |
apply (rule meet_cong_r[simplified], fact+) |
63832 | 2592 |
done |
27701 | 2593 |
qed |
2594 |
||
2595 |
(* |
|
2596 |
lemma (in gcd_condition_monoid) asc_cong_gcd_l [intro]: |
|
2597 |
assumes carr: "b \<in> carrier G" |
|
2598 |
shows "asc_cong (\<lambda>a. somegcd G a b)" |
|
2599 |
using carr |
|
2600 |
unfolding CONG_def |
|
2601 |
by clarsimp (blast intro: gcd_cong_l) |
|
2602 |
||
2603 |
lemma (in gcd_condition_monoid) asc_cong_gcd_r [intro]: |
|
2604 |
assumes carr: "a \<in> carrier G" |
|
2605 |
shows "asc_cong (\<lambda>b. somegcd G a b)" |
|
2606 |
using carr |
|
2607 |
unfolding CONG_def |
|
2608 |
by clarsimp (blast intro: gcd_cong_r) |
|
2609 |
||
63832 | 2610 |
lemmas (in gcd_condition_monoid) asc_cong_gcd_split [simp] = |
27701 | 2611 |
assoc_split[OF _ asc_cong_gcd_l] assoc_split[OF _ asc_cong_gcd_r] |
2612 |
*) |
|
2613 |
||
2614 |
lemma (in gcd_condition_monoid) gcdI: |
|
2615 |
assumes dvd: "a divides b" "a divides c" |
|
2616 |
and others: "\<forall>y\<in>carrier G. y divides b \<and> y divides c \<longrightarrow> y divides a" |
|
2617 |
and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and ccarr: "c \<in> carrier G" |
|
2618 |
shows "a \<sim> somegcd G b c" |
|
63832 | 2619 |
apply (simp add: somegcd_def) |
2620 |
apply (rule someI2_ex) |
|
2621 |
apply (rule exI[of _ a], simp add: isgcd_def) |
|
2622 |
apply (simp add: assms) |
|
2623 |
apply (simp add: isgcd_def assms, clarify) |
|
2624 |
apply (insert assms, blast intro: associatedI) |
|
2625 |
done |
|
27701 | 2626 |
|
2627 |
lemma (in gcd_condition_monoid) gcdI2: |
|
63832 | 2628 |
assumes "a gcdof b c" and "a \<in> carrier G" and "b \<in> carrier G" and "c \<in> carrier G" |
27701 | 2629 |
shows "a \<sim> somegcd G b c" |
63832 | 2630 |
using assms unfolding isgcd_def by (blast intro: gcdI) |
27701 | 2631 |
|
2632 |
lemma (in gcd_condition_monoid) SomeGcd_ex: |
|
2633 |
assumes "finite A" "A \<subseteq> carrier G" "A \<noteq> {}" |
|
2634 |
shows "\<exists>x\<in> carrier G. x = SomeGcd G A" |
|
2635 |
proof - |
|
63832 | 2636 |
interpret weak_lower_semilattice "division_rel G" |
2637 |
by simp |
|
27701 | 2638 |
show ?thesis |
2639 |
apply (simp add: SomeGcd_def) |
|
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2640 |
apply (rule finite_inf_closed[simplified], fact+) |
63832 | 2641 |
done |
27701 | 2642 |
qed |
2643 |
||
2644 |
lemma (in gcd_condition_monoid) gcd_assoc: |
|
2645 |
assumes carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
|
2646 |
shows "somegcd G (somegcd G a b) c \<sim> somegcd G a (somegcd G b c)" |
|
2647 |
proof - |
|
63832 | 2648 |
interpret weak_lower_semilattice "division_rel G" |
2649 |
by simp |
|
27701 | 2650 |
show ?thesis |
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2651 |
apply (subst (2 3) somegcd_meet, (simp add: carr)+) |
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2652 |
apply (simp add: somegcd_meet carr) |
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
2653 |
apply (rule weak_meet_assoc[simplified], fact+) |
63832 | 2654 |
done |
27701 | 2655 |
qed |
2656 |
||
2657 |
lemma (in gcd_condition_monoid) gcd_mult: |
|
2658 |
assumes acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and ccarr: "c \<in> carrier G" |
|
2659 |
shows "c \<otimes> somegcd G a b \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)" |
|
2660 |
proof - (* following Jacobson, Basic Algebra, p.140 *) |
|
2661 |
let ?d = "somegcd G a b" |
|
2662 |
let ?e = "somegcd G (c \<otimes> a) (c \<otimes> b)" |
|
2663 |
note carr[simp] = acarr bcarr ccarr |
|
2664 |
have dcarr: "?d \<in> carrier G" by simp |
|
2665 |
have ecarr: "?e \<in> carrier G" by simp |
|
2666 |
note carr = carr dcarr ecarr |
|
2667 |
||
2668 |
have "?d divides a" by (simp add: gcd_divides_l) |
|
63832 | 2669 |
then have cd'ca: "c \<otimes> ?d divides (c \<otimes> a)" by (simp add: divides_mult_lI) |
27701 | 2670 |
|
2671 |
have "?d divides b" by (simp add: gcd_divides_r) |
|
63832 | 2672 |
then have cd'cb: "c \<otimes> ?d divides (c \<otimes> b)" by (simp add: divides_mult_lI) |
2673 |
||
2674 |
from cd'ca cd'cb have cd'e: "c \<otimes> ?d divides ?e" |
|
2675 |
by (rule gcd_divides) simp_all |
|
2676 |
then obtain u where ucarr[simp]: "u \<in> carrier G" and e_cdu: "?e = c \<otimes> ?d \<otimes> u" |
|
63847 | 2677 |
by blast |
27701 | 2678 |
|
2679 |
note carr = carr ucarr |
|
2680 |
||
63832 | 2681 |
have "?e divides c \<otimes> a" by (rule gcd_divides_l) simp_all |
2682 |
then obtain x where xcarr: "x \<in> carrier G" and ca_ex: "c \<otimes> a = ?e \<otimes> x" |
|
63847 | 2683 |
by blast |
63832 | 2684 |
with e_cdu have ca_cdux: "c \<otimes> a = c \<otimes> ?d \<otimes> u \<otimes> x" |
2685 |
by simp |
|
2686 |
||
2687 |
from ca_cdux xcarr have "c \<otimes> a = c \<otimes> (?d \<otimes> u \<otimes> x)" |
|
2688 |
by (simp add: m_assoc) |
|
2689 |
then have "a = ?d \<otimes> u \<otimes> x" |
|
2690 |
by (rule l_cancel[of c a]) (simp add: xcarr)+ |
|
2691 |
then have du'a: "?d \<otimes> u divides a" |
|
2692 |
by (rule dividesI[OF xcarr]) |
|
2693 |
||
2694 |
have "?e divides c \<otimes> b" by (intro gcd_divides_r) simp_all |
|
2695 |
then obtain x where xcarr: "x \<in> carrier G" and cb_ex: "c \<otimes> b = ?e \<otimes> x" |
|
63847 | 2696 |
by blast |
63832 | 2697 |
with e_cdu have cb_cdux: "c \<otimes> b = c \<otimes> ?d \<otimes> u \<otimes> x" |
2698 |
by simp |
|
2699 |
||
2700 |
from cb_cdux xcarr have "c \<otimes> b = c \<otimes> (?d \<otimes> u \<otimes> x)" |
|
2701 |
by (simp add: m_assoc) |
|
2702 |
with xcarr have "b = ?d \<otimes> u \<otimes> x" |
|
2703 |
by (intro l_cancel[of c b]) simp_all |
|
2704 |
then have du'b: "?d \<otimes> u divides b" |
|
2705 |
by (intro dividesI[OF xcarr]) |
|
2706 |
||
2707 |
from du'a du'b carr have du'd: "?d \<otimes> u divides ?d" |
|
2708 |
by (intro gcd_divides) simp_all |
|
2709 |
then have uunit: "u \<in> Units G" |
|
27701 | 2710 |
proof (elim dividesE) |
2711 |
fix v |
|
2712 |
assume vcarr[simp]: "v \<in> carrier G" |
|
2713 |
assume d: "?d = ?d \<otimes> u \<otimes> v" |
|
2714 |
have "?d \<otimes> \<one> = ?d \<otimes> u \<otimes> v" by simp fact |
|
2715 |
also have "?d \<otimes> u \<otimes> v = ?d \<otimes> (u \<otimes> v)" by (simp add: m_assoc) |
|
2716 |
finally have "?d \<otimes> \<one> = ?d \<otimes> (u \<otimes> v)" . |
|
63832 | 2717 |
then have i2: "\<one> = u \<otimes> v" by (rule l_cancel) simp_all |
2718 |
then have i1: "\<one> = v \<otimes> u" by (simp add: m_comm) |
|
2719 |
from vcarr i1[symmetric] i2[symmetric] show "u \<in> Units G" |
|
2720 |
by (auto simp: Units_def) |
|
27701 | 2721 |
qed |
2722 |
||
63832 | 2723 |
from e_cdu uunit have "somegcd G (c \<otimes> a) (c \<otimes> b) \<sim> c \<otimes> somegcd G a b" |
2724 |
by (intro associatedI2[of u]) simp_all |
|
2725 |
from this[symmetric] show "c \<otimes> somegcd G a b \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)" |
|
2726 |
by simp |
|
27701 | 2727 |
qed |
2728 |
||
2729 |
lemma (in monoid) assoc_subst: |
|
2730 |
assumes ab: "a \<sim> b" |
|
63832 | 2731 |
and cP: "\<forall>a b. a \<in> carrier G \<and> b \<in> carrier G \<and> a \<sim> b |
2732 |
\<longrightarrow> f a \<in> carrier G \<and> f b \<in> carrier G \<and> f a \<sim> f b" |
|
27701 | 2733 |
and carr: "a \<in> carrier G" "b \<in> carrier G" |
2734 |
shows "f a \<sim> f b" |
|
2735 |
using assms by auto |
|
2736 |
||
2737 |
lemma (in gcd_condition_monoid) relprime_mult: |
|
63832 | 2738 |
assumes abrelprime: "somegcd G a b \<sim> \<one>" |
2739 |
and acrelprime: "somegcd G a c \<sim> \<one>" |
|
27701 | 2740 |
and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" |
2741 |
shows "somegcd G a (b \<otimes> c) \<sim> \<one>" |
|
2742 |
proof - |
|
2743 |
have "c = c \<otimes> \<one>" by simp |
|
2744 |
also from abrelprime[symmetric] |
|
63832 | 2745 |
have "\<dots> \<sim> c \<otimes> somegcd G a b" |
2746 |
by (rule assoc_subst) (simp add: mult_cong_r)+ |
|
2747 |
also have "\<dots> \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)" |
|
2748 |
by (rule gcd_mult) fact+ |
|
2749 |
finally have c: "c \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)" |
|
2750 |
by simp |
|
2751 |
||
2752 |
from carr have a: "a \<sim> somegcd G a (c \<otimes> a)" |
|
2753 |
by (fast intro: gcdI divides_prod_l) |
|
2754 |
||
2755 |
have "somegcd G a (b \<otimes> c) \<sim> somegcd G a (c \<otimes> b)" |
|
2756 |
by (simp add: m_comm) |
|
2757 |
also from a have "\<dots> \<sim> somegcd G (somegcd G a (c \<otimes> a)) (c \<otimes> b)" |
|
2758 |
by (rule assoc_subst) (simp add: gcd_cong_l)+ |
|
2759 |
also from gcd_assoc have "\<dots> \<sim> somegcd G a (somegcd G (c \<otimes> a) (c \<otimes> b))" |
|
2760 |
by (rule assoc_subst) simp+ |
|
2761 |
also from c[symmetric] have "\<dots> \<sim> somegcd G a c" |
|
2762 |
by (rule assoc_subst) (simp add: gcd_cong_r)+ |
|
27701 | 2763 |
also note acrelprime |
63832 | 2764 |
finally show "somegcd G a (b \<otimes> c) \<sim> \<one>" |
2765 |
by simp |
|
27701 | 2766 |
qed |
2767 |
||
63832 | 2768 |
lemma (in gcd_condition_monoid) primeness_condition: "primeness_condition_monoid G" |
2769 |
apply unfold_locales |
|
2770 |
apply (rule primeI) |
|
2771 |
apply (elim irreducibleE, assumption) |
|
27701 | 2772 |
proof - |
2773 |
fix p a b |
|
2774 |
assume pcarr: "p \<in> carrier G" and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" |
|
2775 |
and pirr: "irreducible G p" |
|
2776 |
and pdvdab: "p divides a \<otimes> b" |
|
63832 | 2777 |
from pirr have pnunit: "p \<notin> Units G" |
2778 |
and r[rule_format]: "\<forall>b. b \<in> carrier G \<and> properfactor G b p \<longrightarrow> b \<in> Units G" |
|
2779 |
by (fast elim: irreducibleE)+ |
|
27701 | 2780 |
|
2781 |
show "p divides a \<or> p divides b" |
|
2782 |
proof (rule ccontr, clarsimp) |
|
2783 |
assume npdvda: "\<not> p divides a" |
|
63832 | 2784 |
with pcarr acarr have "\<one> \<sim> somegcd G p a" |
2785 |
apply (intro gcdI, simp, simp, simp) |
|
2786 |
apply (fast intro: unit_divides) |
|
2787 |
apply (fast intro: unit_divides) |
|
2788 |
apply (clarsimp simp add: Unit_eq_dividesone[symmetric]) |
|
2789 |
apply (rule r, rule, assumption) |
|
2790 |
apply (rule properfactorI, assumption) |
|
63846 | 2791 |
proof |
27701 | 2792 |
fix y |
2793 |
assume ycarr: "y \<in> carrier G" |
|
2794 |
assume "p divides y" |
|
2795 |
also assume "y divides a" |
|
63832 | 2796 |
finally have "p divides a" |
2797 |
by (simp add: pcarr ycarr acarr) |
|
2798 |
with npdvda show False .. |
|
2799 |
qed simp_all |
|
2800 |
with pcarr acarr have pa: "somegcd G p a \<sim> \<one>" |
|
2801 |
by (fast intro: associated_sym[of "\<one>"] gcd_closed) |
|
27701 | 2802 |
|
2803 |
assume npdvdb: "\<not> p divides b" |
|
63832 | 2804 |
with pcarr bcarr have "\<one> \<sim> somegcd G p b" |
2805 |
apply (intro gcdI, simp, simp, simp) |
|
2806 |
apply (fast intro: unit_divides) |
|
2807 |
apply (fast intro: unit_divides) |
|
2808 |
apply (clarsimp simp add: Unit_eq_dividesone[symmetric]) |
|
2809 |
apply (rule r, rule, assumption) |
|
2810 |
apply (rule properfactorI, assumption) |
|
63846 | 2811 |
proof |
27701 | 2812 |
fix y |
2813 |
assume ycarr: "y \<in> carrier G" |
|
2814 |
assume "p divides y" |
|
2815 |
also assume "y divides b" |
|
2816 |
finally have "p divides b" by (simp add: pcarr ycarr bcarr) |
|
2817 |
with npdvdb |
|
63832 | 2818 |
show "False" .. |
2819 |
qed simp_all |
|
2820 |
with pcarr bcarr have pb: "somegcd G p b \<sim> \<one>" |
|
2821 |
by (fast intro: associated_sym[of "\<one>"] gcd_closed) |
|
2822 |
||
2823 |
from pcarr acarr bcarr pdvdab have "p gcdof p (a \<otimes> b)" |
|
2824 |
by (fast intro: isgcd_divides_l) |
|
2825 |
with pcarr acarr bcarr have "p \<sim> somegcd G p (a \<otimes> b)" |
|
2826 |
by (fast intro: gcdI2) |
|
2827 |
also from pa pb pcarr acarr bcarr have "somegcd G p (a \<otimes> b) \<sim> \<one>" |
|
2828 |
by (rule relprime_mult) |
|
2829 |
finally have "p \<sim> \<one>" |
|
2830 |
by (simp add: pcarr acarr bcarr) |
|
2831 |
with pcarr have "p \<in> Units G" |
|
2832 |
by (fast intro: assoc_unit_l) |
|
2833 |
with pnunit show False .. |
|
27701 | 2834 |
qed |
2835 |
qed |
|
2836 |
||
29237 | 2837 |
sublocale gcd_condition_monoid \<subseteq> primeness_condition_monoid |
27701 | 2838 |
by (rule primeness_condition) |
2839 |
||
2840 |
||
61382 | 2841 |
subsubsection \<open>Divisor chain condition\<close> |
27701 | 2842 |
|
2843 |
lemma (in divisor_chain_condition_monoid) wfactors_exist: |
|
2844 |
assumes acarr: "a \<in> carrier G" |
|
2845 |
shows "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" |
|
2846 |
proof - |
|
2847 |
have r[rule_format]: "a \<in> carrier G \<longrightarrow> (\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a)" |
|
63832 | 2848 |
proof (rule wf_induct[OF division_wellfounded]) |
27701 | 2849 |
fix x |
2850 |
assume ih: "\<forall>y. (y, x) \<in> {(x, y). x \<in> carrier G \<and> y \<in> carrier G \<and> properfactor G x y} |
|
2851 |
\<longrightarrow> y \<in> carrier G \<longrightarrow> (\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as y)" |
|
2852 |
||
2853 |
show "x \<in> carrier G \<longrightarrow> (\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as x)" |
|
63832 | 2854 |
apply clarify |
2855 |
apply (cases "x \<in> Units G") |
|
2856 |
apply (rule exI[of _ "[]"], simp) |
|
2857 |
apply (cases "irreducible G x") |
|
2858 |
apply (rule exI[of _ "[x]"], simp add: wfactors_def) |
|
27701 | 2859 |
proof - |
2860 |
assume xcarr: "x \<in> carrier G" |
|
2861 |
and xnunit: "x \<notin> Units G" |
|
2862 |
and xnirr: "\<not> irreducible G x" |
|
63832 | 2863 |
then have "\<exists>y. y \<in> carrier G \<and> properfactor G y x \<and> y \<notin> Units G" |
2864 |
apply - |
|
2865 |
apply (rule ccontr) |
|
2866 |
apply simp |
|
27701 | 2867 |
apply (subgoal_tac "irreducible G x", simp) |
2868 |
apply (rule irreducibleI, simp, simp) |
|
63832 | 2869 |
done |
2870 |
then obtain y where ycarr: "y \<in> carrier G" and ynunit: "y \<notin> Units G" |
|
2871 |
and pfyx: "properfactor G y x" |
|
63847 | 2872 |
by blast |
63832 | 2873 |
|
2874 |
have ih': "\<And>y. \<lbrakk>y \<in> carrier G; properfactor G y x\<rbrakk> |
|
2875 |
\<Longrightarrow> \<exists>as. set as \<subseteq> carrier G \<and> wfactors G as y" |
|
2876 |
by (rule ih[rule_format, simplified]) (simp add: xcarr)+ |
|
2877 |
||
63847 | 2878 |
from ih' [OF ycarr pfyx] |
2879 |
obtain ys where yscarr: "set ys \<subseteq> carrier G" and yfs: "wfactors G ys y" |
|
2880 |
by blast |
|
63832 | 2881 |
|
2882 |
from pfyx have "y divides x" and nyx: "\<not> y \<sim> x" |
|
2883 |
by (fast elim: properfactorE2)+ |
|
2884 |
then obtain z where zcarr: "z \<in> carrier G" and x: "x = y \<otimes> z" |
|
63847 | 2885 |
by blast |
63832 | 2886 |
|
2887 |
from zcarr ycarr have "properfactor G z x" |
|
27701 | 2888 |
apply (subst x) |
2889 |
apply (intro properfactorI3[of _ _ y]) |
|
63832 | 2890 |
apply (simp add: m_comm) |
2891 |
apply (simp add: ynunit)+ |
|
2892 |
done |
|
63847 | 2893 |
from ih' [OF zcarr this] |
2894 |
obtain zs where zscarr: "set zs \<subseteq> carrier G" and zfs: "wfactors G zs z" |
|
2895 |
by blast |
|
63832 | 2896 |
from yscarr zscarr have xscarr: "set (ys@zs) \<subseteq> carrier G" |
2897 |
by simp |
|
2898 |
from yfs zfs ycarr zcarr yscarr zscarr have "wfactors G (ys@zs) (y\<otimes>z)" |
|
2899 |
by (rule wfactors_mult) |
|
2900 |
then have "wfactors G (ys@zs) x" |
|
2901 |
by (simp add: x) |
|
2902 |
with xscarr show "\<exists>xs. set xs \<subseteq> carrier G \<and> wfactors G xs x" |
|
2903 |
by fast |
|
27701 | 2904 |
qed |
2905 |
qed |
|
63832 | 2906 |
from acarr show ?thesis by (rule r) |
27701 | 2907 |
qed |
2908 |
||
2909 |
||
61382 | 2910 |
subsubsection \<open>Primeness condition\<close> |
27701 | 2911 |
|
2912 |
lemma (in comm_monoid_cancel) multlist_prime_pos: |
|
2913 |
assumes carr: "a \<in> carrier G" "set as \<subseteq> carrier G" |
|
2914 |
and aprime: "prime G a" |
|
2915 |
and "a divides (foldr (op \<otimes>) as \<one>)" |
|
2916 |
shows "\<exists>i<length as. a divides (as!i)" |
|
2917 |
proof - |
|
63832 | 2918 |
have r[rule_format]: "set as \<subseteq> carrier G \<and> a divides (foldr (op \<otimes>) as \<one>) |
2919 |
\<longrightarrow> (\<exists>i. i < length as \<and> a divides (as!i))" |
|
27701 | 2920 |
apply (induct as) |
2921 |
apply clarsimp defer 1 |
|
2922 |
apply clarsimp defer 1 |
|
2923 |
proof - |
|
2924 |
assume "a divides \<one>" |
|
63832 | 2925 |
with carr have "a \<in> Units G" |
2926 |
by (fast intro: divides_unit[of a \<one>]) |
|
2927 |
with aprime show False |
|
2928 |
by (elim primeE, simp) |
|
27701 | 2929 |
next |
2930 |
fix aa as |
|
2931 |
assume ih[rule_format]: "a divides foldr op \<otimes> as \<one> \<longrightarrow> (\<exists>i<length as. a divides as ! i)" |
|
2932 |
and carr': "aa \<in> carrier G" "set as \<subseteq> carrier G" |
|
2933 |
and "a divides aa \<otimes> foldr op \<otimes> as \<one>" |
|
63832 | 2934 |
with carr aprime have "a divides aa \<or> a divides foldr op \<otimes> as \<one>" |
2935 |
by (intro prime_divides) simp+ |
|
2936 |
then show "\<exists>i<Suc (length as). a divides (aa # as) ! i" |
|
2937 |
proof |
|
27701 | 2938 |
assume "a divides aa" |
63832 | 2939 |
then have p1: "a divides (aa#as)!0" by simp |
27701 | 2940 |
have "0 < Suc (length as)" by simp |
63832 | 2941 |
with p1 show ?thesis by fast |
2942 |
next |
|
27701 | 2943 |
assume "a divides foldr op \<otimes> as \<one>" |
63847 | 2944 |
from ih [OF this] obtain i where "a divides as ! i" and len: "i < length as" by auto |
63832 | 2945 |
then have p1: "a divides (aa#as) ! (Suc i)" by simp |
27701 | 2946 |
from len have "Suc i < Suc (length as)" by simp |
63832 | 2947 |
with p1 show ?thesis by force |
2948 |
qed |
|
27701 | 2949 |
qed |
63832 | 2950 |
from assms show ?thesis |
2951 |
by (intro r) auto |
|
27701 | 2952 |
qed |
2953 |
||
2954 |
lemma (in primeness_condition_monoid) wfactors_unique__hlp_induct: |
|
63832 | 2955 |
"\<forall>a as'. a \<in> carrier G \<and> set as \<subseteq> carrier G \<and> set as' \<subseteq> carrier G \<and> |
27701 | 2956 |
wfactors G as a \<and> wfactors G as' a \<longrightarrow> essentially_equal G as as'" |
46129 | 2957 |
proof (induct as) |
63832 | 2958 |
case Nil |
2959 |
show ?case |
|
2960 |
proof auto |
|
46129 | 2961 |
fix a as' |
2962 |
assume a: "a \<in> carrier G" |
|
2963 |
assume "wfactors G [] a" |
|
2964 |
then obtain "\<one> \<sim> a" by (auto elim: wfactorsE) |
|
2965 |
with a have "a \<in> Units G" by (auto intro: assoc_unit_r) |
|
2966 |
moreover assume "wfactors G as' a" |
|
2967 |
moreover assume "set as' \<subseteq> carrier G" |
|
2968 |
ultimately have "as' = []" by (rule unit_wfactors_empty) |
|
2969 |
then show "essentially_equal G [] as'" by simp |
|
2970 |
qed |
|
2971 |
next |
|
63832 | 2972 |
case (Cons ah as) |
2973 |
then show ?case |
|
2974 |
proof clarsimp |
|
46129 | 2975 |
fix a as' |
63832 | 2976 |
assume ih [rule_format]: |
46129 | 2977 |
"\<forall>a as'. a \<in> carrier G \<and> set as' \<subseteq> carrier G \<and> wfactors G as a \<and> |
2978 |
wfactors G as' a \<longrightarrow> essentially_equal G as as'" |
|
2979 |
and acarr: "a \<in> carrier G" and ahcarr: "ah \<in> carrier G" |
|
2980 |
and ascarr: "set as \<subseteq> carrier G" and as'carr: "set as' \<subseteq> carrier G" |
|
2981 |
and afs: "wfactors G (ah # as) a" |
|
2982 |
and afs': "wfactors G as' a" |
|
63832 | 2983 |
then have ahdvda: "ah divides a" |
63847 | 2984 |
by (intro wfactors_dividesI[of "ah#as" "a"]) simp_all |
63832 | 2985 |
then obtain a' where a'carr: "a' \<in> carrier G" and a: "a = ah \<otimes> a'" |
63847 | 2986 |
by blast |
46129 | 2987 |
have a'fs: "wfactors G as a'" |
2988 |
apply (rule wfactorsE[OF afs], rule wfactorsI, simp) |
|
63847 | 2989 |
apply (simp add: a) |
2990 |
apply (insert ascarr a'carr) |
|
46129 | 2991 |
apply (intro assoc_l_cancel[of ah _ a'] multlist_closed ahcarr, assumption+) |
2992 |
done |
|
63832 | 2993 |
from afs have ahirr: "irreducible G ah" |
2994 |
by (elim wfactorsE) simp |
|
2995 |
with ascarr have ahprime: "prime G ah" |
|
2996 |
by (intro irreducible_prime ahcarr) |
|
46129 | 2997 |
|
2998 |
note carr [simp] = acarr ahcarr ascarr as'carr a'carr |
|
2999 |
||
3000 |
note ahdvda |
|
63832 | 3001 |
also from afs' have "a divides (foldr (op \<otimes>) as' \<one>)" |
46129 | 3002 |
by (elim wfactorsE associatedE, simp) |
63832 | 3003 |
finally have "ah divides (foldr (op \<otimes>) as' \<one>)" |
3004 |
by simp |
|
3005 |
with ahprime have "\<exists>i<length as'. ah divides as'!i" |
|
63847 | 3006 |
by (intro multlist_prime_pos) simp_all |
63832 | 3007 |
then obtain i where len: "i<length as'" and ahdvd: "ah divides as'!i" |
63847 | 3008 |
by blast |
46129 | 3009 |
from afs' carr have irrasi: "irreducible G (as'!i)" |
27701 | 3010 |
by (fast intro: nth_mem[OF len] elim: wfactorsE) |
63832 | 3011 |
from len carr have asicarr[simp]: "as'!i \<in> carrier G" |
3012 |
unfolding set_conv_nth by force |
|
46129 | 3013 |
note carr = carr asicarr |
3014 |
||
63847 | 3015 |
from ahdvd obtain x where "x \<in> carrier G" and asi: "as'!i = ah \<otimes> x" |
3016 |
by blast |
|
63832 | 3017 |
with carr irrasi[simplified asi] have asiah: "as'!i \<sim> ah" |
3018 |
apply - |
|
46129 | 3019 |
apply (elim irreducible_prodE[of "ah" "x"], assumption+) |
3020 |
apply (rule associatedI2[of x], assumption+) |
|
3021 |
apply (rule irreducibleE[OF ahirr], simp) |
|
3022 |
done |
|
3023 |
||
3024 |
note setparts = set_take_subset[of i as'] set_drop_subset[of "Suc i" as'] |
|
3025 |
note partscarr [simp] = setparts[THEN subset_trans[OF _ as'carr]] |
|
3026 |
note carr = carr partscarr |
|
3027 |
||
3028 |
have "\<exists>aa_1. aa_1 \<in> carrier G \<and> wfactors G (take i as') aa_1" |
|
3029 |
apply (intro wfactors_prod_exists) |
|
63832 | 3030 |
using setparts afs' |
3031 |
apply (fast elim: wfactorsE) |
|
3032 |
apply simp |
|
3033 |
done |
|
3034 |
then obtain aa_1 where aa1carr: "aa_1 \<in> carrier G" and aa1fs: "wfactors G (take i as') aa_1" |
|
3035 |
by auto |
|
46129 | 3036 |
|
3037 |
have "\<exists>aa_2. aa_2 \<in> carrier G \<and> wfactors G (drop (Suc i) as') aa_2" |
|
3038 |
apply (intro wfactors_prod_exists) |
|
63832 | 3039 |
using setparts afs' |
3040 |
apply (fast elim: wfactorsE) |
|
3041 |
apply simp |
|
3042 |
done |
|
3043 |
then obtain aa_2 where aa2carr: "aa_2 \<in> carrier G" |
|
3044 |
and aa2fs: "wfactors G (drop (Suc i) as') aa_2" |
|
3045 |
by auto |
|
46129 | 3046 |
|
3047 |
note carr = carr aa1carr[simp] aa2carr[simp] |
|
3048 |
||
3049 |
from aa1fs aa2fs |
|
63832 | 3050 |
have v1: "wfactors G (take i as' @ drop (Suc i) as') (aa_1 \<otimes> aa_2)" |
27701 | 3051 |
by (intro wfactors_mult, simp+) |
63832 | 3052 |
then have v1': "wfactors G (as'!i # take i as' @ drop (Suc i) as') (as'!i \<otimes> (aa_1 \<otimes> aa_2))" |
27701 | 3053 |
apply (intro wfactors_mult_single) |
3054 |
using setparts afs' |
|
63832 | 3055 |
apply (fast intro: nth_mem[OF len] elim: wfactorsE) |
3056 |
apply simp_all |
|
3057 |
done |
|
3058 |
||
3059 |
from aa2carr carr aa1fs aa2fs have "wfactors G (as'!i # drop (Suc i) as') (as'!i \<otimes> aa_2)" |
|
3060 |
by (metis irrasi wfactors_mult_single) |
|
46129 | 3061 |
with len carr aa1carr aa2carr aa1fs |
63832 | 3062 |
have v2: "wfactors G (take i as' @ as'!i # drop (Suc i) as') (aa_1 \<otimes> (as'!i \<otimes> aa_2))" |
46129 | 3063 |
apply (intro wfactors_mult) |
3064 |
apply fast |
|
3065 |
apply (simp, (fast intro: nth_mem[OF len])?)+ |
|
63832 | 3066 |
done |
3067 |
||
3068 |
from len have as': "as' = (take i as' @ as'!i # drop (Suc i) as')" |
|
58247
98d0f85d247f
enamed drop_Suc_conv_tl and nth_drop' to Cons_nth_drop_Suc
nipkow
parents:
57865
diff
changeset
|
3069 |
by (simp add: Cons_nth_drop_Suc) |
63832 | 3070 |
with carr have eer: "essentially_equal G (take i as' @ as'!i # drop (Suc i) as') as'" |
27701 | 3071 |
by simp |
63832 | 3072 |
with v2 afs' carr aa1carr aa2carr nth_mem[OF len] have "aa_1 \<otimes> (as'!i \<otimes> aa_2) \<sim> a" |
3073 |
by (metis as' ee_wfactorsD m_closed) |
|
3074 |
then have t1: "as'!i \<otimes> (aa_1 \<otimes> aa_2) \<sim> a" |
|
55242
413ec965f95d
Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents:
53374
diff
changeset
|
3075 |
by (metis aa1carr aa2carr asicarr m_lcomm) |
63832 | 3076 |
from carr asiah have "ah \<otimes> (aa_1 \<otimes> aa_2) \<sim> as'!i \<otimes> (aa_1 \<otimes> aa_2)" |
55242
413ec965f95d
Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents:
53374
diff
changeset
|
3077 |
by (metis associated_sym m_closed mult_cong_l) |
46129 | 3078 |
also note t1 |
63832 | 3079 |
finally have "ah \<otimes> (aa_1 \<otimes> aa_2) \<sim> a" by simp |
3080 |
||
3081 |
with carr aa1carr aa2carr a'carr nth_mem[OF len] have a': "aa_1 \<otimes> aa_2 \<sim> a'" |
|
27701 | 3082 |
by (simp add: a, fast intro: assoc_l_cancel[of ah _ a']) |
3083 |
||
46129 | 3084 |
note v1 |
3085 |
also note a' |
|
63832 | 3086 |
finally have "wfactors G (take i as' @ drop (Suc i) as') a'" |
3087 |
by simp |
|
3088 |
||
3089 |
from a'fs this carr have "essentially_equal G as (take i as' @ drop (Suc i) as')" |
|
27701 | 3090 |
by (intro ih[of a']) simp |
63832 | 3091 |
then have ee1: "essentially_equal G (ah # as) (ah # take i as' @ drop (Suc i) as')" |
3092 |
by (elim essentially_equalE) (fastforce intro: essentially_equalI) |
|
3093 |
||
3094 |
from carr have ee2: "essentially_equal G (ah # take i as' @ drop (Suc i) as') |
|
46129 | 3095 |
(as' ! i # take i as' @ drop (Suc i) as')" |
3096 |
proof (intro essentially_equalI) |
|
3097 |
show "ah # take i as' @ drop (Suc i) as' <~~> ah # take i as' @ drop (Suc i) as'" |
|
27701 | 3098 |
by simp |
46129 | 3099 |
next |
3100 |
show "ah # take i as' @ drop (Suc i) as' [\<sim>] as' ! i # take i as' @ drop (Suc i) as'" |
|
63832 | 3101 |
by (simp add: list_all2_append) (simp add: asiah[symmetric]) |
46129 | 3102 |
qed |
3103 |
||
3104 |
note ee1 |
|
3105 |
also note ee2 |
|
3106 |
also have "essentially_equal G (as' ! i # take i as' @ drop (Suc i) as') |
|
3107 |
(take i as' @ as' ! i # drop (Suc i) as')" |
|
3108 |
apply (intro essentially_equalI) |
|
63832 | 3109 |
apply (subgoal_tac "as' ! i # take i as' @ drop (Suc i) as' <~~> |
3110 |
take i as' @ as' ! i # drop (Suc i) as'") |
|
57865 | 3111 |
apply simp |
46129 | 3112 |
apply (rule perm_append_Cons) |
3113 |
apply simp |
|
57865 | 3114 |
done |
63832 | 3115 |
finally have "essentially_equal G (ah # as) (take i as' @ as' ! i # drop (Suc i) as')" |
3116 |
by simp |
|
3117 |
then show "essentially_equal G (ah # as) as'" |
|
3118 |
by (subst as') |
|
27701 | 3119 |
qed |
3120 |
qed |
|
3121 |
||
3122 |
lemma (in primeness_condition_monoid) wfactors_unique: |
|
3123 |
assumes "wfactors G as a" "wfactors G as' a" |
|
3124 |
and "a \<in> carrier G" "set as \<subseteq> carrier G" "set as' \<subseteq> carrier G" |
|
3125 |
shows "essentially_equal G as as'" |
|
63832 | 3126 |
by (rule wfactors_unique__hlp_induct[rule_format, of a]) (simp add: assms) |
27701 | 3127 |
|
3128 |
||
61382 | 3129 |
subsubsection \<open>Application to factorial monoids\<close> |
3130 |
||
3131 |
text \<open>Number of factors for wellfoundedness\<close> |
|
27701 | 3132 |
|
63832 | 3133 |
definition factorcount :: "_ \<Rightarrow> 'a \<Rightarrow> nat" |
3134 |
where "factorcount G a = |
|
3135 |
(THE c. \<forall>as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> c = length as)" |
|
27701 | 3136 |
|
3137 |
lemma (in monoid) ee_length: |
|
3138 |
assumes ee: "essentially_equal G as bs" |
|
3139 |
shows "length as = length bs" |
|
63832 | 3140 |
by (rule essentially_equalE[OF ee]) (metis list_all2_conv_all_nth perm_length) |
27701 | 3141 |
|
3142 |
lemma (in factorial_monoid) factorcount_exists: |
|
3143 |
assumes carr[simp]: "a \<in> carrier G" |
|
63832 | 3144 |
shows "\<exists>c. \<forall>as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> c = length as" |
27701 | 3145 |
proof - |
63832 | 3146 |
have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" |
3147 |
by (intro wfactors_exist) simp |
|
3148 |
then obtain as where ascarr[simp]: "set as \<subseteq> carrier G" and afs: "wfactors G as a" |
|
3149 |
by (auto simp del: carr) |
|
3150 |
have "\<forall>as'. set as' \<subseteq> carrier G \<and> wfactors G as' a \<longrightarrow> length as = length as'" |
|
36278 | 3151 |
by (metis afs ascarr assms ee_length wfactors_unique) |
63832 | 3152 |
then show "\<exists>c. \<forall>as'. set as' \<subseteq> carrier G \<and> wfactors G as' a \<longrightarrow> c = length as'" .. |
27701 | 3153 |
qed |
3154 |
||
3155 |
lemma (in factorial_monoid) factorcount_unique: |
|
3156 |
assumes afs: "wfactors G as a" |
|
3157 |
and acarr[simp]: "a \<in> carrier G" and ascarr[simp]: "set as \<subseteq> carrier G" |
|
3158 |
shows "factorcount G a = length as" |
|
3159 |
proof - |
|
63832 | 3160 |
have "\<exists>ac. \<forall>as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> ac = length as" |
3161 |
by (rule factorcount_exists) simp |
|
3162 |
then obtain ac where alen: "\<forall>as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> ac = length as" |
|
3163 |
by auto |
|
27701 | 3164 |
have ac: "ac = factorcount G a" |
3165 |
apply (simp add: factorcount_def) |
|
3166 |
apply (rule theI2) |
|
3167 |
apply (rule alen) |
|
55242
413ec965f95d
Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents:
53374
diff
changeset
|
3168 |
apply (metis afs alen ascarr)+ |
63832 | 3169 |
done |
3170 |
from ascarr afs have "ac = length as" |
|
3171 |
by (iprover intro: alen[rule_format]) |
|
3172 |
with ac show ?thesis |
|
3173 |
by simp |
|
27701 | 3174 |
qed |
3175 |
||
3176 |
lemma (in factorial_monoid) divides_fcount: |
|
3177 |
assumes dvd: "a divides b" |
|
63832 | 3178 |
and acarr: "a \<in> carrier G" |
3179 |
and bcarr:"b \<in> carrier G" |
|
3180 |
shows "factorcount G a \<le> factorcount G b" |
|
3181 |
proof (rule dividesE[OF dvd]) |
|
27701 | 3182 |
fix c |
63832 | 3183 |
from assms have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" |
63847 | 3184 |
by blast |
63832 | 3185 |
then obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" |
63847 | 3186 |
by blast |
63832 | 3187 |
with acarr have fca: "factorcount G a = length as" |
3188 |
by (intro factorcount_unique) |
|
27701 | 3189 |
|
3190 |
assume ccarr: "c \<in> carrier G" |
|
63832 | 3191 |
then have "\<exists>cs. set cs \<subseteq> carrier G \<and> wfactors G cs c" |
63847 | 3192 |
by blast |
63832 | 3193 |
then obtain cs where cscarr: "set cs \<subseteq> carrier G" and cfs: "wfactors G cs c" |
63847 | 3194 |
by blast |
27701 | 3195 |
|
3196 |
note [simp] = acarr bcarr ccarr ascarr cscarr |
|
3197 |
||
3198 |
assume b: "b = a \<otimes> c" |
|
63832 | 3199 |
from afs cfs have "wfactors G (as@cs) (a \<otimes> c)" |
3200 |
by (intro wfactors_mult) simp_all |
|
3201 |
with b have "wfactors G (as@cs) b" |
|
3202 |
by simp |
|
3203 |
then have "factorcount G b = length (as@cs)" |
|
3204 |
by (intro factorcount_unique) simp_all |
|
3205 |
then have "factorcount G b = length as + length cs" |
|
3206 |
by simp |
|
3207 |
with fca show ?thesis |
|
3208 |
by simp |
|
27701 | 3209 |
qed |
3210 |
||
3211 |
lemma (in factorial_monoid) associated_fcount: |
|
63832 | 3212 |
assumes acarr: "a \<in> carrier G" |
3213 |
and bcarr: "b \<in> carrier G" |
|
27701 | 3214 |
and asc: "a \<sim> b" |
3215 |
shows "factorcount G a = factorcount G b" |
|
63832 | 3216 |
apply (rule associatedE[OF asc]) |
3217 |
apply (drule divides_fcount[OF _ acarr bcarr]) |
|
3218 |
apply (drule divides_fcount[OF _ bcarr acarr]) |
|
3219 |
apply simp |
|
3220 |
done |
|
27701 | 3221 |
|
3222 |
lemma (in factorial_monoid) properfactor_fcount: |
|
3223 |
assumes acarr: "a \<in> carrier G" and bcarr:"b \<in> carrier G" |
|
3224 |
and pf: "properfactor G a b" |
|
3225 |
shows "factorcount G a < factorcount G b" |
|
63832 | 3226 |
proof (rule properfactorE[OF pf], elim dividesE) |
27701 | 3227 |
fix c |
63832 | 3228 |
from assms have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" |
63847 | 3229 |
by blast |
63832 | 3230 |
then obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" |
63847 | 3231 |
by blast |
63832 | 3232 |
with acarr have fca: "factorcount G a = length as" |
3233 |
by (intro factorcount_unique) |
|
27701 | 3234 |
|
3235 |
assume ccarr: "c \<in> carrier G" |
|
63832 | 3236 |
then have "\<exists>cs. set cs \<subseteq> carrier G \<and> wfactors G cs c" |
63847 | 3237 |
by blast |
63832 | 3238 |
then obtain cs where cscarr: "set cs \<subseteq> carrier G" and cfs: "wfactors G cs c" |
63847 | 3239 |
by blast |
27701 | 3240 |
|
3241 |
assume b: "b = a \<otimes> c" |
|
3242 |
||
63832 | 3243 |
have "wfactors G (as@cs) (a \<otimes> c)" |
3244 |
by (rule wfactors_mult) fact+ |
|
3245 |
with b have "wfactors G (as@cs) b" |
|
3246 |
by simp |
|
3247 |
with ascarr cscarr bcarr have "factorcount G b = length (as@cs)" |
|
3248 |
by (simp add: factorcount_unique) |
|
3249 |
then have fcb: "factorcount G b = length as + length cs" |
|
3250 |
by simp |
|
27701 | 3251 |
|
3252 |
assume nbdvda: "\<not> b divides a" |
|
3253 |
have "c \<notin> Units G" |
|
63846 | 3254 |
proof |
27701 | 3255 |
assume cunit:"c \<in> Units G" |
63832 | 3256 |
have "b \<otimes> inv c = a \<otimes> c \<otimes> inv c" |
3257 |
by (simp add: b) |
|
3258 |
also from ccarr acarr cunit have "\<dots> = a \<otimes> (c \<otimes> inv c)" |
|
3259 |
by (fast intro: m_assoc) |
|
3260 |
also from ccarr cunit have "\<dots> = a \<otimes> \<one>" by simp |
|
3261 |
also from acarr have "\<dots> = a" by simp |
|
27701 | 3262 |
finally have "a = b \<otimes> inv c" by simp |
63832 | 3263 |
with ccarr cunit have "b divides a" |
3264 |
by (fast intro: dividesI[of "inv c"]) |
|
27701 | 3265 |
with nbdvda show False by simp |
3266 |
qed |
|
3267 |
with cfs have "length cs > 0" |
|
36278 | 3268 |
apply - |
3269 |
apply (rule ccontr, simp) |
|
3270 |
apply (metis Units_one_closed ccarr cscarr l_one one_closed properfactorI3 properfactor_fmset unit_wfactors) |
|
3271 |
done |
|
63832 | 3272 |
with fca fcb show ?thesis |
3273 |
by simp |
|
27701 | 3274 |
qed |
3275 |
||
29237 | 3276 |
sublocale factorial_monoid \<subseteq> divisor_chain_condition_monoid |
63832 | 3277 |
apply unfold_locales |
3278 |
apply (rule wfUNIVI) |
|
3279 |
apply (rule measure_induct[of "factorcount G"]) |
|
3280 |
apply simp |
|
3281 |
apply (metis properfactor_fcount) |
|
3282 |
done |
|
27701 | 3283 |
|
29237 | 3284 |
sublocale factorial_monoid \<subseteq> primeness_condition_monoid |
63633 | 3285 |
by standard (rule irreducible_prime) |
27701 | 3286 |
|
3287 |
||
63832 | 3288 |
lemma (in factorial_monoid) primeness_condition: "primeness_condition_monoid G" .. |
3289 |
||
3290 |
lemma (in factorial_monoid) gcd_condition [simp]: "gcd_condition_monoid G" |
|
61169 | 3291 |
by standard (rule gcdof_exists) |
27701 | 3292 |
|
29237 | 3293 |
sublocale factorial_monoid \<subseteq> gcd_condition_monoid |
61169 | 3294 |
by standard (rule gcdof_exists) |
27701 | 3295 |
|
63832 | 3296 |
lemma (in factorial_monoid) division_weak_lattice [simp]: "weak_lattice (division_rel G)" |
27701 | 3297 |
proof - |
63832 | 3298 |
interpret weak_lower_semilattice "division_rel G" |
3299 |
by simp |
|
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27701
diff
changeset
|
3300 |
show "weak_lattice (division_rel G)" |
63832 | 3301 |
proof (unfold_locales, simp_all) |
27701 | 3302 |
fix x y |
3303 |
assume carr: "x \<in> carrier G" "y \<in> carrier G" |
|
63847 | 3304 |
from lcmof_exists [OF this] obtain z where zcarr: "z \<in> carrier G" and isgcd: "z lcmof x y" |
3305 |
by blast |
|
63832 | 3306 |
with carr have "least (division_rel G) z (Upper (division_rel G) {x, y})" |
3307 |
by (simp add: lcmof_leastUpper[symmetric]) |
|
3308 |
then show "\<exists>z. least (division_rel G) z (Upper (division_rel G) {x, y})" |
|
63847 | 3309 |
by blast |
27701 | 3310 |
qed |
3311 |
qed |
|
3312 |
||
3313 |
||
61382 | 3314 |
subsection \<open>Factoriality Theorems\<close> |
27701 | 3315 |
|
3316 |
theorem factorial_condition_one: (* Jacobson theorem 2.21 *) |
|
63847 | 3317 |
"divisor_chain_condition_monoid G \<and> primeness_condition_monoid G \<longleftrightarrow> factorial_monoid G" |
3318 |
proof (rule iffI, clarify) |
|
27701 | 3319 |
assume dcc: "divisor_chain_condition_monoid G" |
63832 | 3320 |
and pc: "primeness_condition_monoid G" |
29237 | 3321 |
interpret divisor_chain_condition_monoid "G" by (rule dcc) |
3322 |
interpret primeness_condition_monoid "G" by (rule pc) |
|
27701 | 3323 |
show "factorial_monoid G" |
63832 | 3324 |
by (fast intro: factorial_monoidI wfactors_exist wfactors_unique) |
27701 | 3325 |
next |
63847 | 3326 |
assume "factorial_monoid G" |
3327 |
then interpret factorial_monoid "G" . |
|
27701 | 3328 |
show "divisor_chain_condition_monoid G \<and> primeness_condition_monoid G" |
63832 | 3329 |
by rule unfold_locales |
27701 | 3330 |
qed |
3331 |
||
3332 |
theorem factorial_condition_two: (* Jacobson theorem 2.22 *) |
|
63847 | 3333 |
"divisor_chain_condition_monoid G \<and> gcd_condition_monoid G \<longleftrightarrow> factorial_monoid G" |
3334 |
proof (rule iffI, clarify) |
|
57865 | 3335 |
assume dcc: "divisor_chain_condition_monoid G" |
63832 | 3336 |
and gc: "gcd_condition_monoid G" |
29237 | 3337 |
interpret divisor_chain_condition_monoid "G" by (rule dcc) |
3338 |
interpret gcd_condition_monoid "G" by (rule gc) |
|
27701 | 3339 |
show "factorial_monoid G" |
63832 | 3340 |
by (simp add: factorial_condition_one[symmetric], rule, unfold_locales) |
27701 | 3341 |
next |
63847 | 3342 |
assume "factorial_monoid G" |
3343 |
then interpret factorial_monoid "G" . |
|
27701 | 3344 |
show "divisor_chain_condition_monoid G \<and> gcd_condition_monoid G" |
63832 | 3345 |
by rule unfold_locales |
27701 | 3346 |
qed |
3347 |
||
3348 |
end |