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src/HOL/Algebra/Divisibility.thy

author | ballarin |

Fri, 01 Aug 2008 18:10:52 +0200 | |

changeset 27717 | 21bbd410ba04 |

parent 27713 | 95b36bfe7fc4 |

child 28599 | 12d914277b8d |

permissions | -rw-r--r-- |

Generalised polynomial lemmas from cring to ring.

(* Title: Divisibility in monoids and rings Id: $Id$ Author: Clemens Ballarin, started 18 July 2008 Based on work by Stephan Hohe. *) theory Divisibility imports Permutation Coset Group begin section {* Factorial Monoids *} subsection {* Monoids with Cancellation Law *} locale monoid_cancel = monoid + 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" 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" lemma (in monoid) monoid_cancelI: 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" 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" shows "monoid_cancel G" by unfold_locales fact+ lemma (in monoid_cancel) is_monoid_cancel: "monoid_cancel G" by intro_locales interpretation group \<subseteq> monoid_cancel by unfold_locales simp+ locale comm_monoid_cancel = monoid_cancel + comm_monoid lemma comm_monoid_cancelI: includes comm_monoid 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" shows "comm_monoid_cancel G" apply unfold_locales apply (subgoal_tac "a \<otimes> c = b \<otimes> c") apply (iprover intro: cancel) apply (simp add: m_comm) apply (iprover intro: cancel) done lemma (in comm_monoid_cancel) is_comm_monoid_cancel: "comm_monoid_cancel G" by intro_locales interpretation comm_group \<subseteq> comm_monoid_cancel by unfold_locales subsection {* Products of Units in Monoids *} lemma (in monoid) Units_m_closed[simp, intro]: assumes h1unit: "h1 \<in> Units G" and h2unit: "h2 \<in> Units G" shows "h1 \<otimes> h2 \<in> Units G" unfolding Units_def using assms apply safe apply fast apply (intro bexI[of _ "inv h2 \<otimes> inv h1"], safe) apply (simp add: m_assoc Units_closed) apply (simp add: m_assoc[symmetric] Units_closed Units_l_inv) apply (simp add: m_assoc Units_closed) apply (simp add: m_assoc[symmetric] Units_closed Units_r_inv) apply fast done lemma (in monoid) prod_unit_l: assumes abunit[simp]: "a \<otimes> b \<in> Units G" and aunit[simp]: "a \<in> Units G" and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" shows "b \<in> Units G" proof - have c: "inv (a \<otimes> b) \<otimes> a \<in> carrier G" by simp have "(inv (a \<otimes> b) \<otimes> a) \<otimes> b = inv (a \<otimes> b) \<otimes> (a \<otimes> b)" by (simp add: m_assoc) also have "\<dots> = \<one>" by (simp add: Units_l_inv) finally have li: "(inv (a \<otimes> b) \<otimes> a) \<otimes> b = \<one>" . have "\<one> = inv a \<otimes> a" by (simp add: Units_l_inv[symmetric]) also have "\<dots> = inv a \<otimes> \<one> \<otimes> a" by simp also have "\<dots> = inv a \<otimes> ((a \<otimes> b) \<otimes> inv (a \<otimes> b)) \<otimes> a" by (simp add: Units_r_inv[OF abunit, symmetric] del: Units_r_inv) also have "\<dots> = ((inv a \<otimes> a) \<otimes> b) \<otimes> inv (a \<otimes> b) \<otimes> a" by (simp add: m_assoc del: Units_l_inv) also have "\<dots> = b \<otimes> inv (a \<otimes> b) \<otimes> a" by (simp add: Units_l_inv) also have "\<dots> = b \<otimes> (inv (a \<otimes> b) \<otimes> a)" by (simp add: m_assoc) finally have ri: "b \<otimes> (inv (a \<otimes> b) \<otimes> a) = \<one> " by simp from c li ri show "b \<in> Units G" by (simp add: Units_def, fast) qed lemma (in monoid) prod_unit_r: assumes abunit[simp]: "a \<otimes> b \<in> Units G" and bunit[simp]: "b \<in> Units G" and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" shows "a \<in> Units G" proof - have c: "b \<otimes> inv (a \<otimes> b) \<in> carrier G" by simp have "a \<otimes> (b \<otimes> inv (a \<otimes> b)) = (a \<otimes> b) \<otimes> inv (a \<otimes> b)" by (simp add: m_assoc del: Units_r_inv) also have "\<dots> = \<one>" by simp finally have li: "a \<otimes> (b \<otimes> inv (a \<otimes> b)) = \<one>" . have "\<one> = b \<otimes> inv b" by (simp add: Units_r_inv[symmetric]) also have "\<dots> = b \<otimes> \<one> \<otimes> inv b" by simp also have "\<dots> = b \<otimes> (inv (a \<otimes> b) \<otimes> (a \<otimes> b)) \<otimes> inv b" by (simp add: Units_l_inv[OF abunit, symmetric] del: Units_l_inv) also have "\<dots> = (b \<otimes> inv (a \<otimes> b) \<otimes> a) \<otimes> (b \<otimes> inv b)" by (simp add: m_assoc del: Units_l_inv) also have "\<dots> = b \<otimes> inv (a \<otimes> b) \<otimes> a" by simp finally have ri: "(b \<otimes> inv (a \<otimes> b)) \<otimes> a = \<one> " by simp from c li ri show "a \<in> Units G" by (simp add: Units_def, fast) qed lemma (in comm_monoid) unit_factor: assumes abunit: "a \<otimes> b \<in> Units G" and [simp]: "a \<in> carrier G" "b \<in> carrier G" shows "a \<in> Units G" using abunit[simplified Units_def] proof clarsimp fix i assume [simp]: "i \<in> carrier G" and li: "i \<otimes> (a \<otimes> b) = \<one>" and ri: "a \<otimes> b \<otimes> i = \<one>" have carr': "b \<otimes> i \<in> carrier G" by simp have "(b \<otimes> i) \<otimes> a = (i \<otimes> b) \<otimes> a" by (simp add: m_comm) also have "\<dots> = i \<otimes> (b \<otimes> a)" by (simp add: m_assoc) also have "\<dots> = i \<otimes> (a \<otimes> b)" by (simp add: m_comm) also note li finally have li': "(b \<otimes> i) \<otimes> a = \<one>" . have "a \<otimes> (b \<otimes> i) = a \<otimes> b \<otimes> i" by (simp add: m_assoc) also note ri finally have ri': "a \<otimes> (b \<otimes> i) = \<one>" . from carr' li' ri' show "a \<in> Units G" by (simp add: Units_def, fast) qed subsection {* Divisibility and Association *} subsubsection {* Function definitions *} constdefs (structure G) factor :: "[_, 'a, 'a] \<Rightarrow> bool" (infix "divides\<index>" 65) "a divides b == \<exists>c\<in>carrier G. b = a \<otimes> c" constdefs (structure G) associated :: "[_, 'a, 'a] => bool" (infix "\<sim>\<index>" 55) "a \<sim> b == a divides b \<and> b divides a" abbreviation "division_rel G == \<lparr>carrier = carrier G, eq = op \<sim>\<^bsub>G\<^esub>, le = op divides\<^bsub>G\<^esub>\<rparr>" constdefs (structure G) properfactor :: "[_, 'a, 'a] \<Rightarrow> bool" "properfactor G a b == a divides b \<and> \<not>(b divides a)" constdefs (structure G) irreducible :: "[_, 'a] \<Rightarrow> bool" "irreducible G a == a \<notin> Units G \<and> (\<forall>b\<in>carrier G. properfactor G b a \<longrightarrow> b \<in> Units G)" constdefs (structure G) prime :: "[_, 'a] \<Rightarrow> bool" "prime G p == p \<notin> Units G \<and> (\<forall>a\<in>carrier G. \<forall>b\<in>carrier G. p divides (a \<otimes> b) \<longrightarrow> p divides a \<or> p divides b)" subsubsection {* Divisibility *} lemma dividesI: fixes G (structure) assumes carr: "c \<in> carrier G" and p: "b = a \<otimes> c" shows "a divides b" unfolding factor_def using assms by fast lemma dividesI' [intro]: fixes G (structure) assumes p: "b = a \<otimes> c" and carr: "c \<in> carrier G" shows "a divides b" using assms by (fast intro: dividesI) lemma dividesD: fixes G (structure) assumes "a divides b" shows "\<exists>c\<in>carrier G. b = a \<otimes> c" using assms unfolding factor_def by fast lemma dividesE [elim]: fixes G (structure) assumes d: "a divides b" and elim: "\<And>c. \<lbrakk>b = a \<otimes> c; c \<in> carrier G\<rbrakk> \<Longrightarrow> P" shows "P" proof - from dividesD[OF d] obtain c where "c\<in>carrier G" and "b = a \<otimes> c" by auto thus "P" by (elim elim) qed lemma (in monoid) divides_refl[simp, intro!]: assumes carr: "a \<in> carrier G" shows "a divides a" apply (intro dividesI[of "\<one>"]) apply (simp, simp add: carr) done lemma (in monoid) divides_trans [trans]: assumes dvds: "a divides b" "b divides c" and acarr: "a \<in> carrier G" shows "a divides c" using dvds[THEN dividesD] by (blast intro: dividesI m_assoc acarr) lemma (in monoid) divides_mult_lI [intro]: assumes ab: "a divides b" and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "(c \<otimes> a) divides (c \<otimes> b)" using ab apply (elim dividesE, simp add: m_assoc[symmetric] carr) apply (fast intro: dividesI) done lemma (in monoid_cancel) divides_mult_l [simp]: assumes carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "(c \<otimes> a) divides (c \<otimes> b) = a divides b" apply safe apply (elim dividesE, intro dividesI, assumption) apply (rule l_cancel[of c]) apply (simp add: m_assoc carr)+ apply (fast intro: divides_mult_lI carr) done lemma (in comm_monoid) divides_mult_rI [intro]: assumes ab: "a divides b" and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "(a \<otimes> c) divides (b \<otimes> c)" using carr ab apply (simp add: m_comm[of a c] m_comm[of b c]) apply (rule divides_mult_lI, assumption+) done lemma (in comm_monoid_cancel) divides_mult_r [simp]: assumes carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "(a \<otimes> c) divides (b \<otimes> c) = a divides b" using carr by (simp add: m_comm[of a c] m_comm[of b c]) lemma (in monoid) divides_prod_r: assumes ab: "a divides b" and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "a divides (b \<otimes> c)" using ab carr by (fast intro: m_assoc) lemma (in comm_monoid) divides_prod_l: assumes carr[intro]: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" and ab: "a divides b" shows "a divides (c \<otimes> b)" using ab carr apply (simp add: m_comm[of c b]) apply (fast intro: divides_prod_r) done lemma (in monoid) unit_divides: assumes uunit: "u \<in> Units G" and acarr: "a \<in> carrier G" shows "u divides a" proof (intro dividesI[of "(inv u) \<otimes> a"], fast intro: uunit acarr) from uunit acarr have xcarr: "inv u \<otimes> a \<in> carrier G" by fast from uunit acarr have "u \<otimes> (inv u \<otimes> a) = (u \<otimes> inv u) \<otimes> a" by (fast intro: m_assoc[symmetric]) also have "\<dots> = \<one> \<otimes> a" by (simp add: Units_r_inv[OF uunit]) also from acarr have "\<dots> = a" by simp finally show "a = u \<otimes> (inv u \<otimes> a)" .. qed lemma (in comm_monoid) divides_unit: assumes udvd: "a divides u" and carr: "a \<in> carrier G" "u \<in> Units G" shows "a \<in> Units G" using udvd carr by (blast intro: unit_factor) lemma (in comm_monoid) Unit_eq_dividesone: assumes ucarr: "u \<in> carrier G" shows "u \<in> Units G = u divides \<one>" using ucarr by (fast dest: divides_unit intro: unit_divides) subsubsection {* Association *} lemma associatedI: fixes G (structure) assumes "a divides b" "b divides a" shows "a \<sim> b" using assms by (simp add: associated_def) lemma (in monoid) associatedI2: assumes uunit[simp]: "u \<in> Units G" and a: "a = b \<otimes> u" and bcarr[simp]: "b \<in> carrier G" shows "a \<sim> b" using uunit bcarr unfolding a apply (intro associatedI) apply (rule dividesI[of "inv u"], simp) apply (simp add: m_assoc Units_closed Units_r_inv) apply fast done lemma (in monoid) associatedI2': assumes a: "a = b \<otimes> u" and uunit: "u \<in> Units G" and bcarr: "b \<in> carrier G" shows "a \<sim> b" using assms by (intro associatedI2) lemma associatedD: fixes G (structure) assumes "a \<sim> b" shows "a divides b" using assms by (simp add: associated_def) lemma (in monoid_cancel) associatedD2: assumes assoc: "a \<sim> b" and carr: "a \<in> carrier G" "b \<in> carrier G" shows "\<exists>u\<in>Units G. a = b \<otimes> u" using assoc unfolding associated_def proof clarify assume "b divides a" hence "\<exists>u\<in>carrier G. a = b \<otimes> u" by (rule dividesD) from this obtain u where ucarr: "u \<in> carrier G" and a: "a = b \<otimes> u" by auto assume "a divides b" hence "\<exists>u'\<in>carrier G. b = a \<otimes> u'" by (rule dividesD) from this obtain u' where u'carr: "u' \<in> carrier G" and b: "b = a \<otimes> u'" by auto note carr = carr ucarr u'carr from carr have "a \<otimes> \<one> = a" by simp also have "\<dots> = b \<otimes> u" by (simp add: a) also have "\<dots> = a \<otimes> u' \<otimes> u" by (simp add: b) also from carr have "\<dots> = a \<otimes> (u' \<otimes> u)" by (simp add: m_assoc) finally have "a \<otimes> \<one> = a \<otimes> (u' \<otimes> u)" . with carr have u1: "\<one> = u' \<otimes> u" by (fast dest: l_cancel) from carr have "b \<otimes> \<one> = b" by simp also have "\<dots> = a \<otimes> u'" by (simp add: b) also have "\<dots> = b \<otimes> u \<otimes> u'" by (simp add: a) also from carr have "\<dots> = b \<otimes> (u \<otimes> u')" by (simp add: m_assoc) finally have "b \<otimes> \<one> = b \<otimes> (u \<otimes> u')" . with carr have u2: "\<one> = u \<otimes> u'" by (fast dest: l_cancel) from u'carr u1[symmetric] u2[symmetric] have "\<exists>u'\<in>carrier G. u' \<otimes> u = \<one> \<and> u \<otimes> u' = \<one>" by fast hence "u \<in> Units G" by (simp add: Units_def ucarr) from ucarr this a show "\<exists>u\<in>Units G. a = b \<otimes> u" by fast qed lemma associatedE: fixes G (structure) assumes assoc: "a \<sim> b" and e: "\<lbrakk>a divides b; b divides a\<rbrakk> \<Longrightarrow> P" shows "P" proof - from assoc have "a divides b" "b divides a" by (simp add: associated_def)+ thus "P" by (elim e) qed lemma (in monoid_cancel) associatedE2: assumes assoc: "a \<sim> b" and e: "\<And>u. \<lbrakk>a = b \<otimes> u; u \<in> Units G\<rbrakk> \<Longrightarrow> P" and carr: "a \<in> carrier G" "b \<in> carrier G" shows "P" proof - from assoc and carr have "\<exists>u\<in>Units G. a = b \<otimes> u" by (rule associatedD2) from this obtain u where "u \<in> Units G" "a = b \<otimes> u" by auto thus "P" by (elim e) qed lemma (in monoid) associated_refl [simp, intro!]: assumes "a \<in> carrier G" shows "a \<sim> a" using assms by (fast intro: associatedI) lemma (in monoid) associated_sym [sym]: assumes "a \<sim> b" and "a \<in> carrier G" "b \<in> carrier G" shows "b \<sim> a" using assms by (iprover intro: associatedI elim: associatedE) lemma (in monoid) associated_trans [trans]: assumes "a \<sim> b" "b \<sim> c" and "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "a \<sim> c" using assms by (iprover intro: associatedI divides_trans elim: associatedE) lemma (in monoid) division_equiv [intro, simp]: "equivalence (division_rel G)" apply unfold_locales apply simp_all apply (rule associated_sym, assumption+) apply (iprover intro: associated_trans) done subsubsection {* Division and associativity *} lemma divides_antisym: fixes G (structure) assumes "a divides b" "b divides a" and "a \<in> carrier G" "b \<in> carrier G" shows "a \<sim> b" using assms by (fast intro: associatedI) lemma (in monoid) divides_cong_l [trans]: assumes xx': "x \<sim> x'" and xdvdy: "x' divides y" and carr [simp]: "x \<in> carrier G" "x' \<in> carrier G" "y \<in> carrier G" shows "x divides y" proof - from xx' have "x divides x'" by (simp add: associatedD) also note xdvdy finally show "x divides y" by simp qed lemma (in monoid) divides_cong_r [trans]: assumes xdvdy: "x divides y" and yy': "y \<sim> y'" and carr[simp]: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G" shows "x divides y'" proof - note xdvdy also from yy' have "y divides y'" by (simp add: associatedD) finally show "x divides y'" by simp qed lemma (in monoid) division_weak_partial_order [simp, intro!]: "weak_partial_order (division_rel G)" apply unfold_locales apply simp_all apply (simp add: associated_sym) apply (blast intro: associated_trans) apply (simp add: divides_antisym) apply (blast intro: divides_trans) apply (blast intro: divides_cong_l divides_cong_r associated_sym) done subsubsection {* Multiplication and associativity *} lemma (in monoid_cancel) mult_cong_r: assumes "b \<sim> b'" and carr: "a \<in> carrier G" "b \<in> carrier G" "b' \<in> carrier G" shows "a \<otimes> b \<sim> a \<otimes> b'" using assms apply (elim associatedE2, intro associatedI2) apply (auto intro: m_assoc[symmetric]) done lemma (in comm_monoid_cancel) mult_cong_l: assumes "a \<sim> a'" and carr: "a \<in> carrier G" "a' \<in> carrier G" "b \<in> carrier G" shows "a \<otimes> b \<sim> a' \<otimes> b" using assms apply (elim associatedE2, intro associatedI2) apply assumption apply (simp add: m_assoc Units_closed) apply (simp add: m_comm Units_closed) apply simp+ done lemma (in monoid_cancel) assoc_l_cancel: assumes carr: "a \<in> carrier G" "b \<in> carrier G" "b' \<in> carrier G" and "a \<otimes> b \<sim> a \<otimes> b'" shows "b \<sim> b'" using assms apply (elim associatedE2, intro associatedI2) apply assumption apply (rule l_cancel[of a]) apply (simp add: m_assoc Units_closed) apply fast+ done lemma (in comm_monoid_cancel) assoc_r_cancel: assumes "a \<otimes> b \<sim> a' \<otimes> b" and carr: "a \<in> carrier G" "a' \<in> carrier G" "b \<in> carrier G" shows "a \<sim> a'" using assms apply (elim associatedE2, intro associatedI2) apply assumption apply (rule r_cancel[of a b]) apply (simp add: m_assoc Units_closed) apply (simp add: m_comm Units_closed) apply fast+ done subsubsection {* Units *} lemma (in monoid_cancel) assoc_unit_l [trans]: assumes asc: "a \<sim> b" and bunit: "b \<in> Units G" and carr: "a \<in> carrier G" shows "a \<in> Units G" using assms by (fast elim: associatedE2) lemma (in monoid_cancel) assoc_unit_r [trans]: assumes aunit: "a \<in> Units G" and asc: "a \<sim> b" and bcarr: "b \<in> carrier G" shows "b \<in> Units G" using aunit bcarr associated_sym[OF asc] by (blast intro: assoc_unit_l) lemma (in comm_monoid) Units_cong: assumes aunit: "a \<in> Units G" and asc: "a \<sim> b" and bcarr: "b \<in> carrier G" shows "b \<in> Units G" using assms by (blast intro: divides_unit elim: associatedE) lemma (in monoid) Units_assoc: assumes units: "a \<in> Units G" "b \<in> Units G" shows "a \<sim> b" using units by (fast intro: associatedI unit_divides) lemma (in monoid) Units_are_ones: "Units G {.=}\<^bsub>(division_rel G)\<^esub> {\<one>}" apply (simp add: set_eq_def elem_def, rule, simp_all) proof clarsimp fix a assume aunit: "a \<in> Units G" show "a \<sim> \<one>" apply (rule associatedI) apply (fast intro: dividesI[of "inv a"] aunit Units_r_inv[symmetric]) apply (fast intro: dividesI[of "a"] l_one[symmetric] Units_closed[OF aunit]) done next have "\<one> \<in> Units G" by simp moreover have "\<one> \<sim> \<one>" by simp ultimately show "\<exists>a \<in> Units G. \<one> \<sim> a" by fast qed lemma (in comm_monoid) Units_Lower: "Units G = Lower (division_rel G) (carrier G)" apply (simp add: Units_def Lower_def) apply (rule, rule) apply clarsimp apply (rule unit_divides) apply (unfold Units_def, fast) apply assumption apply clarsimp proof - fix x assume xcarr: "x \<in> carrier G" assume r[rule_format]: "\<forall>y. y \<in> carrier G \<longrightarrow> x divides y" have "\<one> \<in> carrier G" by simp hence "x divides \<one>" by (rule r) hence "\<exists>x'\<in>carrier G. \<one> = x \<otimes> x'" by (rule dividesE, fast) from this obtain x' where x'carr: "x' \<in> carrier G" and xx': "\<one> = x \<otimes> x'" by auto note xx' also with xcarr x'carr have "\<dots> = x' \<otimes> x" by (simp add: m_comm) finally have "\<one> = x' \<otimes> x" . from x'carr xx'[symmetric] this[symmetric] show "\<exists>y\<in>carrier G. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast qed subsubsection {* Proper factors *} lemma properfactorI: fixes G (structure) assumes "a divides b" and "\<not>(b divides a)" shows "properfactor G a b" using assms unfolding properfactor_def by simp lemma properfactorI2: fixes G (structure) assumes advdb: "a divides b" and neq: "\<not>(a \<sim> b)" shows "properfactor G a b" apply (rule properfactorI, rule advdb) proof (rule ccontr, simp) assume "b divides a" with advdb have "a \<sim> b" by (rule associatedI) with neq show "False" by fast qed lemma (in comm_monoid_cancel) properfactorI3: assumes p: "p = a \<otimes> b" and nunit: "b \<notin> Units G" and carr: "a \<in> carrier G" "b \<in> carrier G" "p \<in> carrier G" shows "properfactor G a p" unfolding p using carr apply (intro properfactorI, fast) proof (clarsimp, elim dividesE) fix c assume ccarr: "c \<in> carrier G" note [simp] = carr ccarr have "a \<otimes> \<one> = a" by simp also assume "a = a \<otimes> b \<otimes> c" also have "\<dots> = a \<otimes> (b \<otimes> c)" by (simp add: m_assoc) finally have "a \<otimes> \<one> = a \<otimes> (b \<otimes> c)" . hence rinv: "\<one> = b \<otimes> c" by (intro l_cancel[of "a" "\<one>" "b \<otimes> c"], simp+) also have "\<dots> = c \<otimes> b" by (simp add: m_comm) finally have linv: "\<one> = c \<otimes> b" . from ccarr linv[symmetric] rinv[symmetric] have "b \<in> Units G" unfolding Units_def by fastsimp with nunit show "False" .. qed lemma properfactorE: fixes G (structure) assumes pf: "properfactor G a b" and r: "\<lbrakk>a divides b; \<not>(b divides a)\<rbrakk> \<Longrightarrow> P" shows "P" using pf unfolding properfactor_def by (fast intro: r) lemma properfactorE2: fixes G (structure) assumes pf: "properfactor G a b" and elim: "\<lbrakk>a divides b; \<not>(a \<sim> b)\<rbrakk> \<Longrightarrow> P" shows "P" using pf unfolding properfactor_def by (fast elim: elim associatedE) lemma (in monoid) properfactor_unitE: assumes uunit: "u \<in> Units G" and pf: "properfactor G a u" and acarr: "a \<in> carrier G" shows "P" using pf unit_divides[OF uunit acarr] by (fast elim: properfactorE) lemma (in monoid) properfactor_divides: assumes pf: "properfactor G a b" shows "a divides b" using pf by (elim properfactorE) lemma (in monoid) properfactor_trans1 [trans]: assumes dvds: "a divides b" "properfactor G b c" and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "properfactor G a c" using dvds carr apply (elim properfactorE, intro properfactorI) apply (iprover intro: divides_trans)+ done lemma (in monoid) properfactor_trans2 [trans]: assumes dvds: "properfactor G a b" "b divides c" and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "properfactor G a c" using dvds carr apply (elim properfactorE, intro properfactorI) apply (iprover intro: divides_trans)+ done lemma properfactor_lless: fixes G (structure) shows "properfactor G = lless (division_rel G)" apply (rule ext) apply (rule ext) apply rule apply (fastsimp elim: properfactorE2 intro: weak_llessI) apply (fastsimp elim: weak_llessE intro: properfactorI2) done lemma (in monoid) properfactor_cong_l [trans]: assumes x'x: "x' \<sim> x" and pf: "properfactor G x y" and carr: "x \<in> carrier G" "x' \<in> carrier G" "y \<in> carrier G" shows "properfactor G x' y" using pf unfolding properfactor_lless proof - interpret weak_partial_order ["division_rel G"] .. from x'x have "x' .=\<^bsub>division_rel G\<^esub> x" by simp also assume "x \<sqsubset>\<^bsub>division_rel G\<^esub> y" finally show "x' \<sqsubset>\<^bsub>division_rel G\<^esub> y" by (simp add: carr) qed lemma (in monoid) properfactor_cong_r [trans]: assumes pf: "properfactor G x y" and yy': "y \<sim> y'" and carr: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G" shows "properfactor G x y'" using pf unfolding properfactor_lless proof - interpret weak_partial_order ["division_rel G"] .. assume "x \<sqsubset>\<^bsub>division_rel G\<^esub> y" also from yy' have "y .=\<^bsub>division_rel G\<^esub> y'" by simp finally show "x \<sqsubset>\<^bsub>division_rel G\<^esub> y'" by (simp add: carr) qed lemma (in monoid_cancel) properfactor_mult_lI [intro]: assumes ab: "properfactor G a b" and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "properfactor G (c \<otimes> a) (c \<otimes> b)" using ab carr by (fastsimp elim: properfactorE intro: properfactorI) lemma (in monoid_cancel) properfactor_mult_l [simp]: assumes carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "properfactor G (c \<otimes> a) (c \<otimes> b) = properfactor G a b" using carr by (fastsimp elim: properfactorE intro: properfactorI) lemma (in comm_monoid_cancel) properfactor_mult_rI [intro]: assumes ab: "properfactor G a b" and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "properfactor G (a \<otimes> c) (b \<otimes> c)" using ab carr by (fastsimp elim: properfactorE intro: properfactorI) lemma (in comm_monoid_cancel) properfactor_mult_r [simp]: assumes carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "properfactor G (a \<otimes> c) (b \<otimes> c) = properfactor G a b" using carr by (fastsimp elim: properfactorE intro: properfactorI) lemma (in monoid) properfactor_prod_r: assumes ab: "properfactor G a b" and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "properfactor G a (b \<otimes> c)" by (intro properfactor_trans2[OF ab] divides_prod_r, simp+) lemma (in comm_monoid) properfactor_prod_l: assumes ab: "properfactor G a b" and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "properfactor G a (c \<otimes> b)" by (intro properfactor_trans2[OF ab] divides_prod_l, simp+) subsection {* Irreducible Elements and Primes *} subsubsection {* Irreducible elements *} lemma irreducibleI: fixes G (structure) assumes "a \<notin> Units G" and "\<And>b. \<lbrakk>b \<in> carrier G; properfactor G b a\<rbrakk> \<Longrightarrow> b \<in> Units G" shows "irreducible G a" using assms unfolding irreducible_def by blast lemma irreducibleE: fixes G (structure) assumes irr: "irreducible G a" 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" shows "P" using assms unfolding irreducible_def by blast lemma irreducibleD: fixes G (structure) assumes irr: "irreducible G a" and pf: "properfactor G b a" and bcarr: "b \<in> carrier G" shows "b \<in> Units G" using assms by (fast elim: irreducibleE) lemma (in monoid_cancel) irreducible_cong [trans]: assumes irred: "irreducible G a" and aa': "a \<sim> a'" and carr[simp]: "a \<in> carrier G" "a' \<in> carrier G" shows "irreducible G a'" using assms apply (elim irreducibleE, intro irreducibleI) apply simp_all proof clarify assume "a' \<in> Units G" also note aa'[symmetric] finally have aunit: "a \<in> Units G" by simp assume "a \<notin> Units G" with aunit show "False" by fast next fix b assume r[rule_format]: "\<forall>b. b \<in> carrier G \<and> properfactor G b a \<longrightarrow> b \<in> Units G" and bcarr[simp]: "b \<in> carrier G" assume "properfactor G b a'" also note aa'[symmetric] finally have "properfactor G b a" by simp with bcarr show "b \<in> Units G" by (fast intro: r) qed lemma (in monoid) irreducible_prod_rI: assumes airr: "irreducible G a" and bunit: "b \<in> Units G" and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" shows "irreducible G (a \<otimes> b)" using airr carr bunit apply (elim irreducibleE, intro irreducibleI, clarify) apply (subgoal_tac "a \<in> Units G", simp) apply (intro prod_unit_r[of a b] carr bunit, assumption) proof - fix c assume [simp]: "c \<in> carrier G" and r[rule_format]: "\<forall>b. b \<in> carrier G \<and> properfactor G b a \<longrightarrow> b \<in> Units G" assume "properfactor G c (a \<otimes> b)" also have "a \<otimes> b \<sim> a" by (intro associatedI2[OF bunit], simp+) finally have pfa: "properfactor G c a" by simp show "c \<in> Units G" by (rule r, simp add: pfa) qed lemma (in comm_monoid) irreducible_prod_lI: assumes birr: "irreducible G b" and aunit: "a \<in> Units G" and carr [simp]: "a \<in> carrier G" "b \<in> carrier G" shows "irreducible G (a \<otimes> b)" apply (subst m_comm, simp+) apply (intro irreducible_prod_rI assms) done lemma (in comm_monoid_cancel) irreducible_prodE [elim]: assumes irr: "irreducible G (a \<otimes> b)" and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" and e1: "\<lbrakk>irreducible G a; b \<in> Units G\<rbrakk> \<Longrightarrow> P" and e2: "\<lbrakk>a \<in> Units G; irreducible G b\<rbrakk> \<Longrightarrow> P" shows "P" using irr proof (elim irreducibleE) assume abnunit: "a \<otimes> b \<notin> Units G" and isunit[rule_format]: "\<forall>ba. ba \<in> carrier G \<and> properfactor G ba (a \<otimes> b) \<longrightarrow> ba \<in> Units G" show "P" proof (cases "a \<in> Units G") assume aunit: "a \<in> Units G" have "irreducible G b" apply (rule irreducibleI) proof (rule ccontr, simp) assume "b \<in> Units G" with aunit have "(a \<otimes> b) \<in> Units G" by fast with abnunit show "False" .. next fix c assume ccarr: "c \<in> carrier G" and "properfactor G c b" hence "properfactor G c (a \<otimes> b)" by (simp add: properfactor_prod_l[of c b a]) from ccarr this show "c \<in> Units G" by (fast intro: isunit) qed from aunit this show "P" by (rule e2) next assume anunit: "a \<notin> Units G" with carr have "properfactor G b (b \<otimes> a)" by (fast intro: properfactorI3) hence bf: "properfactor G b (a \<otimes> b)" by (subst m_comm[of a b], simp+) hence bunit: "b \<in> Units G" by (intro isunit, simp) have "irreducible G a" apply (rule irreducibleI) proof (rule ccontr, simp) assume "a \<in> Units G" with bunit have "(a \<otimes> b) \<in> Units G" by fast with abnunit show "False" .. next fix c assume ccarr: "c \<in> carrier G" and "properfactor G c a" hence "properfactor G c (a \<otimes> b)" by (simp add: properfactor_prod_r[of c a b]) from ccarr this show "c \<in> Units G" by (fast intro: isunit) qed from this bunit show "P" by (rule e1) qed qed subsubsection {* Prime elements *} lemma primeI: fixes G (structure) assumes "p \<notin> Units G" 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" shows "prime G p" using assms unfolding prime_def by blast lemma primeE: fixes G (structure) assumes pprime: "prime G p" and e: "\<lbrakk>p \<notin> Units G; \<forall>a\<in>carrier G. \<forall>b\<in>carrier G. p divides a \<otimes> b \<longrightarrow> p divides a \<or> p divides b\<rbrakk> \<Longrightarrow> P" shows "P" using pprime unfolding prime_def by (blast dest: e) lemma (in comm_monoid_cancel) prime_divides: assumes carr: "a \<in> carrier G" "b \<in> carrier G" and pprime: "prime G p" and pdvd: "p divides a \<otimes> b" shows "p divides a \<or> p divides b" using assms by (blast elim: primeE) lemma (in monoid_cancel) prime_cong [trans]: assumes pprime: "prime G p" and pp': "p \<sim> p'" and carr[simp]: "p \<in> carrier G" "p' \<in> carrier G" shows "prime G p'" using pprime apply (elim primeE, intro primeI) proof clarify assume pnunit: "p \<notin> Units G" assume "p' \<in> Units G" also note pp'[symmetric] finally have "p \<in> Units G" by simp with pnunit show False .. next fix a b assume r[rule_format]: "\<forall>a\<in>carrier G. \<forall>b\<in>carrier G. p divides a \<otimes> b \<longrightarrow> p divides a \<or> p divides b" assume p'dvd: "p' divides a \<otimes> b" and carr'[simp]: "a \<in> carrier G" "b \<in> carrier G" note pp' also note p'dvd finally have "p divides a \<otimes> b" by simp hence "p divides a \<or> p divides b" by (intro r, simp+) moreover { note pp'[symmetric] also assume "p divides a" finally have "p' divides a" by simp hence "p' divides a \<or> p' divides b" by simp } moreover { note pp'[symmetric] also assume "p divides b" finally have "p' divides b" by simp hence "p' divides a \<or> p' divides b" by simp } ultimately show "p' divides a \<or> p' divides b" by fast qed subsection {* Factorization and Factorial Monoids *} subsubsection {* Function definitions *} constdefs (structure G) factors :: "[_, 'a list, 'a] \<Rightarrow> bool" "factors G fs a == (\<forall>x \<in> (set fs). irreducible G x) \<and> foldr (op \<otimes>) fs \<one> = a" wfactors ::"[_, 'a list, 'a] \<Rightarrow> bool" "wfactors G fs a == (\<forall>x \<in> (set fs). irreducible G x) \<and> foldr (op \<otimes>) fs \<one> \<sim> a" abbreviation list_assoc :: "('a,_) monoid_scheme \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "[\<sim>]\<index>" 44) where "list_assoc G == list_all2 (op \<sim>\<^bsub>G\<^esub>)" constdefs (structure G) essentially_equal :: "[_, 'a list, 'a list] \<Rightarrow> bool" "essentially_equal G fs1 fs2 == (\<exists>fs1'. fs1 <~~> fs1' \<and> fs1' [\<sim>] fs2)" locale factorial_monoid = comm_monoid_cancel + 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" and factors_unique: "\<lbrakk>factors G fs a; factors G fs' a; a \<in> carrier G; a \<notin> Units G; set fs \<subseteq> carrier G; set fs' \<subseteq> carrier G\<rbrakk> \<Longrightarrow> essentially_equal G fs fs'" subsubsection {* Comparing lists of elements *} text {* Association on lists *} lemma (in monoid) listassoc_refl [simp, intro]: assumes "set as \<subseteq> carrier G" shows "as [\<sim>] as" using assms by (induct as) simp+ lemma (in monoid) listassoc_sym [sym]: assumes "as [\<sim>] bs" and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" shows "bs [\<sim>] as" using assms proof (induct as arbitrary: bs, simp) case Cons thus ?case apply (induct bs, simp) apply clarsimp apply (iprover intro: associated_sym) done qed lemma (in monoid) listassoc_trans [trans]: assumes "as [\<sim>] bs" and "bs [\<sim>] cs" and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" and "set cs \<subseteq> carrier G" shows "as [\<sim>] cs" using assms apply (simp add: list_all2_conv_all_nth set_conv_nth, safe) apply (rule associated_trans) apply (subgoal_tac "as ! i \<sim> bs ! i", assumption) apply (simp, simp) apply blast+ done lemma (in monoid_cancel) irrlist_listassoc_cong: assumes "\<forall>a\<in>set as. irreducible G a" and "as [\<sim>] bs" and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" shows "\<forall>a\<in>set bs. irreducible G a" using assms apply (clarsimp simp add: list_all2_conv_all_nth set_conv_nth) apply (blast intro: irreducible_cong) done text {* Permutations *} lemma perm_map [intro]: assumes p: "a <~~> b" shows "map f a <~~> map f b" using p by induct auto lemma perm_map_switch: assumes m: "map f a = map f b" and p: "b <~~> c" shows "\<exists>d. a <~~> d \<and> map f d = map f c" using p m by (induct arbitrary: a) (simp, force, force, blast) lemma (in monoid) perm_assoc_switch: assumes a:"as [\<sim>] bs" and p: "bs <~~> cs" shows "\<exists>bs'. as <~~> bs' \<and> bs' [\<sim>] cs" using p a apply (induct bs cs arbitrary: as, simp) apply (clarsimp simp add: list_all2_Cons2, blast) apply (clarsimp simp add: list_all2_Cons2) apply blast apply blast done lemma (in monoid) perm_assoc_switch_r: assumes p: "as <~~> bs" and a:"bs [\<sim>] cs" shows "\<exists>bs'. as [\<sim>] bs' \<and> bs' <~~> cs" using p a apply (induct as bs arbitrary: cs, simp) apply (clarsimp simp add: list_all2_Cons1, blast) apply (clarsimp simp add: list_all2_Cons1) apply blast apply blast done declare perm_sym [sym] lemma perm_setP: assumes perm: "as <~~> bs" and as: "P (set as)" shows "P (set bs)" proof - from perm have "multiset_of as = multiset_of bs" by (simp add: multiset_of_eq_perm) hence "set as = set bs" by (rule multiset_of_eq_setD) with as show "P (set bs)" by simp qed lemmas (in monoid) perm_closed = perm_setP[of _ _ "\<lambda>as. as \<subseteq> carrier G"] lemmas (in monoid) irrlist_perm_cong = perm_setP[of _ _ "\<lambda>as. \<forall>a\<in>as. irreducible G a"] text {* Essentially equal factorizations *} lemma (in monoid) essentially_equalI: assumes ex: "fs1 <~~> fs1'" "fs1' [\<sim>] fs2" shows "essentially_equal G fs1 fs2" using ex unfolding essentially_equal_def by fast lemma (in monoid) essentially_equalE: assumes ee: "essentially_equal G fs1 fs2" and e: "\<And>fs1'. \<lbrakk>fs1 <~~> fs1'; fs1' [\<sim>] fs2\<rbrakk> \<Longrightarrow> P" shows "P" using ee unfolding essentially_equal_def by (fast intro: e) lemma (in monoid) ee_refl [simp,intro]: assumes carr: "set as \<subseteq> carrier G" shows "essentially_equal G as as" using carr by (fast intro: essentially_equalI) lemma (in monoid) ee_sym [sym]: assumes ee: "essentially_equal G as bs" and carr: "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" shows "essentially_equal G bs as" using ee proof (elim essentially_equalE) fix fs assume "as <~~> fs" "fs [\<sim>] bs" hence "\<exists>fs'. as [\<sim>] fs' \<and> fs' <~~> bs" by (rule perm_assoc_switch_r) from this obtain fs' where a: "as [\<sim>] fs'" and p: "fs' <~~> bs" by auto from p have "bs <~~> fs'" by (rule perm_sym) with a[symmetric] carr show ?thesis by (iprover intro: essentially_equalI perm_closed) qed lemma (in monoid) ee_trans [trans]: assumes ab: "essentially_equal G as bs" and bc: "essentially_equal G bs cs" and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G" and cscarr: "set cs \<subseteq> carrier G" shows "essentially_equal G as cs" using ab bc proof (elim essentially_equalE) fix abs bcs assume "abs [\<sim>] bs" and pb: "bs <~~> bcs" hence "\<exists>bs'. abs <~~> bs' \<and> bs' [\<sim>] bcs" by (rule perm_assoc_switch) from this obtain bs' where p: "abs <~~> bs'" and a: "bs' [\<sim>] bcs" by auto assume "as <~~> abs" with p have pp: "as <~~> bs'" by fast from pp ascarr have c1: "set bs' \<subseteq> carrier G" by (rule perm_closed) from pb bscarr have c2: "set bcs \<subseteq> carrier G" by (rule perm_closed) note a also assume "bcs [\<sim>] cs" finally (listassoc_trans) have"bs' [\<sim>] cs" by (simp add: c1 c2 cscarr) with pp show ?thesis by (rule essentially_equalI) qed subsubsection {* Properties of lists of elements *} text {* Multiplication of factors in a list *} lemma (in monoid) multlist_closed [simp, intro]: assumes ascarr: "set fs \<subseteq> carrier G" shows "foldr (op \<otimes>) fs \<one> \<in> carrier G" by (insert ascarr, induct fs, simp+) lemma (in comm_monoid) multlist_dividesI (*[intro]*): assumes "f \<in> set fs" and "f \<in> carrier G" and "set fs \<subseteq> carrier G" shows "f divides (foldr (op \<otimes>) fs \<one>)" using assms apply (induct fs) apply simp apply (case_tac "f = a", simp) apply (fast intro: dividesI) apply clarsimp apply (elim dividesE, intro dividesI) defer 1 apply (simp add: m_comm) apply (simp add: m_assoc[symmetric]) apply (simp add: m_comm) apply simp done lemma (in comm_monoid_cancel) multlist_listassoc_cong: assumes "fs [\<sim>] fs'" and "set fs \<subseteq> carrier G" and "set fs' \<subseteq> carrier G" shows "foldr (op \<otimes>) fs \<one> \<sim> foldr (op \<otimes>) fs' \<one>" using assms proof (induct fs arbitrary: fs', simp) case (Cons a as fs') thus ?case apply (induct fs', simp) proof clarsimp fix b bs assume "a \<sim> b" and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and ascarr: "set as \<subseteq> carrier G" hence p: "a \<otimes> foldr op \<otimes> as \<one> \<sim> b \<otimes> foldr op \<otimes> as \<one>" by (fast intro: mult_cong_l) also assume "as [\<sim>] bs" and bscarr: "set bs \<subseteq> carrier G" 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>" hence "foldr op \<otimes> as \<one> \<sim> foldr op \<otimes> bs \<one>" by simp with ascarr bscarr bcarr have "b \<otimes> foldr op \<otimes> as \<one> \<sim> b \<otimes> foldr op \<otimes> bs \<one>" by (fast intro: mult_cong_r) finally show "a \<otimes> foldr op \<otimes> as \<one> \<sim> b \<otimes> foldr op \<otimes> bs \<one>" by (simp add: ascarr bscarr acarr bcarr) qed qed lemma (in comm_monoid) multlist_perm_cong: assumes prm: "as <~~> bs" and ascarr: "set as \<subseteq> carrier G" shows "foldr (op \<otimes>) as \<one> = foldr (op \<otimes>) bs \<one>" using prm ascarr apply (induct, simp, clarsimp simp add: m_ac, clarsimp) proof clarsimp fix xs ys zs assume "xs <~~> ys" "set xs \<subseteq> carrier G" hence "set ys \<subseteq> carrier G" by (rule perm_closed) moreover assume "set ys \<subseteq> carrier G \<Longrightarrow> foldr op \<otimes> ys \<one> = foldr op \<otimes> zs \<one>" ultimately show "foldr op \<otimes> ys \<one> = foldr op \<otimes> zs \<one>" by simp qed lemma (in comm_monoid_cancel) multlist_ee_cong: assumes "essentially_equal G fs fs'" and "set fs \<subseteq> carrier G" and "set fs' \<subseteq> carrier G" shows "foldr (op \<otimes>) fs \<one> \<sim> foldr (op \<otimes>) fs' \<one>" using assms apply (elim essentially_equalE) apply (simp add: multlist_perm_cong multlist_listassoc_cong perm_closed) done subsubsection {* Factorization in irreducible elements *} lemma wfactorsI: includes (struct G) assumes "\<forall>f\<in>set fs. irreducible G f" and "foldr (op \<otimes>) fs \<one> \<sim> a" shows "wfactors G fs a" using assms unfolding wfactors_def by simp lemma wfactorsE: includes (struct G) assumes wf: "wfactors G fs a" and e: "\<lbrakk>\<forall>f\<in>set fs. irreducible G f; foldr (op \<otimes>) fs \<one> \<sim> a\<rbrakk> \<Longrightarrow> P" shows "P" using wf unfolding wfactors_def by (fast dest: e) lemma (in monoid) factorsI: includes (struct G) assumes "\<forall>f\<in>set fs. irreducible G f" and "foldr (op \<otimes>) fs \<one> = a" shows "factors G fs a" using assms unfolding factors_def by simp lemma factorsE: includes (struct G) assumes f: "factors G fs a" and e: "\<lbrakk>\<forall>f\<in>set fs. irreducible G f; foldr (op \<otimes>) fs \<one> = a\<rbrakk> \<Longrightarrow> P" shows "P" using f unfolding factors_def by (simp add: e) lemma (in monoid) factors_wfactors: assumes "factors G as a" and "set as \<subseteq> carrier G" shows "wfactors G as a" using assms by (blast elim: factorsE intro: wfactorsI) lemma (in monoid) wfactors_factors: assumes "wfactors G as a" and "set as \<subseteq> carrier G" shows "\<exists>a'. factors G as a' \<and> a' \<sim> a" using assms by (blast elim: wfactorsE intro: factorsI) lemma (in monoid) factors_closed [dest]: assumes "factors G fs a" and "set fs \<subseteq> carrier G" shows "a \<in> carrier G" using assms by (elim factorsE, clarsimp) lemma (in monoid) nunit_factors: assumes anunit: "a \<notin> Units G" and fs: "factors G as a" shows "length as > 0" apply (insert fs, elim factorsE) proof (cases "length as = 0") assume "length as = 0" hence fold: "foldr op \<otimes> as \<one> = \<one>" by force assume "foldr op \<otimes> as \<one> = a" with fold have "a = \<one>" by simp then have "a \<in> Units G" by fast with anunit have "False" by simp thus ?thesis .. qed simp lemma (in monoid) unit_wfactors [simp]: assumes aunit: "a \<in> Units G" shows "wfactors G [] a" using aunit by (intro wfactorsI) (simp, simp add: Units_assoc) lemma (in comm_monoid_cancel) unit_wfactors_empty: assumes aunit: "a \<in> Units G" and wf: "wfactors G fs a" and carr[simp]: "set fs \<subseteq> carrier G" shows "fs = []" proof (rule ccontr, cases fs, simp) fix f fs' assume fs: "fs = f # fs'" from carr have fcarr[simp]: "f \<in> carrier G" and carr'[simp]: "set fs' \<subseteq> carrier G" by (simp add: fs)+ from fs wf have "irreducible G f" by (simp add: wfactors_def) hence fnunit: "f \<notin> Units G" by (fast elim: irreducibleE) from fs wf have a: "f \<otimes> foldr (op \<otimes>) fs' \<one> \<sim> a" by (simp add: wfactors_def) note aunit also from fs wf have a: "f \<otimes> foldr (op \<otimes>) fs' \<one> \<sim> a" by (simp add: wfactors_def) have "a \<sim> f \<otimes> foldr (op \<otimes>) fs' \<one>" by (simp add: Units_closed[OF aunit] a[symmetric]) finally have "f \<otimes> foldr (op \<otimes>) fs' \<one> \<in> Units G" by simp hence "f \<in> Units G" by (intro unit_factor[of f], simp+) with fnunit show "False" by simp qed text {* Comparing wfactors *} lemma (in comm_monoid_cancel) wfactors_listassoc_cong_l: assumes fact: "wfactors G fs a" and asc: "fs [\<sim>] fs'" and carr: "a \<in> carrier G" "set fs \<subseteq> carrier G" "set fs' \<subseteq> carrier G" shows "wfactors G fs' a" using fact apply (elim wfactorsE, intro wfactorsI) proof - assume "\<forall>f\<in>set fs. irreducible G f" also note asc finally (irrlist_listassoc_cong) show "\<forall>f\<in>set fs'. irreducible G f" by (simp add: carr) next from asc[symmetric] have "foldr op \<otimes> fs' \<one> \<sim> foldr op \<otimes> fs \<one>" by (simp add: multlist_listassoc_cong carr) also assume "foldr op \<otimes> fs \<one> \<sim> a" finally show "foldr op \<otimes> fs' \<one> \<sim> a" by (simp add: carr) qed lemma (in comm_monoid) wfactors_perm_cong_l: assumes "wfactors G fs a" and "fs <~~> fs'" and "set fs \<subseteq> carrier G" shows "wfactors G fs' a" using assms apply (elim wfactorsE, intro wfactorsI) apply (rule irrlist_perm_cong, assumption+) apply (simp add: multlist_perm_cong[symmetric]) done lemma (in comm_monoid_cancel) wfactors_ee_cong_l [trans]: assumes ee: "essentially_equal G as bs" and bfs: "wfactors G bs b" and carr: "b \<in> carrier G" "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" shows "wfactors G as b" using ee proof (elim essentially_equalE) fix fs assume prm: "as <~~> fs" with carr have fscarr: "set fs \<subseteq> carrier G" by (simp add: perm_closed) note bfs also assume [symmetric]: "fs [\<sim>] bs" also (wfactors_listassoc_cong_l) note prm[symmetric] finally (wfactors_perm_cong_l) show "wfactors G as b" by (simp add: carr fscarr) qed lemma (in monoid) wfactors_cong_r [trans]: assumes fac: "wfactors G fs a" and aa': "a \<sim> a'" and carr[simp]: "a \<in> carrier G" "a' \<in> carrier G" "set fs \<subseteq> carrier G" shows "wfactors G fs a'" using fac proof (elim wfactorsE, intro wfactorsI) assume "foldr op \<otimes> fs \<one> \<sim> a" also note aa' finally show "foldr op \<otimes> fs \<one> \<sim> a'" by simp qed subsubsection {* Essentially equal factorizations *} lemma (in comm_monoid_cancel) unitfactor_ee: assumes uunit: "u \<in> Units G" and carr: "set as \<subseteq> carrier G" shows "essentially_equal G (as[0 := (as!0 \<otimes> u)]) as" (is "essentially_equal G ?as' as") using assms apply (intro essentially_equalI[of _ ?as'], simp) apply (cases as, simp) apply (clarsimp, fast intro: associatedI2[of u]) done lemma (in comm_monoid_cancel) factors_cong_unit: assumes uunit: "u \<in> Units G" and anunit: "a \<notin> Units G" and afs: "factors G as a" and ascarr: "set as \<subseteq> carrier G" shows "factors G (as[0 := (as!0 \<otimes> u)]) (a \<otimes> u)" (is "factors G ?as' ?a'") using assms apply (elim factorsE, clarify) apply (cases as) apply (simp add: nunit_factors) apply clarsimp apply (elim factorsE, intro factorsI) apply (clarsimp, fast intro: irreducible_prod_rI) apply (simp add: m_ac Units_closed) done lemma (in comm_monoid) perm_wfactorsD: assumes prm: "as <~~> bs" and afs: "wfactors G as a" and bfs: "wfactors G bs b" and [simp]: "a \<in> carrier G" "b \<in> carrier G" and ascarr[simp]: "set as \<subseteq> carrier G" shows "a \<sim> b" using afs bfs proof (elim wfactorsE) from prm have [simp]: "set bs \<subseteq> carrier G" by (simp add: perm_closed) assume "foldr op \<otimes> as \<one> \<sim> a" hence "a \<sim> foldr op \<otimes> as \<one>" by (rule associated_sym, simp+) also from prm have "foldr op \<otimes> as \<one> = foldr op \<otimes> bs \<one>" by (rule multlist_perm_cong, simp) also assume "foldr op \<otimes> bs \<one> \<sim> b" finally show "a \<sim> b" by simp qed lemma (in comm_monoid_cancel) listassoc_wfactorsD: assumes assoc: "as [\<sim>] bs" and afs: "wfactors G as a" and bfs: "wfactors G bs b" and [simp]: "a \<in> carrier G" "b \<in> carrier G" and [simp]: "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" shows "a \<sim> b" using afs bfs proof (elim wfactorsE) assume "foldr op \<otimes> as \<one> \<sim> a" hence "a \<sim> foldr op \<otimes> as \<one>" by (rule associated_sym, simp+) also from assoc have "foldr op \<otimes> as \<one> \<sim> foldr op \<otimes> bs \<one>" by (rule multlist_listassoc_cong, simp+) also assume "foldr op \<otimes> bs \<one> \<sim> b" finally show "a \<sim> b" by simp qed lemma (in comm_monoid_cancel) ee_wfactorsD: assumes ee: "essentially_equal G as bs" and afs: "wfactors G as a" and bfs: "wfactors G bs b" and [simp]: "a \<in> carrier G" "b \<in> carrier G" and ascarr[simp]: "set as \<subseteq> carrier G" and bscarr[simp]: "set bs \<subseteq> carrier G" shows "a \<sim> b" using ee proof (elim essentially_equalE) fix fs assume prm: "as <~~> fs" hence as'carr[simp]: "set fs \<subseteq> carrier G" by (simp add: perm_closed) from afs prm have afs': "wfactors G fs a" by (rule wfactors_perm_cong_l, simp) assume "fs [\<sim>] bs" from this afs' bfs show "a \<sim> b" by (rule listassoc_wfactorsD, simp+) qed lemma (in comm_monoid_cancel) ee_factorsD: assumes ee: "essentially_equal G as bs" and afs: "factors G as a" and bfs:"factors G bs b" and "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" shows "a \<sim> b" using assms by (blast intro: factors_wfactors dest: ee_wfactorsD) lemma (in factorial_monoid) ee_factorsI: assumes ab: "a \<sim> b" and afs: "factors G as a" and anunit: "a \<notin> Units G" and bfs: "factors G bs b" and bnunit: "b \<notin> Units G" and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G" shows "essentially_equal G as bs" proof - note carr[simp] = factors_closed[OF afs ascarr] ascarr[THEN subsetD] factors_closed[OF bfs bscarr] bscarr[THEN subsetD] from ab carr have "\<exists>u\<in>Units G. a = b \<otimes> u" by (fast elim: associatedE2) from this obtain u where uunit: "u \<in> Units G" and a: "a = b \<otimes> u" by auto from uunit bscarr have ee: "essentially_equal G (bs[0 := (bs!0 \<otimes> u)]) bs" (is "essentially_equal G ?bs' bs") by (rule unitfactor_ee) from bscarr uunit have bs'carr: "set ?bs' \<subseteq> carrier G" by (cases bs) (simp add: Units_closed)+ from uunit bnunit bfs bscarr have fac: "factors G ?bs' (b \<otimes> u)" by (rule factors_cong_unit) from afs fac[simplified a[symmetric]] ascarr bs'carr anunit have "essentially_equal G as ?bs'" by (blast intro: factors_unique) also note ee finally show "essentially_equal G as bs" by (simp add: ascarr bscarr bs'carr) qed lemma (in factorial_monoid) ee_wfactorsI: assumes asc: "a \<sim> b" and asf: "wfactors G as a" and bsf: "wfactors G bs b" and acarr[simp]: "a \<in> carrier G" and bcarr[simp]: "b \<in> carrier G" and ascarr[simp]: "set as \<subseteq> carrier G" and bscarr[simp]: "set bs \<subseteq> carrier G" shows "essentially_equal G as bs" using assms proof (cases "a \<in> Units G") assume aunit: "a \<in> Units G" also note asc finally have bunit: "b \<in> Units G" by simp from aunit asf ascarr have e: "as = []" by (rule unit_wfactors_empty) from bunit bsf bscarr have e': "bs = []" by (rule unit_wfactors_empty) have "essentially_equal G [] []" by (fast intro: essentially_equalI) thus ?thesis by (simp add: e e') next assume anunit: "a \<notin> Units G" have bnunit: "b \<notin> Units G" proof clarify assume "b \<in> Units G" also note asc[symmetric] finally have "a \<in> Units G" by simp with anunit show "False" .. qed have "\<exists>a'. factors G as a' \<and> a' \<sim> a" by (rule wfactors_factors[OF asf ascarr]) from this obtain a' where fa': "factors G as a'" and a': "a' \<sim> a" by auto from fa' ascarr have a'carr[simp]: "a' \<in> carrier G" by fast have a'nunit: "a' \<notin> Units G" proof (clarify) assume "a' \<in> Units G" also note a' finally have "a \<in> Units G" by simp with anunit show "False" .. qed have "\<exists>b'. factors G bs b' \<and> b' \<sim> b" by (rule wfactors_factors[OF bsf bscarr]) from this obtain b' where fb': "factors G bs b'" and b': "b' \<sim> b" by auto from fb' bscarr have b'carr[simp]: "b' \<in> carrier G" by fast have b'nunit: "b' \<notin> Units G" proof (clarify) assume "b' \<in> Units G" also note b' finally have "b \<in> Units G" by simp with bnunit show "False" .. qed note a' also note asc also note b'[symmetric] finally have "a' \<sim> b'" by simp from this fa' a'nunit fb' b'nunit ascarr bscarr show "essentially_equal G as bs" by (rule ee_factorsI) qed lemma (in factorial_monoid) ee_wfactors: assumes asf: "wfactors G as a" and bsf: "wfactors G bs b" and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G" shows asc: "a \<sim> b = essentially_equal G as bs" using assms by (fast intro: ee_wfactorsI ee_wfactorsD) lemma (in factorial_monoid) wfactors_exist [intro, simp]: assumes acarr[simp]: "a \<in> carrier G" shows "\<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs a" proof (cases "a \<in> Units G") assume "a \<in> Units G" hence "wfactors G [] a" by (rule unit_wfactors) thus ?thesis by (intro exI) force next assume "a \<notin> Units G" hence "\<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs a" by (intro factors_exist acarr) from this obtain fs where fscarr: "set fs \<subseteq> carrier G" and f: "factors G fs a" by auto from f have "wfactors G fs a" by (rule factors_wfactors) fact from fscarr this show ?thesis by fast qed lemma (in monoid) wfactors_prod_exists [intro, simp]: assumes "\<forall>a \<in> set as. irreducible G a" and "set as \<subseteq> carrier G" shows "\<exists>a. a \<in> carrier G \<and> wfactors G as a" unfolding wfactors_def using assms by blast lemma (in factorial_monoid) wfactors_unique: assumes "wfactors G fs a" and "wfactors G fs' a" and "a \<in> carrier G" and "set fs \<subseteq> carrier G" and "set fs' \<subseteq> carrier G" shows "essentially_equal G fs fs'" using assms by (fast intro: ee_wfactorsI[of a a]) lemma (in monoid) factors_mult_single: assumes "irreducible G a" and "factors G fb b" and "a \<in> carrier G" shows "factors G (a # fb) (a \<otimes> b)" using assms unfolding factors_def by simp lemma (in monoid_cancel) wfactors_mult_single: assumes f: "irreducible G a" "wfactors G fb b" "a \<in> carrier G" "b \<in> carrier G" "set fb \<subseteq> carrier G" shows "wfactors G (a # fb) (a \<otimes> b)" using assms unfolding wfactors_def by (simp add: mult_cong_r) lemma (in monoid) factors_mult: assumes factors: "factors G fa a" "factors G fb b" and ascarr: "set fa \<subseteq> carrier G" and bscarr:"set fb \<subseteq> carrier G" shows "factors G (fa @ fb) (a \<otimes> b)" using assms unfolding factors_def apply (safe, force) apply (induct fa) apply simp apply (simp add: m_assoc) done lemma (in comm_monoid_cancel) wfactors_mult [intro]: assumes asf: "wfactors G as a" and bsf:"wfactors G bs b" and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and ascarr: "set as \<subseteq> carrier G" and bscarr:"set bs \<subseteq> carrier G" shows "wfactors G (as @ bs) (a \<otimes> b)" apply (insert wfactors_factors[OF asf ascarr]) apply (insert wfactors_factors[OF bsf bscarr]) proof (clarsimp) fix a' b' assume asf': "factors G as a'" and a'a: "a' \<sim> a" and bsf': "factors G bs b'" and b'b: "b' \<sim> b" from asf' have a'carr: "a' \<in> carrier G" by (rule factors_closed) fact from bsf' have b'carr: "b' \<in> carrier G" by (rule factors_closed) fact note carr = acarr bcarr a'carr b'carr ascarr bscarr from asf' bsf' have "factors G (as @ bs) (a' \<otimes> b')" by (rule factors_mult) fact+ with carr have abf': "wfactors G (as @ bs) (a' \<otimes> b')" by (intro factors_wfactors) simp+ also from b'b carr have trb: "a' \<otimes> b' \<sim> a' \<otimes> b" by (intro mult_cong_r) also from a'a carr have tra: "a' \<otimes> b \<sim> a \<otimes> b" by (intro mult_cong_l) finally show "wfactors G (as @ bs) (a \<otimes> b)" by (simp add: carr) qed lemma (in comm_monoid) factors_dividesI: assumes "factors G fs a" and "f \<in> set fs" and "set fs \<subseteq> carrier G" shows "f divides a" using assms by (fast elim: factorsE intro: multlist_dividesI) lemma (in comm_monoid) wfactors_dividesI: assumes p: "wfactors G fs a" and fscarr: "set fs \<subseteq> carrier G" and acarr: "a \<in> carrier G" and f: "f \<in> set fs" shows "f divides a" apply (insert wfactors_factors[OF p fscarr], clarsimp) proof - fix a' assume fsa': "factors G fs a'" and a'a: "a' \<sim> a" with fscarr have a'carr: "a' \<in> carrier G" by (simp add: factors_closed) from fsa' fscarr f have "f divides a'" by (fast intro: factors_dividesI) also note a'a finally show "f divides a" by (simp add: f fscarr[THEN subsetD] acarr a'carr) qed subsubsection {* Factorial monoids and wfactors *} lemma (in comm_monoid_cancel) factorial_monoidI: assumes wfactors_exists: "\<And>a. a \<in> carrier G \<Longrightarrow> \<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs a" and wfactors_unique: "\<And>a fs fs'. \<lbrakk>a \<in> carrier G; set fs \<subseteq> carrier G; set fs' \<subseteq> carrier G; wfactors G fs a; wfactors G fs' a\<rbrakk> \<Longrightarrow> essentially_equal G fs fs'" shows "factorial_monoid G" proof (unfold_locales) fix a assume acarr: "a \<in> carrier G" and anunit: "a \<notin> Units G" from wfactors_exists[OF acarr] obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" by auto from afs ascarr have "\<exists>a'. factors G as a' \<and> a' \<sim> a" by (rule wfactors_factors) from this obtain a' where afs': "factors G as a'" and a'a: "a' \<sim> a" by auto from afs' ascarr have a'carr: "a' \<in> carrier G" by fast have a'nunit: "a' \<notin> Units G" proof clarify assume "a' \<in> Units G" also note a'a finally have "a \<in> Units G" by (simp add: acarr) with anunit show "False" .. qed from a'carr acarr a'a have "\<exists>u. u \<in> Units G \<and> a' = a \<otimes> u" by (blast elim: associatedE2) from this obtain u where uunit: "u \<in> Units G" and a': "a' = a \<otimes> u" by auto note [simp] = acarr Units_closed[OF uunit] Units_inv_closed[OF uunit] have "a = a \<otimes> \<one>" by simp also have "\<dots> = a \<otimes> (u \<otimes> inv u)" by (simp add: Units_r_inv uunit) also have "\<dots> = a' \<otimes> inv u" by (simp add: m_assoc[symmetric] a'[symmetric]) finally have a: "a = a' \<otimes> inv u" . from ascarr uunit have cr: "set (as[0:=(as!0 \<otimes> inv u)]) \<subseteq> carrier G" by (cases as, clarsimp+) from afs' uunit a'nunit acarr ascarr have "factors G (as[0:=(as!0 \<otimes> inv u)]) a" by (simp add: a factors_cong_unit) with cr show "\<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs a" by fast qed (blast intro: factors_wfactors wfactors_unique) subsection {* Factorizations as Multisets *} text {* Gives useful operations like intersection *} (* FIXME: use class_of x instead of closure_of {x} *) abbreviation "assocs G x == eq_closure_of (division_rel G) {x}" constdefs (structure G) "fmset G as \<equiv> multiset_of (map (\<lambda>a. assocs G a) as)" text {* Helper lemmas *} lemma (in monoid) assocs_repr_independence: assumes "y \<in> assocs G x" and "x \<in> carrier G" shows "assocs G x = assocs G y" using assms apply safe apply (elim closure_ofE2, intro closure_ofI2[of _ _ y]) apply (clarsimp, iprover intro: associated_trans associated_sym, simp+) apply (elim closure_ofE2, intro closure_ofI2[of _ _ x]) apply (clarsimp, iprover intro: associated_trans, simp+) done lemma (in monoid) assocs_self: assumes "x \<in> carrier G" shows "x \<in> assocs G x" using assms by (fastsimp intro: closure_ofI2) lemma (in monoid) assocs_repr_independenceD: assumes repr: "assocs G x = assocs G y" and ycarr: "y \<in> carrier G" shows "y \<in> assocs G x" unfolding repr using ycarr by (intro assocs_self) lemma (in comm_monoid) assocs_assoc: assumes "a \<in> assocs G b" and "b \<in> carrier G" shows "a \<sim> b" using assms by (elim closure_ofE2, simp) lemmas (in comm_monoid) assocs_eqD = assocs_repr_independenceD[THEN assocs_assoc] subsubsection {* Comparing multisets *} lemma (in monoid) fmset_perm_cong: assumes prm: "as <~~> bs" shows "fmset G as = fmset G bs" using perm_map[OF prm] by (simp add: multiset_of_eq_perm fmset_def) lemma (in comm_monoid_cancel) eqc_listassoc_cong: assumes "as [\<sim>] bs" and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" shows "map (assocs G) as = map (assocs G) bs" using assms apply (induct as arbitrary: bs, simp) apply (clarsimp simp add: Cons_eq_map_conv list_all2_Cons1, safe) apply (clarsimp elim!: closure_ofE2) defer 1 apply (clarsimp elim!: closure_ofE2) defer 1 proof - fix a x z assume carr[simp]: "a \<in> carrier G" "x \<in> carrier G" "z \<in> carrier G" assume "x \<sim> a" also assume "a \<sim> z" finally have "x \<sim> z" by simp with carr show "x \<in> assocs G z" by (intro closure_ofI2) simp+ next fix a x z assume carr[simp]: "a \<in> carrier G" "x \<in> carrier G" "z \<in> carrier G" assume "x \<sim> z" also assume [symmetric]: "a \<sim> z" finally have "x \<sim> a" by simp with carr show "x \<in> assocs G a" by (intro closure_ofI2) simp+ qed lemma (in comm_monoid_cancel) fmset_listassoc_cong: assumes "as [\<sim>] bs" and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" shows "fmset G as = fmset G bs" using assms unfolding fmset_def by (simp add: eqc_listassoc_cong) lemma (in comm_monoid_cancel) ee_fmset: assumes ee: "essentially_equal G as bs" and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G" shows "fmset G as = fmset G bs" using ee proof (elim essentially_equalE) fix as' assume prm: "as <~~> as'" from prm ascarr have as'carr: "set as' \<subseteq> carrier G" by (rule perm_closed) from prm have "fmset G as = fmset G as'" by (rule fmset_perm_cong) also assume "as' [\<sim>] bs" with as'carr bscarr have "fmset G as' = fmset G bs" by (simp add: fmset_listassoc_cong) finally show "fmset G as = fmset G bs" . qed lemma (in monoid_cancel) fmset_ee__hlp_induct: assumes prm: "cas <~~> cbs" and cdef: "cas = map (assocs G) as" "cbs = map (assocs G) bs" shows "\<forall>as bs. (cas <~~> cbs \<and> cas = map (assocs G) as \<and> cbs = map (assocs G) bs) \<longrightarrow> (\<exists>as'. as <~~> as' \<and> map (assocs G) as' = cbs)" apply (rule perm.induct[of cas cbs], rule prm) apply safe apply simp_all apply (simp add: map_eq_Cons_conv, blast) apply force proof - fix ys as bs assume p1: "map (assocs G) as <~~> ys" and r1[rule_format]: "\<forall>asa bs. map (assocs G) as = map (assocs G) asa \<and> ys = map (assocs G) bs \<longrightarrow> (\<exists>as'. asa <~~> as' \<and> map (assocs G) as' = map (assocs G) bs)" and p2: "ys <~~> map (assocs G) bs" and r2[rule_format]: "\<forall>as bsa. ys = map (assocs G) as \<and> map (assocs G) bs = map (assocs G) bsa \<longrightarrow> (\<exists>as'. as <~~> as' \<and> map (assocs G) as' = map (assocs G) bsa)" and p3: "map (assocs G) as <~~> map (assocs G) bs" from p1 have "multiset_of (map (assocs G) as) = multiset_of ys" by (simp add: multiset_of_eq_perm) hence setys: "set (map (assocs G) as) = set ys" by (rule multiset_of_eq_setD) have "set (map (assocs G) as) = { assocs G x | x. x \<in> set as}" by clarsimp fast with setys have "set ys \<subseteq> { assocs G x | x. x \<in> set as}" by simp hence "\<exists>yy. ys = map (assocs G) yy" apply (induct ys, simp, clarsimp) proof - fix yy x show "\<exists>yya. (assocs G x) # map (assocs G) yy = map (assocs G) yya" by (rule exI[of _ "x#yy"], simp) qed from this obtain yy where ys: "ys = map (assocs G) yy" by auto from p1 ys have "\<exists>as'. as <~~> as' \<and> map (assocs G) as' = map (assocs G) yy" by (intro r1, simp) from this obtain as' where asas': "as <~~> as'" and as'yy: "map (assocs G) as' = map (assocs G) yy" by auto from p2 ys have "\<exists>as'. yy <~~> as' \<and> map (assocs G) as' = map (assocs G) bs" by (intro r2, simp) from this obtain as'' where yyas'': "yy <~~> as''" and as''bs: "map (assocs G) as'' = map (assocs G) bs" by auto from as'yy and yyas'' have "\<exists>cs. as' <~~> cs \<and> map (assocs G) cs = map (assocs G) as''" by (rule perm_map_switch) from this obtain cs where as'cs: "as' <~~> cs" and csas'': "map (assocs G) cs = map (assocs G) as''" by auto from asas' and as'cs have ascs: "as <~~> cs" by fast from csas'' and as''bs have "map (assocs G) cs = map (assocs G) bs" by simp from ascs and this show "\<exists>as'. as <~~> as' \<and> map (assocs G) as' = map (assocs G) bs" by fast qed lemma (in comm_monoid_cancel) fmset_ee: assumes mset: "fmset G as = fmset G bs" and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G" shows "essentially_equal G as bs" proof - from mset have mpp: "map (assocs G) as <~~> map (assocs G) bs" by (simp add: fmset_def multiset_of_eq_perm) have "\<exists>cas. cas = map (assocs G) as" by simp from this obtain cas where cas: "cas = map (assocs G) as" by simp have "\<exists>cbs. cbs = map (assocs G) bs" by simp from this obtain cbs where cbs: "cbs = map (assocs G) bs" by simp from cas cbs mpp have [rule_format]: "\<forall>as bs. (cas <~~> cbs \<and> cas = map (assocs G) as \<and> cbs = map (assocs G) bs) \<longrightarrow> (\<exists>as'. as <~~> as' \<and> map (assocs G) as' = cbs)" by (intro fmset_ee__hlp_induct, simp+) with mpp cas cbs have "\<exists>as'. as <~~> as' \<and> map (assocs G) as' = map (assocs G) bs" by simp from this obtain as' where tp: "as <~~> as'" and tm: "map (assocs G) as' = map (assocs G) bs" by auto from tm have lene: "length as' = length bs" by (rule map_eq_imp_length_eq) from tp have "set as = set as'" by (simp add: multiset_of_eq_perm multiset_of_eq_setD) with ascarr have as'carr: "set as' \<subseteq> carrier G" by simp from tm as'carr[THEN subsetD] bscarr[THEN subsetD] have "as' [\<sim>] bs" by (induct as' arbitrary: bs) (simp, fastsimp dest: assocs_eqD[THEN associated_sym]) from tp and this show "essentially_equal G as bs" by (fast intro: essentially_equalI) qed lemma (in comm_monoid_cancel) ee_is_fmset: assumes "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" shows "essentially_equal G as bs = (fmset G as = fmset G bs)" using assms by (fast intro: ee_fmset fmset_ee) subsubsection {* Interpreting multisets as factorizations *} lemma (in monoid) mset_fmsetEx: assumes elems: "\<And>X. X \<in> set_of Cs \<Longrightarrow> \<exists>x. P x \<and> X = assocs G x" shows "\<exists>cs. (\<forall>c \<in> set cs. P c) \<and> fmset G cs = Cs" proof - have "\<exists>Cs'. Cs = multiset_of Cs'" by (rule surjE[OF surj_multiset_of], fast) from this obtain Cs' where Cs: "Cs = multiset_of Cs'" by auto have "\<exists>cs. (\<forall>c \<in> set cs. P c) \<and> multiset_of (map (assocs G) cs) = Cs" using elems unfolding Cs apply (induct Cs', simp) apply clarsimp apply (subgoal_tac "\<exists>cs. (\<forall>x\<in>set cs. P x) \<and> multiset_of (map (assocs G) cs) = multiset_of Cs'") proof clarsimp fix a Cs' cs assume ih: "\<And>X. X = a \<or> X \<in> set Cs' \<Longrightarrow> \<exists>x. P x \<and> X = assocs G x" and csP: "\<forall>x\<in>set cs. P x" and mset: "multiset_of (map (assocs G) cs) = multiset_of Cs'" from ih have "\<exists>x. P x \<and> a = assocs G x" by fast from this obtain c where cP: "P c" and a: "a = assocs G c" by auto from cP csP have tP: "\<forall>x\<in>set (c#cs). P x" by simp from mset a have "multiset_of (map (assocs G) (c#cs)) = multiset_of Cs' + {#a#}" by simp from tP this show "\<exists>cs. (\<forall>x\<in>set cs. P x) \<and> multiset_of (map (assocs G) cs) = multiset_of Cs' + {#a#}" by fast qed simp thus ?thesis by (simp add: fmset_def) qed lemma (in monoid) mset_wfactorsEx: assumes elems: "\<And>X. X \<in> set_of Cs \<Longrightarrow> \<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" shows "\<exists>c cs. c \<in> carrier G \<and> set cs \<subseteq> carrier G \<and> wfactors G cs c \<and> fmset G cs = Cs" proof - have "\<exists>cs. (\<forall>c\<in>set cs. c \<in> carrier G \<and> irreducible G c) \<and> fmset G cs = Cs" by (intro mset_fmsetEx, rule elems) from this obtain cs where p[rule_format]: "\<forall>c\<in>set cs. c \<in> carrier G \<and> irreducible G c" and Cs[symmetric]: "fmset G cs = Cs" by auto from p have cscarr: "set cs \<subseteq> carrier G" by fast from p have "\<exists>c. c \<in> carrier G \<and> wfactors G cs c" by (intro wfactors_prod_exists) fast+ from this obtain c where ccarr: "c \<in> carrier G" and cfs: "wfactors G cs c" by auto with cscarr Cs show ?thesis by fast qed subsubsection {* Multiplication on multisets *} lemma (in factorial_monoid) mult_wfactors_fmset: assumes afs: "wfactors G as a" and bfs: "wfactors G bs b" and cfs: "wfactors G cs (a \<otimes> b)" and carr: "a \<in> carrier G" "b \<in> carrier G" "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" "set cs \<subseteq> carrier G" shows "fmset G cs = fmset G as + fmset G bs" proof - from assms have "wfactors G (as @ bs) (a \<otimes> b)" by (intro wfactors_mult) with carr cfs have "essentially_equal G cs (as@bs)" by (intro ee_wfactorsI[of "a\<otimes>b" "a\<otimes>b"], simp+) with carr have "fmset G cs = fmset G (as@bs)" by (intro ee_fmset, simp+) also have "fmset G (as@bs) = fmset G as + fmset G bs" by (simp add: fmset_def) finally show "fmset G cs = fmset G as + fmset G bs" . qed lemma (in factorial_monoid) mult_factors_fmset: assumes afs: "factors G as a" and bfs: "factors G bs b" and cfs: "factors G cs (a \<otimes> b)" and "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" "set cs \<subseteq> carrier G" shows "fmset G cs = fmset G as + fmset G bs" using assms by (blast intro: factors_wfactors mult_wfactors_fmset) lemma (in comm_monoid_cancel) fmset_wfactors_mult: assumes mset: "fmset G cs = fmset G as + fmset G bs" and carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" "set cs \<subseteq> carrier G" and fs: "wfactors G as a" "wfactors G bs b" "wfactors G cs c" shows "c \<sim> a \<otimes> b" proof - from carr fs have m: "wfactors G (as @ bs) (a \<otimes> b)" by (intro wfactors_mult) from mset have "fmset G cs = fmset G (as@bs)" by (simp add: fmset_def) then have "essentially_equal G cs (as@bs)" by (rule fmset_ee) (simp add: carr)+ then show "c \<sim> a \<otimes> b" by (rule ee_wfactorsD[of "cs" "as@bs"]) (simp add: assms m)+ qed subsubsection {* Divisibility on multisets *} lemma (in factorial_monoid) divides_fmsubset: assumes ab: "a divides b" and afs: "wfactors G as a" and bfs: "wfactors G bs b" and carr: "a \<in> carrier G" "b \<in> carrier G" "set as \<subseteq> carrier G" "set bs \<subseteq> carrier G" shows "fmset G as \<le># fmset G bs" using ab proof (elim dividesE) fix c assume ccarr: "c \<in> carrier G" hence "\<exists>cs. set cs \<subseteq> carrier G \<and> wfactors G cs c" by (rule wfactors_exist) from this obtain cs where cscarr: "set cs \<subseteq> carrier G" and cfs: "wfactors G cs c" by auto note carr = carr ccarr cscarr assume "b = a \<otimes> c" with afs bfs cfs carr have "fmset G bs = fmset G as + fmset G cs" by (intro mult_wfactors_fmset[OF afs cfs]) simp+ thus ?thesis by simp qed lemma (in comm_monoid_cancel) fmsubset_divides: assumes msubset: "fmset G as \<le># fmset G bs" and afs: "wfactors G as a" and bfs: "wfactors G bs b" and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G" shows "a divides b" proof - from afs have airr: "\<forall>a \<in> set as. irreducible G a" by (fast elim: wfactorsE) from bfs have birr: "\<forall>b \<in> set bs. irreducible G b" by (fast elim: wfactorsE) 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" proof (intro mset_wfactorsEx, simp) fix X assume "count (fmset G as) X < count (fmset G bs) X" hence "0 < count (fmset G bs) X" by simp hence "X \<in> set_of (fmset G bs)" by simp hence "X \<in> set (map (assocs G) bs)" by (simp add: fmset_def) hence "\<exists>x. x \<in> set bs \<and> X = assocs G x" by (induct bs) auto from this obtain x where xbs: "x \<in> set bs" and X: "X = assocs G x" by auto with bscarr have xcarr: "x \<in> carrier G" by fast from xbs birr have xirr: "irreducible G x" by simp from xcarr and xirr and X show "\<exists>x. x \<in> carrier G \<and> irreducible G x \<and> X = assocs G x" by fast qed from this obtain c cs where ccarr: "c \<in> carrier G" and cscarr: "set cs \<subseteq> carrier G" and csf: "wfactors G cs c" and csmset: "fmset G cs = fmset G bs - fmset G as" by auto from csmset msubset have "fmset G bs = fmset G as + fmset G cs" by (simp add: multiset_eq_conv_count_eq mset_le_def) hence basc: "b \<sim> a \<otimes> c" by (rule fmset_wfactors_mult) fact+ thus ?thesis proof (elim associatedE2) fix u assume "u \<in> Units G" "b = a \<otimes> c \<otimes> u" with acarr ccarr show "a divides b" by (fast intro: dividesI[of "c \<otimes> u"] m_assoc) qed (simp add: acarr bcarr ccarr)+ qed lemma (in factorial_monoid) divides_as_fmsubset: assumes "wfactors G as a" and "wfactors G bs b" and "a \<in> carrier G" and "b \<in> carrier G" and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" shows "a divides b = (fmset G as \<le># fmset G bs)" using assms by (blast intro: divides_fmsubset fmsubset_divides) text {* Proper factors on multisets *} lemma (in factorial_monoid) fmset_properfactor: assumes asubb: "fmset G as \<le># fmset G bs" and anb: "fmset G as \<noteq> fmset G bs" and "wfactors G as a" and "wfactors G bs b" and "a \<in> carrier G" and "b \<in> carrier G" and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" shows "properfactor G a b" apply (rule properfactorI) apply (rule fmsubset_divides[of as bs], fact+) proof assume "b divides a" hence "fmset G bs \<le># fmset G as" by (rule divides_fmsubset) fact+ with asubb have "fmset G as = fmset G bs" by (simp add: mset_le_antisym) with anb show "False" .. qed lemma (in factorial_monoid) properfactor_fmset: assumes pf: "properfactor G a b" and "wfactors G as a" and "wfactors G bs b" and "a \<in> carrier G" and "b \<in> carrier G" and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" shows "fmset G as \<le># fmset G bs \<and> fmset G as \<noteq> fmset G bs" using pf apply (elim properfactorE) apply rule apply (intro divides_fmsubset, assumption) apply (rule assms)+ proof assume bna: "\<not> b divides a" assume "fmset G as = fmset G bs" then have "essentially_equal G as bs" by (rule fmset_ee) fact+ hence "a \<sim> b" by (rule ee_wfactorsD[of as bs]) fact+ hence "b divides a" by (elim associatedE) with bna show "False" .. qed subsection {* Irreducible Elements are Prime *} lemma (in factorial_monoid) irreducible_is_prime: assumes pirr: "irreducible G p" and pcarr: "p \<in> carrier G" shows "prime G p" using pirr proof (elim irreducibleE, intro primeI) fix a b assume acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and pdvdab: "p divides (a \<otimes> b)" and pnunit: "p \<notin> Units G" assume irreduc[rule_format]: "\<forall>b. b \<in> carrier G \<and> properfactor G b p \<longrightarrow> b \<in> Units G" from pdvdab have "\<exists>c\<in>carrier G. a \<otimes> b = p \<otimes> c" by (rule dividesD) from this obtain c where ccarr: "c \<in> carrier G" and abpc: "a \<otimes> b = p \<otimes> c" by auto from acarr have "\<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs a" by (rule wfactors_exist) from this obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" by auto from bcarr have "\<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs b" by (rule wfactors_exist) from this obtain bs where bscarr: "set bs \<subseteq> carrier G" and bfs: "wfactors G bs b" by auto from ccarr have "\<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs c" by (rule wfactors_exist) from this obtain cs where cscarr: "set cs \<subseteq> carrier G" and cfs: "wfactors G cs c" by auto note carr[simp] = pcarr acarr bcarr ccarr ascarr bscarr cscarr from afs and bfs have abfs: "wfactors G (as @ bs) (a \<otimes> b)" by (rule wfactors_mult) fact+ from pirr cfs have pcfs: "wfactors G (p # cs) (p \<otimes> c)" by (rule wfactors_mult_single) fact+ with abpc have abfs': "wfactors G (p # cs) (a \<otimes> b)" by simp from abfs' abfs have "essentially_equal G (p # cs) (as @ bs)" by (rule wfactors_unique) simp+ hence "\<exists>ds. p # cs <~~> ds \<and> ds [\<sim>] (as @ bs)" by (fast elim: essentially_equalE) from this obtain ds where "p # cs <~~> ds" and dsassoc: "ds [\<sim>] (as @ bs)" by auto then have "p \<in> set ds" by (simp add: perm_set_eq[symmetric]) with dsassoc have "\<exists>p'. p' \<in> set (as@bs) \<and> p \<sim> p'" unfolding list_all2_conv_all_nth set_conv_nth by force from this obtain p' where "p' \<in> set (as@bs)" and pp': "p \<sim> p'" by auto hence "p' \<in> set as \<or> p' \<in> set bs" by simp moreover { assume p'elem: "p' \<in> set as" with ascarr have [simp]: "p' \<in> carrier G" by fast note pp' also from afs have "p' divides a" by (rule wfactors_dividesI) fact+ finally have "p divides a" by simp } moreover { assume p'elem: "p' \<in> set bs" with bscarr have [simp]: "p' \<in> carrier G" by fast note pp' also from bfs have "p' divides b" by (rule wfactors_dividesI) fact+ finally have "p divides b" by simp } ultimately show "p divides a \<or> p divides b" by fast qed --"A version using @{const factors}, more complicated" lemma (in factorial_monoid) factors_irreducible_is_prime: assumes pirr: "irreducible G p" and pcarr: "p \<in> carrier G" shows "prime G p" using pirr apply (elim irreducibleE, intro primeI) apply assumption proof - fix a b assume acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and pdvdab: "p divides (a \<otimes> b)" assume irreduc[rule_format]: "\<forall>b. b \<in> carrier G \<and> properfactor G b p \<longrightarrow> b \<in> Units G" from pdvdab have "\<exists>c\<in>carrier G. a \<otimes> b = p \<otimes> c" by (rule dividesD) from this obtain c where ccarr: "c \<in> carrier G" and abpc: "a \<otimes> b = p \<otimes> c" by auto note [simp] = pcarr acarr bcarr ccarr show "p divides a \<or> p divides b" proof (cases "a \<in> Units G") assume aunit: "a \<in> Units G" note pdvdab also have "a \<otimes> b = b \<otimes> a" by (simp add: m_comm) also from aunit have bab: "b \<otimes> a \<sim> b" by (intro associatedI2[of "a"], simp+) finally have "p divides b" by simp thus "p divides a \<or> p divides b" .. next assume anunit: "a \<notin> Units G" show "p divides a \<or> p divides b" proof (cases "b \<in> Units G") assume bunit: "b \<in> Units G" note pdvdab also from bunit have baa: "a \<otimes> b \<sim> a" by (intro associatedI2[of "b"], simp+) finally have "p divides a" by simp thus "p divides a \<or> p divides b" .. next assume bnunit: "b \<notin> Units G" have cnunit: "c \<notin> Units G" proof (rule ccontr, simp) assume cunit: "c \<in> Units G" from bnunit have "properfactor G a (a \<otimes> b)" by (intro properfactorI3[of _ _ b], simp+) also note abpc also from cunit have "p \<otimes> c \<sim> p" by (intro associatedI2[of c], simp+) finally have "properfactor G a p" by simp with acarr have "a \<in> Units G" by (fast intro: irreduc) with anunit show "False" .. qed have abnunit: "a \<otimes> b \<notin> Units G" proof clarsimp assume abunit: "a \<otimes> b \<in> Units G" hence "a \<in> Units G" by (rule unit_factor) fact+ with anunit show "False" .. qed from acarr anunit have "\<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs a" by (rule factors_exist) then obtain as where ascarr: "set as \<subseteq> carrier G" and afac: "factors G as a" by auto from bcarr bnunit have "\<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs b" by (rule factors_exist) then obtain bs where bscarr: "set bs \<subseteq> carrier G" and bfac: "factors G bs b" by auto from ccarr cnunit have "\<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs c" by (rule factors_exist) then obtain cs where cscarr: "set cs \<subseteq> carrier G" and cfac: "factors G cs c" by auto note [simp] = ascarr bscarr cscarr from afac and bfac have abfac: "factors G (as @ bs) (a \<otimes> b)" by (rule factors_mult) fact+ from pirr cfac have pcfac: "factors G (p # cs) (p \<otimes> c)" by (rule factors_mult_single) fact+ with abpc have abfac': "factors G (p # cs) (a \<otimes> b)" by simp from abfac' abfac have "essentially_equal G (p # cs) (as @ bs)" by (rule factors_unique) (fact | simp)+ hence "\<exists>ds. p # cs <~~> ds \<and> ds [\<sim>] (as @ bs)" by (fast elim: essentially_equalE) from this obtain ds where "p # cs <~~> ds" and dsassoc: "ds [\<sim>] (as @ bs)" by auto then have "p \<in> set ds" by (simp add: perm_set_eq[symmetric]) with dsassoc have "\<exists>p'. p' \<in> set (as@bs) \<and> p \<sim> p'" unfolding list_all2_conv_all_nth set_conv_nth by force from this obtain p' where "p' \<in> set (as@bs)" and pp': "p \<sim> p'" by auto hence "p' \<in> set as \<or> p' \<in> set bs" by simp moreover { assume p'elem: "p' \<in> set as" with ascarr have [simp]: "p' \<in> carrier G" by fast note pp' also from afac p'elem have "p' divides a" by (rule factors_dividesI) fact+ finally have "p divides a" by simp } moreover { assume p'elem: "p' \<in> set bs" with bscarr have [simp]: "p' \<in> carrier G" by fast note pp' also from bfac have "p' divides b" by (rule factors_dividesI) fact+ finally have "p divides b" by simp } ultimately show "p divides a \<or> p divides b" by fast qed qed qed subsection {* Greatest Common Divisors and Lowest Common Multiples *} subsubsection {* Definitions *} constdefs (structure G) isgcd :: "[('a,_) monoid_scheme, 'a, 'a, 'a] \<Rightarrow> bool" ("(_ gcdof\<index> _ _)" [81,81,81] 80) "x gcdof a b \<equiv> x divides a \<and> x divides b \<and> (\<forall>y\<in>carrier G. (y divides a \<and> y divides b \<longrightarrow> y divides x))" islcm :: "[_, 'a, 'a, 'a] \<Rightarrow> bool" ("(_ lcmof\<index> _ _)" [81,81,81] 80) "x lcmof a b \<equiv> a divides x \<and> b divides x \<and> (\<forall>y\<in>carrier G. (a divides y \<and> b divides y \<longrightarrow> x divides y))" constdefs (structure G) somegcd :: "('a,_) monoid_scheme \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" "somegcd G a b == SOME x. x \<in> carrier G \<and> x gcdof a b" somelcm :: "('a,_) monoid_scheme \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" "somelcm G a b == SOME x. x \<in> carrier G \<and> x lcmof a b" constdefs (structure G) "SomeGcd G A == inf (division_rel G) A" locale gcd_condition_monoid = comm_monoid_cancel + 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" locale primeness_condition_monoid = comm_monoid_cancel + assumes irreducible_prime: "\<lbrakk>a \<in> carrier G; irreducible G a\<rbrakk> \<Longrightarrow> prime G a" locale divisor_chain_condition_monoid = comm_monoid_cancel + assumes division_wellfounded: "wf {(x, y). x \<in> carrier G \<and> y \<in> carrier G \<and> properfactor G x y}" subsubsection {* Connections to \texttt{Lattice.thy} *} lemma gcdof_greatestLower: fixes G (structure) assumes carr[simp]: "a \<in> carrier G" "b \<in> carrier G" shows "(x \<in> carrier G \<and> x gcdof a b) = greatest (division_rel G) x (Lower (division_rel G) {a, b})" unfolding isgcd_def greatest_def Lower_def elem_def proof (simp, safe) fix xa assume r1[rule_format]: "\<forall>x. (x = a \<or> x = b) \<and> x \<in> carrier G \<longrightarrow> xa divides x" assume r2[rule_format]: "\<forall>y\<in>carrier G. y divides a \<and> y divides b \<longrightarrow> y divides x" assume "xa \<in> carrier G" "x divides a" "x divides b" with carr show "xa divides x" by (fast intro: r1 r2) next fix a' y assume r1[rule_format]: "\<forall>xa\<in>{l. \<forall>x. (x = a \<or> x = b) \<and> x \<in> carrier G \<longrightarrow> l divides x} \<inter> carrier G. xa divides x" assume "y \<in> carrier G" "y divides a" "y divides b" with carr show "y divides x" by (fast intro: r1) qed (simp, simp) lemma lcmof_leastUpper: fixes G (structure) assumes carr[simp]: "a \<in> carrier G" "b \<in> carrier G" shows "(x \<in> carrier G \<and> x lcmof a b) = least (division_rel G) x (Upper (division_rel G) {a, b})" unfolding islcm_def least_def Upper_def elem_def proof (simp, safe) fix xa assume r1[rule_format]: "\<forall>x. (x = a \<or> x = b) \<and> x \<in> carrier G \<longrightarrow> x divides xa" assume r2[rule_format]: "\<forall>y\<in>carrier G. a divides y \<and> b divides y \<longrightarrow> x divides y" assume "xa \<in> carrier G" "a divides x" "b divides x" with carr show "x divides xa" by (fast intro: r1 r2) next fix a' y assume r1[rule_format]: "\<forall>xa\<in>{l. \<forall>x. (x = a \<or> x = b) \<and> x \<in> carrier G \<longrightarrow> x divides l} \<inter> carrier G. x divides xa" assume "y \<in> carrier G" "a divides y" "b divides y" with carr show "x divides y" by (fast intro: r1) qed (simp, simp) lemma somegcd_meet: fixes G (structure) assumes carr: "a \<in> carrier G" "b \<in> carrier G" shows "somegcd G a b = meet (division_rel G) a b" unfolding somegcd_def meet_def inf_def by (simp add: gcdof_greatestLower[OF carr]) lemma (in monoid) isgcd_divides_l: assumes "a divides b" and "a \<in> carrier G" "b \<in> carrier G" shows "a gcdof a b" using assms unfolding isgcd_def by fast lemma (in monoid) isgcd_divides_r: assumes "b divides a" and "a \<in> carrier G" "b \<in> carrier G" shows "b gcdof a b" using assms unfolding isgcd_def by fast subsubsection {* Existence of gcd and lcm *} lemma (in factorial_monoid) gcdof_exists: assumes acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" shows "\<exists>c. c \<in> carrier G \<and> c gcdof a b" proof - from acarr have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" by (rule wfactors_exist) from this obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" by auto from afs have airr: "\<forall>a \<in> set as. irreducible G a" by (fast elim: wfactorsE) from bcarr have "\<exists>bs. set bs \<subseteq> carrier G \<and> wfactors G bs b" by (rule wfactors_exist) from this obtain bs where bscarr: "set bs \<subseteq> carrier G" and bfs: "wfactors G bs b" by auto from bfs have birr: "\<forall>b \<in> set bs. irreducible G b" by (fast elim: wfactorsE) 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 as #\<inter> fmset G bs" proof (intro mset_wfactorsEx) fix X assume "X \<in> set_of (fmset G as #\<inter> fmset G bs)" hence "X \<in> set_of (fmset G as)" by (simp add: multiset_inter_def) hence "X \<in> set (map (assocs G) as)" by (simp add: fmset_def) hence "\<exists>x. X = assocs G x \<and> x \<in> set as" by (induct as) auto from this obtain x where X: "X = assocs G x" and xas: "x \<in> set as" by auto with ascarr have xcarr: "x \<in> carrier G" by fast from xas airr have xirr: "irreducible G x" by simp from xcarr and xirr and X show "\<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" by fast qed from this obtain c cs where ccarr: "c \<in> carrier G" and cscarr: "set cs \<subseteq> carrier G" and csirr: "wfactors G cs c" and csmset: "fmset G cs = fmset G as #\<inter> fmset G bs" by auto have "c gcdof a b" proof (simp add: isgcd_def, safe) from csmset have "fmset G cs \<le># fmset G as" by (simp add: multiset_inter_def mset_le_def) thus "c divides a" by (rule fmsubset_divides) fact+ next from csmset have "fmset G cs \<le># fmset G bs" by (simp add: multiset_inter_def mset_le_def, force) thus "c divides b" by (rule fmsubset_divides) fact+ next fix y assume ycarr: "y \<in> carrier G" hence "\<exists>ys. set ys \<subseteq> carrier G \<and> wfactors G ys y" by (rule wfactors_exist) from this obtain ys where yscarr: "set ys \<subseteq> carrier G" and yfs: "wfactors G ys y" by auto assume "y divides a" hence ya: "fmset G ys \<le># fmset G as" by (rule divides_fmsubset) fact+ assume "y divides b" hence yb: "fmset G ys \<le># fmset G bs" by (rule divides_fmsubset) fact+ from ya yb csmset have "fmset G ys \<le># fmset G cs" by (simp add: mset_le_def multiset_inter_count) thus "y divides c" by (rule fmsubset_divides) fact+ qed with ccarr show "\<exists>c. c \<in> carrier G \<and> c gcdof a b" by fast qed lemma (in factorial_monoid) lcmof_exists: assumes acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" shows "\<exists>c. c \<in> carrier G \<and> c lcmof a b" proof - from acarr have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" by (rule wfactors_exist) from this obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" by auto from afs have airr: "\<forall>a \<in> set as. irreducible G a" by (fast elim: wfactorsE) from bcarr have "\<exists>bs. set bs \<subseteq> carrier G \<and> wfactors G bs b" by (rule wfactors_exist) from this obtain bs where bscarr: "set bs \<subseteq> carrier G" and bfs: "wfactors G bs b" by auto from bfs have birr: "\<forall>b \<in> set bs. irreducible G b" by (fast elim: wfactorsE) 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 as - fmset G bs) + fmset G bs" proof (intro mset_wfactorsEx) fix X assume "X \<in> set_of ((fmset G as - fmset G bs) + fmset G bs)" hence "X \<in> set_of (fmset G as) \<or> X \<in> set_of (fmset G bs)" by (cases "X :# fmset G bs", simp, simp) moreover { assume "X \<in> set_of (fmset G as)" hence "X \<in> set (map (assocs G) as)" by (simp add: fmset_def) hence "\<exists>x. x \<in> set as \<and> X = assocs G x" by (induct as) auto from this obtain x where xas: "x \<in> set as" and X: "X = assocs G x" by auto with ascarr have xcarr: "x \<in> carrier G" by fast from xas airr have xirr: "irreducible G x" by simp from xcarr and xirr and X have "\<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" by fast } moreover { assume "X \<in> set_of (fmset G bs)" hence "X \<in> set (map (assocs G) bs)" by (simp add: fmset_def) hence "\<exists>x. x \<in> set bs \<and> X = assocs G x" by (induct as) auto from this obtain x where xbs: "x \<in> set bs" and X: "X = assocs G x" by auto with bscarr have xcarr: "x \<in> carrier G" by fast from xbs birr have xirr: "irreducible G x" by simp from xcarr and xirr and X have "\<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" by fast } ultimately show "\<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" by fast qed from this obtain c cs where ccarr: "c \<in> carrier G" and cscarr: "set cs \<subseteq> carrier G" and csirr: "wfactors G cs c" and csmset: "fmset G cs = fmset G as - fmset G bs + fmset G bs" by auto have "c lcmof a b" proof (simp add: islcm_def, safe) from csmset have "fmset G as \<le># fmset G cs" by (simp add: mset_le_def, force) thus "a divides c" by (rule fmsubset_divides) fact+ next from csmset have "fmset G bs \<le># fmset G cs" by (simp add: mset_le_def) thus "b divides c" by (rule fmsubset_divides) fact+ next fix y assume ycarr: "y \<in> carrier G" hence "\<exists>ys. set ys \<subseteq> carrier G \<and> wfactors G ys y" by (rule wfactors_exist) from this obtain ys where yscarr: "set ys \<subseteq> carrier G" and yfs: "wfactors G ys y" by auto assume "a divides y" hence ya: "fmset G as \<le># fmset G ys" by (rule divides_fmsubset) fact+ assume "b divides y" hence yb: "fmset G bs \<le># fmset G ys" by (rule divides_fmsubset) fact+ from ya yb csmset have "fmset G cs \<le># fmset G ys" apply (simp add: mset_le_def, clarify) apply (case_tac "count (fmset G as) a < count (fmset G bs) a") apply simp apply simp done thus "c divides y" by (rule fmsubset_divides) fact+ qed with ccarr show "\<exists>c. c \<in> carrier G \<and> c lcmof a b" by fast qed subsection {* Conditions for Factoriality *} subsubsection {* Gcd condition *} lemma (in gcd_condition_monoid) division_weak_lower_semilattice [simp]: shows "weak_lower_semilattice (division_rel G)" proof - interpret weak_partial_order ["division_rel G"] .. show ?thesis apply (unfold_locales, simp_all) proof - fix x y assume carr: "x \<in> carrier G" "y \<in> carrier G" hence "\<exists>z. z \<in> carrier G \<and> z gcdof x y" by (rule gcdof_exists) from this obtain z where zcarr: "z \<in> carrier G" and isgcd: "z gcdof x y" by auto with carr have "greatest (division_rel G) z (Lower (division_rel G) {x, y})" by (subst gcdof_greatestLower[symmetric], simp+) thus "\<exists>z. greatest (division_rel G) z (Lower (division_rel G) {x, y})" by fast qed qed lemma (in gcd_condition_monoid) gcdof_cong_l: assumes a'a: "a' \<sim> a" and agcd: "a gcdof b c" and a'carr: "a' \<in> carrier G" and carr': "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "a' gcdof b c" proof - note carr = a'carr carr' interpret weak_lower_semilattice ["division_rel G"] by simp have "a' \<in> carrier G \<and> a' gcdof b c" apply (simp add: gcdof_greatestLower carr') apply (subst greatest_Lower_cong_l[of _ a]) apply (simp add: a'a) apply (simp add: carr) apply (simp add: carr) apply (simp add: carr) apply (simp add: gcdof_greatestLower[symmetric] agcd carr) done thus ?thesis .. qed lemma (in gcd_condition_monoid) gcd_closed [simp]: assumes carr: "a \<in> carrier G" "b \<in> carrier G" shows "somegcd G a b \<in> carrier G" proof - interpret weak_lower_semilattice ["division_rel G"] by simp show ?thesis apply (simp add: somegcd_meet[OF carr]) apply (rule meet_closed[simplified], fact+) done qed lemma (in gcd_condition_monoid) gcd_isgcd: assumes carr: "a \<in> carrier G" "b \<in> carrier G" shows "(somegcd G a b) gcdof a b" proof - interpret weak_lower_semilattice ["division_rel G"] by simp from carr have "somegcd G a b \<in> carrier G \<and> (somegcd G a b) gcdof a b" apply (subst gcdof_greatestLower, simp, simp) apply (simp add: somegcd_meet[OF carr] meet_def) apply (rule inf_of_two_greatest[simplified], assumption+) done thus "(somegcd G a b) gcdof a b" by simp qed lemma (in gcd_condition_monoid) gcd_exists: assumes carr: "a \<in> carrier G" "b \<in> carrier G" shows "\<exists>x\<in>carrier G. x = somegcd G a b" proof - interpret weak_lower_semilattice ["division_rel G"] by simp show ?thesis apply (simp add: somegcd_meet[OF carr]) apply (rule meet_closed[simplified], fact+) done qed lemma (in gcd_condition_monoid) gcd_divides_l: assumes carr: "a \<in> carrier G" "b \<in> carrier G" shows "(somegcd G a b) divides a" proof - interpret weak_lower_semilattice ["division_rel G"] by simp show ?thesis apply (simp add: somegcd_meet[OF carr]) apply (rule meet_left[simplified], fact+) done qed lemma (in gcd_condition_monoid) gcd_divides_r: assumes carr: "a \<in> carrier G" "b \<in> carrier G" shows "(somegcd G a b) divides b" proof - interpret weak_lower_semilattice ["division_rel G"] by simp show ?thesis apply (simp add: somegcd_meet[OF carr]) apply (rule meet_right[simplified], fact+) done qed lemma (in gcd_condition_monoid) gcd_divides: assumes sub: "z divides x" "z divides y" and L: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" shows "z divides (somegcd G x y)" proof - interpret weak_lower_semilattice ["division_rel G"] by simp show ?thesis apply (simp add: somegcd_meet L) apply (rule meet_le[simplified], fact+) done qed lemma (in gcd_condition_monoid) gcd_cong_l: assumes xx': "x \<sim> x'" and carr: "x \<in> carrier G" "x' \<in> carrier G" "y \<in> carrier G" shows "somegcd G x y \<sim> somegcd G x' y" proof - interpret weak_lower_semilattice ["division_rel G"] by simp show ?thesis apply (simp add: somegcd_meet carr) apply (rule meet_cong_l[simplified], fact+) done qed lemma (in gcd_condition_monoid) gcd_cong_r: assumes carr: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G" and yy': "y \<sim> y'" shows "somegcd G x y \<sim> somegcd G x y'" proof - interpret weak_lower_semilattice ["division_rel G"] by simp show ?thesis apply (simp add: somegcd_meet carr) apply (rule meet_cong_r[simplified], fact+) done qed (* lemma (in gcd_condition_monoid) asc_cong_gcd_l [intro]: assumes carr: "b \<in> carrier G" shows "asc_cong (\<lambda>a. somegcd G a b)" using carr unfolding CONG_def by clarsimp (blast intro: gcd_cong_l) lemma (in gcd_condition_monoid) asc_cong_gcd_r [intro]: assumes carr: "a \<in> carrier G" shows "asc_cong (\<lambda>b. somegcd G a b)" using carr unfolding CONG_def by clarsimp (blast intro: gcd_cong_r) lemmas (in gcd_condition_monoid) asc_cong_gcd_split [simp] = assoc_split[OF _ asc_cong_gcd_l] assoc_split[OF _ asc_cong_gcd_r] *) lemma (in gcd_condition_monoid) gcdI: assumes dvd: "a divides b" "a divides c" and others: "\<forall>y\<in>carrier G. y divides b \<and> y divides c \<longrightarrow> y divides a" and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and ccarr: "c \<in> carrier G" shows "a \<sim> somegcd G b c" apply (simp add: somegcd_def) apply (rule someI2_ex) apply (rule exI[of _ a], simp add: isgcd_def) apply (simp add: assms) apply (simp add: isgcd_def assms, clarify) apply (insert assms, blast intro: associatedI) done lemma (in gcd_condition_monoid) gcdI2: assumes "a gcdof b c" and "a \<in> carrier G" and bcarr: "b \<in> carrier G" and ccarr: "c \<in> carrier G" shows "a \<sim> somegcd G b c" using assms unfolding isgcd_def by (blast intro: gcdI) lemma (in gcd_condition_monoid) SomeGcd_ex: assumes "finite A" "A \<subseteq> carrier G" "A \<noteq> {}" shows "\<exists>x\<in> carrier G. x = SomeGcd G A" proof - interpret weak_lower_semilattice ["division_rel G"] by simp show ?thesis apply (simp add: SomeGcd_def) apply (rule finite_inf_closed[simplified], fact+) done qed lemma (in gcd_condition_monoid) gcd_assoc: assumes carr: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "somegcd G (somegcd G a b) c \<sim> somegcd G a (somegcd G b c)" proof - interpret weak_lower_semilattice ["division_rel G"] by simp show ?thesis apply (subst (2 3) somegcd_meet, (simp add: carr)+) apply (simp add: somegcd_meet carr) apply (rule weak_meet_assoc[simplified], fact+) done qed lemma (in gcd_condition_monoid) gcd_mult: assumes acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and ccarr: "c \<in> carrier G" shows "c \<otimes> somegcd G a b \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)" proof - (* following Jacobson, Basic Algebra, p.140 *) let ?d = "somegcd G a b" let ?e = "somegcd G (c \<otimes> a) (c \<otimes> b)" note carr[simp] = acarr bcarr ccarr have dcarr: "?d \<in> carrier G" by simp have ecarr: "?e \<in> carrier G" by simp note carr = carr dcarr ecarr have "?d divides a" by (simp add: gcd_divides_l) hence cd'ca: "c \<otimes> ?d divides (c \<otimes> a)" by (simp add: divides_mult_lI) have "?d divides b" by (simp add: gcd_divides_r) hence cd'cb: "c \<otimes> ?d divides (c \<otimes> b)" by (simp add: divides_mult_lI) from cd'ca cd'cb have cd'e: "c \<otimes> ?d divides ?e" by (rule gcd_divides) simp+ hence "\<exists>u. u \<in> carrier G \<and> ?e = c \<otimes> ?d \<otimes> u" by (elim dividesE, fast) from this obtain u where ucarr[simp]: "u \<in> carrier G" and e_cdu: "?e = c \<otimes> ?d \<otimes> u" by auto note carr = carr ucarr have "?e divides c \<otimes> a" by (rule gcd_divides_l) simp+ hence "\<exists>x. x \<in> carrier G \<and> c \<otimes> a = ?e \<otimes> x" by (elim dividesE, fast) from this obtain x where xcarr: "x \<in> carrier G" and ca_ex: "c \<otimes> a = ?e \<otimes> x" by auto with e_cdu have ca_cdux: "c \<otimes> a = c \<otimes> ?d \<otimes> u \<otimes> x" by simp from ca_cdux xcarr have "c \<otimes> a = c \<otimes> (?d \<otimes> u \<otimes> x)" by (simp add: m_assoc) then have "a = ?d \<otimes> u \<otimes> x" by (rule l_cancel[of c a]) (simp add: xcarr)+ hence du'a: "?d \<otimes> u divides a" by (rule dividesI[OF xcarr]) have "?e divides c \<otimes> b" by (intro gcd_divides_r, simp+) hence "\<exists>x. x \<in> carrier G \<and> c \<otimes> b = ?e \<otimes> x" by (elim dividesE, fast) from this obtain x where xcarr: "x \<in> carrier G" and cb_ex: "c \<otimes> b = ?e \<otimes> x" by auto with e_cdu have cb_cdux: "c \<otimes> b = c \<otimes> ?d \<otimes> u \<otimes> x" by simp from cb_cdux xcarr have "c \<otimes> b = c \<otimes> (?d \<otimes> u \<otimes> x)" by (simp add: m_assoc) with xcarr have "b = ?d \<otimes> u \<otimes> x" by (intro l_cancel[of c b], simp+) hence du'b: "?d \<otimes> u divides b" by (intro dividesI[OF xcarr]) from du'a du'b carr have du'd: "?d \<otimes> u divides ?d" by (intro gcd_divides, simp+) hence uunit: "u \<in> Units G" proof (elim dividesE) fix v assume vcarr[simp]: "v \<in> carrier G" assume d: "?d = ?d \<otimes> u \<otimes> v" have "?d \<otimes> \<one> = ?d \<otimes> u \<otimes> v" by simp fact also have "?d \<otimes> u \<otimes> v = ?d \<otimes> (u \<otimes> v)" by (simp add: m_assoc) finally have "?d \<otimes> \<one> = ?d \<otimes> (u \<otimes> v)" . hence i2: "\<one> = u \<otimes> v" by (rule l_cancel) simp+ hence i1: "\<one> = v \<otimes> u" by (simp add: m_comm) from vcarr i1[symmetric] i2[symmetric] show "u \<in> Units G" by (unfold Units_def, simp, fast) qed from e_cdu uunit have "somegcd G (c \<otimes> a) (c \<otimes> b) \<sim> c \<otimes> somegcd G a b" by (intro associatedI2[of u], simp+) from this[symmetric] show "c \<otimes> somegcd G a b \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)" by simp qed lemma (in monoid) assoc_subst: assumes ab: "a \<sim> b" and cP: "ALL a b. a : carrier G & b : carrier G & a \<sim> b --> f a : carrier G & f b : carrier G & f a \<sim> f b" and carr: "a \<in> carrier G" "b \<in> carrier G" shows "f a \<sim> f b" using assms by auto lemma (in gcd_condition_monoid) relprime_mult: assumes abrelprime: "somegcd G a b \<sim> \<one>" and acrelprime: "somegcd G a c \<sim> \<one>" and carr[simp]: "a \<in> carrier G" "b \<in> carrier G" "c \<in> carrier G" shows "somegcd G a (b \<otimes> c) \<sim> \<one>" proof - have "c = c \<otimes> \<one>" by simp also from abrelprime[symmetric] have "\<dots> \<sim> c \<otimes> somegcd G a b" by (rule assoc_subst) (simp add: mult_cong_r)+ also have "\<dots> \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)" by (rule gcd_mult) fact+ finally have c: "c \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)" by simp from carr have a: "a \<sim> somegcd G a (c \<otimes> a)" by (fast intro: gcdI divides_prod_l) have "somegcd G a (b \<otimes> c) \<sim> somegcd G a (c \<otimes> b)" by (simp add: m_comm) also from a have "\<dots> \<sim> somegcd G (somegcd G a (c \<otimes> a)) (c \<otimes> b)" by (rule assoc_subst) (simp add: gcd_cong_l)+ also from gcd_assoc have "\<dots> \<sim> somegcd G a (somegcd G (c \<otimes> a) (c \<otimes> b))" by (rule assoc_subst) simp+ also from c[symmetric] have "\<dots> \<sim> somegcd G a c" by (rule assoc_subst) (simp add: gcd_cong_r)+ also note acrelprime finally show "somegcd G a (b \<otimes> c) \<sim> \<one>" by simp qed lemma (in gcd_condition_monoid) primeness_condition: "primeness_condition_monoid G" apply unfold_locales apply (rule primeI) apply (elim irreducibleE, assumption) proof - fix p a b assume pcarr: "p \<in> carrier G" and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and pirr: "irreducible G p" and pdvdab: "p divides a \<otimes> b" from pirr have pnunit: "p \<notin> Units G" and r[rule_format]: "\<forall>b. b \<in> carrier G \<and> properfactor G b p \<longrightarrow> b \<in> Units G" by - (fast elim: irreducibleE)+ show "p divides a \<or> p divides b" proof (rule ccontr, clarsimp) assume npdvda: "\<not> p divides a" with pcarr acarr have "\<one> \<sim> somegcd G p a" apply (intro gcdI, simp, simp, simp) apply (fast intro: unit_divides) apply (fast intro: unit_divides) apply (clarsimp simp add: Unit_eq_dividesone[symmetric]) apply (rule r, rule, assumption) apply (rule properfactorI, assumption) proof (rule ccontr, simp) fix y assume ycarr: "y \<in> carrier G" assume "p divides y" also assume "y divides a" finally have "p divides a" by (simp add: pcarr ycarr acarr) with npdvda show "False" .. qed simp+ with pcarr acarr have pa: "somegcd G p a \<sim> \<one>" by (fast intro: associated_sym[of "\<one>"] gcd_closed) assume npdvdb: "\<not> p divides b" with pcarr bcarr have "\<one> \<sim> somegcd G p b" apply (intro gcdI, simp, simp, simp) apply (fast intro: unit_divides) apply (fast intro: unit_divides) apply (clarsimp simp add: Unit_eq_dividesone[symmetric]) apply (rule r, rule, assumption) apply (rule properfactorI, assumption) proof (rule ccontr, simp) fix y assume ycarr: "y \<in> carrier G" assume "p divides y" also assume "y divides b" finally have "p divides b" by (simp add: pcarr ycarr bcarr) with npdvdb show "False" .. qed simp+ with pcarr bcarr have pb: "somegcd G p b \<sim> \<one>" by (fast intro: associated_sym[of "\<one>"] gcd_closed) from pcarr acarr bcarr pdvdab have "p gcdof p (a \<otimes> b)" by (fast intro: isgcd_divides_l) with pcarr acarr bcarr have "p \<sim> somegcd G p (a \<otimes> b)" by (fast intro: gcdI2) also from pa pb pcarr acarr bcarr have "somegcd G p (a \<otimes> b) \<sim> \<one>" by (rule relprime_mult) finally have "p \<sim> \<one>" by (simp add: pcarr acarr bcarr) with pcarr have "p \<in> Units G" by (fast intro: assoc_unit_l) with pnunit show "False" .. qed qed interpretation gcd_condition_monoid \<subseteq> primeness_condition_monoid by (rule primeness_condition) subsubsection {* Divisor chain condition *} lemma (in divisor_chain_condition_monoid) wfactors_exist: assumes acarr: "a \<in> carrier G" shows "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" proof - have r[rule_format]: "a \<in> carrier G \<longrightarrow> (\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a)" apply (rule wf_induct[OF division_wellfounded]) proof - fix x assume ih: "\<forall>y. (y, x) \<in> {(x, y). x \<in> carrier G \<and> y \<in> carrier G \<and> properfactor G x y} \<longrightarrow> y \<in> carrier G \<longrightarrow> (\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as y)" show "x \<in> carrier G \<longrightarrow> (\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as x)" apply clarify apply (cases "x \<in> Units G") apply (rule exI[of _ "[]"], simp) apply (cases "irreducible G x") apply (rule exI[of _ "[x]"], simp add: wfactors_def) proof - assume xcarr: "x \<in> carrier G" and xnunit: "x \<notin> Units G" and xnirr: "\<not> irreducible G x" hence "\<exists>y. y \<in> carrier G \<and> properfactor G y x \<and> y \<notin> Units G" apply - apply (rule ccontr, simp) apply (subgoal_tac "irreducible G x", simp) apply (rule irreducibleI, simp, simp) done from this obtain y where ycarr: "y \<in> carrier G" and ynunit: "y \<notin> Units G" and pfyx: "properfactor G y x" by auto have ih': "\<And>y. \<lbrakk>y \<in> carrier G; properfactor G y x\<rbrakk> \<Longrightarrow> \<exists>as. set as \<subseteq> carrier G \<and> wfactors G as y" by (rule ih[rule_format, simplified]) (simp add: xcarr)+ from ycarr pfyx have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as y" by (rule ih') from this obtain ys where yscarr: "set ys \<subseteq> carrier G" and yfs: "wfactors G ys y" by auto from pfyx have "y divides x" and nyx: "\<not> y \<sim> x" by - (fast elim: properfactorE2)+ hence "\<exists>z. z \<in> carrier G \<and> x = y \<otimes> z" by (fast elim: dividesE) from this obtain z where zcarr: "z \<in> carrier G" and x: "x = y \<otimes> z" by auto from zcarr ycarr have "properfactor G z x" apply (subst x) apply (intro properfactorI3[of _ _ y]) apply (simp add: m_comm) apply (simp add: ynunit)+ done with zcarr have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as z" by (rule ih') from this obtain zs where zscarr: "set zs \<subseteq> carrier G" and zfs: "wfactors G zs z" by auto from yscarr zscarr have xscarr: "set (ys@zs) \<subseteq> carrier G" by simp from yfs zfs ycarr zcarr yscarr zscarr have "wfactors G (ys@zs) (y\<otimes>z)" by (rule wfactors_mult) hence "wfactors G (ys@zs) x" by (simp add: x) from xscarr this show "\<exists>xs. set xs \<subseteq> carrier G \<and> wfactors G xs x" by fast qed qed from acarr show ?thesis by (rule r) qed subsubsection {* Primeness condition *} lemma (in comm_monoid_cancel) multlist_prime_pos: assumes carr: "a \<in> carrier G" "set as \<subseteq> carrier G" and aprime: "prime G a" and "a divides (foldr (op \<otimes>) as \<one>)" shows "\<exists>i<length as. a divides (as!i)" proof - have r[rule_format]: "set as \<subseteq> carrier G \<and> a divides (foldr (op \<otimes>) as \<one>) \<longrightarrow> (\<exists>i. i < length as \<and> a divides (as!i))" apply (induct as) apply clarsimp defer 1 apply clarsimp defer 1 proof - assume "a divides \<one>" with carr have "a \<in> Units G" by (fast intro: divides_unit[of a \<one>]) with aprime show "False" by (elim primeE, simp) next fix aa as assume ih[rule_format]: "a divides foldr op \<otimes> as \<one> \<longrightarrow> (\<exists>i<length as. a divides as ! i)" and carr': "aa \<in> carrier G" "set as \<subseteq> carrier G" and "a divides aa \<otimes> foldr op \<otimes> as \<one>" with carr aprime have "a divides aa \<or> a divides foldr op \<otimes> as \<one>" by (intro prime_divides) simp+ moreover { assume "a divides aa" hence p1: "a divides (aa#as)!0" by simp have "0 < Suc (length as)" by simp with p1 have "\<exists>i<Suc (length as). a divides (aa # as) ! i" by fast } moreover { assume "a divides foldr op \<otimes> as \<one>" hence "\<exists>i. i < length as \<and> a divides as ! i" by (rule ih) from this obtain i where "a divides as ! i" and len: "i < length as" by auto hence p1: "a divides (aa#as) ! (Suc i)" by simp from len have "Suc i < Suc (length as)" by simp with p1 have "\<exists>i<Suc (length as). a divides (aa # as) ! i" by force } ultimately show "\<exists>i<Suc (length as). a divides (aa # as) ! i" by fast qed from assms show ?thesis by (intro r, safe) qed lemma (in primeness_condition_monoid) wfactors_unique__hlp_induct: "\<forall>a as'. a \<in> carrier G \<and> set as \<subseteq> carrier G \<and> set as' \<subseteq> carrier G \<and> wfactors G as a \<and> wfactors G as' a \<longrightarrow> essentially_equal G as as'" apply (induct as) apply clarsimp defer 1 apply clarsimp defer 1 proof - fix a as' assume acarr: "a \<in> carrier G" and "wfactors G [] a" hence aunit: "a \<in> Units G" apply (elim wfactorsE) apply (simp, rule assoc_unit_r[of "\<one>"], simp+) done assume "set as' \<subseteq> carrier G" "wfactors G as' a" with aunit have "as' = []" by (intro unit_wfactors_empty[of a]) thus "essentially_equal G [] as'" by simp next fix a as ah as' assume ih[rule_format]: "\<forall>a as'. a \<in> carrier G \<and> set as' \<subseteq> carrier G \<and> wfactors G as a \<and> wfactors G as' a \<longrightarrow> essentially_equal G as as'" and acarr: "a \<in> carrier G" and ahcarr: "ah \<in> carrier G" and ascarr: "set as \<subseteq> carrier G" and as'carr: "set as' \<subseteq> carrier G" and afs: "wfactors G (ah # as) a" and afs': "wfactors G as' a" hence ahdvda: "ah divides a" by (intro wfactors_dividesI[of "ah#as" "a"], simp+) hence "\<exists>a'\<in> carrier G. a = ah \<otimes> a'" by (fast elim: dividesE) from this obtain a' where a'carr: "a' \<in> carrier G" and a: "a = ah \<otimes> a'" by auto have a'fs: "wfactors G as a'" apply (rule wfactorsE[OF afs], rule wfactorsI, simp) apply (simp add: a, insert ascarr a'carr) apply (intro assoc_l_cancel[of ah _ a'] multlist_closed ahcarr, assumption+) done from afs have ahirr: "irreducible G ah" by (elim wfactorsE, simp) with ascarr have ahprime: "prime G ah" by (intro irreducible_prime ahcarr) note carr [simp] = acarr ahcarr ascarr as'carr a'carr note ahdvda also from afs' have "a divides (foldr (op \<otimes>) as' \<one>)" by (elim wfactorsE associatedE, simp) finally have "ah divides (foldr (op \<otimes>) as' \<one>)" by simp with ahprime have "\<exists>i<length as'. ah divides as'!i" by (intro multlist_prime_pos, simp+) from this obtain i where len: "i<length as'" and ahdvd: "ah divides as'!i" by auto from afs' carr have irrasi: "irreducible G (as'!i)" by (fast intro: nth_mem[OF len] elim: wfactorsE) from len carr have asicarr[simp]: "as'!i \<in> carrier G" by (unfold set_conv_nth, force) note carr = carr asicarr from ahdvd have "\<exists>x \<in> carrier G. as'!i = ah \<otimes> x" by (fast elim: dividesE) from this obtain x where "x \<in> carrier G" and asi: "as'!i = ah \<otimes> x" by auto with carr irrasi[simplified asi] have asiah: "as'!i \<sim> ah" apply - apply (elim irreducible_prodE[of "ah" "x"], assumption+) apply (rule associatedI2[of x], assumption+) apply (rule irreducibleE[OF ahirr], simp) done note setparts = set_take_subset[of i as'] set_drop_subset[of "Suc i" as'] note partscarr [simp] = setparts[THEN subset_trans[OF _ as'carr]] note carr = carr partscarr have "\<exists>aa_1. aa_1 \<in> carrier G \<and> wfactors G (take i as') aa_1" apply (intro wfactors_prod_exists) using setparts afs' by (fast elim: wfactorsE, simp) from this obtain aa_1 where aa1carr: "aa_1 \<in> carrier G" and aa1fs: "wfactors G (take i as') aa_1" by auto have "\<exists>aa_2. aa_2 \<in> carrier G \<and> wfactors G (drop (Suc i) as') aa_2" apply (intro wfactors_prod_exists) using setparts afs' by (fast elim: wfactorsE, simp) from this obtain aa_2 where aa2carr: "aa_2 \<in> carrier G" and aa2fs: "wfactors G (drop (Suc i) as') aa_2" by auto note carr = carr aa1carr[simp] aa2carr[simp] from aa1fs aa2fs have v1: "wfactors G (take i as' @ drop (Suc i) as') (aa_1 \<otimes> aa_2)" by (intro wfactors_mult, simp+) hence v1': "wfactors G (as'!i # take i as' @ drop (Suc i) as') (as'!i \<otimes> (aa_1 \<otimes> aa_2))" apply (intro wfactors_mult_single) using setparts afs' by (fast intro: nth_mem[OF len] elim: wfactorsE, simp+) from aa2carr carr aa1fs aa2fs have "wfactors G (as'!i # drop (Suc i) as') (as'!i \<otimes> aa_2)" apply (intro wfactors_mult_single) apply (rule wfactorsE[OF afs'], fast intro: nth_mem[OF len]) apply (fast intro: nth_mem[OF len]) apply fast apply fast apply assumption done with len carr aa1carr aa2carr aa1fs have v2: "wfactors G (take i as' @ as'!i # drop (Suc i) as') (aa_1 \<otimes> (as'!i \<otimes> aa_2))" apply (intro wfactors_mult) apply fast apply (simp, (fast intro: nth_mem[OF len])?)+ done from len have as': "as' = (take i as' @ as'!i # drop (Suc i) as')" by (simp add: drop_Suc_conv_tl) with carr have eer: "essentially_equal G (take i as' @ as'!i # drop (Suc i) as') as'" by simp with v2 afs' carr aa1carr aa2carr nth_mem[OF len] have "aa_1 \<otimes> (as'!i \<otimes> aa_2) \<sim> a" apply (intro ee_wfactorsD[of "take i as' @ as'!i # drop (Suc i) as'" "as'"]) apply fast+ apply (simp, fast) done then have t1: "as'!i \<otimes> (aa_1 \<otimes> aa_2) \<sim> a" apply (simp add: m_assoc[symmetric]) apply (simp add: m_comm) done from carr asiah have "ah \<otimes> (aa_1 \<otimes> aa_2) \<sim> as'!i \<otimes> (aa_1 \<otimes> aa_2)" apply (intro mult_cong_l) apply (fast intro: associated_sym, simp+) done also note t1 finally have "ah \<otimes> (aa_1 \<otimes> aa_2) \<sim> a" by simp with carr aa1carr aa2carr a'carr nth_mem[OF len] have a': "aa_1 \<otimes> aa_2 \<sim> a'" by (simp add: a, fast intro: assoc_l_cancel[of ah _ a']) note v1 also note a' finally have "wfactors G (take i as' @ drop (Suc i) as') a'" by simp from a'fs this carr have "essentially_equal G as (take i as' @ drop (Suc i) as')" by (intro ih[of a']) simp hence ee1: "essentially_equal G (ah # as) (ah # take i as' @ drop (Suc i) as')" apply (elim essentially_equalE) apply (fastsimp intro: essentially_equalI) done from carr have ee2: "essentially_equal G (ah # take i as' @ drop (Suc i) as') (as' ! i # take i as' @ drop (Suc i) as')" proof (intro essentially_equalI) show "ah # take i as' @ drop (Suc i) as' <~~> ah # take i as' @ drop (Suc i) as'" by simp next show "ah # take i as' @ drop (Suc i) as' [\<sim>] as' ! i # take i as' @ drop (Suc i) as'" apply (simp add: list_all2_append) apply (simp add: asiah[symmetric] ahcarr asicarr) done qed note ee1 also note ee2 also have "essentially_equal G (as' ! i # take i as' @ drop (Suc i) as') (take i as' @ as' ! i # drop (Suc i) as')" apply (intro essentially_equalI) apply (subgoal_tac "as' ! i # take i as' @ drop (Suc i) as' <~~> take i as' @ as' ! i # drop (Suc i) as'") apply simp apply (rule perm_append_Cons) apply simp done finally have "essentially_equal G (ah # as) (take i as' @ as' ! i # drop (Suc i) as')" by simp thus "essentially_equal G (ah # as) as'" by (subst as', assumption) qed lemma (in primeness_condition_monoid) wfactors_unique: assumes "wfactors G as a" "wfactors G as' a" and "a \<in> carrier G" "set as \<subseteq> carrier G" "set as' \<subseteq> carrier G" shows "essentially_equal G as as'" apply (rule wfactors_unique__hlp_induct[rule_format, of a]) apply (simp add: assms) done subsubsection {* Application to factorial monoids *} text {* Number of factors for wellfoundedness *} constdefs factorcount :: "_ \<Rightarrow> 'a \<Rightarrow> nat" "factorcount G a == THE c. (ALL as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> c = length as)" lemma (in monoid) ee_length: assumes ee: "essentially_equal G as bs" shows "length as = length bs" apply (rule essentially_equalE[OF ee]) apply (subgoal_tac "length as = length fs1'") apply (simp add: list_all2_lengthD) apply (simp add: perm_length) done lemma (in factorial_monoid) factorcount_exists: assumes carr[simp]: "a \<in> carrier G" shows "EX c. ALL as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> c = length as" proof - have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" by (intro wfactors_exist, simp) from this obtain as where ascarr[simp]: "set as \<subseteq> carrier G" and afs: "wfactors G as a" by (auto simp del: carr) have "ALL as'. set as' \<subseteq> carrier G \<and> wfactors G as' a \<longrightarrow> length as = length as'" proof clarify fix as' assume [simp]: "set as' \<subseteq> carrier G" and bfs: "wfactors G as' a" from afs bfs have "essentially_equal G as as'" by (intro ee_wfactorsI[of a a as as'], simp+) thus "length as = length as'" by (rule ee_length) qed thus "EX c. ALL as'. set as' \<subseteq> carrier G \<and> wfactors G as' a \<longrightarrow> c = length as'" .. qed lemma (in factorial_monoid) factorcount_unique: assumes afs: "wfactors G as a" and acarr[simp]: "a \<in> carrier G" and ascarr[simp]: "set as \<subseteq> carrier G" shows "factorcount G a = length as" proof - have "EX ac. ALL as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> ac = length as" by (rule factorcount_exists, simp) from this obtain ac where alen: "ALL as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> ac = length as" by auto have ac: "ac = factorcount G a" apply (simp add: factorcount_def) apply (rule theI2) apply (rule alen) apply (elim allE[of _ "as"], rule allE[OF alen, of "as"], simp add: ascarr afs) apply (elim allE[of _ "as"], rule allE[OF alen, of "as"], simp add: ascarr afs) done from ascarr afs have "ac = length as" by (iprover intro: alen[rule_format]) with ac show ?thesis by simp qed lemma (in factorial_monoid) divides_fcount: assumes dvd: "a divides b" and acarr: "a \<in> carrier G" and bcarr:"b \<in> carrier G" shows "factorcount G a <= factorcount G b" apply (rule dividesE[OF dvd]) proof - fix c from assms have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" by fast from this obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" by auto with acarr have fca: "factorcount G a = length as" by (intro factorcount_unique) assume ccarr: "c \<in> carrier G" hence "\<exists>cs. set cs \<subseteq> carrier G \<and> wfactors G cs c" by fast from this obtain cs where cscarr: "set cs \<subseteq> carrier G" and cfs: "wfactors G cs c" by auto note [simp] = acarr bcarr ccarr ascarr cscarr assume b: "b = a \<otimes> c" from afs cfs have "wfactors G (as@cs) (a \<otimes> c)" by (intro wfactors_mult, simp+) with b have "wfactors G (as@cs) b" by simp hence "factorcount G b = length (as@cs)" by (intro factorcount_unique, simp+) hence "factorcount G b = length as + length cs" by simp with fca show ?thesis by simp qed lemma (in factorial_monoid) associated_fcount: assumes acarr: "a \<in> carrier G" and bcarr:"b \<in> carrier G" and asc: "a \<sim> b" shows "factorcount G a = factorcount G b" apply (rule associatedE[OF asc]) apply (drule divides_fcount[OF _ acarr bcarr]) apply (drule divides_fcount[OF _ bcarr acarr]) apply simp done lemma (in factorial_monoid) properfactor_fcount: assumes acarr: "a \<in> carrier G" and bcarr:"b \<in> carrier G" and pf: "properfactor G a b" shows "factorcount G a < factorcount G b" apply (rule properfactorE[OF pf], elim dividesE) proof - fix c from assms have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" by fast from this obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" by auto with acarr have fca: "factorcount G a = length as" by (intro factorcount_unique) assume ccarr: "c \<in> carrier G" hence "\<exists>cs. set cs \<subseteq> carrier G \<and> wfactors G cs c" by fast from this obtain cs where cscarr: "set cs \<subseteq> carrier G" and cfs: "wfactors G cs c" by auto assume b: "b = a \<otimes> c" have "wfactors G (as@cs) (a \<otimes> c)" by (rule wfactors_mult) fact+ with b have "wfactors G (as@cs) b" by simp with ascarr cscarr bcarr have "factorcount G b = length (as@cs)" by (simp add: factorcount_unique) hence fcb: "factorcount G b = length as + length cs" by simp assume nbdvda: "\<not> b divides a" have "c \<notin> Units G" proof (rule ccontr, simp) assume cunit:"c \<in> Units G" have "b \<otimes> inv c = a \<otimes> c \<otimes> inv c" by (simp add: b) also with ccarr acarr cunit have "\<dots> = a \<otimes> (c \<otimes> inv c)" by (fast intro: m_assoc) also with ccarr cunit have "\<dots> = a \<otimes> \<one>" by (simp add: Units_r_inv) also with acarr have "\<dots> = a" by simp finally have "a = b \<otimes> inv c" by simp with ccarr cunit have "b divides a" by (fast intro: dividesI[of "inv c"]) with nbdvda show False by simp qed with cfs have "length cs > 0" apply - apply (rule ccontr, simp) proof - assume "wfactors G [] c" hence "\<one> \<sim> c" by (elim wfactorsE, simp) with ccarr have cunit: "c \<in> Units G" by (intro assoc_unit_r[of "\<one>" "c"], simp+) assume "c \<notin> Units G" with cunit show "False" by simp qed with fca fcb show ?thesis by simp qed interpretation factorial_monoid \<subseteq> divisor_chain_condition_monoid apply unfold_locales apply (rule wfUNIVI) apply (rule measure_induct[of "factorcount G"]) apply simp (* slow *) (* [1]Applying congruence rule: \<lbrakk>factorcount G y < factorcount G xa \<equiv> ?P'; ?P' \<Longrightarrow> P y \<equiv> ?Q'\<rbrakk> \<Longrightarrow> factorcount G y < factorcount G xa \<longrightarrow> P y \<equiv> ?P' \<longrightarrow> ?Q' trace_simp_depth_limit exceeded! *) proof - fix P x assume r1[rule_format]: "\<forall>y. (\<forall>z. z \<in> carrier G \<and> y \<in> carrier G \<and> properfactor G z y \<longrightarrow> P z) \<longrightarrow> P y" and r2[rule_format]: "\<forall>y. factorcount G y < factorcount G x \<longrightarrow> P y" show "P x" apply (rule r1) apply (rule r2) apply (rule properfactor_fcount, simp+) done qed interpretation factorial_monoid \<subseteq> primeness_condition_monoid by (unfold_locales, rule irreducible_is_prime) lemma (in factorial_monoid) primeness_condition: shows "primeness_condition_monoid G" by unfold_locales lemma (in factorial_monoid) gcd_condition [simp]: shows "gcd_condition_monoid G" by (unfold_locales, rule gcdof_exists) interpretation factorial_monoid \<subseteq> gcd_condition_monoid by (unfold_locales, rule gcdof_exists) lemma (in factorial_monoid) division_weak_lattice [simp]: shows "weak_lattice (division_rel G)" proof - interpret weak_lower_semilattice ["division_rel G"] by simp show "weak_lattice (division_rel G)" apply (unfold_locales, simp_all) proof - fix x y assume carr: "x \<in> carrier G" "y \<in> carrier G" hence "\<exists>z. z \<in> carrier G \<and> z lcmof x y" by (rule lcmof_exists) from this obtain z where zcarr: "z \<in> carrier G" and isgcd: "z lcmof x y" by auto with carr have "least (division_rel G) z (Upper (division_rel G) {x, y})" by (simp add: lcmof_leastUpper[symmetric]) thus "\<exists>z. least (division_rel G) z (Upper (division_rel G) {x, y})" by fast qed qed subsection {* Factoriality Theorems *} theorem factorial_condition_one: (* Jacobson theorem 2.21 *) shows "(divisor_chain_condition_monoid G \<and> primeness_condition_monoid G) = factorial_monoid G" apply rule proof clarify assume dcc: "divisor_chain_condition_monoid G" and pc: "primeness_condition_monoid G" interpret divisor_chain_condition_monoid ["G"] by (rule dcc) interpret primeness_condition_monoid ["G"] by (rule pc) show "factorial_monoid G" by (fast intro: factorial_monoidI wfactors_exist wfactors_unique) next assume fm: "factorial_monoid G" interpret factorial_monoid ["G"] by (rule fm) show "divisor_chain_condition_monoid G \<and> primeness_condition_monoid G" by rule unfold_locales qed theorem factorial_condition_two: (* Jacobson theorem 2.22 *) shows "(divisor_chain_condition_monoid G \<and> gcd_condition_monoid G) = factorial_monoid G" apply rule proof clarify assume dcc: "divisor_chain_condition_monoid G" and gc: "gcd_condition_monoid G" interpret divisor_chain_condition_monoid ["G"] by (rule dcc) interpret gcd_condition_monoid ["G"] by (rule gc) show "factorial_monoid G" by (simp add: factorial_condition_one[symmetric], rule, unfold_locales) next assume fm: "factorial_monoid G" interpret factorial_monoid ["G"] by (rule fm) show "divisor_chain_condition_monoid G \<and> gcd_condition_monoid G" by rule unfold_locales qed end