(* Title: HOL/Algebra/Divisibility.thy Author: Clemens Ballarin Author: Stephan Hohe *) section ‹Divisibility in monoids and rings› theory Divisibility imports "HOL-Library.Permutation" Coset Group begin section ‹Factorial Monoids› subsection ‹Monoids with Cancellation Law› locale monoid_cancel = monoid + assumes l_cancel: "⟦c ⊗ a = c ⊗ b; a ∈ carrier G; b ∈ carrier G; c ∈ carrier G⟧ ⟹ a = b" and r_cancel: "⟦a ⊗ c = b ⊗ c; a ∈ carrier G; b ∈ carrier G; c ∈ carrier G⟧ ⟹ a = b" lemma (in monoid) monoid_cancelI: assumes l_cancel: "⋀a b c. ⟦c ⊗ a = c ⊗ b; a ∈ carrier G; b ∈ carrier G; c ∈ carrier G⟧ ⟹ a = b" and r_cancel: "⋀a b c. ⟦a ⊗ c = b ⊗ c; a ∈ carrier G; b ∈ carrier G; c ∈ carrier G⟧ ⟹ a = b" shows "monoid_cancel G" by standard fact+ lemma (in monoid_cancel) is_monoid_cancel: "monoid_cancel G" .. sublocale group ⊆ monoid_cancel by standard simp_all locale comm_monoid_cancel = monoid_cancel + comm_monoid lemma comm_monoid_cancelI: fixes G (structure) assumes "comm_monoid G" assumes cancel: "⋀a b c. ⟦a ⊗ c = b ⊗ c; a ∈ carrier G; b ∈ carrier G; c ∈ carrier G⟧ ⟹ a = b" shows "comm_monoid_cancel G" proof - interpret comm_monoid G by fact show "comm_monoid_cancel G" by unfold_locales (metis assms(2) m_ac(2))+ qed lemma (in comm_monoid_cancel) is_comm_monoid_cancel: "comm_monoid_cancel G" by intro_locales sublocale comm_group ⊆ comm_monoid_cancel .. subsection ‹Products of Units in Monoids› lemma (in monoid) prod_unit_l: assumes abunit[simp]: "a ⊗ b ∈ Units G" and aunit[simp]: "a ∈ Units G" and carr[simp]: "a ∈ carrier G" "b ∈ carrier G" shows "b ∈ Units G" proof - have c: "inv (a ⊗ b) ⊗ a ∈ carrier G" by simp have "(inv (a ⊗ b) ⊗ a) ⊗ b = inv (a ⊗ b) ⊗ (a ⊗ b)" by (simp add: m_assoc) also have "… = 𝟭" by simp finally have li: "(inv (a ⊗ b) ⊗ a) ⊗ b = 𝟭" . have "𝟭 = inv a ⊗ a" by (simp add: Units_l_inv[symmetric]) also have "… = inv a ⊗ 𝟭 ⊗ a" by simp also have "… = inv a ⊗ ((a ⊗ b) ⊗ inv (a ⊗ b)) ⊗ a" by (simp add: Units_r_inv[OF abunit, symmetric] del: Units_r_inv) also have "… = ((inv a ⊗ a) ⊗ b) ⊗ inv (a ⊗ b) ⊗ a" by (simp add: m_assoc del: Units_l_inv) also have "… = b ⊗ inv (a ⊗ b) ⊗ a" by simp also have "… = b ⊗ (inv (a ⊗ b) ⊗ a)" by (simp add: m_assoc) finally have ri: "b ⊗ (inv (a ⊗ b) ⊗ a) = 𝟭 " by simp from c li ri show "b ∈ Units G" by (auto simp: Units_def) qed lemma (in monoid) prod_unit_r: assumes abunit[simp]: "a ⊗ b ∈ Units G" and bunit[simp]: "b ∈ Units G" and carr[simp]: "a ∈ carrier G" "b ∈ carrier G" shows "a ∈ Units G" proof - have c: "b ⊗ inv (a ⊗ b) ∈ carrier G" by simp have "a ⊗ (b ⊗ inv (a ⊗ b)) = (a ⊗ b) ⊗ inv (a ⊗ b)" by (simp add: m_assoc del: Units_r_inv) also have "… = 𝟭" by simp finally have li: "a ⊗ (b ⊗ inv (a ⊗ b)) = 𝟭" . have "𝟭 = b ⊗ inv b" by (simp add: Units_r_inv[symmetric]) also have "… = b ⊗ 𝟭 ⊗ inv b" by simp also have "… = b ⊗ (inv (a ⊗ b) ⊗ (a ⊗ b)) ⊗ inv b" by (simp add: Units_l_inv[OF abunit, symmetric] del: Units_l_inv) also have "… = (b ⊗ inv (a ⊗ b) ⊗ a) ⊗ (b ⊗ inv b)" by (simp add: m_assoc del: Units_l_inv) also have "… = b ⊗ inv (a ⊗ b) ⊗ a" by simp finally have ri: "(b ⊗ inv (a ⊗ b)) ⊗ a = 𝟭 " by simp from c li ri show "a ∈ Units G" by (auto simp: Units_def) qed lemma (in comm_monoid) unit_factor: assumes abunit: "a ⊗ b ∈ Units G" and [simp]: "a ∈ carrier G" "b ∈ carrier G" shows "a ∈ Units G" using abunit[simplified Units_def] proof clarsimp fix i assume [simp]: "i ∈ carrier G" have carr': "b ⊗ i ∈ carrier G" by simp have "(b ⊗ i) ⊗ a = (i ⊗ b) ⊗ a" by (simp add: m_comm) also have "… = i ⊗ (b ⊗ a)" by (simp add: m_assoc) also have "… = i ⊗ (a ⊗ b)" by (simp add: m_comm) also assume "i ⊗ (a ⊗ b) = 𝟭" finally have li': "(b ⊗ i) ⊗ a = 𝟭" . have "a ⊗ (b ⊗ i) = a ⊗ b ⊗ i" by (simp add: m_assoc) also assume "a ⊗ b ⊗ i = 𝟭" finally have ri': "a ⊗ (b ⊗ i) = 𝟭" . from carr' li' ri' show "a ∈ Units G" by (simp add: Units_def, fast) qed subsection ‹Divisibility and Association› subsubsection ‹Function definitions› definition factor :: "[_, 'a, 'a] ⇒ bool" (infix "dividesı" 65) where "a divides⇘_{G⇙}b ⟷ (∃c∈carrier G. b = a ⊗⇘_{G⇙}c)" definition associated :: "[_, 'a, 'a] ⇒ bool" (infix "∼ı" 55) where "a ∼⇘_{G⇙}b ⟷ a divides⇘_{G⇙}b ∧ b divides⇘_{G⇙}a" abbreviation "division_rel G ≡ ⦇carrier = carrier G, eq = (∼⇘_{G⇙}), le = (divides⇘_{G⇙})⦈" definition properfactor :: "[_, 'a, 'a] ⇒ bool" where "properfactor G a b ⟷ a divides⇘_{G⇙}b ∧ ¬(b divides⇘_{G⇙}a)" definition irreducible :: "[_, 'a] ⇒ bool" where "irreducible G a ⟷ a ∉ Units G ∧ (∀b∈carrier G. properfactor G b a ⟶ b ∈ Units G)" definition prime :: "[_, 'a] ⇒ bool" where "prime G p ⟷ p ∉ Units G ∧ (∀a∈carrier G. ∀b∈carrier G. p divides⇘_{G⇙}(a ⊗⇘_{G⇙}b) ⟶ p divides⇘_{G⇙}a ∨ p divides⇘_{G⇙}b)" subsubsection ‹Divisibility› lemma dividesI: fixes G (structure) assumes carr: "c ∈ carrier G" and p: "b = a ⊗ c" shows "a divides b" unfolding factor_def using assms by fast lemma dividesI' [intro]: fixes G (structure) assumes p: "b = a ⊗ c" and carr: "c ∈ carrier G" shows "a divides b" using assms by (fast intro: dividesI) lemma dividesD: fixes G (structure) assumes "a divides b" shows "∃c∈carrier G. b = a ⊗ c" using assms unfolding factor_def by fast lemma dividesE [elim]: fixes G (structure) assumes d: "a divides b" and elim: "⋀c. ⟦b = a ⊗ c; c ∈ carrier G⟧ ⟹ P" shows "P" proof - from dividesD[OF d] obtain c where "c ∈ carrier G" and "b = a ⊗ c" by auto then show P by (elim elim) qed lemma (in monoid) divides_refl[simp, intro!]: assumes carr: "a ∈ carrier G" shows "a divides a" by (intro dividesI[of "𝟭"]) (simp_all add: carr) lemma (in monoid) divides_trans [trans]: assumes dvds: "a divides b" "b divides c" and acarr: "a ∈ 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 "a divides b" "a ∈ carrier G" "c ∈ carrier G" shows "(c ⊗ a) divides (c ⊗ b)" by (metis assms factor_def m_assoc) lemma (in monoid_cancel) divides_mult_l [simp]: assumes carr: "a ∈ carrier G" "b ∈ carrier G" "c ∈ carrier G" shows "(c ⊗ a) divides (c ⊗ b) = a divides b" proof show "c ⊗ a divides c ⊗ b ⟹ a divides b" using carr monoid.m_assoc monoid_axioms monoid_cancel.l_cancel monoid_cancel_axioms by fastforce show "a divides b ⟹ c ⊗ a divides c ⊗ b" using carr(1) carr(3) by blast qed lemma (in comm_monoid) divides_mult_rI [intro]: assumes ab: "a divides b" and carr: "a ∈ carrier G" "b ∈ carrier G" "c ∈ carrier G" shows "(a ⊗ c) divides (b ⊗ c)" using carr ab by (metis divides_mult_lI m_comm) lemma (in comm_monoid_cancel) divides_mult_r [simp]: assumes carr: "a ∈ carrier G" "b ∈ carrier G" "c ∈ carrier G" shows "(a ⊗ c) divides (b ⊗ 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 ∈ carrier G" "c ∈ carrier G" shows "a divides (b ⊗ c)" using ab carr by (fast intro: m_assoc) lemma (in comm_monoid) divides_prod_l: assumes "a ∈ carrier G" "b ∈ carrier G" "c ∈ carrier G" "a divides b" shows "a divides (c ⊗ b)" using assms by (simp add: divides_prod_r m_comm) lemma (in monoid) unit_divides: assumes uunit: "u ∈ Units G" and acarr: "a ∈ carrier G" shows "u divides a" proof (intro dividesI[of "(inv u) ⊗ a"], fast intro: uunit acarr) from uunit acarr have xcarr: "inv u ⊗ a ∈ carrier G" by fast from uunit acarr have "u ⊗ (inv u ⊗ a) = (u ⊗ inv u) ⊗ a" by (fast intro: m_assoc[symmetric]) also have "… = 𝟭 ⊗ a" by (simp add: Units_r_inv[OF uunit]) also from acarr have "… = a" by simp finally show "a = u ⊗ (inv u ⊗ a)" .. qed lemma (in comm_monoid) divides_unit: assumes udvd: "a divides u" and carr: "a ∈ carrier G" "u ∈ Units G" shows "a ∈ Units G" using udvd carr by (blast intro: unit_factor) lemma (in comm_monoid) Unit_eq_dividesone: assumes ucarr: "u ∈ carrier G" shows "u ∈ Units G = u divides 𝟭" 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 ∼ b" using assms by (simp add: associated_def) lemma (in monoid) associatedI2: assumes uunit[simp]: "u ∈ Units G" and a: "a = b ⊗ u" and bcarr: "b ∈ carrier G" shows "a ∼ b" using uunit bcarr unfolding a apply (intro associatedI) apply (metis Units_closed divides_mult_lI one_closed r_one unit_divides) by blast lemma (in monoid) associatedI2': assumes "a = b ⊗ u" and "u ∈ Units G" and "b ∈ carrier G" shows "a ∼ b" using assms by (intro associatedI2) lemma associatedD: fixes G (structure) assumes "a ∼ b" shows "a divides b" using assms by (simp add: associated_def) lemma (in monoid_cancel) associatedD2: assumes assoc: "a ∼ b" and carr: "a ∈ carrier G" "b ∈ carrier G" shows "∃u∈Units G. a = b ⊗ u" using assoc unfolding associated_def proof clarify assume "b divides a" then obtain u where ucarr: "u ∈ carrier G" and a: "a = b ⊗ u" by (rule dividesE) assume "a divides b" then obtain u' where u'carr: "u' ∈ carrier G" and b: "b = a ⊗ u'" by (rule dividesE) note carr = carr ucarr u'carr from carr have "a ⊗ 𝟭 = a" by simp also have "… = b ⊗ u" by (simp add: a) also have "… = a ⊗ u' ⊗ u" by (simp add: b) also from carr have "… = a ⊗ (u' ⊗ u)" by (simp add: m_assoc) finally have "a ⊗ 𝟭 = a ⊗ (u' ⊗ u)" . with carr have u1: "𝟭 = u' ⊗ u" by (fast dest: l_cancel) from carr have "b ⊗ 𝟭 = b" by simp also have "… = a ⊗ u'" by (simp add: b) also have "… = b ⊗ u ⊗ u'" by (simp add: a) also from carr have "… = b ⊗ (u ⊗ u')" by (simp add: m_assoc) finally have "b ⊗ 𝟭 = b ⊗ (u ⊗ u')" . with carr have u2: "𝟭 = u ⊗ u'" by (fast dest: l_cancel) from u'carr u1[symmetric] u2[symmetric] have "∃u'∈carrier G. u' ⊗ u = 𝟭 ∧ u ⊗ u' = 𝟭" by fast then have "u ∈ Units G" by (simp add: Units_def ucarr) with ucarr a show "∃u∈Units G. a = b ⊗ u" by fast qed lemma associatedE: fixes G (structure) assumes assoc: "a ∼ b" and e: "⟦a divides b; b divides a⟧ ⟹ P" shows "P" proof - from assoc have "a divides b" "b divides a" by (simp_all add: associated_def) then show P by (elim e) qed lemma (in monoid_cancel) associatedE2: assumes assoc: "a ∼ b" and e: "⋀u. ⟦a = b ⊗ u; u ∈ Units G⟧ ⟹ P" and carr: "a ∈ carrier G" "b ∈ carrier G" shows "P" proof - from assoc and carr have "∃u∈Units G. a = b ⊗ u" by (rule associatedD2) then obtain u where "u ∈ Units G" "a = b ⊗ u" by auto then show P by (elim e) qed lemma (in monoid) associated_refl [simp, intro!]: assumes "a ∈ carrier G" shows "a ∼ a" using assms by (fast intro: associatedI) lemma (in monoid) associated_sym [sym]: assumes "a ∼ b" shows "b ∼ a" using assms by (iprover intro: associatedI elim: associatedE) lemma (in monoid) associated_trans [trans]: assumes "a ∼ b" "b ∼ c" and "a ∈ carrier G" "c ∈ carrier G" shows "a ∼ 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 (metis associated_def) apply (iprover intro: associated_trans) done subsubsection ‹Division and associativity› lemmas divides_antisym = associatedI lemma (in monoid) divides_cong_l [trans]: assumes "x ∼ x'" "x' divides y" "x ∈ carrier G" shows "x divides y" by (meson assms associatedD divides_trans) lemma (in monoid) divides_cong_r [trans]: assumes "x divides y" "y ∼ y'" "x ∈ carrier G" shows "x divides y'" by (meson assms associatedD divides_trans) lemma (in monoid) division_weak_partial_order [simp, intro!]: "weak_partial_order (division_rel G)" apply unfold_locales apply (simp_all add: associated_sym divides_antisym) apply (metis associated_trans) apply (metis divides_trans) by (meson associated_def divides_trans) subsubsection ‹Multiplication and associativity› lemma (in monoid_cancel) mult_cong_r: assumes "b ∼ b'" "a ∈ carrier G" "b ∈ carrier G" "b' ∈ carrier G" shows "a ⊗ b ∼ a ⊗ b'" by (meson assms associated_def divides_mult_lI) lemma (in comm_monoid_cancel) mult_cong_l: assumes "a ∼ a'" "a ∈ carrier G" "a' ∈ carrier G" "b ∈ carrier G" shows "a ⊗ b ∼ a' ⊗ b" using assms m_comm mult_cong_r by auto lemma (in monoid_cancel) assoc_l_cancel: assumes "a ∈ carrier G" "b ∈ carrier G" "b' ∈ carrier G" "a ⊗ b ∼ a ⊗ b'" shows "b ∼ b'" by (meson assms associated_def divides_mult_l) lemma (in comm_monoid_cancel) assoc_r_cancel: assumes "a ⊗ b ∼ a' ⊗ b" "a ∈ carrier G" "a' ∈ carrier G" "b ∈ carrier G" shows "a ∼ a'" using assms assoc_l_cancel m_comm by presburger subsubsection ‹Units› lemma (in monoid_cancel) assoc_unit_l [trans]: assumes "a ∼ b" and "b ∈ Units G" and "a ∈ carrier G" shows "a ∈ Units G" using assms by (fast elim: associatedE2) lemma (in monoid_cancel) assoc_unit_r [trans]: assumes aunit: "a ∈ Units G" and asc: "a ∼ b" and bcarr: "b ∈ carrier G" shows "b ∈ Units G" using aunit bcarr associated_sym[OF asc] by (blast intro: assoc_unit_l) lemma (in comm_monoid) Units_cong: assumes aunit: "a ∈ Units G" and asc: "a ∼ b" and bcarr: "b ∈ carrier G" shows "b ∈ Units G" using assms by (blast intro: divides_unit elim: associatedE) lemma (in monoid) Units_assoc: assumes units: "a ∈ Units G" "b ∈ Units G" shows "a ∼ b" using units by (fast intro: associatedI unit_divides) lemma (in monoid) Units_are_ones: "Units G {.=}⇘_{(division_rel G)⇙}{𝟭}" proof - have "a .∈⇘_{division_rel G⇙}{𝟭}" if "a ∈ Units G" for a proof - have "a ∼ 𝟭" by (rule associatedI) (simp_all add: Units_closed that unit_divides) then show ?thesis by (simp add: elem_def) qed moreover have "𝟭 .∈⇘_{division_rel G⇙}Units G" by (simp add: equivalence.mem_imp_elem) ultimately show ?thesis by (auto simp: set_eq_def) qed lemma (in comm_monoid) Units_Lower: "Units G = Lower (division_rel G) (carrier G)" apply (auto simp add: Units_def Lower_def) apply (metis Units_one_closed unit_divides unit_factor) apply (metis Unit_eq_dividesone Units_r_inv_ex m_ac(2) one_closed) done lemma (in monoid_cancel) associated_iff: assumes "a ∈ carrier G" "b ∈ carrier G" shows "a ∼ b ⟷ (∃c ∈ Units G. a = b ⊗ c)" using assms associatedI2' associatedD2 by auto subsubsection ‹Proper factors› lemma properfactorI: fixes G (structure) assumes "a divides b" and "¬(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: "¬(a ∼ b)" shows "properfactor G a b" proof (rule properfactorI, rule advdb, rule notI) assume "b divides a" with advdb have "a ∼ b" by (rule associatedI) with neq show "False" by fast qed lemma (in comm_monoid_cancel) properfactorI3: assumes p: "p = a ⊗ b" and nunit: "b ∉ Units G" and carr: "a ∈ carrier G" "b ∈ carrier G" shows "properfactor G a p" unfolding p using carr apply (intro properfactorI, fast) proof (clarsimp, elim dividesE) fix c assume ccarr: "c ∈ carrier G" note [simp] = carr ccarr have "a ⊗ 𝟭 = a" by simp also assume "a = a ⊗ b ⊗ c" also have "… = a ⊗ (b ⊗ c)" by (simp add: m_assoc) finally have "a ⊗ 𝟭 = a ⊗ (b ⊗ c)" . then have rinv: "𝟭 = b ⊗ c" by (intro l_cancel[of "a" "𝟭" "b ⊗ c"], simp+) also have "… = c ⊗ b" by (simp add: m_comm) finally have linv: "𝟭 = c ⊗ b" . from ccarr linv[symmetric] rinv[symmetric] have "b ∈ Units G" unfolding Units_def by fastforce with nunit show False .. qed lemma properfactorE: fixes G (structure) assumes pf: "properfactor G a b" and r: "⟦a divides b; ¬(b divides a)⟧ ⟹ 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: "⟦a divides b; ¬(a ∼ b)⟧ ⟹ P" shows "P" using pf unfolding properfactor_def by (fast elim: elim associatedE) lemma (in monoid) properfactor_unitE: assumes uunit: "u ∈ Units G" and pf: "properfactor G a u" and acarr: "a ∈ 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 ∈ carrier G" "c ∈ 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 ∈ carrier G" "b ∈ 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)" by (force simp: lless_def properfactor_def associated_def) lemma (in monoid) properfactor_cong_l [trans]: assumes x'x: "x' ∼ x" and pf: "properfactor G x y" and carr: "x ∈ carrier G" "x' ∈ carrier G" "y ∈ 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' .=⇘_{division_rel G⇙}x" by simp also assume "x ⊏⇘_{division_rel G⇙}y" finally show "x' ⊏⇘_{division_rel G⇙}y" by (simp add: carr) qed lemma (in monoid) properfactor_cong_r [trans]: assumes pf: "properfactor G x y" and yy': "y ∼ y'" and carr: "x ∈ carrier G" "y ∈ carrier G" "y' ∈ carrier G" shows "properfactor G x y'" using pf unfolding properfactor_lless proof - interpret weak_partial_order "division_rel G" .. assume "x ⊏⇘_{division_rel G⇙}y" also from yy' have "y .=⇘_{division_rel G⇙}y'" by simp finally show "x ⊏⇘_{division_rel G⇙}y'" by (simp add: carr) qed lemma (in monoid_cancel) properfactor_mult_lI [intro]: assumes ab: "properfactor G a b" and carr: "a ∈ carrier G" "c ∈ carrier G" shows "properfactor G (c ⊗ a) (c ⊗ b)" using ab carr by (fastforce elim: properfactorE intro: properfactorI) lemma (in monoid_cancel) properfactor_mult_l [simp]: assumes carr: "a ∈ carrier G" "b ∈ carrier G" "c ∈ carrier G" shows "properfactor G (c ⊗ a) (c ⊗ b) = properfactor G a b" using carr by (fastforce elim: properfactorE intro: properfactorI) lemma (in comm_monoid_cancel) properfactor_mult_rI [intro]: assumes ab: "properfactor G a b" and carr: "a ∈ carrier G" "c ∈ carrier G" shows "properfactor G (a ⊗ c) (b ⊗ c)" using ab carr by (fastforce elim: properfactorE intro: properfactorI) lemma (in comm_monoid_cancel) properfactor_mult_r [simp]: assumes carr: "a ∈ carrier G" "b ∈ carrier G" "c ∈ carrier G" shows "properfactor G (a ⊗ c) (b ⊗ c) = properfactor G a b" using carr by (fastforce elim: properfactorE intro: properfactorI) lemma (in monoid) properfactor_prod_r: assumes ab: "properfactor G a b" and carr[simp]: "a ∈ carrier G" "b ∈ carrier G" "c ∈ carrier G" shows "properfactor G a (b ⊗ c)" by (intro properfactor_trans2[OF ab] divides_prod_r) simp_all lemma (in comm_monoid) properfactor_prod_l: assumes ab: "properfactor G a b" and carr[simp]: "a ∈ carrier G" "b ∈ carrier G" "c ∈ carrier G" shows "properfactor G a (c ⊗ b)" by (intro properfactor_trans2[OF ab] divides_prod_l) simp_all subsection ‹Irreducible Elements and Primes› subsubsection ‹Irreducible elements› lemma irreducibleI: fixes G (structure) assumes "a ∉ Units G" and "⋀b. ⟦b ∈ carrier G; properfactor G b a⟧ ⟹ b ∈ 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: "⟦a ∉ Units G; ∀b. b ∈ carrier G ∧ properfactor G b a ⟶ b ∈ Units G⟧ ⟹ 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 ∈ carrier G" shows "b ∈ Units G" using assms by (fast elim: irreducibleE) lemma (in monoid_cancel) irreducible_cong [trans]: assumes irred: "irreducible G a" and aa': "a ∼ a'" "a ∈ carrier G" "a' ∈ carrier G" shows "irreducible G a'" using assms apply (auto simp: irreducible_def assoc_unit_l) apply (metis aa' associated_sym properfactor_cong_r) done lemma (in monoid) irreducible_prod_rI: assumes airr: "irreducible G a" and bunit: "b ∈ Units G" and carr[simp]: "a ∈ carrier G" "b ∈ carrier G" shows "irreducible G (a ⊗ b)" using airr carr bunit apply (elim irreducibleE, intro irreducibleI) using prod_unit_r apply blast using associatedI2' properfactor_cong_r by auto lemma (in comm_monoid) irreducible_prod_lI: assumes birr: "irreducible G b" and aunit: "a ∈ Units G" and carr [simp]: "a ∈ carrier G" "b ∈ carrier G" shows "irreducible G (a ⊗ b)" by (metis aunit birr carr irreducible_prod_rI m_comm) lemma (in comm_monoid_cancel) irreducible_prodE [elim]: assumes irr: "irreducible G (a ⊗ b)" and carr[simp]: "a ∈ carrier G" "b ∈ carrier G" and e1: "⟦irreducible G a; b ∈ Units G⟧ ⟹ P" and e2: "⟦a ∈ Units G; irreducible G b⟧ ⟹ P" shows P using irr proof (elim irreducibleE) assume abnunit: "a ⊗ b ∉ Units G" and isunit[rule_format]: "∀ba. ba ∈ carrier G ∧ properfactor G ba (a ⊗ b) ⟶ ba ∈ Units G" show P proof (cases "a ∈ Units G") case aunit: True have "irreducible G b" proof (rule irreducibleI, rule notI) assume "b ∈ Units G" with aunit have "(a ⊗ b) ∈ Units G" by fast with abnunit show "False" .. next fix c assume ccarr: "c ∈ carrier G" and "properfactor G c b" then have "properfactor G c (a ⊗ b)" by (simp add: properfactor_prod_l[of c b a]) with ccarr show "c ∈ Units G" by (fast intro: isunit) qed with aunit show "P" by (rule e2) next case anunit: False with carr have "properfactor G b (b ⊗ a)" by (fast intro: properfactorI3) then have bf: "properfactor G b (a ⊗ b)" by (subst m_comm[of a b], simp+) then have bunit: "b ∈ Units G" by (intro isunit, simp) have "irreducible G a" proof (rule irreducibleI, rule notI) assume "a ∈ Units G" with bunit have "(a ⊗ b) ∈ Units G" by fast with abnunit show "False" .. next fix c assume ccarr: "c ∈ carrier G" and "properfactor G c a" then have "properfactor G c (a ⊗ b)" by (simp add: properfactor_prod_r[of c a b]) with ccarr show "c ∈ 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 ∉ Units G" and "⋀a b. ⟦a ∈ carrier G; b ∈ carrier G; p divides (a ⊗ b)⟧ ⟹ p divides a ∨ 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: "⟦p ∉ Units G; ∀a∈carrier G. ∀b∈carrier G. p divides a ⊗ b ⟶ p divides a ∨ p divides b⟧ ⟹ P" shows "P" using pprime unfolding prime_def by (blast dest: e) lemma (in comm_monoid_cancel) prime_divides: assumes carr: "a ∈ carrier G" "b ∈ carrier G" and pprime: "prime G p" and pdvd: "p divides a ⊗ b" shows "p divides a ∨ p divides b" using assms by (blast elim: primeE) lemma (in monoid_cancel) prime_cong [trans]: assumes "prime G p" and pp': "p ∼ p'" "p ∈ carrier G" "p' ∈ carrier G" shows "prime G p'" using assms apply (auto simp: prime_def assoc_unit_l) apply (metis pp' associated_sym divides_cong_l) done (*by Paulo EmÃlio de Vilhena*) lemma (in comm_monoid_cancel) prime_irreducible: assumes "prime G p" shows "irreducible G p" proof (rule irreducibleI) show "p ∉ Units G" using assms unfolding prime_def by simp next fix b assume A: "b ∈ carrier G" "properfactor G b p" then obtain c where c: "c ∈ carrier G" "p = b ⊗ c" unfolding properfactor_def factor_def by auto hence "p divides c" using A assms unfolding prime_def properfactor_def by auto then obtain b' where b': "b' ∈ carrier G" "c = p ⊗ b'" unfolding factor_def by auto hence "𝟭 = b ⊗ b'" by (metis A(1) l_cancel m_closed m_lcomm one_closed r_one c) thus "b ∈ Units G" using A(1) Units_one_closed b'(1) unit_factor by presburger qed subsection ‹Factorization and Factorial Monoids› subsubsection ‹Function definitions› definition factors :: "[_, 'a list, 'a] ⇒ bool" where "factors G fs a ⟷ (∀x ∈ (set fs). irreducible G x) ∧ foldr (⊗⇘_{G⇙}) fs 𝟭⇘_{G⇙}= a" definition wfactors ::"[_, 'a list, 'a] ⇒ bool" where "wfactors G fs a ⟷ (∀x ∈ (set fs). irreducible G x) ∧ foldr (⊗⇘_{G⇙}) fs 𝟭⇘_{G⇙}∼⇘_{G⇙}a" abbreviation list_assoc :: "('a,_) monoid_scheme ⇒ 'a list ⇒ 'a list ⇒ bool" (infix "[∼]ı" 44) where "list_assoc G ≡ list_all2 (∼⇘_{G⇙})" definition essentially_equal :: "[_, 'a list, 'a list] ⇒ bool" where "essentially_equal G fs1 fs2 ⟷ (∃fs1'. fs1 <~~> fs1' ∧ fs1' [∼]⇘_{G⇙}fs2)" locale factorial_monoid = comm_monoid_cancel + assumes factors_exist: "⟦a ∈ carrier G; a ∉ Units G⟧ ⟹ ∃fs. set fs ⊆ carrier G ∧ factors G fs a" and factors_unique: "⟦factors G fs a; factors G fs' a; a ∈ carrier G; a ∉ Units G; set fs ⊆ carrier G; set fs' ⊆ carrier G⟧ ⟹ essentially_equal G fs fs'" subsubsection ‹Comparing lists of elements› text ‹Association on lists› lemma (in monoid) listassoc_refl [simp, intro]: assumes "set as ⊆ carrier G" shows "as [∼] as" using assms by (induct as) simp_all lemma (in monoid) listassoc_sym [sym]: assumes "as [∼] bs" and "set as ⊆ carrier G" and "set bs ⊆ carrier G" shows "bs [∼] as" using assms proof (induction as arbitrary: bs) case Cons then show ?case by (induction bs) (use associated_sym in auto) qed auto lemma (in monoid) listassoc_trans [trans]: assumes "as [∼] bs" and "bs [∼] cs" and "set as ⊆ carrier G" and "set bs ⊆ carrier G" and "set cs ⊆ carrier G" shows "as [∼] cs" using assms apply (simp add: list_all2_conv_all_nth set_conv_nth, safe) by (metis (mono_tags, lifting) associated_trans nth_mem subsetCE) lemma (in monoid_cancel) irrlist_listassoc_cong: assumes "∀a∈set as. irreducible G a" and "as [∼] bs" and "set as ⊆ carrier G" and "set bs ⊆ carrier G" shows "∀a∈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 "∃d. a <~~> d ∧ 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 [∼] bs" and p: "bs <~~> cs" shows "∃bs'. as <~~> bs' ∧ bs' [∼] cs" using p a proof (induction bs cs arbitrary: as) case (swap y x l) then show ?case by (metis (no_types, hide_lams) list_all2_Cons2 perm.swap) next case (Cons xs ys z) then show ?case by (metis list_all2_Cons2 perm.Cons) next case (trans xs ys zs) then show ?case by (meson perm.trans) qed auto lemma (in monoid) perm_assoc_switch_r: assumes p: "as <~~> bs" and a:"bs [∼] cs" shows "∃bs'. as [∼] bs' ∧ bs' <~~> cs" using p a proof (induction as bs arbitrary: cs) case Nil then show ?case by auto next case (swap y x l) then show ?case by (metis (no_types, hide_lams) list_all2_Cons1 perm.swap) next case (Cons xs ys z) then show ?case by (metis list_all2_Cons1 perm.Cons) next case (trans xs ys zs) then show ?case by (blast intro: elim: ) qed declare perm_sym [sym] lemma perm_setP: assumes perm: "as <~~> bs" and as: "P (set as)" shows "P (set bs)" proof - from perm have "mset as = mset bs" by (simp add: mset_eq_perm) then have "set as = set bs" by (rule mset_eq_setD) with as show "P (set bs)" by simp qed lemmas (in monoid) perm_closed = perm_setP[of _ _ "λas. as ⊆ carrier G"] lemmas (in monoid) irrlist_perm_cong = perm_setP[of _ _ "λas. ∀a∈as. irreducible G a"] text ‹Essentially equal factorizations› lemma (in monoid) essentially_equalI: assumes ex: "fs1 <~~> fs1'" "fs1' [∼] 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: "⋀fs1'. ⟦fs1 <~~> fs1'; fs1' [∼] fs2⟧ ⟹ P" shows "P" using ee unfolding essentially_equal_def by (fast intro: e) lemma (in monoid) ee_refl [simp,intro]: assumes carr: "set as ⊆ 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 ⊆ carrier G" "set bs ⊆ carrier G" shows "essentially_equal G bs as" using ee proof (elim essentially_equalE) fix fs assume "as <~~> fs" "fs [∼] bs" from perm_assoc_switch_r [OF this] obtain fs' where a: "as [∼] fs'" and p: "fs' <~~> bs" by blast 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 ⊆ carrier G" and bscarr: "set bs ⊆ carrier G" and cscarr: "set cs ⊆ carrier G" shows "essentially_equal G as cs" using ab bc proof (elim essentially_equalE) fix abs bcs assume "abs [∼] bs" and pb: "bs <~~> bcs" from perm_assoc_switch [OF this] obtain bs' where p: "abs <~~> bs'" and a: "bs' [∼] bcs" by blast assume "as <~~> abs" with p have pp: "as <~~> bs'" by fast from pp ascarr have c1: "set bs' ⊆ carrier G" by (rule perm_closed) from pb bscarr have c2: "set bcs ⊆ carrier G" by (rule perm_closed) assume "bcs [∼] cs" then have "bs' [∼] cs" using a c1 c2 cscarr listassoc_trans by blast 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 ⊆ carrier G" shows "foldr (⊗) fs 𝟭 ∈ carrier G" using ascarr by (induct fs) simp_all lemma (in comm_monoid) multlist_dividesI: assumes "f ∈ set fs" and "set fs ⊆ carrier G" shows "f divides (foldr (⊗) fs 𝟭)" using assms proof (induction fs) case (Cons a fs) then have f: "f ∈ carrier G" by blast show ?case proof (cases "f = a") case True then show ?thesis using Cons.prems by auto next case False with Cons show ?thesis by clarsimp (metis f divides_prod_l multlist_closed) qed qed auto lemma (in comm_monoid_cancel) multlist_listassoc_cong: assumes "fs [∼] fs'" and "set fs ⊆ carrier G" and "set fs' ⊆ carrier G" shows "foldr (⊗) fs 𝟭 ∼ foldr (⊗) fs' 𝟭" using assms proof (induct fs arbitrary: fs') case (Cons a as fs') then show ?case proof (induction fs') case (Cons b bs) then have p: "a ⊗ foldr (⊗) as 𝟭 ∼ b ⊗ foldr (⊗) as 𝟭" by (simp add: mult_cong_l) then have "foldr (⊗) as 𝟭 ∼ foldr (⊗) bs 𝟭" using Cons by auto with Cons have "b ⊗ foldr (⊗) as 𝟭 ∼ b ⊗ foldr (⊗) bs 𝟭" by (simp add: mult_cong_r) then show ?case using Cons.prems(3) Cons.prems(4) monoid.associated_trans monoid_axioms p by force qed auto qed auto lemma (in comm_monoid) multlist_perm_cong: assumes prm: "as <~~> bs" and ascarr: "set as ⊆ carrier G" shows "foldr (⊗) as 𝟭 = foldr (⊗) bs 𝟭" using prm ascarr proof induction case (swap y x l) then show ?case by (simp add: m_lcomm) next case (trans xs ys zs) then show ?case using perm_closed by auto qed auto lemma (in comm_monoid_cancel) multlist_ee_cong: assumes "essentially_equal G fs fs'" and "set fs ⊆ carrier G" and "set fs' ⊆ carrier G" shows "foldr (⊗) fs 𝟭 ∼ foldr (⊗) fs' 𝟭" 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: fixes G (structure) assumes "∀f∈set fs. irreducible G f" and "foldr (⊗) fs 𝟭 ∼ a" shows "wfactors G fs a" using assms unfolding wfactors_def by simp lemma wfactorsE: fixes G (structure) assumes wf: "wfactors G fs a" and e: "⟦∀f∈set fs. irreducible G f; foldr (⊗) fs 𝟭 ∼ a⟧ ⟹ P" shows "P" using wf unfolding wfactors_def by (fast dest: e) lemma (in monoid) factorsI: assumes "∀f∈set fs. irreducible G f" and "foldr (⊗) fs 𝟭 = a" shows "factors G fs a" using assms unfolding factors_def by simp lemma factorsE: fixes G (structure) assumes f: "factors G fs a" and e: "⟦∀f∈set fs. irreducible G f; foldr (⊗) fs 𝟭 = a⟧ ⟹ 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 ⊆ 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 ⊆ carrier G" shows "∃a'. factors G as a' ∧ a' ∼ a" using assms by (blast elim: wfactorsE intro: factorsI) lemma (in monoid) factors_closed [dest]: assumes "factors G fs a" and "set fs ⊆ carrier G" shows "a ∈ carrier G" using assms by (elim factorsE, clarsimp) lemma (in monoid) nunit_factors: assumes anunit: "a ∉ Units G" and fs: "factors G as a" shows "length as > 0" proof - from anunit Units_one_closed have "a ≠ 𝟭" by auto with fs show ?thesis by (auto elim: factorsE) qed lemma (in monoid) unit_wfactors [simp]: assumes aunit: "a ∈ 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 ∈ Units G" and wf: "wfactors G fs a" and carr[simp]: "set fs ⊆ carrier G" shows "fs = []" proof (cases fs) case Nil then show ?thesis . next case fs: (Cons f fs') from carr have fcarr[simp]: "f ∈ carrier G" and carr'[simp]: "set fs' ⊆ carrier G" by (simp_all add: fs) from fs wf have "irreducible G f" by (simp add: wfactors_def) then have fnunit: "f ∉ Units G" by (fast elim: irreducibleE) from fs wf have a: "f ⊗ foldr (⊗) fs' 𝟭 ∼ a" by (simp add: wfactors_def) note aunit also from fs wf have a: "f ⊗ foldr (⊗) fs' 𝟭 ∼ a" by (simp add: wfactors_def) have "a ∼ f ⊗ foldr (⊗) fs' 𝟭" by (simp add: Units_closed[OF aunit] a[symmetric]) finally have "f ⊗ foldr (⊗) fs' 𝟭 ∈ Units G" by simp then have "f ∈ Units G" by (intro unit_factor[of f], simp+) with fnunit show ?thesis by contradiction qed text ‹Comparing wfactors› lemma (in comm_monoid_cancel) wfactors_listassoc_cong_l: assumes fact: "wfactors G fs a" and asc: "fs [∼] fs'" and carr: "a ∈ carrier G" "set fs ⊆ carrier G" "set fs' ⊆ carrier G" shows "wfactors G fs' a" proof - { from asc[symmetric] have "foldr (⊗) fs' 𝟭 ∼ foldr (⊗) fs 𝟭" by (simp add: multlist_listassoc_cong carr) also assume "foldr (⊗) fs 𝟭 ∼ a" finally have "foldr (⊗) fs' 𝟭 ∼ a" by (simp add: carr) } then show ?thesis using fact by (meson asc carr(2) carr(3) irrlist_listassoc_cong wfactors_def) qed lemma (in comm_monoid) wfactors_perm_cong_l: assumes "wfactors G fs a" and "fs <~~> fs'" and "set fs ⊆ carrier G" shows "wfactors G fs' a" using assms irrlist_perm_cong multlist_perm_cong wfactors_def by fastforce 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 ∈ carrier G" "set as ⊆ carrier G" "set bs ⊆ 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 ⊆ carrier G" by (simp add: perm_closed) note bfs also assume [symmetric]: "fs [∼] 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 ∼ a'" and carr[simp]: "a ∈ carrier G" "a' ∈ carrier G" "set fs ⊆ carrier G" shows "wfactors G fs a'" using fac proof (elim wfactorsE, intro wfactorsI) assume "foldr (⊗) fs 𝟭 ∼ a" also note aa' finally show "foldr (⊗) fs 𝟭 ∼ a'" by simp qed subsubsection ‹Essentially equal factorizations› lemma (in comm_monoid_cancel) unitfactor_ee: assumes uunit: "u ∈ Units G" and carr: "set as ⊆ carrier G" shows "essentially_equal G (as[0 := (as!0 ⊗ u)]) as" (is "essentially_equal G ?as' as") proof - have "as[0 := as ! 0 ⊗ u] [∼] as" proof (cases as) case (Cons a as') then show ?thesis using associatedI2 carr uunit by auto qed auto then show ?thesis using essentially_equal_def by blast qed lemma (in comm_monoid_cancel) factors_cong_unit: assumes u: "u ∈ Units G" and a: "a ∉ Units G" and afs: "factors G as a" and ascarr: "set as ⊆ carrier G" shows "factors G (as[0 := (as!0 ⊗ u)]) (a ⊗ u)" (is "factors G ?as' ?a'") proof (cases as) case Nil then show ?thesis using afs a nunit_factors by auto next case (Cons b bs) have *: "∀f∈set as. irreducible G f" "foldr (⊗) as 𝟭 = a" using afs by (auto simp: factors_def) show ?thesis proof (intro factorsI) show "foldr (⊗) (as[0 := as ! 0 ⊗ u]) 𝟭 = a ⊗ u" using Cons u ascarr * by (auto simp add: m_ac Units_closed) show "∀f∈set (as[0 := as ! 0 ⊗ u]). irreducible G f" using Cons u ascarr * by (force intro: irreducible_prod_rI) qed qed 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 ∈ carrier G" "b ∈ carrier G" and ascarr [simp]: "set as ⊆ carrier G" shows "a ∼ b" using afs bfs proof (elim wfactorsE) from prm have [simp]: "set bs ⊆ carrier G" by (simp add: perm_closed) assume "foldr (⊗) as 𝟭 ∼ a" then have "a ∼ foldr (⊗) as 𝟭" by (simp add: associated_sym) also from prm have "foldr (⊗) as 𝟭 = foldr (⊗) bs 𝟭" by (rule multlist_perm_cong, simp) also assume "foldr (⊗) bs 𝟭 ∼ b" finally show "a ∼ b" by simp qed lemma (in comm_monoid_cancel) listassoc_wfactorsD: assumes assoc: "as [∼] bs" and afs: "wfactors G as a" and bfs: "wfactors G bs b" and [simp]: "a ∈ carrier G" "b ∈ carrier G" and [simp]: "set as ⊆ carrier G" "set bs ⊆ carrier G" shows "a ∼ b" using afs bfs proof (elim wfactorsE) assume "foldr (⊗) as 𝟭 ∼ a" then have "a ∼ foldr (⊗) as 𝟭" by (simp add: associated_sym) also from assoc have "foldr (⊗) as 𝟭 ∼ foldr (⊗) bs 𝟭" by (rule multlist_listassoc_cong, simp+) also assume "foldr (⊗) bs 𝟭 ∼ b" finally show "a ∼ 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 ∈ carrier G" "b ∈ carrier G" and ascarr[simp]: "set as ⊆ carrier G" and bscarr[simp]: "set bs ⊆ carrier G" shows "a ∼ b" using ee proof (elim essentially_equalE) fix fs assume prm: "as <~~> fs" then have as'carr[simp]: "set fs ⊆ 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 [∼] bs" from this afs' bfs show "a ∼ b" by (rule listassoc_wfactorsD) simp_all 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 ⊆ carrier G" "set bs ⊆ carrier G" shows "a ∼ b" using assms by (blast intro: factors_wfactors dest: ee_wfactorsD) lemma (in factorial_monoid) ee_factorsI: assumes ab: "a ∼ b" and afs: "factors G as a" and anunit: "a ∉ Units G" and bfs: "factors G bs b" and bnunit: "b ∉ Units G" and ascarr: "set as ⊆ carrier G" and bscarr: "set bs ⊆ 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 obtain u where uunit: "u ∈ Units G" and a: "a = b ⊗ u" by (elim associatedE2) from uunit bscarr have ee: "essentially_equal G (bs[0 := (bs!0 ⊗ u)]) bs" (is "essentially_equal G ?bs' bs") by (rule unitfactor_ee) from bscarr uunit have bs'carr: "set ?bs' ⊆ carrier G" by (cases bs) (simp_all add: Units_closed) from uunit bnunit bfs bscarr have fac: "factors G ?bs' (b ⊗ 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 ∼ b" and asf: "wfactors G as a" and bsf: "wfactors G bs b" and acarr[simp]: "a ∈ carrier G" and bcarr[simp]: "b ∈ carrier G" and ascarr[simp]: "set as ⊆ carrier G" and bscarr[simp]: "set bs ⊆ carrier G" shows "essentially_equal G as bs" using assms proof (cases "a ∈ Units G") case aunit: True also note asc finally have bunit: "b ∈ 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) then show ?thesis by (simp add: e e') next case anunit: False have bnunit: "b ∉ Units G" proof clarify assume "b ∈ Units G" also note asc[symmetric] finally have "a ∈ Units G" by simp with anunit show False .. qed from wfactors_factors[OF asf ascarr] obtain a' where fa': "factors G as a'" and a': "a' ∼ a" by blast from fa' ascarr have a'carr[simp]: "a' ∈ carrier G" by fast have a'nunit: "a' ∉ Units G" proof clarify assume "a' ∈ Units G" also note a' finally have "a ∈ Units G" by simp with anunit show "False" .. qed from wfactors_factors[OF bsf bscarr] obtain b' where fb': "factors G bs b'" and b': "b' ∼ b" by blast from fb' bscarr have b'carr[simp]: "b' ∈ carrier G" by fast have b'nunit: "b' ∉ Units G" proof clarify assume "b' ∈ Units G" also note b' finally have "b ∈ Units G" by simp with bnunit show False .. qed note a' also note asc also note b'[symmetric] finally have "a' ∼ 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 ∈ carrier G" and bcarr: "b ∈ carrier G" and ascarr: "set as ⊆ carrier G" and bscarr: "set bs ⊆ carrier G" shows asc: "a ∼ 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 ∈ carrier G" shows "∃fs. set fs ⊆ carrier G ∧ wfactors G fs a" proof (cases "a ∈ Units G") case True then have "wfactors G [] a" by (rule unit_wfactors) then show ?thesis by (intro exI) force next case False with factors_exist [OF acarr] obtain fs where fscarr: "set fs ⊆ carrier G" and f: "factors G fs a" by blast from f have "wfactors G fs a" by (rule factors_wfactors) fact with fscarr show ?thesis by fast qed lemma (in monoid) wfactors_prod_exists [intro, simp]: assumes "∀a ∈ set as. irreducible G a" and "set as ⊆ carrier G" shows "∃a. a ∈ carrier G ∧ 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 ∈ carrier G" and "set fs ⊆ carrier G" and "set fs' ⊆ 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 ∈ carrier G" shows "factors G (a # fb) (a ⊗ 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 ∈ carrier G" "b ∈ carrier G" "set fb ⊆ carrier G" shows "wfactors G (a # fb) (a ⊗ 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 ⊆ carrier G" and bscarr: "set fb ⊆ carrier G" shows "factors G (fa @ fb) (a ⊗ b)" proof - have "foldr (⊗) (fa @ fb) 𝟭 = foldr (⊗) fa 𝟭 ⊗ foldr (⊗) fb 𝟭" if "set fa ⊆ carrier G" "Ball (set fa) (irreducible G)" using that bscarr by (induct fa) (simp_all add: m_assoc) then show ?thesis using assms unfolding factors_def by force qed lemma (in comm_monoid_cancel) wfactors_mult [intro]: assumes asf: "wfactors G as a" and bsf:"wfactors G bs b" and acarr: "a ∈ carrier G" and bcarr: "b ∈ carrier G" and ascarr: "set as ⊆ carrier G" and bscarr:"set bs ⊆ carrier G" shows "wfactors G (as @ bs) (a ⊗ b)" using wfactors_factors[OF asf ascarr] and wfactors_factors[OF bsf bscarr] proof clarsimp fix a' b' assume asf': "factors G as a'" and a'a: "a' ∼ a" and bsf': "factors G bs b'" and b'b: "b' ∼ b" from asf' have a'carr: "a' ∈ carrier G" by (rule factors_closed) fact from bsf' have b'carr: "b' ∈ 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' ⊗ b')" by (rule factors_mult) fact+ with carr have abf': "wfactors G (as @ bs) (a' ⊗ b')" by (intro factors_wfactors) simp_all also from b'b carr have trb: "a' ⊗ b' ∼ a' ⊗ b" by (intro mult_cong_r) also from a'a carr have tra: "a' ⊗ b ∼ a ⊗ b" by (intro mult_cong_l) finally show "wfactors G (as @ bs) (a ⊗ b)" by (simp add: carr) qed lemma (in comm_monoid) factors_dividesI: assumes "factors G fs a" and "f ∈ set fs" and "set fs ⊆ 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 ⊆ carrier G" and acarr: "a ∈ carrier G" and f: "f ∈ set fs" shows "f divides a" using wfactors_factors[OF p fscarr] proof clarsimp fix a' assume fsa': "factors G fs a'" and a'a: "a' ∼ a" with fscarr have a'carr: "a' ∈ 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: "⋀a. ⟦ a ∈ carrier G; a ∉ Units G ⟧ ⟹ ∃fs. set fs ⊆ carrier G ∧ wfactors G fs a" and wfactors_unique: "⋀a fs fs'. ⟦a ∈ carrier G; set fs ⊆ carrier G; set fs' ⊆ carrier G; wfactors G fs a; wfactors G fs' a⟧ ⟹ essentially_equal G fs fs'" shows "factorial_monoid G" proof fix a assume acarr: "a ∈ carrier G" and anunit: "a ∉ Units G" from wfactors_exists[OF acarr anunit] obtain as where ascarr: "set as ⊆ carrier G" and afs: "wfactors G as a" by blast from wfactors_factors [OF afs ascarr] obtain a' where afs': "factors G as a'" and a'a: "a' ∼ a" by blast from afs' ascarr have a'carr: "a' ∈ carrier G" by fast have a'nunit: "a' ∉ Units G" proof clarify assume "a' ∈ Units G" also note a'a finally have "a ∈ Units G" by (simp add: acarr) with anunit show False .. qed from a'carr acarr a'a obtain u where uunit: "u ∈ Units G" and a': "a' = a ⊗ u" by (blast elim: associatedE2) note [simp] = acarr Units_closed[OF uunit] Units_inv_closed[OF uunit] have "a = a ⊗ 𝟭" by simp also have "… = a ⊗ (u ⊗ inv u)" by (simp add: uunit) also have "… = a' ⊗ inv u" by (simp add: m_assoc[symmetric] a'[symmetric]) finally have a: "a = a' ⊗ inv u" . from ascarr uunit have cr: "set (as[0:=(as!0 ⊗ inv u)]) ⊆ carrier G" by (cases as) auto from afs' uunit a'nunit acarr ascarr have "factors G (as[0:=(as!0 ⊗ inv u)]) a" by (simp add: a factors_cong_unit) with cr show "∃fs. set fs ⊆ carrier G ∧ 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}" definition "fmset G as = mset (map (λa. assocs G a) as)" text ‹Helper lemmas› lemma (in monoid) assocs_repr_independence: assumes "y ∈ assocs G x" "x ∈ carrier G" shows "assocs G x = assocs G y" using assms by (simp add: eq_closure_of_def elem_def) (use associated_sym associated_trans in ‹blast+›) lemma (in monoid) assocs_self: assumes "x ∈ carrier G" shows "x ∈ assocs G x" using assms by (fastforce intro: closure_ofI2) lemma (in monoid) assocs_repr_independenceD: assumes repr: "assocs G x = assocs G y" and ycarr: "y ∈ carrier G" shows "y ∈ assocs G x" unfolding repr using ycarr by (intro assocs_self) lemma (in comm_monoid) assocs_assoc: assumes "a ∈ assocs G b" "b ∈ carrier G" shows "a ∼ 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] unfolding mset_eq_perm fmset_def by blast lemma (in comm_monoid_cancel) eqc_listassoc_cong: assumes "as [∼] bs" and "set as ⊆ carrier G" and "set bs ⊆ carrier G" shows "map (assocs G) as = map (assocs G) bs" using assms proof (induction as arbitrary: bs) case Nil then show ?case by simp next case (Cons a as) then show ?case proof (clarsimp simp add: Cons_eq_map_conv list_all2_Cons1) fix z zs assume zzs: "a ∈ carrier G" "set as ⊆ carrier G" "bs = z # zs" "a ∼ z" "as [∼] zs" "z ∈ carrier G" "set zs ⊆ carrier G" then show "assocs G a = assocs G z" apply (simp add: eq_closure_of_def elem_def) using ‹a ∈ carrier G› ‹z ∈ carrier G› ‹a ∼ z› associated_sym associated_trans by blast+ qed qed lemma (in comm_monoid_cancel) fmset_listassoc_cong: assumes "as [∼] bs" and "set as ⊆ carrier G" and "set bs ⊆ 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 ⊆ carrier G" and bscarr: "set bs ⊆ 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' ⊆ carrier G" by (rule perm_closed) from prm have "fmset G as = fmset G as'" by (rule fmset_perm_cong) also assume "as' [∼] 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_aux: assumes "cas <~~> cbs" "cas = map (assocs G) as" "cbs = map (assocs G) bs" shows "∃as'. as <~~> as' ∧ map (assocs G) as' = cbs" using assms proof (induction cas cbs arbitrary: as bs rule: perm.induct) case (Cons xs ys z) then show ?case by (clarsimp simp add: map_eq_Cons_conv) blast next case (trans xs ys zs) then obtain as' where " as <~~> as' ∧ map (assocs G) as' = ys" by (metis (no_types, lifting) ex_map_conv mset_eq_perm set_mset_mset) then show ?case using trans.IH(2) trans.prems(2) by blast qed auto lemma (in comm_monoid_cancel) fmset_ee: assumes mset: "fmset G as = fmset G bs" and ascarr: "set as ⊆ carrier G" and bscarr: "set bs ⊆ carrier G" shows "essentially_equal G as bs" proof - from mset have "map (assocs G) as <~~> map (assocs G) bs" by (simp add: fmset_def mset_eq_perm del: mset_map) then obtain as' where tp: "as <~~> as'" and tm: "map (assocs G) as' = map (assocs G) bs" using fmset_ee_aux by blast with ascarr have as'carr: "set as' ⊆ carrier G" using perm_closed by blast from tm as'carr[THEN subsetD] bscarr[THEN subsetD] have "as' [∼] bs" by (induct as' arbitrary: bs) (simp, fastforce dest: assocs_eqD[THEN associated_sym]) with tp show "essentially_equal G as bs" by (fast intro: essentially_equalI) qed lemma (in comm_monoid_cancel) ee_is_fmset: assumes "set as ⊆ carrier G" and "set bs ⊆ 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: "⋀X. X ∈ set_mset Cs ⟹ ∃x. P x ∧ X = assocs G x" shows "∃cs. (∀c ∈ set cs. P c) ∧ fmset G cs = Cs" proof - from surjE[OF surj_mset] obtain Cs' where Cs: "Cs = mset Cs'" by blast have "∃cs. (∀c ∈ set cs. P c) ∧ mset (map (assocs G) cs) = Cs" using elems unfolding Cs proof (induction Cs') case (Cons a Cs') then obtain c cs where csP: "∀x∈set cs. P x" and mset: "mset (map (assocs G) cs) = mset Cs'" and cP: "P c" and a: "a = assocs G c" by force then have tP: "∀x∈set (c#cs). P x" by simp show ?case using tP mset a by fastforce qed auto then show ?thesis by (simp add: fmset_def) qed lemma (in monoid) mset_wfactorsEx: assumes elems: "⋀X. X ∈ set_mset Cs ⟹ ∃x. (x ∈ carrier G ∧ irreducible G x) ∧ X = assocs G x" shows "∃c cs. c ∈ carrier G ∧ set cs ⊆ carrier G ∧ wfactors G cs c ∧ fmset G cs = Cs" proof - have "∃cs. (∀c∈set cs. c ∈ carrier G ∧ irreducible G c) ∧ fmset G cs = Cs" by (intro mset_fmsetEx, rule elems) then obtain cs where p[rule_format]: "∀c∈set cs. c ∈ carrier G ∧ irreducible G c" and Cs[symmetric]: "fmset G cs = Cs" by auto from p have cscarr: "set cs ⊆ carrier G" by fast from p have "∃c. c ∈ carrier G ∧ wfactors G cs c" by (intro wfactors_prod_exists) auto then obtain c where ccarr: "c ∈ 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 ⊗ b)" and carr: "a ∈ carrier G" "b ∈ carrier G" "set as ⊆ carrier G" "set bs ⊆ carrier G" "set cs ⊆ carrier G" shows "fmset G cs = fmset G as + fmset G bs" proof - from assms have "wfactors G (as @ bs) (a ⊗ b)" by (intro wfactors_mult) with carr cfs have "essentially_equal G cs (as@bs)" by (intro ee_wfactorsI[of "a⊗b" "a⊗b"]) simp_all with carr have "fmset G cs = fmset G (as@bs)" by (intro ee_fmset) simp_all 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 ⊗ b)" and "set as ⊆ carrier G" "set bs ⊆ carrier G" "set cs ⊆ 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 ∈ carrier G" "b ∈ carrier G" "c ∈ carrier G" "set as ⊆ carrier G" "set bs ⊆ carrier G" "set cs ⊆ carrier G" and fs: "wfactors G as a" "wfactors G bs b" "wfactors G cs c" shows "c ∼ a ⊗ b" proof - from carr fs have m: "wfactors G (as @ bs) (a ⊗ 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_all add: carr) then show "c ∼ a ⊗ b" by (rule ee_wfactorsD[of "cs" "as@bs"]) (simp_all 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 ∈ carrier G" "b ∈ carrier G" "set as ⊆ carrier G" "set bs ⊆ carrier G" shows "fmset G as ⊆# fmset G bs" using ab proof (elim dividesE) fix c assume ccarr: "c ∈ carrier G" from wfactors_exist [OF this] obtain cs where cscarr: "set cs ⊆ carrier G" and cfs: "wfactors G cs c" by blast note carr = carr ccarr cscarr assume "b = a ⊗ 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_all then show ?thesis by simp qed lemma (in comm_monoid_cancel) fmsubset_divides: assumes msubset: "fmset G as ⊆# fmset G bs" and afs: "wfactors G as a" and bfs: "wfactors G bs b" and acarr: "a ∈ carrier G" and bcarr: "b ∈ carrier G" and ascarr: "set as ⊆ carrier G" and bscarr: "set bs ⊆ carrier G" shows "a divides b" proof - from afs have airr: "∀a ∈ set as. irreducible G a" by (fast elim: wfactorsE) from bfs have birr: "∀b ∈ set bs. irreducible G b" by (fast elim: wfactorsE) have "∃c cs. c ∈ carrier G ∧ set cs ⊆ carrier G ∧ wfactors G cs c ∧ fmset G cs = fmset G bs - fmset G as" proof (intro mset_wfactorsEx, simp) fix X assume "X ∈# fmset G bs - fmset G as" then have "X ∈# fmset G bs" by (rule in_diffD) then have "X ∈ set (map (assocs G) bs)" by (simp add: fmset_def) then have "∃x. x ∈ set bs ∧ X = assocs G x" by (induct bs) auto then obtain x where xbs: "x ∈ set bs" and X: "X = assocs G x" by auto with bscarr have xcarr: "x ∈ carrier G" by fast from xbs birr have xirr: "irreducible G x" by simp from xcarr and xirr and X show "∃x. x ∈ carrier G ∧ irreducible G x ∧ X = assocs G x" by fast qed then obtain c cs where ccarr: "c ∈ carrier G" and cscarr: "set cs ⊆ 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_iff subseteq_mset_def) then have basc: "b ∼ a ⊗ c" by (rule fmset_wfactors_mult) fact+ then show ?thesis proof (elim associatedE2) fix u assume "u ∈ Units G" "b = a ⊗ c ⊗ u" with acarr ccarr show "a divides b" by (fast intro: dividesI[of "c ⊗ u"] m_assoc) qed (simp_all add: acarr bcarr ccarr) qed lemma (in factorial_monoid) divides_as_fmsubset: assumes "wfactors G as a" and "wfactors G bs b" and "a ∈ carrier G" and "b ∈ carrier G" and "set as ⊆ carrier G" and "set bs ⊆ carrier G" shows "a divides b = (fmset G as ⊆# 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 ⊆# fmset G bs" and anb: "fmset G as ≠ fmset G bs" and "wfactors G as a" and "wfactors G bs b" and "a ∈ carrier G" and "b ∈ carrier G" and "set as ⊆ carrier G" and "set bs ⊆ carrier G" shows "properfactor G a b" proof (rule properfactorI) show "a divides b" using assms asubb fmsubset_divides by blast show "¬ b divides a" by (meson anb assms asubb factorial_monoid.divides_fmsubset factorial_monoid_axioms subset_mset.antisym) 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 ∈ carrier G" and "b ∈ carrier G" and "set as ⊆ carrier G" and "set bs ⊆ carrier G" shows "fmset G as ⊆# fmset G bs ∧ fmset G as ≠ fmset G bs" using pf apply safe apply (meson assms divides_as_fmsubset monoid.properfactor_divides monoid_axioms) by (meson assms associated_def comm_monoid_cancel.ee_wfactorsD comm_monoid_cancel.fmset_ee factorial_monoid_axioms factorial_monoid_def properfactorE) subsection ‹Irreducible Elements are Prime› lemma (in factorial_monoid) irreducible_prime: assumes pirr: "irreducible G p" and pcarr: "p ∈ carrier G" shows "prime G p" using pirr proof (elim irreducibleE, intro primeI) fix a b assume acarr: "a ∈ carrier G" and bcarr: "b ∈ carrier G" and pdvdab: "p divides (a ⊗ b)" and pnunit: "p ∉ Units G" assume irreduc[rule_format]: "∀b. b ∈ carrier G ∧ properfactor G b p ⟶ b ∈ Units G" from pdvdab obtain c where ccarr: "c ∈ carrier G" and abpc: "a ⊗ b = p ⊗ c" by (rule dividesE) obtain as where ascarr: "set as ⊆ carrier G" and afs: "wfactors G as a" using wfactors_exist [OF acarr] by blast obtain bs where bscarr: "set bs ⊆ carrier G" and bfs: "wfactors G bs b" using wfactors_exist [OF bcarr] by blast obtain cs where cscarr: "set cs ⊆ carrier G" and cfs: "wfactors G cs c" using wfactors_exist [OF ccarr] by blast note carr[simp] = pcarr acarr bcarr ccarr ascarr bscarr cscarr from pirr cfs abpc have "wfactors G (p # cs) (a ⊗ b)" by (simp add: wfactors_mult_single) moreover have "wfactors G (as @ bs) (a ⊗ b)" by (rule wfactors_mult [OF afs bfs]) fact+ ultimately have "essentially_equal G (p # cs) (as @ bs)" by (rule wfactors_unique) simp+ then obtain ds where "p # cs <~~> ds" and dsassoc: "ds [∼] (as @ bs)" by (fast elim: essentially_equalE) then have "p ∈ set ds" by (simp add: perm_set_eq[symmetric]) with dsassoc obtain p' where "p' ∈ set (as@bs)" and pp': "p ∼ p'" unfolding list_all2_conv_all_nth set_conv_nth by force then consider "p' ∈ set as" | "p' ∈ set bs" by auto then show "p divides a ∨ p divides b" using afs bfs divides_cong_l pp' wfactors_dividesI by (meson acarr ascarr bcarr bscarr pcarr) qed ― ‹A version using @{const factors}, more complicated› lemma (in factorial_monoid) factors_irreducible_prime: assumes pirr: "irreducible G p" and pcarr: "p ∈ carrier G" shows "prime G p" proof (rule primeI) show "p ∉ Units G" by (meson irreducibleE pirr) have irreduc: "⋀b. ⟦b ∈ carrier G; properfactor G b p⟧ ⟹ b ∈ Units G" using pirr by (auto simp: irreducible_def) show "p divides a ∨ p divides b" if acarr: "a ∈ carrier G" and bcarr: "b ∈ carrier G" and pdvdab: "p divides (a ⊗ b)" for a b proof - from pdvdab obtain c where ccarr: "c ∈ carrier G" and abpc: "a ⊗ b = p ⊗ c" by (rule dividesE) note [simp] = pcarr acarr bcarr ccarr show "p divides a ∨ p divides b" proof (cases "a ∈ Units G") case True then have "p divides b" by (metis acarr associatedI2' associated_def bcarr divides_trans m_comm pcarr pdvdab) then show ?thesis .. next case anunit: False show ?thesis proof (cases "b ∈ Units G") case True then have "p divides a" by (meson acarr bcarr divides_unit irreducible_prime pcarr pdvdab pirr prime_def) then show ?thesis .. next case bnunit: False then have cnunit: "c ∉ Units G" by (metis abpc acarr anunit bcarr ccarr irreducible_prodE irreducible_prod_rI pcarr pirr) then have abnunit: "a ⊗ b ∉ Units G" using acarr anunit bcarr unit_factor by blast obtain as where ascarr: "set as ⊆ carrier G" and afac: "factors G as a" using factors_exist [OF acarr anunit] by blast obtain bs where bscarr: "set bs ⊆ carrier G" and bfac: "factors G bs b" using factors_exist [OF bcarr bnunit] by blast obtain cs where cscarr: "set cs ⊆ carrier G" and cfac: "factors G cs c" using factors_exist [OF ccarr cnunit] by auto note [simp] = ascarr bscarr cscarr from pirr cfac abpc have abfac': "factors G (p # cs) (a ⊗ b)" by (simp add: factors_mult_single) from afac and bfac have "factors G (as @ bs) (a ⊗ b)" by (rule factors_mult) fact+ with abfac' have "essentially_equal G (p # cs) (as @ bs)" using abnunit factors_unique by auto then obtain ds where "p # cs <~~> ds" and dsassoc: "ds [∼] (as @ bs)" by (fast elim: essentially_equalE) then have "p ∈ set ds" by (simp add: perm_set_eq[symmetric]) with dsassoc obtain p' where "p' ∈ set (as@bs)" and pp': "p ∼ p'" unfolding list_all2_conv_all_nth set_conv_nth by force then consider "p' ∈ set as" | "p' ∈ set bs" by auto then show "p divides a ∨ p divides b" by (meson afac bfac divides_cong_l factors_dividesI pp' ascarr bscarr pcarr) qed qed qed qed subsection ‹Greatest Common Divisors and Lowest Common Multiples› subsubsection ‹Definitions› definition isgcd :: "[('a,_) monoid_scheme, 'a, 'a, 'a] ⇒ bool" ("(_ gcdofı _ _)" [81,81,81] 80) where "x gcdof⇘_{G⇙}a b ⟷ x divides⇘_{G⇙}a ∧ x divides⇘_{G⇙}b ∧ (∀y∈carrier G. (y divides⇘_{G⇙}a ∧ y divides⇘_{G⇙}b ⟶ y divides⇘_{G⇙}x))" definition islcm :: "[_, 'a, 'a, 'a] ⇒ bool" ("(_ lcmofı _ _)" [81,81,81] 80) where "x lcmof⇘_{G⇙}a b ⟷ a divides⇘_{G⇙}x ∧ b divides⇘_{G⇙}x ∧ (∀y∈carrier G. (a divides⇘_{G⇙}y ∧ b divides⇘_{G⇙}y ⟶ x divides⇘_{G⇙}y))" definition somegcd :: "('a,_) monoid_scheme ⇒ 'a ⇒ 'a ⇒ 'a" where "somegcd G a b = (SOME x. x ∈ carrier G ∧ x gcdof⇘_{G⇙}a b)" definition somelcm :: "('a,_) monoid_scheme ⇒ 'a ⇒ 'a ⇒ 'a" where "somelcm G a b = (SOME x. x ∈ carrier G ∧ x lcmof⇘_{G⇙}a b)" definition "SomeGcd G A = inf (division_rel G) A" locale gcd_condition_monoid = comm_monoid_cancel + assumes gcdof_exists: "⟦a ∈ carrier G; b ∈ carrier G⟧ ⟹ ∃c. c ∈ carrier G ∧ c gcdof a b" locale primeness_condition_monoid = comm_monoid_cancel + assumes irreducible_prime: "⟦a ∈ carrier G; irreducible G a⟧ ⟹ prime G a" locale divisor_chain_condition_monoid = comm_monoid_cancel + assumes division_wellfounded: "wf {(x, y). x ∈ carrier G ∧ y ∈ carrier G ∧ properfactor G x y}" subsubsection ‹Connections to \texttt{Lattice.thy}› lemma gcdof_greatestLower: fixes G (structure) assumes carr[simp]: "a ∈ carrier G" "b ∈ carrier G" shows "(x ∈ carrier G ∧ x gcdof a b) = greatest (division_rel G) x (Lower (division_rel G) {a, b})" by (auto simp: isgcd_def greatest_def Lower_def elem_def) lemma lcmof_leastUpper: fixes G (structure) assumes carr[simp]: "a ∈ carrier G" "b ∈ carrier G" shows "(x ∈ carrier G ∧ x lcmof a b) = least (division_rel G) x (Upper (division_rel G) {a, b})" by (auto simp: islcm_def least_def Upper_def elem_def) lemma somegcd_meet: fixes G (structure) assumes carr: "a ∈ carrier G" "b ∈ carrier G" shows "somegcd G a b = meet (division_rel G) a b" by (simp add: somegcd_def meet_def inf_def gcdof_greatestLower[OF carr]) lemma (in monoid) isgcd_divides_l: assumes "a divides b" and "a ∈ carrier G" "b ∈ 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 ∈ carrier G" "b ∈ 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 ∈ carrier G" and bcarr: "b ∈ carrier G" shows "∃c. c ∈ carrier G ∧ c gcdof a b" proof - from wfactors_exist [OF acarr] obtain as where ascarr: "set as ⊆ carrier G" and afs: "wfactors G as a" by blast from afs have airr: "∀a ∈ set as. irreducible G a" by (fast elim: wfactorsE) from wfactors_exist [OF bcarr] obtain bs where bscarr: "set bs ⊆ carrier G" and bfs: "wfactors G bs b" by blast from bfs have birr: "∀b ∈ set bs. irreducible G b" by (fast elim: wfactorsE) have "∃c cs. c ∈ carrier G ∧ set cs ⊆ carrier G ∧ wfactors G cs c ∧ fmset G cs = fmset G as ∩# fmset G bs" proof (intro mset_wfactorsEx) fix X assume "X ∈# fmset G as ∩# fmset G bs" then have "X ∈# fmset G as" by simp then have "X ∈ set (map (assocs G) as)" by (simp add: fmset_def) then have "∃x. X = assocs G x ∧ x ∈ set as" by (induct as) auto then obtain x where X: "X = assocs G x" and xas: "x ∈ set as" by blast with ascarr have xcarr: "x ∈ carrier G" by blast from xas airr have xirr: "irreducible G x" by simp from xcarr and xirr and X show "∃x. (x ∈ carrier G ∧ irreducible G x) ∧ X = assocs G x" by blast qed then obtain c cs where ccarr: "c ∈ carrier G" and cscarr: "set cs ⊆ carrier G" and csirr: "wfactors G cs c" and csmset: "fmset G cs = fmset G as ∩# fmset G bs" by auto have "c gcdof a b" proof (simp add: isgcd_def, safe) from csmset have "fmset G cs ⊆# fmset G as" by (simp add: multiset_inter_def subset_mset_def) then show "c divides a" by (rule fmsubset_divides) fact+ next from csmset have "fmset G cs ⊆# fmset G bs" by (simp add: multiset_inter_def subseteq_mset_def, force) then show "c divides b" by (rule fmsubset_divides) fact+ next fix y assume "y ∈ carrier G" from wfactors_exist [OF this] obtain ys where yscarr: "set ys ⊆ carrier G" and yfs: "wfactors G ys y" by blast assume "y divides a" then have ya: "fmset G ys ⊆# fmset G as" by (rule divides_fmsubset) fact+ assume "y divides b" then have yb: "fmset G ys ⊆# fmset G bs" by (rule divides_fmsubset) fact+ from ya yb csmset have "fmset G ys ⊆# fmset G cs" by (simp add: subset_mset_def) then show "y divides c" by (rule fmsubset_divides) fact+ qed with ccarr show "∃c. c ∈ carrier G ∧ c gcdof a b" by fast qed lemma (in factorial_monoid) lcmof_exists: assumes acarr: "a ∈ carrier G" and bcarr: "b ∈ carrier G" shows "∃c. c ∈ carrier G ∧ c lcmof a b" proof - from wfactors_exist [OF acarr] obtain as where ascarr: "set as ⊆ carrier G" and afs: "wfactors G as a" by blast from afs have airr: "∀a ∈ set as. irreducible G a" by (fast elim: wfactorsE) from wfactors_exist [OF bcarr] obtain bs where bscarr: "set bs ⊆ carrier G" and bfs: "wfactors G bs b" by blast from bfs have birr: "∀b ∈ set bs. irreducible G b" by (fast elim: wfactorsE) have "∃c cs. c ∈ carrier G ∧ set cs ⊆ carrier G ∧ wfactors G cs c ∧ fmset G cs = (fmset G as - fmset G bs) + fmset G bs" proof (intro mset_wfactorsEx) fix X assume "X ∈# (fmset G as - fmset G bs) + fmset G bs" then have "X ∈# fmset G as ∨ X ∈# fmset G bs" by (auto dest: in_diffD) then consider "X ∈ set_mset (fmset G as)" | "X ∈ set_mset (fmset G bs)" by fast then show "∃x. (x ∈ carrier G ∧ irreducible G x) ∧ X = assocs G x" proof cases case 1 then have "X ∈ set (map (assocs G) as)" by (simp add: fmset_def) then have "∃x. x ∈ set as ∧ X = assocs G x" by (induct as) auto then obtain x where xas: "x ∈ set as" and X: "X = assocs G x" by auto with ascarr have xcarr: "x ∈ carrier G" by fast from xas airr have xirr: "irreducible G x" by simp from xcarr and xirr and X show ?thesis by fast next case 2 then have "X ∈ set (map (assocs G) bs)" by (simp add: fmset_def) then have "∃x. x ∈ set bs ∧ X = assocs G x" by (induct as) auto then obtain x where xbs: "x ∈ set bs" and X: "X = assocs G x" by auto with bscarr have xcarr: "x ∈ carrier G" by fast from xbs birr have xirr: "irreducible G x" by simp from xcarr and xirr and X show ?thesis by fast qed qed then obtain c cs where ccarr: "c ∈ carrier G" and cscarr: "set cs ⊆ 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 ⊆# fmset G cs" by (simp add: subseteq_mset_def, force) then show "a divides c" by (rule fmsubset_divides) fact+ next from csmset have "fmset G bs ⊆# fmset G cs" by (simp add: subset_mset_def) then show "b divides c" by (rule fmsubset_divides) fact+ next fix y assume "y ∈ carrier G" from wfactors_exist [OF this] obtain ys where yscarr: "set ys ⊆ carrier G" and yfs: "wfactors G ys y" by blast assume "a divides y" then have ya: "fmset G as ⊆# fmset G ys" by (rule divides_fmsubset) fact+ assume "b divides y" then have yb: "fmset G bs ⊆# fmset G ys" by (rule divides_fmsubset) fact+ from ya yb csmset have "fmset G cs ⊆# fmset G ys" using subset_eq_diff_conv subset_mset.le_diff_conv2 by fastforce then show "c divides y" by (rule fmsubset_divides) fact+ qed with ccarr show "∃c. c ∈ carrier G ∧ c lcmof a b" by fast qed subsection ‹Conditions for Factoriality› subsubsection ‹Gcd condition› lemma (in gcd_condition_monoid) division_weak_lower_semilattice [simp]: "weak_lower_semilattice (division_rel G)" proof - interpret weak_partial_order "division_rel G" .. show ?thesis proof (unfold_locales, simp_all) fix x y assume carr: "x ∈ carrier G" "y ∈ carrier G" from gcdof_exists [OF this] obtain z where zcarr: "z ∈ carrier G" and isgcd: "z gcdof x y" by blast with carr have "greatest (division_rel G) z (Lower (division_rel G) {x, y})" by (subst gcdof_greatestLower[symmetric], simp+) then show "∃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' ∼ a" and agcd: "a gcdof b c" and a'carr: "a' ∈ carrier G" and carr': "a ∈ carrier G" "b ∈ carrier G" "c ∈ carrier G" shows "a' gcdof b c" proof - note carr = a'carr carr' interpret weak_lower_semilattice "division_rel G" by simp have "is_glb (division_rel G) a' {b, c}" by (subst greatest_Lower_cong_l[of _ a]) (simp_all add: a'a carr gcdof_greatestLower[symmetric] agcd) then have "a' ∈ carrier G ∧ a' gcdof b c" by (simp add: gcdof_greatestLower carr') then show ?thesis .. qed lemma (in gcd_condition_monoid) gcd_closed [simp]: assumes carr: "a ∈ carrier G" "b ∈ carrier G" shows "somegcd G a b ∈ 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 ∈ carrier G" "b ∈ 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 ∈ carrier G ∧ (somegcd G a b) gcdof a b" by (simp add: gcdof_greatestLower inf_of_two_greatest meet_def somegcd_meet) then show "(somegcd G a b) gcdof a b" by simp qed lemma (in gcd_condition_monoid) gcd_exists: assumes carr: "a ∈ carrier G" "b ∈ carrier G" shows "∃x∈carrier G. x = somegcd G a b" proof - interpret weak_lower_semilattice "division_rel G" by simp show ?thesis by (metis carr(1) carr(2) gcd_closed) qed lemma (in gcd_condition_monoid) gcd_divides_l: assumes carr: "a ∈ carrier G" "b ∈ carrier G" shows "(somegcd G a b) divides a" proof - interpret weak_lower_semilattice "division_rel G" by simp show ?thesis by (metis carr(1) carr(2) gcd_isgcd isgcd_def) qed lemma (in gcd_condition_monoid) gcd_divides_r: assumes carr: "a ∈ carrier G" "b ∈ carrier G" shows "(somegcd G a b) divides b" proof - interpret weak_lower_semilattice "division_rel G" by simp show ?thesis by (metis carr gcd_isgcd isgcd_def) qed lemma (in gcd_condition_monoid) gcd_divides: assumes sub: "z divides x" "z divides y" and L: "x ∈ carrier G" "y ∈ carrier G" "z ∈ carrier G" shows "z divides (somegcd G x y)" proof - interpret weak_lower_semilattice "division_rel G" by simp show ?thesis by (metis gcd_isgcd isgcd_def assms) qed lemma (in gcd_condition_monoid) gcd_cong_l: assumes xx': "x ∼ x'" and carr: "x ∈ carrier G" "x' ∈ carrier G" "y ∈ carrier G" shows "somegcd G x y ∼ 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 ∈ carrier G" "y ∈ carrier G" "y' ∈ carrier G" and yy': "y ∼ y'" shows "somegcd G x y ∼ 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 ∈ carrier G" shows "asc_cong (λa. 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 ∈ carrier G" shows "asc_cong (λb. 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: "⋀y. ⟦y∈carrier G; y divides b; y divides c⟧ ⟹ y divides a" and acarr: "a ∈ carrier G" and bcarr: "b ∈ carrier G" and ccarr: "c ∈ carrier G" shows "a ∼ somegcd G b c" proof - have "∃a. a ∈ carrier G ∧ a gcdof b c" by (simp add: bcarr ccarr gcdof_exists) moreover have "⋀x. x ∈ carrier G ∧ x gcdof b c ⟹ a ∼ x" by (simp add: acarr associated_def dvd isgcd_def others) ultimately show ?thesis unfolding somegcd_def by (blast intro: someI2_ex) qed lemma (in gcd_condition_monoid) gcdI2: assumes "a gcdof b c" and "a ∈ carrier G" and "b ∈ carrier G" and "c ∈ carrier G" shows "a ∼ somegcd G b c" using assms unfolding isgcd_def by (simp add: gcdI) lemma (in gcd_condition_monoid) SomeGcd_ex: assumes "finite A" "A ⊆ carrier G" "A ≠ {}" shows "∃x∈ 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 ∈ carrier G" "b ∈ carrier G" "c ∈ carrier G" shows "somegcd G (somegcd G a b) c ∼ somegcd G a (somegcd G b c)" proof - interpret weak_lower_semilattice "division_rel G" by simp show ?thesis unfolding associated_def by (meson carr divides_trans gcd_divides gcd_divides_l gcd_divides_r gcd_exists) qed lemma (in gcd_condition_monoid) gcd_mult: assumes acarr: "a ∈ carrier G" and bcarr: "b ∈ carrier G" and ccarr: "c ∈ carrier G" shows "c ⊗ somegcd G a b ∼ somegcd G (c ⊗ a) (c ⊗ b)" proof - (* following Jacobson, Basic Algebra, p.140 *) let ?d = "somegcd G a b" let ?e = "somegcd G (c ⊗ a) (c ⊗ b)" note carr[simp] = acarr bcarr ccarr have dcarr: "?d ∈ carrier G" by simp have ecarr: "?e ∈ carrier G" by simp note carr = carr dcarr ecarr have "?d divides a" by (simp add: gcd_divides_l) then have cd'ca: "c ⊗ ?d divides (c ⊗ a)" by (simp add: divides_mult_lI) have "?d divides b" by (simp add: gcd_divides_r) then have cd'cb: "c ⊗ ?d divides (c ⊗ b)" by (simp add: divides_mult_lI) from cd'ca cd'cb have cd'e: "c ⊗ ?d divides ?e" by (rule gcd_divides) simp_all then obtain u where ucarr[simp]: "u ∈ carrier G" and e_cdu: "?e = c ⊗ ?d ⊗ u" by blast note carr = carr ucarr have "?e divides c ⊗ a" by (rule gcd_divides_l) simp_all then obtain x where xcarr: "x ∈ carrier G" and ca_ex: "c ⊗ a = ?e ⊗ x" by blast with e_cdu have ca_cdux: "c ⊗ a = c ⊗ ?d ⊗ u ⊗ x" by simp from ca_cdux xcarr have "c ⊗ a = c ⊗ (?d ⊗ u ⊗ x)" by (simp add: m_assoc) then have "a = ?d ⊗ u ⊗ x" by (rule l_cancel[of c a]) (simp add: xcarr)+ then have du'a: "?d ⊗ u divides a" by (rule dividesI[OF xcarr]) have "?e divides c ⊗ b" by (intro gcd_divides_r) simp_all then obtain x where xcarr: "x ∈ carrier G" and cb_ex: "c ⊗ b = ?e ⊗ x" by blast with e_cdu have cb_cdux: "c ⊗ b = c ⊗ ?d ⊗ u ⊗ x" by simp from cb_cdux xcarr have "c ⊗ b = c ⊗ (?d ⊗ u ⊗ x)" by (simp add: m_assoc) with xcarr have "b = ?d ⊗ u ⊗ x" by (intro l_cancel[of c b]) simp_all then have du'b: "?d ⊗ u divides b" by (intro dividesI[OF xcarr]) from du'a du'b carr have du'd: "?d ⊗ u divides ?d" by (intro gcd_divides) simp_all then have uunit: "u ∈ Units G" proof (elim dividesE) fix v assume vcarr[simp]: "v ∈ carrier G" assume d: "?d = ?d ⊗ u ⊗ v" have "?d ⊗ 𝟭 = ?d ⊗ u ⊗ v" by simp fact also have "?d ⊗ u ⊗ v = ?d ⊗ (u ⊗ v)" by (simp add: m_assoc) finally have "?d ⊗ 𝟭 = ?d ⊗ (u ⊗ v)" . then have i2: "𝟭 = u ⊗ v" by (rule l_cancel) simp_all then have i1: "𝟭 = v ⊗ u" by (simp add: m_comm) from vcarr i1[symmetric] i2[symmetric] show "u ∈ Units G" by (auto simp: Units_def) qed from e_cdu uunit have "somegcd G (c ⊗ a) (c ⊗ b) ∼ c ⊗ somegcd G a b" by (intro associatedI2[of u]) simp_all from this[symmetric] show "c ⊗ somegcd G a b ∼ somegcd G (c ⊗ a) (c ⊗ b)" by simp qed lemma (in monoid) assoc_subst: assumes ab: "a ∼ b" and cP: "∀a b. a ∈ carrier G ∧ b ∈ carrier G ∧ a ∼ b ⟶ f a ∈ carrier G ∧ f b ∈ carrier G ∧ f a ∼ f b" and carr: "a ∈ carrier G" "b ∈ carrier G" shows "f a ∼ f b" using assms by auto lemma (in gcd_condition_monoid) relprime_mult: assumes abrelprime: "somegcd G a b ∼ 𝟭" and acrelprime: "somegcd G a c ∼ 𝟭" and carr[simp]: "a ∈ carrier G" "b ∈ carrier G" "c ∈ carrier G" shows "somegcd G a (b ⊗ c) ∼ 𝟭" proof - have "c = c ⊗ 𝟭" by simp also from abrelprime[symmetric] have "… ∼ c ⊗ somegcd G a b" by (rule assoc_subst) (simp add: mult_cong_r)+ also have "… ∼ somegcd G (c ⊗ a) (c ⊗ b)" by (rule gcd_mult) fact+ finally have c: "c ∼ somegcd G (c ⊗ a) (c ⊗ b)" by simp from carr have a: "a ∼ somegcd G a (c ⊗ a)" by (fast intro: gcdI divides_prod_l) have "somegcd G a (b ⊗ c) ∼ somegcd G a (c ⊗ b)" by (simp add: m_comm) also from a have "… ∼ somegcd G (somegcd G a (c ⊗ a)) (c ⊗ b)" by (rule assoc_subst) (simp add: gcd_cong_l)+ also from gcd_assoc have "… ∼ somegcd G a (somegcd G (c ⊗ a) (c ⊗ b))" by (rule assoc_subst) simp+ also from c[symmetric] have "… ∼ somegcd G a c" by (rule assoc_subst) (simp add: gcd_cong_r)+ also note acrelprime finally show "somegcd G a (b ⊗ c) ∼ 𝟭" by simp qed lemma (in gcd_condition_monoid) primeness_condition: "primeness_condition_monoid G" proof - have *: "p divides a ∨ p divides b" if pcarr[simp]: "p ∈ carrier G" and acarr[simp]: "a ∈ carrier G" and bcarr[simp]: "b ∈ carrier G" and pirr: "irreducible G p" and pdvdab: "p divides a ⊗ b" for p a b proof - from pirr have pnunit: "p ∉ Units G" and r: "⋀b. ⟦b ∈ carrier G; properfactor G b p⟧ ⟹ b ∈ Units G" by (fast elim: irreducibleE)+ show "p divides a ∨ p divides b" proof (rule ccontr, clarsimp) assume npdvda: "¬ p divides a" and npdvdb: "¬ p divides b" have "𝟭 ∼ somegcd G p a" proof (intro gcdI unit_divides) show "⋀y. ⟦y ∈ carrier G; y divides p; y divides a⟧ ⟹ y ∈ Units G" by (meson divides_trans npdvda pcarr properfactorI r) qed auto with pcarr acarr have pa: "somegcd G p a ∼ 𝟭" by (fast intro: associated_sym[of "𝟭"] gcd_closed) have "𝟭 ∼ somegcd G p b" proof (intro gcdI unit_divides) show "⋀y. ⟦y ∈ carrier G; y divides p; y divides b⟧ ⟹ y ∈ Units G" by (meson divides_trans npdvdb pcarr properfactorI r) qed auto with pcarr bcarr have pb: "somegcd G p b ∼ 𝟭" by (fast intro: associated_sym[of "𝟭"] gcd_closed) have "p ∼ somegcd G p (a ⊗ b)" using pdvdab by (simp add: gcdI2 isgcd_divides_l) also from pa pb pcarr acarr bcarr have "somegcd G p (a ⊗ b) ∼ 𝟭" by (rule relprime_mult) finally have "p ∼ 𝟭" by simp with pcarr have "p ∈ Units G" by (fast intro: assoc_unit_l) with pnunit show False .. qed qed show ?thesis by unfold_locales (metis * primeI irreducibleE) qed sublocale gcd_condition_monoid ⊆ primeness_condition_monoid by (rule primeness_condition) subsubsection ‹Divisor chain condition› lemma (in divisor_chain_condition_monoid) wfactors_exist: assumes acarr: "a ∈ carrier G" shows "∃as. set as ⊆ carrier G ∧ wfactors G as a" proof - have r: "a ∈ carrier G ⟹ (∃as. set as ⊆ carrier G ∧ wfactors G as a)" using division_wellfounded proof (induction rule: wf_induct_rule) case (less x) then have xcarr: "x ∈ carrier G" by auto show ?case proof (cases "x ∈ Units G") case True then show ?thesis by (metis bot.extremum list.set(1) unit_wfactors) next case xnunit: False show ?thesis proof (cases "irreducible G x") case True then show ?thesis by (rule_tac x="[x]" in exI) (simp add: wfactors_def xcarr) next case False then obtain y where ycarr: "y ∈ carrier G" and ynunit: "y ∉ Units G" and pfyx: "properfactor G y x" by (meson irreducible_def xnunit) obtain ys where yscarr: "set ys ⊆ carrier G" and yfs: "wfactors G ys y" using less ycarr pfyx by blast then obtain z where zcarr: "z ∈ carrier G" and x: "x = y ⊗ z" by (meson dividesE pfyx properfactorE2) from zcarr ycarr have "properfactor G z x" using m_comm properfactorI3 x ynunit by blast with less zcarr obtain zs where zscarr: "set zs ⊆ carrier G" and zfs: "wfactors G zs z" by blast from yscarr zscarr have xscarr: "set (ys@zs) ⊆ carrier G" by simp have "wfactors G (ys@zs) (y⊗z)" using xscarr ycarr yfs zcarr zfs by auto then have "wfactors G (ys@zs) x" by (simp add: x) with xscarr show "∃xs. set xs ⊆ carrier G ∧ wfactors G xs x" by fast qed qed qed from acarr show ?thesis by (rule r) qed subsubsection ‹Primeness condition› lemma (in comm_monoid_cancel) multlist_prime_pos: assumes aprime: "prime G a" and carr: "a ∈ carrier G" and as: "set as ⊆ carrier G" "a divides (foldr (⊗) as 𝟭)" shows "∃i<length as. a divides (as!i)" using as proof (induction as) case Nil then show ?case by simp (meson Units_one_closed aprime carr divides_unit primeE) next case (Cons x as) then have "x ∈ carrier G" "set as ⊆ carrier G" and "a divides x ⊗ foldr (⊗) as 𝟭" by (auto simp: ) with carr aprime have "a divides x ∨ a divides foldr (⊗) as 𝟭" by (intro prime_divides) simp+ then show ?case using Cons.IH Cons.prems(1) by force qed lemma (in primeness_condition_monoid) wfactors_unique__hlp_induct: "∀a as'. a ∈ carrier G ∧ set as ⊆ carrier G ∧ set as' ⊆ carrier G ∧ wfactors G as a ∧ wfactors G as' a ⟶ essentially_equal G as as'" proof (induct as) case Nil show ?case apply (clarsimp simp: wfactors_def) by (metis Units_one_closed assoc_unit_r list_update_nonempty unit_wfactors_empty unitfactor_ee wfactorsI) next case (Cons ah as) then show ?case proof clarsimp fix a as' assume ih [rule_format]: "∀a as'. a ∈ carrier G ∧ set as' ⊆ carrier G ∧ wfactors G as a ∧ wfactors G as' a ⟶ essentially_equal G as as'" and acarr: "a ∈ carrier G" and ahcarr: "ah ∈ carrier G" and ascarr: "set as ⊆ carrier G" and as'carr: "set as' ⊆ carrier G" and afs: "wfactors G (ah # as) a" and afs': "wfactors G as' a" then have ahdvda: "ah divides a" by (intro wfactors_dividesI[of "ah#as" "a"]) simp_all then obtain a' where a'carr: "a' ∈ carrier G" and a: "a = ah ⊗ a'" by blast have a'fs: "wfactors G as a'" apply (rule wfactorsE[OF afs], rule wfactorsI, simp) by (metis a a'carr ahcarr ascarr assoc_l_cancel factorsI factors_def factors_mult_single list.set_intros(1) list.set_intros(2) multlist_closed) 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 (⊗) as' 𝟭)" by (elim wfactorsE associatedE, simp) finally have "ah divides (foldr (⊗) as' 𝟭)" by simp with ahprime have "∃i<length as'. ah divides as'!i" by (intro multlist_prime_pos) simp_all then obtain i where len: "i<length as'" and ahdvd: "ah divides as'!i" by blast 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 ∈ carrier G" unfolding set_conv_nth by force note carr = carr asicarr from ahdvd obtain x where "x ∈ carrier G" and asi: "as'!i = ah ⊗ x" by blast with carr irrasi[simplified asi] have asiah: "as'!i ∼ ah" by (metis ahprime associatedI2 irreducible_prodE primeE) 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 "∃aa_1. aa_1 ∈ carrier G ∧ wfactors G (take i as') aa_1" by (meson afs' in_set_takeD partscarr(1) wfactorsE wfactors_prod_exists) then obtain aa_1 where aa1carr: "aa_1 ∈ carrier G" and aa1fs: "wfactors G (take i as') aa_1" by auto have "∃aa_2. aa_2 ∈ carrier G ∧ wfactors G (drop (Suc i) as') aa_2" by (meson afs' in_set_dropD partscarr(2) wfactors_def wfactors_prod_exists) then obtain aa_2 where aa2carr: "aa_2 ∈ 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 ⊗ aa_2)" by (intro wfactors_mult, simp+) then have v1': "wfactors G (as'!i # take i as' @ drop (Suc i) as') (as'!i ⊗ (aa_1 ⊗ aa_2))" using irrasi wfactors_mult_single by auto from aa2carr carr aa1fs aa2fs have "wfactors G (as'!i # drop (Suc i) as') (as'!i ⊗ aa_2)" by (metis irrasi wfactors_mult_single) with len carr aa1carr aa2carr aa1fs have v2: "wfactors G (take i as' @ as'!i # drop (Suc i) as') (aa_1 ⊗ (as'!i ⊗ aa_2))" using wfactors_mult by auto from len have as': "as' = (take i as' @ as'!i # drop (Suc i) as')" by (simp add: Cons_nth_drop_Suc) 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 ⊗ (as'!i ⊗ aa_2) ∼ a" by (metis as' ee_wfactorsD m_closed) then have t1: "as'!i ⊗ (aa_1 ⊗ aa_2) ∼ a" by (metis aa1carr aa2carr asicarr m_lcomm) from carr asiah have "ah ⊗ (aa_1 ⊗ aa_2) ∼ as'!i ⊗ (aa_1 ⊗ aa_2)" by (metis associated_sym m_closed mult_cong_l) also note t1 finally have "ah ⊗ (aa_1 ⊗ aa_2) ∼ a" by simp with carr aa1carr aa2carr a'carr nth_mem[OF len] have a': "aa_1 ⊗ aa_2 ∼ 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 then have ee1: "essentially_equal G (ah # as) (ah # take i as' @ drop (Suc i) as')" by (elim essentially_equalE) (fastforce intro: essentially_equalI) 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' [∼] as' ! i # take i as' @ drop (Suc i) as'" by (simp add: list_all2_append) (simp add: asiah[symmetric]) 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')" by (metis as' as'carr listassoc_refl essentially_equalI perm_append_Cons) finally have "essentially_equal G (ah # as) (take i as' @ as' ! i # drop (Suc i) as')" by simp then show "essentially_equal G (ah # as) as'" by (subst as') qed qed lemma (in primeness_condition_monoid) wfactors_unique: assumes "wfactors G as a" "wfactors G as' a" and "a ∈ carrier G" "set as ⊆ carrier G" "set as' ⊆ carrier G" shows "essentially_equal G as as'" by (rule wfactors_unique__hlp_induct[rule_format, of a]) (simp add: assms) subsubsection ‹Application to factorial monoids› text ‹Number of factors for wellfoundedness› definition factorcount :: "_ ⇒ 'a ⇒ nat" where "factorcount G a = (THE c. ∀as. set as ⊆ carrier G ∧ wfactors G as a ⟶ c = length as)" lemma (in monoid) ee_length: assumes ee: "essentially_equal G as bs" shows "length as = length bs" by (rule essentially_equalE[OF ee]) (metis list_all2_conv_all_nth perm_length) lemma (in factorial_monoid) factorcount_exists: assumes carr[simp]: "a ∈ carrier G" shows "∃c. ∀as. set as ⊆ carrier G ∧ wfactors G as a ⟶ c = length as" proof - have "∃as. set as ⊆ carrier G ∧ wfactors G as a" by (intro wfactors_exist) simp then obtain as where ascarr[simp]: "set as ⊆ carrier G" and afs: "wfactors G as a" by (auto simp del: carr) have "∀as'. set as' ⊆ carrier G ∧ wfactors G as' a ⟶ length as = length as'" by (metis afs ascarr assms ee_length wfactors_unique) then show "∃c. ∀as'. set as' ⊆ carrier G ∧ wfactors G as' a ⟶ c = length as'" .. qed lemma (in factorial_monoid) factorcount_unique: assumes afs: "wfactors G as a" and acarr[simp]: "a ∈ carrier G" and ascarr: "set as ⊆ carrier G" shows "factorcount G a = length as" proof - have "∃ac. ∀as. set as ⊆ carrier G ∧ wfactors G as a ⟶ ac = length as" by (rule factorcount_exists) simp then obtain ac where alen: "∀as. set as ⊆ carrier G ∧ wfactors G as a ⟶ ac = length as" by auto then have ac: "ac = factorcount G a" unfolding factorcount_def using ascarr by (blast intro: theI2 afs) from ascarr afs have "ac = length as" by (simp add: alen) with ac show ?thesis by simp qed lemma (in factorial_monoid) divides_fcount: assumes dvd: "a divides b" and acarr: "a ∈ carrier G" and bcarr:"b ∈ carrier G" shows "factorcount G a ≤ factorcount G b" proof (rule dividesE[OF dvd]) fix c from assms have "∃as. set as ⊆ carrier G ∧ wfactors G as a" by blast then obtain as where ascarr: "set as ⊆ carrier G" and afs: "wfactors G as a" by blast with acarr have fca: "factorcount G a = length as" by (intro factorcount_unique) assume ccarr: "c ∈ carrier G" then have "∃cs. set cs ⊆ carrier G ∧ wfactors G cs c" by blast then obtain cs where cscarr: "set cs ⊆ carrier G" and cfs: "wfactors G cs c" by blast note [simp] = acarr bcarr ccarr ascarr cscarr assume b: "b = a ⊗ c" from afs cfs have "wfactors G (as@cs) (a ⊗ c)" by (intro wfactors_mult) simp_all with b have "wfactors G (as@cs) b" by simp then have "factorcount G b = length (as@cs)" by (intro factorcount_unique) simp_all then have "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 ∈ carrier G" and bcarr: "b ∈ carrier G" and asc: "a ∼ b" shows "factorcount G a = factorcount G b" using assms by (auto simp: associated_def factorial_monoid.divides_fcount factorial_monoid_axioms le_antisym) lemma (in factorial_monoid) properfactor_fcount: assumes acarr: "a ∈ carrier G" and bcarr:"b ∈ carrier G" and pf: "properfactor G a b" shows "factorcount G a < factorcount G b" proof (rule properfactorE[OF pf], elim dividesE) fix c from assms have "∃as. set as ⊆ carrier G ∧ wfactors G as a" by blast then obtain as where ascarr: "set as ⊆ carrier G" and afs: "wfactors G as a" by blast with acarr have fca: "factorcount G a = length as" by (intro factorcount_unique) assume ccarr: "c ∈ carrier G" then have "∃cs. set cs ⊆ carrier G ∧ wfactors G cs c" by blast then obtain cs where cscarr: "set cs ⊆ carrier G" and cfs: "wfactors G cs c" by blast assume b: "b = a ⊗ c" have "wfactors G (as@cs) (a ⊗ 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) then have fcb: "factorcount G b = length as + length cs" by simp assume nbdvda: "¬ b divides a" have "c ∉ Units G" proof assume cunit:"c ∈ Units G" have "b ⊗ inv c = a ⊗ c ⊗ inv c" by (simp add: b) also from ccarr acarr cunit have "… = a ⊗ (c ⊗ inv c)" by (fast intro: m_assoc) also from ccarr cunit have "… = a ⊗ 𝟭" by simp also from acarr have "… = a" by simp finally have "a = b ⊗ 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" by (metis Units_one_closed assoc_unit_r ccarr foldr.simps(1) id_apply length_greater_0_conv wfactors_def) with fca fcb show ?thesis by simp qed sublocale factorial_monoid ⊆ divisor_chain_condition_monoid apply unfold_locales apply (rule wfUNIVI) apply (rule measure_induct[of "factorcount G"]) apply simp apply (metis properfactor_fcount) done sublocale factorial_monoid ⊆ primeness_condition_monoid by standard (rule irreducible_prime) lemma (in factorial_monoid) primeness_condition: "primeness_condition_monoid G" .. lemma (in factorial_monoid) gcd_condition [simp]: "gcd_condition_monoid G" by standard (rule gcdof_exists) sublocale factorial_monoid ⊆ gcd_condition_monoid by standard (rule gcdof_exists) lemma (in factorial_monoid) division_weak_lattice [simp]: "weak_lattice (division_rel G)" proof - interpret weak_lower_semilattice "division_rel G" by simp show "weak_lattice (division_rel G)" proof (unfold_locales, simp_all) fix x y assume carr: "x ∈ carrier G" "y ∈ carrier G" from lcmof_exists [OF this] obtain z where zcarr: "z ∈ carrier G" and isgcd: "z lcmof x y" by blast with carr have "least (division_rel G) z (Upper (division_rel G) {x, y})" by (simp add: lcmof_leastUpper[symmetric]) then show "∃z. least (division_rel G) z (Upper (division_rel G) {x, y})" by blast qed qed subsection ‹Factoriality Theorems› theorem factorial_condition_one: (* Jacobson theorem 2.21 *) "divisor_chain_condition_monoid G ∧ primeness_condition_monoid G ⟷ factorial_monoid G" proof (rule iffI, 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 "factorial_monoid G" then interpret factorial_monoid "G" . show "divisor_chain_condition_monoid G ∧ primeness_condition_monoid G" by rule unfold_locales qed theorem factorial_condition_two: (* Jacobson theorem 2.22 *) "divisor_chain_condition_monoid G ∧ gcd_condition_monoid G ⟷ factorial_monoid G" proof (rule iffI, 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 "factorial_monoid G" then interpret factorial_monoid "G" . show "divisor_chain_condition_monoid G ∧ gcd_condition_monoid G" by rule unfold_locales qed end