src/HOL/Algebra/Divisibility.thy
 author nipkow Fri, 13 Nov 2009 14:14:04 +0100 changeset 33657 a4179bf442d1 parent 32960 69916a850301 child 35272 c283ae736bea permissions -rw-r--r--
renamed lemmas "anti_sym" -> "antisym"
```
(*
Title:     Divisibility in monoids and rings
Author:    Clemens Ballarin, started 18 July 2008

Based on work by Stephan Hohe.
*)

theory Divisibility
imports Permutation Coset Group
begin

section {* Factorial Monoids *}

subsection {* Monoids with Cancellation Law *}

locale monoid_cancel = monoid +
assumes l_cancel:
"\<lbrakk>c \<otimes> a = c \<otimes> b; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b"
and r_cancel:
"\<lbrakk>a \<otimes> c = b \<otimes> c; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b"

lemma (in monoid) monoid_cancelI:
assumes l_cancel:
"\<And>a b c. \<lbrakk>c \<otimes> a = c \<otimes> b; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b"
and r_cancel:
"\<And>a b c. \<lbrakk>a \<otimes> c = b \<otimes> c; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b"
shows "monoid_cancel G"
proof qed fact+

lemma (in monoid_cancel) is_monoid_cancel:
"monoid_cancel G"
..

sublocale group \<subseteq> monoid_cancel
proof qed simp+

locale comm_monoid_cancel = monoid_cancel + comm_monoid

lemma comm_monoid_cancelI:
fixes G (structure)
assumes "comm_monoid G"
assumes cancel:
"\<And>a b c. \<lbrakk>a \<otimes> c = b \<otimes> c; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b"
shows "comm_monoid_cancel G"
proof -
interpret comm_monoid G by fact
show "comm_monoid_cancel G"
apply unfold_locales
apply (subgoal_tac "a \<otimes> c = b \<otimes> c")
apply (iprover intro: cancel)
apply (iprover intro: cancel)
done
qed

lemma (in comm_monoid_cancel) is_comm_monoid_cancel:
"comm_monoid_cancel G"
by intro_locales

sublocale comm_group \<subseteq> comm_monoid_cancel
..

subsection {* Products of Units in Monoids *}

lemma (in monoid) Units_m_closed[simp, intro]:
assumes h1unit: "h1 \<in> Units G" and h2unit: "h2 \<in> Units G"
shows "h1 \<otimes> h2 \<in> Units G"
unfolding Units_def
using assms
apply safe
apply fast
apply (intro bexI[of _ "inv h2 \<otimes> inv h1"], safe)
apply (simp add: m_assoc[symmetric] Units_closed Units_l_inv)
apply (simp add: m_assoc[symmetric] Units_closed Units_r_inv)
apply fast
done

lemma (in monoid) prod_unit_l:
assumes abunit[simp]: "a \<otimes> b \<in> Units G" and aunit[simp]: "a \<in> Units G"
and carr[simp]: "a \<in> carrier G"  "b \<in> carrier G"
shows "b \<in> Units G"
proof -
have c: "inv (a \<otimes> b) \<otimes> a \<in> carrier G" by simp

have "(inv (a \<otimes> b) \<otimes> a) \<otimes> b = inv (a \<otimes> b) \<otimes> (a \<otimes> b)" by (simp add: m_assoc)
also have "\<dots> = \<one>" by (simp add: Units_l_inv)
finally have li: "(inv (a \<otimes> b) \<otimes> a) \<otimes> b = \<one>" .

have "\<one> = inv a \<otimes> a" by (simp add: Units_l_inv[symmetric])
also have "\<dots> = inv a \<otimes> \<one> \<otimes> a" by simp
also have "\<dots> = inv a \<otimes> ((a \<otimes> b) \<otimes> inv (a \<otimes> b)) \<otimes> a"
by (simp add: Units_r_inv[OF abunit, symmetric] del: Units_r_inv)
also have "\<dots> = ((inv a \<otimes> a) \<otimes> b) \<otimes> inv (a \<otimes> b) \<otimes> a"
by (simp add: m_assoc del: Units_l_inv)
also have "\<dots> = b \<otimes> inv (a \<otimes> b) \<otimes> a" by (simp add: Units_l_inv)
also have "\<dots> = b \<otimes> (inv (a \<otimes> b) \<otimes> a)" by (simp add: m_assoc)
finally have ri: "b \<otimes> (inv (a \<otimes> b) \<otimes> a) = \<one> " by simp

from c li ri
show "b \<in> Units G" by (simp add: Units_def, fast)
qed

lemma (in monoid) prod_unit_r:
assumes abunit[simp]: "a \<otimes> b \<in> Units G" and bunit[simp]: "b \<in> Units G"
and carr[simp]: "a \<in> carrier G"  "b \<in> carrier G"
shows "a \<in> Units G"
proof -
have c: "b \<otimes> inv (a \<otimes> b) \<in> carrier G" by simp

have "a \<otimes> (b \<otimes> inv (a \<otimes> b)) = (a \<otimes> b) \<otimes> inv (a \<otimes> b)"
by (simp add: m_assoc del: Units_r_inv)
also have "\<dots> = \<one>" by simp
finally have li: "a \<otimes> (b \<otimes> inv (a \<otimes> b)) = \<one>" .

have "\<one> = b \<otimes> inv b" by (simp add: Units_r_inv[symmetric])
also have "\<dots> = b \<otimes> \<one> \<otimes> inv b" by simp
also have "\<dots> = b \<otimes> (inv (a \<otimes> b) \<otimes> (a \<otimes> b)) \<otimes> inv b"
by (simp add: Units_l_inv[OF abunit, symmetric] del: Units_l_inv)
also have "\<dots> = (b \<otimes> inv (a \<otimes> b) \<otimes> a) \<otimes> (b \<otimes> inv b)"
by (simp add: m_assoc del: Units_l_inv)
also have "\<dots> = b \<otimes> inv (a \<otimes> b) \<otimes> a" by simp
finally have ri: "(b \<otimes> inv (a \<otimes> b)) \<otimes> a = \<one> " by simp

from c li ri
show "a \<in> Units G" by (simp add: Units_def, fast)
qed

lemma (in comm_monoid) unit_factor:
assumes abunit: "a \<otimes> b \<in> Units G"
and [simp]: "a \<in> carrier G"  "b \<in> carrier G"
shows "a \<in> Units G"
using abunit[simplified Units_def]
proof clarsimp
fix i
assume [simp]: "i \<in> carrier G"
and li: "i \<otimes> (a \<otimes> b) = \<one>"
and ri: "a \<otimes> b \<otimes> i = \<one>"

have carr': "b \<otimes> i \<in> carrier G" by simp

have "(b \<otimes> i) \<otimes> a = (i \<otimes> b) \<otimes> a" by (simp add: m_comm)
also have "\<dots> = i \<otimes> (b \<otimes> a)" by (simp add: m_assoc)
also have "\<dots> = i \<otimes> (a \<otimes> b)" by (simp add: m_comm)
also note li
finally have li': "(b \<otimes> i) \<otimes> a = \<one>" .

have "a \<otimes> (b \<otimes> i) = a \<otimes> b \<otimes> i" by (simp add: m_assoc)
also note ri
finally have ri': "a \<otimes> (b \<otimes> i) = \<one>" .

from carr' li' ri'
show "a \<in> Units G" by (simp add: Units_def, fast)
qed

subsection {* Divisibility and Association *}

subsubsection {* Function definitions *}

constdefs (structure G)
factor :: "[_, 'a, 'a] \<Rightarrow> bool" (infix "divides\<index>" 65)
"a divides b == \<exists>c\<in>carrier G. b = a \<otimes> c"

constdefs (structure G)
associated :: "[_, 'a, 'a] => bool" (infix "\<sim>\<index>" 55)
"a \<sim> b == a divides b \<and> b divides a"

abbreviation
"division_rel G == \<lparr>carrier = carrier G, eq = op \<sim>\<^bsub>G\<^esub>, le = op divides\<^bsub>G\<^esub>\<rparr>"

constdefs (structure G)
properfactor :: "[_, 'a, 'a] \<Rightarrow> bool"
"properfactor G a b == a divides b \<and> \<not>(b divides a)"

constdefs (structure G)
irreducible :: "[_, 'a] \<Rightarrow> bool"
"irreducible G a == a \<notin> Units G \<and> (\<forall>b\<in>carrier G. properfactor G b a \<longrightarrow> b \<in> Units G)"

constdefs (structure G)
prime :: "[_, 'a] \<Rightarrow> bool"
"prime G p == p \<notin> Units G \<and>
(\<forall>a\<in>carrier G. \<forall>b\<in>carrier G. p divides (a \<otimes> b) \<longrightarrow> p divides a \<or> p divides b)"

subsubsection {* Divisibility *}

lemma dividesI:
fixes G (structure)
assumes carr: "c \<in> carrier G"
and p: "b = a \<otimes> c"
shows "a divides b"
unfolding factor_def
using assms by fast

lemma dividesI' [intro]:
fixes G (structure)
assumes p: "b = a \<otimes> c"
and carr: "c \<in> carrier G"
shows "a divides b"
using assms
by (fast intro: dividesI)

lemma dividesD:
fixes G (structure)
assumes "a divides b"
shows "\<exists>c\<in>carrier G. b = a \<otimes> c"
using assms
unfolding factor_def
by fast

lemma dividesE [elim]:
fixes G (structure)
assumes d: "a divides b"
and elim: "\<And>c. \<lbrakk>b = a \<otimes> c; c \<in> carrier G\<rbrakk> \<Longrightarrow> P"
shows "P"
proof -
from dividesD[OF d]
obtain c
where "c\<in>carrier G"
and "b = a \<otimes> c"
by auto
thus "P" by (elim elim)
qed

lemma (in monoid) divides_refl[simp, intro!]:
assumes carr: "a \<in> carrier G"
shows "a divides a"
apply (intro dividesI[of "\<one>"])
done

lemma (in monoid) divides_trans [trans]:
assumes dvds: "a divides b"  "b divides c"
and acarr: "a \<in> carrier G"
shows "a divides c"
using dvds[THEN dividesD]
by (blast intro: dividesI m_assoc acarr)

lemma (in monoid) divides_mult_lI [intro]:
assumes ab: "a divides b"
and carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "(c \<otimes> a) divides (c \<otimes> b)"
using ab
apply (elim dividesE, simp add: m_assoc[symmetric] carr)
apply (fast intro: dividesI)
done

lemma (in monoid_cancel) divides_mult_l [simp]:
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "(c \<otimes> a) divides (c \<otimes> b) = a divides b"
apply safe
apply (elim dividesE, intro dividesI, assumption)
apply (rule l_cancel[of c])
apply (fast intro: divides_mult_lI carr)
done

lemma (in comm_monoid) divides_mult_rI [intro]:
assumes ab: "a divides b"
and carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "(a \<otimes> c) divides (b \<otimes> c)"
using carr ab
apply (simp add: m_comm[of a c] m_comm[of b c])
apply (rule divides_mult_lI, assumption+)
done

lemma (in comm_monoid_cancel) divides_mult_r [simp]:
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "(a \<otimes> c) divides (b \<otimes> c) = a divides b"
using carr
by (simp add: m_comm[of a c] m_comm[of b c])

lemma (in monoid) divides_prod_r:
assumes ab: "a divides b"
and carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "a divides (b \<otimes> c)"
using ab carr
by (fast intro: m_assoc)

lemma (in comm_monoid) divides_prod_l:
assumes carr[intro]: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
and ab: "a divides b"
shows "a divides (c \<otimes> b)"
using ab carr
apply (simp add: m_comm[of c b])
apply (fast intro: divides_prod_r)
done

lemma (in monoid) unit_divides:
assumes uunit: "u \<in> Units G"
and acarr: "a \<in> carrier G"
shows "u divides a"
proof (intro dividesI[of "(inv u) \<otimes> a"], fast intro: uunit acarr)
from uunit acarr
have xcarr: "inv u \<otimes> a \<in> carrier G" by fast

from uunit acarr
have "u \<otimes> (inv u \<otimes> a) = (u \<otimes> inv u) \<otimes> a" by (fast intro: m_assoc[symmetric])
also have "\<dots> = \<one> \<otimes> a" by (simp add: Units_r_inv[OF uunit])
also from acarr
have "\<dots> = a" by simp
finally
show "a = u \<otimes> (inv u \<otimes> a)" ..
qed

lemma (in comm_monoid) divides_unit:
assumes udvd: "a divides u"
and  carr: "a \<in> carrier G"  "u \<in> Units G"
shows "a \<in> Units G"
using udvd carr
by (blast intro: unit_factor)

lemma (in comm_monoid) Unit_eq_dividesone:
assumes ucarr: "u \<in> carrier G"
shows "u \<in> Units G = u divides \<one>"
using ucarr
by (fast dest: divides_unit intro: unit_divides)

subsubsection {* Association *}

lemma associatedI:
fixes G (structure)
assumes "a divides b"  "b divides a"
shows "a \<sim> b"
using assms

lemma (in monoid) associatedI2:
assumes uunit[simp]: "u \<in> Units G"
and a: "a = b \<otimes> u"
and bcarr[simp]: "b \<in> carrier G"
shows "a \<sim> b"
using uunit bcarr
unfolding a
apply (intro associatedI)
apply (rule dividesI[of "inv u"], simp)
apply (simp add: m_assoc Units_closed Units_r_inv)
apply fast
done

lemma (in monoid) associatedI2':
assumes a: "a = b \<otimes> u"
and uunit: "u \<in> Units G"
and bcarr: "b \<in> carrier G"
shows "a \<sim> b"
using assms by (intro associatedI2)

lemma associatedD:
fixes G (structure)
assumes "a \<sim> b"
shows "a divides b"
using assms by (simp add: associated_def)

lemma (in monoid_cancel) associatedD2:
assumes assoc: "a \<sim> b"
and carr: "a \<in> carrier G"  "b \<in> carrier G"
shows "\<exists>u\<in>Units G. a = b \<otimes> u"
using assoc
unfolding associated_def
proof clarify
assume "b divides a"
hence "\<exists>u\<in>carrier G. a = b \<otimes> u" by (rule dividesD)
from this obtain u
where ucarr: "u \<in> carrier G" and a: "a = b \<otimes> u"
by auto

assume "a divides b"
hence "\<exists>u'\<in>carrier G. b = a \<otimes> u'" by (rule dividesD)
from this obtain u'
where u'carr: "u' \<in> carrier G" and b: "b = a \<otimes> u'"
by auto
note carr = carr ucarr u'carr

from carr
have "a \<otimes> \<one> = a" by simp
also have "\<dots> = b \<otimes> u" by (simp add: a)
also have "\<dots> = a \<otimes> u' \<otimes> u" by (simp add: b)
also from carr
have "\<dots> = a \<otimes> (u' \<otimes> u)" by (simp add: m_assoc)
finally
have "a \<otimes> \<one> = a \<otimes> (u' \<otimes> u)" .
with carr
have u1: "\<one> = u' \<otimes> u" by (fast dest: l_cancel)

from carr
have "b \<otimes> \<one> = b" by simp
also have "\<dots> = a \<otimes> u'" by (simp add: b)
also have "\<dots> = b \<otimes> u \<otimes> u'" by (simp add: a)
also from carr
have "\<dots> = b \<otimes> (u \<otimes> u')" by (simp add: m_assoc)
finally
have "b \<otimes> \<one> = b \<otimes> (u \<otimes> u')" .
with carr
have u2: "\<one> = u \<otimes> u'" by (fast dest: l_cancel)

from u'carr u1[symmetric] u2[symmetric]
have "\<exists>u'\<in>carrier G. u' \<otimes> u = \<one> \<and> u \<otimes> u' = \<one>" by fast
hence "u \<in> Units G" by (simp add: Units_def ucarr)

from ucarr this a
show "\<exists>u\<in>Units G. a = b \<otimes> u" by fast
qed

lemma associatedE:
fixes G (structure)
assumes assoc: "a \<sim> b"
and e: "\<lbrakk>a divides b; b divides a\<rbrakk> \<Longrightarrow> P"
shows "P"
proof -
from assoc
have "a divides b"  "b divides a"
thus "P" by (elim e)
qed

lemma (in monoid_cancel) associatedE2:
assumes assoc: "a \<sim> b"
and e: "\<And>u. \<lbrakk>a = b \<otimes> u; u \<in> Units G\<rbrakk> \<Longrightarrow> P"
and carr: "a \<in> carrier G"  "b \<in> carrier G"
shows "P"
proof -
from assoc and carr
have "\<exists>u\<in>Units G. a = b \<otimes> u" by (rule associatedD2)
from this obtain u
where "u \<in> Units G"  "a = b \<otimes> u"
by auto
thus "P" by (elim e)
qed

lemma (in monoid) associated_refl [simp, intro!]:
assumes "a \<in> carrier G"
shows "a \<sim> a"
using assms
by (fast intro: associatedI)

lemma (in monoid) associated_sym [sym]:
assumes "a \<sim> b"
and "a \<in> carrier G"  "b \<in> carrier G"
shows "b \<sim> a"
using assms
by (iprover intro: associatedI elim: associatedE)

lemma (in monoid) associated_trans [trans]:
assumes "a \<sim> b"  "b \<sim> c"
and "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "a \<sim> c"
using assms
by (iprover intro: associatedI divides_trans elim: associatedE)

lemma (in monoid) division_equiv [intro, simp]:
"equivalence (division_rel G)"
apply unfold_locales
apply simp_all
apply (rule associated_sym, assumption+)
apply (iprover intro: associated_trans)
done

subsubsection {* Division and associativity *}

lemma divides_antisym:
fixes G (structure)
assumes "a divides b"  "b divides a"
and "a \<in> carrier G"  "b \<in> carrier G"
shows "a \<sim> b"
using assms
by (fast intro: associatedI)

lemma (in monoid) divides_cong_l [trans]:
assumes xx': "x \<sim> x'"
and xdvdy: "x' divides y"
and carr [simp]: "x \<in> carrier G"  "x' \<in> carrier G"  "y \<in> carrier G"
shows "x divides y"
proof -
from xx'
have "x divides x'" by (simp add: associatedD)
also note xdvdy
finally
show "x divides y" by simp
qed

lemma (in monoid) divides_cong_r [trans]:
assumes xdvdy: "x divides y"
and yy': "y \<sim> y'"
and carr[simp]: "x \<in> carrier G"  "y \<in> carrier G"  "y' \<in> carrier G"
shows "x divides y'"
proof -
note xdvdy
also from yy'
have "y divides y'" by (simp add: associatedD)
finally
show "x divides y'" by simp
qed

lemma (in monoid) division_weak_partial_order [simp, intro!]:
"weak_partial_order (division_rel G)"
apply unfold_locales
apply simp_all
apply (blast intro: associated_trans)
apply (blast intro: divides_trans)
apply (blast intro: divides_cong_l divides_cong_r associated_sym)
done

subsubsection {* Multiplication and associativity *}

lemma (in monoid_cancel) mult_cong_r:
assumes "b \<sim> b'"
and carr: "a \<in> carrier G"  "b \<in> carrier G"  "b' \<in> carrier G"
shows "a \<otimes> b \<sim> a \<otimes> b'"
using assms
apply (elim associatedE2, intro associatedI2)
apply (auto intro: m_assoc[symmetric])
done

lemma (in comm_monoid_cancel) mult_cong_l:
assumes "a \<sim> a'"
and carr: "a \<in> carrier G"  "a' \<in> carrier G"  "b \<in> carrier G"
shows "a \<otimes> b \<sim> a' \<otimes> b"
using assms
apply (elim associatedE2, intro associatedI2)
apply assumption
apply simp+
done

lemma (in monoid_cancel) assoc_l_cancel:
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"  "b' \<in> carrier G"
and "a \<otimes> b \<sim> a \<otimes> b'"
shows "b \<sim> b'"
using assms
apply (elim associatedE2, intro associatedI2)
apply assumption
apply (rule l_cancel[of a])
apply fast+
done

lemma (in comm_monoid_cancel) assoc_r_cancel:
assumes "a \<otimes> b \<sim> a' \<otimes> b"
and carr: "a \<in> carrier G"  "a' \<in> carrier G"  "b \<in> carrier G"
shows "a \<sim> a'"
using assms
apply (elim associatedE2, intro associatedI2)
apply assumption
apply (rule r_cancel[of a b])
apply fast+
done

subsubsection {* Units *}

lemma (in monoid_cancel) assoc_unit_l [trans]:
assumes asc: "a \<sim> b" and bunit: "b \<in> Units G"
and carr: "a \<in> carrier G"
shows "a \<in> Units G"
using assms
by (fast elim: associatedE2)

lemma (in monoid_cancel) assoc_unit_r [trans]:
assumes aunit: "a \<in> Units G" and asc: "a \<sim> b"
and bcarr: "b \<in> carrier G"
shows "b \<in> Units G"
using aunit bcarr associated_sym[OF asc]
by (blast intro: assoc_unit_l)

lemma (in comm_monoid) Units_cong:
assumes aunit: "a \<in> Units G" and asc: "a \<sim> b"
and bcarr: "b \<in> carrier G"
shows "b \<in> Units G"
using assms
by (blast intro: divides_unit elim: associatedE)

lemma (in monoid) Units_assoc:
assumes units: "a \<in> Units G"  "b \<in> Units G"
shows "a \<sim> b"
using units
by (fast intro: associatedI unit_divides)

lemma (in monoid) Units_are_ones:
"Units G {.=}\<^bsub>(division_rel G)\<^esub> {\<one>}"
apply (simp add: set_eq_def elem_def, rule, simp_all)
proof clarsimp
fix a
assume aunit: "a \<in> Units G"
show "a \<sim> \<one>"
apply (rule associatedI)
apply (fast intro: dividesI[of "inv a"] aunit Units_r_inv[symmetric])
apply (fast intro: dividesI[of "a"] l_one[symmetric] Units_closed[OF aunit])
done
next
have "\<one> \<in> Units G" by simp
moreover have "\<one> \<sim> \<one>" by simp
ultimately show "\<exists>a \<in> Units G. \<one> \<sim> a" by fast
qed

lemma (in comm_monoid) Units_Lower:
"Units G = Lower (division_rel G) (carrier G)"
apply (rule, rule)
apply clarsimp
apply (rule unit_divides)
apply (unfold Units_def, fast)
apply assumption
apply clarsimp
proof -
fix x
assume xcarr: "x \<in> carrier G"
assume r[rule_format]: "\<forall>y. y \<in> carrier G \<longrightarrow> x divides y"
have "\<one> \<in> carrier G" by simp
hence "x divides \<one>" by (rule r)
hence "\<exists>x'\<in>carrier G. \<one> = x \<otimes> x'" by (rule dividesE, fast)
from this obtain x'
where x'carr: "x' \<in> carrier G"
and xx': "\<one> = x \<otimes> x'"
by auto

note xx'
also with xcarr x'carr
have "\<dots> = x' \<otimes> x" by (simp add: m_comm)
finally
have "\<one> = x' \<otimes> x" .

from x'carr xx'[symmetric] this[symmetric]
show "\<exists>y\<in>carrier G. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
qed

subsubsection {* Proper factors *}

lemma properfactorI:
fixes G (structure)
assumes "a divides b"
and "\<not>(b divides a)"
shows "properfactor G a b"
using assms
unfolding properfactor_def
by simp

lemma properfactorI2:
fixes G (structure)
and neq: "\<not>(a \<sim> b)"
shows "properfactor G a b"
proof (rule ccontr, simp)
assume "b divides a"
with advdb have "a \<sim> b" by (rule associatedI)
with neq show "False" by fast
qed

lemma (in comm_monoid_cancel) properfactorI3:
assumes p: "p = a \<otimes> b"
and nunit: "b \<notin> Units G"
and carr: "a \<in> carrier G"  "b \<in> carrier G"  "p \<in> carrier G"
shows "properfactor G a p"
unfolding p
using carr
apply (intro properfactorI, fast)
proof (clarsimp, elim dividesE)
fix c
assume ccarr: "c \<in> carrier G"
note [simp] = carr ccarr

have "a \<otimes> \<one> = a" by simp
also assume "a = a \<otimes> b \<otimes> c"
also have "\<dots> = a \<otimes> (b \<otimes> c)" by (simp add: m_assoc)
finally have "a \<otimes> \<one> = a \<otimes> (b \<otimes> c)" .

hence rinv: "\<one> = b \<otimes> c" by (intro l_cancel[of "a" "\<one>" "b \<otimes> c"], simp+)
also have "\<dots> = c \<otimes> b" by (simp add: m_comm)
finally have linv: "\<one> = c \<otimes> b" .

from ccarr linv[symmetric] rinv[symmetric]
have "b \<in> Units G" unfolding Units_def by fastsimp
with nunit
show "False" ..
qed

lemma properfactorE:
fixes G (structure)
assumes pf: "properfactor G a b"
and r: "\<lbrakk>a divides b; \<not>(b divides a)\<rbrakk> \<Longrightarrow> P"
shows "P"
using pf
unfolding properfactor_def
by (fast intro: r)

lemma properfactorE2:
fixes G (structure)
assumes pf: "properfactor G a b"
and elim: "\<lbrakk>a divides b; \<not>(a \<sim> b)\<rbrakk> \<Longrightarrow> P"
shows "P"
using pf
unfolding properfactor_def
by (fast elim: elim associatedE)

lemma (in monoid) properfactor_unitE:
assumes uunit: "u \<in> Units G"
and pf: "properfactor G a u"
and acarr: "a \<in> carrier G"
shows "P"
using pf unit_divides[OF uunit acarr]
by (fast elim: properfactorE)

lemma (in monoid) properfactor_divides:
assumes pf: "properfactor G a b"
shows "a divides b"
using pf
by (elim properfactorE)

lemma (in monoid) properfactor_trans1 [trans]:
assumes dvds: "a divides b"  "properfactor G b c"
and carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "properfactor G a c"
using dvds carr
apply (elim properfactorE, intro properfactorI)
apply (iprover intro: divides_trans)+
done

lemma (in monoid) properfactor_trans2 [trans]:
assumes dvds: "properfactor G a b"  "b divides c"
and carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "properfactor G a c"
using dvds carr
apply (elim properfactorE, intro properfactorI)
apply (iprover intro: divides_trans)+
done

lemma properfactor_lless:
fixes G (structure)
shows "properfactor G = lless (division_rel G)"
apply (rule ext) apply (rule ext) apply rule
apply (fastsimp elim: properfactorE2 intro: weak_llessI)
apply (fastsimp elim: weak_llessE intro: properfactorI2)
done

lemma (in monoid) properfactor_cong_l [trans]:
assumes x'x: "x' \<sim> x"
and pf: "properfactor G x y"
and carr: "x \<in> carrier G"  "x' \<in> carrier G"  "y \<in> carrier G"
shows "properfactor G x' y"
using pf
unfolding properfactor_lless
proof -
interpret weak_partial_order "division_rel G" ..
from x'x
have "x' .=\<^bsub>division_rel G\<^esub> x" by simp
also assume "x \<sqsubset>\<^bsub>division_rel G\<^esub> y"
finally
show "x' \<sqsubset>\<^bsub>division_rel G\<^esub> y" by (simp add: carr)
qed

lemma (in monoid) properfactor_cong_r [trans]:
assumes pf: "properfactor G x y"
and yy': "y \<sim> y'"
and carr: "x \<in> carrier G"  "y \<in> carrier G"  "y' \<in> carrier G"
shows "properfactor G x y'"
using pf
unfolding properfactor_lless
proof -
interpret weak_partial_order "division_rel G" ..
assume "x \<sqsubset>\<^bsub>division_rel G\<^esub> y"
also from yy'
have "y .=\<^bsub>division_rel G\<^esub> y'" by simp
finally
show "x \<sqsubset>\<^bsub>division_rel G\<^esub> y'" by (simp add: carr)
qed

lemma (in monoid_cancel) properfactor_mult_lI [intro]:
assumes ab: "properfactor G a b"
and carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "properfactor G (c \<otimes> a) (c \<otimes> b)"
using ab carr
by (fastsimp elim: properfactorE intro: properfactorI)

lemma (in monoid_cancel) properfactor_mult_l [simp]:
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "properfactor G (c \<otimes> a) (c \<otimes> b) = properfactor G a b"
using carr
by (fastsimp elim: properfactorE intro: properfactorI)

lemma (in comm_monoid_cancel) properfactor_mult_rI [intro]:
assumes ab: "properfactor G a b"
and carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "properfactor G (a \<otimes> c) (b \<otimes> c)"
using ab carr
by (fastsimp elim: properfactorE intro: properfactorI)

lemma (in comm_monoid_cancel) properfactor_mult_r [simp]:
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "properfactor G (a \<otimes> c) (b \<otimes> c) = properfactor G a b"
using carr
by (fastsimp elim: properfactorE intro: properfactorI)

lemma (in monoid) properfactor_prod_r:
assumes ab: "properfactor G a b"
and carr[simp]: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "properfactor G a (b \<otimes> c)"
by (intro properfactor_trans2[OF ab] divides_prod_r, simp+)

lemma (in comm_monoid) properfactor_prod_l:
assumes ab: "properfactor G a b"
and carr[simp]: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "properfactor G a (c \<otimes> b)"
by (intro properfactor_trans2[OF ab] divides_prod_l, simp+)

subsection {* Irreducible Elements and Primes *}

subsubsection {* Irreducible elements *}

lemma irreducibleI:
fixes G (structure)
assumes "a \<notin> Units G"
and "\<And>b. \<lbrakk>b \<in> carrier G; properfactor G b a\<rbrakk> \<Longrightarrow> b \<in> Units G"
shows "irreducible G a"
using assms
unfolding irreducible_def
by blast

lemma irreducibleE:
fixes G (structure)
assumes irr: "irreducible G a"
and elim: "\<lbrakk>a \<notin> Units G; \<forall>b. b \<in> carrier G \<and> properfactor G b a \<longrightarrow> b \<in> Units G\<rbrakk> \<Longrightarrow> P"
shows "P"
using assms
unfolding irreducible_def
by blast

lemma irreducibleD:
fixes G (structure)
assumes irr: "irreducible G a"
and pf: "properfactor G b a"
and bcarr: "b \<in> carrier G"
shows "b \<in> Units G"
using assms
by (fast elim: irreducibleE)

lemma (in monoid_cancel) irreducible_cong [trans]:
assumes irred: "irreducible G a"
and aa': "a \<sim> a'"
and carr[simp]: "a \<in> carrier G"  "a' \<in> carrier G"
shows "irreducible G a'"
using assms
apply (elim irreducibleE, intro irreducibleI)
apply simp_all
proof clarify
assume "a' \<in> Units G"
also note aa'[symmetric]
finally have aunit: "a \<in> Units G" by simp

assume "a \<notin> Units G"
with aunit
show "False" by fast
next
fix b
assume r[rule_format]: "\<forall>b. b \<in> carrier G \<and> properfactor G b a \<longrightarrow> b \<in> Units G"
and bcarr[simp]: "b \<in> carrier G"
assume "properfactor G b a'"
also note aa'[symmetric]
finally
have "properfactor G b a" by simp

with bcarr
show "b \<in> Units G" by (fast intro: r)
qed

lemma (in monoid) irreducible_prod_rI:
assumes airr: "irreducible G a"
and bunit: "b \<in> Units G"
and carr[simp]: "a \<in> carrier G"  "b \<in> carrier G"
shows "irreducible G (a \<otimes> b)"
using airr carr bunit
apply (elim irreducibleE, intro irreducibleI, clarify)
apply (subgoal_tac "a \<in> Units G", simp)
apply (intro prod_unit_r[of a b] carr bunit, assumption)
proof -
fix c
assume [simp]: "c \<in> carrier G"
and r[rule_format]: "\<forall>b. b \<in> carrier G \<and> properfactor G b a \<longrightarrow> b \<in> Units G"
assume "properfactor G c (a \<otimes> b)"
also have "a \<otimes> b \<sim> a" by (intro associatedI2[OF bunit], simp+)
finally
have pfa: "properfactor G c a" by simp
show "c \<in> Units G" by (rule r, simp add: pfa)
qed

lemma (in comm_monoid) irreducible_prod_lI:
assumes birr: "irreducible G b"
and aunit: "a \<in> Units G"
and carr [simp]: "a \<in> carrier G"  "b \<in> carrier G"
shows "irreducible G (a \<otimes> b)"
apply (subst m_comm, simp+)
apply (intro irreducible_prod_rI assms)
done

lemma (in comm_monoid_cancel) irreducible_prodE [elim]:
assumes irr: "irreducible G (a \<otimes> b)"
and carr[simp]: "a \<in> carrier G"  "b \<in> carrier G"
and e1: "\<lbrakk>irreducible G a; b \<in> Units G\<rbrakk> \<Longrightarrow> P"
and e2: "\<lbrakk>a \<in> Units G; irreducible G b\<rbrakk> \<Longrightarrow> P"
shows "P"
using irr
proof (elim irreducibleE)
assume abnunit: "a \<otimes> b \<notin> Units G"
and isunit[rule_format]: "\<forall>ba. ba \<in> carrier G \<and> properfactor G ba (a \<otimes> b) \<longrightarrow> ba \<in> Units G"

show "P"
proof (cases "a \<in> Units G")
assume aunit: "a \<in>  Units G"

have "irreducible G b"
apply (rule irreducibleI)
proof (rule ccontr, simp)
assume "b \<in> Units G"
with aunit have "(a \<otimes> b) \<in> Units G" by fast
with abnunit show "False" ..
next
fix c
assume ccarr: "c \<in> carrier G"
and "properfactor G c b"
hence "properfactor G c (a \<otimes> b)" by (simp add: properfactor_prod_l[of c b a])
from ccarr this show "c \<in> Units G" by (fast intro: isunit)
qed

from aunit this show "P" by (rule e2)
next
assume anunit: "a \<notin> Units G"
with carr have "properfactor G b (b \<otimes> a)" by (fast intro: properfactorI3)
hence bf: "properfactor G b (a \<otimes> b)" by (subst m_comm[of a b], simp+)
hence bunit: "b \<in> Units G" by (intro isunit, simp)

have "irreducible G a"
apply (rule irreducibleI)
proof (rule ccontr, simp)
assume "a \<in> Units G"
with bunit have "(a \<otimes> b) \<in> Units G" by fast
with abnunit show "False" ..
next
fix c
assume ccarr: "c \<in> carrier G"
and "properfactor G c a"
hence "properfactor G c (a \<otimes> b)" by (simp add: properfactor_prod_r[of c a b])
from ccarr this show "c \<in> Units G" by (fast intro: isunit)
qed

from this bunit show "P" by (rule e1)
qed
qed

subsubsection {* Prime elements *}

lemma primeI:
fixes G (structure)
assumes "p \<notin> Units G"
and "\<And>a b. \<lbrakk>a \<in> carrier G; b \<in> carrier G; p divides (a \<otimes> b)\<rbrakk> \<Longrightarrow> p divides a \<or> p divides b"
shows "prime G p"
using assms
unfolding prime_def
by blast

lemma primeE:
fixes G (structure)
assumes pprime: "prime G p"
and e: "\<lbrakk>p \<notin> Units G; \<forall>a\<in>carrier G. \<forall>b\<in>carrier G.
p divides a \<otimes> b \<longrightarrow> p divides a \<or> p divides b\<rbrakk> \<Longrightarrow> P"
shows "P"
using pprime
unfolding prime_def
by (blast dest: e)

lemma (in comm_monoid_cancel) prime_divides:
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"
and pprime: "prime G p"
and pdvd: "p divides a \<otimes> b"
shows "p divides a \<or> p divides b"
using assms
by (blast elim: primeE)

lemma (in monoid_cancel) prime_cong [trans]:
assumes pprime: "prime G p"
and pp': "p \<sim> p'"
and carr[simp]: "p \<in> carrier G"  "p' \<in> carrier G"
shows "prime G p'"
using pprime
apply (elim primeE, intro primeI)
proof clarify
assume pnunit: "p \<notin> Units G"
assume "p' \<in> Units G"
also note pp'[symmetric]
finally
have "p \<in> Units G" by simp
with pnunit
show False ..
next
fix a b
assume r[rule_format]:
"\<forall>a\<in>carrier G. \<forall>b\<in>carrier G. p divides a \<otimes> b \<longrightarrow> p divides a \<or> p divides b"
assume p'dvd: "p' divides a \<otimes> b"
and carr'[simp]: "a \<in> carrier G"  "b \<in> carrier G"

note pp'
also note p'dvd
finally
have "p divides a \<otimes> b" by simp
hence "p divides a \<or> p divides b" by (intro r, simp+)
moreover {
note pp'[symmetric]
also assume "p divides a"
finally
have "p' divides a" by simp
hence "p' divides a \<or> p' divides b" by simp
}
moreover {
note pp'[symmetric]
also assume "p divides b"
finally
have "p' divides b" by simp
hence "p' divides a \<or> p' divides b" by simp
}
ultimately
show "p' divides a \<or> p' divides b" by fast
qed

subsection {* Factorization and Factorial Monoids *}

subsubsection {* Function definitions *}

constdefs (structure G)
factors :: "[_, 'a list, 'a] \<Rightarrow> bool"
"factors G fs a == (\<forall>x \<in> (set fs). irreducible G x) \<and> foldr (op \<otimes>) fs \<one> = a"

wfactors ::"[_, 'a list, 'a] \<Rightarrow> bool"
"wfactors G fs a == (\<forall>x \<in> (set fs). irreducible G x) \<and> foldr (op \<otimes>) fs \<one> \<sim> a"

abbreviation
list_assoc :: "('a,_) monoid_scheme \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "[\<sim>]\<index>" 44) where
"list_assoc G == list_all2 (op \<sim>\<^bsub>G\<^esub>)"

constdefs (structure G)
essentially_equal :: "[_, 'a list, 'a list] \<Rightarrow> bool"
"essentially_equal G fs1 fs2 == (\<exists>fs1'. fs1 <~~> fs1' \<and> fs1' [\<sim>] fs2)"

locale factorial_monoid = comm_monoid_cancel +
assumes factors_exist:
"\<lbrakk>a \<in> carrier G; a \<notin> Units G\<rbrakk> \<Longrightarrow> \<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs a"
and factors_unique:
"\<lbrakk>factors G fs a; factors G fs' a; a \<in> carrier G; a \<notin> Units G;
set fs \<subseteq> carrier G; set fs' \<subseteq> carrier G\<rbrakk> \<Longrightarrow> essentially_equal G fs fs'"

subsubsection {* Comparing lists of elements *}

text {* Association on lists *}

lemma (in monoid) listassoc_refl [simp, intro]:
assumes "set as \<subseteq> carrier G"
shows "as [\<sim>] as"
using assms
by (induct as) simp+

lemma (in monoid) listassoc_sym [sym]:
assumes "as [\<sim>] bs"
and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G"
shows "bs [\<sim>] as"
using assms
proof (induct as arbitrary: bs, simp)
case Cons
thus ?case
apply (induct bs, simp)
apply clarsimp
apply (iprover intro: associated_sym)
done
qed

lemma (in monoid) listassoc_trans [trans]:
assumes "as [\<sim>] bs" and "bs [\<sim>] cs"
and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G" and "set cs \<subseteq> carrier G"
shows "as [\<sim>] cs"
using assms
apply (simp add: list_all2_conv_all_nth set_conv_nth, safe)
apply (rule associated_trans)
apply (subgoal_tac "as ! i \<sim> bs ! i", assumption)
apply (simp, simp)
apply blast+
done

lemma (in monoid_cancel) irrlist_listassoc_cong:
assumes "\<forall>a\<in>set as. irreducible G a"
and "as [\<sim>] bs"
and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G"
shows "\<forall>a\<in>set bs. irreducible G a"
using assms
apply (clarsimp simp add: list_all2_conv_all_nth set_conv_nth)
apply (blast intro: irreducible_cong)
done

text {* Permutations *}

lemma perm_map [intro]:
assumes p: "a <~~> b"
shows "map f a <~~> map f b"
using p
by induct auto

lemma perm_map_switch:
assumes m: "map f a = map f b" and p: "b <~~> c"
shows "\<exists>d. a <~~> d \<and> map f d = map f c"
using p m
by (induct arbitrary: a) (simp, force, force, blast)

lemma (in monoid) perm_assoc_switch:
assumes a:"as [\<sim>] bs" and p: "bs <~~> cs"
shows "\<exists>bs'. as <~~> bs' \<and> bs' [\<sim>] cs"
using p a
apply (induct bs cs arbitrary: as, simp)
apply (clarsimp simp add: list_all2_Cons2, blast)
apply blast
apply blast
done

lemma (in monoid) perm_assoc_switch_r:
assumes p: "as <~~> bs" and a:"bs [\<sim>] cs"
shows "\<exists>bs'. as [\<sim>] bs' \<and> bs' <~~> cs"
using p a
apply (induct as bs arbitrary: cs, simp)
apply (clarsimp simp add: list_all2_Cons1, blast)
apply blast
apply blast
done

declare perm_sym [sym]

lemma perm_setP:
assumes perm: "as <~~> bs"
and as: "P (set as)"
shows "P (set bs)"
proof -
from perm
have "multiset_of as = multiset_of bs"
hence "set as = set bs" by (rule multiset_of_eq_setD)
with as
show "P (set bs)" by simp
qed

lemmas (in monoid) perm_closed =
perm_setP[of _ _ "\<lambda>as. as \<subseteq> carrier G"]

lemmas (in monoid) irrlist_perm_cong =
perm_setP[of _ _ "\<lambda>as. \<forall>a\<in>as. irreducible G a"]

text {* Essentially equal factorizations *}

lemma (in monoid) essentially_equalI:
assumes ex: "fs1 <~~> fs1'"  "fs1' [\<sim>] fs2"
shows "essentially_equal G fs1 fs2"
using ex
unfolding essentially_equal_def
by fast

lemma (in monoid) essentially_equalE:
assumes ee: "essentially_equal G fs1 fs2"
and e: "\<And>fs1'. \<lbrakk>fs1 <~~> fs1'; fs1' [\<sim>] fs2\<rbrakk> \<Longrightarrow> P"
shows "P"
using ee
unfolding essentially_equal_def
by (fast intro: e)

lemma (in monoid) ee_refl [simp,intro]:
assumes carr: "set as \<subseteq> carrier G"
shows "essentially_equal G as as"
using carr
by (fast intro: essentially_equalI)

lemma (in monoid) ee_sym [sym]:
assumes ee: "essentially_equal G as bs"
and carr: "set as \<subseteq> carrier G"  "set bs \<subseteq> carrier G"
shows "essentially_equal G bs as"
using ee
proof (elim essentially_equalE)
fix fs
assume "as <~~> fs"  "fs [\<sim>] bs"
hence "\<exists>fs'. as [\<sim>] fs' \<and> fs' <~~> bs" by (rule perm_assoc_switch_r)
from this obtain fs'
where a: "as [\<sim>] fs'" and p: "fs' <~~> bs"
by auto
from p have "bs <~~> fs'" by (rule perm_sym)
with a[symmetric] carr
show ?thesis
by (iprover intro: essentially_equalI perm_closed)
qed

lemma (in monoid) ee_trans [trans]:
assumes ab: "essentially_equal G as bs" and bc: "essentially_equal G bs cs"
and ascarr: "set as \<subseteq> carrier G"
and bscarr: "set bs \<subseteq> carrier G"
and cscarr: "set cs \<subseteq> carrier G"
shows "essentially_equal G as cs"
using ab bc
proof (elim essentially_equalE)
fix abs bcs
assume  "abs [\<sim>] bs" and pb: "bs <~~> bcs"
hence "\<exists>bs'. abs <~~> bs' \<and> bs' [\<sim>] bcs" by (rule perm_assoc_switch)
from this obtain bs'
where p: "abs <~~> bs'" and a: "bs' [\<sim>] bcs"
by auto

assume "as <~~> abs"
with p
have pp: "as <~~> bs'" by fast

from pp ascarr have c1: "set bs' \<subseteq> carrier G" by (rule perm_closed)
from pb bscarr have c2: "set bcs \<subseteq> carrier G" by (rule perm_closed)
note a
also assume "bcs [\<sim>] cs"
finally (listassoc_trans) have"bs' [\<sim>] cs" by (simp add: c1 c2 cscarr)

with pp
show ?thesis
by (rule essentially_equalI)
qed

subsubsection {* Properties of lists of elements *}

text {* Multiplication of factors in a list *}

lemma (in monoid) multlist_closed [simp, intro]:
assumes ascarr: "set fs \<subseteq> carrier G"
shows "foldr (op \<otimes>) fs \<one> \<in> carrier G"
by (insert ascarr, induct fs, simp+)

lemma  (in comm_monoid) multlist_dividesI (*[intro]*):
assumes "f \<in> set fs" and "f \<in> carrier G" and "set fs \<subseteq> carrier G"
shows "f divides (foldr (op \<otimes>) fs \<one>)"
using assms
apply (induct fs)
apply simp
apply (case_tac "f = a", simp)
apply (fast intro: dividesI)
apply clarsimp
apply (elim dividesE, intro dividesI)
defer 1
apply simp
done

lemma (in comm_monoid_cancel) multlist_listassoc_cong:
assumes "fs [\<sim>] fs'"
and "set fs \<subseteq> carrier G" and "set fs' \<subseteq> carrier G"
shows "foldr (op \<otimes>) fs \<one> \<sim> foldr (op \<otimes>) fs' \<one>"
using assms
proof (induct fs arbitrary: fs', simp)
case (Cons a as fs')
thus ?case
apply (induct fs', simp)
proof clarsimp
fix b bs
assume "a \<sim> b"
and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G"
and ascarr: "set as \<subseteq> carrier G"
hence p: "a \<otimes> foldr op \<otimes> as \<one> \<sim> b \<otimes> foldr op \<otimes> as \<one>"
by (fast intro: mult_cong_l)
also
assume "as [\<sim>] bs"
and bscarr: "set bs \<subseteq> carrier G"
and "\<And>fs'. \<lbrakk>as [\<sim>] fs'; set fs' \<subseteq> carrier G\<rbrakk> \<Longrightarrow> foldr op \<otimes> as \<one> \<sim> foldr op \<otimes> fs' \<one>"
hence "foldr op \<otimes> as \<one> \<sim> foldr op \<otimes> bs \<one>" by simp
with ascarr bscarr bcarr
have "b \<otimes> foldr op \<otimes> as \<one> \<sim> b \<otimes> foldr op \<otimes> bs \<one>"
by (fast intro: mult_cong_r)
finally
show "a \<otimes> foldr op \<otimes> as \<one> \<sim> b \<otimes> foldr op \<otimes> bs \<one>"
by (simp add: ascarr bscarr acarr bcarr)
qed
qed

lemma (in comm_monoid) multlist_perm_cong:
assumes prm: "as <~~> bs"
and ascarr: "set as \<subseteq> carrier G"
shows "foldr (op \<otimes>) as \<one> = foldr (op \<otimes>) bs \<one>"
using prm ascarr
apply (induct, simp, clarsimp simp add: m_ac, clarsimp)
proof clarsimp
fix xs ys zs
assume "xs <~~> ys"  "set xs \<subseteq> carrier G"
hence "set ys \<subseteq> carrier G" by (rule perm_closed)
moreover assume "set ys \<subseteq> carrier G \<Longrightarrow> foldr op \<otimes> ys \<one> = foldr op \<otimes> zs \<one>"
ultimately show "foldr op \<otimes> ys \<one> = foldr op \<otimes> zs \<one>" by simp
qed

lemma (in comm_monoid_cancel) multlist_ee_cong:
assumes "essentially_equal G fs fs'"
and "set fs \<subseteq> carrier G" and "set fs' \<subseteq> carrier G"
shows "foldr (op \<otimes>) fs \<one> \<sim> foldr (op \<otimes>) fs' \<one>"
using assms
apply (elim essentially_equalE)
apply (simp add: multlist_perm_cong multlist_listassoc_cong perm_closed)
done

subsubsection {* Factorization in irreducible elements *}

lemma wfactorsI:
fixes G (structure)
assumes "\<forall>f\<in>set fs. irreducible G f"
and "foldr (op \<otimes>) fs \<one> \<sim> a"
shows "wfactors G fs a"
using assms
unfolding wfactors_def
by simp

lemma wfactorsE:
fixes G (structure)
assumes wf: "wfactors G fs a"
and e: "\<lbrakk>\<forall>f\<in>set fs. irreducible G f; foldr (op \<otimes>) fs \<one> \<sim> a\<rbrakk> \<Longrightarrow> P"
shows "P"
using wf
unfolding wfactors_def
by (fast dest: e)

lemma (in monoid) factorsI:
assumes "\<forall>f\<in>set fs. irreducible G f"
and "foldr (op \<otimes>) fs \<one> = a"
shows "factors G fs a"
using assms
unfolding factors_def
by simp

lemma factorsE:
fixes G (structure)
assumes f: "factors G fs a"
and e: "\<lbrakk>\<forall>f\<in>set fs. irreducible G f; foldr (op \<otimes>) fs \<one> = a\<rbrakk> \<Longrightarrow> P"
shows "P"
using f
unfolding factors_def

lemma (in monoid) factors_wfactors:
assumes "factors G as a" and "set as \<subseteq> carrier G"
shows "wfactors G as a"
using assms
by (blast elim: factorsE intro: wfactorsI)

lemma (in monoid) wfactors_factors:
assumes "wfactors G as a" and "set as \<subseteq> carrier G"
shows "\<exists>a'. factors G as a' \<and> a' \<sim> a"
using assms
by (blast elim: wfactorsE intro: factorsI)

lemma (in monoid) factors_closed [dest]:
assumes "factors G fs a" and "set fs \<subseteq> carrier G"
shows "a \<in> carrier G"
using assms
by (elim factorsE, clarsimp)

lemma (in monoid) nunit_factors:
assumes anunit: "a \<notin> Units G"
and fs: "factors G as a"
shows "length as > 0"
apply (insert fs, elim factorsE)
proof (cases "length as = 0")
assume "length as = 0"
hence fold: "foldr op \<otimes> as \<one> = \<one>" by force

assume "foldr op \<otimes> as \<one> = a"
with fold
have "a = \<one>" by simp
then have "a \<in> Units G" by fast
with anunit
have "False" by simp
thus ?thesis ..
qed simp

lemma (in monoid) unit_wfactors [simp]:
assumes aunit: "a \<in> Units G"
shows "wfactors G [] a"
using aunit
by (intro wfactorsI) (simp, simp add: Units_assoc)

lemma (in comm_monoid_cancel) unit_wfactors_empty:
assumes aunit: "a \<in> Units G"
and wf: "wfactors G fs a"
and carr[simp]: "set fs \<subseteq> carrier G"
shows "fs = []"
proof (rule ccontr, cases fs, simp)
fix f fs'
assume fs: "fs = f # fs'"

from carr
have fcarr[simp]: "f \<in> carrier G"
and carr'[simp]: "set fs' \<subseteq> carrier G"

from fs wf
have "irreducible G f" by (simp add: wfactors_def)
hence fnunit: "f \<notin> Units G" by (fast elim: irreducibleE)

from fs wf
have a: "f \<otimes> foldr (op \<otimes>) fs' \<one> \<sim> a" by (simp add: wfactors_def)

note aunit
also from fs wf
have a: "f \<otimes> foldr (op \<otimes>) fs' \<one> \<sim> a" by (simp add: wfactors_def)
have "a \<sim> f \<otimes> foldr (op \<otimes>) fs' \<one>"
by (simp add: Units_closed[OF aunit] a[symmetric])
finally
have "f \<otimes> foldr (op \<otimes>) fs' \<one> \<in> Units G" by simp
hence "f \<in> Units G" by (intro unit_factor[of f], simp+)

with fnunit show "False" by simp
qed

text {* Comparing wfactors *}

lemma (in comm_monoid_cancel) wfactors_listassoc_cong_l:
assumes fact: "wfactors G fs a"
and asc: "fs [\<sim>] fs'"
and carr: "a \<in> carrier G"  "set fs \<subseteq> carrier G"  "set fs' \<subseteq> carrier G"
shows "wfactors G fs' a"
using fact
apply (elim wfactorsE, intro wfactorsI)
proof -
assume "\<forall>f\<in>set fs. irreducible G f"
also note asc
finally (irrlist_listassoc_cong)
show "\<forall>f\<in>set fs'. irreducible G f" by (simp add: carr)
next
from asc[symmetric]
have "foldr op \<otimes> fs' \<one> \<sim> foldr op \<otimes> fs \<one>"
also assume "foldr op \<otimes> fs \<one> \<sim> a"
finally
show "foldr op \<otimes> fs' \<one> \<sim> a" by (simp add: carr)
qed

lemma (in comm_monoid) wfactors_perm_cong_l:
assumes "wfactors G fs a"
and "fs <~~> fs'"
and "set fs \<subseteq> carrier G"
shows "wfactors G fs' a"
using assms
apply (elim wfactorsE, intro wfactorsI)
apply (rule irrlist_perm_cong, assumption+)
done

lemma (in comm_monoid_cancel) wfactors_ee_cong_l [trans]:
assumes ee: "essentially_equal G as bs"
and bfs: "wfactors G bs b"
and carr: "b \<in> carrier G"  "set as \<subseteq> carrier G"  "set bs \<subseteq> carrier G"
shows "wfactors G as b"
using ee
proof (elim essentially_equalE)
fix fs
assume prm: "as <~~> fs"
with carr
have fscarr: "set fs \<subseteq> carrier G" by (simp add: perm_closed)

note bfs
also assume [symmetric]: "fs [\<sim>] bs"
also (wfactors_listassoc_cong_l)
note prm[symmetric]
finally (wfactors_perm_cong_l)
show "wfactors G as b" by (simp add: carr fscarr)
qed

lemma (in monoid) wfactors_cong_r [trans]:
assumes fac: "wfactors G fs a" and aa': "a \<sim> a'"
and carr[simp]: "a \<in> carrier G"  "a' \<in> carrier G"  "set fs \<subseteq> carrier G"
shows "wfactors G fs a'"
using fac
proof (elim wfactorsE, intro wfactorsI)
assume "foldr op \<otimes> fs \<one> \<sim> a" also note aa'
finally show "foldr op \<otimes> fs \<one> \<sim> a'" by simp
qed

subsubsection {* Essentially equal factorizations *}

lemma (in comm_monoid_cancel) unitfactor_ee:
assumes uunit: "u \<in> Units G"
and carr: "set as \<subseteq> carrier G"
shows "essentially_equal G (as[0 := (as!0 \<otimes> u)]) as" (is "essentially_equal G ?as' as")
using assms
apply (intro essentially_equalI[of _ ?as'], simp)
apply (cases as, simp)
apply (clarsimp, fast intro: associatedI2[of u])
done

lemma (in comm_monoid_cancel) factors_cong_unit:
assumes uunit: "u \<in> Units G" and anunit: "a \<notin> Units G"
and afs: "factors G as a"
and ascarr: "set as \<subseteq> carrier G"
shows "factors G (as[0 := (as!0 \<otimes> u)]) (a \<otimes> u)" (is "factors G ?as' ?a'")
using assms
apply (elim factorsE, clarify)
apply (cases as)
apply clarsimp
apply (elim factorsE, intro factorsI)
apply (clarsimp, fast intro: irreducible_prod_rI)
done

lemma (in comm_monoid) perm_wfactorsD:
assumes prm: "as <~~> bs"
and afs: "wfactors G as a" and bfs: "wfactors G bs b"
and [simp]: "a \<in> carrier G"  "b \<in> carrier G"
and ascarr[simp]: "set as \<subseteq> carrier G"
shows "a \<sim> b"
using afs bfs
proof (elim wfactorsE)
from prm have [simp]: "set bs \<subseteq> carrier G" by (simp add: perm_closed)
assume "foldr op \<otimes> as \<one> \<sim> a"
hence "a \<sim> foldr op \<otimes> as \<one>" by (rule associated_sym, simp+)
also from prm
have "foldr op \<otimes> as \<one> = foldr op \<otimes> bs \<one>" by (rule multlist_perm_cong, simp)
also assume "foldr op \<otimes> bs \<one> \<sim> b"
finally
show "a \<sim> b" by simp
qed

lemma (in comm_monoid_cancel) listassoc_wfactorsD:
assumes assoc: "as [\<sim>] bs"
and afs: "wfactors G as a" and bfs: "wfactors G bs b"
and [simp]: "a \<in> carrier G"  "b \<in> carrier G"
and [simp]: "set as \<subseteq> carrier G"  "set bs \<subseteq> carrier G"
shows "a \<sim> b"
using afs bfs
proof (elim wfactorsE)
assume "foldr op \<otimes> as \<one> \<sim> a"
hence "a \<sim> foldr op \<otimes> as \<one>" by (rule associated_sym, simp+)
also from assoc
have "foldr op \<otimes> as \<one> \<sim> foldr op \<otimes> bs \<one>" by (rule multlist_listassoc_cong, simp+)
also assume "foldr op \<otimes> bs \<one> \<sim> b"
finally
show "a \<sim> b" by simp
qed

lemma (in comm_monoid_cancel) ee_wfactorsD:
assumes ee: "essentially_equal G as bs"
and afs: "wfactors G as a" and bfs: "wfactors G bs b"
and [simp]: "a \<in> carrier G"  "b \<in> carrier G"
and ascarr[simp]: "set as \<subseteq> carrier G" and bscarr[simp]: "set bs \<subseteq> carrier G"
shows "a \<sim> b"
using ee
proof (elim essentially_equalE)
fix fs
assume prm: "as <~~> fs"
hence as'carr[simp]: "set fs \<subseteq> carrier G" by (simp add: perm_closed)
from afs prm
have afs': "wfactors G fs a" by (rule wfactors_perm_cong_l, simp)
assume "fs [\<sim>] bs"
from this afs' bfs
show "a \<sim> b" by (rule listassoc_wfactorsD, simp+)
qed

lemma (in comm_monoid_cancel) ee_factorsD:
assumes ee: "essentially_equal G as bs"
and afs: "factors G as a" and bfs:"factors G bs b"
and "set as \<subseteq> carrier G"  "set bs \<subseteq> carrier G"
shows "a \<sim> b"
using assms
by (blast intro: factors_wfactors dest: ee_wfactorsD)

lemma (in factorial_monoid) ee_factorsI:
assumes ab: "a \<sim> b"
and afs: "factors G as a" and anunit: "a \<notin> Units G"
and bfs: "factors G bs b" and bnunit: "b \<notin> Units G"
and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G"
shows "essentially_equal G as bs"
proof -
note carr[simp] = factors_closed[OF afs ascarr] ascarr[THEN subsetD]
factors_closed[OF bfs bscarr] bscarr[THEN subsetD]

from ab carr
have "\<exists>u\<in>Units G. a = b \<otimes> u" by (fast elim: associatedE2)
from this obtain u
where uunit: "u \<in> Units G"
and a: "a = b \<otimes> u" by auto

from uunit bscarr
have ee: "essentially_equal G (bs[0 := (bs!0 \<otimes> u)]) bs"
(is "essentially_equal G ?bs' bs")
by (rule unitfactor_ee)

from bscarr uunit
have bs'carr: "set ?bs' \<subseteq> carrier G"
by (cases bs) (simp add: Units_closed)+

from uunit bnunit bfs bscarr
have fac: "factors G ?bs' (b \<otimes> u)"
by (rule factors_cong_unit)

from afs fac[simplified a[symmetric]] ascarr bs'carr anunit
have "essentially_equal G as ?bs'"
by (blast intro: factors_unique)
also note ee
finally
show "essentially_equal G as bs" by (simp add: ascarr bscarr bs'carr)
qed

lemma (in factorial_monoid) ee_wfactorsI:
assumes asc: "a \<sim> b"
and asf: "wfactors G as a" and bsf: "wfactors G bs b"
and acarr[simp]: "a \<in> carrier G" and bcarr[simp]: "b \<in> carrier G"
and ascarr[simp]: "set as \<subseteq> carrier G" and bscarr[simp]: "set bs \<subseteq> carrier G"
shows "essentially_equal G as bs"
using assms
proof (cases "a \<in> Units G")
assume aunit: "a \<in> Units G"
also note asc
finally have bunit: "b \<in> Units G" by simp

from aunit asf ascarr
have e: "as = []" by (rule unit_wfactors_empty)
from bunit bsf bscarr
have e': "bs = []" by (rule unit_wfactors_empty)

have "essentially_equal G [] []"
by (fast intro: essentially_equalI)
thus ?thesis by (simp add: e e')
next
assume anunit: "a \<notin> Units G"
have bnunit: "b \<notin> Units G"
proof clarify
assume "b \<in> Units G"
also note asc[symmetric]
finally have "a \<in> Units G" by simp
with anunit
show "False" ..
qed

have "\<exists>a'. factors G as a' \<and> a' \<sim> a" by (rule wfactors_factors[OF asf ascarr])
from this obtain a'
where fa': "factors G as a'"
and a': "a' \<sim> a"
by auto
from fa' ascarr
have a'carr[simp]: "a' \<in> carrier G" by fast

have a'nunit: "a' \<notin> Units G"
proof (clarify)
assume "a' \<in> Units G"
also note a'
finally have "a \<in> Units G" by simp
with anunit
show "False" ..
qed

have "\<exists>b'. factors G bs b' \<and> b' \<sim> b" by (rule wfactors_factors[OF bsf bscarr])
from this obtain b'
where fb': "factors G bs b'"
and b': "b' \<sim> b"
by auto
from fb' bscarr
have b'carr[simp]: "b' \<in> carrier G" by fast

have b'nunit: "b' \<notin> Units G"
proof (clarify)
assume "b' \<in> Units G"
also note b'
finally have "b \<in> Units G" by simp
with bnunit
show "False" ..
qed

note a'
also note asc
also note b'[symmetric]
finally
have "a' \<sim> b'" by simp

from this fa' a'nunit fb' b'nunit ascarr bscarr
show "essentially_equal G as bs"
by (rule ee_factorsI)
qed

lemma (in factorial_monoid) ee_wfactors:
assumes asf: "wfactors G as a"
and bsf: "wfactors G bs b"
and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G"
and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G"
shows asc: "a \<sim> b = essentially_equal G as bs"
using assms
by (fast intro: ee_wfactorsI ee_wfactorsD)

lemma (in factorial_monoid) wfactors_exist [intro, simp]:
assumes acarr[simp]: "a \<in> carrier G"
shows "\<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs a"
proof (cases "a \<in> Units G")
assume "a \<in> Units G"
hence "wfactors G [] a" by (rule unit_wfactors)
thus ?thesis by (intro exI) force
next
assume "a \<notin> Units G"
hence "\<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs a" by (intro factors_exist acarr)
from this obtain fs
where fscarr: "set fs \<subseteq> carrier G"
and f: "factors G fs a"
by auto
from f have "wfactors G fs a" by (rule factors_wfactors) fact
from fscarr this
show ?thesis by fast
qed

lemma (in monoid) wfactors_prod_exists [intro, simp]:
assumes "\<forall>a \<in> set as. irreducible G a" and "set as \<subseteq> carrier G"
shows "\<exists>a. a \<in> carrier G \<and> wfactors G as a"
unfolding wfactors_def
using assms
by blast

lemma (in factorial_monoid) wfactors_unique:
assumes "wfactors G fs a" and "wfactors G fs' a"
and "a \<in> carrier G"
and "set fs \<subseteq> carrier G" and "set fs' \<subseteq> carrier G"
shows "essentially_equal G fs fs'"
using assms
by (fast intro: ee_wfactorsI[of a a])

lemma (in monoid) factors_mult_single:
assumes "irreducible G a" and "factors G fb b" and "a \<in> carrier G"
shows "factors G (a # fb) (a \<otimes> b)"
using assms
unfolding factors_def
by simp

lemma (in monoid_cancel) wfactors_mult_single:
assumes f: "irreducible G a"  "wfactors G fb b"
"a \<in> carrier G"  "b \<in> carrier G"  "set fb \<subseteq> carrier G"
shows "wfactors G (a # fb) (a \<otimes> b)"
using assms
unfolding wfactors_def

lemma (in monoid) factors_mult:
assumes factors: "factors G fa a"  "factors G fb b"
and ascarr: "set fa \<subseteq> carrier G" and bscarr:"set fb \<subseteq> carrier G"
shows "factors G (fa @ fb) (a \<otimes> b)"
using assms
unfolding factors_def
apply (safe, force)
apply (induct fa)
apply simp
done

lemma (in comm_monoid_cancel) wfactors_mult [intro]:
assumes asf: "wfactors G as a" and bsf:"wfactors G bs b"
and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G"
and ascarr: "set as \<subseteq> carrier G" and bscarr:"set bs \<subseteq> carrier G"
shows "wfactors G (as @ bs) (a \<otimes> b)"
apply (insert wfactors_factors[OF asf ascarr])
apply (insert wfactors_factors[OF bsf bscarr])
proof (clarsimp)
fix a' b'
assume asf': "factors G as a'" and a'a: "a' \<sim> a"
and bsf': "factors G bs b'" and b'b: "b' \<sim> b"
from asf' have a'carr: "a' \<in> carrier G" by (rule factors_closed) fact
from bsf' have b'carr: "b' \<in> carrier G" by (rule factors_closed) fact

note carr = acarr bcarr a'carr b'carr ascarr bscarr

from asf' bsf'
have "factors G (as @ bs) (a' \<otimes> b')" by (rule factors_mult) fact+

with carr
have abf': "wfactors G (as @ bs) (a' \<otimes> b')" by (intro factors_wfactors) simp+
also from b'b carr
have trb: "a' \<otimes> b' \<sim> a' \<otimes> b" by (intro mult_cong_r)
also from a'a carr
have tra: "a' \<otimes> b \<sim> a \<otimes> b" by (intro mult_cong_l)
finally
show "wfactors G (as @ bs) (a \<otimes> b)"
qed

lemma (in comm_monoid) factors_dividesI:
assumes "factors G fs a" and "f \<in> set fs"
and "set fs \<subseteq> carrier G"
shows "f divides a"
using assms
by (fast elim: factorsE intro: multlist_dividesI)

lemma (in comm_monoid) wfactors_dividesI:
assumes p: "wfactors G fs a"
and fscarr: "set fs \<subseteq> carrier G" and acarr: "a \<in> carrier G"
and f: "f \<in> set fs"
shows "f divides a"
apply (insert wfactors_factors[OF p fscarr], clarsimp)
proof -
fix a'
assume fsa': "factors G fs a'"
and a'a: "a' \<sim> a"
with fscarr
have a'carr: "a' \<in> carrier G" by (simp add: factors_closed)

from fsa' fscarr f
have "f divides a'" by (fast intro: factors_dividesI)
also note a'a
finally
show "f divides a" by (simp add: f fscarr[THEN subsetD] acarr a'carr)
qed

subsubsection {* Factorial monoids and wfactors *}

lemma (in comm_monoid_cancel) factorial_monoidI:
assumes wfactors_exists:
"\<And>a. a \<in> carrier G \<Longrightarrow> \<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs a"
and wfactors_unique:
"\<And>a fs fs'. \<lbrakk>a \<in> carrier G; set fs \<subseteq> carrier G; set fs' \<subseteq> carrier G;
wfactors G fs a; wfactors G fs' a\<rbrakk> \<Longrightarrow> essentially_equal G fs fs'"
shows "factorial_monoid G"
proof
fix a
assume acarr: "a \<in> carrier G" and anunit: "a \<notin> Units G"

from wfactors_exists[OF acarr]
obtain as
where ascarr: "set as \<subseteq> carrier G"
and afs: "wfactors G as a"
by auto
from afs ascarr
have "\<exists>a'. factors G as a' \<and> a' \<sim> a" by (rule wfactors_factors)
from this obtain a'
where afs': "factors G as a'"
and a'a: "a' \<sim> a"
by auto
from afs' ascarr
have a'carr: "a' \<in> carrier G" by fast
have a'nunit: "a' \<notin> Units G"
proof clarify
assume "a' \<in> Units G"
also note a'a
finally have "a \<in> Units G" by (simp add: acarr)
with anunit
show "False" ..
qed

from a'carr acarr a'a
have "\<exists>u. u \<in> Units G \<and> a' = a \<otimes> u" by (blast elim: associatedE2)
from this obtain  u
where uunit: "u \<in> Units G"
and a': "a' = a \<otimes> u"
by auto

note [simp] = acarr Units_closed[OF uunit] Units_inv_closed[OF uunit]

have "a = a \<otimes> \<one>" by simp
also have "\<dots> = a \<otimes> (u \<otimes> inv u)" by (simp add: Units_r_inv uunit)
also have "\<dots> = a' \<otimes> inv u" by (simp add: m_assoc[symmetric] a'[symmetric])
finally
have a: "a = a' \<otimes> inv u" .

from ascarr uunit
have cr: "set (as[0:=(as!0 \<otimes> inv u)]) \<subseteq> carrier G"
by (cases as, clarsimp+)

from afs' uunit a'nunit acarr ascarr
have "factors G (as[0:=(as!0 \<otimes> inv u)]) a"

with cr
show "\<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs a" by fast
qed (blast intro: factors_wfactors wfactors_unique)

subsection {* Factorizations as Multisets *}

text {* Gives useful operations like intersection *}

(* FIXME: use class_of x instead of closure_of {x} *)

abbreviation
"assocs G x == eq_closure_of (division_rel G) {x}"

constdefs (structure G)
"fmset G as \<equiv> multiset_of (map (\<lambda>a. assocs G a) as)"

text {* Helper lemmas *}

lemma (in monoid) assocs_repr_independence:
assumes "y \<in> assocs G x"
and "x \<in> carrier G"
shows "assocs G x = assocs G y"
using assms
apply safe
apply (elim closure_ofE2, intro closure_ofI2[of _ _ y])
apply (clarsimp, iprover intro: associated_trans associated_sym, simp+)
apply (elim closure_ofE2, intro closure_ofI2[of _ _ x])
apply (clarsimp, iprover intro: associated_trans, simp+)
done

lemma (in monoid) assocs_self:
assumes "x \<in> carrier G"
shows "x \<in> assocs G x"
using assms
by (fastsimp intro: closure_ofI2)

lemma (in monoid) assocs_repr_independenceD:
assumes repr: "assocs G x = assocs G y"
and ycarr: "y \<in> carrier G"
shows "y \<in> assocs G x"
unfolding repr
using ycarr
by (intro assocs_self)

lemma (in comm_monoid) assocs_assoc:
assumes "a \<in> assocs G b"
and "b \<in> carrier G"
shows "a \<sim> b"
using assms
by (elim closure_ofE2, simp)

lemmas (in comm_monoid) assocs_eqD =
assocs_repr_independenceD[THEN assocs_assoc]

subsubsection {* Comparing multisets *}

lemma (in monoid) fmset_perm_cong:
assumes prm: "as <~~> bs"
shows "fmset G as = fmset G bs"
using perm_map[OF prm]

lemma (in comm_monoid_cancel) eqc_listassoc_cong:
assumes "as [\<sim>] bs"
and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G"
shows "map (assocs G) as = map (assocs G) bs"
using assms
apply (induct as arbitrary: bs, simp)
apply (clarsimp simp add: Cons_eq_map_conv list_all2_Cons1, safe)
apply (clarsimp elim!: closure_ofE2) defer 1
apply (clarsimp elim!: closure_ofE2) defer 1
proof -
fix a x z
assume carr[simp]: "a \<in> carrier G"  "x \<in> carrier G"  "z \<in> carrier G"
assume "x \<sim> a"
also assume "a \<sim> z"
finally have "x \<sim> z" by simp
with carr
show "x \<in> assocs G z"
by (intro closure_ofI2) simp+
next
fix a x z
assume carr[simp]: "a \<in> carrier G"  "x \<in> carrier G"  "z \<in> carrier G"
assume "x \<sim> z"
also assume [symmetric]: "a \<sim> z"
finally have "x \<sim> a" by simp
with carr
show "x \<in> assocs G a"
by (intro closure_ofI2) simp+
qed

lemma (in comm_monoid_cancel) fmset_listassoc_cong:
assumes "as [\<sim>] bs"
and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G"
shows "fmset G as = fmset G bs"
using assms
unfolding fmset_def

lemma (in comm_monoid_cancel) ee_fmset:
assumes ee: "essentially_equal G as bs"
and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G"
shows "fmset G as = fmset G bs"
using ee
proof (elim essentially_equalE)
fix as'
assume prm: "as <~~> as'"
from prm ascarr
have as'carr: "set as' \<subseteq> carrier G" by (rule perm_closed)

from prm
have "fmset G as = fmset G as'" by (rule fmset_perm_cong)
also assume "as' [\<sim>] bs"
with as'carr bscarr
have "fmset G as' = fmset G bs" by (simp add: fmset_listassoc_cong)
finally
show "fmset G as = fmset G bs" .
qed

lemma (in monoid_cancel) fmset_ee__hlp_induct:
assumes prm: "cas <~~> cbs"
and cdef: "cas = map (assocs G) as"  "cbs = map (assocs G) bs"
shows "\<forall>as bs. (cas <~~> cbs \<and> cas = map (assocs G) as \<and>
cbs = map (assocs G) bs) \<longrightarrow> (\<exists>as'. as <~~> as' \<and> map (assocs G) as' = cbs)"
apply (rule perm.induct[of cas cbs], rule prm)
apply safe apply simp_all
apply force
proof -
fix ys as bs
assume p1: "map (assocs G) as <~~> ys"
and r1[rule_format]:
"\<forall>asa bs. map (assocs G) as = map (assocs G) asa \<and>
ys = map (assocs G) bs
\<longrightarrow> (\<exists>as'. asa <~~> as' \<and> map (assocs G) as' = map (assocs G) bs)"
and p2: "ys <~~> map (assocs G) bs"
and r2[rule_format]:
"\<forall>as bsa. ys = map (assocs G) as \<and>
map (assocs G) bs = map (assocs G) bsa
\<longrightarrow> (\<exists>as'. as <~~> as' \<and> map (assocs G) as' = map (assocs G) bsa)"
and p3: "map (assocs G) as <~~> map (assocs G) bs"

from p1
have "multiset_of (map (assocs G) as) = multiset_of ys"
hence setys: "set (map (assocs G) as) = set ys" by (rule multiset_of_eq_setD)

have "set (map (assocs G) as) = { assocs G x | x. x \<in> set as}" by clarsimp fast
with setys have "set ys \<subseteq> { assocs G x | x. x \<in> set as}" by simp
hence "\<exists>yy. ys = map (assocs G) yy"
apply (induct ys, simp, clarsimp)
proof -
fix yy x
show "\<exists>yya. (assocs G x) # map (assocs G) yy =
map (assocs G) yya"
by (rule exI[of _ "x#yy"], simp)
qed
from this obtain yy
where ys: "ys = map (assocs G) yy"
by auto

from p1 ys
have "\<exists>as'. as <~~> as' \<and> map (assocs G) as' = map (assocs G) yy"
by (intro r1, simp)
from this obtain as'
where asas': "as <~~> as'"
and as'yy: "map (assocs G) as' = map (assocs G) yy"
by auto

from p2 ys
have "\<exists>as'. yy <~~> as' \<and> map (assocs G) as' = map (assocs G) bs"
by (intro r2, simp)
from this obtain as''
where yyas'': "yy <~~> as''"
and as''bs: "map (assocs G) as'' = map (assocs G) bs"
by auto

from as'yy and yyas''
have "\<exists>cs. as' <~~> cs \<and> map (assocs G) cs = map (assocs G) as''"
by (rule perm_map_switch)
from this obtain cs
where as'cs: "as' <~~> cs"
and csas'': "map (assocs G) cs = map (assocs G) as''"
by auto

from asas' and as'cs
have ascs: "as <~~> cs" by fast
from csas'' and as''bs
have "map (assocs G) cs = map (assocs G) bs" by simp
from ascs and this
show "\<exists>as'. as <~~> as' \<and> map (assocs G) as' = map (assocs G) bs" by fast
qed

lemma (in comm_monoid_cancel) fmset_ee:
assumes mset: "fmset G as = fmset G bs"
and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G"
shows "essentially_equal G as bs"
proof -
from mset
have mpp: "map (assocs G) as <~~> map (assocs G) bs"

have "\<exists>cas. cas = map (assocs G) as" by simp
from this obtain cas where cas: "cas = map (assocs G) as" by simp

have "\<exists>cbs. cbs = map (assocs G) bs" by simp
from this obtain cbs where cbs: "cbs = map (assocs G) bs" by simp

from cas cbs mpp
have [rule_format]:
"\<forall>as bs. (cas <~~> cbs \<and> cas = map (assocs G) as \<and>
cbs = map (assocs G) bs)
\<longrightarrow> (\<exists>as'. as <~~> as' \<and> map (assocs G) as' = cbs)"
by (intro fmset_ee__hlp_induct, simp+)
with mpp cas cbs
have "\<exists>as'. as <~~> as' \<and> map (assocs G) as' = map (assocs G) bs"
by simp

from this obtain as'
where tp: "as <~~> as'"
and tm: "map (assocs G) as' = map (assocs G) bs"
by auto
from tm have lene: "length as' = length bs" by (rule map_eq_imp_length_eq)
from tp have "set as = set as'" by (simp add: multiset_of_eq_perm multiset_of_eq_setD)
with ascarr
have as'carr: "set as' \<subseteq> carrier G" by simp

from tm as'carr[THEN subsetD] bscarr[THEN subsetD]
have "as' [\<sim>] bs"
by (induct as' arbitrary: bs) (simp, fastsimp dest: assocs_eqD[THEN associated_sym])

from tp and this
show "essentially_equal G as bs" by (fast intro: essentially_equalI)
qed

lemma (in comm_monoid_cancel) ee_is_fmset:
assumes "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G"
shows "essentially_equal G as bs = (fmset G as = fmset G bs)"
using assms
by (fast intro: ee_fmset fmset_ee)

subsubsection {* Interpreting multisets as factorizations *}

lemma (in monoid) mset_fmsetEx:
assumes elems: "\<And>X. X \<in> set_of Cs \<Longrightarrow> \<exists>x. P x \<and> X = assocs G x"
shows "\<exists>cs. (\<forall>c \<in> set cs. P c) \<and> fmset G cs = Cs"
proof -
have "\<exists>Cs'. Cs = multiset_of Cs'"
by (rule surjE[OF surj_multiset_of], fast)
from this obtain Cs'
where Cs: "Cs = multiset_of Cs'"
by auto

have "\<exists>cs. (\<forall>c \<in> set cs. P c) \<and> multiset_of (map (assocs G) cs) = Cs"
using elems
unfolding Cs
apply (induct Cs', simp)
apply clarsimp
apply (subgoal_tac "\<exists>cs. (\<forall>x\<in>set cs. P x) \<and>
multiset_of (map (assocs G) cs) = multiset_of Cs'")
proof clarsimp
fix a Cs' cs
assume ih: "\<And>X. X = a \<or> X \<in> set Cs' \<Longrightarrow> \<exists>x. P x \<and> X = assocs G x"
and csP: "\<forall>x\<in>set cs. P x"
and mset: "multiset_of (map (assocs G) cs) = multiset_of Cs'"
from ih
have "\<exists>x. P x \<and> a = assocs G x" by fast
from this obtain c
where cP: "P c"
and a: "a = assocs G c"
by auto
from cP csP
have tP: "\<forall>x\<in>set (c#cs). P x" by simp
from mset a
have "multiset_of (map (assocs G) (c#cs)) = multiset_of Cs' + {#a#}" by simp
from tP this
show "\<exists>cs. (\<forall>x\<in>set cs. P x) \<and>
multiset_of (map (assocs G) cs) =
multiset_of Cs' + {#a#}" by fast
qed simp
thus ?thesis by (simp add: fmset_def)
qed

lemma (in monoid) mset_wfactorsEx:
assumes elems: "\<And>X. X \<in> set_of Cs
\<Longrightarrow> \<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x"
shows "\<exists>c cs. c \<in> carrier G \<and> set cs \<subseteq> carrier G \<and> wfactors G cs c \<and> fmset G cs = Cs"
proof -
have "\<exists>cs. (\<forall>c\<in>set cs. c \<in> carrier G \<and> irreducible G c) \<and> fmset G cs = Cs"
by (intro mset_fmsetEx, rule elems)
from this obtain cs
where p[rule_format]: "\<forall>c\<in>set cs. c \<in> carrier G \<and> irreducible G c"
and Cs[symmetric]: "fmset G cs = Cs"
by auto

from p
have cscarr: "set cs \<subseteq> carrier G" by fast

from p
have "\<exists>c. c \<in> carrier G \<and> wfactors G cs c"
by (intro wfactors_prod_exists) fast+
from this obtain c
where ccarr: "c \<in> carrier G"
and cfs: "wfactors G cs c"
by auto

with cscarr Cs
show ?thesis by fast
qed

subsubsection {* Multiplication on multisets *}

lemma (in factorial_monoid) mult_wfactors_fmset:
assumes afs: "wfactors G as a" and bfs: "wfactors G bs b" and cfs: "wfactors G cs (a \<otimes> b)"
and carr: "a \<in> carrier G"  "b \<in> carrier G"
"set as \<subseteq> carrier G"  "set bs \<subseteq> carrier G"  "set cs \<subseteq> carrier G"
shows "fmset G cs = fmset G as + fmset G bs"
proof -
from assms
have "wfactors G (as @ bs) (a \<otimes> b)" by (intro wfactors_mult)
with carr cfs
have "essentially_equal G cs (as@bs)" by (intro ee_wfactorsI[of "a\<otimes>b" "a\<otimes>b"], simp+)
with carr
have "fmset G cs = fmset G (as@bs)" by (intro ee_fmset, simp+)
also have "fmset G (as@bs) = fmset G as + fmset G bs" by (simp add: fmset_def)
finally show "fmset G cs = fmset G as + fmset G bs" .
qed

lemma (in factorial_monoid) mult_factors_fmset:
assumes afs: "factors G as a" and bfs: "factors G bs b" and cfs: "factors G cs (a \<otimes> b)"
and "set as \<subseteq> carrier G"  "set bs \<subseteq> carrier G"  "set cs \<subseteq> carrier G"
shows "fmset G cs = fmset G as + fmset G bs"
using assms
by (blast intro: factors_wfactors mult_wfactors_fmset)

lemma (in comm_monoid_cancel) fmset_wfactors_mult:
assumes mset: "fmset G cs = fmset G as + fmset G bs"
and carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
"set as \<subseteq> carrier G"  "set bs \<subseteq> carrier G"  "set cs \<subseteq> carrier G"
and fs: "wfactors G as a"  "wfactors G bs b"  "wfactors G cs c"
shows "c \<sim> a \<otimes> b"
proof -
from carr fs
have m: "wfactors G (as @ bs) (a \<otimes> b)" by (intro wfactors_mult)

from mset
have "fmset G cs = fmset G (as@bs)" by (simp add: fmset_def)
then have "essentially_equal G cs (as@bs)" by (rule fmset_ee) (simp add: carr)+
then show "c \<sim> a \<otimes> b" by (rule ee_wfactorsD[of "cs" "as@bs"]) (simp add: assms m)+
qed

subsubsection {* Divisibility on multisets *}

lemma (in factorial_monoid) divides_fmsubset:
assumes ab: "a divides b"
and afs: "wfactors G as a" and bfs: "wfactors G bs b"
and carr: "a \<in> carrier G"  "b \<in> carrier G"  "set as \<subseteq> carrier G"  "set bs \<subseteq> carrier G"
shows "fmset G as \<le># fmset G bs"
using ab
proof (elim dividesE)
fix c
assume ccarr: "c \<in> carrier G"
hence "\<exists>cs. set cs \<subseteq> carrier G \<and> wfactors G cs c" by (rule wfactors_exist)
from this obtain cs
where cscarr: "set cs \<subseteq> carrier G"
and cfs: "wfactors G cs c" by auto
note carr = carr ccarr cscarr

assume "b = a \<otimes> c"
with afs bfs cfs carr
have "fmset G bs = fmset G as + fmset G cs"
by (intro mult_wfactors_fmset[OF afs cfs]) simp+

thus ?thesis by simp
qed

lemma (in comm_monoid_cancel) fmsubset_divides:
assumes msubset: "fmset G as \<le># fmset G bs"
and afs: "wfactors G as a" and bfs: "wfactors G bs b"
and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G"
and ascarr: "set as \<subseteq> carrier G" and bscarr: "set bs \<subseteq> carrier G"
shows "a divides b"
proof -
from afs have airr: "\<forall>a \<in> set as. irreducible G a" by (fast elim: wfactorsE)
from bfs have birr: "\<forall>b \<in> set bs. irreducible G b" by (fast elim: wfactorsE)

have "\<exists>c cs. c \<in> carrier G \<and> set cs \<subseteq> carrier G \<and> wfactors G cs c \<and> fmset G cs = fmset G bs - fmset G as"
proof (intro mset_wfactorsEx, simp)
fix X
assume "count (fmset G as) X < count (fmset G bs) X"
hence "0 < count (fmset G bs) X" by simp
hence "X \<in> set_of (fmset G bs)" by simp
hence "X \<in> set (map (assocs G) bs)" by (simp add: fmset_def)
hence "\<exists>x. x \<in> set bs \<and> X = assocs G x" by (induct bs) auto
from this obtain x
where xbs: "x \<in> set bs"
and X: "X = assocs G x"
by auto

with bscarr have xcarr: "x \<in> carrier G" by fast
from xbs birr have xirr: "irreducible G x" by simp

from xcarr and xirr and X
show "\<exists>x. x \<in> carrier G \<and> irreducible G x \<and> X = assocs G x" by fast
qed
from this obtain c cs
where ccarr: "c \<in> carrier G"
and cscarr: "set cs \<subseteq> carrier G"
and csf: "wfactors G cs c"
and csmset: "fmset G cs = fmset G bs - fmset G as" by auto

from csmset msubset
have "fmset G bs = fmset G as + fmset G cs"
hence basc: "b \<sim> a \<otimes> c"
by (rule fmset_wfactors_mult) fact+

thus ?thesis
proof (elim associatedE2)
fix u
assume "u \<in> Units G"  "b = a \<otimes> c \<otimes> u"
with acarr ccarr
show "a divides b" by (fast intro: dividesI[of "c \<otimes> u"] m_assoc)
qed (simp add: acarr bcarr ccarr)+
qed

lemma (in factorial_monoid) divides_as_fmsubset:
assumes "wfactors G as a" and "wfactors G bs b"
and "a \<in> carrier G" and "b \<in> carrier G"
and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G"
shows "a divides b = (fmset G as \<le># fmset G bs)"
using assms
by (blast intro: divides_fmsubset fmsubset_divides)

text {* Proper factors on multisets *}

lemma (in factorial_monoid) fmset_properfactor:
assumes asubb: "fmset G as \<le># fmset G bs"
and anb: "fmset G as \<noteq> fmset G bs"
and "wfactors G as a" and "wfactors G bs b"
and "a \<in> carrier G" and "b \<in> carrier G"
and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G"
shows "properfactor G a b"
apply (rule properfactorI)
apply (rule fmsubset_divides[of as bs], fact+)
proof
assume "b divides a"
hence "fmset G bs \<le># fmset G as"
by (rule divides_fmsubset) fact+
with asubb
have "fmset G as = fmset G bs" by (simp add: mset_le_antisym)
with anb
show "False" ..
qed

lemma (in factorial_monoid) properfactor_fmset:
assumes pf: "properfactor G a b"
and "wfactors G as a" and "wfactors G bs b"
and "a \<in> carrier G" and "b \<in> carrier G"
and "set as \<subseteq> carrier G" and "set bs \<subseteq> carrier G"
shows "fmset G as \<le># fmset G bs \<and> fmset G as \<noteq> fmset G bs"
using pf
apply (elim properfactorE)
apply rule
apply (intro divides_fmsubset, assumption)
apply (rule assms)+
proof
assume bna: "\<not> b divides a"
assume "fmset G as = fmset G bs"
then have "essentially_equal G as bs" by (rule fmset_ee) fact+
hence "a \<sim> b" by (rule ee_wfactorsD[of as bs]) fact+
hence "b divides a" by (elim associatedE)
with bna
show "False" ..
qed

subsection {* Irreducible Elements are Prime *}

lemma (in factorial_monoid) irreducible_is_prime:
assumes pirr: "irreducible G p"
and pcarr: "p \<in> carrier G"
shows "prime G p"
using pirr
proof (elim irreducibleE, intro primeI)
fix a b
assume acarr: "a \<in> carrier G"  and bcarr: "b \<in> carrier G"
and pdvdab: "p divides (a \<otimes> b)"
and pnunit: "p \<notin> Units G"
assume irreduc[rule_format]:
"\<forall>b. b \<in> carrier G \<and> properfactor G b p \<longrightarrow> b \<in> Units G"
from pdvdab
have "\<exists>c\<in>carrier G. a \<otimes> b = p \<otimes> c" by (rule dividesD)
from this obtain c
where ccarr: "c \<in> carrier G"
and abpc: "a \<otimes> b = p \<otimes> c"
by auto

from acarr have "\<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs a" by (rule wfactors_exist)
from this obtain as where ascarr: "set as \<subseteq> carrier G" and afs: "wfactors G as a" by auto

from bcarr have "\<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs b" by (rule wfactors_exist)
from this obtain bs where bscarr: "set bs \<subseteq> carrier G" and bfs: "wfactors G bs b" by auto

from ccarr have "\<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs c" by (rule wfactors_exist)
from this obtain cs where cscarr: "set cs \<subseteq> carrier G" and cfs: "wfactors G cs c" by auto

note carr[simp] = pcarr acarr bcarr ccarr ascarr bscarr cscarr

from afs and bfs
have abfs: "wfactors G (as @ bs) (a \<otimes> b)" by (rule wfactors_mult) fact+

from pirr cfs
have pcfs: "wfactors G (p # cs) (p \<otimes> c)" by (rule wfactors_mult_single) fact+
with abpc
have abfs': "wfactors G (p # cs) (a \<otimes> b)" by simp

from abfs' abfs
have "essentially_equal G (p # cs) (as @ bs)"
by (rule wfactors_unique) simp+

hence "\<exists>ds. p # cs <~~> ds \<and> ds [\<sim>] (as @ bs)"
by (fast elim: essentially_equalE)
from this obtain ds
where "p # cs <~~> ds"
and dsassoc: "ds [\<sim>] (as @ bs)"
by auto

then have "p \<in> set ds"
with dsassoc
have "\<exists>p'. p' \<in> set (as@bs) \<and> p \<sim> p'"
unfolding list_all2_conv_all_nth set_conv_nth
by force

from this obtain p'
where "p' \<in> set (as@bs)"
and pp': "p \<sim> p'"
by auto

hence "p' \<in> set as \<or> p' \<in> set bs" by simp
moreover
{
assume p'elem: "p' \<in> set as"
with ascarr have [simp]: "p' \<in> carrier G" by fast

note pp'
also from afs
have "p' divides a" by (rule wfactors_dividesI) fact+
finally
have "p divides a" by simp
}
moreover
{
assume p'elem: "p' \<in> set bs"
with bscarr have [simp]: "p' \<in> carrier G" by fast

note pp'
also from bfs
have "p' divides b" by (rule wfactors_dividesI) fact+
finally
have "p divides b" by simp
}
ultimately
show "p divides a \<or> p divides b" by fast
qed

--"A version using @{const factors}, more complicated"
lemma (in factorial_monoid) factors_irreducible_is_prime:
assumes pirr: "irreducible G p"
and pcarr: "p \<in> carrier G"
shows "prime G p"
using pirr
apply (elim irreducibleE, intro primeI)
apply assumption
proof -
fix a b
assume acarr: "a \<in> carrier G"
and bcarr: "b \<in> carrier G"
and pdvdab: "p divides (a \<otimes> b)"
assume irreduc[rule_format]:
"\<forall>b. b \<in> carrier G \<and> properfactor G b p \<longrightarrow> b \<in> Units G"
from pdvdab
have "\<exists>c\<in>carrier G. a \<otimes> b = p \<otimes> c" by (rule dividesD)
from this obtain c
where ccarr: "c \<in> carrier G"
and abpc: "a \<otimes> b = p \<otimes> c"
by auto
note [simp] = pcarr acarr bcarr ccarr

show "p divides a \<or> p divides b"
proof (cases "a \<in> Units G")
assume aunit: "a \<in> Units G"

note pdvdab
also have "a \<otimes> b = b \<otimes> a" by (simp add: m_comm)
also from aunit
have bab: "b \<otimes> a \<sim> b"
by (intro associatedI2[of "a"], simp+)
finally
have "p divides b" by simp
thus "p divides a \<or> p divides b" ..
next
assume anunit: "a \<notin> Units G"

show "p divides a \<or> p divides b"
proof (cases "b \<in> Units G")
assume bunit: "b \<in> Units G"

note pdvdab
also from bunit
have baa: "a \<otimes> b \<sim> a"
by (intro associatedI2[of "b"], simp+)
finally
have "p divides a" by simp
thus "p divides a \<or> p divides b" ..
next
assume bnunit: "b \<notin> Units G"

have cnunit: "c \<notin> Units G"
proof (rule ccontr, simp)
assume cunit: "c \<in> Units G"
from bnunit
have "properfactor G a (a \<otimes> b)"
by (intro properfactorI3[of _ _ b], simp+)
also note abpc
also from cunit
have "p \<otimes> c \<sim> p"
by (intro associatedI2[of c], simp+)
finally
have "properfactor G a p" by simp

with acarr
have "a \<in> Units G" by (fast intro: irreduc)
with anunit
show "False" ..
qed

have abnunit: "a \<otimes> b \<notin> Units G"
proof clarsimp
assume abunit: "a \<otimes> b \<in> Units G"
hence "a \<in> Units G" by (rule unit_factor) fact+
with anunit
show "False" ..
qed

from acarr anunit have "\<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs a" by (rule factors_exist)
then obtain as where ascarr: "set as \<subseteq> carrier G" and afac: "factors G as a" by auto

from bcarr bnunit have "\<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs b" by (rule factors_exist)
then obtain bs where bscarr: "set bs \<subseteq> carrier G" and bfac: "factors G bs b" by auto

from ccarr cnunit have "\<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs c" by (rule factors_exist)
then obtain cs where cscarr: "set cs \<subseteq> carrier G" and cfac: "factors G cs c" by auto

note [simp] = ascarr bscarr cscarr

from afac and bfac
have abfac: "factors G (as @ bs) (a \<otimes> b)" by (rule factors_mult) fact+

from pirr cfac
have pcfac: "factors G (p # cs) (p \<otimes> c)" by (rule factors_mult_single) fact+
with abpc
have abfac': "factors G (p # cs) (a \<otimes> b)" by simp

from abfac' abfac
have "essentially_equal G (p # cs) (as @ bs)"
by (rule factors_unique) (fact | simp)+

hence "\<exists>ds. p # cs <~~> ds \<and> ds [\<sim>] (as @ bs)"
by (fast elim: essentially_equalE)
from this obtain ds
where "p # cs <~~> ds"
and dsassoc: "ds [\<sim>] (as @ bs)"
by auto

then have "p \<in> set ds"
with dsassoc
have "\<exists>p'. p' \<in> set (as@bs) \<and> p \<sim> p'"
unfolding list_all2_conv_all_nth set_conv_nth
by force

from this obtain p'
where "p' \<in> set (as@bs)"
and pp': "p \<sim> p'" by auto

hence "p' \<in> set as \<or> p' \<in> set bs" by simp
moreover
{
assume p'elem: "p' \<in> set as"
with ascarr have [simp]: "p' \<in> carrier G" by fast

note pp'
also from afac p'elem
have "p' divides a" by (rule factors_dividesI) fact+
finally
have "p divides a" by simp
}
moreover
{
assume p'elem: "p' \<in> set bs"
with bscarr have [simp]: "p' \<in> carrier G" by fast

note pp'
also from bfac
have "p' divides b" by (rule factors_dividesI) fact+
finally have "p divides b" by simp
}
ultimately
show "p divides a \<or> p divides b" by fast
qed
qed
qed

subsection {* Greatest Common Divisors and Lowest Common Multiples *}

subsubsection {* Definitions *}

constdefs (structure G)
isgcd :: "[('a,_) monoid_scheme, 'a, 'a, 'a] \<Rightarrow> bool"  ("(_ gcdof\<index> _ _)" [81,81,81] 80)
"x gcdof a b \<equiv> x divides a \<and> x divides b \<and>
(\<forall>y\<in>carrier G. (y divides a \<and> y divides b \<longrightarrow> y divides x))"

islcm :: "[_, 'a, 'a, 'a] \<Rightarrow> bool"  ("(_ lcmof\<index> _ _)" [81,81,81] 80)
"x lcmof a b \<equiv> a divides x \<and> b divides x \<and>
(\<forall>y\<in>carrier G. (a divides y \<and> b divides y \<longrightarrow> x divides y))"

constdefs (structure G)
somegcd :: "('a,_) monoid_scheme \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"
"somegcd G a b == SOME x. x \<in> carrier G \<and> x gcdof a b"

somelcm :: "('a,_) monoid_scheme \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"
"somelcm G a b == SOME x. x \<in> carrier G \<and> x lcmof a b"

constdefs (structure G)
"SomeGcd G A == inf (division_rel G) A"

locale gcd_condition_monoid = comm_monoid_cancel +
assumes gcdof_exists:
"\<lbrakk>a \<in> carrier G; b \<in> carrier G\<rbrakk> \<Longrightarrow> \<exists>c. c \<in> carrier G \<and> c gcdof a b"

locale primeness_condition_monoid = comm_monoid_cancel +
assumes irreducible_prime:
"\<lbrakk>a \<in> carrier G; irreducible G a\<rbrakk> \<Longrightarrow> prime G a"

locale divisor_chain_condition_monoid = comm_monoid_cancel +
assumes division_wellfounded:
"wf {(x, y). x \<in> carrier G \<and> y \<in> carrier G \<and> properfactor G x y}"

subsubsection {* Connections to \texttt{Lattice.thy} *}

lemma gcdof_greatestLower:
fixes G (structure)
assumes carr[simp]: "a \<in> carrier G"  "b \<in> carrier G"
shows "(x \<in> carrier G \<and> x gcdof a b) =
greatest (division_rel G) x (Lower (division_rel G) {a, b})"
unfolding isgcd_def greatest_def Lower_def elem_def
by auto

lemma lcmof_leastUpper:
fixes G (structure)
assumes carr[simp]: "a \<in> carrier G"  "b \<in> carrier G"
shows "(x \<in> carrier G \<and> x lcmof a b) =
least (division_rel G) x (Upper (division_rel G) {a, b})"
unfolding islcm_def least_def Upper_def elem_def
by auto

lemma somegcd_meet:
fixes G (structure)
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"
shows "somegcd G a b = meet (division_rel G) a b"
unfolding somegcd_def meet_def inf_def

lemma (in monoid) isgcd_divides_l:
assumes "a divides b"
and "a \<in> carrier G"  "b \<in> carrier G"
shows "a gcdof a b"
using assms
unfolding isgcd_def
by fast

lemma (in monoid) isgcd_divides_r:
assumes "b divides a"
and "a \<in> carrier G"  "b \<in> carrier G"
shows "b gcdof a b"
using assms
unfolding isgcd_def
by fast

subsubsection {* Existence of gcd and lcm *}

lemma (in factorial_monoid) gcdof_exists:
assumes acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G"
shows "\<exists>c. c \<in> carrier G \<and> c gcdof a b"
proof -
from acarr have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" by (rule wfactors_exist)
from this obtain as
where ascarr: "set as \<subseteq> carrier G"
and afs: "wfactors G as a"
by auto
from afs have airr: "\<forall>a \<in> set as. irreducible G a" by (fast elim: wfactorsE)

from bcarr have "\<exists>bs. set bs \<subseteq> carrier G \<and> wfactors G bs b" by (rule wfactors_exist)
from this obtain bs
where bscarr: "set bs \<subseteq> carrier G"
and bfs: "wfactors G bs b"
by auto
from bfs have birr: "\<forall>b \<in> set bs. irreducible G b" by (fast elim: wfactorsE)

have "\<exists>c cs. c \<in> carrier G \<and> set cs \<subseteq> carrier G \<and> wfactors G cs c \<and>
fmset G cs = fmset G as #\<inter> fmset G bs"
proof (intro mset_wfactorsEx)
fix X
assume "X \<in> set_of (fmset G as #\<inter> fmset G bs)"
hence "X \<in> set_of (fmset G as)" by (simp add: multiset_inter_def)
hence "X \<in> set (map (assocs G) as)" by (simp add: fmset_def)
hence "\<exists>x. X = assocs G x \<and> x \<in> set as" by (induct as) auto
from this obtain x
where X: "X = assocs G x"
and xas: "x \<in> set as"
by auto
with ascarr have xcarr: "x \<in> carrier G" by fast
from xas airr have xirr: "irreducible G x" by simp

from xcarr and xirr and X
show "\<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" by fast
qed

from this obtain c cs
where ccarr: "c \<in> carrier G"
and cscarr: "set cs \<subseteq> carrier G"
and csirr: "wfactors G cs c"
and csmset: "fmset G cs = fmset G as #\<inter> fmset G bs" by auto

have "c gcdof a b"
from csmset
have "fmset G cs \<le># fmset G as"
thus "c divides a" by (rule fmsubset_divides) fact+
next
from csmset
have "fmset G cs \<le># fmset G bs"
by (simp add: multiset_inter_def mset_le_def, force)
thus "c divides b" by (rule fmsubset_divides) fact+
next
fix y
assume ycarr: "y \<in> carrier G"
hence "\<exists>ys. set ys \<subseteq> carrier G \<and> wfactors G ys y" by (rule wfactors_exist)
from this obtain ys
where yscarr: "set ys \<subseteq> carrier G"
and yfs: "wfactors G ys y"
by auto

assume "y divides a"
hence ya: "fmset G ys \<le># fmset G as" by (rule divides_fmsubset) fact+

assume "y divides b"
hence yb: "fmset G ys \<le># fmset G bs" by (rule divides_fmsubset) fact+

from ya yb csmset
have "fmset G ys \<le># fmset G cs" by (simp add: mset_le_def multiset_inter_count)
thus "y divides c" by (rule fmsubset_divides) fact+
qed

with ccarr
show "\<exists>c. c \<in> carrier G \<and> c gcdof a b" by fast
qed

lemma (in factorial_monoid) lcmof_exists:
assumes acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G"
shows "\<exists>c. c \<in> carrier G \<and> c lcmof a b"
proof -
from acarr have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" by (rule wfactors_exist)
from this obtain as
where ascarr: "set as \<subseteq> carrier G"
and afs: "wfactors G as a"
by auto
from afs have airr: "\<forall>a \<in> set as. irreducible G a" by (fast elim: wfactorsE)

from bcarr have "\<exists>bs. set bs \<subseteq> carrier G \<and> wfactors G bs b" by (rule wfactors_exist)
from this obtain bs
where bscarr: "set bs \<subseteq> carrier G"
and bfs: "wfactors G bs b"
by auto
from bfs have birr: "\<forall>b \<in> set bs. irreducible G b" by (fast elim: wfactorsE)

have "\<exists>c cs. c \<in> carrier G \<and> set cs \<subseteq> carrier G \<and> wfactors G cs c \<and>
fmset G cs = (fmset G as - fmset G bs) + fmset G bs"
proof (intro mset_wfactorsEx)
fix X
assume "X \<in> set_of ((fmset G as - fmset G bs) + fmset G bs)"
hence "X \<in> set_of (fmset G as) \<or> X \<in> set_of (fmset G bs)"
by (cases "X :# fmset G bs", simp, simp)
moreover
{
assume "X \<in> set_of (fmset G as)"
hence "X \<in> set (map (assocs G) as)" by (simp add: fmset_def)
hence "\<exists>x. x \<in> set as \<and> X = assocs G x" by (induct as) auto
from this obtain x
where xas: "x \<in> set as"
and X: "X = assocs G x" by auto

with ascarr have xcarr: "x \<in> carrier G" by fast
from xas airr have xirr: "irreducible G x" by simp

from xcarr and xirr and X
have "\<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" by fast
}
moreover
{
assume "X \<in> set_of (fmset G bs)"
hence "X \<in> set (map (assocs G) bs)" by (simp add: fmset_def)
hence "\<exists>x. x \<in> set bs \<and> X = assocs G x" by (induct as) auto
from this obtain x
where xbs: "x \<in> set bs"
and X: "X = assocs G x" by auto

with bscarr have xcarr: "x \<in> carrier G" by fast
from xbs birr have xirr: "irreducible G x" by simp

from xcarr and xirr and X
have "\<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" by fast
}
ultimately
show "\<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" by fast
qed

from this obtain c cs
where ccarr: "c \<in> carrier G"
and cscarr: "set cs \<subseteq> carrier G"
and csirr: "wfactors G cs c"
and csmset: "fmset G cs = fmset G as - fmset G bs + fmset G bs" by auto

have "c lcmof a b"
from csmset have "fmset G as \<le># fmset G cs" by (simp add: mset_le_def, force)
thus "a divides c" by (rule fmsubset_divides) fact+
next
from csmset have "fmset G bs \<le># fmset G cs" by (simp add: mset_le_def)
thus "b divides c" by (rule fmsubset_divides) fact+
next
fix y
assume ycarr: "y \<in> carrier G"
hence "\<exists>ys. set ys \<subseteq> carrier G \<and> wfactors G ys y" by (rule wfactors_exist)
from this obtain ys
where yscarr: "set ys \<subseteq> carrier G"
and yfs: "wfactors G ys y"
by auto

assume "a divides y"
hence ya: "fmset G as \<le># fmset G ys" by (rule divides_fmsubset) fact+

assume "b divides y"
hence yb: "fmset G bs \<le># fmset G ys" by (rule divides_fmsubset) fact+

from ya yb csmset
have "fmset G cs \<le># fmset G ys"
apply (case_tac "count (fmset G as) a < count (fmset G bs) a")
apply simp
apply simp
done
thus "c divides y" by (rule fmsubset_divides) fact+
qed

with ccarr
show "\<exists>c. c \<in> carrier G \<and> c lcmof a b" by fast
qed

subsection {* Conditions for Factoriality *}

subsubsection {* Gcd condition *}

lemma (in gcd_condition_monoid) division_weak_lower_semilattice [simp]:
shows "weak_lower_semilattice (division_rel G)"
proof -
interpret weak_partial_order "division_rel G" ..
show ?thesis
apply (unfold_locales, simp_all)
proof -
fix x y
assume carr: "x \<in> carrier G"  "y \<in> carrier G"
hence "\<exists>z. z \<in> carrier G \<and> z gcdof x y" by (rule gcdof_exists)
from this obtain z
where zcarr: "z \<in> carrier G"
and isgcd: "z gcdof x y"
by auto
with carr
have "greatest (division_rel G) z (Lower (division_rel G) {x, y})"
by (subst gcdof_greatestLower[symmetric], simp+)
thus "\<exists>z. greatest (division_rel G) z (Lower (division_rel G) {x, y})" by fast
qed
qed

lemma (in gcd_condition_monoid) gcdof_cong_l:
assumes a'a: "a' \<sim> a"
and agcd: "a gcdof b c"
and a'carr: "a' \<in> carrier G" and carr': "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "a' gcdof b c"
proof -
note carr = a'carr carr'
interpret weak_lower_semilattice "division_rel G" by simp
have "a' \<in> carrier G \<and> a' gcdof b c"
apply (subst greatest_Lower_cong_l[of _ a])
apply (simp add: gcdof_greatestLower[symmetric] agcd carr)
done
thus ?thesis ..
qed

lemma (in gcd_condition_monoid) gcd_closed [simp]:
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"
shows "somegcd G a b \<in> carrier G"
proof -
interpret weak_lower_semilattice "division_rel G" by simp
show ?thesis
apply (rule meet_closed[simplified], fact+)
done
qed

lemma (in gcd_condition_monoid) gcd_isgcd:
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"
shows "(somegcd G a b) gcdof a b"
proof -
interpret weak_lower_semilattice "division_rel G" by simp
from carr
have "somegcd G a b \<in> carrier G \<and> (somegcd G a b) gcdof a b"
apply (subst gcdof_greatestLower, simp, simp)
apply (simp add: somegcd_meet[OF carr] meet_def)
apply (rule inf_of_two_greatest[simplified], assumption+)
done
thus "(somegcd G a b) gcdof a b" by simp
qed

lemma (in gcd_condition_monoid) gcd_exists:
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"
shows "\<exists>x\<in>carrier G. x = somegcd G a b"
proof -
interpret weak_lower_semilattice "division_rel G" by simp
show ?thesis
apply (rule meet_closed[simplified], fact+)
done
qed

lemma (in gcd_condition_monoid) gcd_divides_l:
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"
shows "(somegcd G a b) divides a"
proof -
interpret weak_lower_semilattice "division_rel G" by simp
show ?thesis
apply (rule meet_left[simplified], fact+)
done
qed

lemma (in gcd_condition_monoid) gcd_divides_r:
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"
shows "(somegcd G a b) divides b"
proof -
interpret weak_lower_semilattice "division_rel G" by simp
show ?thesis
apply (rule meet_right[simplified], fact+)
done
qed

lemma (in gcd_condition_monoid) gcd_divides:
assumes sub: "z divides x"  "z divides y"
and L: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
shows "z divides (somegcd G x y)"
proof -
interpret weak_lower_semilattice "division_rel G" by simp
show ?thesis
apply (rule meet_le[simplified], fact+)
done
qed

lemma (in gcd_condition_monoid) gcd_cong_l:
assumes xx': "x \<sim> x'"
and carr: "x \<in> carrier G"  "x' \<in> carrier G"  "y \<in> carrier G"
shows "somegcd G x y \<sim> somegcd G x' y"
proof -
interpret weak_lower_semilattice "division_rel G" by simp
show ?thesis
apply (rule meet_cong_l[simplified], fact+)
done
qed

lemma (in gcd_condition_monoid) gcd_cong_r:
assumes carr: "x \<in> carrier G"  "y \<in> carrier G"  "y' \<in> carrier G"
and yy': "y \<sim> y'"
shows "somegcd G x y \<sim> somegcd G x y'"
proof -
interpret weak_lower_semilattice "division_rel G" by simp
show ?thesis
apply (rule meet_cong_r[simplified], fact+)
done
qed

(*
lemma (in gcd_condition_monoid) asc_cong_gcd_l [intro]:
assumes carr: "b \<in> carrier G"
shows "asc_cong (\<lambda>a. somegcd G a b)"
using carr
unfolding CONG_def
by clarsimp (blast intro: gcd_cong_l)

lemma (in gcd_condition_monoid) asc_cong_gcd_r [intro]:
assumes carr: "a \<in> carrier G"
shows "asc_cong (\<lambda>b. somegcd G a b)"
using carr
unfolding CONG_def
by clarsimp (blast intro: gcd_cong_r)

lemmas (in gcd_condition_monoid) asc_cong_gcd_split [simp] =
assoc_split[OF _ asc_cong_gcd_l] assoc_split[OF _ asc_cong_gcd_r]
*)

lemma (in gcd_condition_monoid) gcdI:
assumes dvd: "a divides b"  "a divides c"
and others: "\<forall>y\<in>carrier G. y divides b \<and> y divides c \<longrightarrow> y divides a"
and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and ccarr: "c \<in> carrier G"
shows "a \<sim> somegcd G b c"
apply (rule someI2_ex)
apply (rule exI[of _ a], simp add: isgcd_def)
apply (simp add: isgcd_def assms, clarify)
apply (insert assms, blast intro: associatedI)
done

lemma (in gcd_condition_monoid) gcdI2:
assumes "a gcdof b c"
and "a \<in> carrier G" and bcarr: "b \<in> carrier G" and ccarr: "c \<in> carrier G"
shows "a \<sim> somegcd G b c"
using assms
unfolding isgcd_def
by (blast intro: gcdI)

lemma (in gcd_condition_monoid) SomeGcd_ex:
assumes "finite A"  "A \<subseteq> carrier G"  "A \<noteq> {}"
shows "\<exists>x\<in> carrier G. x = SomeGcd G A"
proof -
interpret weak_lower_semilattice "division_rel G" by simp
show ?thesis
apply (rule finite_inf_closed[simplified], fact+)
done
qed

lemma (in gcd_condition_monoid) gcd_assoc:
assumes carr: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "somegcd G (somegcd G a b) c \<sim> somegcd G a (somegcd G b c)"
proof -
interpret weak_lower_semilattice "division_rel G" by simp
show ?thesis
apply (subst (2 3) somegcd_meet, (simp add: carr)+)
apply (rule weak_meet_assoc[simplified], fact+)
done
qed

lemma (in gcd_condition_monoid) gcd_mult:
assumes acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G" and ccarr: "c \<in> carrier G"
shows "c \<otimes> somegcd G a b \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)"
proof - (* following Jacobson, Basic Algebra, p.140 *)
let ?d = "somegcd G a b"
let ?e = "somegcd G (c \<otimes> a) (c \<otimes> b)"
note carr[simp] = acarr bcarr ccarr
have dcarr: "?d \<in> carrier G" by simp
have ecarr: "?e \<in> carrier G" by simp
note carr = carr dcarr ecarr

have "?d divides a" by (simp add: gcd_divides_l)
hence cd'ca: "c \<otimes> ?d divides (c \<otimes> a)" by (simp add: divides_mult_lI)

have "?d divides b" by (simp add: gcd_divides_r)
hence cd'cb: "c \<otimes> ?d divides (c \<otimes> b)" by (simp add: divides_mult_lI)

from cd'ca cd'cb
have cd'e: "c \<otimes> ?d divides ?e"
by (rule gcd_divides) simp+

hence "\<exists>u. u \<in> carrier G \<and> ?e = c \<otimes> ?d \<otimes> u"
by (elim dividesE, fast)
from this obtain u
where ucarr[simp]: "u \<in> carrier G"
and e_cdu: "?e = c \<otimes> ?d \<otimes> u"
by auto

note carr = carr ucarr

have "?e divides c \<otimes> a" by (rule gcd_divides_l) simp+
hence "\<exists>x. x \<in> carrier G \<and> c \<otimes> a = ?e \<otimes> x"
by (elim dividesE, fast)
from this obtain x
where xcarr: "x \<in> carrier G"
and ca_ex: "c \<otimes> a = ?e \<otimes> x"
by auto
with e_cdu
have ca_cdux: "c \<otimes> a = c \<otimes> ?d \<otimes> u \<otimes> x" by simp

from ca_cdux xcarr
have "c \<otimes> a = c \<otimes> (?d \<otimes> u \<otimes> x)" by (simp add: m_assoc)
then have "a = ?d \<otimes> u \<otimes> x" by (rule l_cancel[of c a]) (simp add: xcarr)+
hence du'a: "?d \<otimes> u divides a" by (rule dividesI[OF xcarr])

have "?e divides c \<otimes> b" by (intro gcd_divides_r, simp+)
hence "\<exists>x. x \<in> carrier G \<and> c \<otimes> b = ?e \<otimes> x"
by (elim dividesE, fast)
from this obtain x
where xcarr: "x \<in> carrier G"
and cb_ex: "c \<otimes> b = ?e \<otimes> x"
by auto
with e_cdu
have cb_cdux: "c \<otimes> b = c \<otimes> ?d \<otimes> u \<otimes> x" by simp

from cb_cdux xcarr
have "c \<otimes> b = c \<otimes> (?d \<otimes> u \<otimes> x)" by (simp add: m_assoc)
with xcarr
have "b = ?d \<otimes> u \<otimes> x" by (intro l_cancel[of c b], simp+)
hence du'b: "?d \<otimes> u divides b" by (intro dividesI[OF xcarr])

from du'a du'b carr
have du'd: "?d \<otimes> u divides ?d"
by (intro gcd_divides, simp+)
hence uunit: "u \<in> Units G"
proof (elim dividesE)
fix v
assume vcarr[simp]: "v \<in> carrier G"
assume d: "?d = ?d \<otimes> u \<otimes> v"
have "?d \<otimes> \<one> = ?d \<otimes> u \<otimes> v" by simp fact
also have "?d \<otimes> u \<otimes> v = ?d \<otimes> (u \<otimes> v)" by (simp add: m_assoc)
finally have "?d \<otimes> \<one> = ?d \<otimes> (u \<otimes> v)" .
hence i2: "\<one> = u \<otimes> v" by (rule l_cancel) simp+
hence i1: "\<one> = v \<otimes> u" by (simp add: m_comm)
from vcarr i1[symmetric] i2[symmetric]
show "u \<in> Units G"
by (unfold Units_def, simp, fast)
qed

from e_cdu uunit
have "somegcd G (c \<otimes> a) (c \<otimes> b) \<sim> c \<otimes> somegcd G a b"
by (intro associatedI2[of u], simp+)
from this[symmetric]
show "c \<otimes> somegcd G a b \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)" by simp
qed

lemma (in monoid) assoc_subst:
assumes ab: "a \<sim> b"
and cP: "ALL a b. a : carrier G & b : carrier G & a \<sim> b
--> f a : carrier G & f b : carrier G & f a \<sim> f b"
and carr: "a \<in> carrier G"  "b \<in> carrier G"
shows "f a \<sim> f b"
using assms by auto

lemma (in gcd_condition_monoid) relprime_mult:
assumes abrelprime: "somegcd G a b \<sim> \<one>" and acrelprime: "somegcd G a c \<sim> \<one>"
and carr[simp]: "a \<in> carrier G"  "b \<in> carrier G"  "c \<in> carrier G"
shows "somegcd G a (b \<otimes> c) \<sim> \<one>"
proof -
have "c = c \<otimes> \<one>" by simp
also from abrelprime[symmetric]
have "\<dots> \<sim> c \<otimes> somegcd G a b"
by (rule assoc_subst) (simp add: mult_cong_r)+
also have "\<dots> \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)" by (rule gcd_mult) fact+
finally
have c: "c \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)" by simp

from carr
have a: "a \<sim> somegcd G a (c \<otimes> a)"
by (fast intro: gcdI divides_prod_l)

have "somegcd G a (b \<otimes> c) \<sim> somegcd G a (c \<otimes> b)" by (simp add: m_comm)
also from a
have "\<dots> \<sim> somegcd G (somegcd G a (c \<otimes> a)) (c \<otimes> b)"
by (rule assoc_subst) (simp add: gcd_cong_l)+
also from gcd_assoc
have "\<dots> \<sim> somegcd G a (somegcd G (c \<otimes> a) (c \<otimes> b))"
by (rule assoc_subst) simp+
also from c[symmetric]
have "\<dots> \<sim> somegcd G a c"
by (rule assoc_subst) (simp add: gcd_cong_r)+
also note acrelprime
finally
show "somegcd G a (b \<otimes> c) \<sim> \<one>" by simp
qed

lemma (in gcd_condition_monoid) primeness_condition:
"primeness_condition_monoid G"
apply unfold_locales
apply (rule primeI)
apply (elim irreducibleE, assumption)
proof -
fix p a b
assume pcarr: "p \<in> carrier G" and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G"
and pirr: "irreducible G p"
and pdvdab: "p divides a \<otimes> b"
from pirr
have pnunit: "p \<notin> Units G"
and r[rule_format]: "\<forall>b. b \<in> carrier G \<and> properfactor G b p \<longrightarrow> b \<in> Units G"
by - (fast elim: irreducibleE)+

show "p divides a \<or> p divides b"
proof (rule ccontr, clarsimp)
assume npdvda: "\<not> p divides a"
with pcarr acarr
have "\<one> \<sim> somegcd G p a"
apply (intro gcdI, simp, simp, simp)
apply (fast intro: unit_divides)
apply (fast intro: unit_divides)
apply (rule r, rule, assumption)
apply (rule properfactorI, assumption)
proof (rule ccontr, simp)
fix y
assume ycarr: "y \<in> carrier G"
assume "p divides y"
also assume "y divides a"
finally
have "p divides a" by (simp add: pcarr ycarr acarr)
with npdvda
show "False" ..
qed simp+
with pcarr acarr
have pa: "somegcd G p a \<sim> \<one>" by (fast intro: associated_sym[of "\<one>"] gcd_closed)

assume npdvdb: "\<not> p divides b"
with pcarr bcarr
have "\<one> \<sim> somegcd G p b"
apply (intro gcdI, simp, simp, simp)
apply (fast intro: unit_divides)
apply (fast intro: unit_divides)
apply (rule r, rule, assumption)
apply (rule properfactorI, assumption)
proof (rule ccontr, simp)
fix y
assume ycarr: "y \<in> carrier G"
assume "p divides y"
also assume "y divides b"
finally have "p divides b" by (simp add: pcarr ycarr bcarr)
with npdvdb
show "False" ..
qed simp+
with pcarr bcarr
have pb: "somegcd G p b \<sim> \<one>" by (fast intro: associated_sym[of "\<one>"] gcd_closed)

from pcarr acarr bcarr pdvdab
have "p gcdof p (a \<otimes> b)" by (fast intro: isgcd_divides_l)

with pcarr acarr bcarr
have "p \<sim> somegcd G p (a \<otimes> b)" by (fast intro: gcdI2)
also from pa pb pcarr acarr bcarr
have "somegcd G p (a \<otimes> b) \<sim> \<one>" by (rule relprime_mult)
finally have "p \<sim> \<one>" by (simp add: pcarr acarr bcarr)

with pcarr
have "p \<in> Units G" by (fast intro: assoc_unit_l)
with pnunit
show "False" ..
qed
qed

sublocale gcd_condition_monoid \<subseteq> primeness_condition_monoid
by (rule primeness_condition)

subsubsection {* Divisor chain condition *}

lemma (in divisor_chain_condition_monoid) wfactors_exist:
assumes acarr: "a \<in> carrier G"
shows "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a"
proof -
have r[rule_format]: "a \<in> carrier G \<longrightarrow> (\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a)"
apply (rule wf_induct[OF division_wellfounded])
proof -
fix x
assume ih: "\<forall>y. (y, x) \<in> {(x, y). x \<in> carrier G \<and> y \<in> carrier G \<and> properfactor G x y}
\<longrightarrow> y \<in> carrier G \<longrightarrow> (\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as y)"

show "x \<in> carrier G \<longrightarrow> (\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as x)"
apply clarify
apply (cases "x \<in> Units G")
apply (rule exI[of _ "[]"], simp)
apply (cases "irreducible G x")
apply (rule exI[of _ "[x]"], simp add: wfactors_def)
proof -
assume xcarr: "x \<in> carrier G"
and xnunit: "x \<notin> Units G"
and xnirr: "\<not> irreducible G x"
hence "\<exists>y. y \<in> carrier G \<and> properfactor G y x \<and> y \<notin> Units G"
apply - apply (rule ccontr, simp)
apply (subgoal_tac "irreducible G x", simp)
apply (rule irreducibleI, simp, simp)
done
from this obtain y
where ycarr: "y \<in> carrier G"
and ynunit: "y \<notin> Units G"
and pfyx: "properfactor G y x"
by auto

have ih':
"\<And>y. \<lbrakk>y \<in> carrier G; properfactor G y x\<rbrakk>
\<Longrightarrow> \<exists>as. set as \<subseteq> carrier G \<and> wfactors G as y"
by (rule ih[rule_format, simplified]) (simp add: xcarr)+

from ycarr pfyx
have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as y"
by (rule ih')
from this obtain ys
where yscarr: "set ys \<subseteq> carrier G"
and yfs: "wfactors G ys y"
by auto

from pfyx
have "y divides x"
and nyx: "\<not> y \<sim> x"
by - (fast elim: properfactorE2)+
hence "\<exists>z. z \<in> carrier G \<and> x = y \<otimes> z"
by (fast elim: dividesE)

from this obtain z
where zcarr: "z \<in> carrier G"
and x: "x = y \<otimes> z"
by auto

from zcarr ycarr
have "properfactor G z x"
apply (subst x)
apply (intro properfactorI3[of _ _ y])
done
with zcarr
have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as z"
by (rule ih')
from this obtain zs
where zscarr: "set zs \<subseteq> carrier G"
and zfs: "wfactors G zs z"
by auto

from yscarr zscarr
have xscarr: "set (ys@zs) \<subseteq> carrier G" by simp
from yfs zfs ycarr zcarr yscarr zscarr
have "wfactors G (ys@zs) (y\<otimes>z)" by (rule wfactors_mult)
hence "wfactors G (ys@zs) x" by (simp add: x)

from xscarr this
show "\<exists>xs. set xs \<subseteq> carrier G \<and> wfactors G xs x" by fast
qed
qed

from acarr
show ?thesis by (rule r)
qed

subsubsection {* Primeness condition *}

lemma (in comm_monoid_cancel) multlist_prime_pos:
assumes carr: "a \<in> carrier G"  "set as \<subseteq> carrier G"
and aprime: "prime G a"
and "a divides (foldr (op \<otimes>) as \<one>)"
shows "\<exists>i<length as. a divides (as!i)"
proof -
have r[rule_format]:
"set as \<subseteq> carrier G \<and> a divides (foldr (op \<otimes>) as \<one>)
\<longrightarrow> (\<exists>i. i < length as \<and> a divides (as!i))"
apply (induct as)
apply clarsimp defer 1
apply clarsimp defer 1
proof -
assume "a divides \<one>"
with carr
have "a \<in> Units G"
by (fast intro: divides_unit[of a \<one>])
with aprime
show "False" by (elim primeE, simp)
next
fix aa as
assume ih[rule_format]: "a divides foldr op \<otimes> as \<one> \<longrightarrow> (\<exists>i<length as. a divides as ! i)"
and carr': "aa \<in> carrier G"  "set as \<subseteq> carrier G"
and "a divides aa \<otimes> foldr op \<otimes> as \<one>"
with carr aprime
have "a divides aa \<or> a divides foldr op \<otimes> as \<one>"
by (intro prime_divides) simp+
moreover {
assume "a divides aa"
hence p1: "a divides (aa#as)!0" by simp
have "0 < Suc (length as)" by simp
with p1 have "\<exists>i<Suc (length as). a divides (aa # as) ! i" by fast
}
moreover {
assume "a divides foldr op \<otimes> as \<one>"
hence "\<exists>i. i < length as \<and> a divides as ! i" by (rule ih)
from this obtain i where "a divides as ! i" and len: "i < length as" by auto
hence p1: "a divides (aa#as) ! (Suc i)" by simp
from len have "Suc i < Suc (length as)" by simp
with p1 have "\<exists>i<Suc (length as). a divides (aa # as) ! i" by force
}
ultimately
show "\<exists>i<Suc (length as). a divides (aa # as) ! i" by fast
qed

from assms
show ?thesis
by (intro r, safe)
qed

lemma (in primeness_condition_monoid) wfactors_unique__hlp_induct:
"\<forall>a as'. a \<in> carrier G \<and> set as \<subseteq> carrier G \<and> set as' \<subseteq> carrier G \<and>
wfactors G as a \<and> wfactors G as' a \<longrightarrow> essentially_equal G as as'"
apply (induct as)
apply clarsimp defer 1
apply clarsimp defer 1
proof -
fix a as'
assume acarr: "a \<in> carrier G"
and "wfactors G [] a"
hence aunit: "a \<in> Units G"
apply (elim wfactorsE)
apply (simp, rule assoc_unit_r[of "\<one>"], simp+)
done

assume "set as' \<subseteq> carrier G"  "wfactors G as' a"
with aunit
have "as' = []"
by (intro unit_wfactors_empty[of a])
thus "essentially_equal G [] as'" by simp
next
fix a as ah as'
assume ih[rule_format]:
"\<forall>a as'. a \<in> carrier G \<and> set as' \<subseteq> carrier G \<and>
wfactors G as a \<and> wfactors G as' a \<longrightarrow> essentially_equal G as as'"
and acarr: "a \<in> carrier G" and ahcarr: "ah \<in> carrier G"
and ascarr: "set as \<subseteq> carrier G" and as'carr: "set as' \<subseteq> carrier G"
and afs: "wfactors G (ah # as) a"
and afs': "wfactors G as' a"
hence ahdvda: "ah divides a"
by (intro wfactors_dividesI[of "ah#as" "a"], simp+)
hence "\<exists>a'\<in> carrier G. a = ah \<otimes> a'" by (fast elim: dividesE)
from this obtain a'
where a'carr: "a' \<in> carrier G"
and a: "a = ah \<otimes> a'"
by auto
have a'fs: "wfactors G as a'"
apply (rule wfactorsE[OF afs], rule wfactorsI, simp)
apply (simp add: a, insert ascarr a'carr)
apply (intro assoc_l_cancel[of ah _ a'] multlist_closed ahcarr, assumption+)
done

from afs have ahirr: "irreducible G ah" by (elim wfactorsE, simp)
with ascarr have ahprime: "prime G ah" by (intro irreducible_prime ahcarr)

note carr [simp] = acarr ahcarr ascarr as'carr a'carr

note ahdvda
also from afs'
have "a divides (foldr (op \<otimes>) as' \<one>)"
by (elim wfactorsE associatedE, simp)
finally have "ah divides (foldr (op \<otimes>) as' \<one>)" by simp

with ahprime
have "\<exists>i<length as'. ah divides as'!i"
by (intro multlist_prime_pos, simp+)
from this obtain i
where len: "i<length as'" and ahdvd: "ah divides as'!i"
by auto
from afs' carr have irrasi: "irreducible G (as'!i)"
by (fast intro: nth_mem[OF len] elim: wfactorsE)
from len carr
have asicarr[simp]: "as'!i \<in> carrier G" by (unfold set_conv_nth, force)
note carr = carr asicarr

from ahdvd have "\<exists>x \<in> carrier G. as'!i = ah \<otimes> x" by (fast elim: dividesE)
from this obtain x where "x \<in> carrier G" and asi: "as'!i = ah \<otimes> x" by auto

with carr irrasi[simplified asi]
have asiah: "as'!i \<sim> ah" apply -
apply (elim irreducible_prodE[of "ah" "x"], assumption+)
apply (rule associatedI2[of x], assumption+)
apply (rule irreducibleE[OF ahirr], simp)
done

note setparts = set_take_subset[of i as'] set_drop_subset[of "Suc i" as']
note partscarr [simp] = setparts[THEN subset_trans[OF _ as'carr]]
note carr = carr partscarr

have "\<exists>aa_1. aa_1 \<in> carrier G \<and> wfactors G (take i as') aa_1"
apply (intro wfactors_prod_exists)
using setparts afs' by (fast elim: wfactorsE, simp)
from this obtain aa_1
where aa1carr: "aa_1 \<in> carrier G"
and aa1fs: "wfactors G (take i as') aa_1"
by auto

have "\<exists>aa_2. aa_2 \<in> carrier G \<and> wfactors G (drop (Suc i) as') aa_2"
apply (intro wfactors_prod_exists)
using setparts afs' by (fast elim: wfactorsE, simp)
from this obtain aa_2
where aa2carr: "aa_2 \<in> carrier G"
and aa2fs: "wfactors G (drop (Suc i) as') aa_2"
by auto

note carr = carr aa1carr[simp] aa2carr[simp]

from aa1fs aa2fs
have v1: "wfactors G (take i as' @ drop (Suc i) as') (aa_1 \<otimes> aa_2)"
by (intro wfactors_mult, simp+)
hence v1': "wfactors G (as'!i # take i as' @ drop (Suc i) as') (as'!i \<otimes> (aa_1 \<otimes> aa_2))"
apply (intro wfactors_mult_single)
using setparts afs'
by (fast intro: nth_mem[OF len] elim: wfactorsE, simp+)

from aa2carr carr aa1fs aa2fs
have "wfactors G (as'!i # drop (Suc i) as') (as'!i \<otimes> aa_2)"
apply (intro wfactors_mult_single)
apply (rule wfactorsE[OF afs'], fast intro: nth_mem[OF len])
apply (fast intro: nth_mem[OF len])
apply fast
apply fast
apply assumption
done
with len carr aa1carr aa2carr aa1fs
have v2: "wfactors G (take i as' @ as'!i # drop (Suc i) as') (aa_1 \<otimes> (as'!i \<otimes> aa_2))"
apply (intro wfactors_mult)
apply fast
apply (simp, (fast intro: nth_mem[OF len])?)+
done

from len
have as': "as' = (take i as' @ as'!i # drop (Suc i) as')"
with carr
have eer: "essentially_equal G (take i as' @ as'!i # drop (Suc i) as') as'"
by simp

with v2 afs' carr aa1carr aa2carr nth_mem[OF len]
have "aa_1 \<otimes> (as'!i \<otimes> aa_2) \<sim> a"
apply (intro ee_wfactorsD[of "take i as' @ as'!i # drop (Suc i) as'"  "as'"])
apply fast+
apply (simp, fast)
done
then
have t1: "as'!i \<otimes> (aa_1 \<otimes> aa_2) \<sim> a"
done
from carr asiah
have "ah \<otimes> (aa_1 \<otimes> aa_2) \<sim> as'!i \<otimes> (aa_1 \<otimes> aa_2)"
apply (intro mult_cong_l)
apply (fast intro: associated_sym, simp+)
done
also note t1
finally
have "ah \<otimes> (aa_1 \<otimes> aa_2) \<sim> a" by simp

with carr aa1carr aa2carr a'carr nth_mem[OF len]
have a': "aa_1 \<otimes> aa_2 \<sim> a'"
by (simp add: a, fast intro: assoc_l_cancel[of ah _ a'])

note v1
also note a'
finally have "wfactors G (take i as' @ drop (Suc i) as') a'" by simp

from a'fs this carr
have "essentially_equal G as (take i as' @ drop (Suc i) as')"
by (intro ih[of a']) simp

hence ee1: "essentially_equal G (ah # as) (ah # take i as' @ drop (Suc i) as')"
apply (elim essentially_equalE) apply (fastsimp intro: essentially_equalI)
done

from carr
have ee2: "essentially_equal G (ah # take i as' @ drop (Suc i) as')
(as' ! i # take i as' @ drop (Suc i) as')"
proof (intro essentially_equalI)
show "ah # take i as' @ drop (Suc i) as' <~~> ah # take i as' @ drop (Suc i) as'"
by simp
next show "ah # take i as' @ drop (Suc i) as' [\<sim>]
as' ! i # take i as' @ drop (Suc i) as'"
apply (simp add: asiah[symmetric] ahcarr asicarr)
done
qed

note ee1
also note ee2
also have "essentially_equal G (as' ! i # take i as' @ drop (Suc i) as')
(take i as' @ as' ! i # drop (Suc i) as')"
apply (intro essentially_equalI)
apply (subgoal_tac "as' ! i # take i as' @ drop (Suc i) as' <~~>
take i as' @ as' ! i # drop (Suc i) as'")
apply simp
apply (rule perm_append_Cons)
apply simp
done
finally
have "essentially_equal G (ah # as)
(take i as' @ as' ! i # drop (Suc i) as')" by simp

thus "essentially_equal G (ah # as) as'" by (subst as', assumption)
qed

lemma (in primeness_condition_monoid) wfactors_unique:
assumes "wfactors G as a"  "wfactors G as' a"
and "a \<in> carrier G"  "set as \<subseteq> carrier G"  "set as' \<subseteq> carrier G"
shows "essentially_equal G as as'"
apply (rule wfactors_unique__hlp_induct[rule_format, of a])
done

subsubsection {* Application to factorial monoids *}

text {* Number of factors for wellfoundedness *}

constdefs
factorcount :: "_ \<Rightarrow> 'a \<Rightarrow> nat"
"factorcount G a == THE c. (ALL as. set as \<subseteq> carrier G \<and>
wfactors G as a \<longrightarrow> c = length as)"

lemma (in monoid) ee_length:
assumes ee: "essentially_equal G as bs"
shows "length as = length bs"
apply (rule essentially_equalE[OF ee])
apply (subgoal_tac "length as = length fs1'")
done

lemma (in factorial_monoid) factorcount_exists:
assumes carr[simp]: "a \<in> carrier G"
shows "EX c. ALL as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> c = length as"
proof -
have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" by (intro wfactors_exist, simp)
from this obtain as
where ascarr[simp]: "set as \<subseteq> carrier G"
and afs: "wfactors G as a"
by (auto simp del: carr)

have "ALL as'. set as' \<subseteq> carrier G \<and> wfactors G as' a \<longrightarrow> length as = length as'"
proof clarify
fix as'
assume [simp]: "set as' \<subseteq> carrier G"
and bfs: "wfactors G as' a"
from afs bfs
have "essentially_equal G as as'"
by (intro ee_wfactorsI[of a a as as'], simp+)
thus "length as = length as'" by (rule ee_length)
qed
thus "EX c. ALL as'. set as' \<subseteq> carrier G \<and> wfactors G as' a \<longrightarrow> c = length as'" ..
qed

lemma (in factorial_monoid) factorcount_unique:
assumes afs: "wfactors G as a"
and acarr[simp]: "a \<in> carrier G" and ascarr[simp]: "set as \<subseteq> carrier G"
shows "factorcount G a = length as"
proof -
have "EX ac. ALL as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> ac = length as" by (rule factorcount_exists, simp)
from this obtain ac where
alen: "ALL as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> ac = length as"
by auto
have ac: "ac = factorcount G a"
apply (rule theI2)
apply (rule alen)
apply (elim allE[of _ "as"], rule allE[OF alen, of "as"], simp add: ascarr afs)
apply (elim allE[of _ "as"], rule allE[OF alen, of "as"], simp add: ascarr afs)
done

from ascarr afs have "ac = length as" by (iprover intro: alen[rule_format])
with ac show ?thesis by simp
qed

lemma (in factorial_monoid) divides_fcount:
assumes dvd: "a divides b"
and acarr: "a \<in> carrier G" and bcarr:"b \<in> carrier G"
shows "factorcount G a <= factorcount G b"
apply (rule dividesE[OF dvd])
proof -
fix c
from assms
have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" by fast
from this obtain as
where ascarr: "set as \<subseteq> carrier G"
and afs: "wfactors G as a"
by auto
with acarr have fca: "factorcount G a = length as" by (intro factorcount_unique)

assume ccarr: "c \<in> carrier G"
hence "\<exists>cs. set cs \<subseteq> carrier G \<and> wfactors G cs c" by fast
from this obtain cs
where cscarr: "set cs \<subseteq> carrier G"
and cfs: "wfactors G cs c"
by auto

note [simp] = acarr bcarr ccarr ascarr cscarr

assume b: "b = a \<otimes> c"
from afs cfs
have "wfactors G (as@cs) (a \<otimes> c)" by (intro wfactors_mult, simp+)
with b have "wfactors G (as@cs) b" by simp
hence "factorcount G b = length (as@cs)" by (intro factorcount_unique, simp+)
hence "factorcount G b = length as + length cs" by simp
with fca show ?thesis by simp
qed

lemma (in factorial_monoid) associated_fcount:
assumes acarr: "a \<in> carrier G" and bcarr:"b \<in> carrier G"
and asc: "a \<sim> b"
shows "factorcount G a = factorcount G b"
apply (rule associatedE[OF asc])
apply (drule divides_fcount[OF _ acarr bcarr])
apply (drule divides_fcount[OF _ bcarr acarr])
apply simp
done

lemma (in factorial_monoid) properfactor_fcount:
assumes acarr: "a \<in> carrier G" and bcarr:"b \<in> carrier G"
and pf: "properfactor G a b"
shows "factorcount G a < factorcount G b"
apply (rule properfactorE[OF pf], elim dividesE)
proof -
fix c
from assms
have "\<exists>as. set as \<subseteq> carrier G \<and> wfactors G as a" by fast
from this obtain as
where ascarr: "set as \<subseteq> carrier G"
and afs: "wfactors G as a"
by auto
with acarr have fca: "factorcount G a = length as" by (intro factorcount_unique)

assume ccarr: "c \<in> carrier G"
hence "\<exists>cs. set cs \<subseteq> carrier G \<and> wfactors G cs c" by fast
from this obtain cs
where cscarr: "set cs \<subseteq> carrier G"
and cfs: "wfactors G cs c"
by auto

assume b: "b = a \<otimes> c"

have "wfactors G (as@cs) (a \<otimes> c)" by (rule wfactors_mult) fact+
with b
have "wfactors G (as@cs) b" by simp
with ascarr cscarr bcarr
have "factorcount G b = length (as@cs)" by (simp add: factorcount_unique)
hence fcb: "factorcount G b = length as + length cs" by simp

assume nbdvda: "\<not> b divides a"
have "c \<notin> Units G"
proof (rule ccontr, simp)
assume cunit:"c \<in> Units G"

have "b \<otimes> inv c = a \<otimes> c \<otimes> inv c" by (simp add: b)
also with ccarr acarr cunit
have "\<dots> = a \<otimes> (c \<otimes> inv c)" by (fast intro: m_assoc)
also with ccarr cunit
have "\<dots> = a \<otimes> \<one>" by (simp add: Units_r_inv)
also with acarr
have "\<dots> = a" by simp
finally have "a = b \<otimes> inv c" by simp
with ccarr cunit
have "b divides a" by (fast intro: dividesI[of "inv c"])
with nbdvda show False by simp
qed

with cfs have "length cs > 0"
apply -
apply (rule ccontr, simp)
proof -
assume "wfactors G [] c"
hence "\<one> \<sim> c" by (elim wfactorsE, simp)
with ccarr
have cunit: "c \<in> Units G" by (intro assoc_unit_r[of "\<one>" "c"], simp+)
assume "c \<notin> Units G"
with cunit show "False" by simp
qed

with fca fcb show ?thesis by simp
qed

sublocale factorial_monoid \<subseteq> divisor_chain_condition_monoid
apply unfold_locales
apply (rule wfUNIVI)
apply (rule measure_induct[of "factorcount G"])
apply simp (* slow *) (*
[1]Applying congruence rule:
\<lbrakk>factorcount G y < factorcount G xa \<equiv> ?P'; ?P' \<Longrightarrow> P y \<equiv> ?Q'\<rbrakk> \<Longrightarrow> factorcount G y < factorcount G xa \<longrightarrow> P y \<equiv> ?P' \<longrightarrow> ?Q'

trace_simp_depth_limit exceeded!
*)
proof -
fix P x
assume r1[rule_format]:
"\<forall>y. (\<forall>z. z \<in> carrier G \<and> y \<in> carrier G \<and> properfactor G z y \<longrightarrow> P z) \<longrightarrow> P y"
and r2[rule_format]: "\<forall>y. factorcount G y < factorcount G x \<longrightarrow> P y"
show "P x"
apply (rule r1)
apply (rule r2)
apply (rule properfactor_fcount, simp+)
done
qed

sublocale factorial_monoid \<subseteq> primeness_condition_monoid
proof qed (rule irreducible_is_prime)

lemma (in factorial_monoid) primeness_condition:
shows "primeness_condition_monoid G"
..

lemma (in factorial_monoid) gcd_condition [simp]:
shows "gcd_condition_monoid G"
proof qed (rule gcdof_exists)

sublocale factorial_monoid \<subseteq> gcd_condition_monoid
proof qed (rule gcdof_exists)

lemma (in factorial_monoid) division_weak_lattice [simp]:
shows "weak_lattice (division_rel G)"
proof -
interpret weak_lower_semilattice "division_rel G" by simp

show "weak_lattice (division_rel G)"
apply (unfold_locales, simp_all)
proof -
fix x y
assume carr: "x \<in> carrier G"  "y \<in> carrier G"

hence "\<exists>z. z \<in> carrier G \<and> z lcmof x y" by (rule lcmof_exists)
from this obtain z
where zcarr: "z \<in> carrier G"
and isgcd: "z lcmof x y"
by auto
with carr
have "least (division_rel G) z (Upper (division_rel G) {x, y})"
thus "\<exists>z. least (division_rel G) z (Upper (division_rel G) {x, y})" by fast
qed
qed

subsection {* Factoriality Theorems *}

theorem factorial_condition_one: (* Jacobson theorem 2.21 *)
shows "(divisor_chain_condition_monoid G \<and> primeness_condition_monoid G) =
factorial_monoid G"
apply rule
proof clarify
assume dcc: "divisor_chain_condition_monoid G"
and pc: "primeness_condition_monoid G"
interpret divisor_chain_condition_monoid "G" by (rule dcc)
interpret primeness_condition_monoid "G" by (rule pc)

show "factorial_monoid G"
by (fast intro: factorial_monoidI wfactors_exist wfactors_unique)
next
assume fm: "factorial_monoid G"
interpret factorial_monoid "G" by (rule fm)
show "divisor_chain_condition_monoid G \<and> primeness_condition_monoid G"
by rule unfold_locales
qed

theorem factorial_condition_two: (* Jacobson theorem 2.22 *)
shows "(divisor_chain_condition_monoid G \<and> gcd_condition_monoid G) = factorial_monoid G"
apply rule
proof clarify
assume dcc: "divisor_chain_condition_monoid G"
and gc: "gcd_condition_monoid G"
interpret divisor_chain_condition_monoid "G" by (rule dcc)
interpret gcd_condition_monoid "G" by (rule gc)
show "factorial_monoid G"
by (simp add: factorial_condition_one[symmetric], rule, unfold_locales)
next
assume fm: "factorial_monoid G"
interpret factorial_monoid "G" by (rule fm)
show "divisor_chain_condition_monoid G \<and> gcd_condition_monoid G"
by rule unfold_locales
qed

end
```