--- a/src/HOL/Algebra/QuotRing.thy Sat Jul 21 23:25:22 2018 +0200
+++ b/src/HOL/Algebra/QuotRing.thy Sun Jul 22 13:29:24 2018 +0200
@@ -16,61 +16,34 @@
where "rcoset_mult R I A B = (\<Union>a\<in>A. \<Union>b\<in>B. I +>\<^bsub>R\<^esub> (a \<otimes>\<^bsub>R\<^esub> b))"
-text \<open>@{const "rcoset_mult"} fulfils the properties required by
- congruences\<close>
+text \<open>@{const "rcoset_mult"} fulfils the properties required by congruences\<close>
lemma (in ideal) rcoset_mult_add:
- "x \<in> carrier R \<Longrightarrow> y \<in> carrier R \<Longrightarrow> [mod I:] (I +> x) \<Otimes> (I +> y) = I +> (x \<otimes> y)"
- apply rule
- apply (rule, simp add: rcoset_mult_def, clarsimp)
- defer 1
- apply (rule, simp add: rcoset_mult_def)
- defer 1
+ assumes "x \<in> carrier R" "y \<in> carrier R"
+ shows "[mod I:] (I +> x) \<Otimes> (I +> y) = I +> (x \<otimes> y)"
proof -
- fix z x' y'
- assume carr: "x \<in> carrier R" "y \<in> carrier R"
- and x'rcos: "x' \<in> I +> x"
- and y'rcos: "y' \<in> I +> y"
- and zrcos: "z \<in> I +> x' \<otimes> y'"
-
- from x'rcos have "\<exists>h\<in>I. x' = h \<oplus> x"
- by (simp add: a_r_coset_def r_coset_def)
- then obtain hx where hxI: "hx \<in> I" and x': "x' = hx \<oplus> x"
- by fast+
-
- from y'rcos have "\<exists>h\<in>I. y' = h \<oplus> y"
- by (simp add: a_r_coset_def r_coset_def)
- then obtain hy where hyI: "hy \<in> I" and y': "y' = hy \<oplus> y"
- by fast+
-
- from zrcos have "\<exists>h\<in>I. z = h \<oplus> (x' \<otimes> y')"
- by (simp add: a_r_coset_def r_coset_def)
- then obtain hz where hzI: "hz \<in> I" and z: "z = hz \<oplus> (x' \<otimes> y')"
- by fast+
-
- note carr = carr hxI[THEN a_Hcarr] hyI[THEN a_Hcarr] hzI[THEN a_Hcarr]
-
- from z have "z = hz \<oplus> (x' \<otimes> y')" .
- also from x' y' have "\<dots> = hz \<oplus> ((hx \<oplus> x) \<otimes> (hy \<oplus> y))" by simp
- also from carr have "\<dots> = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" by algebra
- finally have z2: "z = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" .
-
- from hxI hyI hzI carr have "hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy \<in> I"
- by (simp add: I_l_closed I_r_closed)
-
- with z2 have "\<exists>h\<in>I. z = h \<oplus> x \<otimes> y" by fast
- then show "z \<in> I +> x \<otimes> y" by (simp add: a_r_coset_def r_coset_def)
-next
- fix z
- assume xcarr: "x \<in> carrier R"
- and ycarr: "y \<in> carrier R"
- and zrcos: "z \<in> I +> x \<otimes> y"
- from xcarr have xself: "x \<in> I +> x" by (intro a_rcos_self)
- from ycarr have yself: "y \<in> I +> y" by (intro a_rcos_self)
- show "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b"
- using xself and yself and zrcos by fast
+ have 1: "z \<in> I +> x \<otimes> y"
+ if x'rcos: "x' \<in> I +> x" and y'rcos: "y' \<in> I +> y" and zrcos: "z \<in> I +> x' \<otimes> y'" for z x' y'
+ proof -
+ from that
+ obtain hx hy hz where hxI: "hx \<in> I" and x': "x' = hx \<oplus> x" and hyI: "hy \<in> I" and y': "y' = hy \<oplus> y"
+ and hzI: "hz \<in> I" and z: "z = hz \<oplus> (x' \<otimes> y')"
+ by (auto simp: a_r_coset_def r_coset_def)
+ note carr = assms hxI[THEN a_Hcarr] hyI[THEN a_Hcarr] hzI[THEN a_Hcarr]
+ from z x' y' have "z = hz \<oplus> ((hx \<oplus> x) \<otimes> (hy \<oplus> y))" by simp
+ also from carr have "\<dots> = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" by algebra
+ finally have z2: "z = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" .
+ from hxI hyI hzI carr have "hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy \<in> I"
+ by (simp add: I_l_closed I_r_closed)
+ with z2 show ?thesis
+ by (auto simp add: a_r_coset_def r_coset_def)
+ qed
+ have 2: "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b" if "z \<in> I +> x \<otimes> y" for z
+ using assms a_rcos_self that by blast
+ show ?thesis
+ unfolding rcoset_mult_def using assms
+ by (auto simp: intro!: 1 2)
qed
-
subsection \<open>Quotient Ring Definition\<close>
definition FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring"
@@ -79,59 +52,30 @@
\<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, mult = rcoset_mult R I,
one = (I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>), zero = I, add = set_add R\<rparr>"
+lemmas FactRing_simps = FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric]
subsection \<open>Factorization over General Ideals\<close>
text \<open>The quotient is a ring\<close>
lemma (in ideal) quotient_is_ring: "ring (R Quot I)"
-apply (rule ringI)
- \<comment> \<open>abelian group\<close>
- apply (rule comm_group_abelian_groupI)
- apply (simp add: FactRing_def)
- apply (rule a_factorgroup_is_comm_group[unfolded A_FactGroup_def'])
- \<comment> \<open>mult monoid\<close>
- apply (rule monoidI)
- apply (simp_all add: FactRing_def A_RCOSETS_def RCOSETS_def
- a_r_coset_def[symmetric])
- \<comment> \<open>mult closed\<close>
- apply (clarify)
- apply (simp add: rcoset_mult_add, fast)
- \<comment> \<open>mult \<open>one_closed\<close>\<close>
- apply force
- \<comment> \<open>mult assoc\<close>
- apply clarify
- apply (simp add: rcoset_mult_add m_assoc)
- \<comment> \<open>mult one\<close>
- apply clarify
- apply (simp add: rcoset_mult_add)
- apply clarify
- apply (simp add: rcoset_mult_add)
- \<comment> \<open>distr\<close>
- apply clarify
- apply (simp add: rcoset_mult_add a_rcos_sum l_distr)
-apply clarify
-apply (simp add: rcoset_mult_add a_rcos_sum r_distr)
-done
+proof (rule ringI)
+ show "abelian_group (R Quot I)"
+ apply (rule comm_group_abelian_groupI)
+ apply (simp add: FactRing_def a_factorgroup_is_comm_group[unfolded A_FactGroup_def'])
+ done
+ show "Group.monoid (R Quot I)"
+ by (rule monoidI)
+ (auto simp add: FactRing_simps rcoset_mult_add m_assoc)
+qed (auto simp: FactRing_simps rcoset_mult_add a_rcos_sum l_distr r_distr)
text \<open>This is a ring homomorphism\<close>
lemma (in ideal) rcos_ring_hom: "((+>) I) \<in> ring_hom R (R Quot I)"
-apply (rule ring_hom_memI)
- apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
- apply (simp add: FactRing_def rcoset_mult_add)
- apply (simp add: FactRing_def a_rcos_sum)
-apply (simp add: FactRing_def)
-done
+ by (simp add: ring_hom_memI FactRing_def a_rcosetsI[OF a_subset] rcoset_mult_add a_rcos_sum)
lemma (in ideal) rcos_ring_hom_ring: "ring_hom_ring R (R Quot I) ((+>) I)"
-apply (rule ring_hom_ringI)
- apply (rule is_ring, rule quotient_is_ring)
- apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
- apply (simp add: FactRing_def rcoset_mult_add)
- apply (simp add: FactRing_def a_rcos_sum)
-apply (simp add: FactRing_def)
-done
+ by (simp add: local.ring_axioms quotient_is_ring rcos_ring_hom ring_hom_ringI2)
text \<open>The quotient of a cring is also commutative\<close>
lemma (in ideal) quotient_is_cring:
@@ -140,12 +84,9 @@
proof -
interpret cring R by fact
show ?thesis
- apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro)
- apply (rule quotient_is_ring)
+ apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro quotient_is_ring)
apply (rule ring.axioms[OF quotient_is_ring])
- apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric])
- apply clarify
- apply (simp add: rcoset_mult_add m_comm)
+ apply (auto simp add: FactRing_simps rcoset_mult_add m_comm)
done
qed
@@ -169,37 +110,19 @@
text \<open>The quotient ring generated by a prime ideal is a domain\<close>
lemma (in primeideal) quotient_is_domain: "domain (R Quot I)"
- apply (rule domain.intro)
- apply (rule quotient_is_cring, rule is_cring)
- apply (rule domain_axioms.intro)
- apply (simp add: FactRing_def) defer 1
- apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarify)
- apply (simp add: rcoset_mult_add) defer 1
-proof (rule ccontr, clarsimp)
- assume "I +> \<one> = I"
- then have "\<one> \<in> I" by (simp only: a_coset_join1 one_closed a_subgroup)
- then have "carrier R \<subseteq> I" by (subst one_imp_carrier, simp, fast)
- with a_subset have "I = carrier R" by fast
- with I_notcarr show False by fast
-next
- fix x y
- assume carr: "x \<in> carrier R" "y \<in> carrier R"
+proof -
+ have 1: "I +> \<one> = I \<Longrightarrow> False"
+ using I_notcarr a_rcos_self one_imp_carrier by blast
+ have 2: "I +> x = I"
+ if carr: "x \<in> carrier R" "y \<in> carrier R"
and a: "I +> x \<otimes> y = I"
- and b: "I +> y \<noteq> I"
-
- have ynI: "y \<notin> I"
- proof (rule ccontr, simp)
- assume "y \<in> I"
- then have "I +> y = I" by (rule a_rcos_const)
- with b show False by simp
- qed
-
- from carr have "x \<otimes> y \<in> I +> x \<otimes> y" by (simp add: a_rcos_self)
- then have xyI: "x \<otimes> y \<in> I" by (simp add: a)
-
- from xyI and carr have xI: "x \<in> I \<or> y \<in> I" by (simp add: I_prime)
- with ynI have "x \<in> I" by fast
- then show "I +> x = I" by (rule a_rcos_const)
+ and b: "I +> y \<noteq> I" for x y
+ by (metis I_prime a a_rcos_const a_rcos_self b m_closed that)
+ show ?thesis
+ apply (intro domain.intro quotient_is_cring is_cring domain_axioms.intro)
+ apply (metis "1" FactRing_def monoid.simps(2) ring.simps(1))
+ apply (simp add: FactRing_simps)
+ by (metis "2" rcoset_mult_add)
qed
text \<open>Generating right cosets of a prime ideal is a homomorphism
@@ -210,88 +133,58 @@
subsection \<open>Factorization over Maximal Ideals\<close>
-text \<open>In a commutative ring, the quotient ring over a maximal ideal
- is a field.
- The proof follows ``W. Adkins, S. Weintraub: Algebra --
- An Approach via Module Theory''\<close>
-lemma (in maximalideal) quotient_is_field:
+text \<open>In a commutative ring, the quotient ring over a maximal ideal is a field.
+ The proof follows ``W. Adkins, S. Weintraub: Algebra -- An Approach via Module Theory''\<close>
+proposition (in maximalideal) quotient_is_field:
assumes "cring R"
shows "field (R Quot I)"
proof -
interpret cring R by fact
- show ?thesis
- apply (intro cring.cring_fieldI2)
- apply (rule quotient_is_cring, rule is_cring)
- defer 1
- apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarsimp)
- apply (simp add: rcoset_mult_add) defer 1
- proof (rule ccontr, simp)
- \<comment> \<open>Quotient is not empty\<close>
+ have 1: "\<zero>\<^bsub>R Quot I\<^esub> \<noteq> \<one>\<^bsub>R Quot I\<^esub>" \<comment> \<open>Quotient is not empty\<close>
+ proof
assume "\<zero>\<^bsub>R Quot I\<^esub> = \<one>\<^bsub>R Quot I\<^esub>"
then have II1: "I = I +> \<one>" by (simp add: FactRing_def)
- from a_rcos_self[OF one_closed] have "\<one> \<in> I"
- by (simp add: II1[symmetric])
- then have "I = carrier R" by (rule one_imp_carrier)
+ then have "I = carrier R"
+ using a_rcos_self one_imp_carrier by blast
with I_notcarr show False by simp
- next
+ qed
+ have 2: "\<exists>y\<in>carrier R. I +> a \<otimes> y = I +> \<one>" if IanI: "I +> a \<noteq> I" and acarr: "a \<in> carrier R" for a
\<comment> \<open>Existence of Inverse\<close>
- fix a
- assume IanI: "I +> a \<noteq> I" and acarr: "a \<in> carrier R"
-
+ proof -
\<comment> \<open>Helper ideal \<open>J\<close>\<close>
define J :: "'a set" where "J = (carrier R #> a) <+> I"
have idealJ: "ideal J R"
- apply (unfold J_def, rule add_ideals)
- apply (simp only: cgenideal_eq_rcos[symmetric], rule cgenideal_ideal, rule acarr)
- apply (rule is_ideal)
- done
-
- \<comment> \<open>Showing @{term "J"} not smaller than @{term "I"}\<close>
+ using J_def acarr add_ideals cgenideal_eq_rcos cgenideal_ideal is_ideal by auto
have IinJ: "I \<subseteq> J"
- proof (rule, simp add: J_def r_coset_def set_add_defs)
+ proof (clarsimp simp: J_def r_coset_def set_add_defs)
fix x
assume xI: "x \<in> I"
- have Zcarr: "\<zero> \<in> carrier R" by fast
- from xI[THEN a_Hcarr] acarr
- have "x = \<zero> \<otimes> a \<oplus> x" by algebra
- with Zcarr and xI show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast
- qed
-
- \<comment> \<open>Showing @{term "J \<noteq> I"}\<close>
- have anI: "a \<notin> I"
- proof (rule ccontr, simp)
- assume "a \<in> I"
- then have "I +> a = I" by (rule a_rcos_const)
- with IanI show False by simp
+ have "x = \<zero> \<otimes> a \<oplus> x"
+ by (simp add: acarr xI)
+ with xI show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast
qed
-
- have aJ: "a \<in> J"
- proof (simp add: J_def r_coset_def set_add_defs)
- from acarr
- have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra
- with one_closed and additive_subgroup.zero_closed[OF is_additive_subgroup]
- show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast
+ have JnI: "J \<noteq> I"
+ proof -
+ have "a \<notin> I"
+ using IanI a_rcos_const by blast
+ moreover have "a \<in> J"
+ proof (simp add: J_def r_coset_def set_add_defs)
+ from acarr
+ have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra
+ with one_closed and additive_subgroup.zero_closed[OF is_additive_subgroup]
+ show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast
+ qed
+ ultimately show ?thesis by blast
qed
-
- from aJ and anI have JnI: "J \<noteq> I" by fast
-
- \<comment> \<open>Deducing @{term "J = carrier R"} because @{term "I"} is maximal\<close>
- from idealJ and IinJ have "J = I \<or> J = carrier R"
- proof (rule I_maximal, unfold J_def)
- have "carrier R #> a \<subseteq> carrier R"
- using subset_refl acarr by (rule r_coset_subset_G)
- then show "carrier R #> a <+> I \<subseteq> carrier R"
- using a_subset by (rule set_add_closed)
- qed
-
- with JnI have Jcarr: "J = carrier R" by simp
+ then have Jcarr: "J = carrier R"
+ using I_maximal IinJ additive_subgroup.a_subset idealJ ideal_def by blast
\<comment> \<open>Calculating an inverse for @{term "a"}\<close>
from one_closed[folded Jcarr]
- have "\<exists>r\<in>carrier R. \<exists>i\<in>I. \<one> = r \<otimes> a \<oplus> i"
- by (simp add: J_def r_coset_def set_add_defs)
- then obtain r i where rcarr: "r \<in> carrier R"
- and iI: "i \<in> I" and one: "\<one> = r \<otimes> a \<oplus> i" by fast
+ obtain r i where rcarr: "r \<in> carrier R"
+ and iI: "i \<in> I" and one: "\<one> = r \<otimes> a \<oplus> i"
+ by (auto simp add: J_def r_coset_def set_add_defs)
+
from one and rcarr and acarr and iI[THEN a_Hcarr]
have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra
@@ -305,6 +198,10 @@
from rcarr and this[symmetric]
show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast
qed
+ show ?thesis
+ apply (intro cring.cring_fieldI2 quotient_is_cring is_cring 1)
+ apply (clarsimp simp add: FactRing_simps rcoset_mult_add 2)
+ done
qed
@@ -916,7 +813,7 @@
proof -
fix X assume "X \<in> carrier (R Quot (a_kernel R S h))"
thus "\<exists>x \<in> carrier R. X = (a_kernel R S h) +> x"
- unfolding FactRing_def RCOSETS_def A_RCOSETS_def by (simp add: a_r_coset_def)
+ unfolding FactRing_simps by (simp add: a_r_coset_def)
qed
lemma (in ring_hom_ring) the_elem_wf:
@@ -989,7 +886,7 @@
by (metis image_iff)
hence "the_elem (h ` ((a_kernel R S h) +> x)) = y" by simp
moreover have "(a_kernel R S h) +> x \<in> carrier (R Quot (a_kernel R S h))"
- unfolding FactRing_def RCOSETS_def A_RCOSETS_def by (auto simp add: x a_r_coset_def)
+ unfolding FactRing_simps by (auto simp add: x a_r_coset_def)
ultimately show "y \<in> (\<lambda>X. (the_elem (h ` X))) ` carrier (R Quot (a_kernel R S h))" by blast
qed
qed