--- a/src/HOL/Algebra/QuotRing.thy Mon Sep 19 22:48:05 2011 +0200
+++ b/src/HOL/Algebra/QuotRing.thy Mon Sep 19 23:18:18 2011 +0200
@@ -10,8 +10,7 @@
subsection {* Multiplication on Cosets *}
-definition
- rcoset_mult :: "[('a, _) ring_scheme, 'a set, 'a set, 'a set] \<Rightarrow> 'a set"
+definition rcoset_mult :: "[('a, _) ring_scheme, 'a set, 'a set, 'a set] \<Rightarrow> 'a set"
("[mod _:] _ \<Otimes>\<index> _" [81,81,81] 80)
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))"
@@ -19,86 +18,71 @@
text {* @{const "rcoset_mult"} fulfils the properties required by
congruences *}
lemma (in ideal) rcoset_mult_add:
- "\<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<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
+ "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
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'"
+ 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 x'rcos
- have "\<exists>h\<in>I. x' = h \<oplus> x" by (simp add: a_r_coset_def r_coset_def)
- from this 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)
- from this
- obtain hy
- where hyI: "hy \<in> I"
- and y': "y' = hy \<oplus> y"
- 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)
- from this
- obtain hz
- where hzI: "hz \<in> I"
- and z: "z = hz \<oplus> (x' \<otimes> 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" .
+ 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)
+ 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)
- from this and z2
- have "\<exists>h\<in>I. z = h \<oplus> x \<otimes> y" by fast
- thus "z \<in> I +> x \<otimes> y" by (simp add: a_r_coset_def r_coset_def)
+ 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)
-
- from xself and yself and zrcos
- show "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b" by fast
+ 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
qed
subsection {* Quotient Ring Definition *}
-definition
- FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring" (infixl "Quot" 65)
+definition FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring"
+ (infixl "Quot" 65)
where "FactRing R I =
- \<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>"
+ \<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>"
subsection {* Factorization over General Ideals *}
text {* The quotient is a ring *}
-lemma (in ideal) quotient_is_ring:
- shows "ring (R Quot I)"
+lemma (in ideal) quotient_is_ring: "ring (R Quot I)"
apply (rule ringI)
--{* abelian group *}
apply (rule comm_group_abelian_groupI)
@@ -112,15 +96,15 @@
apply (clarify)
apply (simp add: rcoset_mult_add, fast)
--{* mult @{text one_closed} *}
- apply (force intro: one_closed)
+ apply force
--{* mult assoc *}
apply clarify
apply (simp add: rcoset_mult_add m_assoc)
--{* mult one *}
apply clarify
- apply (simp add: rcoset_mult_add l_one)
+ apply (simp add: rcoset_mult_add)
apply clarify
- apply (simp add: rcoset_mult_add r_one)
+ apply (simp add: rcoset_mult_add)
--{* distr *}
apply clarify
apply (simp add: rcoset_mult_add a_rcos_sum l_distr)
@@ -131,8 +115,7 @@
text {* This is a ring homomorphism *}
-lemma (in ideal) rcos_ring_hom:
- "(op +> I) \<in> ring_hom R (R Quot I)"
+lemma (in ideal) rcos_ring_hom: "(op +> 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)
@@ -140,8 +123,7 @@
apply (simp add: FactRing_def)
done
-lemma (in ideal) rcos_ring_hom_ring:
- "ring_hom_ring R (R Quot I) (op +> I)"
+lemma (in ideal) rcos_ring_hom_ring: "ring_hom_ring R (R Quot I) (op +> I)"
apply (rule ring_hom_ringI)
apply (rule is_ring, rule quotient_is_ring)
apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
@@ -156,13 +138,14 @@
shows "cring (R Quot I)"
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 (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)
-done
+ show ?thesis
+ apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro)
+ apply (rule 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)
+ done
qed
text {* Cosets as a ring homomorphism on crings *}
@@ -171,65 +154,57 @@
shows "ring_hom_cring R (R Quot I) (op +> I)"
proof -
interpret cring R by fact
- show ?thesis apply (rule ring_hom_cringI)
- apply (rule rcos_ring_hom_ring)
- apply (rule is_cring)
-apply (rule quotient_is_cring)
-apply (rule is_cring)
-done
+ show ?thesis
+ apply (rule ring_hom_cringI)
+ apply (rule rcos_ring_hom_ring)
+ apply (rule is_cring)
+ apply (rule quotient_is_cring)
+ apply (rule is_cring)
+ done
qed
subsection {* Factorization over Prime Ideals *}
text {* The quotient ring generated by a prime ideal is a domain *}
-lemma (in primeideal) quotient_is_domain:
- shows "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
+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"
- hence "\<one> \<in> I" by (simp only: a_coset_join1 one_closed a_subgroup)
- hence "carrier R \<subseteq> I" by (subst one_imp_carrier, simp, fast)
- from this and a_subset
- have "I = carrier R" by fast
- from this and I_notcarr
- show "False" by fast
+ 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"
- and a: "I +> x \<otimes> y = I"
- and b: "I +> y \<noteq> I"
+ 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"
- hence "I +> y = I" by (rule a_rcos_const)
- from this and b
- show "False" by simp
+ 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)
- from this
- have xyI: "x \<otimes> y \<in> I" by (simp add: a)
+ 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)
- from this and ynI
- have "x \<in> I" by fast
- thus "I +> x = I" by (rule a_rcos_const)
+ 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)
qed
text {* Generating right cosets of a prime ideal is a homomorphism
on commutative rings *}
-lemma (in primeideal) rcos_ring_hom_cring:
- shows "ring_hom_cring R (R Quot I) (op +> I)"
-by (rule rcos_ring_hom_cring, rule is_cring)
+lemma (in primeideal) rcos_ring_hom_cring: "ring_hom_cring R (R Quot I) (op +> I)"
+ by (rule rcos_ring_hom_cring) (rule is_cring)
subsection {* Factorization over Maximal Ideals *}
@@ -243,106 +218,92 @@
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)
- --{* Quotient is not empty *}
- assume "\<zero>\<^bsub>R Quot I\<^esub> = \<one>\<^bsub>R Quot I\<^esub>"
- hence 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])
- hence "I = carrier R" by (rule one_imp_carrier)
- from this and I_notcarr
- show "False" by simp
-next
- --{* Existence of Inverse *}
- fix a
- assume IanI: "I +> a \<noteq> I"
- and acarr: "a \<in> carrier R"
+ 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)
+ --{* Quotient is not empty *}
+ 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)
+ with I_notcarr show False by simp
+ next
+ --{* Existence of Inverse *}
+ fix a
+ assume IanI: "I +> a \<noteq> I" and acarr: "a \<in> carrier R"
- --{* Helper ideal @{text "J"} *}
- def J \<equiv> "(carrier R #> a) <+> I :: 'a set"
- 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
+ --{* Helper ideal @{text "J"} *}
+ def J \<equiv> "(carrier R #> a) <+> I :: 'a set"
+ 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
- --{* Showing @{term "J"} not smaller than @{term "I"} *}
- have IinJ: "I \<subseteq> J"
- proof (rule, simp add: 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
-
- from Zcarr and xI and this
- show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast
- qed
-
- --{* Showing @{term "J \<noteq> I"} *}
- have anI: "a \<notin> I"
- proof (rule ccontr, simp)
- assume "a \<in> I"
- hence "I +> a = I" by (rule a_rcos_const)
- from this and IanI
- show "False" by simp
- qed
+ --{* Showing @{term "J"} not smaller than @{term "I"} *}
+ have IinJ: "I \<subseteq> J"
+ proof (rule, simp add: 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
- 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
- from one_closed and additive_subgroup.zero_closed[OF is_additive_subgroup] and this
- show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast
- qed
-
- from aJ and anI
- have JnI: "J \<noteq> I" by fast
+ --{* Showing @{term "J \<noteq> I"} *}
+ 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
+ qed
- --{* Deducing @{term "J = carrier R"} because @{term "I"} is maximal *}
- 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)
- from this and a_subset
- show "carrier R #> a <+> I \<subseteq> carrier R" by (rule set_add_closed)
- 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
+ qed
- from this and JnI
- have Jcarr: "J = carrier R" by simp
+ from aJ and anI have JnI: "J \<noteq> I" by fast
- --{* Calculating an inverse for @{term "a"} *}
- 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)
- from this
- obtain r i
- where rcarr: "r \<in> carrier R"
- and iI: "i \<in> I"
- and one: "\<one> = r \<otimes> a \<oplus> i"
- by fast
- from one and rcarr and acarr and iI[THEN a_Hcarr]
- have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra
+ --{* Deducing @{term "J = carrier R"} because @{term "I"} is maximal *}
+ 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
- --{* Lifting to cosets *}
- from iI
- have "\<ominus>i \<oplus> \<one> \<in> I +> \<one>"
- by (intro a_rcosI, simp, intro a_subset, simp)
- from this and rai1
- have "a \<otimes> r \<in> I +> \<one>" by simp
- from this have "I +> \<one> = I +> a \<otimes> r"
- by (rule a_repr_independence, simp) (rule a_subgroup)
+ --{* Calculating an inverse for @{term "a"} *}
+ 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
+ from one and rcarr and acarr and iI[THEN a_Hcarr]
+ have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra
- from rcarr and this[symmetric]
- show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast
-qed
+ --{* Lifting to cosets *}
+ from iI have "\<ominus>i \<oplus> \<one> \<in> I +> \<one>"
+ by (intro a_rcosI, simp, intro a_subset, simp)
+ with rai1 have "a \<otimes> r \<in> I +> \<one>" by simp
+ then have "I +> \<one> = I +> a \<otimes> r"
+ by (rule a_repr_independence, simp) (rule a_subgroup)
+
+ from rcarr and this[symmetric]
+ show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast
+ qed
qed
end