# HG changeset patch
# User wenzelm
# Date 1316467098 -7200
# Node ID 0d2d59525912ab2ea428ba07e30ec66f138da4ec
# Parent  5bd261075711e5c5858d65b892beada8715de587
tuned proofs;

diff -r 5bd261075711 -r 0d2d59525912 src/HOL/Algebra/QuotRing.thy
--- 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