author wenzelm Sat Sep 03 22:05:25 2011 +0200 (2011-09-03 ago) changeset 44677 3fb27b19e058 parent 44676 7de87f1ae965 child 44682 e5ba1c0b8cac
misc tuning and simplification of proofs;
 src/HOL/Algebra/Ideal.thy file | annotate | diff | revisions src/HOL/Algebra/Ring.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/Algebra/Ideal.thy	Sat Sep 03 21:15:35 2011 +0200
1.2 +++ b/src/HOL/Algebra/Ideal.thy	Sat Sep 03 22:05:25 2011 +0200
1.3 @@ -14,25 +14,24 @@
1.4
1.5  locale ideal = additive_subgroup I R + ring R for I and R (structure) +
1.6    assumes I_l_closed: "\<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes> a \<in> I"
1.7 -      and I_r_closed: "\<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"
1.8 +    and I_r_closed: "\<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"
1.9
1.10  sublocale ideal \<subseteq> abelian_subgroup I R
1.11 -apply (intro abelian_subgroupI3 abelian_group.intro)
1.12 -  apply (rule ideal.axioms, rule ideal_axioms)
1.13 - apply (rule abelian_group.axioms, rule ring.axioms, rule ideal.axioms, rule ideal_axioms)
1.14 -apply (rule abelian_group.axioms, rule ring.axioms, rule ideal.axioms, rule ideal_axioms)
1.15 -done
1.16 +  apply (intro abelian_subgroupI3 abelian_group.intro)
1.17 +    apply (rule ideal.axioms, rule ideal_axioms)
1.18 +   apply (rule abelian_group.axioms, rule ring.axioms, rule ideal.axioms, rule ideal_axioms)
1.19 +  apply (rule abelian_group.axioms, rule ring.axioms, rule ideal.axioms, rule ideal_axioms)
1.20 +  done
1.21
1.22 -lemma (in ideal) is_ideal:
1.23 -  "ideal I R"
1.24 -by (rule ideal_axioms)
1.25 +lemma (in ideal) is_ideal: "ideal I R"
1.26 +  by (rule ideal_axioms)
1.27
1.28  lemma idealI:
1.29    fixes R (structure)
1.30    assumes "ring R"
1.31    assumes a_subgroup: "subgroup I \<lparr>carrier = carrier R, mult = add R, one = zero R\<rparr>"
1.32 -      and I_l_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes> a \<in> I"
1.33 -      and I_r_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"
1.34 +    and I_l_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes> a \<in> I"
1.35 +    and I_r_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"
1.36    shows "ideal I R"
1.37  proof -
1.38    interpret ring R by fact
1.39 @@ -47,19 +46,16 @@
1.40
1.41  subsubsection (in ring) {* Ideals Generated by a Subset of @{term "carrier R"} *}
1.42
1.43 -definition
1.44 -  genideal :: "('a, 'b) ring_scheme \<Rightarrow> 'a set \<Rightarrow> 'a set"  ("Idl\<index> _"  79)
1.45 +definition genideal :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set"  ("Idl\<index> _"  79)
1.46    where "genideal R S = Inter {I. ideal I R \<and> S \<subseteq> I}"
1.47
1.48 -
1.49  subsubsection {* Principal Ideals *}
1.50
1.51  locale principalideal = ideal +
1.52    assumes generate: "\<exists>i \<in> carrier R. I = Idl {i}"
1.53
1.54 -lemma (in principalideal) is_principalideal:
1.55 -  shows "principalideal I R"
1.56 -by (rule principalideal_axioms)
1.57 +lemma (in principalideal) is_principalideal: "principalideal I R"
1.58 +  by (rule principalideal_axioms)
1.59
1.60  lemma principalidealI:
1.61    fixes R (structure)
1.62 @@ -68,7 +64,9 @@
1.63    shows "principalideal I R"
1.64  proof -
1.65    interpret ideal I R by fact
1.66 -  show ?thesis  by (intro principalideal.intro principalideal_axioms.intro) (rule is_ideal, rule generate)
1.67 +  show ?thesis
1.68 +    by (intro principalideal.intro principalideal_axioms.intro)
1.69 +      (rule is_ideal, rule generate)
1.70  qed
1.71
1.72
1.73 @@ -76,22 +74,22 @@
1.74
1.75  locale maximalideal = ideal +
1.76    assumes I_notcarr: "carrier R \<noteq> I"
1.77 -      and I_maximal: "\<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R"
1.78 +    and I_maximal: "\<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R"
1.79
1.80 -lemma (in maximalideal) is_maximalideal:
1.81 - shows "maximalideal I R"
1.82 -by (rule maximalideal_axioms)
1.83 +lemma (in maximalideal) is_maximalideal: "maximalideal I R"
1.84 +  by (rule maximalideal_axioms)
1.85
1.86  lemma maximalidealI:
1.87    fixes R
1.88    assumes "ideal I R"
1.89    assumes I_notcarr: "carrier R \<noteq> I"
1.90 -     and I_maximal: "\<And>J. \<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R"
1.91 +    and I_maximal: "\<And>J. \<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R"
1.92    shows "maximalideal I R"
1.93  proof -
1.94    interpret ideal I R by fact
1.95 -  show ?thesis by (intro maximalideal.intro maximalideal_axioms.intro)
1.96 -    (rule is_ideal, rule I_notcarr, rule I_maximal)
1.97 +  show ?thesis
1.98 +    by (intro maximalideal.intro maximalideal_axioms.intro)
1.99 +      (rule is_ideal, rule I_notcarr, rule I_maximal)
1.100  qed
1.101
1.102
1.103 @@ -99,24 +97,24 @@
1.104
1.105  locale primeideal = ideal + cring +
1.106    assumes I_notcarr: "carrier R \<noteq> I"
1.107 -      and I_prime: "\<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I"
1.108 +    and I_prime: "\<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I"
1.109
1.110 -lemma (in primeideal) is_primeideal:
1.111 - shows "primeideal I R"
1.112 -by (rule primeideal_axioms)
1.113 +lemma (in primeideal) is_primeideal: "primeideal I R"
1.114 +  by (rule primeideal_axioms)
1.115
1.116  lemma primeidealI:
1.117    fixes R (structure)
1.118    assumes "ideal I R"
1.119    assumes "cring R"
1.120    assumes I_notcarr: "carrier R \<noteq> I"
1.121 -      and I_prime: "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I"
1.122 +    and I_prime: "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I"
1.123    shows "primeideal I R"
1.124  proof -
1.125    interpret ideal I R by fact
1.126    interpret cring R by fact
1.127 -  show ?thesis by (intro primeideal.intro primeideal_axioms.intro)
1.128 -    (rule is_ideal, rule is_cring, rule I_notcarr, rule I_prime)
1.129 +  show ?thesis
1.130 +    by (intro primeideal.intro primeideal_axioms.intro)
1.131 +      (rule is_ideal, rule is_cring, rule I_notcarr, rule I_prime)
1.132  qed
1.133
1.134  lemma primeidealI2:
1.135 @@ -124,9 +122,9 @@
1.137    assumes "cring R"
1.138    assumes I_l_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes> a \<in> I"
1.139 -      and I_r_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"
1.140 -      and I_notcarr: "carrier R \<noteq> I"
1.141 -      and I_prime: "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I"
1.142 +    and I_r_closed: "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes> x \<in> I"
1.143 +    and I_notcarr: "carrier R \<noteq> I"
1.144 +    and I_prime: "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I"
1.145    shows "primeideal I R"
1.146  proof -
1.147    interpret additive_subgroup I R by fact
1.148 @@ -144,31 +142,27 @@
1.149
1.150  subsection {* Special Ideals *}
1.151
1.152 -lemma (in ring) zeroideal:
1.153 -  shows "ideal {\<zero>} R"
1.154 -apply (intro idealI subgroup.intro)
1.155 -      apply (rule is_ring)
1.156 -     apply simp+
1.157 -  apply (fold a_inv_def, simp)
1.158 - apply simp+
1.159 -done
1.160 +lemma (in ring) zeroideal: "ideal {\<zero>} R"
1.161 +  apply (intro idealI subgroup.intro)
1.162 +        apply (rule is_ring)
1.163 +       apply simp+
1.164 +    apply (fold a_inv_def, simp)
1.165 +   apply simp+
1.166 +  done
1.167
1.168 -lemma (in ring) oneideal:
1.169 -  shows "ideal (carrier R) R"
1.170 +lemma (in ring) oneideal: "ideal (carrier R) R"
1.171    by (rule idealI) (auto intro: is_ring add.subgroupI)
1.172
1.173 -lemma (in "domain") zeroprimeideal:
1.174 - shows "primeideal {\<zero>} R"
1.175 -apply (intro primeidealI)
1.176 -   apply (rule zeroideal)
1.177 -  apply (rule domain.axioms, rule domain_axioms)
1.178 - defer 1
1.179 - apply (simp add: integral)
1.180 +lemma (in "domain") zeroprimeideal: "primeideal {\<zero>} R"
1.181 +  apply (intro primeidealI)
1.182 +     apply (rule zeroideal)
1.183 +    apply (rule domain.axioms, rule domain_axioms)
1.184 +   defer 1
1.185 +   apply (simp add: integral)
1.186  proof (rule ccontr, simp)
1.187    assume "carrier R = {\<zero>}"
1.188 -  from this have "\<one> = \<zero>" by (rule one_zeroI)
1.189 -  from this and one_not_zero
1.190 -      show "False" by simp
1.191 +  then have "\<one> = \<zero>" by (rule one_zeroI)
1.192 +  with one_not_zero show False by simp
1.193  qed
1.194
1.195
1.196 @@ -177,22 +171,20 @@
1.197  lemma (in ideal) one_imp_carrier:
1.198    assumes I_one_closed: "\<one> \<in> I"
1.199    shows "I = carrier R"
1.200 -apply (rule)
1.201 -apply (rule)
1.202 -apply (rule a_Hcarr, simp)
1.203 +  apply (rule)
1.204 +  apply (rule)
1.205 +  apply (rule a_Hcarr, simp)
1.206  proof
1.207    fix x
1.208    assume xcarr: "x \<in> carrier R"
1.209 -  from I_one_closed and this
1.210 -      have "x \<otimes> \<one> \<in> I" by (intro I_l_closed)
1.211 -  from this and xcarr
1.212 -      show "x \<in> I" by simp
1.213 +  with I_one_closed have "x \<otimes> \<one> \<in> I" by (intro I_l_closed)
1.214 +  with xcarr show "x \<in> I" by simp
1.215  qed
1.216
1.217  lemma (in ideal) Icarr:
1.218    assumes iI: "i \<in> I"
1.219    shows "i \<in> carrier R"
1.220 -using iI by (rule a_Hcarr)
1.221 +  using iI by (rule a_Hcarr)
1.222
1.223
1.224  subsection {* Intersection of Ideals *}
1.225 @@ -207,17 +199,17 @@
1.226    interpret ideal I R by fact
1.227    interpret ideal J R by fact
1.228    show ?thesis
1.229 -apply (intro idealI subgroup.intro)
1.230 -      apply (rule is_ring)
1.231 -     apply (force simp add: a_subset)
1.232 -    apply (simp add: a_inv_def[symmetric])
1.233 -   apply simp
1.234 -  apply (simp add: a_inv_def[symmetric])
1.235 - apply (clarsimp, rule)
1.236 -  apply (fast intro: ideal.I_l_closed ideal.intro assms)+
1.237 -apply (clarsimp, rule)
1.238 - apply (fast intro: ideal.I_r_closed ideal.intro assms)+
1.239 -done
1.240 +    apply (intro idealI subgroup.intro)
1.241 +          apply (rule is_ring)
1.242 +         apply (force simp add: a_subset)
1.243 +        apply (simp add: a_inv_def[symmetric])
1.244 +       apply simp
1.245 +      apply (simp add: a_inv_def[symmetric])
1.246 +     apply (clarsimp, rule)
1.247 +      apply (fast intro: ideal.I_l_closed ideal.intro assms)+
1.248 +    apply (clarsimp, rule)
1.249 +     apply (fast intro: ideal.I_r_closed ideal.intro assms)+
1.250 +    done
1.251  qed
1.252
1.253  text {* The intersection of any Number of Ideals is again
1.254 @@ -226,26 +218,25 @@
1.255    assumes Sideals: "\<And>I. I \<in> S \<Longrightarrow> ideal I R"
1.256      and notempty: "S \<noteq> {}"
1.257    shows "ideal (Inter S) R"
1.258 -apply (unfold_locales)
1.260 -      apply rule unfolding mem_Collect_eq defer 1
1.261 -      apply rule defer 1
1.262 -      apply rule defer 1
1.263 -      apply (fold a_inv_def, rule) defer 1
1.264 -      apply rule defer 1
1.265 -      apply rule defer 1
1.266 +  apply (unfold_locales)
1.267 +  apply (simp_all add: Inter_eq)
1.268 +        apply rule unfolding mem_Collect_eq defer 1
1.269 +        apply rule defer 1
1.270 +        apply rule defer 1
1.271 +        apply (fold a_inv_def, rule) defer 1
1.272 +        apply rule defer 1
1.273 +        apply rule defer 1
1.274  proof -
1.275    fix x y
1.276    assume "\<forall>I\<in>S. x \<in> I"
1.277 -  hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
1.278 +  then have xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
1.279    assume "\<forall>I\<in>S. y \<in> I"
1.280 -  hence yI: "\<And>I. I \<in> S \<Longrightarrow> y \<in> I" by simp
1.281 +  then have yI: "\<And>I. I \<in> S \<Longrightarrow> y \<in> I" by simp
1.282
1.283    fix J
1.284    assume JS: "J \<in> S"
1.285    interpret ideal J R by (rule Sideals[OF JS])
1.286 -  from xI[OF JS] and yI[OF JS]
1.287 -      show "x \<oplus> y \<in> J" by (rule a_closed)
1.288 +  from xI[OF JS] and yI[OF JS] show "x \<oplus> y \<in> J" by (rule a_closed)
1.289  next
1.290    fix J
1.291    assume JS: "J \<in> S"
1.292 @@ -254,50 +245,47 @@
1.293  next
1.294    fix x
1.295    assume "\<forall>I\<in>S. x \<in> I"
1.296 -  hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
1.297 +  then have xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
1.298
1.299    fix J
1.300    assume JS: "J \<in> S"
1.301    interpret ideal J R by (rule Sideals[OF JS])
1.302
1.303 -  from xI[OF JS]
1.304 -      show "\<ominus> x \<in> J" by (rule a_inv_closed)
1.305 +  from xI[OF JS] show "\<ominus> x \<in> J" by (rule a_inv_closed)
1.306  next
1.307    fix x y
1.308    assume "\<forall>I\<in>S. x \<in> I"
1.309 -  hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
1.310 +  then have xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
1.311    assume ycarr: "y \<in> carrier R"
1.312
1.313    fix J
1.314    assume JS: "J \<in> S"
1.315    interpret ideal J R by (rule Sideals[OF JS])
1.316
1.317 -  from xI[OF JS] and ycarr
1.318 -      show "y \<otimes> x \<in> J" by (rule I_l_closed)
1.319 +  from xI[OF JS] and ycarr show "y \<otimes> x \<in> J" by (rule I_l_closed)
1.320  next
1.321    fix x y
1.322    assume "\<forall>I\<in>S. x \<in> I"
1.323 -  hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
1.324 +  then have xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
1.325    assume ycarr: "y \<in> carrier R"
1.326
1.327    fix J
1.328    assume JS: "J \<in> S"
1.329    interpret ideal J R by (rule Sideals[OF JS])
1.330
1.331 -  from xI[OF JS] and ycarr
1.332 -      show "x \<otimes> y \<in> J" by (rule I_r_closed)
1.333 +  from xI[OF JS] and ycarr show "x \<otimes> y \<in> J" by (rule I_r_closed)
1.334  next
1.335    fix x
1.336    assume "\<forall>I\<in>S. x \<in> I"
1.337 -  hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
1.338 +  then have xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
1.339
1.340    from notempty have "\<exists>I0. I0 \<in> S" by blast
1.341 -  from this obtain I0 where I0S: "I0 \<in> S" by auto
1.342 +  then obtain I0 where I0S: "I0 \<in> S" by auto
1.343
1.344    interpret ideal I0 R by (rule Sideals[OF I0S])
1.345
1.346    from xI[OF I0S] have "x \<in> I0" .
1.347 -  from this and a_subset show "x \<in> carrier R" by fast
1.348 +  with a_subset show "x \<in> carrier R" by fast
1.349  next
1.350
1.351  qed
1.352 @@ -309,40 +297,41 @@
1.353    assumes idealI: "ideal I R"
1.354        and idealJ: "ideal J R"
1.355    shows "ideal (I <+> J) R"
1.356 -apply (rule ideal.intro)
1.358 -   apply (intro ideal.axioms[OF idealI])
1.359 -  apply (intro ideal.axioms[OF idealJ])
1.360 - apply (rule is_ring)
1.361 -apply (rule ideal_axioms.intro)
1.364 +  apply (rule ideal.intro)
1.366 +     apply (intro ideal.axioms[OF idealI])
1.367 +    apply (intro ideal.axioms[OF idealJ])
1.368 +   apply (rule is_ring)
1.369 +  apply (rule ideal_axioms.intro)
1.372  proof -
1.373    fix x i j
1.374    assume xcarr: "x \<in> carrier R"
1.375 -     and iI: "i \<in> I"
1.376 -     and jJ: "j \<in> J"
1.377 +    and iI: "i \<in> I"
1.378 +    and jJ: "j \<in> J"
1.379    from xcarr ideal.Icarr[OF idealI iI] ideal.Icarr[OF idealJ jJ]
1.380 -      have c: "(i \<oplus> j) \<otimes> x = (i \<otimes> x) \<oplus> (j \<otimes> x)" by algebra
1.381 -  from xcarr and iI
1.382 -      have a: "i \<otimes> x \<in> I" by (simp add: ideal.I_r_closed[OF idealI])
1.383 -  from xcarr and jJ
1.384 -      have b: "j \<otimes> x \<in> J" by (simp add: ideal.I_r_closed[OF idealJ])
1.385 -  from a b c
1.386 -      show "\<exists>ha\<in>I. \<exists>ka\<in>J. (i \<oplus> j) \<otimes> x = ha \<oplus> ka" by fast
1.387 +  have c: "(i \<oplus> j) \<otimes> x = (i \<otimes> x) \<oplus> (j \<otimes> x)"
1.388 +    by algebra
1.389 +  from xcarr and iI have a: "i \<otimes> x \<in> I"
1.390 +    by (simp add: ideal.I_r_closed[OF idealI])
1.391 +  from xcarr and jJ have b: "j \<otimes> x \<in> J"
1.392 +    by (simp add: ideal.I_r_closed[OF idealJ])
1.393 +  from a b c show "\<exists>ha\<in>I. \<exists>ka\<in>J. (i \<oplus> j) \<otimes> x = ha \<oplus> ka"
1.394 +    by fast
1.395  next
1.396    fix x i j
1.397    assume xcarr: "x \<in> carrier R"
1.398 -     and iI: "i \<in> I"
1.399 -     and jJ: "j \<in> J"
1.400 +    and iI: "i \<in> I"
1.401 +    and jJ: "j \<in> J"
1.402    from xcarr ideal.Icarr[OF idealI iI] ideal.Icarr[OF idealJ jJ]
1.403 -      have c: "x \<otimes> (i \<oplus> j) = (x \<otimes> i) \<oplus> (x \<otimes> j)" by algebra
1.404 -  from xcarr and iI
1.405 -      have a: "x \<otimes> i \<in> I" by (simp add: ideal.I_l_closed[OF idealI])
1.406 -  from xcarr and jJ
1.407 -      have b: "x \<otimes> j \<in> J" by (simp add: ideal.I_l_closed[OF idealJ])
1.408 -  from a b c
1.409 -      show "\<exists>ha\<in>I. \<exists>ka\<in>J. x \<otimes> (i \<oplus> j) = ha \<oplus> ka" by fast
1.410 +  have c: "x \<otimes> (i \<oplus> j) = (x \<otimes> i) \<oplus> (x \<otimes> j)" by algebra
1.411 +  from xcarr and iI have a: "x \<otimes> i \<in> I"
1.412 +    by (simp add: ideal.I_l_closed[OF idealI])
1.413 +  from xcarr and jJ have b: "x \<otimes> j \<in> J"
1.414 +    by (simp add: ideal.I_l_closed[OF idealJ])
1.415 +  from a b c show "\<exists>ha\<in>I. \<exists>ka\<in>J. x \<otimes> (i \<oplus> j) = ha \<oplus> ka"
1.416 +    by fast
1.417  qed
1.418
1.419
1.420 @@ -361,87 +350,74 @@
1.421  lemma (in ring) genideal_self:
1.422    assumes "S \<subseteq> carrier R"
1.423    shows "S \<subseteq> Idl S"
1.424 -unfolding genideal_def
1.425 -by fast
1.426 +  unfolding genideal_def by fast
1.427
1.428  lemma (in ring) genideal_self':
1.429    assumes carr: "i \<in> carrier R"
1.430    shows "i \<in> Idl {i}"
1.431  proof -
1.432 -  from carr
1.433 -      have "{i} \<subseteq> Idl {i}" by (fast intro!: genideal_self)
1.434 -  thus "i \<in> Idl {i}" by fast
1.435 +  from carr have "{i} \<subseteq> Idl {i}" by (fast intro!: genideal_self)
1.436 +  then show "i \<in> Idl {i}" by fast
1.437  qed
1.438
1.439  text {* @{term genideal} generates the minimal ideal *}
1.440  lemma (in ring) genideal_minimal:
1.441    assumes a: "ideal I R"
1.442 -      and b: "S \<subseteq> I"
1.443 +    and b: "S \<subseteq> I"
1.444    shows "Idl S \<subseteq> I"
1.445 -unfolding genideal_def
1.446 -by (rule, elim InterD, simp add: a b)
1.447 +  unfolding genideal_def by rule (elim InterD, simp add: a b)
1.448
1.449  text {* Generated ideals and subsets *}
1.450  lemma (in ring) Idl_subset_ideal:
1.451    assumes Iideal: "ideal I R"
1.452 -      and Hcarr: "H \<subseteq> carrier R"
1.453 +    and Hcarr: "H \<subseteq> carrier R"
1.454    shows "(Idl H \<subseteq> I) = (H \<subseteq> I)"
1.455  proof
1.456    assume a: "Idl H \<subseteq> I"
1.457    from Hcarr have "H \<subseteq> Idl H" by (rule genideal_self)
1.458 -  from this and a
1.459 -      show "H \<subseteq> I" by simp
1.460 +  with a show "H \<subseteq> I" by simp
1.461  next
1.462    fix x
1.463    assume HI: "H \<subseteq> I"
1.464 -
1.465 -  from Iideal and HI
1.466 -      have "I \<in> {I. ideal I R \<and> H \<subseteq> I}" by fast
1.467 -  from this
1.468 -      show "Idl H \<subseteq> I"
1.469 -      unfolding genideal_def
1.470 -      by fast
1.471 +  from Iideal and HI have "I \<in> {I. ideal I R \<and> H \<subseteq> I}" by fast
1.472 +  then show "Idl H \<subseteq> I" unfolding genideal_def by fast
1.473  qed
1.474
1.475  lemma (in ring) subset_Idl_subset:
1.476    assumes Icarr: "I \<subseteq> carrier R"
1.477 -      and HI: "H \<subseteq> I"
1.478 +    and HI: "H \<subseteq> I"
1.479    shows "Idl H \<subseteq> Idl I"
1.480  proof -
1.481 -  from HI and genideal_self[OF Icarr]
1.482 -      have HIdlI: "H \<subseteq> Idl I" by fast
1.483 +  from HI and genideal_self[OF Icarr] have HIdlI: "H \<subseteq> Idl I"
1.484 +    by fast
1.485
1.486 -  from Icarr
1.487 -      have Iideal: "ideal (Idl I) R" by (rule genideal_ideal)
1.488 -  from HI and Icarr
1.489 -      have "H \<subseteq> carrier R" by fast
1.490 -  from Iideal and this
1.491 -      have "(H \<subseteq> Idl I) = (Idl H \<subseteq> Idl I)"
1.492 -      by (rule Idl_subset_ideal[symmetric])
1.493 +  from Icarr have Iideal: "ideal (Idl I) R"
1.494 +    by (rule genideal_ideal)
1.495 +  from HI and Icarr have "H \<subseteq> carrier R"
1.496 +    by fast
1.497 +  with Iideal have "(H \<subseteq> Idl I) = (Idl H \<subseteq> Idl I)"
1.498 +    by (rule Idl_subset_ideal[symmetric])
1.499
1.500 -  from HIdlI and this
1.501 -      show "Idl H \<subseteq> Idl I" by simp
1.502 +  with HIdlI show "Idl H \<subseteq> Idl I" by simp
1.503  qed
1.504
1.505  lemma (in ring) Idl_subset_ideal':
1.506    assumes acarr: "a \<in> carrier R" and bcarr: "b \<in> carrier R"
1.507    shows "(Idl {a} \<subseteq> Idl {b}) = (a \<in> Idl {b})"
1.508 -apply (subst Idl_subset_ideal[OF genideal_ideal[of "{b}"], of "{a}"])
1.509 -  apply (fast intro: bcarr, fast intro: acarr)
1.510 -apply fast
1.511 -done
1.512 +  apply (subst Idl_subset_ideal[OF genideal_ideal[of "{b}"], of "{a}"])
1.513 +    apply (fast intro: bcarr, fast intro: acarr)
1.514 +  apply fast
1.515 +  done
1.516
1.517 -lemma (in ring) genideal_zero:
1.518 -  "Idl {\<zero>} = {\<zero>}"
1.519 -apply rule
1.520 - apply (rule genideal_minimal[OF zeroideal], simp)
1.522 -done
1.523 +lemma (in ring) genideal_zero: "Idl {\<zero>} = {\<zero>}"
1.524 +  apply rule
1.525 +   apply (rule genideal_minimal[OF zeroideal], simp)
1.526 +  apply (simp add: genideal_self')
1.527 +  done
1.528
1.529 -lemma (in ring) genideal_one:
1.530 -  "Idl {\<one>} = carrier R"
1.531 +lemma (in ring) genideal_one: "Idl {\<one>} = carrier R"
1.532  proof -
1.533 -  interpret ideal "Idl {\<one>}" "R" by (rule genideal_ideal, fast intro: one_closed)
1.534 +  interpret ideal "Idl {\<one>}" "R" by (rule genideal_ideal) fast
1.535    show "Idl {\<one>} = carrier R"
1.536    apply (rule, rule a_subset)
1.537    apply (simp add: one_imp_carrier genideal_self')
1.538 @@ -451,77 +427,76 @@
1.539
1.540  text {* Generation of Principal Ideals in Commutative Rings *}
1.541
1.542 -definition
1.543 -  cgenideal :: "('a, 'b) monoid_scheme \<Rightarrow> 'a \<Rightarrow> 'a set"  ("PIdl\<index> _"  79)
1.544 +definition cgenideal :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set"  ("PIdl\<index> _"  79)
1.545    where "cgenideal R a = {x \<otimes>\<^bsub>R\<^esub> a | x. x \<in> carrier R}"
1.546
1.547  text {* genhideal (?) really generates an ideal *}
1.548  lemma (in cring) cgenideal_ideal:
1.549    assumes acarr: "a \<in> carrier R"
1.550    shows "ideal (PIdl a) R"
1.551 -apply (unfold cgenideal_def)
1.552 -apply (rule idealI[OF is_ring])
1.553 -   apply (rule subgroup.intro)
1.554 -      apply (simp_all add: monoid_record_simps)
1.555 -      apply (blast intro: acarr m_closed)
1.556 -      apply clarsimp defer 1
1.557 -      defer 1
1.558 -      apply (fold a_inv_def, clarsimp) defer 1
1.559 -      apply clarsimp defer 1
1.560 -      apply clarsimp defer 1
1.561 +  apply (unfold cgenideal_def)
1.562 +  apply (rule idealI[OF is_ring])
1.563 +     apply (rule subgroup.intro)
1.564 +        apply simp_all
1.565 +        apply (blast intro: acarr)
1.566 +        apply clarsimp defer 1
1.567 +        defer 1
1.568 +        apply (fold a_inv_def, clarsimp) defer 1
1.569 +        apply clarsimp defer 1
1.570 +        apply clarsimp defer 1
1.571  proof -
1.572    fix x y
1.573    assume xcarr: "x \<in> carrier R"
1.574 -     and ycarr: "y \<in> carrier R"
1.575 +    and ycarr: "y \<in> carrier R"
1.576    note carr = acarr xcarr ycarr
1.577
1.578 -  from carr
1.579 -      have "x \<otimes> a \<oplus> y \<otimes> a = (x \<oplus> y) \<otimes> a" by (simp add: l_distr)
1.580 -  from this and carr
1.581 -      show "\<exists>z. x \<otimes> a \<oplus> y \<otimes> a = z \<otimes> a \<and> z \<in> carrier R" by fast
1.582 +  from carr have "x \<otimes> a \<oplus> y \<otimes> a = (x \<oplus> y) \<otimes> a"
1.583 +    by (simp add: l_distr)
1.584 +  with carr show "\<exists>z. x \<otimes> a \<oplus> y \<otimes> a = z \<otimes> a \<and> z \<in> carrier R"
1.585 +    by fast
1.586  next
1.587    from l_null[OF acarr, symmetric] and zero_closed
1.588 -      show "\<exists>x. \<zero> = x \<otimes> a \<and> x \<in> carrier R" by fast
1.589 +  show "\<exists>x. \<zero> = x \<otimes> a \<and> x \<in> carrier R" by fast
1.590  next
1.591    fix x
1.592    assume xcarr: "x \<in> carrier R"
1.593    note carr = acarr xcarr
1.594
1.595 -  from carr
1.596 -      have "\<ominus> (x \<otimes> a) = (\<ominus> x) \<otimes> a" by (simp add: l_minus)
1.597 -  from this and carr
1.598 -      show "\<exists>z. \<ominus> (x \<otimes> a) = z \<otimes> a \<and> z \<in> carrier R" by fast
1.599 +  from carr have "\<ominus> (x \<otimes> a) = (\<ominus> x) \<otimes> a"
1.600 +    by (simp add: l_minus)
1.601 +  with carr show "\<exists>z. \<ominus> (x \<otimes> a) = z \<otimes> a \<and> z \<in> carrier R"
1.602 +    by fast
1.603  next
1.604    fix x y
1.605    assume xcarr: "x \<in> carrier R"
1.606       and ycarr: "y \<in> carrier R"
1.607    note carr = acarr xcarr ycarr
1.608
1.609 -  from carr
1.610 -      have "y \<otimes> a \<otimes> x = (y \<otimes> x) \<otimes> a" by (simp add: m_assoc, simp add: m_comm)
1.611 -  from this and carr
1.612 -      show "\<exists>z. y \<otimes> a \<otimes> x = z \<otimes> a \<and> z \<in> carrier R" by fast
1.613 +  from carr have "y \<otimes> a \<otimes> x = (y \<otimes> x) \<otimes> a"
1.615 +  with carr show "\<exists>z. y \<otimes> a \<otimes> x = z \<otimes> a \<and> z \<in> carrier R"
1.616 +    by fast
1.617  next
1.618    fix x y
1.619    assume xcarr: "x \<in> carrier R"
1.620       and ycarr: "y \<in> carrier R"
1.621    note carr = acarr xcarr ycarr
1.622
1.623 -  from carr
1.624 -      have "x \<otimes> (y \<otimes> a) = (x \<otimes> y) \<otimes> a" by (simp add: m_assoc)
1.625 -  from this and carr
1.626 -      show "\<exists>z. x \<otimes> (y \<otimes> a) = z \<otimes> a \<and> z \<in> carrier R" by fast
1.627 +  from carr have "x \<otimes> (y \<otimes> a) = (x \<otimes> y) \<otimes> a"
1.628 +    by (simp add: m_assoc)
1.629 +  with carr show "\<exists>z. x \<otimes> (y \<otimes> a) = z \<otimes> a \<and> z \<in> carrier R"
1.630 +    by fast
1.631  qed
1.632
1.633  lemma (in ring) cgenideal_self:
1.634    assumes icarr: "i \<in> carrier R"
1.635    shows "i \<in> PIdl i"
1.636 -unfolding cgenideal_def
1.637 +  unfolding cgenideal_def
1.638  proof simp
1.639 -  from icarr
1.640 -      have "i = \<one> \<otimes> i" by simp
1.641 -  from this and icarr
1.642 -      show "\<exists>x. i = x \<otimes> i \<and> x \<in> carrier R" by fast
1.643 +  from icarr have "i = \<one> \<otimes> i"
1.644 +    by simp
1.645 +  with icarr show "\<exists>x. i = x \<otimes> i \<and> x \<in> carrier R"
1.646 +    by fast
1.647  qed
1.648
1.649  text {* @{const "cgenideal"} is minimal *}
1.650 @@ -532,7 +507,8 @@
1.651    shows "PIdl a \<subseteq> J"
1.652  proof -
1.653    interpret ideal J R by fact
1.654 -  show ?thesis unfolding cgenideal_def
1.655 +  show ?thesis
1.656 +    unfolding cgenideal_def
1.657      apply rule
1.658      apply clarify
1.659      using aJ
1.660 @@ -543,30 +519,28 @@
1.661  lemma (in cring) cgenideal_eq_genideal:
1.662    assumes icarr: "i \<in> carrier R"
1.663    shows "PIdl i = Idl {i}"
1.664 -apply rule
1.665 - apply (intro cgenideal_minimal)
1.666 -  apply (rule genideal_ideal, fast intro: icarr)
1.667 - apply (rule genideal_self', fast intro: icarr)
1.668 -apply (intro genideal_minimal)
1.669 - apply (rule cgenideal_ideal [OF icarr])
1.670 -apply (simp, rule cgenideal_self [OF icarr])
1.671 -done
1.672 +  apply rule
1.673 +   apply (intro cgenideal_minimal)
1.674 +    apply (rule genideal_ideal, fast intro: icarr)
1.675 +   apply (rule genideal_self', fast intro: icarr)
1.676 +  apply (intro genideal_minimal)
1.677 +   apply (rule cgenideal_ideal [OF icarr])
1.678 +  apply (simp, rule cgenideal_self [OF icarr])
1.679 +  done
1.680
1.681 -lemma (in cring) cgenideal_eq_rcos:
1.682 - "PIdl i = carrier R #> i"
1.683 -unfolding cgenideal_def r_coset_def
1.684 -by fast
1.685 +lemma (in cring) cgenideal_eq_rcos: "PIdl i = carrier R #> i"
1.686 +  unfolding cgenideal_def r_coset_def by fast
1.687
1.688  lemma (in cring) cgenideal_is_principalideal:
1.689    assumes icarr: "i \<in> carrier R"
1.690    shows "principalideal (PIdl i) R"
1.691 -apply (rule principalidealI)
1.692 -apply (rule cgenideal_ideal [OF icarr])
1.693 +  apply (rule principalidealI)
1.694 +  apply (rule cgenideal_ideal [OF icarr])
1.695  proof -
1.696 -  from icarr
1.697 -      have "PIdl i = Idl {i}" by (rule cgenideal_eq_genideal)
1.698 -  from icarr and this
1.699 -      show "\<exists>i'\<in>carrier R. PIdl i = Idl {i'}" by fast
1.700 +  from icarr have "PIdl i = Idl {i}"
1.701 +    by (rule cgenideal_eq_genideal)
1.702 +  with icarr show "\<exists>i'\<in>carrier R. PIdl i = Idl {i'}"
1.703 +    by fast
1.704  qed
1.705
1.706
1.707 @@ -574,83 +548,79 @@
1.708
1.709  lemma (in ring) union_genideal:
1.710    assumes idealI: "ideal I R"
1.711 -      and idealJ: "ideal J R"
1.712 +    and idealJ: "ideal J R"
1.713    shows "Idl (I \<union> J) = I <+> J"
1.714 -apply rule
1.715 - apply (rule ring.genideal_minimal)
1.716 -   apply (rule is_ring)
1.717 -  apply (rule add_ideals[OF idealI idealJ])
1.718 - apply (rule)
1.719 - apply (simp add: set_add_defs) apply (elim disjE) defer 1 defer 1
1.720 - apply (rule) apply (simp add: set_add_defs genideal_def) apply clarsimp defer 1
1.721 +  apply rule
1.722 +   apply (rule ring.genideal_minimal)
1.723 +     apply (rule is_ring)
1.724 +    apply (rule add_ideals[OF idealI idealJ])
1.725 +   apply (rule)
1.726 +   apply (simp add: set_add_defs) apply (elim disjE) defer 1 defer 1
1.727 +   apply (rule) apply (simp add: set_add_defs genideal_def) apply clarsimp defer 1
1.728  proof -
1.729    fix x
1.730    assume xI: "x \<in> I"
1.731    have ZJ: "\<zero> \<in> J"
1.732 -      by (intro additive_subgroup.zero_closed, rule ideal.axioms[OF idealJ])
1.733 -  from ideal.Icarr[OF idealI xI]
1.734 -      have "x = x \<oplus> \<zero>" by algebra
1.735 -  from xI and ZJ and this
1.736 -      show "\<exists>h\<in>I. \<exists>k\<in>J. x = h \<oplus> k" by fast
1.737 +    by (intro additive_subgroup.zero_closed) (rule ideal.axioms[OF idealJ])
1.738 +  from ideal.Icarr[OF idealI xI] have "x = x \<oplus> \<zero>"
1.739 +    by algebra
1.740 +  with xI and ZJ show "\<exists>h\<in>I. \<exists>k\<in>J. x = h \<oplus> k"
1.741 +    by fast
1.742  next
1.743    fix x
1.744    assume xJ: "x \<in> J"
1.745    have ZI: "\<zero> \<in> I"
1.746 -      by (intro additive_subgroup.zero_closed, rule ideal.axioms[OF idealI])
1.747 -  from ideal.Icarr[OF idealJ xJ]
1.748 -      have "x = \<zero> \<oplus> x" by algebra
1.749 -  from ZI and xJ and this
1.750 -      show "\<exists>h\<in>I. \<exists>k\<in>J. x = h \<oplus> k" by fast
1.751 +    by (intro additive_subgroup.zero_closed, rule ideal.axioms[OF idealI])
1.752 +  from ideal.Icarr[OF idealJ xJ] have "x = \<zero> \<oplus> x"
1.753 +    by algebra
1.754 +  with ZI and xJ show "\<exists>h\<in>I. \<exists>k\<in>J. x = h \<oplus> k"
1.755 +    by fast
1.756  next
1.757    fix i j K
1.758    assume iI: "i \<in> I"
1.759 -     and jJ: "j \<in> J"
1.760 -     and idealK: "ideal K R"
1.761 -     and IK: "I \<subseteq> K"
1.762 -     and JK: "J \<subseteq> K"
1.763 -  from iI and IK
1.764 -     have iK: "i \<in> K" by fast
1.765 -  from jJ and JK
1.766 -     have jK: "j \<in> K" by fast
1.767 -  from iK and jK
1.768 -     show "i \<oplus> j \<in> K" by (intro additive_subgroup.a_closed) (rule ideal.axioms[OF idealK])
1.769 +    and jJ: "j \<in> J"
1.770 +    and idealK: "ideal K R"
1.771 +    and IK: "I \<subseteq> K"
1.772 +    and JK: "J \<subseteq> K"
1.773 +  from iI and IK have iK: "i \<in> K" by fast
1.774 +  from jJ and JK have jK: "j \<in> K" by fast
1.775 +  from iK and jK show "i \<oplus> j \<in> K"
1.776 +    by (intro additive_subgroup.a_closed) (rule ideal.axioms[OF idealK])
1.777  qed
1.778
1.779
1.780  subsection {* Properties of Principal Ideals *}
1.781
1.782  text {* @{text "\<zero>"} generates the zero ideal *}
1.783 -lemma (in ring) zero_genideal:
1.784 -  shows "Idl {\<zero>} = {\<zero>}"
1.785 -apply rule
1.786 -apply (simp add: genideal_minimal zeroideal)
1.787 -apply (fast intro!: genideal_self)
1.788 -done
1.789 +lemma (in ring) zero_genideal: "Idl {\<zero>} = {\<zero>}"
1.790 +  apply rule
1.791 +  apply (simp add: genideal_minimal zeroideal)
1.792 +  apply (fast intro!: genideal_self)
1.793 +  done
1.794
1.795  text {* @{text "\<one>"} generates the unit ideal *}
1.796 -lemma (in ring) one_genideal:
1.797 -  shows "Idl {\<one>} = carrier R"
1.798 +lemma (in ring) one_genideal: "Idl {\<one>} = carrier R"
1.799  proof -
1.800 -  have "\<one> \<in> Idl {\<one>}" by (simp add: genideal_self')
1.801 -  thus "Idl {\<one>} = carrier R" by (intro ideal.one_imp_carrier, fast intro: genideal_ideal)
1.802 +  have "\<one> \<in> Idl {\<one>}"
1.803 +    by (simp add: genideal_self')
1.804 +  then show "Idl {\<one>} = carrier R"
1.805 +    by (intro ideal.one_imp_carrier) (fast intro: genideal_ideal)
1.806  qed
1.807
1.808
1.809  text {* The zero ideal is a principal ideal *}
1.810 -corollary (in ring) zeropideal:
1.811 -  shows "principalideal {\<zero>} R"
1.812 -apply (rule principalidealI)
1.813 - apply (rule zeroideal)
1.814 -apply (blast intro!: zero_closed zero_genideal[symmetric])
1.815 -done
1.816 +corollary (in ring) zeropideal: "principalideal {\<zero>} R"
1.817 +  apply (rule principalidealI)
1.818 +   apply (rule zeroideal)
1.819 +  apply (blast intro!: zero_genideal[symmetric])
1.820 +  done
1.821
1.822  text {* The unit ideal is a principal ideal *}
1.823 -corollary (in ring) onepideal:
1.824 -  shows "principalideal (carrier R) R"
1.825 -apply (rule principalidealI)
1.826 - apply (rule oneideal)
1.827 -apply (blast intro!: one_closed one_genideal[symmetric])
1.828 -done
1.829 +corollary (in ring) onepideal: "principalideal (carrier R) R"
1.830 +  apply (rule principalidealI)
1.831 +   apply (rule oneideal)
1.832 +  apply (blast intro!: one_genideal[symmetric])
1.833 +  done
1.834
1.835
1.836  text {* Every principal ideal is a right coset of the carrier *}
1.837 @@ -659,28 +629,24 @@
1.838    shows "\<exists>x\<in>I. I = carrier R #> x"
1.839  proof -
1.840    interpret cring R by fact
1.841 -  from generate
1.842 -      obtain i
1.843 -        where icarr: "i \<in> carrier R"
1.844 -        and I1: "I = Idl {i}"
1.845 -      by fast+
1.846 +  from generate obtain i where icarr: "i \<in> carrier R" and I1: "I = Idl {i}"
1.847 +    by fast+
1.848
1.849 -  from icarr and genideal_self[of "{i}"]
1.850 -      have "i \<in> Idl {i}" by fast
1.851 -  hence iI: "i \<in> I" by (simp add: I1)
1.852 +  from icarr and genideal_self[of "{i}"] have "i \<in> Idl {i}"
1.853 +    by fast
1.854 +  then have iI: "i \<in> I" by (simp add: I1)
1.855
1.856 -  from I1 icarr
1.857 -      have I2: "I = PIdl i" by (simp add: cgenideal_eq_genideal)
1.858 +  from I1 icarr have I2: "I = PIdl i"
1.859 +    by (simp add: cgenideal_eq_genideal)
1.860
1.861    have "PIdl i = carrier R #> i"
1.862 -      unfolding cgenideal_def r_coset_def
1.863 -      by fast
1.864 +    unfolding cgenideal_def r_coset_def by fast
1.865
1.866 -  from I2 and this
1.867 -      have "I = carrier R #> i" by simp
1.868 +  with I2 have "I = carrier R #> i"
1.869 +    by simp
1.870
1.871 -  from iI and this
1.872 -      show "\<exists>x\<in>I. I = carrier R #> x" by fast
1.873 +  with iI show "\<exists>x\<in>I. I = carrier R #> x"
1.874 +    by fast
1.875  qed
1.876
1.877
1.878 @@ -693,16 +659,16 @@
1.879  proof (rule ccontr, clarsimp)
1.880    interpret cring R by fact
1.881    assume InR: "carrier R \<noteq> I"
1.882 -     and "\<forall>a. a \<in> carrier R \<longrightarrow> (\<forall>b. a \<otimes> b \<in> I \<longrightarrow> b \<in> carrier R \<longrightarrow> a \<in> I \<or> b \<in> I)"
1.883 -  hence I_prime: "\<And> a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I" by simp
1.884 +    and "\<forall>a. a \<in> carrier R \<longrightarrow> (\<forall>b. a \<otimes> b \<in> I \<longrightarrow> b \<in> carrier R \<longrightarrow> a \<in> I \<or> b \<in> I)"
1.885 +  then have I_prime: "\<And> a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I"
1.886 +    by simp
1.887    have "primeideal I R"
1.888 -      apply (rule primeideal.intro [OF is_ideal is_cring])
1.889 -      apply (rule primeideal_axioms.intro)
1.890 -       apply (rule InR)
1.891 -      apply (erule (2) I_prime)
1.892 -      done
1.893 -  from this and notprime
1.894 -      show "False" by simp
1.895 +    apply (rule primeideal.intro [OF is_ideal is_cring])
1.896 +    apply (rule primeideal_axioms.intro)
1.897 +     apply (rule InR)
1.898 +    apply (erule (2) I_prime)
1.899 +    done
1.900 +  with notprime show False by simp
1.901  qed
1.902
1.903  lemma (in ideal) primeidealCE:
1.904 @@ -721,47 +687,44 @@
1.905  lemma (in cring) zeroprimeideal_domainI:
1.906    assumes pi: "primeideal {\<zero>} R"
1.907    shows "domain R"
1.908 -apply (rule domain.intro, rule is_cring)
1.909 -apply (rule domain_axioms.intro)
1.910 +  apply (rule domain.intro, rule is_cring)
1.911 +  apply (rule domain_axioms.intro)
1.912  proof (rule ccontr, simp)
1.913    interpret primeideal "{\<zero>}" "R" by (rule pi)
1.914    assume "\<one> = \<zero>"
1.915 -  hence "carrier R = {\<zero>}" by (rule one_zeroD)
1.916 -  from this[symmetric] and I_notcarr
1.917 -      show "False" by simp
1.918 +  then have "carrier R = {\<zero>}" by (rule one_zeroD)
1.919 +  from this[symmetric] and I_notcarr show False
1.920 +    by simp
1.921  next
1.922    interpret primeideal "{\<zero>}" "R" by (rule pi)
1.923    fix a b
1.924 -  assume ab: "a \<otimes> b = \<zero>"
1.925 -     and carr: "a \<in> carrier R" "b \<in> carrier R"
1.926 -  from ab
1.927 -      have abI: "a \<otimes> b \<in> {\<zero>}" by fast
1.928 -  from carr and this
1.929 -      have "a \<in> {\<zero>} \<or> b \<in> {\<zero>}" by (rule I_prime)
1.930 -  thus "a = \<zero> \<or> b = \<zero>" by simp
1.931 +  assume ab: "a \<otimes> b = \<zero>" and carr: "a \<in> carrier R" "b \<in> carrier R"
1.932 +  from ab have abI: "a \<otimes> b \<in> {\<zero>}"
1.933 +    by fast
1.934 +  with carr have "a \<in> {\<zero>} \<or> b \<in> {\<zero>}"
1.935 +    by (rule I_prime)
1.936 +  then show "a = \<zero> \<or> b = \<zero>" by simp
1.937  qed
1.938
1.939 -corollary (in cring) domain_eq_zeroprimeideal:
1.940 -  shows "domain R = primeideal {\<zero>} R"
1.941 -apply rule
1.942 - apply (erule domain.zeroprimeideal)
1.943 -apply (erule zeroprimeideal_domainI)
1.944 -done
1.945 +corollary (in cring) domain_eq_zeroprimeideal: "domain R = primeideal {\<zero>} R"
1.946 +  apply rule
1.947 +   apply (erule domain.zeroprimeideal)
1.948 +  apply (erule zeroprimeideal_domainI)
1.949 +  done
1.950
1.951
1.952  subsection {* Maximal Ideals *}
1.953
1.954  lemma (in ideal) helper_I_closed:
1.955    assumes carr: "a \<in> carrier R" "x \<in> carrier R" "y \<in> carrier R"
1.956 -      and axI: "a \<otimes> x \<in> I"
1.957 +    and axI: "a \<otimes> x \<in> I"
1.958    shows "a \<otimes> (x \<otimes> y) \<in> I"
1.959  proof -
1.960 -  from axI and carr
1.961 -     have "(a \<otimes> x) \<otimes> y \<in> I" by (simp add: I_r_closed)
1.962 -  also from carr
1.963 -     have "(a \<otimes> x) \<otimes> y = a \<otimes> (x \<otimes> y)" by (simp add: m_assoc)
1.964 -  finally
1.965 -     show "a \<otimes> (x \<otimes> y) \<in> I" .
1.966 +  from axI and carr have "(a \<otimes> x) \<otimes> y \<in> I"
1.967 +    by (simp add: I_r_closed)
1.968 +  also from carr have "(a \<otimes> x) \<otimes> y = a \<otimes> (x \<otimes> y)"
1.969 +    by (simp add: m_assoc)
1.970 +  finally show "a \<otimes> (x \<otimes> y) \<in> I" .
1.971  qed
1.972
1.973  lemma (in ideal) helper_max_prime:
1.974 @@ -787,19 +750,18 @@
1.975      have "\<ominus>(a \<otimes> x) \<in> I" by simp
1.976      also from acarr xcarr
1.977      have "\<ominus>(a \<otimes> x) = a \<otimes> (\<ominus>x)" by algebra
1.978 -    finally
1.979 -    show "a \<otimes> (\<ominus>x) \<in> I" .
1.980 -    from acarr
1.981 -    have "a \<otimes> \<zero> = \<zero>" by simp
1.982 +    finally show "a \<otimes> (\<ominus>x) \<in> I" .
1.983 +    from acarr have "a \<otimes> \<zero> = \<zero>" by simp
1.984    next
1.985      fix x y
1.986      assume xcarr: "x \<in> carrier R"
1.987        and ycarr: "y \<in> carrier R"
1.988        and ayI: "a \<otimes> y \<in> I"
1.989 -    from ayI and acarr xcarr ycarr
1.990 -    have "a \<otimes> (y \<otimes> x) \<in> I" by (simp add: helper_I_closed)
1.991 -    moreover from xcarr ycarr
1.992 -    have "y \<otimes> x = x \<otimes> y" by (simp add: m_comm)
1.993 +    from ayI and acarr xcarr ycarr have "a \<otimes> (y \<otimes> x) \<in> I"
1.994 +      by (simp add: helper_I_closed)
1.995 +    moreover
1.996 +    from xcarr ycarr have "y \<otimes> x = x \<otimes> y"
1.997 +      by (simp add: m_comm)
1.998      ultimately
1.999      show "a \<otimes> (x \<otimes> y) \<in> I" by simp
1.1000    qed
1.1001 @@ -818,40 +780,36 @@
1.1003    proof -
1.1004      assume "\<exists>a b. a \<in> carrier R \<and> b \<in> carrier R \<and> a \<otimes> b \<in> I \<and> a \<notin> I \<and> b \<notin> I"
1.1005 -    from this
1.1006 -    obtain a b
1.1007 -      where acarr: "a \<in> carrier R"
1.1008 -      and bcarr: "b \<in> carrier R"
1.1009 -      and abI: "a \<otimes> b \<in> I"
1.1010 -      and anI: "a \<notin> I"
1.1011 -      and bnI: "b \<notin> I"
1.1012 -      by fast
1.1013 +    then obtain a b where
1.1014 +      acarr: "a \<in> carrier R" and
1.1015 +      bcarr: "b \<in> carrier R" and
1.1016 +      abI: "a \<otimes> b \<in> I" and
1.1017 +      anI: "a \<notin> I" and
1.1018 +      bnI: "b \<notin> I" by fast
1.1019      def J \<equiv> "{x\<in>carrier R. a \<otimes> x \<in> I}"
1.1020
1.1021 -    from is_cring and acarr
1.1022 -    have idealJ: "ideal J R" unfolding J_def by (rule helper_max_prime)
1.1023 +    from is_cring and acarr have idealJ: "ideal J R"
1.1024 +      unfolding J_def by (rule helper_max_prime)
1.1025
1.1026      have IsubJ: "I \<subseteq> J"
1.1027      proof
1.1028        fix x
1.1029        assume xI: "x \<in> I"
1.1030 -      from this and acarr
1.1031 -      have "a \<otimes> x \<in> I" by (intro I_l_closed)
1.1032 -      from xI[THEN a_Hcarr] this
1.1033 -      show "x \<in> J" unfolding J_def by fast
1.1034 +      with acarr have "a \<otimes> x \<in> I"
1.1035 +        by (intro I_l_closed)
1.1036 +      with xI[THEN a_Hcarr] show "x \<in> J"
1.1037 +        unfolding J_def by fast
1.1038      qed
1.1039
1.1040 -    from abI and acarr bcarr
1.1041 -    have "b \<in> J" unfolding J_def by fast
1.1042 -    from bnI and this
1.1043 -    have JnI: "J \<noteq> I" by fast
1.1044 +    from abI and acarr bcarr have "b \<in> J"
1.1045 +      unfolding J_def by fast
1.1046 +    with bnI have JnI: "J \<noteq> I" by fast
1.1047      from acarr
1.1048      have "a = a \<otimes> \<one>" by algebra
1.1049 -    from this and anI
1.1050 -    have "a \<otimes> \<one> \<notin> I" by simp
1.1051 -    from one_closed and this
1.1052 -    have "\<one> \<notin> J" unfolding J_def by fast
1.1053 -    hence Jncarr: "J \<noteq> carrier R" by fast
1.1054 +    with anI have "a \<otimes> \<one> \<notin> I" by simp
1.1055 +    with one_closed have "\<one> \<notin> J"
1.1056 +      unfolding J_def by fast
1.1057 +    then have Jncarr: "J \<noteq> carrier R" by fast
1.1058
1.1059      interpret ideal J R by (rule idealJ)
1.1060
1.1061 @@ -862,8 +820,7 @@
1.1062        apply (rule a_subset)
1.1063        done
1.1064
1.1065 -    from this and JnI and Jncarr
1.1066 -    show "False" by simp
1.1067 +    with JnI and Jncarr show False by simp
1.1068    qed
1.1069  qed
1.1070
1.1071 @@ -873,111 +830,93 @@
1.1072  --"A non-zero cring that has only the two trivial ideals is a field"
1.1073  lemma (in cring) trivialideals_fieldI:
1.1074    assumes carrnzero: "carrier R \<noteq> {\<zero>}"
1.1075 -      and haveideals: "{I. ideal I R} = {{\<zero>}, carrier R}"
1.1076 +    and haveideals: "{I. ideal I R} = {{\<zero>}, carrier R}"
1.1077    shows "field R"
1.1078 -apply (rule cring_fieldI)
1.1079 -apply (rule, rule, rule)
1.1080 - apply (erule Units_closed)
1.1081 -defer 1
1.1082 -  apply rule
1.1083 -defer 1
1.1084 +  apply (rule cring_fieldI)
1.1085 +  apply (rule, rule, rule)
1.1086 +   apply (erule Units_closed)
1.1087 +  defer 1
1.1088 +    apply rule
1.1089 +  defer 1
1.1090  proof (rule ccontr, simp)
1.1091    assume zUnit: "\<zero> \<in> Units R"
1.1092 -  hence a: "\<zero> \<otimes> inv \<zero> = \<one>" by (rule Units_r_inv)
1.1093 -  from zUnit
1.1094 -      have "\<zero> \<otimes> inv \<zero> = \<zero>" by (intro l_null, rule Units_inv_closed)
1.1095 -  from a[symmetric] and this
1.1096 -      have "\<one> = \<zero>" by simp
1.1097 -  hence "carrier R = {\<zero>}" by (rule one_zeroD)
1.1098 -  from this and carrnzero
1.1099 -      show "False" by simp
1.1100 +  then have a: "\<zero> \<otimes> inv \<zero> = \<one>" by (rule Units_r_inv)
1.1101 +  from zUnit have "\<zero> \<otimes> inv \<zero> = \<zero>"
1.1102 +    by (intro l_null) (rule Units_inv_closed)
1.1103 +  with a[symmetric] have "\<one> = \<zero>" by simp
1.1104 +  then have "carrier R = {\<zero>}" by (rule one_zeroD)
1.1105 +  with carrnzero show False by simp
1.1106  next
1.1107    fix x
1.1108    assume xcarr': "x \<in> carrier R - {\<zero>}"
1.1109 -  hence xcarr: "x \<in> carrier R" by fast
1.1110 -  from xcarr'
1.1111 -      have xnZ: "x \<noteq> \<zero>" by fast
1.1112 -  from xcarr
1.1113 -      have xIdl: "ideal (PIdl x) R" by (intro cgenideal_ideal, fast)
1.1114 +  then have xcarr: "x \<in> carrier R" by fast
1.1115 +  from xcarr' have xnZ: "x \<noteq> \<zero>" by fast
1.1116 +  from xcarr have xIdl: "ideal (PIdl x) R"
1.1117 +    by (intro cgenideal_ideal) fast
1.1118
1.1119 -  from xcarr
1.1120 -      have "x \<in> PIdl x" by (intro cgenideal_self, fast)
1.1121 -  from this and xnZ
1.1122 -      have "PIdl x \<noteq> {\<zero>}" by fast
1.1123 -  from haveideals and this
1.1124 -      have "PIdl x = carrier R"
1.1125 -      by (blast intro!: xIdl)
1.1126 -  hence "\<one> \<in> PIdl x" by simp
1.1127 -  hence "\<exists>y. \<one> = y \<otimes> x \<and> y \<in> carrier R" unfolding cgenideal_def by blast
1.1128 -  from this
1.1129 -      obtain y
1.1130 -        where ycarr: " y \<in> carrier R"
1.1131 -        and ylinv: "\<one> = y \<otimes> x"
1.1132 -      by fast+
1.1133 -  from ylinv and xcarr ycarr
1.1134 -      have yrinv: "\<one> = x \<otimes> y" by (simp add: m_comm)
1.1135 +  from xcarr have "x \<in> PIdl x"
1.1136 +    by (intro cgenideal_self) fast
1.1137 +  with xnZ have "PIdl x \<noteq> {\<zero>}" by fast
1.1138 +  with haveideals have "PIdl x = carrier R"
1.1139 +    by (blast intro!: xIdl)
1.1140 +  then have "\<one> \<in> PIdl x" by simp
1.1141 +  then have "\<exists>y. \<one> = y \<otimes> x \<and> y \<in> carrier R"
1.1142 +    unfolding cgenideal_def by blast
1.1143 +  then obtain y where ycarr: " y \<in> carrier R" and ylinv: "\<one> = y \<otimes> x"
1.1144 +    by fast+
1.1145 +  from ylinv and xcarr ycarr have yrinv: "\<one> = x \<otimes> y"
1.1146 +    by (simp add: m_comm)
1.1147    from ycarr and ylinv[symmetric] and yrinv[symmetric]
1.1148 -      have "\<exists>y \<in> carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
1.1149 -  from this and xcarr
1.1150 -      show "x \<in> Units R"
1.1151 -      unfolding Units_def
1.1152 -      by fast
1.1153 +  have "\<exists>y \<in> carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
1.1154 +  with xcarr show "x \<in> Units R"
1.1155 +    unfolding Units_def by fast
1.1156  qed
1.1157
1.1158 -lemma (in field) all_ideals:
1.1159 -  shows "{I. ideal I R} = {{\<zero>}, carrier R}"
1.1160 -apply (rule, rule)
1.1161 +lemma (in field) all_ideals: "{I. ideal I R} = {{\<zero>}, carrier R}"
1.1162 +  apply (rule, rule)
1.1163  proof -
1.1164    fix I
1.1165    assume a: "I \<in> {I. ideal I R}"
1.1166 -  with this
1.1167 -      interpret ideal I R by simp
1.1168 +  then interpret ideal I R by simp
1.1169
1.1170    show "I \<in> {{\<zero>}, carrier R}"
1.1171    proof (cases "\<exists>a. a \<in> I - {\<zero>}")
1.1172 -    assume "\<exists>a. a \<in> I - {\<zero>}"
1.1173 -    from this
1.1174 -        obtain a
1.1175 -          where aI: "a \<in> I"
1.1176 -          and anZ: "a \<noteq> \<zero>"
1.1177 -        by fast+
1.1178 -    from aI[THEN a_Hcarr] anZ
1.1179 -        have aUnit: "a \<in> Units R" by (simp add: field_Units)
1.1180 -    hence a: "a \<otimes> inv a = \<one>" by (rule Units_r_inv)
1.1181 -    from aI and aUnit
1.1182 -        have "a \<otimes> inv a \<in> I" by (simp add: I_r_closed del: Units_r_inv)
1.1183 -    hence oneI: "\<one> \<in> I" by (simp add: a[symmetric])
1.1184 +    case True
1.1185 +    then obtain a where aI: "a \<in> I" and anZ: "a \<noteq> \<zero>"
1.1186 +      by fast+
1.1187 +    from aI[THEN a_Hcarr] anZ have aUnit: "a \<in> Units R"
1.1188 +      by (simp add: field_Units)
1.1189 +    then have a: "a \<otimes> inv a = \<one>" by (rule Units_r_inv)
1.1190 +    from aI and aUnit have "a \<otimes> inv a \<in> I"
1.1191 +      by (simp add: I_r_closed del: Units_r_inv)
1.1192 +    then have oneI: "\<one> \<in> I" by (simp add: a[symmetric])
1.1193
1.1194      have "carrier R \<subseteq> I"
1.1195      proof
1.1196        fix x
1.1197        assume xcarr: "x \<in> carrier R"
1.1198 -      from oneI and this
1.1199 -          have "\<one> \<otimes> x \<in> I" by (rule I_r_closed)
1.1200 -      from this and xcarr
1.1201 -          show "x \<in> I" by simp
1.1202 +      with oneI have "\<one> \<otimes> x \<in> I" by (rule I_r_closed)
1.1203 +      with xcarr show "x \<in> I" by simp
1.1204      qed
1.1205 -    from this and a_subset
1.1206 -        have "I = carrier R" by fast
1.1207 -    thus "I \<in> {{\<zero>}, carrier R}" by fast
1.1208 +    with a_subset have "I = carrier R" by fast
1.1209 +    then show "I \<in> {{\<zero>}, carrier R}" by fast
1.1210    next
1.1211 -    assume "\<not> (\<exists>a. a \<in> I - {\<zero>})"
1.1212 -    hence IZ: "\<And>a. a \<in> I \<Longrightarrow> a = \<zero>" by simp
1.1213 +    case False
1.1214 +    then have IZ: "\<And>a. a \<in> I \<Longrightarrow> a = \<zero>" by simp
1.1215
1.1216      have a: "I \<subseteq> {\<zero>}"
1.1217      proof
1.1218        fix x
1.1219        assume "x \<in> I"
1.1220 -      hence "x = \<zero>" by (rule IZ)
1.1221 -      thus "x \<in> {\<zero>}" by fast
1.1222 +      then have "x = \<zero>" by (rule IZ)
1.1223 +      then show "x \<in> {\<zero>}" by fast
1.1224      qed
1.1225
1.1226      have "\<zero> \<in> I" by simp
1.1227 -    hence "{\<zero>} \<subseteq> I" by fast
1.1228 +    then have "{\<zero>} \<subseteq> I" by fast
1.1229
1.1230 -    from this and a
1.1231 -        have "I = {\<zero>}" by fast
1.1232 -    thus "I \<in> {{\<zero>}, carrier R}" by fast
1.1233 +    with a have "I = {\<zero>}" by fast
1.1234 +    then show "I \<in> {{\<zero>}, carrier R}" by fast
1.1235    qed
1.1236  qed (simp add: zeroideal oneideal)
1.1237
1.1238 @@ -985,52 +924,47 @@
1.1239  lemma (in cring) trivialideals_eq_field:
1.1240    assumes carrnzero: "carrier R \<noteq> {\<zero>}"
1.1241    shows "({I. ideal I R} = {{\<zero>}, carrier R}) = field R"
1.1242 -by (fast intro!: trivialideals_fieldI[OF carrnzero] field.all_ideals)
1.1243 +  by (fast intro!: trivialideals_fieldI[OF carrnzero] field.all_ideals)
1.1244
1.1245
1.1246  text {* Like zeroprimeideal for domains *}
1.1247 -lemma (in field) zeromaximalideal:
1.1248 -  "maximalideal {\<zero>} R"
1.1249 -apply (rule maximalidealI)
1.1250 -  apply (rule zeroideal)
1.1251 +lemma (in field) zeromaximalideal: "maximalideal {\<zero>} R"
1.1252 +  apply (rule maximalidealI)
1.1253 +    apply (rule zeroideal)
1.1254  proof-
1.1255 -  from one_not_zero
1.1256 -      have "\<one> \<notin> {\<zero>}" by simp
1.1257 -  from this and one_closed
1.1258 -      show "carrier R \<noteq> {\<zero>}" by fast
1.1259 +  from one_not_zero have "\<one> \<notin> {\<zero>}" by simp
1.1260 +  with one_closed show "carrier R \<noteq> {\<zero>}" by fast
1.1261  next
1.1262    fix J
1.1263    assume Jideal: "ideal J R"
1.1264 -  hence "J \<in> {I. ideal I R}"
1.1265 -      by fast
1.1266 -
1.1267 -  from this and all_ideals
1.1268 -      show "J = {\<zero>} \<or> J = carrier R" by simp
1.1269 +  then have "J \<in> {I. ideal I R}" by fast
1.1270 +  with all_ideals show "J = {\<zero>} \<or> J = carrier R"
1.1271 +    by simp
1.1272  qed
1.1273
1.1274  lemma (in cring) zeromaximalideal_fieldI:
1.1275    assumes zeromax: "maximalideal {\<zero>} R"
1.1276    shows "field R"
1.1277 -apply (rule trivialideals_fieldI, rule maximalideal.I_notcarr[OF zeromax])
1.1278 -apply rule apply clarsimp defer 1
1.1279 - apply (simp add: zeroideal oneideal)
1.1280 +  apply (rule trivialideals_fieldI, rule maximalideal.I_notcarr[OF zeromax])
1.1281 +  apply rule apply clarsimp defer 1
1.1282 +   apply (simp add: zeroideal oneideal)
1.1283  proof -
1.1284    fix J
1.1285    assume Jn0: "J \<noteq> {\<zero>}"
1.1286 -     and idealJ: "ideal J R"
1.1287 +    and idealJ: "ideal J R"
1.1288    interpret ideal J R by (rule idealJ)
1.1289 -  have "{\<zero>} \<subseteq> J" by (rule ccontr, simp)
1.1290 +  have "{\<zero>} \<subseteq> J" by (rule ccontr) simp
1.1291    from zeromax and idealJ and this and a_subset
1.1292 -      have "J = {\<zero>} \<or> J = carrier R" by (rule maximalideal.I_maximal)
1.1293 -  from this and Jn0
1.1294 -      show "J = carrier R" by simp
1.1295 +  have "J = {\<zero>} \<or> J = carrier R"
1.1296 +    by (rule maximalideal.I_maximal)
1.1297 +  with Jn0 show "J = carrier R"
1.1298 +    by simp
1.1299  qed
1.1300
1.1301 -lemma (in cring) zeromaximalideal_eq_field:
1.1302 -  "maximalideal {\<zero>} R = field R"
1.1303 -apply rule
1.1304 - apply (erule zeromaximalideal_fieldI)
1.1305 -apply (erule field.zeromaximalideal)
1.1306 -done
1.1307 +lemma (in cring) zeromaximalideal_eq_field: "maximalideal {\<zero>} R = field R"
1.1308 +  apply rule
1.1309 +   apply (erule zeromaximalideal_fieldI)
1.1310 +  apply (erule field.zeromaximalideal)
1.1311 +  done
1.1312
1.1313  end
```
```     2.1 --- a/src/HOL/Algebra/Ring.thy	Sat Sep 03 21:15:35 2011 +0200
2.2 +++ b/src/HOL/Algebra/Ring.thy	Sat Sep 03 22:05:25 2011 +0200
2.3 @@ -367,7 +367,7 @@
2.4  proof -
2.5    assume R: "x \<in> carrier R" "y \<in> carrier R"
2.6    then have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = (\<ominus> x \<oplus> x) \<otimes> y" by (simp add: l_distr)
2.7 -  also from R have "... = \<zero>" by (simp add: l_neg l_null)
2.8 +  also from R have "... = \<zero>" by (simp add: l_neg)
2.9    finally have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = \<zero>" .
2.10    with R have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
2.11    with R show ?thesis by (simp add: a_assoc r_neg)
2.12 @@ -378,7 +378,7 @@
2.13  proof -
2.14    assume R: "x \<in> carrier R" "y \<in> carrier R"
2.15    then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr)
2.16 -  also from R have "... = \<zero>" by (simp add: l_neg r_null)
2.17 +  also from R have "... = \<zero>" by (simp add: l_neg)
2.18    finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" .
2.19    with R have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
2.20    with R show ?thesis by (simp add: a_assoc r_neg )
2.21 @@ -464,7 +464,6 @@
2.22  proof -
2.23    interpret ring R by fact
2.24    interpret cring S by fact
2.25 -ML_val {* Algebra.print_structures @{context} *}
2.26    from RS show ?thesis by algebra
2.27  qed
2.28
```