author wenzelm Mon Sep 19 23:18:18 2011 +0200 (2011-09-19) changeset 45005 0d2d59525912 parent 45004 5bd261075711 child 45006 11a542f50fc3
tuned proofs;
```     1.1 --- a/src/HOL/Algebra/QuotRing.thy	Mon Sep 19 22:48:05 2011 +0200
1.2 +++ b/src/HOL/Algebra/QuotRing.thy	Mon Sep 19 23:18:18 2011 +0200
1.3 @@ -10,8 +10,7 @@
1.4
1.5  subsection {* Multiplication on Cosets *}
1.6
1.7 -definition
1.8 -  rcoset_mult :: "[('a, _) ring_scheme, 'a set, 'a set, 'a set] \<Rightarrow> 'a set"
1.9 +definition rcoset_mult :: "[('a, _) ring_scheme, 'a set, 'a set, 'a set] \<Rightarrow> 'a set"
1.10      ("[mod _:] _ \<Otimes>\<index> _" [81,81,81] 80)
1.11    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))"
1.12
1.13 @@ -19,86 +18,71 @@
1.14  text {* @{const "rcoset_mult"} fulfils the properties required by
1.15    congruences *}
1.17 -  "\<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> [mod I:] (I +> x) \<Otimes> (I +> y) = I +> (x \<otimes> y)"
1.18 -apply rule
1.19 -apply (rule, simp add: rcoset_mult_def, clarsimp)
1.20 -defer 1
1.21 -apply (rule, simp add: rcoset_mult_def)
1.22 -defer 1
1.23 +    "x \<in> carrier R \<Longrightarrow> y \<in> carrier R \<Longrightarrow> [mod I:] (I +> x) \<Otimes> (I +> y) = I +> (x \<otimes> y)"
1.24 +  apply rule
1.25 +  apply (rule, simp add: rcoset_mult_def, clarsimp)
1.26 +  defer 1
1.27 +  apply (rule, simp add: rcoset_mult_def)
1.28 +  defer 1
1.29  proof -
1.30    fix z x' y'
1.31    assume carr: "x \<in> carrier R" "y \<in> carrier R"
1.32 -     and x'rcos: "x' \<in> I +> x"
1.33 -     and y'rcos: "y' \<in> I +> y"
1.34 -     and zrcos: "z \<in> I +> x' \<otimes> y'"
1.35 +    and x'rcos: "x' \<in> I +> x"
1.36 +    and y'rcos: "y' \<in> I +> y"
1.37 +    and zrcos: "z \<in> I +> x' \<otimes> y'"
1.38 +
1.39 +  from x'rcos have "\<exists>h\<in>I. x' = h \<oplus> x"
1.40 +    by (simp add: a_r_coset_def r_coset_def)
1.41 +  then obtain hx where hxI: "hx \<in> I" and x': "x' = hx \<oplus> x"
1.42 +    by fast+
1.43
1.44 -  from x'rcos
1.45 -      have "\<exists>h\<in>I. x' = h \<oplus> x" by (simp add: a_r_coset_def r_coset_def)
1.46 -  from this obtain hx
1.47 -      where hxI: "hx \<in> I"
1.48 -      and x': "x' = hx \<oplus> x"
1.49 -      by fast+
1.50 -
1.51 -  from y'rcos
1.52 -      have "\<exists>h\<in>I. y' = h \<oplus> y" by (simp add: a_r_coset_def r_coset_def)
1.53 -  from this
1.54 -      obtain hy
1.55 -      where hyI: "hy \<in> I"
1.56 -      and y': "y' = hy \<oplus> y"
1.57 -      by fast+
1.58 +  from y'rcos have "\<exists>h\<in>I. y' = h \<oplus> y"
1.59 +    by (simp add: a_r_coset_def r_coset_def)
1.60 +  then obtain hy where hyI: "hy \<in> I" and y': "y' = hy \<oplus> y"
1.61 +    by fast+
1.62
1.63 -  from zrcos
1.64 -      have "\<exists>h\<in>I. z = h \<oplus> (x' \<otimes> y')" by (simp add: a_r_coset_def r_coset_def)
1.65 -  from this
1.66 -      obtain hz
1.67 -      where hzI: "hz \<in> I"
1.68 -      and z: "z = hz \<oplus> (x' \<otimes> y')"
1.69 -      by fast+
1.70 +  from zrcos have "\<exists>h\<in>I. z = h \<oplus> (x' \<otimes> y')"
1.71 +    by (simp add: a_r_coset_def r_coset_def)
1.72 +  then obtain hz where hzI: "hz \<in> I" and z: "z = hz \<oplus> (x' \<otimes> y')"
1.73 +    by fast+
1.74
1.75    note carr = carr hxI[THEN a_Hcarr] hyI[THEN a_Hcarr] hzI[THEN a_Hcarr]
1.76
1.77    from z have "z = hz \<oplus> (x' \<otimes> y')" .
1.78 -  also from x' y'
1.79 -      have "\<dots> = hz \<oplus> ((hx \<oplus> x) \<otimes> (hy \<oplus> y))" by simp
1.80 -  also from carr
1.81 -      have "\<dots> = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" by algebra
1.82 -  finally
1.83 -      have z2: "z = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" .
1.84 +  also from x' y' have "\<dots> = hz \<oplus> ((hx \<oplus> x) \<otimes> (hy \<oplus> y))" by simp
1.85 +  also from carr have "\<dots> = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" by algebra
1.86 +  finally have z2: "z = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" .
1.87
1.88 -  from hxI hyI hzI carr
1.89 -      have "hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy \<in> I"  by (simp add: I_l_closed I_r_closed)
1.90 +  from hxI hyI hzI carr have "hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy \<in> I"
1.91 +    by (simp add: I_l_closed I_r_closed)
1.92
1.93 -  from this and z2
1.94 -      have "\<exists>h\<in>I. z = h \<oplus> x \<otimes> y" by fast
1.95 -  thus "z \<in> I +> x \<otimes> y" by (simp add: a_r_coset_def r_coset_def)
1.96 +  with z2 have "\<exists>h\<in>I. z = h \<oplus> x \<otimes> y" by fast
1.97 +  then show "z \<in> I +> x \<otimes> y" by (simp add: a_r_coset_def r_coset_def)
1.98  next
1.99    fix z
1.100    assume xcarr: "x \<in> carrier R"
1.101 -     and ycarr: "y \<in> carrier R"
1.102 -     and zrcos: "z \<in> I +> x \<otimes> y"
1.103 -  from xcarr
1.104 -      have xself: "x \<in> I +> x" by (intro a_rcos_self)
1.105 -  from ycarr
1.106 -      have yself: "y \<in> I +> y" by (intro a_rcos_self)
1.107 -
1.108 -  from xself and yself and zrcos
1.109 -      show "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b" by fast
1.110 +    and ycarr: "y \<in> carrier R"
1.111 +    and zrcos: "z \<in> I +> x \<otimes> y"
1.112 +  from xcarr have xself: "x \<in> I +> x" by (intro a_rcos_self)
1.113 +  from ycarr have yself: "y \<in> I +> y" by (intro a_rcos_self)
1.114 +  show "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b"
1.115 +    using xself and yself and zrcos by fast
1.116  qed
1.117
1.118
1.119  subsection {* Quotient Ring Definition *}
1.120
1.121 -definition
1.122 -  FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring"  (infixl "Quot" 65)
1.123 +definition FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring"
1.124 +    (infixl "Quot" 65)
1.125    where "FactRing R I =
1.126 -    \<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>"
1.127 +    \<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, mult = rcoset_mult R I,
1.128 +      one = (I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>), zero = I, add = set_add R\<rparr>"
1.129
1.130
1.131  subsection {* Factorization over General Ideals *}
1.132
1.133  text {* The quotient is a ring *}
1.134 -lemma (in ideal) quotient_is_ring:
1.135 -  shows "ring (R Quot I)"
1.136 +lemma (in ideal) quotient_is_ring: "ring (R Quot I)"
1.137  apply (rule ringI)
1.138     --{* abelian group *}
1.139     apply (rule comm_group_abelian_groupI)
1.140 @@ -112,15 +96,15 @@
1.141        apply (clarify)
1.143       --{* mult @{text one_closed} *}
1.144 -     apply (force intro: one_closed)
1.145 +     apply force
1.146      --{* mult assoc *}
1.147      apply clarify
1.149     --{* mult one *}
1.150     apply clarify
1.153    apply clarify
1.156   --{* distr *}
1.157   apply clarify
1.159 @@ -131,8 +115,7 @@
1.160
1.161  text {* This is a ring homomorphism *}
1.162
1.163 -lemma (in ideal) rcos_ring_hom:
1.164 -  "(op +> I) \<in> ring_hom R (R Quot I)"
1.165 +lemma (in ideal) rcos_ring_hom: "(op +> I) \<in> ring_hom R (R Quot I)"
1.166  apply (rule ring_hom_memI)
1.167     apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
1.169 @@ -140,8 +123,7 @@
1.171  done
1.172
1.173 -lemma (in ideal) rcos_ring_hom_ring:
1.174 -  "ring_hom_ring R (R Quot I) (op +> I)"
1.175 +lemma (in ideal) rcos_ring_hom_ring: "ring_hom_ring R (R Quot I) (op +> I)"
1.176  apply (rule ring_hom_ringI)
1.177       apply (rule is_ring, rule quotient_is_ring)
1.178     apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
1.179 @@ -156,13 +138,14 @@
1.180    shows "cring (R Quot I)"
1.181  proof -
1.182    interpret cring R by fact
1.183 -  show ?thesis apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro)
1.184 -  apply (rule quotient_is_ring)
1.185 - apply (rule ring.axioms[OF quotient_is_ring])
1.186 -apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric])
1.187 -apply clarify
1.189 -done
1.190 +  show ?thesis
1.191 +    apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro)
1.192 +      apply (rule quotient_is_ring)
1.193 +     apply (rule ring.axioms[OF quotient_is_ring])
1.194 +    apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric])
1.195 +    apply clarify
1.197 +    done
1.198  qed
1.199
1.200  text {* Cosets as a ring homomorphism on crings *}
1.201 @@ -171,65 +154,57 @@
1.202    shows "ring_hom_cring R (R Quot I) (op +> I)"
1.203  proof -
1.204    interpret cring R by fact
1.205 -  show ?thesis apply (rule ring_hom_cringI)
1.206 -  apply (rule rcos_ring_hom_ring)
1.207 - apply (rule is_cring)
1.208 -apply (rule quotient_is_cring)
1.209 -apply (rule is_cring)
1.210 -done
1.211 +  show ?thesis
1.212 +    apply (rule ring_hom_cringI)
1.213 +      apply (rule rcos_ring_hom_ring)
1.214 +     apply (rule is_cring)
1.215 +    apply (rule quotient_is_cring)
1.216 +   apply (rule is_cring)
1.217 +   done
1.218  qed
1.219
1.220
1.221  subsection {* Factorization over Prime Ideals *}
1.222
1.223  text {* The quotient ring generated by a prime ideal is a domain *}
1.224 -lemma (in primeideal) quotient_is_domain:
1.225 -  shows "domain (R Quot I)"
1.226 -apply (rule domain.intro)
1.227 - apply (rule quotient_is_cring, rule is_cring)
1.228 -apply (rule domain_axioms.intro)
1.229 - apply (simp add: FactRing_def) defer 1
1.230 - apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarify)
1.232 +lemma (in primeideal) quotient_is_domain: "domain (R Quot I)"
1.233 +  apply (rule domain.intro)
1.234 +   apply (rule quotient_is_cring, rule is_cring)
1.235 +  apply (rule domain_axioms.intro)
1.236 +   apply (simp add: FactRing_def) defer 1
1.237 +    apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarify)
1.239  proof (rule ccontr, clarsimp)
1.240    assume "I +> \<one> = I"
1.241 -  hence "\<one> \<in> I" by (simp only: a_coset_join1 one_closed a_subgroup)
1.242 -  hence "carrier R \<subseteq> I" by (subst one_imp_carrier, simp, fast)
1.243 -  from this and a_subset
1.244 -      have "I = carrier R" by fast
1.245 -  from this and I_notcarr
1.246 -      show "False" by fast
1.247 +  then have "\<one> \<in> I" by (simp only: a_coset_join1 one_closed a_subgroup)
1.248 +  then have "carrier R \<subseteq> I" by (subst one_imp_carrier, simp, fast)
1.249 +  with a_subset have "I = carrier R" by fast
1.250 +  with I_notcarr show False by fast
1.251  next
1.252    fix x y
1.253    assume carr: "x \<in> carrier R" "y \<in> carrier R"
1.254 -     and a: "I +> x \<otimes> y = I"
1.255 -     and b: "I +> y \<noteq> I"
1.256 +    and a: "I +> x \<otimes> y = I"
1.257 +    and b: "I +> y \<noteq> I"
1.258
1.259    have ynI: "y \<notin> I"
1.260    proof (rule ccontr, simp)
1.261      assume "y \<in> I"
1.262 -    hence "I +> y = I" by (rule a_rcos_const)
1.263 -    from this and b
1.264 -        show "False" by simp
1.265 +    then have "I +> y = I" by (rule a_rcos_const)
1.266 +    with b show False by simp
1.267    qed
1.268
1.269 -  from carr
1.270 -      have "x \<otimes> y \<in> I +> x \<otimes> y" by (simp add: a_rcos_self)
1.271 -  from this
1.272 -      have xyI: "x \<otimes> y \<in> I" by (simp add: a)
1.273 +  from carr have "x \<otimes> y \<in> I +> x \<otimes> y" by (simp add: a_rcos_self)
1.274 +  then have xyI: "x \<otimes> y \<in> I" by (simp add: a)
1.275
1.276 -  from xyI and carr
1.277 -      have xI: "x \<in> I \<or> y \<in> I" by (simp add: I_prime)
1.278 -  from this and ynI
1.279 -      have "x \<in> I" by fast
1.280 -  thus "I +> x = I" by (rule a_rcos_const)
1.281 +  from xyI and carr have xI: "x \<in> I \<or> y \<in> I" by (simp add: I_prime)
1.282 +  with ynI have "x \<in> I" by fast
1.283 +  then show "I +> x = I" by (rule a_rcos_const)
1.284  qed
1.285
1.286  text {* Generating right cosets of a prime ideal is a homomorphism
1.287          on commutative rings *}
1.288 -lemma (in primeideal) rcos_ring_hom_cring:
1.289 -  shows "ring_hom_cring R (R Quot I) (op +> I)"
1.290 -by (rule rcos_ring_hom_cring, rule is_cring)
1.291 +lemma (in primeideal) rcos_ring_hom_cring: "ring_hom_cring R (R Quot I) (op +> I)"
1.292 +  by (rule rcos_ring_hom_cring) (rule is_cring)
1.293
1.294
1.295  subsection {* Factorization over Maximal Ideals *}
1.296 @@ -243,106 +218,92 @@
1.297    shows "field (R Quot I)"
1.298  proof -
1.299    interpret cring R by fact
1.300 -  show ?thesis apply (intro cring.cring_fieldI2)
1.301 -  apply (rule quotient_is_cring, rule is_cring)
1.302 - defer 1
1.303 - apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarsimp)
1.305 -proof (rule ccontr, simp)
1.306 -  --{* Quotient is not empty *}
1.307 -  assume "\<zero>\<^bsub>R Quot I\<^esub> = \<one>\<^bsub>R Quot I\<^esub>"
1.308 -  hence II1: "I = I +> \<one>" by (simp add: FactRing_def)
1.309 -  from a_rcos_self[OF one_closed]
1.310 -  have "\<one> \<in> I" by (simp add: II1[symmetric])
1.311 -  hence "I = carrier R" by (rule one_imp_carrier)
1.312 -  from this and I_notcarr
1.313 -  show "False" by simp
1.314 -next
1.315 -  --{* Existence of Inverse *}
1.316 -  fix a
1.317 -  assume IanI: "I +> a \<noteq> I"
1.318 -    and acarr: "a \<in> carrier R"
1.319 +  show ?thesis
1.320 +    apply (intro cring.cring_fieldI2)
1.321 +      apply (rule quotient_is_cring, rule is_cring)
1.322 +     defer 1
1.323 +     apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarsimp)
1.325 +  proof (rule ccontr, simp)
1.326 +    --{* Quotient is not empty *}
1.327 +    assume "\<zero>\<^bsub>R Quot I\<^esub> = \<one>\<^bsub>R Quot I\<^esub>"
1.328 +    then have II1: "I = I +> \<one>" by (simp add: FactRing_def)
1.329 +    from a_rcos_self[OF one_closed] have "\<one> \<in> I"
1.330 +      by (simp add: II1[symmetric])
1.331 +    then have "I = carrier R" by (rule one_imp_carrier)
1.332 +    with I_notcarr show False by simp
1.333 +  next
1.334 +    --{* Existence of Inverse *}
1.335 +    fix a
1.336 +    assume IanI: "I +> a \<noteq> I" and acarr: "a \<in> carrier R"
1.337
1.338 -  --{* Helper ideal @{text "J"} *}
1.339 -  def J \<equiv> "(carrier R #> a) <+> I :: 'a set"
1.340 -  have idealJ: "ideal J R"
1.341 -    apply (unfold J_def, rule add_ideals)
1.342 -     apply (simp only: cgenideal_eq_rcos[symmetric], rule cgenideal_ideal, rule acarr)
1.343 -    apply (rule is_ideal)
1.344 -    done
1.345 +    --{* Helper ideal @{text "J"} *}
1.346 +    def J \<equiv> "(carrier R #> a) <+> I :: 'a set"
1.347 +    have idealJ: "ideal J R"
1.348 +      apply (unfold J_def, rule add_ideals)
1.349 +       apply (simp only: cgenideal_eq_rcos[symmetric], rule cgenideal_ideal, rule acarr)
1.350 +      apply (rule is_ideal)
1.351 +      done
1.352
1.353 -  --{* Showing @{term "J"} not smaller than @{term "I"} *}
1.354 -  have IinJ: "I \<subseteq> J"
1.356 -    fix x
1.357 -    assume xI: "x \<in> I"
1.358 -    have Zcarr: "\<zero> \<in> carrier R" by fast
1.359 -    from xI[THEN a_Hcarr] acarr
1.360 -    have "x = \<zero> \<otimes> a \<oplus> x" by algebra
1.361 -
1.362 -    from Zcarr and xI and this
1.363 -    show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast
1.364 -  qed
1.365 -
1.366 -  --{* Showing @{term "J \<noteq> I"} *}
1.367 -  have anI: "a \<notin> I"
1.368 -  proof (rule ccontr, simp)
1.369 -    assume "a \<in> I"
1.370 -    hence "I +> a = I" by (rule a_rcos_const)
1.371 -    from this and IanI
1.372 -    show "False" by simp
1.373 -  qed
1.374 +    --{* Showing @{term "J"} not smaller than @{term "I"} *}
1.375 +    have IinJ: "I \<subseteq> J"
1.377 +      fix x
1.378 +      assume xI: "x \<in> I"
1.379 +      have Zcarr: "\<zero> \<in> carrier R" by fast
1.380 +      from xI[THEN a_Hcarr] acarr
1.381 +      have "x = \<zero> \<otimes> a \<oplus> x" by algebra
1.382 +      with Zcarr and xI show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast
1.383 +    qed
1.384
1.385 -  have aJ: "a \<in> J"
1.387 -    from acarr
1.388 -    have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra
1.390 -    show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast
1.391 -  qed
1.392 -
1.393 -  from aJ and anI
1.394 -  have JnI: "J \<noteq> I" by fast
1.395 +    --{* Showing @{term "J \<noteq> I"} *}
1.396 +    have anI: "a \<notin> I"
1.397 +    proof (rule ccontr, simp)
1.398 +      assume "a \<in> I"
1.399 +      then have "I +> a = I" by (rule a_rcos_const)
1.400 +      with IanI show False by simp
1.401 +    qed
1.402
1.403 -  --{* Deducing @{term "J = carrier R"} because @{term "I"} is maximal *}
1.404 -  from idealJ and IinJ
1.405 -  have "J = I \<or> J = carrier R"
1.406 -  proof (rule I_maximal, unfold J_def)
1.407 -    have "carrier R #> a \<subseteq> carrier R"
1.408 -      using subset_refl acarr
1.409 -      by (rule r_coset_subset_G)
1.410 -    from this and a_subset
1.411 -    show "carrier R #> a <+> I \<subseteq> carrier R" by (rule set_add_closed)
1.412 -  qed
1.413 +    have aJ: "a \<in> J"
1.415 +      from acarr
1.416 +      have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra
1.418 +      show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast
1.419 +    qed
1.420
1.421 -  from this and JnI
1.422 -  have Jcarr: "J = carrier R" by simp
1.423 +    from aJ and anI have JnI: "J \<noteq> I" by fast
1.424
1.425 -  --{* Calculating an inverse for @{term "a"} *}
1.426 -  from one_closed[folded Jcarr]
1.427 -  have "\<exists>r\<in>carrier R. \<exists>i\<in>I. \<one> = r \<otimes> a \<oplus> i"
1.429 -  from this
1.430 -  obtain r i
1.431 -    where rcarr: "r \<in> carrier R"
1.432 -      and iI: "i \<in> I"
1.433 -      and one: "\<one> = r \<otimes> a \<oplus> i"
1.434 -    by fast
1.435 -  from one and rcarr and acarr and iI[THEN a_Hcarr]
1.436 -  have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra
1.437 +    --{* Deducing @{term "J = carrier R"} because @{term "I"} is maximal *}
1.438 +    from idealJ and IinJ have "J = I \<or> J = carrier R"
1.439 +    proof (rule I_maximal, unfold J_def)
1.440 +      have "carrier R #> a \<subseteq> carrier R"
1.441 +        using subset_refl acarr by (rule r_coset_subset_G)
1.442 +      then show "carrier R #> a <+> I \<subseteq> carrier R"
1.443 +        using a_subset by (rule set_add_closed)
1.444 +    qed
1.445 +
1.446 +    with JnI have Jcarr: "J = carrier R" by simp
1.447
1.448 -  --{* Lifting to cosets *}
1.449 -  from iI
1.450 -  have "\<ominus>i \<oplus> \<one> \<in> I +> \<one>"
1.451 -    by (intro a_rcosI, simp, intro a_subset, simp)
1.452 -  from this and rai1
1.453 -  have "a \<otimes> r \<in> I +> \<one>" by simp
1.454 -  from this have "I +> \<one> = I +> a \<otimes> r"
1.455 -    by (rule a_repr_independence, simp) (rule a_subgroup)
1.456 +    --{* Calculating an inverse for @{term "a"} *}
1.457 +    from one_closed[folded Jcarr]
1.458 +    have "\<exists>r\<in>carrier R. \<exists>i\<in>I. \<one> = r \<otimes> a \<oplus> i"
1.460 +    then obtain r i where rcarr: "r \<in> carrier R"
1.461 +      and iI: "i \<in> I" and one: "\<one> = r \<otimes> a \<oplus> i" by fast
1.462 +    from one and rcarr and acarr and iI[THEN a_Hcarr]
1.463 +    have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra
1.464
1.465 -  from rcarr and this[symmetric]
1.466 -  show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast
1.467 -qed
1.468 +    --{* Lifting to cosets *}
1.469 +    from iI have "\<ominus>i \<oplus> \<one> \<in> I +> \<one>"
1.470 +      by (intro a_rcosI, simp, intro a_subset, simp)
1.471 +    with rai1 have "a \<otimes> r \<in> I +> \<one>" by simp
1.472 +    then have "I +> \<one> = I +> a \<otimes> r"
1.473 +      by (rule a_repr_independence, simp) (rule a_subgroup)
1.474 +
1.475 +    from rcarr and this[symmetric]
1.476 +    show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast
1.477 +  qed
1.478  qed
1.479
1.480  end
```