src/HOL/Algebra/RingHom.thy
 author ballarin Tue Dec 16 21:10:53 2008 +0100 (2008-12-16) changeset 29237 e90d9d51106b parent 28823 dcbef866c9e2 child 29240 bb81c3709fb6 permissions -rw-r--r--
More porting to new locales.
```     1 (*
```
```     2   Title:     HOL/Algebra/RingHom.thy
```
```     3   Author:    Stephan Hohe, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 theory RingHom
```
```     7 imports Ideal
```
```     8 begin
```
```     9
```
```    10 section {* Homomorphisms of Non-Commutative Rings *}
```
```    11
```
```    12 text {* Lifting existing lemmas in a @{text ring_hom_ring} locale *}
```
```    13 locale ring_hom_ring = R: ring R + S: ring S +
```
```    14   fixes h
```
```    15   assumes homh: "h \<in> ring_hom R S"
```
```    16   notes hom_mult [simp] = ring_hom_mult [OF homh]
```
```    17     and hom_one [simp] = ring_hom_one [OF homh]
```
```    18
```
```    19 sublocale ring_hom_cring \<subseteq> ring_hom_ring
```
```    20   proof qed (rule homh)
```
```    21
```
```    22 sublocale ring_hom_ring \<subseteq> abelian_group_hom R S
```
```    23 apply (rule abelian_group_homI)
```
```    24   apply (rule R.is_abelian_group)
```
```    25  apply (rule S.is_abelian_group)
```
```    26 apply (intro group_hom.intro group_hom_axioms.intro)
```
```    27   apply (rule R.a_group)
```
```    28  apply (rule S.a_group)
```
```    29 apply (insert homh, unfold hom_def ring_hom_def)
```
```    30 apply simp
```
```    31 done
```
```    32
```
```    33 lemma (in ring_hom_ring) is_ring_hom_ring:
```
```    34   "ring_hom_ring R S h"
```
```    35   by (rule ring_hom_ring_axioms)
```
```    36
```
```    37 lemma ring_hom_ringI:
```
```    38   fixes R (structure) and S (structure)
```
```    39   assumes "ring R" "ring S"
```
```    40   assumes (* morphism: "h \<in> carrier R \<rightarrow> carrier S" *)
```
```    41           hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
```
```    42       and compatible_mult: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
```
```    43       and compatible_add: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
```
```    44       and compatible_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
```
```    45   shows "ring_hom_ring R S h"
```
```    46 proof -
```
```    47   interpret ring R by fact
```
```    48   interpret ring S by fact
```
```    49   show ?thesis apply unfold_locales
```
```    50 apply (unfold ring_hom_def, safe)
```
```    51    apply (simp add: hom_closed Pi_def)
```
```    52   apply (erule (1) compatible_mult)
```
```    53  apply (erule (1) compatible_add)
```
```    54 apply (rule compatible_one)
```
```    55 done
```
```    56 qed
```
```    57
```
```    58 lemma ring_hom_ringI2:
```
```    59   assumes "ring R" "ring S"
```
```    60   assumes h: "h \<in> ring_hom R S"
```
```    61   shows "ring_hom_ring R S h"
```
```    62 proof -
```
```    63   interpret R!: ring R by fact
```
```    64   interpret S!: ring S by fact
```
```    65   show ?thesis apply (intro ring_hom_ring.intro ring_hom_ring_axioms.intro)
```
```    66     apply (rule R.is_ring)
```
```    67     apply (rule S.is_ring)
```
```    68     apply (rule h)
```
```    69     done
```
```    70 qed
```
```    71
```
```    72 lemma ring_hom_ringI3:
```
```    73   fixes R (structure) and S (structure)
```
```    74   assumes "abelian_group_hom R S h" "ring R" "ring S"
```
```    75   assumes compatible_mult: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
```
```    76       and compatible_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
```
```    77   shows "ring_hom_ring R S h"
```
```    78 proof -
```
```    79   interpret abelian_group_hom R S h by fact
```
```    80   interpret R!: ring R by fact
```
```    81   interpret S!: ring S by fact
```
```    82   show ?thesis apply (intro ring_hom_ring.intro ring_hom_ring_axioms.intro, rule R.is_ring, rule S.is_ring)
```
```    83     apply (insert group_hom.homh[OF a_group_hom])
```
```    84     apply (unfold hom_def ring_hom_def, simp)
```
```    85     apply safe
```
```    86     apply (erule (1) compatible_mult)
```
```    87     apply (rule compatible_one)
```
```    88     done
```
```    89 qed
```
```    90
```
```    91 lemma ring_hom_cringI:
```
```    92   assumes "ring_hom_ring R S h" "cring R" "cring S"
```
```    93   shows "ring_hom_cring R S h"
```
```    94 proof -
```
```    95   interpret ring_hom_ring R S h by fact
```
```    96   interpret R!: cring R by fact
```
```    97   interpret S!: cring S by fact
```
```    98   show ?thesis by (intro ring_hom_cring.intro ring_hom_cring_axioms.intro)
```
```    99     (rule R.is_cring, rule S.is_cring, rule homh)
```
```   100 qed
```
```   101
```
```   102 subsection {* The Kernel of a Ring Homomorphism *}
```
```   103
```
```   104 --"the kernel of a ring homomorphism is an ideal"
```
```   105 lemma (in ring_hom_ring) kernel_is_ideal:
```
```   106   shows "ideal (a_kernel R S h) R"
```
```   107 apply (rule idealI)
```
```   108    apply (rule R.is_ring)
```
```   109   apply (rule additive_subgroup.a_subgroup[OF additive_subgroup_a_kernel])
```
```   110  apply (unfold a_kernel_def', simp+)
```
```   111 done
```
```   112
```
```   113 text {* Elements of the kernel are mapped to zero *}
```
```   114 lemma (in abelian_group_hom) kernel_zero [simp]:
```
```   115   "i \<in> a_kernel R S h \<Longrightarrow> h i = \<zero>\<^bsub>S\<^esub>"
```
```   116 by (simp add: a_kernel_defs)
```
```   117
```
```   118
```
```   119 subsection {* Cosets *}
```
```   120
```
```   121 text {* Cosets of the kernel correspond to the elements of the image of the homomorphism *}
```
```   122 lemma (in ring_hom_ring) rcos_imp_homeq:
```
```   123   assumes acarr: "a \<in> carrier R"
```
```   124       and xrcos: "x \<in> a_kernel R S h +> a"
```
```   125   shows "h x = h a"
```
```   126 proof -
```
```   127   interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal)
```
```   128
```
```   129   from xrcos
```
```   130       have "\<exists>i \<in> a_kernel R S h. x = i \<oplus> a" by (simp add: a_r_coset_defs)
```
```   131   from this obtain i
```
```   132       where iker: "i \<in> a_kernel R S h"
```
```   133         and x: "x = i \<oplus> a"
```
```   134       by fast+
```
```   135   note carr = acarr iker[THEN a_Hcarr]
```
```   136
```
```   137   from x
```
```   138       have "h x = h (i \<oplus> a)" by simp
```
```   139   also from carr
```
```   140       have "\<dots> = h i \<oplus>\<^bsub>S\<^esub> h a" by simp
```
```   141   also from iker
```
```   142       have "\<dots> = \<zero>\<^bsub>S\<^esub> \<oplus>\<^bsub>S\<^esub> h a" by simp
```
```   143   also from carr
```
```   144       have "\<dots> = h a" by simp
```
```   145   finally
```
```   146       show "h x = h a" .
```
```   147 qed
```
```   148
```
```   149 lemma (in ring_hom_ring) homeq_imp_rcos:
```
```   150   assumes acarr: "a \<in> carrier R"
```
```   151       and xcarr: "x \<in> carrier R"
```
```   152       and hx: "h x = h a"
```
```   153   shows "x \<in> a_kernel R S h +> a"
```
```   154 proof -
```
```   155   interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal)
```
```   156
```
```   157   note carr = acarr xcarr
```
```   158   note hcarr = acarr[THEN hom_closed] xcarr[THEN hom_closed]
```
```   159
```
```   160   from hx and hcarr
```
```   161       have a: "h x \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub>h a = \<zero>\<^bsub>S\<^esub>" by algebra
```
```   162   from carr
```
```   163       have "h x \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub>h a = h (x \<oplus> \<ominus>a)" by simp
```
```   164   from a and this
```
```   165       have b: "h (x \<oplus> \<ominus>a) = \<zero>\<^bsub>S\<^esub>" by simp
```
```   166
```
```   167   from carr have "x \<oplus> \<ominus>a \<in> carrier R" by simp
```
```   168   from this and b
```
```   169       have "x \<oplus> \<ominus>a \<in> a_kernel R S h"
```
```   170       unfolding a_kernel_def'
```
```   171       by fast
```
```   172
```
```   173   from this and carr
```
```   174       show "x \<in> a_kernel R S h +> a" by (simp add: a_rcos_module_rev)
```
```   175 qed
```
```   176
```
```   177 corollary (in ring_hom_ring) rcos_eq_homeq:
```
```   178   assumes acarr: "a \<in> carrier R"
```
```   179   shows "(a_kernel R S h) +> a = {x \<in> carrier R. h x = h a}"
```
```   180 apply rule defer 1
```
```   181 apply clarsimp defer 1
```
```   182 proof
```
```   183   interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal)
```
```   184
```
```   185   fix x
```
```   186   assume xrcos: "x \<in> a_kernel R S h +> a"
```
```   187   from acarr and this
```
```   188       have xcarr: "x \<in> carrier R"
```
```   189       by (rule a_elemrcos_carrier)
```
```   190
```
```   191   from xrcos
```
```   192       have "h x = h a" by (rule rcos_imp_homeq[OF acarr])
```
```   193   from xcarr and this
```
```   194       show "x \<in> {x \<in> carrier R. h x = h a}" by fast
```
```   195 next
```
```   196   interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal)
```
```   197
```
```   198   fix x
```
```   199   assume xcarr: "x \<in> carrier R"
```
```   200      and hx: "h x = h a"
```
```   201   from acarr xcarr hx
```
```   202       show "x \<in> a_kernel R S h +> a" by (rule homeq_imp_rcos)
```
```   203 qed
```
```   204
```
```   205 end
```