author | ballarin |
Fri, 19 Dec 2008 14:31:17 +0100 | |
changeset 29246 | 3593802c9cf1 |
parent 29240 | bb81c3709fb6 |
child 30729 | 461ee3e49ad3 |
permissions | -rw-r--r-- |
20318
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Restructured algebra library, added ideals and quotient rings.
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1 |
(* |
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Restructured algebra library, added ideals and quotient rings.
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2 |
Title: HOL/Algebra/RingHom.thy |
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Restructured algebra library, added ideals and quotient rings.
ballarin
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3 |
Author: Stephan Hohe, TU Muenchen |
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Restructured algebra library, added ideals and quotient rings.
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4 |
*) |
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Restructured algebra library, added ideals and quotient rings.
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5 |
|
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Restructured algebra library, added ideals and quotient rings.
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6 |
theory RingHom |
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Restructured algebra library, added ideals and quotient rings.
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7 |
imports Ideal |
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Restructured algebra library, added ideals and quotient rings.
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8 |
begin |
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Restructured algebra library, added ideals and quotient rings.
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9 |
|
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Restructured algebra library, added ideals and quotient rings.
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10 |
section {* Homomorphisms of Non-Commutative Rings *} |
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Restructured algebra library, added ideals and quotient rings.
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11 |
|
21502 | 12 |
text {* Lifting existing lemmas in a @{text ring_hom_ring} locale *} |
29240 | 13 |
locale ring_hom_ring = R: ring R + S: ring S |
14 |
for R (structure) and S (structure) + |
|
29237 | 15 |
fixes h |
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Restructured algebra library, added ideals and quotient rings.
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16 |
assumes homh: "h \<in> ring_hom R S" |
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Restructured algebra library, added ideals and quotient rings.
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17 |
notes hom_mult [simp] = ring_hom_mult [OF homh] |
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Restructured algebra library, added ideals and quotient rings.
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18 |
and hom_one [simp] = ring_hom_one [OF homh] |
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Restructured algebra library, added ideals and quotient rings.
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19 |
|
29246 | 20 |
sublocale ring_hom_cring \<subseteq> ring: ring_hom_ring |
28823 | 21 |
proof qed (rule homh) |
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Restructured algebra library, added ideals and quotient rings.
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22 |
|
29246 | 23 |
sublocale ring_hom_ring \<subseteq> abelian_group: abelian_group_hom R S |
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Restructured algebra library, added ideals and quotient rings.
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24 |
apply (rule abelian_group_homI) |
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Restructured algebra library, added ideals and quotient rings.
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25 |
apply (rule R.is_abelian_group) |
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Restructured algebra library, added ideals and quotient rings.
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26 |
apply (rule S.is_abelian_group) |
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Restructured algebra library, added ideals and quotient rings.
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27 |
apply (intro group_hom.intro group_hom_axioms.intro) |
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Restructured algebra library, added ideals and quotient rings.
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28 |
apply (rule R.a_group) |
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Restructured algebra library, added ideals and quotient rings.
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29 |
apply (rule S.a_group) |
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Restructured algebra library, added ideals and quotient rings.
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30 |
apply (insert homh, unfold hom_def ring_hom_def) |
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Restructured algebra library, added ideals and quotient rings.
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31 |
apply simp |
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Restructured algebra library, added ideals and quotient rings.
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32 |
done |
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Restructured algebra library, added ideals and quotient rings.
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33 |
|
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Restructured algebra library, added ideals and quotient rings.
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34 |
lemma (in ring_hom_ring) is_ring_hom_ring: |
27611 | 35 |
"ring_hom_ring R S h" |
36 |
by (rule ring_hom_ring_axioms) |
|
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Restructured algebra library, added ideals and quotient rings.
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37 |
|
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Restructured algebra library, added ideals and quotient rings.
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38 |
lemma ring_hom_ringI: |
27611 | 39 |
fixes R (structure) and S (structure) |
40 |
assumes "ring R" "ring S" |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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41 |
assumes (* morphism: "h \<in> carrier R \<rightarrow> carrier S" *) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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42 |
hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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43 |
and compatible_mult: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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44 |
and compatible_add: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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45 |
and compatible_one: "h \<one> = \<one>\<^bsub>S\<^esub>" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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46 |
shows "ring_hom_ring R S h" |
27611 | 47 |
proof - |
29237 | 48 |
interpret ring R by fact |
49 |
interpret ring S by fact |
|
27611 | 50 |
show ?thesis apply unfold_locales |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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51 |
apply (unfold ring_hom_def, safe) |
23463 | 52 |
apply (simp add: hom_closed Pi_def) |
53 |
apply (erule (1) compatible_mult) |
|
54 |
apply (erule (1) compatible_add) |
|
55 |
apply (rule compatible_one) |
|
20318
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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56 |
done |
27611 | 57 |
qed |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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58 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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59 |
lemma ring_hom_ringI2: |
27611 | 60 |
assumes "ring R" "ring S" |
23350 | 61 |
assumes h: "h \<in> ring_hom R S" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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62 |
shows "ring_hom_ring R S h" |
27611 | 63 |
proof - |
29237 | 64 |
interpret R!: ring R by fact |
65 |
interpret S!: ring S by fact |
|
27611 | 66 |
show ?thesis apply (intro ring_hom_ring.intro ring_hom_ring_axioms.intro) |
67 |
apply (rule R.is_ring) |
|
68 |
apply (rule S.is_ring) |
|
69 |
apply (rule h) |
|
70 |
done |
|
71 |
qed |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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72 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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73 |
lemma ring_hom_ringI3: |
27611 | 74 |
fixes R (structure) and S (structure) |
75 |
assumes "abelian_group_hom R S h" "ring R" "ring S" |
|
20318
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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76 |
assumes compatible_mult: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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77 |
and compatible_one: "h \<one> = \<one>\<^bsub>S\<^esub>" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
78 |
shows "ring_hom_ring R S h" |
27611 | 79 |
proof - |
29237 | 80 |
interpret abelian_group_hom R S h by fact |
81 |
interpret R!: ring R by fact |
|
82 |
interpret S!: ring S by fact |
|
27611 | 83 |
show ?thesis apply (intro ring_hom_ring.intro ring_hom_ring_axioms.intro, rule R.is_ring, rule S.is_ring) |
84 |
apply (insert group_hom.homh[OF a_group_hom]) |
|
85 |
apply (unfold hom_def ring_hom_def, simp) |
|
86 |
apply safe |
|
87 |
apply (erule (1) compatible_mult) |
|
88 |
apply (rule compatible_one) |
|
89 |
done |
|
90 |
qed |
|
20318
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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91 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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92 |
lemma ring_hom_cringI: |
27611 | 93 |
assumes "ring_hom_ring R S h" "cring R" "cring S" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
94 |
shows "ring_hom_cring R S h" |
27611 | 95 |
proof - |
29237 | 96 |
interpret ring_hom_ring R S h by fact |
97 |
interpret R!: cring R by fact |
|
98 |
interpret S!: cring S by fact |
|
27611 | 99 |
show ?thesis by (intro ring_hom_cring.intro ring_hom_cring_axioms.intro) |
23463 | 100 |
(rule R.is_cring, rule S.is_cring, rule homh) |
27611 | 101 |
qed |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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102 |
|
27717
21bbd410ba04
Generalised polynomial lemmas from cring to ring.
ballarin
parents:
27611
diff
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103 |
subsection {* The Kernel of a Ring Homomorphism *} |
20318
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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104 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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105 |
--"the kernel of a ring homomorphism is an ideal" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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106 |
lemma (in ring_hom_ring) kernel_is_ideal: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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107 |
shows "ideal (a_kernel R S h) R" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
108 |
apply (rule idealI) |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
109 |
apply (rule R.is_ring) |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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110 |
apply (rule additive_subgroup.a_subgroup[OF additive_subgroup_a_kernel]) |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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111 |
apply (unfold a_kernel_def', simp+) |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
112 |
done |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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changeset
|
113 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
114 |
text {* Elements of the kernel are mapped to zero *} |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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115 |
lemma (in abelian_group_hom) kernel_zero [simp]: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
116 |
"i \<in> a_kernel R S h \<Longrightarrow> h i = \<zero>\<^bsub>S\<^esub>" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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117 |
by (simp add: a_kernel_defs) |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
118 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
119 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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changeset
|
120 |
subsection {* Cosets *} |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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121 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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122 |
text {* Cosets of the kernel correspond to the elements of the image of the homomorphism *} |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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123 |
lemma (in ring_hom_ring) rcos_imp_homeq: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
124 |
assumes acarr: "a \<in> carrier R" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
125 |
and xrcos: "x \<in> a_kernel R S h +> a" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
126 |
shows "h x = h a" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
127 |
proof - |
29237 | 128 |
interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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129 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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changeset
|
130 |
from xrcos |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
131 |
have "\<exists>i \<in> a_kernel R S h. x = i \<oplus> a" by (simp add: a_r_coset_defs) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
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132 |
from this obtain i |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
133 |
where iker: "i \<in> a_kernel R S h" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
134 |
and x: "x = i \<oplus> a" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
135 |
by fast+ |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
136 |
note carr = acarr iker[THEN a_Hcarr] |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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changeset
|
137 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
138 |
from x |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
139 |
have "h x = h (i \<oplus> a)" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
140 |
also from carr |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
141 |
have "\<dots> = h i \<oplus>\<^bsub>S\<^esub> h a" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
142 |
also from iker |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
143 |
have "\<dots> = \<zero>\<^bsub>S\<^esub> \<oplus>\<^bsub>S\<^esub> h a" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
144 |
also from carr |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
145 |
have "\<dots> = h a" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
146 |
finally |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
147 |
show "h x = h a" . |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
148 |
qed |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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changeset
|
149 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
150 |
lemma (in ring_hom_ring) homeq_imp_rcos: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
151 |
assumes acarr: "a \<in> carrier R" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
152 |
and xcarr: "x \<in> carrier R" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
153 |
and hx: "h x = h a" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
154 |
shows "x \<in> a_kernel R S h +> a" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
155 |
proof - |
29237 | 156 |
interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
157 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
158 |
note carr = acarr xcarr |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
159 |
note hcarr = acarr[THEN hom_closed] xcarr[THEN hom_closed] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
160 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
161 |
from hx and hcarr |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
162 |
have a: "h x \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub>h a = \<zero>\<^bsub>S\<^esub>" by algebra |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
163 |
from carr |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
164 |
have "h x \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub>h a = h (x \<oplus> \<ominus>a)" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
165 |
from a and this |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
166 |
have b: "h (x \<oplus> \<ominus>a) = \<zero>\<^bsub>S\<^esub>" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
167 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
168 |
from carr have "x \<oplus> \<ominus>a \<in> carrier R" by simp |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
169 |
from this and b |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
170 |
have "x \<oplus> \<ominus>a \<in> a_kernel R S h" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
171 |
unfolding a_kernel_def' |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
172 |
by fast |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
173 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
174 |
from this and carr |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
175 |
show "x \<in> a_kernel R S h +> a" by (simp add: a_rcos_module_rev) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
176 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
177 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
178 |
corollary (in ring_hom_ring) rcos_eq_homeq: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
179 |
assumes acarr: "a \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
180 |
shows "(a_kernel R S h) +> a = {x \<in> carrier R. h x = h a}" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
181 |
apply rule defer 1 |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
182 |
apply clarsimp defer 1 |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
183 |
proof |
29237 | 184 |
interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
185 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
186 |
fix x |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
187 |
assume xrcos: "x \<in> a_kernel R S h +> a" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
188 |
from acarr and this |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
189 |
have xcarr: "x \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
190 |
by (rule a_elemrcos_carrier) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
191 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
192 |
from xrcos |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
193 |
have "h x = h a" by (rule rcos_imp_homeq[OF acarr]) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
194 |
from xcarr and this |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
195 |
show "x \<in> {x \<in> carrier R. h x = h a}" by fast |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
196 |
next |
29237 | 197 |
interpret ideal "a_kernel R S h" "R" by (rule kernel_is_ideal) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
198 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
199 |
fix x |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
200 |
assume xcarr: "x \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
201 |
and hx: "h x = h a" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
202 |
from acarr xcarr hx |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
203 |
show "x \<in> a_kernel R S h +> a" by (rule homeq_imp_rcos) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
204 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
205 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
206 |
end |