author | wenzelm |
Wed, 13 Jun 2007 00:01:41 +0200 | |
changeset 23350 | 50c5b0912a0c |
parent 22265 | 3c5c6bdf61de |
child 23957 | 54fab60ddc97 |
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: The algebraic hierarchy of rings |
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Restructured algebra library, added ideals and quotient rings.
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3 |
Id: $Id$ |
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Restructured algebra library, added ideals and quotient rings.
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4 |
Author: Clemens Ballarin, started 9 December 1996 |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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5 |
Copyright: Clemens Ballarin |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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6 |
*) |
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Restructured algebra library, added ideals and quotient rings.
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7 |
|
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Restructured algebra library, added ideals and quotient rings.
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8 |
theory Ring imports FiniteProduct |
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Restructured algebra library, added ideals and quotient rings.
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9 |
uses ("ringsimp.ML") begin |
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Restructured algebra library, added ideals and quotient rings.
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10 |
|
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Restructured algebra library, added ideals and quotient rings.
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11 |
|
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Restructured algebra library, added ideals and quotient rings.
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12 |
section {* Abelian Groups *} |
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Restructured algebra library, added ideals and quotient rings.
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13 |
|
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Restructured algebra library, added ideals and quotient rings.
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14 |
record 'a ring = "'a monoid" + |
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Restructured algebra library, added ideals and quotient rings.
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15 |
zero :: 'a ("\<zero>\<index>") |
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Restructured algebra library, added ideals and quotient rings.
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16 |
add :: "['a, 'a] => 'a" (infixl "\<oplus>\<index>" 65) |
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Restructured algebra library, added ideals and quotient rings.
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17 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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18 |
text {* Derived operations. *} |
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Restructured algebra library, added ideals and quotient rings.
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19 |
|
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Restructured algebra library, added ideals and quotient rings.
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20 |
constdefs (structure R) |
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Restructured algebra library, added ideals and quotient rings.
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21 |
a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _" [81] 80) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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22 |
"a_inv R == m_inv (| carrier = carrier R, mult = add R, one = zero R |)" |
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Restructured algebra library, added ideals and quotient rings.
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23 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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24 |
a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" (infixl "\<ominus>\<index>" 65) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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25 |
"[| x \<in> carrier R; y \<in> carrier R |] ==> x \<ominus> y == x \<oplus> (\<ominus> y)" |
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Restructured algebra library, added ideals and quotient rings.
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26 |
|
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Restructured algebra library, added ideals and quotient rings.
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27 |
locale abelian_monoid = |
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Restructured algebra library, added ideals and quotient rings.
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28 |
fixes G (structure) |
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Restructured algebra library, added ideals and quotient rings.
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29 |
assumes a_comm_monoid: |
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Restructured algebra library, added ideals and quotient rings.
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30 |
"comm_monoid (| carrier = carrier G, mult = add G, one = zero G |)" |
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Restructured algebra library, added ideals and quotient rings.
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31 |
|
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Restructured algebra library, added ideals and quotient rings.
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32 |
|
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Restructured algebra library, added ideals and quotient rings.
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33 |
text {* |
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Restructured algebra library, added ideals and quotient rings.
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34 |
The following definition is redundant but simple to use. |
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Restructured algebra library, added ideals and quotient rings.
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35 |
*} |
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Restructured algebra library, added ideals and quotient rings.
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36 |
|
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Restructured algebra library, added ideals and quotient rings.
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37 |
locale abelian_group = abelian_monoid + |
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Restructured algebra library, added ideals and quotient rings.
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38 |
assumes a_comm_group: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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39 |
"comm_group (| carrier = carrier G, mult = add G, one = zero G |)" |
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Restructured algebra library, added ideals and quotient rings.
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40 |
|
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Restructured algebra library, added ideals and quotient rings.
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41 |
|
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Restructured algebra library, added ideals and quotient rings.
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42 |
subsection {* Basic Properties *} |
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Restructured algebra library, added ideals and quotient rings.
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43 |
|
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Restructured algebra library, added ideals and quotient rings.
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44 |
lemma abelian_monoidI: |
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Restructured algebra library, added ideals and quotient rings.
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45 |
fixes R (structure) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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46 |
assumes a_closed: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
47 |
"!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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48 |
and zero_closed: "\<zero> \<in> carrier R" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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49 |
and a_assoc: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
50 |
"!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==> |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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51 |
(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
52 |
and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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53 |
and a_comm: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
54 |
"!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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55 |
shows "abelian_monoid R" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
56 |
by (auto intro!: abelian_monoid.intro comm_monoidI intro: prems) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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57 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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58 |
lemma abelian_groupI: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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59 |
fixes R (structure) |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
60 |
assumes a_closed: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
61 |
"!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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62 |
and zero_closed: "zero R \<in> carrier R" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
63 |
and a_assoc: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
64 |
"!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==> |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
65 |
(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
66 |
and a_comm: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
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|
67 |
"!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
68 |
and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
69 |
and l_inv_ex: "!!x. x \<in> carrier R ==> EX y : carrier R. y \<oplus> x = \<zero>" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
70 |
shows "abelian_group R" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
71 |
by (auto intro!: abelian_group.intro abelian_monoidI |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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72 |
abelian_group_axioms.intro comm_monoidI comm_groupI |
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Restructured algebra library, added ideals and quotient rings.
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73 |
intro: prems) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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74 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
75 |
lemma (in abelian_monoid) a_monoid: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
76 |
"monoid (| carrier = carrier G, mult = add G, one = zero G |)" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
77 |
by (rule comm_monoid.axioms, rule a_comm_monoid) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
78 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
79 |
lemma (in abelian_group) a_group: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
80 |
"group (| carrier = carrier G, mult = add G, one = zero G |)" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
81 |
by (simp add: group_def a_monoid) |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
82 |
(simp add: comm_group.axioms group.axioms a_comm_group) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
83 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
84 |
lemmas monoid_record_simps = partial_object.simps monoid.simps |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
85 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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changeset
|
86 |
lemma (in abelian_monoid) a_closed [intro, simp]: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
87 |
"\<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier G" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
88 |
by (rule monoid.m_closed [OF a_monoid, simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
89 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
90 |
lemma (in abelian_monoid) zero_closed [intro, simp]: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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changeset
|
91 |
"\<zero> \<in> carrier G" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
92 |
by (rule monoid.one_closed [OF a_monoid, simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
93 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
94 |
lemma (in abelian_group) a_inv_closed [intro, simp]: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
95 |
"x \<in> carrier G ==> \<ominus> x \<in> carrier G" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
96 |
by (simp add: a_inv_def |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
97 |
group.inv_closed [OF a_group, simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
98 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
99 |
lemma (in abelian_group) minus_closed [intro, simp]: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
100 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y \<in> carrier G" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
101 |
by (simp add: a_minus_def) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
102 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
103 |
lemma (in abelian_group) a_l_cancel [simp]: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
104 |
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
105 |
(x \<oplus> y = x \<oplus> z) = (y = z)" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
106 |
by (rule group.l_cancel [OF a_group, simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
107 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
108 |
lemma (in abelian_group) a_r_cancel [simp]: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
109 |
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
110 |
(y \<oplus> x = z \<oplus> x) = (y = z)" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
111 |
by (rule group.r_cancel [OF a_group, simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset
|
112 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
113 |
lemma (in abelian_monoid) a_assoc: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset
|
114 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
115 |
(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
116 |
by (rule monoid.m_assoc [OF a_monoid, simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
117 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
118 |
lemma (in abelian_monoid) l_zero [simp]: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
119 |
"x \<in> carrier G ==> \<zero> \<oplus> x = x" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
120 |
by (rule monoid.l_one [OF a_monoid, simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
121 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
122 |
lemma (in abelian_group) l_neg: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
123 |
"x \<in> carrier G ==> \<ominus> x \<oplus> x = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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changeset
|
124 |
by (simp add: a_inv_def |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
125 |
group.l_inv [OF a_group, simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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changeset
|
126 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
127 |
lemma (in abelian_monoid) a_comm: |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
128 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x" |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
129 |
by (rule comm_monoid.m_comm [OF a_comm_monoid, |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
130 |
simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
131 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
132 |
lemma (in abelian_monoid) a_lcomm: |
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|
133 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> |
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parents:
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|
134 |
x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
135 |
by (rule comm_monoid.m_lcomm [OF a_comm_monoid, |
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Restructured algebra library, added ideals and quotient rings.
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|
136 |
simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
137 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
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|
138 |
lemma (in abelian_monoid) r_zero [simp]: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
139 |
"x \<in> carrier G ==> x \<oplus> \<zero> = x" |
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parents:
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|
140 |
using monoid.r_one [OF a_monoid] |
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Restructured algebra library, added ideals and quotient rings.
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|
141 |
by simp |
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|
142 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
143 |
lemma (in abelian_group) r_neg: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
144 |
"x \<in> carrier G ==> x \<oplus> (\<ominus> x) = \<zero>" |
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parents:
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|
145 |
using group.r_inv [OF a_group] |
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|
146 |
by (simp add: a_inv_def) |
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|
147 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
148 |
lemma (in abelian_group) minus_zero [simp]: |
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|
149 |
"\<ominus> \<zero> = \<zero>" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
150 |
by (simp add: a_inv_def |
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Restructured algebra library, added ideals and quotient rings.
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|
151 |
group.inv_one [OF a_group, simplified monoid_record_simps]) |
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152 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
153 |
lemma (in abelian_group) minus_minus [simp]: |
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parents:
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|
154 |
"x \<in> carrier G ==> \<ominus> (\<ominus> x) = x" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
155 |
using group.inv_inv [OF a_group, simplified monoid_record_simps] |
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|
156 |
by (simp add: a_inv_def) |
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Restructured algebra library, added ideals and quotient rings.
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|
157 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
158 |
lemma (in abelian_group) a_inv_inj: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
159 |
"inj_on (a_inv G) (carrier G)" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
160 |
using group.inv_inj [OF a_group, simplified monoid_record_simps] |
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161 |
by (simp add: a_inv_def) |
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|
162 |
|
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Restructured algebra library, added ideals and quotient rings.
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|
163 |
lemma (in abelian_group) minus_add: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
164 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> (x \<oplus> y) = \<ominus> x \<oplus> \<ominus> y" |
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parents:
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|
165 |
using comm_group.inv_mult [OF a_comm_group] |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
166 |
by (simp add: a_inv_def) |
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Restructured algebra library, added ideals and quotient rings.
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|
167 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
168 |
lemma (in abelian_group) minus_equality: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
169 |
"[| x \<in> carrier G; y \<in> carrier G; y \<oplus> x = \<zero> |] ==> \<ominus> x = y" |
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|
170 |
using group.inv_equality [OF a_group] |
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Restructured algebra library, added ideals and quotient rings.
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171 |
by (auto simp add: a_inv_def) |
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Restructured algebra library, added ideals and quotient rings.
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|
172 |
|
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Restructured algebra library, added ideals and quotient rings.
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|
173 |
lemma (in abelian_monoid) minus_unique: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
174 |
"[| x \<in> carrier G; y \<in> carrier G; y' \<in> carrier G; |
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Restructured algebra library, added ideals and quotient rings.
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|
175 |
y \<oplus> x = \<zero>; x \<oplus> y' = \<zero> |] ==> y = y'" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
176 |
using monoid.inv_unique [OF a_monoid] |
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Restructured algebra library, added ideals and quotient rings.
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|
177 |
by (simp add: a_inv_def) |
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Restructured algebra library, added ideals and quotient rings.
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|
178 |
|
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Restructured algebra library, added ideals and quotient rings.
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|
179 |
lemmas (in abelian_monoid) a_ac = a_assoc a_comm a_lcomm |
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180 |
|
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|
181 |
text {* Derive an @{text "abelian_group"} from a @{text "comm_group"} *} |
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|
182 |
lemma comm_group_abelian_groupI: |
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Restructured algebra library, added ideals and quotient rings.
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|
183 |
fixes G (structure) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
184 |
assumes cg: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" |
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Restructured algebra library, added ideals and quotient rings.
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|
185 |
shows "abelian_group G" |
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Restructured algebra library, added ideals and quotient rings.
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|
186 |
proof - |
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Restructured algebra library, added ideals and quotient rings.
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|
187 |
interpret comm_group ["\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"] |
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Restructured algebra library, added ideals and quotient rings.
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|
188 |
by (rule cg) |
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Restructured algebra library, added ideals and quotient rings.
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|
189 |
show "abelian_group G" by (unfold_locales) |
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Restructured algebra library, added ideals and quotient rings.
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|
190 |
qed |
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Restructured algebra library, added ideals and quotient rings.
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|
191 |
|
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Restructured algebra library, added ideals and quotient rings.
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|
192 |
|
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Restructured algebra library, added ideals and quotient rings.
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|
193 |
subsection {* Sums over Finite Sets *} |
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194 |
|
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Restructured algebra library, added ideals and quotient rings.
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|
195 |
text {* |
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|
196 |
This definition makes it easy to lift lemmas from @{term finprod}. |
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Restructured algebra library, added ideals and quotient rings.
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|
197 |
*} |
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Restructured algebra library, added ideals and quotient rings.
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|
198 |
|
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Restructured algebra library, added ideals and quotient rings.
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|
199 |
constdefs |
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|
200 |
finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b" |
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Restructured algebra library, added ideals and quotient rings.
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|
201 |
"finsum G f A == finprod (| carrier = carrier G, |
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Restructured algebra library, added ideals and quotient rings.
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|
202 |
mult = add G, one = zero G |) f A" |
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Restructured algebra library, added ideals and quotient rings.
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|
203 |
|
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Restructured algebra library, added ideals and quotient rings.
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|
204 |
syntax |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
205 |
"_finsum" :: "index => idt => 'a set => 'b => 'b" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
206 |
("(3\<Oplus>__:_. _)" [1000, 0, 51, 10] 10) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
207 |
syntax (xsymbols) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
208 |
"_finsum" :: "index => idt => 'a set => 'b => 'b" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
209 |
("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
210 |
syntax (HTML output) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
211 |
"_finsum" :: "index => idt => 'a set => 'b => 'b" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
212 |
("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10) |
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Restructured algebra library, added ideals and quotient rings.
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|
213 |
translations |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
214 |
"\<Oplus>\<index>i:A. b" == "finsum \<struct>\<index> (%i. b) A" |
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Restructured algebra library, added ideals and quotient rings.
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|
215 |
-- {* Beware of argument permutation! *} |
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Restructured algebra library, added ideals and quotient rings.
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|
216 |
|
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Restructured algebra library, added ideals and quotient rings.
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|
217 |
(* |
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Restructured algebra library, added ideals and quotient rings.
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|
218 |
lemmas (in abelian_monoid) finsum_empty [simp] = |
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Restructured algebra library, added ideals and quotient rings.
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|
219 |
comm_monoid.finprod_empty [OF a_comm_monoid, simplified] |
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Restructured algebra library, added ideals and quotient rings.
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|
220 |
is dangeous, because attributes (like simplified) are applied upon opening |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
221 |
the locale, simplified refers to the simpset at that time!!! |
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Restructured algebra library, added ideals and quotient rings.
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|
222 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
223 |
lemmas (in abelian_monoid) finsum_empty [simp] = |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
224 |
abelian_monoid.finprod_empty [OF a_abelian_monoid, folded finsum_def, |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
225 |
simplified monoid_record_simps] |
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Restructured algebra library, added ideals and quotient rings.
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|
226 |
makes the locale slow, because proofs are repeated for every |
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Restructured algebra library, added ideals and quotient rings.
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|
227 |
"lemma (in abelian_monoid)" command. |
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Restructured algebra library, added ideals and quotient rings.
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|
228 |
When lemma is used time in UnivPoly.thy from beginning to UP_cring goes down |
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Restructured algebra library, added ideals and quotient rings.
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|
229 |
from 110 secs to 60 secs. |
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Restructured algebra library, added ideals and quotient rings.
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|
230 |
*) |
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Restructured algebra library, added ideals and quotient rings.
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|
231 |
|
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Restructured algebra library, added ideals and quotient rings.
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|
232 |
lemma (in abelian_monoid) finsum_empty [simp]: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
233 |
"finsum G f {} = \<zero>" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
234 |
by (rule comm_monoid.finprod_empty [OF a_comm_monoid, |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
235 |
folded finsum_def, simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
236 |
|
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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|
237 |
lemma (in abelian_monoid) finsum_insert [simp]: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
238 |
"[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |] |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
239 |
==> finsum G f (insert a F) = f a \<oplus> finsum G f F" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
240 |
by (rule comm_monoid.finprod_insert [OF a_comm_monoid, |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
241 |
folded finsum_def, simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
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|
242 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
243 |
lemma (in abelian_monoid) finsum_zero [simp]: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
244 |
"finite A ==> (\<Oplus>i\<in>A. \<zero>) = \<zero>" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
245 |
by (rule comm_monoid.finprod_one [OF a_comm_monoid, folded finsum_def, |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
246 |
simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
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|
247 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
248 |
lemma (in abelian_monoid) finsum_closed [simp]: |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
249 |
fixes A |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
250 |
assumes fin: "finite A" and f: "f \<in> A -> carrier G" |
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Restructured algebra library, added ideals and quotient rings.
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|
251 |
shows "finsum G f A \<in> carrier G" |
23350 | 252 |
apply (rule comm_monoid.finprod_closed [OF a_comm_monoid, |
20318
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
253 |
folded finsum_def, simplified monoid_record_simps]) |
23350 | 254 |
apply (rule fin) |
255 |
apply (rule f) |
|
256 |
done |
|
20318
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Restructured algebra library, added ideals and quotient rings.
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|
257 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
258 |
lemma (in abelian_monoid) finsum_Un_Int: |
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Restructured algebra library, added ideals and quotient rings.
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|
259 |
"[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==> |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
260 |
finsum G g (A Un B) \<oplus> finsum G g (A Int B) = |
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Restructured algebra library, added ideals and quotient rings.
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|
261 |
finsum G g A \<oplus> finsum G g B" |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
262 |
by (rule comm_monoid.finprod_Un_Int [OF a_comm_monoid, |
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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset
|
263 |
folded finsum_def, simplified monoid_record_simps]) |
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Restructured algebra library, added ideals and quotient rings.
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|
264 |
|
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Restructured algebra library, added ideals and quotient rings.
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parents:
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|
265 |
lemma (in abelian_monoid) finsum_Un_disjoint: |
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Restructured algebra library, added ideals and quotient rings.
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|
266 |
"[| finite A; finite B; A Int B = {}; |
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Restructured algebra library, added ideals and quotient rings.
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|
267 |
g \<in> A -> carrier G; g \<in> B -> carrier G |] |
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Restructured algebra library, added ideals and quotient rings.
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changeset
|
268 |
==> finsum G g (A Un B) = finsum G g A \<oplus> finsum G g B" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
269 |
by (rule comm_monoid.finprod_Un_disjoint [OF a_comm_monoid, |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
270 |
folded finsum_def, simplified monoid_record_simps]) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
271 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
272 |
lemma (in abelian_monoid) finsum_addf: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
273 |
"[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
274 |
finsum G (%x. f x \<oplus> g x) A = (finsum G f A \<oplus> finsum G g A)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
275 |
by (rule comm_monoid.finprod_multf [OF a_comm_monoid, |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
276 |
folded finsum_def, simplified monoid_record_simps]) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
277 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
278 |
lemma (in abelian_monoid) finsum_cong': |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
279 |
"[| A = B; g : B -> carrier G; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
280 |
!!i. i : B ==> f i = g i |] ==> finsum G f A = finsum G g B" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
281 |
by (rule comm_monoid.finprod_cong' [OF a_comm_monoid, |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
282 |
folded finsum_def, simplified monoid_record_simps]) auto |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
283 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
284 |
lemma (in abelian_monoid) finsum_0 [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
285 |
"f : {0::nat} -> carrier G ==> finsum G f {..0} = f 0" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
286 |
by (rule comm_monoid.finprod_0 [OF a_comm_monoid, folded finsum_def, |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
287 |
simplified monoid_record_simps]) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
288 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
289 |
lemma (in abelian_monoid) finsum_Suc [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
290 |
"f : {..Suc n} -> carrier G ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
291 |
finsum G f {..Suc n} = (f (Suc n) \<oplus> finsum G f {..n})" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
292 |
by (rule comm_monoid.finprod_Suc [OF a_comm_monoid, folded finsum_def, |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
293 |
simplified monoid_record_simps]) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
294 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
295 |
lemma (in abelian_monoid) finsum_Suc2: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
296 |
"f : {..Suc n} -> carrier G ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
297 |
finsum G f {..Suc n} = (finsum G (%i. f (Suc i)) {..n} \<oplus> f 0)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
298 |
by (rule comm_monoid.finprod_Suc2 [OF a_comm_monoid, folded finsum_def, |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
299 |
simplified monoid_record_simps]) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
300 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
301 |
lemma (in abelian_monoid) finsum_add [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
302 |
"[| f : {..n} -> carrier G; g : {..n} -> carrier G |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
303 |
finsum G (%i. f i \<oplus> g i) {..n::nat} = |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
304 |
finsum G f {..n} \<oplus> finsum G g {..n}" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
305 |
by (rule comm_monoid.finprod_mult [OF a_comm_monoid, folded finsum_def, |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
306 |
simplified monoid_record_simps]) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
307 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
308 |
lemma (in abelian_monoid) finsum_cong: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
309 |
"[| A = B; f : B -> carrier G; |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
310 |
!!i. i : B =simp=> f i = g i |] ==> finsum G f A = finsum G g B" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
311 |
by (rule comm_monoid.finprod_cong [OF a_comm_monoid, folded finsum_def, |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
312 |
simplified monoid_record_simps]) (auto simp add: simp_implies_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
313 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
314 |
text {*Usually, if this rule causes a failed congruence proof error, |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
315 |
the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown. |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
316 |
Adding @{thm [source] Pi_def} to the simpset is often useful. *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
317 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
318 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
319 |
section {* The Algebraic Hierarchy of Rings *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
320 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
321 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
322 |
subsection {* Basic Definitions *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
323 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
324 |
locale ring = abelian_group R + monoid R + |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
325 |
assumes l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
326 |
==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
327 |
and r_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
328 |
==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
329 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
330 |
locale cring = ring + comm_monoid R |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
331 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
332 |
locale "domain" = cring + |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
333 |
assumes one_not_zero [simp]: "\<one> ~= \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
334 |
and integral: "[| a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
335 |
a = \<zero> | b = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
336 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
337 |
locale field = "domain" + |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
338 |
assumes field_Units: "Units R = carrier R - {\<zero>}" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
339 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
340 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
341 |
subsection {* Rings *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
342 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
343 |
lemma ringI: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
344 |
fixes R (structure) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
345 |
assumes abelian_group: "abelian_group R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
346 |
and monoid: "monoid R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
347 |
and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
348 |
==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
349 |
and r_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
350 |
==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
351 |
shows "ring R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
352 |
by (auto intro: ring.intro |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
353 |
abelian_group.axioms ring_axioms.intro prems) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
354 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
355 |
lemma (in ring) is_abelian_group: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
356 |
"abelian_group R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
357 |
by (auto intro!: abelian_groupI a_assoc a_comm l_neg) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
358 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
359 |
lemma (in ring) is_monoid: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
360 |
"monoid R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
361 |
by (auto intro!: monoidI m_assoc) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
362 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
363 |
lemma (in ring) is_ring: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
364 |
"ring R" |
23350 | 365 |
by fact |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
366 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
367 |
lemmas ring_record_simps = monoid_record_simps ring.simps |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
368 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
369 |
lemma cringI: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
370 |
fixes R (structure) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
371 |
assumes abelian_group: "abelian_group R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
372 |
and comm_monoid: "comm_monoid R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
373 |
and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
374 |
==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
375 |
shows "cring R" |
23350 | 376 |
proof (intro cring.intro ring.intro) |
377 |
show "ring_axioms R" |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
378 |
-- {* Right-distributivity follows from left-distributivity and |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
379 |
commutativity. *} |
23350 | 380 |
proof (rule ring_axioms.intro) |
381 |
fix x y z |
|
382 |
assume R: "x \<in> carrier R" "y \<in> carrier R" "z \<in> carrier R" |
|
383 |
note [simp] = comm_monoid.axioms [OF comm_monoid] |
|
384 |
abelian_group.axioms [OF abelian_group] |
|
385 |
abelian_monoid.a_closed |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
386 |
|
23350 | 387 |
from R have "z \<otimes> (x \<oplus> y) = (x \<oplus> y) \<otimes> z" |
388 |
by (simp add: comm_monoid.m_comm [OF comm_monoid.intro]) |
|
389 |
also from R have "... = x \<otimes> z \<oplus> y \<otimes> z" by (simp add: l_distr) |
|
390 |
also from R have "... = z \<otimes> x \<oplus> z \<otimes> y" |
|
391 |
by (simp add: comm_monoid.m_comm [OF comm_monoid.intro]) |
|
392 |
finally show "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" . |
|
393 |
qed (rule l_distr) |
|
394 |
qed (auto intro: cring.intro |
|
395 |
abelian_group.axioms comm_monoid.axioms ring_axioms.intro prems) |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
396 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
397 |
lemma (in cring) is_comm_monoid: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
398 |
"comm_monoid R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
399 |
by (auto intro!: comm_monoidI m_assoc m_comm) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
400 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
401 |
lemma (in cring) is_cring: |
23350 | 402 |
"cring R" by fact |
403 |
||
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
404 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
405 |
subsubsection {* Normaliser for Rings *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
406 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
407 |
lemma (in abelian_group) r_neg2: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
408 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus> (\<ominus> x \<oplus> y) = y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
409 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
410 |
assume G: "x \<in> carrier G" "y \<in> carrier G" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
411 |
then have "(x \<oplus> \<ominus> x) \<oplus> y = y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
412 |
by (simp only: r_neg l_zero) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
413 |
with G show ?thesis |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
414 |
by (simp add: a_ac) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
415 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
416 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
417 |
lemma (in abelian_group) r_neg1: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
418 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> x \<oplus> (x \<oplus> y) = y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
419 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
420 |
assume G: "x \<in> carrier G" "y \<in> carrier G" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
421 |
then have "(\<ominus> x \<oplus> x) \<oplus> y = y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
422 |
by (simp only: l_neg l_zero) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
423 |
with G show ?thesis by (simp add: a_ac) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
424 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
425 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
426 |
text {* |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
427 |
The following proofs are from Jacobson, Basic Algebra I, pp.~88--89 |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
428 |
*} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
429 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
430 |
lemma (in ring) l_null [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
431 |
"x \<in> carrier R ==> \<zero> \<otimes> x = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
432 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
433 |
assume R: "x \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
434 |
then have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = (\<zero> \<oplus> \<zero>) \<otimes> x" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
435 |
by (simp add: l_distr del: l_zero r_zero) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
436 |
also from R have "... = \<zero> \<otimes> x \<oplus> \<zero>" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
437 |
finally have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = \<zero> \<otimes> x \<oplus> \<zero>" . |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
438 |
with R show ?thesis by (simp del: r_zero) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
439 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
440 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
441 |
lemma (in ring) r_null [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
442 |
"x \<in> carrier R ==> x \<otimes> \<zero> = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
443 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
444 |
assume R: "x \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
445 |
then have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> (\<zero> \<oplus> \<zero>)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
446 |
by (simp add: r_distr del: l_zero r_zero) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
447 |
also from R have "... = x \<otimes> \<zero> \<oplus> \<zero>" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
448 |
finally have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> \<zero> \<oplus> \<zero>" . |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
449 |
with R show ?thesis by (simp del: r_zero) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
450 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
451 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
452 |
lemma (in ring) l_minus: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
453 |
"[| x \<in> carrier R; y \<in> carrier R |] ==> \<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
454 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
455 |
assume R: "x \<in> carrier R" "y \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
456 |
then have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = (\<ominus> x \<oplus> x) \<otimes> y" by (simp add: l_distr) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
457 |
also from R have "... = \<zero>" by (simp add: l_neg l_null) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
458 |
finally have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = \<zero>" . |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
459 |
with R have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp |
21896
9a7949815a84
Experimenting with interpretations of "definition".
ballarin
parents:
20318
diff
changeset
|
460 |
with R show ?thesis by (simp add: a_assoc r_neg) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
461 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
462 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
463 |
lemma (in ring) r_minus: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
464 |
"[| x \<in> carrier R; y \<in> carrier R |] ==> x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
465 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
466 |
assume R: "x \<in> carrier R" "y \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
467 |
then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
468 |
also from R have "... = \<zero>" by (simp add: l_neg r_null) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
469 |
finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" . |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
470 |
with R have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
471 |
with R show ?thesis by (simp add: a_assoc r_neg ) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
472 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
473 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
474 |
lemma (in ring) minus_eq: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
475 |
"[| x \<in> carrier R; y \<in> carrier R |] ==> x \<ominus> y = x \<oplus> \<ominus> y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
476 |
by (simp only: a_minus_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
477 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
478 |
text {* Setup algebra method: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
479 |
compute distributive normal form in locale contexts *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
480 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
481 |
use "ringsimp.ML" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
482 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
483 |
setup Algebra.setup |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
484 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
485 |
lemmas (in ring) ring_simprules |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
486 |
[algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] = |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
487 |
a_closed zero_closed a_inv_closed minus_closed m_closed one_closed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
488 |
a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
489 |
r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
490 |
a_lcomm r_distr l_null r_null l_minus r_minus |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
491 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
492 |
lemmas (in cring) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
493 |
[algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] = |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
494 |
_ |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
495 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
496 |
lemmas (in cring) cring_simprules |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
497 |
[algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] = |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
498 |
a_closed zero_closed a_inv_closed minus_closed m_closed one_closed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
499 |
a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
500 |
r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
501 |
a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
502 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
503 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
504 |
lemma (in cring) nat_pow_zero: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
505 |
"(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
506 |
by (induct n) simp_all |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
507 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
508 |
lemma (in ring) one_zeroD: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
509 |
assumes onezero: "\<one> = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
510 |
shows "carrier R = {\<zero>}" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
511 |
proof (rule, rule) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
512 |
fix x |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
513 |
assume xcarr: "x \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
514 |
from xcarr |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
515 |
have "x = x \<otimes> \<one>" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
516 |
from this and onezero |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
517 |
have "x = x \<otimes> \<zero>" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
518 |
from this and xcarr |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
519 |
have "x = \<zero>" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
520 |
thus "x \<in> {\<zero>}" by fast |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
521 |
qed fast |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
522 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
523 |
lemma (in ring) one_zeroI: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
524 |
assumes carrzero: "carrier R = {\<zero>}" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
525 |
shows "\<one> = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
526 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
527 |
from one_closed and carrzero |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
528 |
show "\<one> = \<zero>" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
529 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
530 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
531 |
lemma (in ring) one_zero: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
532 |
shows "(carrier R = {\<zero>}) = (\<one> = \<zero>)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
533 |
by (rule, erule one_zeroI, erule one_zeroD) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
534 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
535 |
lemma (in ring) one_not_zero: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
536 |
shows "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
537 |
by (simp add: one_zero) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
538 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
539 |
text {* Two examples for use of method algebra *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
540 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
541 |
lemma |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
542 |
includes ring R + cring S |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
543 |
shows "[| a \<in> carrier R; b \<in> carrier R; c \<in> carrier S; d \<in> carrier S |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
544 |
a \<oplus> \<ominus> (a \<oplus> \<ominus> b) = b & c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
545 |
by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
546 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
547 |
lemma |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
548 |
includes cring |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
549 |
shows "[| a \<in> carrier R; b \<in> carrier R |] ==> a \<ominus> (a \<ominus> b) = b" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
550 |
by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
551 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
552 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
553 |
subsubsection {* Sums over Finite Sets *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
554 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
555 |
lemma (in cring) finsum_ldistr: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
556 |
"[| finite A; a \<in> carrier R; f \<in> A -> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
557 |
finsum R f A \<otimes> a = finsum R (%i. f i \<otimes> a) A" |
22265 | 558 |
proof (induct set: finite) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
559 |
case empty then show ?case by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
560 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
561 |
case (insert x F) then show ?case by (simp add: Pi_def l_distr) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
562 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
563 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
564 |
lemma (in cring) finsum_rdistr: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
565 |
"[| finite A; a \<in> carrier R; f \<in> A -> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
566 |
a \<otimes> finsum R f A = finsum R (%i. a \<otimes> f i) A" |
22265 | 567 |
proof (induct set: finite) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
568 |
case empty then show ?case by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
569 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
570 |
case (insert x F) then show ?case by (simp add: Pi_def r_distr) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
571 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
572 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
573 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
574 |
subsection {* Integral Domains *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
575 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
576 |
lemma (in "domain") zero_not_one [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
577 |
"\<zero> ~= \<one>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
578 |
by (rule not_sym) simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
579 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
580 |
lemma (in "domain") integral_iff: (* not by default a simp rule! *) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
581 |
"[| a \<in> carrier R; b \<in> carrier R |] ==> (a \<otimes> b = \<zero>) = (a = \<zero> | b = \<zero>)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
582 |
proof |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
583 |
assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
584 |
then show "a = \<zero> | b = \<zero>" by (simp add: integral) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
585 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
586 |
assume "a \<in> carrier R" "b \<in> carrier R" "a = \<zero> | b = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
587 |
then show "a \<otimes> b = \<zero>" by auto |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
588 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
589 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
590 |
lemma (in "domain") m_lcancel: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
591 |
assumes prem: "a ~= \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
592 |
and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
593 |
shows "(a \<otimes> b = a \<otimes> c) = (b = c)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
594 |
proof |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
595 |
assume eq: "a \<otimes> b = a \<otimes> c" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
596 |
with R have "a \<otimes> (b \<ominus> c) = \<zero>" by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
597 |
with R have "a = \<zero> | (b \<ominus> c) = \<zero>" by (simp add: integral_iff) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
598 |
with prem and R have "b \<ominus> c = \<zero>" by auto |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
599 |
with R have "b = b \<ominus> (b \<ominus> c)" by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
600 |
also from R have "b \<ominus> (b \<ominus> c) = c" by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
601 |
finally show "b = c" . |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
602 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
603 |
assume "b = c" then show "a \<otimes> b = a \<otimes> c" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
604 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
605 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
606 |
lemma (in "domain") m_rcancel: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
607 |
assumes prem: "a ~= \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
608 |
and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
609 |
shows conc: "(b \<otimes> a = c \<otimes> a) = (b = c)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
610 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
611 |
from prem and R have "(a \<otimes> b = a \<otimes> c) = (b = c)" by (rule m_lcancel) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
612 |
with R show ?thesis by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
613 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
614 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
615 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
616 |
subsection {* Fields *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
617 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
618 |
text {* Field would not need to be derived from domain, the properties |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
619 |
for domain follow from the assumptions of field *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
620 |
lemma (in cring) cring_fieldI: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
621 |
assumes field_Units: "Units R = carrier R - {\<zero>}" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
622 |
shows "field R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
623 |
proof unfold_locales |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
624 |
from field_Units |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
625 |
have a: "\<zero> \<notin> Units R" by fast |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
626 |
have "\<one> \<in> Units R" by fast |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
627 |
from this and a |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
628 |
show "\<one> \<noteq> \<zero>" by force |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
629 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
630 |
fix a b |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
631 |
assume acarr: "a \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
632 |
and bcarr: "b \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
633 |
and ab: "a \<otimes> b = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
634 |
show "a = \<zero> \<or> b = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
635 |
proof (cases "a = \<zero>", simp) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
636 |
assume "a \<noteq> \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
637 |
from this and field_Units and acarr |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
638 |
have aUnit: "a \<in> Units R" by fast |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
639 |
from bcarr |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
640 |
have "b = \<one> \<otimes> b" by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
641 |
also from aUnit acarr |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
642 |
have "... = (inv a \<otimes> a) \<otimes> b" by (simp add: Units_l_inv) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
643 |
also from acarr bcarr aUnit[THEN Units_inv_closed] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
644 |
have "... = (inv a) \<otimes> (a \<otimes> b)" by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
645 |
also from ab and acarr bcarr aUnit |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
646 |
have "... = (inv a) \<otimes> \<zero>" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
647 |
also from aUnit[THEN Units_inv_closed] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
648 |
have "... = \<zero>" by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
649 |
finally |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
650 |
have "b = \<zero>" . |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
651 |
thus "a = \<zero> \<or> b = \<zero>" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
652 |
qed |
23350 | 653 |
qed (rule field_Units) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
654 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
655 |
text {* Another variant to show that something is a field *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
656 |
lemma (in cring) cring_fieldI2: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
657 |
assumes notzero: "\<zero> \<noteq> \<one>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
658 |
and invex: "\<And>a. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>\<rbrakk> \<Longrightarrow> \<exists>b\<in>carrier R. a \<otimes> b = \<one>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
659 |
shows "field R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
660 |
apply (rule cring_fieldI, simp add: Units_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
661 |
apply (rule, clarsimp) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
662 |
apply (simp add: notzero) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
663 |
proof (clarsimp) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
664 |
fix x |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
665 |
assume xcarr: "x \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
666 |
and "x \<noteq> \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
667 |
from this |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
668 |
have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
669 |
from this |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
670 |
obtain y |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
671 |
where ycarr: "y \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
672 |
and xy: "x \<otimes> y = \<one>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
673 |
by fast |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
674 |
from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
675 |
from ycarr and this and xy |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
676 |
show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
677 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
678 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
679 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
680 |
subsection {* Morphisms *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
681 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
682 |
constdefs (structure R S) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
683 |
ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
684 |
"ring_hom R S == {h. h \<in> carrier R -> carrier S & |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
685 |
(ALL x y. x \<in> carrier R & y \<in> carrier R --> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
686 |
h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y & h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y) & |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
687 |
h \<one> = \<one>\<^bsub>S\<^esub>}" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
688 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
689 |
lemma ring_hom_memI: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
690 |
fixes R (structure) and S (structure) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
691 |
assumes hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
692 |
and hom_mult: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
693 |
h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
694 |
and hom_add: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
695 |
h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
696 |
and hom_one: "h \<one> = \<one>\<^bsub>S\<^esub>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
697 |
shows "h \<in> ring_hom R S" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
698 |
by (auto simp add: ring_hom_def prems Pi_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
699 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
700 |
lemma ring_hom_closed: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
701 |
"[| h \<in> ring_hom R S; x \<in> carrier R |] ==> h x \<in> carrier S" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
702 |
by (auto simp add: ring_hom_def funcset_mem) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
703 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
704 |
lemma ring_hom_mult: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
705 |
fixes R (structure) and S (structure) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
706 |
shows |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
707 |
"[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
708 |
h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
709 |
by (simp add: ring_hom_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
710 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
711 |
lemma ring_hom_add: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
712 |
fixes R (structure) and S (structure) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
713 |
shows |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
714 |
"[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
715 |
h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
716 |
by (simp add: ring_hom_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
717 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
718 |
lemma ring_hom_one: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
719 |
fixes R (structure) and S (structure) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
720 |
shows "h \<in> ring_hom R S ==> h \<one> = \<one>\<^bsub>S\<^esub>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
721 |
by (simp add: ring_hom_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
722 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
723 |
locale ring_hom_cring = cring R + cring S + |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
724 |
fixes h |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
725 |
assumes homh [simp, intro]: "h \<in> ring_hom R S" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
726 |
notes hom_closed [simp, intro] = ring_hom_closed [OF homh] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
727 |
and hom_mult [simp] = ring_hom_mult [OF homh] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
728 |
and hom_add [simp] = ring_hom_add [OF homh] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
729 |
and hom_one [simp] = ring_hom_one [OF homh] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
730 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
731 |
lemma (in ring_hom_cring) hom_zero [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
732 |
"h \<zero> = \<zero>\<^bsub>S\<^esub>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
733 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
734 |
have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
735 |
by (simp add: hom_add [symmetric] del: hom_add) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
736 |
then show ?thesis by (simp del: S.r_zero) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
737 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
738 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
739 |
lemma (in ring_hom_cring) hom_a_inv [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
740 |
"x \<in> carrier R ==> h (\<ominus> x) = \<ominus>\<^bsub>S\<^esub> h x" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
741 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
742 |
assume R: "x \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
743 |
then have "h x \<oplus>\<^bsub>S\<^esub> h (\<ominus> x) = h x \<oplus>\<^bsub>S\<^esub> (\<ominus>\<^bsub>S\<^esub> h x)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
744 |
by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
745 |
with R show ?thesis by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
746 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
747 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
748 |
lemma (in ring_hom_cring) hom_finsum [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
749 |
"[| finite A; f \<in> A -> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
750 |
h (finsum R f A) = finsum S (h o f) A" |
22265 | 751 |
proof (induct set: finite) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
752 |
case empty then show ?case by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
753 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
754 |
case insert then show ?case by (simp add: Pi_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
755 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
756 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
757 |
lemma (in ring_hom_cring) hom_finprod: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
758 |
"[| finite A; f \<in> A -> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
759 |
h (finprod R f A) = finprod S (h o f) A" |
22265 | 760 |
proof (induct set: finite) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
761 |
case empty then show ?case by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
762 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
763 |
case insert then show ?case by (simp add: Pi_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
764 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
765 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
766 |
declare ring_hom_cring.hom_finprod [simp] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
767 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
768 |
lemma id_ring_hom [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
769 |
"id \<in> ring_hom R R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
770 |
by (auto intro!: ring_hom_memI) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
771 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
772 |
end |