author | wenzelm |
Sat, 20 May 2006 23:36:51 +0200 | |
changeset 19684 | 6101fbebda1d |
parent 16417 | 9bc16273c2d4 |
child 19699 | 1ecda5544e88 |
permissions | -rw-r--r-- |
13813 | 1 |
(* |
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Title: HOL/Algebra/Group.thy |
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Id: $Id$ |
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Author: Clemens Ballarin, started 4 February 2003 |
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Based on work by Florian Kammueller, L C Paulson and Markus Wenzel. |
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*) |
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13949
0ce528cd6f19
HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents:
13944
diff
changeset
|
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header {* Groups *} |
13813 | 10 |
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16417 | 11 |
theory Group imports FuncSet Lattice begin |
13813 | 12 |
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14761 | 13 |
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14963 | 14 |
section {* Monoids and Groups *} |
13936 | 15 |
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13813 | 16 |
text {* |
14963 | 17 |
Definitions follow \cite{Jacobson:1985}. |
13813 | 18 |
*} |
19 |
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20 |
subsection {* Definitions *} |
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21 |
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14963 | 22 |
record 'a monoid = "'a partial_object" + |
23 |
mult :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<otimes>\<index>" 70) |
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24 |
one :: 'a ("\<one>\<index>") |
|
13817 | 25 |
|
14651 | 26 |
constdefs (structure G) |
14852 | 27 |
m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _" [81] 80) |
14651 | 28 |
"inv x == (THE y. y \<in> carrier G & x \<otimes> y = \<one> & y \<otimes> x = \<one>)" |
13936 | 29 |
|
14651 | 30 |
Units :: "_ => 'a set" |
14852 | 31 |
--{*The set of invertible elements*} |
14963 | 32 |
"Units G == {y. y \<in> carrier G & (\<exists>x \<in> carrier G. x \<otimes> y = \<one> & y \<otimes> x = \<one>)}" |
13936 | 33 |
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consts |
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pow :: "[('a, 'm) monoid_scheme, 'a, 'b::number] => 'a" (infixr "'(^')\<index>" 75) |
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19684 | 37 |
defs (unchecked overloaded) |
14693 | 38 |
nat_pow_def: "pow G a n == nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n" |
13936 | 39 |
int_pow_def: "pow G a z == |
14693 | 40 |
let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) |
41 |
in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z)" |
|
13813 | 42 |
|
14963 | 43 |
locale monoid = struct G + |
13813 | 44 |
assumes m_closed [intro, simp]: |
14963 | 45 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> carrier G" |
46 |
and m_assoc: |
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"\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> |
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\<Longrightarrow> (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
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and one_closed [intro, simp]: "\<one> \<in> carrier G" |
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and l_one [simp]: "x \<in> carrier G \<Longrightarrow> \<one> \<otimes> x = x" |
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and r_one [simp]: "x \<in> carrier G \<Longrightarrow> x \<otimes> \<one> = x" |
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13817 | 52 |
|
13936 | 53 |
lemma monoidI: |
14693 | 54 |
includes struct G |
13936 | 55 |
assumes m_closed: |
14693 | 56 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
57 |
and one_closed: "\<one> \<in> carrier G" |
|
13936 | 58 |
and m_assoc: |
59 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
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14693 | 60 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
61 |
and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
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and r_one: "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x" |
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13936 | 63 |
shows "monoid G" |
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by (fast intro!: monoid.intro intro: prems) |
13936 | 65 |
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lemma (in monoid) Units_closed [dest]: |
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"x \<in> Units G ==> x \<in> carrier G" |
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by (unfold Units_def) fast |
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lemma (in monoid) inv_unique: |
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14693 | 71 |
assumes eq: "y \<otimes> x = \<one>" "x \<otimes> y' = \<one>" |
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and G: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G" |
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13936 | 73 |
shows "y = y'" |
74 |
proof - |
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from G eq have "y = y \<otimes> (x \<otimes> y')" by simp |
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also from G have "... = (y \<otimes> x) \<otimes> y'" by (simp add: m_assoc) |
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also from G eq have "... = y'" by simp |
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finally show ?thesis . |
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qed |
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||
13940 | 81 |
lemma (in monoid) Units_one_closed [intro, simp]: |
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"\<one> \<in> Units G" |
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by (unfold Units_def) auto |
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||
13936 | 85 |
lemma (in monoid) Units_inv_closed [intro, simp]: |
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"x \<in> Units G ==> inv x \<in> carrier G" |
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13943 | 87 |
apply (unfold Units_def m_inv_def, auto) |
13936 | 88 |
apply (rule theI2, fast) |
13943 | 89 |
apply (fast intro: inv_unique, fast) |
13936 | 90 |
done |
91 |
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lemma (in monoid) Units_l_inv: |
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"x \<in> Units G ==> inv x \<otimes> x = \<one>" |
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13943 | 94 |
apply (unfold Units_def m_inv_def, auto) |
13936 | 95 |
apply (rule theI2, fast) |
13943 | 96 |
apply (fast intro: inv_unique, fast) |
13936 | 97 |
done |
98 |
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lemma (in monoid) Units_r_inv: |
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"x \<in> Units G ==> x \<otimes> inv x = \<one>" |
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13943 | 101 |
apply (unfold Units_def m_inv_def, auto) |
13936 | 102 |
apply (rule theI2, fast) |
13943 | 103 |
apply (fast intro: inv_unique, fast) |
13936 | 104 |
done |
105 |
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lemma (in monoid) Units_inv_Units [intro, simp]: |
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"x \<in> Units G ==> inv x \<in> Units G" |
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proof - |
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assume x: "x \<in> Units G" |
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show "inv x \<in> Units G" |
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by (auto simp add: Units_def |
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intro: Units_l_inv Units_r_inv x Units_closed [OF x]) |
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qed |
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lemma (in monoid) Units_l_cancel [simp]: |
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"[| x \<in> Units G; y \<in> carrier G; z \<in> carrier G |] ==> |
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(x \<otimes> y = x \<otimes> z) = (y = z)" |
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proof |
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assume eq: "x \<otimes> y = x \<otimes> z" |
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14693 | 120 |
and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G" |
13936 | 121 |
then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z" |
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by (simp add: m_assoc Units_closed) |
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with G show "y = z" by (simp add: Units_l_inv) |
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next |
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assume eq: "y = z" |
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14693 | 126 |
and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G" |
13936 | 127 |
then show "x \<otimes> y = x \<otimes> z" by simp |
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qed |
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lemma (in monoid) Units_inv_inv [simp]: |
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"x \<in> Units G ==> inv (inv x) = x" |
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proof - |
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assume x: "x \<in> Units G" |
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then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x" |
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by (simp add: Units_l_inv Units_r_inv) |
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with x show ?thesis by (simp add: Units_closed) |
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qed |
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lemma (in monoid) inv_inj_on_Units: |
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"inj_on (m_inv G) (Units G)" |
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proof (rule inj_onI) |
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fix x y |
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14693 | 143 |
assume G: "x \<in> Units G" "y \<in> Units G" and eq: "inv x = inv y" |
13936 | 144 |
then have "inv (inv x) = inv (inv y)" by simp |
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with G show "x = y" by simp |
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qed |
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13940 | 148 |
lemma (in monoid) Units_inv_comm: |
149 |
assumes inv: "x \<otimes> y = \<one>" |
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14693 | 150 |
and G: "x \<in> Units G" "y \<in> Units G" |
13940 | 151 |
shows "y \<otimes> x = \<one>" |
152 |
proof - |
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from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed) |
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with G show ?thesis by (simp del: r_one add: m_assoc Units_closed) |
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qed |
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13936 | 157 |
text {* Power *} |
158 |
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lemma (in monoid) nat_pow_closed [intro, simp]: |
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"x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G" |
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by (induct n) (simp_all add: nat_pow_def) |
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lemma (in monoid) nat_pow_0 [simp]: |
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"x (^) (0::nat) = \<one>" |
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by (simp add: nat_pow_def) |
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lemma (in monoid) nat_pow_Suc [simp]: |
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"x (^) (Suc n) = x (^) n \<otimes> x" |
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by (simp add: nat_pow_def) |
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lemma (in monoid) nat_pow_one [simp]: |
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"\<one> (^) (n::nat) = \<one>" |
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by (induct n) simp_all |
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174 |
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lemma (in monoid) nat_pow_mult: |
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"x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)" |
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by (induct m) (simp_all add: m_assoc [THEN sym]) |
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lemma (in monoid) nat_pow_pow: |
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"x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)" |
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by (induct m) (simp, simp add: nat_pow_mult add_commute) |
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182 |
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text {* |
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A group is a monoid all of whose elements are invertible. |
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*} |
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locale group = monoid + |
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assumes Units: "carrier G <= Units G" |
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14761 | 190 |
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lemma (in group) is_group: "group G" |
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by (rule group.intro [OF prems]) |
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||
13936 | 194 |
theorem groupI: |
14693 | 195 |
includes struct G |
13936 | 196 |
assumes m_closed [simp]: |
14693 | 197 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
198 |
and one_closed [simp]: "\<one> \<in> carrier G" |
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13936 | 199 |
and m_assoc: |
200 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
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14693 | 201 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
202 |
and l_one [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
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14963 | 203 |
and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
13936 | 204 |
shows "group G" |
205 |
proof - |
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have l_cancel [simp]: |
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"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
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14693 | 208 |
(x \<otimes> y = x \<otimes> z) = (y = z)" |
13936 | 209 |
proof |
210 |
fix x y z |
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14693 | 211 |
assume eq: "x \<otimes> y = x \<otimes> z" |
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and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
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13936 | 213 |
with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G" |
14693 | 214 |
and l_inv: "x_inv \<otimes> x = \<one>" by fast |
215 |
from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> z" |
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13936 | 216 |
by (simp add: m_assoc) |
217 |
with G show "y = z" by (simp add: l_inv) |
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218 |
next |
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fix x y z |
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assume eq: "y = z" |
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14693 | 221 |
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
222 |
then show "x \<otimes> y = x \<otimes> z" by simp |
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13936 | 223 |
qed |
224 |
have r_one: |
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14693 | 225 |
"!!x. x \<in> carrier G ==> x \<otimes> \<one> = x" |
13936 | 226 |
proof - |
227 |
fix x |
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assume x: "x \<in> carrier G" |
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with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G" |
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14693 | 230 |
and l_inv: "x_inv \<otimes> x = \<one>" by fast |
231 |
from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x" |
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13936 | 232 |
by (simp add: m_assoc [symmetric] l_inv) |
14693 | 233 |
with x xG show "x \<otimes> \<one> = x" by simp |
13936 | 234 |
qed |
235 |
have inv_ex: |
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14963 | 236 |
"!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>" |
13936 | 237 |
proof - |
238 |
fix x |
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239 |
assume x: "x \<in> carrier G" |
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with l_inv_ex obtain y where y: "y \<in> carrier G" |
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14693 | 241 |
and l_inv: "y \<otimes> x = \<one>" by fast |
242 |
from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>" |
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13936 | 243 |
by (simp add: m_assoc [symmetric] l_inv r_one) |
14693 | 244 |
with x y have r_inv: "x \<otimes> y = \<one>" |
13936 | 245 |
by simp |
14963 | 246 |
from x y show "\<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>" |
13936 | 247 |
by (fast intro: l_inv r_inv) |
248 |
qed |
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then have carrier_subset_Units: "carrier G <= Units G" |
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by (unfold Units_def) fast |
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show ?thesis |
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14963 | 252 |
by (fast intro!: group.intro monoid.intro group_axioms.intro |
13936 | 253 |
carrier_subset_Units intro: prems r_one) |
254 |
qed |
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255 |
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256 |
lemma (in monoid) monoid_groupI: |
|
257 |
assumes l_inv_ex: |
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14963 | 258 |
"!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
13936 | 259 |
shows "group G" |
260 |
by (rule groupI) (auto intro: m_assoc l_inv_ex) |
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261 |
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262 |
lemma (in group) Units_eq [simp]: |
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263 |
"Units G = carrier G" |
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264 |
proof |
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265 |
show "Units G <= carrier G" by fast |
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266 |
next |
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show "carrier G <= Units G" by (rule Units) |
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268 |
qed |
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269 |
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lemma (in group) inv_closed [intro, simp]: |
|
271 |
"x \<in> carrier G ==> inv x \<in> carrier G" |
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272 |
using Units_inv_closed by simp |
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273 |
||
14963 | 274 |
lemma (in group) l_inv [simp]: |
13936 | 275 |
"x \<in> carrier G ==> inv x \<otimes> x = \<one>" |
276 |
using Units_l_inv by simp |
|
13813 | 277 |
|
278 |
subsection {* Cancellation Laws and Basic Properties *} |
|
279 |
||
280 |
lemma (in group) l_cancel [simp]: |
|
281 |
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
282 |
(x \<otimes> y = x \<otimes> z) = (y = z)" |
|
13936 | 283 |
using Units_l_inv by simp |
13940 | 284 |
|
14963 | 285 |
lemma (in group) r_inv [simp]: |
13813 | 286 |
"x \<in> carrier G ==> x \<otimes> inv x = \<one>" |
287 |
proof - |
|
288 |
assume x: "x \<in> carrier G" |
|
289 |
then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>" |
|
290 |
by (simp add: m_assoc [symmetric] l_inv) |
|
291 |
with x show ?thesis by (simp del: r_one) |
|
292 |
qed |
|
293 |
||
294 |
lemma (in group) r_cancel [simp]: |
|
295 |
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
296 |
(y \<otimes> x = z \<otimes> x) = (y = z)" |
|
297 |
proof |
|
298 |
assume eq: "y \<otimes> x = z \<otimes> x" |
|
14693 | 299 |
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
13813 | 300 |
then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)" |
14963 | 301 |
by (simp add: m_assoc [symmetric] del: r_inv) |
302 |
with G show "y = z" by simp |
|
13813 | 303 |
next |
304 |
assume eq: "y = z" |
|
14693 | 305 |
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
13813 | 306 |
then show "y \<otimes> x = z \<otimes> x" by simp |
307 |
qed |
|
308 |
||
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
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|
309 |
lemma (in group) inv_one [simp]: |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
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|
310 |
"inv \<one> = \<one>" |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
311 |
proof - |
14963 | 312 |
have "inv \<one> = \<one> \<otimes> (inv \<one>)" by (simp del: r_inv) |
313 |
moreover have "... = \<one>" by simp |
|
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
314 |
finally show ?thesis . |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
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|
315 |
qed |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
316 |
|
13813 | 317 |
lemma (in group) inv_inv [simp]: |
318 |
"x \<in> carrier G ==> inv (inv x) = x" |
|
13936 | 319 |
using Units_inv_inv by simp |
320 |
||
321 |
lemma (in group) inv_inj: |
|
322 |
"inj_on (m_inv G) (carrier G)" |
|
323 |
using inv_inj_on_Units by simp |
|
13813 | 324 |
|
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
325 |
lemma (in group) inv_mult_group: |
13813 | 326 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x" |
327 |
proof - |
|
14693 | 328 |
assume G: "x \<in> carrier G" "y \<in> carrier G" |
13813 | 329 |
then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)" |
14963 | 330 |
by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric]) |
331 |
with G show ?thesis by (simp del: l_inv) |
|
13813 | 332 |
qed |
333 |
||
13940 | 334 |
lemma (in group) inv_comm: |
335 |
"[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>" |
|
14693 | 336 |
by (rule Units_inv_comm) auto |
13940 | 337 |
|
13944 | 338 |
lemma (in group) inv_equality: |
13943 | 339 |
"[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y" |
340 |
apply (simp add: m_inv_def) |
|
341 |
apply (rule the_equality) |
|
14693 | 342 |
apply (simp add: inv_comm [of y x]) |
343 |
apply (rule r_cancel [THEN iffD1], auto) |
|
13943 | 344 |
done |
345 |
||
13936 | 346 |
text {* Power *} |
347 |
||
348 |
lemma (in group) int_pow_def2: |
|
349 |
"a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))" |
|
350 |
by (simp add: int_pow_def nat_pow_def Let_def) |
|
351 |
||
352 |
lemma (in group) int_pow_0 [simp]: |
|
353 |
"x (^) (0::int) = \<one>" |
|
354 |
by (simp add: int_pow_def2) |
|
355 |
||
356 |
lemma (in group) int_pow_one [simp]: |
|
357 |
"\<one> (^) (z::int) = \<one>" |
|
358 |
by (simp add: int_pow_def2) |
|
359 |
||
14963 | 360 |
subsection {* Subgroups *} |
13813 | 361 |
|
14963 | 362 |
locale subgroup = var H + struct G + |
363 |
assumes subset: "H \<subseteq> carrier G" |
|
364 |
and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H" |
|
365 |
and one_closed [simp]: "\<one> \<in> H" |
|
366 |
and m_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> inv x \<in> H" |
|
13813 | 367 |
|
368 |
declare (in subgroup) group.intro [intro] |
|
13949
0ce528cd6f19
HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents:
13944
diff
changeset
|
369 |
|
14963 | 370 |
lemma (in subgroup) mem_carrier [simp]: |
371 |
"x \<in> H \<Longrightarrow> x \<in> carrier G" |
|
372 |
using subset by blast |
|
13813 | 373 |
|
14963 | 374 |
lemma subgroup_imp_subset: |
375 |
"subgroup H G \<Longrightarrow> H \<subseteq> carrier G" |
|
376 |
by (rule subgroup.subset) |
|
377 |
||
378 |
lemma (in subgroup) subgroup_is_group [intro]: |
|
13813 | 379 |
includes group G |
14963 | 380 |
shows "group (G\<lparr>carrier := H\<rparr>)" |
381 |
by (rule groupI) (auto intro: m_assoc l_inv mem_carrier) |
|
13813 | 382 |
|
383 |
text {* |
|
384 |
Since @{term H} is nonempty, it contains some element @{term x}. Since |
|
385 |
it is closed under inverse, it contains @{text "inv x"}. Since |
|
386 |
it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}. |
|
387 |
*} |
|
388 |
||
389 |
lemma (in group) one_in_subset: |
|
390 |
"[| H \<subseteq> carrier G; H \<noteq> {}; \<forall>a \<in> H. inv a \<in> H; \<forall>a\<in>H. \<forall>b\<in>H. a \<otimes> b \<in> H |] |
|
391 |
==> \<one> \<in> H" |
|
392 |
by (force simp add: l_inv) |
|
393 |
||
394 |
text {* A characterization of subgroups: closed, non-empty subset. *} |
|
395 |
||
396 |
lemma (in group) subgroupI: |
|
397 |
assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}" |
|
14963 | 398 |
and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H" |
399 |
and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H" |
|
13813 | 400 |
shows "subgroup H G" |
14963 | 401 |
proof (simp add: subgroup_def prems) |
402 |
show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: prems) |
|
13813 | 403 |
qed |
404 |
||
13936 | 405 |
declare monoid.one_closed [iff] group.inv_closed [simp] |
406 |
monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp] |
|
13813 | 407 |
|
408 |
lemma subgroup_nonempty: |
|
409 |
"~ subgroup {} G" |
|
410 |
by (blast dest: subgroup.one_closed) |
|
411 |
||
412 |
lemma (in subgroup) finite_imp_card_positive: |
|
413 |
"finite (carrier G) ==> 0 < card H" |
|
414 |
proof (rule classical) |
|
14963 | 415 |
assume "finite (carrier G)" "~ 0 < card H" |
416 |
then have "finite H" by (blast intro: finite_subset [OF subset]) |
|
417 |
with prems have "subgroup {} G" by simp |
|
13813 | 418 |
with subgroup_nonempty show ?thesis by contradiction |
419 |
qed |
|
420 |
||
13936 | 421 |
(* |
422 |
lemma (in monoid) Units_subgroup: |
|
423 |
"subgroup (Units G) G" |
|
424 |
*) |
|
425 |
||
13813 | 426 |
subsection {* Direct Products *} |
427 |
||
14963 | 428 |
constdefs |
429 |
DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid" (infixr "\<times>\<times>" 80) |
|
430 |
"G \<times>\<times> H \<equiv> \<lparr>carrier = carrier G \<times> carrier H, |
|
431 |
mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')), |
|
432 |
one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>" |
|
13813 | 433 |
|
14963 | 434 |
lemma DirProd_monoid: |
435 |
includes monoid G + monoid H |
|
436 |
shows "monoid (G \<times>\<times> H)" |
|
437 |
proof - |
|
438 |
from prems |
|
439 |
show ?thesis by (unfold monoid_def DirProd_def, auto) |
|
440 |
qed |
|
13813 | 441 |
|
442 |
||
14963 | 443 |
text{*Does not use the previous result because it's easier just to use auto.*} |
444 |
lemma DirProd_group: |
|
13813 | 445 |
includes group G + group H |
14963 | 446 |
shows "group (G \<times>\<times> H)" |
13936 | 447 |
by (rule groupI) |
14963 | 448 |
(auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv |
449 |
simp add: DirProd_def) |
|
13813 | 450 |
|
14963 | 451 |
lemma carrier_DirProd [simp]: |
452 |
"carrier (G \<times>\<times> H) = carrier G \<times> carrier H" |
|
453 |
by (simp add: DirProd_def) |
|
13944 | 454 |
|
14963 | 455 |
lemma one_DirProd [simp]: |
456 |
"\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)" |
|
457 |
by (simp add: DirProd_def) |
|
13944 | 458 |
|
14963 | 459 |
lemma mult_DirProd [simp]: |
460 |
"(g, h) \<otimes>\<^bsub>(G \<times>\<times> H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')" |
|
461 |
by (simp add: DirProd_def) |
|
13944 | 462 |
|
14963 | 463 |
lemma inv_DirProd [simp]: |
13944 | 464 |
includes group G + group H |
465 |
assumes g: "g \<in> carrier G" |
|
466 |
and h: "h \<in> carrier H" |
|
14963 | 467 |
shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)" |
468 |
apply (rule group.inv_equality [OF DirProd_group]) |
|
13944 | 469 |
apply (simp_all add: prems group_def group.l_inv) |
470 |
done |
|
471 |
||
15696 | 472 |
text{*This alternative proof of the previous result demonstrates interpret. |
15763
b901a127ac73
Interpretation supports statically scoped attributes; documentation.
ballarin
parents:
15696
diff
changeset
|
473 |
It uses @{text Prod.inv_equality} (available after @{text interpret}) |
b901a127ac73
Interpretation supports statically scoped attributes; documentation.
ballarin
parents:
15696
diff
changeset
|
474 |
instead of @{text "group.inv_equality [OF DirProd_group]"}. *} |
14963 | 475 |
lemma |
476 |
includes group G + group H |
|
477 |
assumes g: "g \<in> carrier G" |
|
478 |
and h: "h \<in> carrier H" |
|
479 |
shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)" |
|
480 |
proof - |
|
15696 | 481 |
interpret Prod: group ["G \<times>\<times> H"] |
482 |
by (auto intro: DirProd_group group.intro group.axioms prems) |
|
14963 | 483 |
show ?thesis by (simp add: Prod.inv_equality g h) |
484 |
qed |
|
485 |
||
486 |
||
487 |
subsection {* Homomorphisms and Isomorphisms *} |
|
13813 | 488 |
|
14651 | 489 |
constdefs (structure G and H) |
490 |
hom :: "_ => _ => ('a => 'b) set" |
|
13813 | 491 |
"hom G H == |
492 |
{h. h \<in> carrier G -> carrier H & |
|
14693 | 493 |
(\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y)}" |
13813 | 494 |
|
495 |
lemma hom_mult: |
|
14693 | 496 |
"[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |] |
497 |
==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y" |
|
498 |
by (simp add: hom_def) |
|
13813 | 499 |
|
500 |
lemma hom_closed: |
|
501 |
"[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H" |
|
502 |
by (auto simp add: hom_def funcset_mem) |
|
503 |
||
14761 | 504 |
lemma (in group) hom_compose: |
505 |
"[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I" |
|
506 |
apply (auto simp add: hom_def funcset_compose) |
|
507 |
apply (simp add: compose_def funcset_mem) |
|
13943 | 508 |
done |
509 |
||
14761 | 510 |
|
511 |
subsection {* Isomorphisms *} |
|
512 |
||
14803 | 513 |
constdefs |
514 |
iso :: "_ => _ => ('a => 'b) set" (infixr "\<cong>" 60) |
|
515 |
"G \<cong> H == {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}" |
|
14761 | 516 |
|
14803 | 517 |
lemma iso_refl: "(%x. x) \<in> G \<cong> G" |
14761 | 518 |
by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) |
519 |
||
520 |
lemma (in group) iso_sym: |
|
14803 | 521 |
"h \<in> G \<cong> H \<Longrightarrow> Inv (carrier G) h \<in> H \<cong> G" |
14761 | 522 |
apply (simp add: iso_def bij_betw_Inv) |
523 |
apply (subgoal_tac "Inv (carrier G) h \<in> carrier H \<rightarrow> carrier G") |
|
524 |
prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_Inv]) |
|
525 |
apply (simp add: hom_def bij_betw_def Inv_f_eq funcset_mem f_Inv_f) |
|
526 |
done |
|
527 |
||
528 |
lemma (in group) iso_trans: |
|
14803 | 529 |
"[|h \<in> G \<cong> H; i \<in> H \<cong> I|] ==> (compose (carrier G) i h) \<in> G \<cong> I" |
14761 | 530 |
by (auto simp add: iso_def hom_compose bij_betw_compose) |
531 |
||
14963 | 532 |
lemma DirProd_commute_iso: |
533 |
shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)" |
|
14761 | 534 |
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) |
535 |
||
14963 | 536 |
lemma DirProd_assoc_iso: |
537 |
shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> (G \<times>\<times> H \<times>\<times> I) \<cong> (G \<times>\<times> (H \<times>\<times> I))" |
|
14761 | 538 |
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) |
539 |
||
540 |
||
14963 | 541 |
text{*Basis for homomorphism proofs: we assume two groups @{term G} and |
15076
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14963
diff
changeset
|
542 |
@{term H}, with a homomorphism @{term h} between them*} |
13813 | 543 |
locale group_hom = group G + group H + var h + |
544 |
assumes homh: "h \<in> hom G H" |
|
545 |
notes hom_mult [simp] = hom_mult [OF homh] |
|
546 |
and hom_closed [simp] = hom_closed [OF homh] |
|
547 |
||
548 |
lemma (in group_hom) one_closed [simp]: |
|
549 |
"h \<one> \<in> carrier H" |
|
550 |
by simp |
|
551 |
||
552 |
lemma (in group_hom) hom_one [simp]: |
|
14693 | 553 |
"h \<one> = \<one>\<^bsub>H\<^esub>" |
13813 | 554 |
proof - |
15076
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14963
diff
changeset
|
555 |
have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>" |
13813 | 556 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
557 |
then show ?thesis by (simp del: r_one) |
|
558 |
qed |
|
559 |
||
560 |
lemma (in group_hom) inv_closed [simp]: |
|
561 |
"x \<in> carrier G ==> h (inv x) \<in> carrier H" |
|
562 |
by simp |
|
563 |
||
564 |
lemma (in group_hom) hom_inv [simp]: |
|
14693 | 565 |
"x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)" |
13813 | 566 |
proof - |
567 |
assume x: "x \<in> carrier G" |
|
14693 | 568 |
then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>" |
14963 | 569 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
14693 | 570 |
also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" |
14963 | 571 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
14693 | 572 |
finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" . |
14963 | 573 |
with x show ?thesis by (simp del: H.r_inv) |
13813 | 574 |
qed |
575 |
||
13949
0ce528cd6f19
HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents:
13944
diff
changeset
|
576 |
subsection {* Commutative Structures *} |
13936 | 577 |
|
578 |
text {* |
|
579 |
Naming convention: multiplicative structures that are commutative |
|
580 |
are called \emph{commutative}, additive structures are called |
|
581 |
\emph{Abelian}. |
|
582 |
*} |
|
13813 | 583 |
|
584 |
subsection {* Definition *} |
|
585 |
||
14963 | 586 |
locale comm_monoid = monoid + |
587 |
assumes m_comm: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x" |
|
13813 | 588 |
|
14963 | 589 |
lemma (in comm_monoid) m_lcomm: |
590 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> |
|
13813 | 591 |
x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)" |
592 |
proof - |
|
14693 | 593 |
assume xyz: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
13813 | 594 |
from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc) |
595 |
also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm) |
|
596 |
also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc) |
|
597 |
finally show ?thesis . |
|
598 |
qed |
|
599 |
||
14963 | 600 |
lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm |
13813 | 601 |
|
13936 | 602 |
lemma comm_monoidI: |
14693 | 603 |
includes struct G |
13936 | 604 |
assumes m_closed: |
14693 | 605 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
606 |
and one_closed: "\<one> \<in> carrier G" |
|
13936 | 607 |
and m_assoc: |
608 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 609 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
610 |
and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
|
13936 | 611 |
and m_comm: |
14693 | 612 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
13936 | 613 |
shows "comm_monoid G" |
614 |
using l_one |
|
14963 | 615 |
by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro |
616 |
intro: prems simp: m_closed one_closed m_comm) |
|
13817 | 617 |
|
13936 | 618 |
lemma (in monoid) monoid_comm_monoidI: |
619 |
assumes m_comm: |
|
14693 | 620 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
13936 | 621 |
shows "comm_monoid G" |
622 |
by (rule comm_monoidI) (auto intro: m_assoc m_comm) |
|
14963 | 623 |
|
14693 | 624 |
(*lemma (in comm_monoid) r_one [simp]: |
13817 | 625 |
"x \<in> carrier G ==> x \<otimes> \<one> = x" |
626 |
proof - |
|
627 |
assume G: "x \<in> carrier G" |
|
628 |
then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm) |
|
629 |
also from G have "... = x" by simp |
|
630 |
finally show ?thesis . |
|
14693 | 631 |
qed*) |
14963 | 632 |
|
13936 | 633 |
lemma (in comm_monoid) nat_pow_distr: |
634 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> |
|
635 |
(x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n" |
|
636 |
by (induct n) (simp, simp add: m_ac) |
|
637 |
||
638 |
locale comm_group = comm_monoid + group |
|
639 |
||
640 |
lemma (in group) group_comm_groupI: |
|
641 |
assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> |
|
14693 | 642 |
x \<otimes> y = y \<otimes> x" |
13936 | 643 |
shows "comm_group G" |
14963 | 644 |
by (fast intro: comm_group.intro comm_monoid_axioms.intro |
14761 | 645 |
is_group prems) |
13817 | 646 |
|
13936 | 647 |
lemma comm_groupI: |
14693 | 648 |
includes struct G |
13936 | 649 |
assumes m_closed: |
14693 | 650 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
651 |
and one_closed: "\<one> \<in> carrier G" |
|
13936 | 652 |
and m_assoc: |
653 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 654 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
13936 | 655 |
and m_comm: |
14693 | 656 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
657 |
and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
|
14963 | 658 |
and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
13936 | 659 |
shows "comm_group G" |
660 |
by (fast intro: group.group_comm_groupI groupI prems) |
|
661 |
||
662 |
lemma (in comm_group) inv_mult: |
|
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
663 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y" |
13936 | 664 |
by (simp add: m_ac inv_mult_group) |
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
665 |
|
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
666 |
subsection {* Lattice of subgroups of a group *} |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
667 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
668 |
text_raw {* \label{sec:subgroup-lattice} *} |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
669 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
670 |
theorem (in group) subgroups_partial_order: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
671 |
"partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
672 |
by (rule partial_order.intro) simp_all |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
673 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
674 |
lemma (in group) subgroup_self: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
675 |
"subgroup (carrier G) G" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
676 |
by (rule subgroupI) auto |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
677 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
678 |
lemma (in group) subgroup_imp_group: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
679 |
"subgroup H G ==> group (G(| carrier := H |))" |
14963 | 680 |
using subgroup.subgroup_is_group [OF _ group.intro] . |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
681 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
682 |
lemma (in group) is_monoid [intro, simp]: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
683 |
"monoid G" |
14963 | 684 |
by (auto intro: monoid.intro m_assoc) |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
685 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
686 |
lemma (in group) subgroup_inv_equality: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
687 |
"[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
688 |
apply (rule_tac inv_equality [THEN sym]) |
14761 | 689 |
apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+) |
690 |
apply (rule subsetD [OF subgroup.subset], assumption+) |
|
691 |
apply (rule subsetD [OF subgroup.subset], assumption) |
|
692 |
apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+) |
|
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
693 |
done |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
694 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
695 |
theorem (in group) subgroups_Inter: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
696 |
assumes subgr: "(!!H. H \<in> A ==> subgroup H G)" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
697 |
and not_empty: "A ~= {}" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
698 |
shows "subgroup (\<Inter>A) G" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
699 |
proof (rule subgroupI) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
700 |
from subgr [THEN subgroup.subset] and not_empty |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
701 |
show "\<Inter>A \<subseteq> carrier G" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
702 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
703 |
from subgr [THEN subgroup.one_closed] |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
704 |
show "\<Inter>A ~= {}" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
705 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
706 |
fix x assume "x \<in> \<Inter>A" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
707 |
with subgr [THEN subgroup.m_inv_closed] |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
708 |
show "inv x \<in> \<Inter>A" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
709 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
710 |
fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
711 |
with subgr [THEN subgroup.m_closed] |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
712 |
show "x \<otimes> y \<in> \<Inter>A" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
713 |
qed |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
714 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
715 |
theorem (in group) subgroups_complete_lattice: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
716 |
"complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
717 |
(is "complete_lattice ?L") |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
718 |
proof (rule partial_order.complete_lattice_criterion1) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
719 |
show "partial_order ?L" by (rule subgroups_partial_order) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
720 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
721 |
have "greatest ?L (carrier G) (carrier ?L)" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
722 |
by (unfold greatest_def) (simp add: subgroup.subset subgroup_self) |
14963 | 723 |
then show "\<exists>G. greatest ?L G (carrier ?L)" .. |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
724 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
725 |
fix A |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
726 |
assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
727 |
then have Int_subgroup: "subgroup (\<Inter>A) G" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
728 |
by (fastsimp intro: subgroups_Inter) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
729 |
have "greatest ?L (\<Inter>A) (Lower ?L A)" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
730 |
(is "greatest ?L ?Int _") |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
731 |
proof (rule greatest_LowerI) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
732 |
fix H |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
733 |
assume H: "H \<in> A" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
734 |
with L have subgroupH: "subgroup H G" by auto |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
735 |
from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H") |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
736 |
by (rule subgroup_imp_group) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
737 |
from groupH have monoidH: "monoid ?H" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
738 |
by (rule group.is_monoid) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
739 |
from H have Int_subset: "?Int \<subseteq> H" by fastsimp |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
740 |
then show "le ?L ?Int H" by simp |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
741 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
742 |
fix H |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
743 |
assume H: "H \<in> Lower ?L A" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
744 |
with L Int_subgroup show "le ?L H ?Int" by (fastsimp intro: Inter_greatest) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
745 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
746 |
show "A \<subseteq> carrier ?L" by (rule L) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
747 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
748 |
show "?Int \<in> carrier ?L" by simp (rule Int_subgroup) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
749 |
qed |
14963 | 750 |
then show "\<exists>I. greatest ?L I (Lower ?L A)" .. |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
751 |
qed |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
752 |
|
13813 | 753 |
end |