author paulson Thu May 01 11:54:18 2003 +0200 (2003-05-01) changeset 13944 9b34607cd83e parent 13943 83d842ccd4aa child 13945 5433b2755e98
new proofs about direct products, etc.
 src/HOL/Algebra/Group.thy file | annotate | diff | revisions src/HOL/Algebra/README.html file | annotate | diff | revisions
```     1.1 --- a/src/HOL/Algebra/Group.thy	Thu May 01 10:29:44 2003 +0200
1.2 +++ b/src/HOL/Algebra/Group.thy	Thu May 01 11:54:18 2003 +0200
1.3 @@ -348,7 +348,7 @@
1.4    "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
1.5    by (rule Units_inv_comm) auto
1.6
1.7 -lemma (in group) m_inv_equality:
1.8 +lemma (in group) inv_equality:
1.9       "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"
1.11  apply (rule the_equality)
1.12 @@ -567,6 +567,27 @@
1.13      (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
1.15
1.16 +lemma carrier_DirProdGroup [simp]:
1.17 +     "carrier (G \<times>\<^sub>g H) = carrier G \<times> carrier H"
1.18 +  by (simp add: DirProdGroup_def DirProdSemigroup_def)
1.19 +
1.20 +lemma one_DirProdGroup [simp]:
1.21 +     "one (G \<times>\<^sub>g H) = (one G, one H)"
1.22 +  by (simp add: DirProdGroup_def DirProdSemigroup_def);
1.23 +
1.24 +lemma mult_DirProdGroup [simp]:
1.25 +     "mult (G \<times>\<^sub>g H) (g, h) (g', h') = (mult G g g', mult H h h')"
1.26 +  by (simp add: DirProdGroup_def DirProdSemigroup_def)
1.27 +
1.28 +lemma inv_DirProdGroup [simp]:
1.29 +  includes group G + group H
1.30 +  assumes g: "g \<in> carrier G"
1.31 +      and h: "h \<in> carrier H"
1.32 +  shows "m_inv (G \<times>\<^sub>g H) (g, h) = (m_inv G g, m_inv H h)"
1.33 +  apply (rule group.inv_equality [OF DirProdGroup_group])
1.34 +  apply (simp_all add: prems group_def group.l_inv)
1.35 +  done
1.36 +
1.37  subsection {* Homomorphisms *}
1.38
1.39  constdefs
```
```     2.1 --- a/src/HOL/Algebra/README.html	Thu May 01 10:29:44 2003 +0200
2.2 +++ b/src/HOL/Algebra/README.html	Thu May 01 11:54:18 2003 +0200
2.3 @@ -48,6 +48,36 @@
2.4  <P>[Ballarin1999] Clemens Ballarin, Computer Algebra and Theorem Proving,
2.5    Author's <A HREF="http://iaks-www.ira.uka.de/iaks-calmet/ballarin/publications.html">PhD thesis</A>, 1999.
2.6
2.7 +<H2>GroupTheory -- Group Theory using Locales, including Sylow's Theorem</H2>
2.8 +
2.9 +<P>This directory presents proofs about group theory, by
2.10 +Florian Kammüller.  (Later, Larry Paulson simplified some of the proofs.)
2.11 +These theories use locales and were indeed the original motivation for
2.12 +locales.  However, this treatment of groups must still be regarded as
2.13 +experimental.  We can expect to see refinements in the future.
2.14 +
2.15 +Here is an outline of the directory's contents:
2.16 +
2.17 +<UL> <LI>Theory <A HREF="Group.html"><CODE>Group</CODE></A> defines
2.18 +semigroups, groups, homomorphisms and the subgroup relation.  It also defines
2.19 +the product of two groups.  It defines the factorization of a group and shows
2.20 +that the factorization a normal subgroup is a group.
2.21 +
2.22 +<LI>Theory <A HREF="Bij.html"><CODE>Bij</CODE></A>
2.23 +defines bijections over sets and operations on them and shows that they
2.24 +are a group.  It shows that automorphisms form a group.
2.25 +
2.26 +<LI>Theory <A HREF="Ring.html"><CODE>Ring</CODE></A> defines rings and proves
2.27 +a few basic theorems.  Ring automorphisms are shown to form a group.
2.28 +
2.29 +<LI>Theory <A HREF="Sylow.html"><CODE>Sylow</CODE></A>
2.30 +contains a proof of the first Sylow theorem.
2.31 +
2.32 +<LI>Theory <A HREF="Summation.html"><CODE>Summation</CODE></A> Extends
2.33 +abelian groups by a summation operator for finite sets (provided by
2.34 +Clemens Ballarin).
2.35 +</UL>
2.36 +
2.37  <HR>