src/HOL/Algebra/Group.thy
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(*
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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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theory Group imports FuncSet Lattice begin
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section {* Monoids and Groups *}
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subsection {* Definitions *}
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text {*
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  Definitions follow \cite{Jacobson:1985}.
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*}
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record 'a monoid =  "'a partial_object" +
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  mult    :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<otimes>\<index>" 70)
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  one     :: 'a ("\<one>\<index>")
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constdefs (structure G)
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  m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _" [81] 80)
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  "inv x == (THE y. y \<in> carrier G & x \<otimes> y = \<one> & y \<otimes> x = \<one>)"
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  Units :: "_ => 'a set"
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  --{*The set of invertible elements*}
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  "Units G == {y. y \<in> carrier G & (\<exists>x \<in> carrier G. x \<otimes> y = \<one> & y \<otimes> x = \<one>)}"
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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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defs (overloaded)
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  nat_pow_def: "pow G a n == nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n"
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  int_pow_def: "pow G a z ==
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    let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a)
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    in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z)"
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locale monoid =
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  fixes G (structure)
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  assumes m_closed [intro, simp]:
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         "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> carrier G"
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      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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lemma monoidI:
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  assumes m_closed:
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      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
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    and one_closed: "\<one> \<in> carrier G"
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    and m_assoc:
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      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
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      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
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    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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  shows "monoid G"
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  by (fast intro!: monoid.intro intro: prems)
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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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  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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  shows "y = y'"
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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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lemma (in monoid) Units_m_closed [intro, simp]:
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  assumes x: "x \<in> Units G" and y: "y \<in> Units G"
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  shows "x \<otimes> y \<in> Units G"
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proof -
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  from x obtain x' where x: "x \<in> carrier G" "x' \<in> carrier G" and xinv: "x \<otimes> x' = \<one>" "x' \<otimes> x = \<one>"
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    unfolding Units_def by fast
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  from y obtain y' where y: "y \<in> carrier G" "y' \<in> carrier G" and yinv: "y \<otimes> y' = \<one>" "y' \<otimes> y = \<one>"
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    unfolding Units_def by fast
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  from x y xinv yinv have "y' \<otimes> (x' \<otimes> x) \<otimes> y = \<one>" by simp
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  moreover from x y xinv yinv have "x \<otimes> (y \<otimes> y') \<otimes> x' = \<one>" by simp
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  moreover note x y
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  ultimately show ?thesis unfolding Units_def
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    -- "Must avoid premature use of @{text hyp_subst_tac}."
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    apply (rule_tac CollectI)
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    apply (rule)
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    apply (fast)
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    apply (rule bexI [where x = "y' \<otimes> x'"])
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    apply (auto simp: m_assoc)
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    done
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qed
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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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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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  apply (unfold Units_def m_inv_def, auto)
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  apply (rule theI2, fast)
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   apply (fast intro: inv_unique, fast)
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  done
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lemma (in monoid) Units_l_inv_ex:
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  "x \<in> Units G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
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  by (unfold Units_def) auto
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lemma (in monoid) Units_r_inv_ex:
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  "x \<in> Units G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"
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  by (unfold Units_def) auto
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lemma (in monoid) Units_l_inv [simp]:
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  "x \<in> Units G ==> inv x \<otimes> x = \<one>"
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  apply (unfold Units_def m_inv_def, auto)
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  apply (rule theI2, fast)
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   apply (fast intro: inv_unique, fast)
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  done
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lemma (in monoid) Units_r_inv [simp]:
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  "x \<in> Units G ==> x \<otimes> inv x = \<one>"
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  apply (unfold Units_def m_inv_def, auto)
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  apply (rule theI2, fast)
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   apply (fast intro: inv_unique, fast)
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  done
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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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    and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"
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  then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z"
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   150
    by (simp add: m_assoc Units_closed del: Units_l_inv)
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   151
  with G show "y = z" by (simp add: Units_l_inv)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   152
next
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   153
  assume eq: "y = z"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   154
    and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   155
  then show "x \<otimes> y = x \<otimes> z" by simp
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   156
qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   157
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   158
lemma (in monoid) Units_inv_inv [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   159
  "x \<in> Units G ==> inv (inv x) = x"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   160
proof -
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   161
  assume x: "x \<in> Units G"
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   162
  then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x" by simp
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   163
  with x show ?thesis by (simp add: Units_closed del: Units_l_inv Units_r_inv)
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   164
qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   165
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   166
lemma (in monoid) inv_inj_on_Units:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   167
  "inj_on (m_inv G) (Units G)"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   168
proof (rule inj_onI)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   169
  fix x y
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   170
  assume G: "x \<in> Units G"  "y \<in> Units G" and eq: "inv x = inv y"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   171
  then have "inv (inv x) = inv (inv y)" by simp
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   172
  with G show "x = y" by simp
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   173
qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   174
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   175
lemma (in monoid) Units_inv_comm:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   176
  assumes inv: "x \<otimes> y = \<one>"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   177
    and G: "x \<in> Units G"  "y \<in> Units G"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   178
  shows "y \<otimes> x = \<one>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   179
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   180
  from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   181
  with G show ?thesis by (simp del: r_one add: m_assoc Units_closed)
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   182
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   183
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   184
text {* Power *}
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   185
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   186
lemma (in monoid) nat_pow_closed [intro, simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   187
  "x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   188
  by (induct n) (simp_all add: nat_pow_def)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   189
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   190
lemma (in monoid) nat_pow_0 [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   191
  "x (^) (0::nat) = \<one>"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   192
  by (simp add: nat_pow_def)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   193
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   194
lemma (in monoid) nat_pow_Suc [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   195
  "x (^) (Suc n) = x (^) n \<otimes> x"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   196
  by (simp add: nat_pow_def)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   197
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   198
lemma (in monoid) nat_pow_one [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   199
  "\<one> (^) (n::nat) = \<one>"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   200
  by (induct n) simp_all
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   201
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   202
lemma (in monoid) nat_pow_mult:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   203
  "x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   204
  by (induct m) (simp_all add: m_assoc [THEN sym])
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   205
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   206
lemma (in monoid) nat_pow_pow:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   207
  "x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   208
  by (induct m) (simp, simp add: nat_pow_mult add_commute)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   209
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   210
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   211
(* Jacobson defines submonoid here. *)
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   212
(* Jacobson defines the order of a monoid here. *)
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   213
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   214
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   215
subsection {* Groups *}
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   216
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   217
text {*
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   218
  A group is a monoid all of whose elements are invertible.
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   219
*}
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   220
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   221
locale group = monoid +
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   222
  assumes Units: "carrier G <= Units G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   223
26199
04817a8802f2 explicit referencing of background facts;
wenzelm
parents: 23350
diff changeset
   224
lemma (in group) is_group: "group G" by (rule group_axioms)
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   225
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   226
theorem groupI:
19783
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19699
diff changeset
   227
  fixes G (structure)
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   228
  assumes m_closed [simp]:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   229
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   230
    and one_closed [simp]: "\<one> \<in> carrier G"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   231
    and m_assoc:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   232
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   233
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   234
    and l_one [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   235
    and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   236
  shows "group G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   237
proof -
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   238
  have l_cancel [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   239
    "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   240
    (x \<otimes> y = x \<otimes> z) = (y = z)"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   241
  proof
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   242
    fix x y z
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   243
    assume eq: "x \<otimes> y = x \<otimes> z"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   244
      and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   245
    with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   246
      and l_inv: "x_inv \<otimes> x = \<one>" by fast
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   247
    from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> z"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   248
      by (simp add: m_assoc)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   249
    with G show "y = z" by (simp add: l_inv)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   250
  next
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   251
    fix x y z
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   252
    assume eq: "y = z"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   253
      and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   254
    then show "x \<otimes> y = x \<otimes> z" by simp
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   255
  qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   256
  have r_one:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   257
    "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   258
  proof -
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   259
    fix x
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   260
    assume x: "x \<in> carrier G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   261
    with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   262
      and l_inv: "x_inv \<otimes> x = \<one>" by fast
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   263
    from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   264
      by (simp add: m_assoc [symmetric] l_inv)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   265
    with x xG show "x \<otimes> \<one> = x" by simp
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   266
  qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   267
  have inv_ex:
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   268
    "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   269
  proof -
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   270
    fix x
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   271
    assume x: "x \<in> carrier G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   272
    with l_inv_ex obtain y where y: "y \<in> carrier G"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   273
      and l_inv: "y \<otimes> x = \<one>" by fast
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   274
    from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   275
      by (simp add: m_assoc [symmetric] l_inv r_one)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   276
    with x y have r_inv: "x \<otimes> y = \<one>"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   277
      by simp
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   278
    from x y show "\<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   279
      by (fast intro: l_inv r_inv)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   280
  qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   281
  then have carrier_subset_Units: "carrier G <= Units G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   282
    by (unfold Units_def) fast
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   283
  show ?thesis by unfold_locales (auto simp: r_one m_assoc carrier_subset_Units)
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   284
qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   285
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   286
lemma (in monoid) group_l_invI:
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   287
  assumes l_inv_ex:
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   288
    "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   289
  shows "group G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   290
  by (rule groupI) (auto intro: m_assoc l_inv_ex)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   291
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   292
lemma (in group) Units_eq [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   293
  "Units G = carrier G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   294
proof
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   295
  show "Units G <= carrier G" by fast
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   296
next
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   297
  show "carrier G <= Units G" by (rule Units)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   298
qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   299
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   300
lemma (in group) inv_closed [intro, simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   301
  "x \<in> carrier G ==> inv x \<in> carrier G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   302
  using Units_inv_closed by simp
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   303
19981
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   304
lemma (in group) l_inv_ex [simp]:
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   305
  "x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   306
  using Units_l_inv_ex by simp
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   307
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   308
lemma (in group) r_inv_ex [simp]:
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   309
  "x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   310
  using Units_r_inv_ex by simp
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   311
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   312
lemma (in group) l_inv [simp]:
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   313
  "x \<in> carrier G ==> inv x \<otimes> x = \<one>"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   314
  using Units_l_inv by simp
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   315
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   316
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   317
subsection {* Cancellation Laws and Basic Properties *}
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   318
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   319
lemma (in group) l_cancel [simp]:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   320
  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   321
   (x \<otimes> y = x \<otimes> z) = (y = z)"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   322
  using Units_l_inv by simp
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   323
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   324
lemma (in group) r_inv [simp]:
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   325
  "x \<in> carrier G ==> x \<otimes> inv x = \<one>"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   326
proof -
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   327
  assume x: "x \<in> carrier G"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   328
  then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   329
    by (simp add: m_assoc [symmetric] l_inv)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   330
  with x show ?thesis by (simp del: r_one)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   331
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   332
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   333
lemma (in group) r_cancel [simp]:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   334
  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   335
   (y \<otimes> x = z \<otimes> x) = (y = z)"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   336
proof
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   337
  assume eq: "y \<otimes> x = z \<otimes> x"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   338
    and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   339
  then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)"
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   340
    by (simp add: m_assoc [symmetric] del: r_inv Units_r_inv)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   341
  with G show "y = z" by simp
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   342
next
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   343
  assume eq: "y = z"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   344
    and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   345
  then show "y \<otimes> x = z \<otimes> x" by simp
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   346
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   347
13854
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   348
lemma (in group) inv_one [simp]:
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   349
  "inv \<one> = \<one>"
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   350
proof -
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   351
  have "inv \<one> = \<one> \<otimes> (inv \<one>)" by (simp del: r_inv Units_r_inv)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   352
  moreover have "... = \<one>" by simp
13854
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   353
  finally show ?thesis .
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   354
qed
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   355
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   356
lemma (in group) inv_inv [simp]:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   357
  "x \<in> carrier G ==> inv (inv x) = x"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   358
  using Units_inv_inv by simp
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   359
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   360
lemma (in group) inv_inj:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   361
  "inj_on (m_inv G) (carrier G)"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   362
  using inv_inj_on_Units by simp
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   363
13854
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   364
lemma (in group) inv_mult_group:
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   365
  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   366
proof -
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   367
  assume G: "x \<in> carrier G"  "y \<in> carrier G"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   368
  then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   369
    by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric])
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   370
  with G show ?thesis by (simp del: l_inv Units_l_inv)
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   371
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   372
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   373
lemma (in group) inv_comm:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   374
  "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   375
  by (rule Units_inv_comm) auto
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   376
13944
9b34607cd83e new proofs about direct products, etc.
paulson
parents: 13943
diff changeset
   377
lemma (in group) inv_equality:
13943
83d842ccd4aa moving Bij.thy from GroupTheory to Algebra
paulson
parents: 13940
diff changeset
   378
     "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"
83d842ccd4aa moving Bij.thy from GroupTheory to Algebra
paulson
parents: 13940
diff changeset
   379
apply (simp add: m_inv_def)
83d842ccd4aa moving Bij.thy from GroupTheory to Algebra
paulson
parents: 13940
diff changeset
   380
apply (rule the_equality)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   381
 apply (simp add: inv_comm [of y x])
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   382
apply (rule r_cancel [THEN iffD1], auto)
13943
83d842ccd4aa moving Bij.thy from GroupTheory to Algebra
paulson
parents: 13940
diff changeset
   383
done
83d842ccd4aa moving Bij.thy from GroupTheory to Algebra
paulson
parents: 13940
diff changeset
   384
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   385
text {* Power *}
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   386
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   387
lemma (in group) int_pow_def2:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   388
  "a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   389
  by (simp add: int_pow_def nat_pow_def Let_def)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   390
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   391
lemma (in group) int_pow_0 [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   392
  "x (^) (0::int) = \<one>"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   393
  by (simp add: int_pow_def2)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   394
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   395
lemma (in group) int_pow_one [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   396
  "\<one> (^) (z::int) = \<one>"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   397
  by (simp add: int_pow_def2)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   398
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   399
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   400
subsection {* Subgroups *}
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   401
19783
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19699
diff changeset
   402
locale subgroup =
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19699
diff changeset
   403
  fixes H and G (structure)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   404
  assumes subset: "H \<subseteq> carrier G"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   405
    and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H"
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   406
    and one_closed [simp]: "\<one> \<in> H"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   407
    and m_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> inv x \<in> H"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   408
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   409
lemma (in subgroup) is_subgroup:
26199
04817a8802f2 explicit referencing of background facts;
wenzelm
parents: 23350
diff changeset
   410
  "subgroup H G" by (rule subgroup_axioms)
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   411
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   412
declare (in subgroup) group.intro [intro]
13949
0ce528cd6f19 HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents: 13944
diff changeset
   413
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   414
lemma (in subgroup) mem_carrier [simp]:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   415
  "x \<in> H \<Longrightarrow> x \<in> carrier G"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   416
  using subset by blast
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   417
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   418
lemma subgroup_imp_subset:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   419
  "subgroup H G \<Longrightarrow> H \<subseteq> carrier G"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   420
  by (rule subgroup.subset)
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   421
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   422
lemma (in subgroup) subgroup_is_group [intro]:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   423
  assumes "group G"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   424
  shows "group (G\<lparr>carrier := H\<rparr>)"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   425
proof -
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   426
  interpret group [G] by fact
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   427
  show ?thesis
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   428
    apply (rule monoid.group_l_invI)
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   429
    apply (unfold_locales) [1]
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   430
    apply (auto intro: m_assoc l_inv mem_carrier)
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   431
    done
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   432
qed
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   433
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   434
text {*
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   435
  Since @{term H} is nonempty, it contains some element @{term x}.  Since
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   436
  it is closed under inverse, it contains @{text "inv x"}.  Since
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   437
  it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}.
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   438
*}
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   439
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   440
lemma (in group) one_in_subset:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   441
  "[| 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 |]
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   442
   ==> \<one> \<in> H"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   443
by (force simp add: l_inv)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   444
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   445
text {* A characterization of subgroups: closed, non-empty subset. *}
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   446
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   447
lemma (in group) subgroupI:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   448
  assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   449
    and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   450
    and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   451
  shows "subgroup H G"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   452
proof (simp add: subgroup_def prems)
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   453
  show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: prems)
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   454
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   455
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   456
declare monoid.one_closed [iff] group.inv_closed [simp]
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   457
  monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   458
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   459
lemma subgroup_nonempty:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   460
  "~ subgroup {} G"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   461
  by (blast dest: subgroup.one_closed)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   462
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   463
lemma (in subgroup) finite_imp_card_positive:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   464
  "finite (carrier G) ==> 0 < card H"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   465
proof (rule classical)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   466
  assume "finite (carrier G)" "~ 0 < card H"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   467
  then have "finite H" by (blast intro: finite_subset [OF subset])
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   468
  with prems have "subgroup {} G" by simp
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   469
  with subgroup_nonempty show ?thesis by contradiction
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   470
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   471
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   472
(*
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   473
lemma (in monoid) Units_subgroup:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   474
  "subgroup (Units G) G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   475
*)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   476
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   477
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   478
subsection {* Direct Products *}
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   479
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   480
constdefs
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   481
  DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid"  (infixr "\<times>\<times>" 80)
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   482
  "G \<times>\<times> H \<equiv> \<lparr>carrier = carrier G \<times> carrier H,
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   483
                mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')),
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   484
                one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   485
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   486
lemma DirProd_monoid:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   487
  assumes "monoid G" and "monoid H"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   488
  shows "monoid (G \<times>\<times> H)"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   489
proof -
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   490
  interpret G: monoid [G] by fact
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   491
  interpret H: monoid [H] by fact
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   492
  from prems
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   493
  show ?thesis by (unfold monoid_def DirProd_def, auto) 
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   494
qed
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   495
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   496
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   497
text{*Does not use the previous result because it's easier just to use auto.*}
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   498
lemma DirProd_group:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   499
  assumes "group G" and "group H"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   500
  shows "group (G \<times>\<times> H)"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   501
proof -
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   502
  interpret G: group [G] by fact
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   503
  interpret H: group [H] by fact
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   504
  show ?thesis by (rule groupI)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   505
     (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   506
           simp add: DirProd_def)
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   507
qed
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   508
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   509
lemma carrier_DirProd [simp]:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   510
     "carrier (G \<times>\<times> H) = carrier G \<times> carrier H"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   511
  by (simp add: DirProd_def)
13944
9b34607cd83e new proofs about direct products, etc.
paulson
parents: 13943
diff changeset
   512
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   513
lemma one_DirProd [simp]:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   514
     "\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   515
  by (simp add: DirProd_def)
13944
9b34607cd83e new proofs about direct products, etc.
paulson
parents: 13943
diff changeset
   516
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   517
lemma mult_DirProd [simp]:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   518
     "(g, h) \<otimes>\<^bsub>(G \<times>\<times> H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   519
  by (simp add: DirProd_def)
13944
9b34607cd83e new proofs about direct products, etc.
paulson
parents: 13943
diff changeset
   520
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   521
lemma inv_DirProd [simp]:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   522
  assumes "group G" and "group H"
13944
9b34607cd83e new proofs about direct products, etc.
paulson
parents: 13943
diff changeset
   523
  assumes g: "g \<in> carrier G"
9b34607cd83e new proofs about direct products, etc.
paulson
parents: 13943
diff changeset
   524
      and h: "h \<in> carrier H"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   525
  shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   526
proof -
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   527
  interpret G: group [G] by fact
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   528
  interpret H: group [H] by fact
15696
1da4ce092c0b First release of interpretation commands.
ballarin
parents: 15076
diff changeset
   529
  interpret Prod: group ["G \<times>\<times> H"]
1da4ce092c0b First release of interpretation commands.
ballarin
parents: 15076
diff changeset
   530
    by (auto intro: DirProd_group group.intro group.axioms prems)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   531
  show ?thesis by (simp add: Prod.inv_equality g h)
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   532
qed
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   533
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   534
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   535
subsection {* Homomorphisms and Isomorphisms *}
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   536
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14551
diff changeset
   537
constdefs (structure G and H)
02b8f3bcf7fe improved notation;
wenzelm
parents: 14551
diff changeset
   538
  hom :: "_ => _ => ('a => 'b) set"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   539
  "hom G H ==
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   540
    {h. h \<in> carrier G -> carrier H &
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   541
      (\<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
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   542
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   543
lemma hom_mult:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   544
  "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |]
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   545
   ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   546
  by (simp add: hom_def)
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   547
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   548
lemma hom_closed:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   549
  "[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   550
  by (auto simp add: hom_def funcset_mem)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   551
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   552
lemma (in group) hom_compose:
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   553
     "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   554
apply (auto simp add: hom_def funcset_compose) 
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   555
apply (simp add: compose_def funcset_mem)
13943
83d842ccd4aa moving Bij.thy from GroupTheory to Algebra
paulson
parents: 13940
diff changeset
   556
done
83d842ccd4aa moving Bij.thy from GroupTheory to Algebra
paulson
parents: 13940
diff changeset
   557
14803
f7557773cc87 more group isomorphisms
paulson
parents: 14761
diff changeset
   558
constdefs
f7557773cc87 more group isomorphisms
paulson
parents: 14761
diff changeset
   559
  iso :: "_ => _ => ('a => 'b) set"  (infixr "\<cong>" 60)
f7557773cc87 more group isomorphisms
paulson
parents: 14761
diff changeset
   560
  "G \<cong> H == {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}"
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   561
14803
f7557773cc87 more group isomorphisms
paulson
parents: 14761
diff changeset
   562
lemma iso_refl: "(%x. x) \<in> G \<cong> G"
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   563
by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   564
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   565
lemma (in group) iso_sym:
14803
f7557773cc87 more group isomorphisms
paulson
parents: 14761
diff changeset
   566
     "h \<in> G \<cong> H \<Longrightarrow> Inv (carrier G) h \<in> H \<cong> G"
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   567
apply (simp add: iso_def bij_betw_Inv) 
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   568
apply (subgoal_tac "Inv (carrier G) h \<in> carrier H \<rightarrow> carrier G") 
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   569
 prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_Inv]) 
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   570
apply (simp add: hom_def bij_betw_def Inv_f_eq funcset_mem f_Inv_f) 
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   571
done
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   572
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   573
lemma (in group) iso_trans: 
14803
f7557773cc87 more group isomorphisms
paulson
parents: 14761
diff changeset
   574
     "[|h \<in> G \<cong> H; i \<in> H \<cong> I|] ==> (compose (carrier G) i h) \<in> G \<cong> I"
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   575
by (auto simp add: iso_def hom_compose bij_betw_compose)
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   576
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   577
lemma DirProd_commute_iso:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   578
  shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)"
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   579
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   580
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   581
lemma DirProd_assoc_iso:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   582
  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
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   583
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   584
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   585
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   586
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
   587
  @{term H}, with a homomorphism @{term h} between them*}
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   588
locale group_hom = group G + group H + var h +
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   589
  assumes homh: "h \<in> hom G H"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   590
  notes hom_mult [simp] = hom_mult [OF homh]
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   591
    and hom_closed [simp] = hom_closed [OF homh]
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   592
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   593
lemma (in group_hom) one_closed [simp]:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   594
  "h \<one> \<in> carrier H"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   595
  by simp
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   596
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   597
lemma (in group_hom) hom_one [simp]:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   598
  "h \<one> = \<one>\<^bsub>H\<^esub>"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   599
proof -
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14963
diff changeset
   600
  have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   601
    by (simp add: hom_mult [symmetric] del: hom_mult)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   602
  then show ?thesis by (simp del: r_one)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   603
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   604
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   605
lemma (in group_hom) inv_closed [simp]:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   606
  "x \<in> carrier G ==> h (inv x) \<in> carrier H"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   607
  by simp
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   608
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   609
lemma (in group_hom) hom_inv [simp]:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   610
  "x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   611
proof -
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   612
  assume x: "x \<in> carrier G"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   613
  then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   614
    by (simp add: hom_mult [symmetric] del: hom_mult)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   615
  also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   616
    by (simp add: hom_mult [symmetric] del: hom_mult)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   617
  finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" .
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   618
  with x show ?thesis by (simp del: H.r_inv H.Units_r_inv)
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   619
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   620
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   621
13949
0ce528cd6f19 HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents: 13944
diff changeset
   622
subsection {* Commutative Structures *}
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   623
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   624
text {*
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   625
  Naming convention: multiplicative structures that are commutative
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   626
  are called \emph{commutative}, additive structures are called
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   627
  \emph{Abelian}.
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   628
*}
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   629
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   630
locale comm_monoid = monoid +
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   631
  assumes m_comm: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   632
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   633
lemma (in comm_monoid) m_lcomm:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   634
  "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   635
   x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   636
proof -
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   637
  assume xyz: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   638
  from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   639
  also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   640
  also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   641
  finally show ?thesis .
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   642
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   643
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   644
lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   645
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   646
lemma comm_monoidI:
19783
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19699
diff changeset
   647
  fixes G (structure)
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   648
  assumes m_closed:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   649
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   650
    and one_closed: "\<one> \<in> carrier G"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   651
    and m_assoc:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   652
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   653
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   654
    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   655
    and m_comm:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   656
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   657
  shows "comm_monoid G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   658
  using l_one
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   659
    by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro 
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   660
             intro: prems simp: m_closed one_closed m_comm)
13817
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   661
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   662
lemma (in monoid) monoid_comm_monoidI:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   663
  assumes m_comm:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   664
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   665
  shows "comm_monoid G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   666
  by (rule comm_monoidI) (auto intro: m_assoc m_comm)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   667
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   668
(*lemma (in comm_monoid) r_one [simp]:
13817
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   669
  "x \<in> carrier G ==> x \<otimes> \<one> = x"
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   670
proof -
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   671
  assume G: "x \<in> carrier G"
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   672
  then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   673
  also from G have "... = x" by simp
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   674
  finally show ?thesis .
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   675
qed*)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   676
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   677
lemma (in comm_monoid) nat_pow_distr:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   678
  "[| x \<in> carrier G; y \<in> carrier G |] ==>
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   679
  (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   680
  by (induct n) (simp, simp add: m_ac)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   681
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   682
locale comm_group = comm_monoid + group
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   683
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   684
lemma (in group) group_comm_groupI:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   685
  assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   686
      x \<otimes> y = y \<otimes> x"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   687
  shows "comm_group G"
19984
29bb4659f80a Method intro_locales replaced by intro_locales and unfold_locales.
ballarin
parents: 19981
diff changeset
   688
  by unfold_locales (simp_all add: m_comm)
13817
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   689
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   690
lemma comm_groupI:
19783
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19699
diff changeset
   691
  fixes G (structure)
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   692
  assumes m_closed:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   693
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   694
    and one_closed: "\<one> \<in> carrier G"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   695
    and m_assoc:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   696
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   697
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   698
    and m_comm:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   699
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   700
    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   701
    and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   702
  shows "comm_group G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   703
  by (fast intro: group.group_comm_groupI groupI prems)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   704
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   705
lemma (in comm_group) inv_mult:
13854
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   706
  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   707
  by (simp add: m_ac inv_mult_group)
13854
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   708
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   709
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   710
subsection {* The Lattice of Subgroups of a Group *}
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   711
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   712
text_raw {* \label{sec:subgroup-lattice} *}
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   713
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   714
theorem (in group) subgroups_partial_order:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21041
diff changeset
   715
  "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   716
  by (rule partial_order.intro) simp_all
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   717
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   718
lemma (in group) subgroup_self:
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   719
  "subgroup (carrier G) G"
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   720
  by (rule subgroupI) auto
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   721
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   722
lemma (in group) subgroup_imp_group:
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   723
  "subgroup H G ==> group (G(| carrier := H |))"
26199
04817a8802f2 explicit referencing of background facts;
wenzelm
parents: 23350
diff changeset
   724
  by (erule subgroup.subgroup_is_group) (rule group_axioms)
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   725
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   726
lemma (in group) is_monoid [intro, simp]:
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   727
  "monoid G"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   728
  by (auto intro: monoid.intro m_assoc) 
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   729
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   730
lemma (in group) subgroup_inv_equality:
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   731
  "[| 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
   732
apply (rule_tac inv_equality [THEN sym])
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   733
  apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   734
 apply (rule subsetD [OF subgroup.subset], assumption+)
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   735
apply (rule subsetD [OF subgroup.subset], assumption)
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   736
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
   737
done
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   738
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   739
theorem (in group) subgroups_Inter:
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   740
  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
   741
    and not_empty: "A ~= {}"
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   742
  shows "subgroup (\<Inter>A) G"
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   743
proof (rule subgroupI)
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   744
  from subgr [THEN subgroup.subset] and not_empty
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   745
  show "\<Inter>A \<subseteq> carrier G" by blast
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   746
next
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   747
  from subgr [THEN subgroup.one_closed]
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   748
  show "\<Inter>A ~= {}" by blast
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   749
next
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   750
  fix x assume "x \<in> \<Inter>A"
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   751
  with subgr [THEN subgroup.m_inv_closed]
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   752
  show "inv x \<in> \<Inter>A" by blast
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   753
next
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   754
  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
   755
  with subgr [THEN subgroup.m_closed]
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   756
  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
   757
qed
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   758
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   759
theorem (in group) subgroups_complete_lattice:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21041
diff changeset
   760
  "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21041
diff changeset
   761
    (is "complete_lattice ?L")
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   762
proof (rule partial_order.complete_lattice_criterion1)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21041
diff changeset
   763
  show "partial_order ?L" by (rule subgroups_partial_order)
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   764
next
26805
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   765
  show "\<exists>G. greatest ?L G (carrier ?L)"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   766
  proof
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   767
    show "greatest ?L (carrier G) (carrier ?L)"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   768
      by (unfold greatest_def)
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   769
        (simp add: subgroup.subset subgroup_self)
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   770
  qed
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   771
next
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   772
  fix A
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21041
diff changeset
   773
  assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   774
  then have Int_subgroup: "subgroup (\<Inter>A) G"
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   775
    by (fastsimp intro: subgroups_Inter)
26805
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   776
  show "\<exists>I. greatest ?L I (Lower ?L A)"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   777
  proof
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   778
    show "greatest ?L (\<Inter>A) (Lower ?L A)"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   779
      (is "greatest _ ?Int _")
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   780
    proof (rule greatest_LowerI)
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   781
      fix H
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   782
      assume H: "H \<in> A"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   783
      with L have subgroupH: "subgroup H G" by auto
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   784
      from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   785
	by (rule subgroup_imp_group)
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   786
      from groupH have monoidH: "monoid ?H"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   787
	by (rule group.is_monoid)
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   788
      from H have Int_subset: "?Int \<subseteq> H" by fastsimp
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   789
      then show "le ?L ?Int H" by simp
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   790
    next
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   791
      fix H
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   792
      assume H: "H \<in> Lower ?L A"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   793
      with L Int_subgroup show "le ?L H ?Int"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   794
	by (fastsimp simp: Lower_def intro: Inter_greatest)
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   795
    next
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   796
      show "A \<subseteq> carrier ?L" by (rule L)
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   797
    next
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   798
      show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   799
    qed
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   800
  qed
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   801
qed
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   802
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   803
end