src/HOL/Algebra/Group.thy
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(*
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  Title:  HOL/Algebra/Group.thy
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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
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imports Lattice FuncSet
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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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  fixes G (structure)
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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: assms)
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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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d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   150
  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
   151
    by (simp add: m_assoc Units_closed del: Units_l_inv)
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   152
  with G show "y = z" by (simp add: Units_l_inv)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   153
next
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   154
  assume eq: "y = z"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   155
    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
   156
  then show "x \<otimes> y = x \<otimes> z" by simp
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   157
qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   158
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   159
lemma (in monoid) Units_inv_inv [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   160
  "x \<in> Units G ==> inv (inv x) = x"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   161
proof -
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   162
  assume x: "x \<in> Units G"
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   163
  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
   164
  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
   165
qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   166
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   167
lemma (in monoid) inv_inj_on_Units:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   168
  "inj_on (m_inv G) (Units G)"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   169
proof (rule inj_onI)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   170
  fix x y
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   171
  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
   172
  then have "inv (inv x) = inv (inv y)" by simp
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   173
  with G show "x = y" by simp
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   174
qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   175
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   176
lemma (in monoid) Units_inv_comm:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   177
  assumes inv: "x \<otimes> y = \<one>"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   178
    and G: "x \<in> Units G"  "y \<in> Units G"
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   179
  shows "y \<otimes> x = \<one>"
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   180
proof -
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   181
  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
   182
  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
   183
qed
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   184
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   185
text {* Power *}
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   186
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   187
lemma (in monoid) nat_pow_closed [intro, simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   188
  "x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   189
  by (induct n) (simp_all add: nat_pow_def)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   190
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   191
lemma (in monoid) nat_pow_0 [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   192
  "x (^) (0::nat) = \<one>"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   193
  by (simp add: nat_pow_def)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   194
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   195
lemma (in monoid) nat_pow_Suc [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   196
  "x (^) (Suc n) = x (^) n \<otimes> x"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   197
  by (simp add: nat_pow_def)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   198
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   199
lemma (in monoid) nat_pow_one [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   200
  "\<one> (^) (n::nat) = \<one>"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   201
  by (induct n) simp_all
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   202
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   203
lemma (in monoid) nat_pow_mult:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   204
  "x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   205
  by (induct m) (simp_all add: m_assoc [THEN sym])
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   206
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   207
lemma (in monoid) nat_pow_pow:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   208
  "x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   209
  by (induct m) (simp, simp add: nat_pow_mult add_commute)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   210
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   211
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   212
(* Jacobson defines submonoid here. *)
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   213
(* Jacobson defines the order of a monoid here. *)
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
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   216
subsection {* Groups *}
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   217
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   218
text {*
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   219
  A group is a monoid all of whose elements are invertible.
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   220
*}
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   221
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   222
locale group = monoid +
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   223
  assumes Units: "carrier G <= Units G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   224
26199
04817a8802f2 explicit referencing of background facts;
wenzelm
parents: 23350
diff changeset
   225
lemma (in group) is_group: "group G" by (rule group_axioms)
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   226
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   227
theorem groupI:
19783
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19699
diff changeset
   228
  fixes G (structure)
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   229
  assumes m_closed [simp]:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   230
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   231
    and one_closed [simp]: "\<one> \<in> carrier G"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   232
    and m_assoc:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   233
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   234
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   235
    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
   236
    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
   237
  shows "group G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   238
proof -
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   239
  have l_cancel [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   240
    "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   241
    (x \<otimes> y = x \<otimes> z) = (y = z)"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   242
  proof
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   243
    fix x y z
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   244
    assume eq: "x \<otimes> y = x \<otimes> z"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   245
      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
   246
    with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   247
      and l_inv: "x_inv \<otimes> x = \<one>" by fast
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   248
    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
   249
      by (simp add: m_assoc)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   250
    with G show "y = z" by (simp add: l_inv)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   251
  next
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   252
    fix x y z
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   253
    assume eq: "y = z"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   254
      and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   255
    then show "x \<otimes> y = x \<otimes> z" by simp
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   256
  qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   257
  have r_one:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   258
    "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   259
  proof -
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   260
    fix x
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   261
    assume x: "x \<in> carrier G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   262
    with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   263
      and l_inv: "x_inv \<otimes> x = \<one>" by fast
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   264
    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
   265
      by (simp add: m_assoc [symmetric] l_inv)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   266
    with x xG show "x \<otimes> \<one> = x" by simp
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   267
  qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   268
  have inv_ex:
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   269
    "!!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
   270
  proof -
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   271
    fix x
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   272
    assume x: "x \<in> carrier G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   273
    with l_inv_ex obtain y where y: "y \<in> carrier G"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   274
      and l_inv: "y \<otimes> x = \<one>" by fast
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   275
    from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   276
      by (simp add: m_assoc [symmetric] l_inv r_one)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   277
    with x y have r_inv: "x \<otimes> y = \<one>"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   278
      by simp
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   279
    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
   280
      by (fast intro: l_inv r_inv)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   281
  qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   282
  then have carrier_subset_Units: "carrier G <= Units G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   283
    by (unfold Units_def) fast
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 27714
diff changeset
   284
  show ?thesis proof qed (auto simp: r_one m_assoc carrier_subset_Units)
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   285
qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   286
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   287
lemma (in monoid) group_l_invI:
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   288
  assumes l_inv_ex:
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   289
    "!!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
   290
  shows "group G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   291
  by (rule groupI) (auto intro: m_assoc l_inv_ex)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   292
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   293
lemma (in group) Units_eq [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   294
  "Units G = carrier G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   295
proof
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   296
  show "Units G <= carrier G" by fast
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   297
next
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   298
  show "carrier G <= Units G" by (rule Units)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   299
qed
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   300
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   301
lemma (in group) inv_closed [intro, simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   302
  "x \<in> carrier G ==> inv x \<in> carrier G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   303
  using Units_inv_closed by simp
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   304
19981
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   305
lemma (in group) l_inv_ex [simp]:
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   306
  "x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   307
  using Units_l_inv_ex by simp
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   308
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   309
lemma (in group) r_inv_ex [simp]:
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   310
  "x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   311
  using Units_r_inv_ex by simp
c0f124a0d385 Minor new lemmas.
ballarin
parents: 19931
diff changeset
   312
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   313
lemma (in group) l_inv [simp]:
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   314
  "x \<in> carrier G ==> inv x \<otimes> x = \<one>"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   315
  using Units_l_inv by simp
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   316
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   317
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   318
subsection {* Cancellation Laws and Basic Properties *}
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   319
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   320
lemma (in group) l_cancel [simp]:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   321
  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   322
   (x \<otimes> y = x \<otimes> z) = (y = z)"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   323
  using Units_l_inv by simp
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   324
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   325
lemma (in group) r_inv [simp]:
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   326
  "x \<in> carrier G ==> x \<otimes> inv x = \<one>"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   327
proof -
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   328
  assume x: "x \<in> carrier G"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   329
  then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   330
    by (simp add: m_assoc [symmetric] l_inv)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   331
  with x show ?thesis by (simp del: r_one)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   332
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   333
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   334
lemma (in group) r_cancel [simp]:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   335
  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   336
   (y \<otimes> x = z \<otimes> x) = (y = z)"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   337
proof
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   338
  assume eq: "y \<otimes> x = z \<otimes> x"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   339
    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
   340
  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
   341
    by (simp add: m_assoc [symmetric] del: r_inv Units_r_inv)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   342
  with G show "y = z" by simp
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   343
next
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   344
  assume eq: "y = z"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   345
    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
   346
  then show "y \<otimes> x = z \<otimes> x" by simp
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   347
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   348
13854
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   349
lemma (in group) inv_one [simp]:
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   350
  "inv \<one> = \<one>"
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   351
proof -
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   352
  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
   353
  moreover have "... = \<one>" by simp
13854
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   354
  finally show ?thesis .
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   355
qed
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   356
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   357
lemma (in group) inv_inv [simp]:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   358
  "x \<in> carrier G ==> inv (inv x) = x"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   359
  using Units_inv_inv by simp
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   360
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   361
lemma (in group) inv_inj:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   362
  "inj_on (m_inv G) (carrier G)"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   363
  using inv_inj_on_Units by simp
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   364
13854
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   365
lemma (in group) inv_mult_group:
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   366
  "[| 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
   367
proof -
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   368
  assume G: "x \<in> carrier G"  "y \<in> carrier G"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   369
  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
   370
    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
   371
  with G show ?thesis by (simp del: l_inv Units_l_inv)
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   372
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   373
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   374
lemma (in group) inv_comm:
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   375
  "[| 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
   376
  by (rule Units_inv_comm) auto
13940
c67798653056 HOL-Algebra: New polynomial development added.
ballarin
parents: 13936
diff changeset
   377
13944
9b34607cd83e new proofs about direct products, etc.
paulson
parents: 13943
diff changeset
   378
lemma (in group) inv_equality:
13943
83d842ccd4aa moving Bij.thy from GroupTheory to Algebra
paulson
parents: 13940
diff changeset
   379
     "[|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
   380
apply (simp add: m_inv_def)
83d842ccd4aa moving Bij.thy from GroupTheory to Algebra
paulson
parents: 13940
diff changeset
   381
apply (rule the_equality)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   382
 apply (simp add: inv_comm [of y x])
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   383
apply (rule r_cancel [THEN iffD1], auto)
13943
83d842ccd4aa moving Bij.thy from GroupTheory to Algebra
paulson
parents: 13940
diff changeset
   384
done
83d842ccd4aa moving Bij.thy from GroupTheory to Algebra
paulson
parents: 13940
diff changeset
   385
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   386
text {* Power *}
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   387
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   388
lemma (in group) int_pow_def2:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   389
  "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
   390
  by (simp add: int_pow_def nat_pow_def Let_def)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   391
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   392
lemma (in group) int_pow_0 [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   393
  "x (^) (0::int) = \<one>"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   394
  by (simp add: int_pow_def2)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   395
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   396
lemma (in group) int_pow_one [simp]:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   397
  "\<one> (^) (z::int) = \<one>"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   398
  by (simp add: int_pow_def2)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   399
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   400
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   401
subsection {* Subgroups *}
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   402
19783
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19699
diff changeset
   403
locale subgroup =
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19699
diff changeset
   404
  fixes H and G (structure)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   405
  assumes subset: "H \<subseteq> carrier G"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   406
    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
   407
    and one_closed [simp]: "\<one> \<in> H"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   408
    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
   409
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   410
lemma (in subgroup) is_subgroup:
26199
04817a8802f2 explicit referencing of background facts;
wenzelm
parents: 23350
diff changeset
   411
  "subgroup H G" by (rule subgroup_axioms)
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   412
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   413
declare (in subgroup) group.intro [intro]
13949
0ce528cd6f19 HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents: 13944
diff changeset
   414
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   415
lemma (in subgroup) mem_carrier [simp]:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   416
  "x \<in> H \<Longrightarrow> x \<in> carrier G"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   417
  using subset by blast
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   418
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   419
lemma subgroup_imp_subset:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   420
  "subgroup H G \<Longrightarrow> H \<subseteq> carrier G"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   421
  by (rule subgroup.subset)
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   422
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   423
lemma (in subgroup) subgroup_is_group [intro]:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   424
  assumes "group G"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   425
  shows "group (G\<lparr>carrier := H\<rparr>)"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   426
proof -
29237
e90d9d51106b More porting to new locales.
ballarin
parents: 28823
diff changeset
   427
  interpret group G by fact
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   428
  show ?thesis
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   429
    apply (rule monoid.group_l_invI)
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   430
    apply (unfold_locales) [1]
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   431
    apply (auto intro: m_assoc l_inv mem_carrier)
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   432
    done
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   433
qed
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   434
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   435
text {*
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   436
  Since @{term H} is nonempty, it contains some element @{term x}.  Since
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   437
  it is closed under inverse, it contains @{text "inv x"}.  Since
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   438
  it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}.
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   439
*}
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   440
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   441
lemma (in group) one_in_subset:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   442
  "[| 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
   443
   ==> \<one> \<in> H"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   444
by (force simp add: l_inv)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   445
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   446
text {* A characterization of subgroups: closed, non-empty subset. *}
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   447
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   448
lemma (in group) subgroupI:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   449
  assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   450
    and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   451
    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
   452
  shows "subgroup H G"
27714
27b4d7c01f8b Tuned (for the sake of a meaningless log entry).
ballarin
parents: 27713
diff changeset
   453
proof (simp add: subgroup_def assms)
27b4d7c01f8b Tuned (for the sake of a meaningless log entry).
ballarin
parents: 27713
diff changeset
   454
  show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: assms)
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   455
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   456
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   457
declare monoid.one_closed [iff] group.inv_closed [simp]
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   458
  monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   459
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   460
lemma subgroup_nonempty:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   461
  "~ subgroup {} G"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   462
  by (blast dest: subgroup.one_closed)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   463
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   464
lemma (in subgroup) finite_imp_card_positive:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   465
  "finite (carrier G) ==> 0 < card H"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   466
proof (rule classical)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   467
  assume "finite (carrier G)" "~ 0 < card H"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   468
  then have "finite H" by (blast intro: finite_subset [OF subset])
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   469
  with prems have "subgroup {} G" by simp
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   470
  with subgroup_nonempty show ?thesis by contradiction
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   471
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   472
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   473
(*
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   474
lemma (in monoid) Units_subgroup:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   475
  "subgroup (Units G) G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   476
*)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   477
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   478
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   479
subsection {* Direct Products *}
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   480
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   481
constdefs
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   482
  DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid"  (infixr "\<times>\<times>" 80)
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   483
  "G \<times>\<times> H \<equiv> \<lparr>carrier = carrier G \<times> carrier H,
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   484
                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
   485
                one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   486
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   487
lemma DirProd_monoid:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   488
  assumes "monoid G" and "monoid H"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   489
  shows "monoid (G \<times>\<times> H)"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   490
proof -
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 29240
diff changeset
   491
  interpret G: monoid G by fact
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 29240
diff changeset
   492
  interpret H: monoid H by fact
27714
27b4d7c01f8b Tuned (for the sake of a meaningless log entry).
ballarin
parents: 27713
diff changeset
   493
  from assms
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   494
  show ?thesis by (unfold monoid_def DirProd_def, auto) 
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   495
qed
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   496
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   497
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   498
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
   499
lemma DirProd_group:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   500
  assumes "group G" and "group H"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   501
  shows "group (G \<times>\<times> H)"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   502
proof -
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 29240
diff changeset
   503
  interpret G: group G by fact
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 29240
diff changeset
   504
  interpret H: group H by fact
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   505
  show ?thesis by (rule groupI)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   506
     (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
   507
           simp add: DirProd_def)
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   508
qed
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   509
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   510
lemma carrier_DirProd [simp]:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   511
     "carrier (G \<times>\<times> H) = carrier G \<times> carrier H"
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   512
  by (simp add: DirProd_def)
13944
9b34607cd83e new proofs about direct products, etc.
paulson
parents: 13943
diff changeset
   513
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   514
lemma one_DirProd [simp]:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   515
     "\<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
   516
  by (simp add: DirProd_def)
13944
9b34607cd83e new proofs about direct products, etc.
paulson
parents: 13943
diff changeset
   517
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   518
lemma mult_DirProd [simp]:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   519
     "(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
   520
  by (simp add: DirProd_def)
13944
9b34607cd83e new proofs about direct products, etc.
paulson
parents: 13943
diff changeset
   521
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   522
lemma inv_DirProd [simp]:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 26805
diff changeset
   523
  assumes "group G" and "group H"
13944
9b34607cd83e new proofs about direct products, etc.
paulson
parents: 13943
diff changeset
   524
  assumes g: "g \<in> carrier G"
9b34607cd83e new proofs about direct products, etc.
paulson
parents: 13943
diff changeset
   525
      and h: "h \<in> carrier H"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   526
  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
   527
proof -
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 29240
diff changeset
   528
  interpret G: group G by fact
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 29240
diff changeset
   529
  interpret H: group H by fact
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 29240
diff changeset
   530
  interpret Prod: group "G \<times>\<times> H"
27714
27b4d7c01f8b Tuned (for the sake of a meaningless log entry).
ballarin
parents: 27713
diff changeset
   531
    by (auto intro: DirProd_group group.intro group.axioms assms)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   532
  show ?thesis by (simp add: Prod.inv_equality g h)
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   533
qed
27698
197f0517f0bd Unit_inv_l, Unit_inv_r made [simp].
ballarin
parents: 27611
diff changeset
   534
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   535
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   536
subsection {* Homomorphisms and Isomorphisms *}
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   537
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14551
diff changeset
   538
constdefs (structure G and H)
02b8f3bcf7fe improved notation;
wenzelm
parents: 14551
diff changeset
   539
  hom :: "_ => _ => ('a => 'b) set"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   540
  "hom G H ==
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   541
    {h. h \<in> carrier G -> carrier H &
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   542
      (\<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
   543
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   544
lemma (in group) hom_compose:
31754
b5260f5272a4 tuned FuncSet
nipkow
parents: 31727
diff changeset
   545
  "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"
b5260f5272a4 tuned FuncSet
nipkow
parents: 31727
diff changeset
   546
by (fastsimp simp add: hom_def compose_def)
13943
83d842ccd4aa moving Bij.thy from GroupTheory to Algebra
paulson
parents: 13940
diff changeset
   547
14803
f7557773cc87 more group isomorphisms
paulson
parents: 14761
diff changeset
   548
constdefs
f7557773cc87 more group isomorphisms
paulson
parents: 14761
diff changeset
   549
  iso :: "_ => _ => ('a => 'b) set"  (infixr "\<cong>" 60)
f7557773cc87 more group isomorphisms
paulson
parents: 14761
diff changeset
   550
  "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
   551
14803
f7557773cc87 more group isomorphisms
paulson
parents: 14761
diff changeset
   552
lemma iso_refl: "(%x. x) \<in> G \<cong> G"
31727
2621a957d417 Made Pi_I [simp]
nipkow
parents: 30729
diff changeset
   553
by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   554
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   555
lemma (in group) iso_sym:
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 32989
diff changeset
   556
     "h \<in> G \<cong> H \<Longrightarrow> inv_into (carrier G) h \<in> H \<cong> G"
764547b68538 inv_onto -> inv_into
nipkow
parents: 32989
diff changeset
   557
apply (simp add: iso_def bij_betw_inv_into) 
764547b68538 inv_onto -> inv_into
nipkow
parents: 32989
diff changeset
   558
apply (subgoal_tac "inv_into (carrier G) h \<in> carrier H \<rightarrow> carrier G") 
764547b68538 inv_onto -> inv_into
nipkow
parents: 32989
diff changeset
   559
 prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_inv_into]) 
764547b68538 inv_onto -> inv_into
nipkow
parents: 32989
diff changeset
   560
apply (simp add: hom_def bij_betw_def inv_into_f_eq f_inv_into_f Pi_def)
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   561
done
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   562
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   563
lemma (in group) iso_trans: 
14803
f7557773cc87 more group isomorphisms
paulson
parents: 14761
diff changeset
   564
     "[|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
   565
by (auto simp add: iso_def hom_compose bij_betw_compose)
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   566
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   567
lemma DirProd_commute_iso:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   568
  shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)"
31754
b5260f5272a4 tuned FuncSet
nipkow
parents: 31727
diff changeset
   569
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   570
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   571
lemma DirProd_assoc_iso:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   572
  shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> (G \<times>\<times> H \<times>\<times> I) \<cong> (G \<times>\<times> (H \<times>\<times> I))"
31727
2621a957d417 Made Pi_I [simp]
nipkow
parents: 30729
diff changeset
   573
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   574
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   575
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   576
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
   577
  @{term H}, with a homomorphism @{term h} between them*}
29237
e90d9d51106b More porting to new locales.
ballarin
parents: 28823
diff changeset
   578
locale group_hom = G: group G + H: group H for G (structure) and H (structure) +
e90d9d51106b More porting to new locales.
ballarin
parents: 28823
diff changeset
   579
  fixes h
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   580
  assumes homh: "h \<in> hom G H"
29240
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   581
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   582
lemma (in group_hom) hom_mult [simp]:
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   583
  "[| x \<in> carrier G; y \<in> carrier G |] ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   584
proof -
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   585
  assume "x \<in> carrier G" "y \<in> carrier G"
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   586
  with homh [unfolded hom_def] show ?thesis by simp
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   587
qed
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   588
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   589
lemma (in group_hom) hom_closed [simp]:
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   590
  "x \<in> carrier G ==> h x \<in> carrier H"
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   591
proof -
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   592
  assume "x \<in> carrier G"
31754
b5260f5272a4 tuned FuncSet
nipkow
parents: 31727
diff changeset
   593
  with homh [unfolded hom_def] show ?thesis by auto
29240
bb81c3709fb6 More porting to new locales.
ballarin
parents: 29237
diff changeset
   594
qed
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   595
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   596
lemma (in group_hom) one_closed [simp]:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   597
  "h \<one> \<in> carrier H"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   598
  by simp
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   599
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   600
lemma (in group_hom) hom_one [simp]:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   601
  "h \<one> = \<one>\<^bsub>H\<^esub>"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   602
proof -
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14963
diff changeset
   603
  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
   604
    by (simp add: hom_mult [symmetric] del: hom_mult)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   605
  then show ?thesis by (simp del: r_one)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   606
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   607
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   608
lemma (in group_hom) inv_closed [simp]:
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   609
  "x \<in> carrier G ==> h (inv x) \<in> carrier H"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   610
  by simp
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   611
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   612
lemma (in group_hom) hom_inv [simp]:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   613
  "x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   614
proof -
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   615
  assume x: "x \<in> carrier G"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   616
  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
   617
    by (simp add: hom_mult [symmetric] del: hom_mult)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   618
  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
   619
    by (simp add: hom_mult [symmetric] del: hom_mult)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   620
  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
   621
  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
   622
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   623
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   624
13949
0ce528cd6f19 HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents: 13944
diff changeset
   625
subsection {* Commutative Structures *}
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   626
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   627
text {*
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   628
  Naming convention: multiplicative structures that are commutative
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   629
  are called \emph{commutative}, additive structures are called
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   630
  \emph{Abelian}.
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   631
*}
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
locale comm_monoid = monoid +
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   634
  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
   635
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   636
lemma (in comm_monoid) m_lcomm:
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   637
  "\<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
   638
   x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   639
proof -
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   640
  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
   641
  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
   642
  also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   643
  also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   644
  finally show ?thesis .
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   645
qed
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   646
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   647
lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   648
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   649
lemma comm_monoidI:
19783
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19699
diff changeset
   650
  fixes G (structure)
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   651
  assumes m_closed:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   652
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   653
    and one_closed: "\<one> \<in> carrier G"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   654
    and m_assoc:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   655
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   656
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   657
    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   658
    and m_comm:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   659
      "!!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
   660
  shows "comm_monoid G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   661
  using l_one
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   662
    by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro 
27714
27b4d7c01f8b Tuned (for the sake of a meaningless log entry).
ballarin
parents: 27713
diff changeset
   663
             intro: assms simp: m_closed one_closed m_comm)
13817
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   664
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   665
lemma (in monoid) monoid_comm_monoidI:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   666
  assumes m_comm:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   667
      "!!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
   668
  shows "comm_monoid G"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   669
  by (rule comm_monoidI) (auto intro: m_assoc m_comm)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   670
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   671
(*lemma (in comm_monoid) r_one [simp]:
13817
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   672
  "x \<in> carrier G ==> x \<otimes> \<one> = x"
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   673
proof -
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   674
  assume G: "x \<in> carrier G"
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   675
  then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   676
  also from G have "... = x" by simp
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   677
  finally show ?thesis .
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   678
qed*)
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   679
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   680
lemma (in comm_monoid) nat_pow_distr:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   681
  "[| x \<in> carrier G; y \<in> carrier G |] ==>
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   682
  (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   683
  by (induct n) (simp, simp add: m_ac)
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   684
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   685
locale comm_group = comm_monoid + group
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   686
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   687
lemma (in group) group_comm_groupI:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   688
  assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   689
      x \<otimes> y = y \<otimes> x"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   690
  shows "comm_group G"
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 27714
diff changeset
   691
  proof qed (simp_all add: m_comm)
13817
7e031a968443 Product operator added --- preliminary.
ballarin
parents: 13813
diff changeset
   692
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   693
lemma comm_groupI:
19783
82f365a14960 Improved parameter management of locales.
ballarin
parents: 19699
diff changeset
   694
  fixes G (structure)
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   695
  assumes m_closed:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   696
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   697
    and one_closed: "\<one> \<in> carrier G"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   698
    and m_assoc:
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   699
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   700
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   701
    and m_comm:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   702
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
4deda204e1d8 improved syntax;
wenzelm
parents: 14651
diff changeset
   703
    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   704
    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
   705
  shows "comm_group G"
27714
27b4d7c01f8b Tuned (for the sake of a meaningless log entry).
ballarin
parents: 27713
diff changeset
   706
  by (fast intro: group.group_comm_groupI groupI assms)
13936
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   707
d3671b878828 Greatly extended CRing. Added Module.
ballarin
parents: 13854
diff changeset
   708
lemma (in comm_group) inv_mult:
13854
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   709
  "[| 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
   710
  by (simp add: m_ac inv_mult_group)
13854
91c9ab25fece First distributed version of Group and Ring theory.
ballarin
parents: 13835
diff changeset
   711
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   712
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   713
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
   714
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   715
text_raw {* \label{sec:subgroup-lattice} *}
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   716
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   717
theorem (in group) subgroups_partial_order:
27713
95b36bfe7fc4 New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents: 27698
diff changeset
   718
  "partial_order (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 27714
diff changeset
   719
  proof qed simp_all
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   720
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   721
lemma (in group) subgroup_self:
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   722
  "subgroup (carrier G) G"
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   723
  by (rule subgroupI) auto
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   724
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   725
lemma (in group) subgroup_imp_group:
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   726
  "subgroup H G ==> group (G(| carrier := H |))"
26199
04817a8802f2 explicit referencing of background facts;
wenzelm
parents: 23350
diff changeset
   727
  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
   728
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   729
lemma (in group) is_monoid [intro, simp]:
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   730
  "monoid G"
14963
d584e32f7d46 removal of magmas and semigroups
paulson
parents: 14852
diff changeset
   731
  by (auto intro: monoid.intro m_assoc) 
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   732
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   733
lemma (in group) subgroup_inv_equality:
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   734
  "[| 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
   735
apply (rule_tac inv_equality [THEN sym])
14761
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   736
  apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   737
 apply (rule subsetD [OF subgroup.subset], assumption+)
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   738
apply (rule subsetD [OF subgroup.subset], assumption)
28b5eb4a867f more results about isomorphisms
paulson
parents: 14751
diff changeset
   739
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
   740
done
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   741
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   742
theorem (in group) subgroups_Inter:
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   743
  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
   744
    and not_empty: "A ~= {}"
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   745
  shows "subgroup (\<Inter>A) G"
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   746
proof (rule subgroupI)
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   747
  from subgr [THEN subgroup.subset] and not_empty
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   748
  show "\<Inter>A \<subseteq> carrier G" 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
  from subgr [THEN subgroup.one_closed]
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   751
  show "\<Inter>A ~= {}" by blast
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   752
next
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   753
  fix x assume "x \<in> \<Inter>A"
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   754
  with subgr [THEN subgroup.m_inv_closed]
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   755
  show "inv x \<in> \<Inter>A" by blast
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   756
next
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   757
  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
   758
  with subgr [THEN subgroup.m_closed]
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   759
  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
   760
qed
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   761
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   762
theorem (in group) subgroups_complete_lattice:
27713
95b36bfe7fc4 New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents: 27698
diff changeset
   763
  "complete_lattice (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21041
diff changeset
   764
    (is "complete_lattice ?L")
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   765
proof (rule partial_order.complete_lattice_criterion1)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21041
diff changeset
   766
  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
   767
next
26805
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   768
  show "\<exists>G. greatest ?L G (carrier ?L)"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   769
  proof
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   770
    show "greatest ?L (carrier G) (carrier ?L)"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   771
      by (unfold greatest_def)
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   772
        (simp add: subgroup.subset subgroup_self)
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   773
  qed
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   774
next
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   775
  fix A
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21041
diff changeset
   776
  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
   777
  then have Int_subgroup: "subgroup (\<Inter>A) G"
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   778
    by (fastsimp intro: subgroups_Inter)
26805
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   779
  show "\<exists>I. greatest ?L I (Lower ?L A)"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   780
  proof
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   781
    show "greatest ?L (\<Inter>A) (Lower ?L A)"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   782
      (is "greatest _ ?Int _")
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   783
    proof (rule greatest_LowerI)
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   784
      fix H
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   785
      assume H: "H \<in> A"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   786
      with L have subgroupH: "subgroup H G" by auto
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   787
      from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31754
diff changeset
   788
        by (rule subgroup_imp_group)
26805
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   789
      from groupH have monoidH: "monoid ?H"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31754
diff changeset
   790
        by (rule group.is_monoid)
26805
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   791
      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
   792
      then show "le ?L ?Int H" by simp
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   793
    next
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   794
      fix H
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   795
      assume H: "H \<in> Lower ?L A"
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   796
      with L Int_subgroup show "le ?L H ?Int"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31754
diff changeset
   797
        by (fastsimp simp: Lower_def intro: Inter_greatest)
26805
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   798
    next
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   799
      show "A \<subseteq> carrier ?L" by (rule L)
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   800
    next
27941d7d9a11 Replaced forward proofs of existential statements by backward proofs
berghofe
parents: 26199
diff changeset
   801
      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
   802
    qed
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   803
  qed
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   804
qed
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   805
13813
722593f2f068 New development of algebra: Groups.
ballarin
parents:
diff changeset
   806
end