src/HOL/Algebra/Group.thy
author ballarin
Thu Jul 31 09:49:21 2008 +0200 (2008-07-31)
changeset 27714 27b4d7c01f8b
parent 27713 95b36bfe7fc4
child 28823 dcbef866c9e2
permissions -rw-r--r--
Tuned (for the sake of a meaningless log entry).
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(*
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  Title:  HOL/Algebra/Group.thy
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  Id:     $Id$
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  Author: Clemens Ballarin, started 4 February 2003
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Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.
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*)
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theory Group imports FuncSet Lattice begin
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section {* Monoids and Groups *}
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subsection {* Definitions *}
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text {*
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  Definitions follow \cite{Jacobson:1985}.
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*}
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record 'a monoid =  "'a partial_object" +
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  mult    :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<otimes>\<index>" 70)
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  one     :: 'a ("\<one>\<index>")
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constdefs (structure G)
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  m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _" [81] 80)
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  "inv x == (THE y. y \<in> carrier G & x \<otimes> y = \<one> & y \<otimes> x = \<one>)"
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  Units :: "_ => 'a set"
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  --{*The set of invertible elements*}
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  "Units G == {y. y \<in> carrier G & (\<exists>x \<in> carrier G. x \<otimes> y = \<one> & y \<otimes> x = \<one>)}"
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consts
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  pow :: "[('a, 'm) monoid_scheme, 'a, 'b::number] => 'a" (infixr "'(^')\<index>" 75)
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defs (overloaded)
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  nat_pow_def: "pow G a n == nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n"
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  int_pow_def: "pow G a z ==
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    let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a)
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    in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z)"
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locale monoid =
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  fixes G (structure)
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  assumes m_closed [intro, simp]:
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         "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> carrier G"
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      and m_assoc:
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         "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> 
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          \<Longrightarrow> (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
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      and one_closed [intro, simp]: "\<one> \<in> carrier G"
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      and l_one [simp]: "x \<in> carrier G \<Longrightarrow> \<one> \<otimes> x = x"
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      and r_one [simp]: "x \<in> carrier G \<Longrightarrow> x \<otimes> \<one> = x"
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lemma monoidI:
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  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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  then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z"
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    by (simp add: m_assoc Units_closed del: Units_l_inv)
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  with G show "y = z" by (simp add: Units_l_inv)
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next
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  assume eq: "y = z"
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    and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"
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  then show "x \<otimes> y = x \<otimes> z" by simp
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qed
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lemma (in monoid) Units_inv_inv [simp]:
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  "x \<in> Units G ==> inv (inv x) = x"
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proof -
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  assume x: "x \<in> Units G"
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  then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x" by simp
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  with x show ?thesis by (simp add: Units_closed del: Units_l_inv Units_r_inv)
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qed
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lemma (in monoid) inv_inj_on_Units:
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  "inj_on (m_inv G) (Units G)"
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proof (rule inj_onI)
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  fix x y
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  assume G: "x \<in> Units G"  "y \<in> Units G" and eq: "inv x = inv y"
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  then have "inv (inv x) = inv (inv y)" by simp
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  with G show "x = y" by simp
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qed
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lemma (in monoid) Units_inv_comm:
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  assumes inv: "x \<otimes> y = \<one>"
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    and G: "x \<in> Units G"  "y \<in> Units G"
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  shows "y \<otimes> x = \<one>"
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proof -
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  from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed)
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  with G show ?thesis by (simp del: r_one add: m_assoc Units_closed)
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qed
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text {* Power *}
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lemma (in monoid) nat_pow_closed [intro, simp]:
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  "x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G"
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  by (induct n) (simp_all add: nat_pow_def)
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lemma (in monoid) nat_pow_0 [simp]:
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  "x (^) (0::nat) = \<one>"
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  by (simp add: nat_pow_def)
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lemma (in monoid) nat_pow_Suc [simp]:
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  "x (^) (Suc n) = x (^) n \<otimes> x"
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  by (simp add: nat_pow_def)
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lemma (in monoid) nat_pow_one [simp]:
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  "\<one> (^) (n::nat) = \<one>"
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  by (induct n) simp_all
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lemma (in monoid) nat_pow_mult:
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  "x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)"
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  by (induct m) (simp_all add: m_assoc [THEN sym])
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lemma (in monoid) nat_pow_pow:
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  "x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)"
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  by (induct m) (simp, simp add: nat_pow_mult add_commute)
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(* Jacobson defines submonoid here. *)
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(* Jacobson defines the order of a monoid here. *)
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subsection {* Groups *}
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text {*
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  A group is a monoid all of whose elements are invertible.
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*}
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locale group = monoid +
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  assumes Units: "carrier G <= Units G"
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lemma (in group) is_group: "group G" by (rule group_axioms)
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theorem groupI:
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  fixes G (structure)
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  assumes m_closed [simp]:
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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 [simp]: "\<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 [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
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    and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
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  shows "group G"
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proof -
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  have l_cancel [simp]:
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    "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
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    (x \<otimes> y = x \<otimes> z) = (y = z)"
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  proof
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    fix x y z
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    assume eq: "x \<otimes> y = x \<otimes> z"
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      and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
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    with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
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      and l_inv: "x_inv \<otimes> x = \<one>" by fast
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    from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> z"
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      by (simp add: m_assoc)
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    with G show "y = z" by (simp add: l_inv)
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  next
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    fix x y z
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    assume eq: "y = z"
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      and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
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    then show "x \<otimes> y = x \<otimes> z" by simp
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  qed
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  have r_one:
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    "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"
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  proof -
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    fix x
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    assume x: "x \<in> carrier G"
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    with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
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      and l_inv: "x_inv \<otimes> x = \<one>" by fast
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    from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x"
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      by (simp add: m_assoc [symmetric] l_inv)
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    with x xG show "x \<otimes> \<one> = x" by simp
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  qed
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  have inv_ex:
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    "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"
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  proof -
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    fix x
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    assume x: "x \<in> carrier G"
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    with l_inv_ex obtain y where y: "y \<in> carrier G"
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      and l_inv: "y \<otimes> x = \<one>" by fast
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    from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>"
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      by (simp add: m_assoc [symmetric] l_inv r_one)
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    with x y have r_inv: "x \<otimes> y = \<one>"
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      by simp
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    from x y show "\<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"
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      by (fast intro: l_inv r_inv)
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  qed
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  then have carrier_subset_Units: "carrier G <= Units G"
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    by (unfold Units_def) fast
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  show ?thesis by unfold_locales (auto simp: r_one m_assoc carrier_subset_Units)
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qed
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lemma (in monoid) group_l_invI:
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  assumes l_inv_ex:
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    "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
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  shows "group G"
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  by (rule groupI) (auto intro: m_assoc l_inv_ex)
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lemma (in group) Units_eq [simp]:
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  "Units G = carrier G"
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proof
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  show "Units G <= carrier G" by fast
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next
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  show "carrier G <= Units G" by (rule Units)
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qed
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lemma (in group) inv_closed [intro, simp]:
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  "x \<in> carrier G ==> inv x \<in> carrier G"
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  using Units_inv_closed by simp
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lemma (in group) l_inv_ex [simp]:
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  "x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
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  using Units_l_inv_ex by simp
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ballarin@19981
   308
lemma (in group) r_inv_ex [simp]:
ballarin@19981
   309
  "x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"
ballarin@19981
   310
  using Units_r_inv_ex by simp
ballarin@19981
   311
paulson@14963
   312
lemma (in group) l_inv [simp]:
ballarin@13936
   313
  "x \<in> carrier G ==> inv x \<otimes> x = \<one>"
ballarin@13936
   314
  using Units_l_inv by simp
ballarin@13813
   315
ballarin@20318
   316
ballarin@13813
   317
subsection {* Cancellation Laws and Basic Properties *}
ballarin@13813
   318
ballarin@13813
   319
lemma (in group) l_cancel [simp]:
ballarin@13813
   320
  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
ballarin@13813
   321
   (x \<otimes> y = x \<otimes> z) = (y = z)"
ballarin@13936
   322
  using Units_l_inv by simp
ballarin@13940
   323
paulson@14963
   324
lemma (in group) r_inv [simp]:
ballarin@13813
   325
  "x \<in> carrier G ==> x \<otimes> inv x = \<one>"
ballarin@13813
   326
proof -
ballarin@13813
   327
  assume x: "x \<in> carrier G"
ballarin@13813
   328
  then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>"
ballarin@13813
   329
    by (simp add: m_assoc [symmetric] l_inv)
ballarin@13813
   330
  with x show ?thesis by (simp del: r_one)
ballarin@13813
   331
qed
ballarin@13813
   332
ballarin@13813
   333
lemma (in group) r_cancel [simp]:
ballarin@13813
   334
  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
ballarin@13813
   335
   (y \<otimes> x = z \<otimes> x) = (y = z)"
ballarin@13813
   336
proof
ballarin@13813
   337
  assume eq: "y \<otimes> x = z \<otimes> x"
wenzelm@14693
   338
    and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
ballarin@13813
   339
  then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)"
ballarin@27698
   340
    by (simp add: m_assoc [symmetric] del: r_inv Units_r_inv)
paulson@14963
   341
  with G show "y = z" by simp
ballarin@13813
   342
next
ballarin@13813
   343
  assume eq: "y = z"
wenzelm@14693
   344
    and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
ballarin@13813
   345
  then show "y \<otimes> x = z \<otimes> x" by simp
ballarin@13813
   346
qed
ballarin@13813
   347
ballarin@13854
   348
lemma (in group) inv_one [simp]:
ballarin@13854
   349
  "inv \<one> = \<one>"
ballarin@13854
   350
proof -
ballarin@27698
   351
  have "inv \<one> = \<one> \<otimes> (inv \<one>)" by (simp del: r_inv Units_r_inv)
paulson@14963
   352
  moreover have "... = \<one>" by simp
ballarin@13854
   353
  finally show ?thesis .
ballarin@13854
   354
qed
ballarin@13854
   355
ballarin@13813
   356
lemma (in group) inv_inv [simp]:
ballarin@13813
   357
  "x \<in> carrier G ==> inv (inv x) = x"
ballarin@13936
   358
  using Units_inv_inv by simp
ballarin@13936
   359
ballarin@13936
   360
lemma (in group) inv_inj:
ballarin@13936
   361
  "inj_on (m_inv G) (carrier G)"
ballarin@13936
   362
  using inv_inj_on_Units by simp
ballarin@13813
   363
ballarin@13854
   364
lemma (in group) inv_mult_group:
ballarin@13813
   365
  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"
ballarin@13813
   366
proof -
wenzelm@14693
   367
  assume G: "x \<in> carrier G"  "y \<in> carrier G"
ballarin@13813
   368
  then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)"
paulson@14963
   369
    by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric])
ballarin@27698
   370
  with G show ?thesis by (simp del: l_inv Units_l_inv)
ballarin@13813
   371
qed
ballarin@13813
   372
ballarin@13940
   373
lemma (in group) inv_comm:
ballarin@13940
   374
  "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
wenzelm@14693
   375
  by (rule Units_inv_comm) auto
ballarin@13940
   376
paulson@13944
   377
lemma (in group) inv_equality:
paulson@13943
   378
     "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"
paulson@13943
   379
apply (simp add: m_inv_def)
paulson@13943
   380
apply (rule the_equality)
wenzelm@14693
   381
 apply (simp add: inv_comm [of y x])
wenzelm@14693
   382
apply (rule r_cancel [THEN iffD1], auto)
paulson@13943
   383
done
paulson@13943
   384
ballarin@13936
   385
text {* Power *}
ballarin@13936
   386
ballarin@13936
   387
lemma (in group) int_pow_def2:
ballarin@13936
   388
  "a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))"
ballarin@13936
   389
  by (simp add: int_pow_def nat_pow_def Let_def)
ballarin@13936
   390
ballarin@13936
   391
lemma (in group) int_pow_0 [simp]:
ballarin@13936
   392
  "x (^) (0::int) = \<one>"
ballarin@13936
   393
  by (simp add: int_pow_def2)
ballarin@13936
   394
ballarin@13936
   395
lemma (in group) int_pow_one [simp]:
ballarin@13936
   396
  "\<one> (^) (z::int) = \<one>"
ballarin@13936
   397
  by (simp add: int_pow_def2)
ballarin@13936
   398
ballarin@20318
   399
paulson@14963
   400
subsection {* Subgroups *}
ballarin@13813
   401
ballarin@19783
   402
locale subgroup =
ballarin@19783
   403
  fixes H and G (structure)
paulson@14963
   404
  assumes subset: "H \<subseteq> carrier G"
paulson@14963
   405
    and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H"
ballarin@20318
   406
    and one_closed [simp]: "\<one> \<in> H"
paulson@14963
   407
    and m_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> inv x \<in> H"
ballarin@13813
   408
ballarin@20318
   409
lemma (in subgroup) is_subgroup:
wenzelm@26199
   410
  "subgroup H G" by (rule subgroup_axioms)
ballarin@20318
   411
ballarin@13813
   412
declare (in subgroup) group.intro [intro]
ballarin@13949
   413
paulson@14963
   414
lemma (in subgroup) mem_carrier [simp]:
paulson@14963
   415
  "x \<in> H \<Longrightarrow> x \<in> carrier G"
paulson@14963
   416
  using subset by blast
ballarin@13813
   417
paulson@14963
   418
lemma subgroup_imp_subset:
paulson@14963
   419
  "subgroup H G \<Longrightarrow> H \<subseteq> carrier G"
paulson@14963
   420
  by (rule subgroup.subset)
paulson@14963
   421
paulson@14963
   422
lemma (in subgroup) subgroup_is_group [intro]:
ballarin@27611
   423
  assumes "group G"
ballarin@27611
   424
  shows "group (G\<lparr>carrier := H\<rparr>)"
ballarin@27611
   425
proof -
ballarin@27611
   426
  interpret group [G] by fact
ballarin@27611
   427
  show ?thesis
ballarin@27698
   428
    apply (rule monoid.group_l_invI)
ballarin@27698
   429
    apply (unfold_locales) [1]
ballarin@27698
   430
    apply (auto intro: m_assoc l_inv mem_carrier)
ballarin@27698
   431
    done
ballarin@27611
   432
qed
ballarin@13813
   433
ballarin@13813
   434
text {*
ballarin@13813
   435
  Since @{term H} is nonempty, it contains some element @{term x}.  Since
ballarin@13813
   436
  it is closed under inverse, it contains @{text "inv x"}.  Since
ballarin@13813
   437
  it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}.
ballarin@13813
   438
*}
ballarin@13813
   439
ballarin@13813
   440
lemma (in group) one_in_subset:
ballarin@13813
   441
  "[| H \<subseteq> carrier G; H \<noteq> {}; \<forall>a \<in> H. inv a \<in> H; \<forall>a\<in>H. \<forall>b\<in>H. a \<otimes> b \<in> H |]
ballarin@13813
   442
   ==> \<one> \<in> H"
ballarin@13813
   443
by (force simp add: l_inv)
ballarin@13813
   444
ballarin@13813
   445
text {* A characterization of subgroups: closed, non-empty subset. *}
ballarin@13813
   446
ballarin@13813
   447
lemma (in group) subgroupI:
ballarin@13813
   448
  assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"
paulson@14963
   449
    and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H"
paulson@14963
   450
    and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H"
ballarin@13813
   451
  shows "subgroup H G"
ballarin@27714
   452
proof (simp add: subgroup_def assms)
ballarin@27714
   453
  show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: assms)
ballarin@13813
   454
qed
ballarin@13813
   455
ballarin@13936
   456
declare monoid.one_closed [iff] group.inv_closed [simp]
ballarin@13936
   457
  monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]
ballarin@13813
   458
ballarin@13813
   459
lemma subgroup_nonempty:
ballarin@13813
   460
  "~ subgroup {} G"
ballarin@13813
   461
  by (blast dest: subgroup.one_closed)
ballarin@13813
   462
ballarin@13813
   463
lemma (in subgroup) finite_imp_card_positive:
ballarin@13813
   464
  "finite (carrier G) ==> 0 < card H"
ballarin@13813
   465
proof (rule classical)
paulson@14963
   466
  assume "finite (carrier G)" "~ 0 < card H"
paulson@14963
   467
  then have "finite H" by (blast intro: finite_subset [OF subset])
paulson@14963
   468
  with prems have "subgroup {} G" by simp
ballarin@13813
   469
  with subgroup_nonempty show ?thesis by contradiction
ballarin@13813
   470
qed
ballarin@13813
   471
ballarin@13936
   472
(*
ballarin@13936
   473
lemma (in monoid) Units_subgroup:
ballarin@13936
   474
  "subgroup (Units G) G"
ballarin@13936
   475
*)
ballarin@13936
   476
ballarin@20318
   477
ballarin@13813
   478
subsection {* Direct Products *}
ballarin@13813
   479
paulson@14963
   480
constdefs
paulson@14963
   481
  DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid"  (infixr "\<times>\<times>" 80)
paulson@14963
   482
  "G \<times>\<times> H \<equiv> \<lparr>carrier = carrier G \<times> carrier H,
paulson@14963
   483
                mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')),
paulson@14963
   484
                one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"
ballarin@13813
   485
paulson@14963
   486
lemma DirProd_monoid:
ballarin@27611
   487
  assumes "monoid G" and "monoid H"
paulson@14963
   488
  shows "monoid (G \<times>\<times> H)"
paulson@14963
   489
proof -
ballarin@27611
   490
  interpret G: monoid [G] by fact
ballarin@27611
   491
  interpret H: monoid [H] by fact
ballarin@27714
   492
  from assms
paulson@14963
   493
  show ?thesis by (unfold monoid_def DirProd_def, auto) 
paulson@14963
   494
qed
ballarin@13813
   495
ballarin@13813
   496
paulson@14963
   497
text{*Does not use the previous result because it's easier just to use auto.*}
paulson@14963
   498
lemma DirProd_group:
ballarin@27611
   499
  assumes "group G" and "group H"
paulson@14963
   500
  shows "group (G \<times>\<times> H)"
ballarin@27611
   501
proof -
ballarin@27611
   502
  interpret G: group [G] by fact
ballarin@27611
   503
  interpret H: group [H] by fact
ballarin@27611
   504
  show ?thesis by (rule groupI)
paulson@14963
   505
     (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
paulson@14963
   506
           simp add: DirProd_def)
ballarin@27611
   507
qed
ballarin@13813
   508
paulson@14963
   509
lemma carrier_DirProd [simp]:
paulson@14963
   510
     "carrier (G \<times>\<times> H) = carrier G \<times> carrier H"
paulson@14963
   511
  by (simp add: DirProd_def)
paulson@13944
   512
paulson@14963
   513
lemma one_DirProd [simp]:
paulson@14963
   514
     "\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)"
paulson@14963
   515
  by (simp add: DirProd_def)
paulson@13944
   516
paulson@14963
   517
lemma mult_DirProd [simp]:
paulson@14963
   518
     "(g, h) \<otimes>\<^bsub>(G \<times>\<times> H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')"
paulson@14963
   519
  by (simp add: DirProd_def)
paulson@13944
   520
paulson@14963
   521
lemma inv_DirProd [simp]:
ballarin@27611
   522
  assumes "group G" and "group H"
paulson@13944
   523
  assumes g: "g \<in> carrier G"
paulson@13944
   524
      and h: "h \<in> carrier H"
paulson@14963
   525
  shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"
ballarin@27611
   526
proof -
ballarin@27611
   527
  interpret G: group [G] by fact
ballarin@27611
   528
  interpret H: group [H] by fact
ballarin@15696
   529
  interpret Prod: group ["G \<times>\<times> H"]
ballarin@27714
   530
    by (auto intro: DirProd_group group.intro group.axioms assms)
paulson@14963
   531
  show ?thesis by (simp add: Prod.inv_equality g h)
paulson@14963
   532
qed
ballarin@27698
   533
paulson@14963
   534
paulson@14963
   535
subsection {* Homomorphisms and Isomorphisms *}
ballarin@13813
   536
wenzelm@14651
   537
constdefs (structure G and H)
wenzelm@14651
   538
  hom :: "_ => _ => ('a => 'b) set"
ballarin@13813
   539
  "hom G H ==
ballarin@13813
   540
    {h. h \<in> carrier G -> carrier H &
wenzelm@14693
   541
      (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y)}"
ballarin@13813
   542
ballarin@13813
   543
lemma hom_mult:
wenzelm@14693
   544
  "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |]
wenzelm@14693
   545
   ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
wenzelm@14693
   546
  by (simp add: hom_def)
ballarin@13813
   547
ballarin@13813
   548
lemma hom_closed:
ballarin@13813
   549
  "[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"
ballarin@13813
   550
  by (auto simp add: hom_def funcset_mem)
ballarin@13813
   551
paulson@14761
   552
lemma (in group) hom_compose:
paulson@14761
   553
     "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"
paulson@14761
   554
apply (auto simp add: hom_def funcset_compose) 
paulson@14761
   555
apply (simp add: compose_def funcset_mem)
paulson@13943
   556
done
paulson@13943
   557
paulson@14803
   558
constdefs
paulson@14803
   559
  iso :: "_ => _ => ('a => 'b) set"  (infixr "\<cong>" 60)
paulson@14803
   560
  "G \<cong> H == {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}"
paulson@14761
   561
paulson@14803
   562
lemma iso_refl: "(%x. x) \<in> G \<cong> G"
paulson@14761
   563
by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
paulson@14761
   564
paulson@14761
   565
lemma (in group) iso_sym:
paulson@14803
   566
     "h \<in> G \<cong> H \<Longrightarrow> Inv (carrier G) h \<in> H \<cong> G"
paulson@14761
   567
apply (simp add: iso_def bij_betw_Inv) 
paulson@14761
   568
apply (subgoal_tac "Inv (carrier G) h \<in> carrier H \<rightarrow> carrier G") 
paulson@14761
   569
 prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_Inv]) 
paulson@14761
   570
apply (simp add: hom_def bij_betw_def Inv_f_eq funcset_mem f_Inv_f) 
paulson@14761
   571
done
paulson@14761
   572
paulson@14761
   573
lemma (in group) iso_trans: 
paulson@14803
   574
     "[|h \<in> G \<cong> H; i \<in> H \<cong> I|] ==> (compose (carrier G) i h) \<in> G \<cong> I"
paulson@14761
   575
by (auto simp add: iso_def hom_compose bij_betw_compose)
paulson@14761
   576
paulson@14963
   577
lemma DirProd_commute_iso:
paulson@14963
   578
  shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)"
paulson@14761
   579
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
paulson@14761
   580
paulson@14963
   581
lemma DirProd_assoc_iso:
paulson@14963
   582
  shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> (G \<times>\<times> H \<times>\<times> I) \<cong> (G \<times>\<times> (H \<times>\<times> I))"
paulson@14761
   583
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
paulson@14761
   584
paulson@14761
   585
paulson@14963
   586
text{*Basis for homomorphism proofs: we assume two groups @{term G} and
ballarin@15076
   587
  @{term H}, with a homomorphism @{term h} between them*}
ballarin@13813
   588
locale group_hom = group G + group H + var h +
ballarin@13813
   589
  assumes homh: "h \<in> hom G H"
ballarin@13813
   590
  notes hom_mult [simp] = hom_mult [OF homh]
ballarin@13813
   591
    and hom_closed [simp] = hom_closed [OF homh]
ballarin@13813
   592
ballarin@13813
   593
lemma (in group_hom) one_closed [simp]:
ballarin@13813
   594
  "h \<one> \<in> carrier H"
ballarin@13813
   595
  by simp
ballarin@13813
   596
ballarin@13813
   597
lemma (in group_hom) hom_one [simp]:
wenzelm@14693
   598
  "h \<one> = \<one>\<^bsub>H\<^esub>"
ballarin@13813
   599
proof -
ballarin@15076
   600
  have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>"
ballarin@13813
   601
    by (simp add: hom_mult [symmetric] del: hom_mult)
ballarin@13813
   602
  then show ?thesis by (simp del: r_one)
ballarin@13813
   603
qed
ballarin@13813
   604
ballarin@13813
   605
lemma (in group_hom) inv_closed [simp]:
ballarin@13813
   606
  "x \<in> carrier G ==> h (inv x) \<in> carrier H"
ballarin@13813
   607
  by simp
ballarin@13813
   608
ballarin@13813
   609
lemma (in group_hom) hom_inv [simp]:
wenzelm@14693
   610
  "x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"
ballarin@13813
   611
proof -
ballarin@13813
   612
  assume x: "x \<in> carrier G"
wenzelm@14693
   613
  then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>"
paulson@14963
   614
    by (simp add: hom_mult [symmetric] del: hom_mult)
wenzelm@14693
   615
  also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)"
paulson@14963
   616
    by (simp add: hom_mult [symmetric] del: hom_mult)
wenzelm@14693
   617
  finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" .
ballarin@27698
   618
  with x show ?thesis by (simp del: H.r_inv H.Units_r_inv)
ballarin@13813
   619
qed
ballarin@13813
   620
ballarin@20318
   621
ballarin@13949
   622
subsection {* Commutative Structures *}
ballarin@13936
   623
ballarin@13936
   624
text {*
ballarin@13936
   625
  Naming convention: multiplicative structures that are commutative
ballarin@13936
   626
  are called \emph{commutative}, additive structures are called
ballarin@13936
   627
  \emph{Abelian}.
ballarin@13936
   628
*}
ballarin@13813
   629
paulson@14963
   630
locale comm_monoid = monoid +
paulson@14963
   631
  assumes m_comm: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x"
ballarin@13813
   632
paulson@14963
   633
lemma (in comm_monoid) m_lcomm:
paulson@14963
   634
  "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
ballarin@13813
   635
   x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
ballarin@13813
   636
proof -
wenzelm@14693
   637
  assume xyz: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
ballarin@13813
   638
  from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)
ballarin@13813
   639
  also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)
ballarin@13813
   640
  also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)
ballarin@13813
   641
  finally show ?thesis .
ballarin@13813
   642
qed
ballarin@13813
   643
paulson@14963
   644
lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm
ballarin@13813
   645
ballarin@13936
   646
lemma comm_monoidI:
ballarin@19783
   647
  fixes G (structure)
ballarin@13936
   648
  assumes m_closed:
wenzelm@14693
   649
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
wenzelm@14693
   650
    and one_closed: "\<one> \<in> carrier G"
ballarin@13936
   651
    and m_assoc:
ballarin@13936
   652
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
wenzelm@14693
   653
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
wenzelm@14693
   654
    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
ballarin@13936
   655
    and m_comm:
wenzelm@14693
   656
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
ballarin@13936
   657
  shows "comm_monoid G"
ballarin@13936
   658
  using l_one
paulson@14963
   659
    by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro 
ballarin@27714
   660
             intro: assms simp: m_closed one_closed m_comm)
ballarin@13817
   661
ballarin@13936
   662
lemma (in monoid) monoid_comm_monoidI:
ballarin@13936
   663
  assumes m_comm:
wenzelm@14693
   664
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
ballarin@13936
   665
  shows "comm_monoid G"
ballarin@13936
   666
  by (rule comm_monoidI) (auto intro: m_assoc m_comm)
paulson@14963
   667
wenzelm@14693
   668
(*lemma (in comm_monoid) r_one [simp]:
ballarin@13817
   669
  "x \<in> carrier G ==> x \<otimes> \<one> = x"
ballarin@13817
   670
proof -
ballarin@13817
   671
  assume G: "x \<in> carrier G"
ballarin@13817
   672
  then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)
ballarin@13817
   673
  also from G have "... = x" by simp
ballarin@13817
   674
  finally show ?thesis .
wenzelm@14693
   675
qed*)
paulson@14963
   676
ballarin@13936
   677
lemma (in comm_monoid) nat_pow_distr:
ballarin@13936
   678
  "[| x \<in> carrier G; y \<in> carrier G |] ==>
ballarin@13936
   679
  (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"
ballarin@13936
   680
  by (induct n) (simp, simp add: m_ac)
ballarin@13936
   681
ballarin@13936
   682
locale comm_group = comm_monoid + group
ballarin@13936
   683
ballarin@13936
   684
lemma (in group) group_comm_groupI:
ballarin@13936
   685
  assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>
wenzelm@14693
   686
      x \<otimes> y = y \<otimes> x"
ballarin@13936
   687
  shows "comm_group G"
ballarin@19984
   688
  by unfold_locales (simp_all add: m_comm)
ballarin@13817
   689
ballarin@13936
   690
lemma comm_groupI:
ballarin@19783
   691
  fixes G (structure)
ballarin@13936
   692
  assumes m_closed:
wenzelm@14693
   693
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
wenzelm@14693
   694
    and one_closed: "\<one> \<in> carrier G"
ballarin@13936
   695
    and m_assoc:
ballarin@13936
   696
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
wenzelm@14693
   697
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
ballarin@13936
   698
    and m_comm:
wenzelm@14693
   699
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
wenzelm@14693
   700
    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
paulson@14963
   701
    and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
ballarin@13936
   702
  shows "comm_group G"
ballarin@27714
   703
  by (fast intro: group.group_comm_groupI groupI assms)
ballarin@13936
   704
ballarin@13936
   705
lemma (in comm_group) inv_mult:
ballarin@13854
   706
  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"
ballarin@13936
   707
  by (simp add: m_ac inv_mult_group)
ballarin@13854
   708
ballarin@20318
   709
ballarin@20318
   710
subsection {* The Lattice of Subgroups of a Group *}
ballarin@14751
   711
ballarin@14751
   712
text_raw {* \label{sec:subgroup-lattice} *}
ballarin@14751
   713
ballarin@14751
   714
theorem (in group) subgroups_partial_order:
ballarin@27713
   715
  "partial_order (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"
ballarin@27713
   716
  by unfold_locales simp_all
ballarin@14751
   717
ballarin@14751
   718
lemma (in group) subgroup_self:
ballarin@14751
   719
  "subgroup (carrier G) G"
ballarin@14751
   720
  by (rule subgroupI) auto
ballarin@14751
   721
ballarin@14751
   722
lemma (in group) subgroup_imp_group:
ballarin@14751
   723
  "subgroup H G ==> group (G(| carrier := H |))"
wenzelm@26199
   724
  by (erule subgroup.subgroup_is_group) (rule group_axioms)
ballarin@14751
   725
ballarin@14751
   726
lemma (in group) is_monoid [intro, simp]:
ballarin@14751
   727
  "monoid G"
paulson@14963
   728
  by (auto intro: monoid.intro m_assoc) 
ballarin@14751
   729
ballarin@14751
   730
lemma (in group) subgroup_inv_equality:
ballarin@14751
   731
  "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"
ballarin@14751
   732
apply (rule_tac inv_equality [THEN sym])
paulson@14761
   733
  apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)
paulson@14761
   734
 apply (rule subsetD [OF subgroup.subset], assumption+)
paulson@14761
   735
apply (rule subsetD [OF subgroup.subset], assumption)
paulson@14761
   736
apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+)
ballarin@14751
   737
done
ballarin@14751
   738
ballarin@14751
   739
theorem (in group) subgroups_Inter:
ballarin@14751
   740
  assumes subgr: "(!!H. H \<in> A ==> subgroup H G)"
ballarin@14751
   741
    and not_empty: "A ~= {}"
ballarin@14751
   742
  shows "subgroup (\<Inter>A) G"
ballarin@14751
   743
proof (rule subgroupI)
ballarin@14751
   744
  from subgr [THEN subgroup.subset] and not_empty
ballarin@14751
   745
  show "\<Inter>A \<subseteq> carrier G" by blast
ballarin@14751
   746
next
ballarin@14751
   747
  from subgr [THEN subgroup.one_closed]
ballarin@14751
   748
  show "\<Inter>A ~= {}" by blast
ballarin@14751
   749
next
ballarin@14751
   750
  fix x assume "x \<in> \<Inter>A"
ballarin@14751
   751
  with subgr [THEN subgroup.m_inv_closed]
ballarin@14751
   752
  show "inv x \<in> \<Inter>A" by blast
ballarin@14751
   753
next
ballarin@14751
   754
  fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"
ballarin@14751
   755
  with subgr [THEN subgroup.m_closed]
ballarin@14751
   756
  show "x \<otimes> y \<in> \<Inter>A" by blast
ballarin@14751
   757
qed
ballarin@14751
   758
ballarin@14751
   759
theorem (in group) subgroups_complete_lattice:
ballarin@27713
   760
  "complete_lattice (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"
ballarin@22063
   761
    (is "complete_lattice ?L")
ballarin@14751
   762
proof (rule partial_order.complete_lattice_criterion1)
ballarin@22063
   763
  show "partial_order ?L" by (rule subgroups_partial_order)
ballarin@14751
   764
next
berghofe@26805
   765
  show "\<exists>G. greatest ?L G (carrier ?L)"
berghofe@26805
   766
  proof
berghofe@26805
   767
    show "greatest ?L (carrier G) (carrier ?L)"
berghofe@26805
   768
      by (unfold greatest_def)
berghofe@26805
   769
        (simp add: subgroup.subset subgroup_self)
berghofe@26805
   770
  qed
ballarin@14751
   771
next
ballarin@14751
   772
  fix A
ballarin@22063
   773
  assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"
ballarin@14751
   774
  then have Int_subgroup: "subgroup (\<Inter>A) G"
ballarin@14751
   775
    by (fastsimp intro: subgroups_Inter)
berghofe@26805
   776
  show "\<exists>I. greatest ?L I (Lower ?L A)"
berghofe@26805
   777
  proof
berghofe@26805
   778
    show "greatest ?L (\<Inter>A) (Lower ?L A)"
berghofe@26805
   779
      (is "greatest _ ?Int _")
berghofe@26805
   780
    proof (rule greatest_LowerI)
berghofe@26805
   781
      fix H
berghofe@26805
   782
      assume H: "H \<in> A"
berghofe@26805
   783
      with L have subgroupH: "subgroup H G" by auto
berghofe@26805
   784
      from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
berghofe@26805
   785
	by (rule subgroup_imp_group)
berghofe@26805
   786
      from groupH have monoidH: "monoid ?H"
berghofe@26805
   787
	by (rule group.is_monoid)
berghofe@26805
   788
      from H have Int_subset: "?Int \<subseteq> H" by fastsimp
berghofe@26805
   789
      then show "le ?L ?Int H" by simp
berghofe@26805
   790
    next
berghofe@26805
   791
      fix H
berghofe@26805
   792
      assume H: "H \<in> Lower ?L A"
berghofe@26805
   793
      with L Int_subgroup show "le ?L H ?Int"
berghofe@26805
   794
	by (fastsimp simp: Lower_def intro: Inter_greatest)
berghofe@26805
   795
    next
berghofe@26805
   796
      show "A \<subseteq> carrier ?L" by (rule L)
berghofe@26805
   797
    next
berghofe@26805
   798
      show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
berghofe@26805
   799
    qed
ballarin@14751
   800
  qed
ballarin@14751
   801
qed
ballarin@14751
   802
ballarin@13813
   803
end