src/HOL/Algebra/Group.thy
author hoelzl
Tue Nov 05 09:44:57 2013 +0100 (2013-11-05)
changeset 54257 5c7a3b6b05a9
parent 47108 2a1953f0d20d
child 55415 05f5fdb8d093
permissions -rw-r--r--
generalize SUP and INF to the syntactic type classes Sup and Inf
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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 "~~/src/HOL/Library/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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definition
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  m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _" [81] 80)
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  where "inv\<^bsub>G\<^esub> x = (THE y. y \<in> carrier G & x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> & y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)"
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definition
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  Units :: "_ => 'a set"
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  --{*The set of invertible elements*}
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  where "Units G = {y. y \<in> carrier G & (\<exists>x \<in> carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> & y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)}"
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consts
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  pow :: "[('a, 'm) monoid_scheme, 'a, 'b::semiring_1] => 'a"  (infixr "'(^')\<index>" 75)
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overloading nat_pow == "pow :: [_, 'a, nat] => 'a"
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begin
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  definition "nat_pow G a n = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n"
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end
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overloading int_pow == "pow :: [_, 'a, int] => 'a"
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begin
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  definition "int_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 z < 0 then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z))"
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end
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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
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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 default (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)
ballarin@13936
   304
qed
ballarin@13936
   305
ballarin@13936
   306
lemma (in group) inv_closed [intro, simp]:
ballarin@13936
   307
  "x \<in> carrier G ==> inv x \<in> carrier G"
ballarin@13936
   308
  using Units_inv_closed by simp
ballarin@13936
   309
ballarin@19981
   310
lemma (in group) l_inv_ex [simp]:
ballarin@19981
   311
  "x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
ballarin@19981
   312
  using Units_l_inv_ex by simp
ballarin@19981
   313
ballarin@19981
   314
lemma (in group) r_inv_ex [simp]:
ballarin@19981
   315
  "x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"
ballarin@19981
   316
  using Units_r_inv_ex by simp
ballarin@19981
   317
paulson@14963
   318
lemma (in group) l_inv [simp]:
ballarin@13936
   319
  "x \<in> carrier G ==> inv x \<otimes> x = \<one>"
ballarin@13936
   320
  using Units_l_inv by simp
ballarin@13813
   321
ballarin@20318
   322
ballarin@13813
   323
subsection {* Cancellation Laws and Basic Properties *}
ballarin@13813
   324
ballarin@13813
   325
lemma (in group) l_cancel [simp]:
ballarin@13813
   326
  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
ballarin@13813
   327
   (x \<otimes> y = x \<otimes> z) = (y = z)"
ballarin@13936
   328
  using Units_l_inv by simp
ballarin@13940
   329
paulson@14963
   330
lemma (in group) r_inv [simp]:
ballarin@13813
   331
  "x \<in> carrier G ==> x \<otimes> inv x = \<one>"
ballarin@13813
   332
proof -
ballarin@13813
   333
  assume x: "x \<in> carrier G"
ballarin@13813
   334
  then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>"
wenzelm@44472
   335
    by (simp add: m_assoc [symmetric])
ballarin@13813
   336
  with x show ?thesis by (simp del: r_one)
ballarin@13813
   337
qed
ballarin@13813
   338
ballarin@13813
   339
lemma (in group) r_cancel [simp]:
ballarin@13813
   340
  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
ballarin@13813
   341
   (y \<otimes> x = z \<otimes> x) = (y = z)"
ballarin@13813
   342
proof
ballarin@13813
   343
  assume eq: "y \<otimes> x = z \<otimes> x"
wenzelm@14693
   344
    and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
ballarin@13813
   345
  then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)"
ballarin@27698
   346
    by (simp add: m_assoc [symmetric] del: r_inv Units_r_inv)
paulson@14963
   347
  with G show "y = z" by simp
ballarin@13813
   348
next
ballarin@13813
   349
  assume eq: "y = z"
wenzelm@14693
   350
    and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
ballarin@13813
   351
  then show "y \<otimes> x = z \<otimes> x" by simp
ballarin@13813
   352
qed
ballarin@13813
   353
ballarin@13854
   354
lemma (in group) inv_one [simp]:
ballarin@13854
   355
  "inv \<one> = \<one>"
ballarin@13854
   356
proof -
ballarin@27698
   357
  have "inv \<one> = \<one> \<otimes> (inv \<one>)" by (simp del: r_inv Units_r_inv)
paulson@14963
   358
  moreover have "... = \<one>" by simp
ballarin@13854
   359
  finally show ?thesis .
ballarin@13854
   360
qed
ballarin@13854
   361
ballarin@13813
   362
lemma (in group) inv_inv [simp]:
ballarin@13813
   363
  "x \<in> carrier G ==> inv (inv x) = x"
ballarin@13936
   364
  using Units_inv_inv by simp
ballarin@13936
   365
ballarin@13936
   366
lemma (in group) inv_inj:
ballarin@13936
   367
  "inj_on (m_inv G) (carrier G)"
ballarin@13936
   368
  using inv_inj_on_Units by simp
ballarin@13813
   369
ballarin@13854
   370
lemma (in group) inv_mult_group:
ballarin@13813
   371
  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"
ballarin@13813
   372
proof -
wenzelm@14693
   373
  assume G: "x \<in> carrier G"  "y \<in> carrier G"
ballarin@13813
   374
  then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)"
wenzelm@44472
   375
    by (simp add: m_assoc) (simp add: m_assoc [symmetric])
ballarin@27698
   376
  with G show ?thesis by (simp del: l_inv Units_l_inv)
ballarin@13813
   377
qed
ballarin@13813
   378
ballarin@13940
   379
lemma (in group) inv_comm:
ballarin@13940
   380
  "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
wenzelm@14693
   381
  by (rule Units_inv_comm) auto
ballarin@13940
   382
paulson@13944
   383
lemma (in group) inv_equality:
paulson@13943
   384
     "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"
paulson@13943
   385
apply (simp add: m_inv_def)
paulson@13943
   386
apply (rule the_equality)
wenzelm@14693
   387
 apply (simp add: inv_comm [of y x])
wenzelm@14693
   388
apply (rule r_cancel [THEN iffD1], auto)
paulson@13943
   389
done
paulson@13943
   390
ballarin@13936
   391
text {* Power *}
ballarin@13936
   392
ballarin@13936
   393
lemma (in group) int_pow_def2:
huffman@46559
   394
  "a (^) (z::int) = (if z < 0 then inv (a (^) (nat (-z))) else a (^) (nat z))"
ballarin@13936
   395
  by (simp add: int_pow_def nat_pow_def Let_def)
ballarin@13936
   396
ballarin@13936
   397
lemma (in group) int_pow_0 [simp]:
ballarin@13936
   398
  "x (^) (0::int) = \<one>"
ballarin@13936
   399
  by (simp add: int_pow_def2)
ballarin@13936
   400
ballarin@13936
   401
lemma (in group) int_pow_one [simp]:
ballarin@13936
   402
  "\<one> (^) (z::int) = \<one>"
ballarin@13936
   403
  by (simp add: int_pow_def2)
ballarin@13936
   404
ballarin@20318
   405
paulson@14963
   406
subsection {* Subgroups *}
ballarin@13813
   407
ballarin@19783
   408
locale subgroup =
ballarin@19783
   409
  fixes H and G (structure)
paulson@14963
   410
  assumes subset: "H \<subseteq> carrier G"
paulson@14963
   411
    and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H"
ballarin@20318
   412
    and one_closed [simp]: "\<one> \<in> H"
paulson@14963
   413
    and m_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> inv x \<in> H"
ballarin@13813
   414
ballarin@20318
   415
lemma (in subgroup) is_subgroup:
wenzelm@26199
   416
  "subgroup H G" by (rule subgroup_axioms)
ballarin@20318
   417
ballarin@13813
   418
declare (in subgroup) group.intro [intro]
ballarin@13949
   419
paulson@14963
   420
lemma (in subgroup) mem_carrier [simp]:
paulson@14963
   421
  "x \<in> H \<Longrightarrow> x \<in> carrier G"
paulson@14963
   422
  using subset by blast
ballarin@13813
   423
paulson@14963
   424
lemma subgroup_imp_subset:
paulson@14963
   425
  "subgroup H G \<Longrightarrow> H \<subseteq> carrier G"
paulson@14963
   426
  by (rule subgroup.subset)
paulson@14963
   427
paulson@14963
   428
lemma (in subgroup) subgroup_is_group [intro]:
ballarin@27611
   429
  assumes "group G"
ballarin@27611
   430
  shows "group (G\<lparr>carrier := H\<rparr>)"
ballarin@27611
   431
proof -
ballarin@29237
   432
  interpret group G by fact
ballarin@27611
   433
  show ?thesis
ballarin@27698
   434
    apply (rule monoid.group_l_invI)
ballarin@27698
   435
    apply (unfold_locales) [1]
ballarin@27698
   436
    apply (auto intro: m_assoc l_inv mem_carrier)
ballarin@27698
   437
    done
ballarin@27611
   438
qed
ballarin@13813
   439
ballarin@13813
   440
text {*
ballarin@13813
   441
  Since @{term H} is nonempty, it contains some element @{term x}.  Since
ballarin@13813
   442
  it is closed under inverse, it contains @{text "inv x"}.  Since
ballarin@13813
   443
  it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}.
ballarin@13813
   444
*}
ballarin@13813
   445
ballarin@13813
   446
lemma (in group) one_in_subset:
ballarin@13813
   447
  "[| 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
   448
   ==> \<one> \<in> H"
wenzelm@44472
   449
by force
ballarin@13813
   450
ballarin@13813
   451
text {* A characterization of subgroups: closed, non-empty subset. *}
ballarin@13813
   452
ballarin@13813
   453
lemma (in group) subgroupI:
ballarin@13813
   454
  assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"
paulson@14963
   455
    and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H"
paulson@14963
   456
    and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H"
ballarin@13813
   457
  shows "subgroup H G"
ballarin@27714
   458
proof (simp add: subgroup_def assms)
ballarin@27714
   459
  show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: assms)
ballarin@13813
   460
qed
ballarin@13813
   461
ballarin@13936
   462
declare monoid.one_closed [iff] group.inv_closed [simp]
ballarin@13936
   463
  monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]
ballarin@13813
   464
ballarin@13813
   465
lemma subgroup_nonempty:
ballarin@13813
   466
  "~ subgroup {} G"
ballarin@13813
   467
  by (blast dest: subgroup.one_closed)
ballarin@13813
   468
ballarin@13813
   469
lemma (in subgroup) finite_imp_card_positive:
ballarin@13813
   470
  "finite (carrier G) ==> 0 < card H"
ballarin@13813
   471
proof (rule classical)
wenzelm@41528
   472
  assume "finite (carrier G)" and a: "~ 0 < card H"
paulson@14963
   473
  then have "finite H" by (blast intro: finite_subset [OF subset])
wenzelm@41528
   474
  with is_subgroup a have "subgroup {} G" by simp
ballarin@13813
   475
  with subgroup_nonempty show ?thesis by contradiction
ballarin@13813
   476
qed
ballarin@13813
   477
ballarin@13936
   478
(*
ballarin@13936
   479
lemma (in monoid) Units_subgroup:
ballarin@13936
   480
  "subgroup (Units G) G"
ballarin@13936
   481
*)
ballarin@13936
   482
ballarin@20318
   483
ballarin@13813
   484
subsection {* Direct Products *}
ballarin@13813
   485
wenzelm@35848
   486
definition
wenzelm@35848
   487
  DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid" (infixr "\<times>\<times>" 80) where
wenzelm@35848
   488
  "G \<times>\<times> H =
wenzelm@35848
   489
    \<lparr>carrier = carrier G \<times> carrier H,
wenzelm@35848
   490
     mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')),
wenzelm@35848
   491
     one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"
ballarin@13813
   492
paulson@14963
   493
lemma DirProd_monoid:
ballarin@27611
   494
  assumes "monoid G" and "monoid H"
paulson@14963
   495
  shows "monoid (G \<times>\<times> H)"
paulson@14963
   496
proof -
wenzelm@30729
   497
  interpret G: monoid G by fact
wenzelm@30729
   498
  interpret H: monoid H by fact
ballarin@27714
   499
  from assms
paulson@14963
   500
  show ?thesis by (unfold monoid_def DirProd_def, auto) 
paulson@14963
   501
qed
ballarin@13813
   502
ballarin@13813
   503
paulson@14963
   504
text{*Does not use the previous result because it's easier just to use auto.*}
paulson@14963
   505
lemma DirProd_group:
ballarin@27611
   506
  assumes "group G" and "group H"
paulson@14963
   507
  shows "group (G \<times>\<times> H)"
ballarin@27611
   508
proof -
wenzelm@30729
   509
  interpret G: group G by fact
wenzelm@30729
   510
  interpret H: group H by fact
ballarin@27611
   511
  show ?thesis by (rule groupI)
paulson@14963
   512
     (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
paulson@14963
   513
           simp add: DirProd_def)
ballarin@27611
   514
qed
ballarin@13813
   515
paulson@14963
   516
lemma carrier_DirProd [simp]:
paulson@14963
   517
     "carrier (G \<times>\<times> H) = carrier G \<times> carrier H"
paulson@14963
   518
  by (simp add: DirProd_def)
paulson@13944
   519
paulson@14963
   520
lemma one_DirProd [simp]:
paulson@14963
   521
     "\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)"
paulson@14963
   522
  by (simp add: DirProd_def)
paulson@13944
   523
paulson@14963
   524
lemma mult_DirProd [simp]:
paulson@14963
   525
     "(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
   526
  by (simp add: DirProd_def)
paulson@13944
   527
paulson@14963
   528
lemma inv_DirProd [simp]:
ballarin@27611
   529
  assumes "group G" and "group H"
paulson@13944
   530
  assumes g: "g \<in> carrier G"
paulson@13944
   531
      and h: "h \<in> carrier H"
paulson@14963
   532
  shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"
ballarin@27611
   533
proof -
wenzelm@30729
   534
  interpret G: group G by fact
wenzelm@30729
   535
  interpret H: group H by fact
wenzelm@30729
   536
  interpret Prod: group "G \<times>\<times> H"
ballarin@27714
   537
    by (auto intro: DirProd_group group.intro group.axioms assms)
paulson@14963
   538
  show ?thesis by (simp add: Prod.inv_equality g h)
paulson@14963
   539
qed
ballarin@27698
   540
paulson@14963
   541
paulson@14963
   542
subsection {* Homomorphisms and Isomorphisms *}
ballarin@13813
   543
wenzelm@35847
   544
definition
wenzelm@35847
   545
  hom :: "_ => _ => ('a => 'b) set" where
wenzelm@35848
   546
  "hom G H =
ballarin@13813
   547
    {h. h \<in> carrier G -> carrier H &
wenzelm@14693
   548
      (\<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
   549
paulson@14761
   550
lemma (in group) hom_compose:
nipkow@31754
   551
  "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"
nipkow@44890
   552
by (fastforce simp add: hom_def compose_def)
paulson@13943
   553
wenzelm@35848
   554
definition
wenzelm@35848
   555
  iso :: "_ => _ => ('a => 'b) set" (infixr "\<cong>" 60)
wenzelm@35848
   556
  where "G \<cong> H = {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}"
paulson@14761
   557
paulson@14803
   558
lemma iso_refl: "(%x. x) \<in> G \<cong> G"
nipkow@31727
   559
by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)
paulson@14761
   560
paulson@14761
   561
lemma (in group) iso_sym:
nipkow@33057
   562
     "h \<in> G \<cong> H \<Longrightarrow> inv_into (carrier G) h \<in> H \<cong> G"
nipkow@33057
   563
apply (simp add: iso_def bij_betw_inv_into) 
nipkow@33057
   564
apply (subgoal_tac "inv_into (carrier G) h \<in> carrier H \<rightarrow> carrier G") 
nipkow@33057
   565
 prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_inv_into]) 
nipkow@33057
   566
apply (simp add: hom_def bij_betw_def inv_into_f_eq f_inv_into_f Pi_def)
paulson@14761
   567
done
paulson@14761
   568
paulson@14761
   569
lemma (in group) iso_trans: 
paulson@14803
   570
     "[|h \<in> G \<cong> H; i \<in> H \<cong> I|] ==> (compose (carrier G) i h) \<in> G \<cong> I"
paulson@14761
   571
by (auto simp add: iso_def hom_compose bij_betw_compose)
paulson@14761
   572
paulson@14963
   573
lemma DirProd_commute_iso:
paulson@14963
   574
  shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)"
nipkow@31754
   575
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)
paulson@14761
   576
paulson@14963
   577
lemma DirProd_assoc_iso:
paulson@14963
   578
  shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> (G \<times>\<times> H \<times>\<times> I) \<cong> (G \<times>\<times> (H \<times>\<times> I))"
nipkow@31727
   579
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)
paulson@14761
   580
paulson@14761
   581
paulson@14963
   582
text{*Basis for homomorphism proofs: we assume two groups @{term G} and
ballarin@15076
   583
  @{term H}, with a homomorphism @{term h} between them*}
ballarin@29237
   584
locale group_hom = G: group G + H: group H for G (structure) and H (structure) +
ballarin@29237
   585
  fixes h
ballarin@13813
   586
  assumes homh: "h \<in> hom G H"
ballarin@29240
   587
ballarin@29240
   588
lemma (in group_hom) hom_mult [simp]:
ballarin@29240
   589
  "[| x \<in> carrier G; y \<in> carrier G |] ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
ballarin@29240
   590
proof -
ballarin@29240
   591
  assume "x \<in> carrier G" "y \<in> carrier G"
ballarin@29240
   592
  with homh [unfolded hom_def] show ?thesis by simp
ballarin@29240
   593
qed
ballarin@29240
   594
ballarin@29240
   595
lemma (in group_hom) hom_closed [simp]:
ballarin@29240
   596
  "x \<in> carrier G ==> h x \<in> carrier H"
ballarin@29240
   597
proof -
ballarin@29240
   598
  assume "x \<in> carrier G"
nipkow@31754
   599
  with homh [unfolded hom_def] show ?thesis by auto
ballarin@29240
   600
qed
ballarin@13813
   601
ballarin@13813
   602
lemma (in group_hom) one_closed [simp]:
ballarin@13813
   603
  "h \<one> \<in> carrier H"
ballarin@13813
   604
  by simp
ballarin@13813
   605
ballarin@13813
   606
lemma (in group_hom) hom_one [simp]:
wenzelm@14693
   607
  "h \<one> = \<one>\<^bsub>H\<^esub>"
ballarin@13813
   608
proof -
ballarin@15076
   609
  have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>"
ballarin@13813
   610
    by (simp add: hom_mult [symmetric] del: hom_mult)
ballarin@13813
   611
  then show ?thesis by (simp del: r_one)
ballarin@13813
   612
qed
ballarin@13813
   613
ballarin@13813
   614
lemma (in group_hom) inv_closed [simp]:
ballarin@13813
   615
  "x \<in> carrier G ==> h (inv x) \<in> carrier H"
ballarin@13813
   616
  by simp
ballarin@13813
   617
ballarin@13813
   618
lemma (in group_hom) hom_inv [simp]:
wenzelm@14693
   619
  "x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"
ballarin@13813
   620
proof -
ballarin@13813
   621
  assume x: "x \<in> carrier G"
wenzelm@14693
   622
  then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>"
paulson@14963
   623
    by (simp add: hom_mult [symmetric] del: hom_mult)
wenzelm@14693
   624
  also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)"
paulson@14963
   625
    by (simp add: hom_mult [symmetric] del: hom_mult)
wenzelm@14693
   626
  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
   627
  with x show ?thesis by (simp del: H.r_inv H.Units_r_inv)
ballarin@13813
   628
qed
ballarin@13813
   629
ballarin@20318
   630
ballarin@13949
   631
subsection {* Commutative Structures *}
ballarin@13936
   632
ballarin@13936
   633
text {*
ballarin@13936
   634
  Naming convention: multiplicative structures that are commutative
ballarin@13936
   635
  are called \emph{commutative}, additive structures are called
ballarin@13936
   636
  \emph{Abelian}.
ballarin@13936
   637
*}
ballarin@13813
   638
paulson@14963
   639
locale comm_monoid = monoid +
paulson@14963
   640
  assumes m_comm: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x"
ballarin@13813
   641
paulson@14963
   642
lemma (in comm_monoid) m_lcomm:
paulson@14963
   643
  "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
ballarin@13813
   644
   x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
ballarin@13813
   645
proof -
wenzelm@14693
   646
  assume xyz: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
ballarin@13813
   647
  from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)
ballarin@13813
   648
  also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)
ballarin@13813
   649
  also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)
ballarin@13813
   650
  finally show ?thesis .
ballarin@13813
   651
qed
ballarin@13813
   652
paulson@14963
   653
lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm
ballarin@13813
   654
ballarin@13936
   655
lemma comm_monoidI:
ballarin@19783
   656
  fixes G (structure)
ballarin@13936
   657
  assumes m_closed:
wenzelm@14693
   658
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
wenzelm@14693
   659
    and one_closed: "\<one> \<in> carrier G"
ballarin@13936
   660
    and m_assoc:
ballarin@13936
   661
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
wenzelm@14693
   662
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
wenzelm@14693
   663
    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
ballarin@13936
   664
    and m_comm:
wenzelm@14693
   665
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
ballarin@13936
   666
  shows "comm_monoid G"
ballarin@13936
   667
  using l_one
paulson@14963
   668
    by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro 
ballarin@27714
   669
             intro: assms simp: m_closed one_closed m_comm)
ballarin@13817
   670
ballarin@13936
   671
lemma (in monoid) monoid_comm_monoidI:
ballarin@13936
   672
  assumes m_comm:
wenzelm@14693
   673
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
ballarin@13936
   674
  shows "comm_monoid G"
ballarin@13936
   675
  by (rule comm_monoidI) (auto intro: m_assoc m_comm)
paulson@14963
   676
wenzelm@14693
   677
(*lemma (in comm_monoid) r_one [simp]:
ballarin@13817
   678
  "x \<in> carrier G ==> x \<otimes> \<one> = x"
ballarin@13817
   679
proof -
ballarin@13817
   680
  assume G: "x \<in> carrier G"
ballarin@13817
   681
  then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)
ballarin@13817
   682
  also from G have "... = x" by simp
ballarin@13817
   683
  finally show ?thesis .
wenzelm@14693
   684
qed*)
paulson@14963
   685
ballarin@13936
   686
lemma (in comm_monoid) nat_pow_distr:
ballarin@13936
   687
  "[| x \<in> carrier G; y \<in> carrier G |] ==>
ballarin@13936
   688
  (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"
ballarin@13936
   689
  by (induct n) (simp, simp add: m_ac)
ballarin@13936
   690
ballarin@13936
   691
locale comm_group = comm_monoid + group
ballarin@13936
   692
ballarin@13936
   693
lemma (in group) group_comm_groupI:
ballarin@13936
   694
  assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>
wenzelm@14693
   695
      x \<otimes> y = y \<otimes> x"
ballarin@13936
   696
  shows "comm_group G"
wenzelm@44655
   697
  by default (simp_all add: m_comm)
ballarin@13817
   698
ballarin@13936
   699
lemma comm_groupI:
ballarin@19783
   700
  fixes G (structure)
ballarin@13936
   701
  assumes m_closed:
wenzelm@14693
   702
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
wenzelm@14693
   703
    and one_closed: "\<one> \<in> carrier G"
ballarin@13936
   704
    and m_assoc:
ballarin@13936
   705
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
wenzelm@14693
   706
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
ballarin@13936
   707
    and m_comm:
wenzelm@14693
   708
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
wenzelm@14693
   709
    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
paulson@14963
   710
    and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
ballarin@13936
   711
  shows "comm_group G"
ballarin@27714
   712
  by (fast intro: group.group_comm_groupI groupI assms)
ballarin@13936
   713
ballarin@13936
   714
lemma (in comm_group) inv_mult:
ballarin@13854
   715
  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"
ballarin@13936
   716
  by (simp add: m_ac inv_mult_group)
ballarin@13854
   717
ballarin@20318
   718
ballarin@20318
   719
subsection {* The Lattice of Subgroups of a Group *}
ballarin@14751
   720
ballarin@14751
   721
text_raw {* \label{sec:subgroup-lattice} *}
ballarin@14751
   722
ballarin@14751
   723
theorem (in group) subgroups_partial_order:
ballarin@27713
   724
  "partial_order (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"
wenzelm@44655
   725
  by default simp_all
ballarin@14751
   726
ballarin@14751
   727
lemma (in group) subgroup_self:
ballarin@14751
   728
  "subgroup (carrier G) G"
ballarin@14751
   729
  by (rule subgroupI) auto
ballarin@14751
   730
ballarin@14751
   731
lemma (in group) subgroup_imp_group:
ballarin@14751
   732
  "subgroup H G ==> group (G(| carrier := H |))"
wenzelm@26199
   733
  by (erule subgroup.subgroup_is_group) (rule group_axioms)
ballarin@14751
   734
ballarin@14751
   735
lemma (in group) is_monoid [intro, simp]:
ballarin@14751
   736
  "monoid G"
paulson@14963
   737
  by (auto intro: monoid.intro m_assoc) 
ballarin@14751
   738
ballarin@14751
   739
lemma (in group) subgroup_inv_equality:
ballarin@14751
   740
  "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"
ballarin@14751
   741
apply (rule_tac inv_equality [THEN sym])
paulson@14761
   742
  apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)
paulson@14761
   743
 apply (rule subsetD [OF subgroup.subset], assumption+)
paulson@14761
   744
apply (rule subsetD [OF subgroup.subset], assumption)
paulson@14761
   745
apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+)
ballarin@14751
   746
done
ballarin@14751
   747
ballarin@14751
   748
theorem (in group) subgroups_Inter:
ballarin@14751
   749
  assumes subgr: "(!!H. H \<in> A ==> subgroup H G)"
ballarin@14751
   750
    and not_empty: "A ~= {}"
ballarin@14751
   751
  shows "subgroup (\<Inter>A) G"
ballarin@14751
   752
proof (rule subgroupI)
ballarin@14751
   753
  from subgr [THEN subgroup.subset] and not_empty
ballarin@14751
   754
  show "\<Inter>A \<subseteq> carrier G" by blast
ballarin@14751
   755
next
ballarin@14751
   756
  from subgr [THEN subgroup.one_closed]
ballarin@14751
   757
  show "\<Inter>A ~= {}" by blast
ballarin@14751
   758
next
ballarin@14751
   759
  fix x assume "x \<in> \<Inter>A"
ballarin@14751
   760
  with subgr [THEN subgroup.m_inv_closed]
ballarin@14751
   761
  show "inv x \<in> \<Inter>A" by blast
ballarin@14751
   762
next
ballarin@14751
   763
  fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"
ballarin@14751
   764
  with subgr [THEN subgroup.m_closed]
ballarin@14751
   765
  show "x \<otimes> y \<in> \<Inter>A" by blast
ballarin@14751
   766
qed
ballarin@14751
   767
ballarin@14751
   768
theorem (in group) subgroups_complete_lattice:
ballarin@27713
   769
  "complete_lattice (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"
ballarin@22063
   770
    (is "complete_lattice ?L")
ballarin@14751
   771
proof (rule partial_order.complete_lattice_criterion1)
ballarin@22063
   772
  show "partial_order ?L" by (rule subgroups_partial_order)
ballarin@14751
   773
next
wenzelm@46008
   774
  have "greatest ?L (carrier G) (carrier ?L)"
wenzelm@46008
   775
    by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)
wenzelm@46008
   776
  then show "\<exists>G. greatest ?L G (carrier ?L)" ..
ballarin@14751
   777
next
ballarin@14751
   778
  fix A
ballarin@22063
   779
  assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"
ballarin@14751
   780
  then have Int_subgroup: "subgroup (\<Inter>A) G"
nipkow@44890
   781
    by (fastforce intro: subgroups_Inter)
wenzelm@46008
   782
  have "greatest ?L (\<Inter>A) (Lower ?L A)" (is "greatest _ ?Int _")
wenzelm@46008
   783
  proof (rule greatest_LowerI)
wenzelm@46008
   784
    fix H
wenzelm@46008
   785
    assume H: "H \<in> A"
wenzelm@46008
   786
    with L have subgroupH: "subgroup H G" by auto
wenzelm@46008
   787
    from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
wenzelm@46008
   788
      by (rule subgroup_imp_group)
wenzelm@46008
   789
    from groupH have monoidH: "monoid ?H"
wenzelm@46008
   790
      by (rule group.is_monoid)
wenzelm@46008
   791
    from H have Int_subset: "?Int \<subseteq> H" by fastforce
wenzelm@46008
   792
    then show "le ?L ?Int H" by simp
wenzelm@46008
   793
  next
wenzelm@46008
   794
    fix H
wenzelm@46008
   795
    assume H: "H \<in> Lower ?L A"
wenzelm@46008
   796
    with L Int_subgroup show "le ?L H ?Int"
wenzelm@46008
   797
      by (fastforce simp: Lower_def intro: Inter_greatest)
wenzelm@46008
   798
  next
wenzelm@46008
   799
    show "A \<subseteq> carrier ?L" by (rule L)
wenzelm@46008
   800
  next
wenzelm@46008
   801
    show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
ballarin@14751
   802
  qed
wenzelm@46008
   803
  then show "\<exists>I. greatest ?L I (Lower ?L A)" ..
ballarin@14751
   804
qed
ballarin@14751
   805
ballarin@13813
   806
end