src/HOL/Algebra/Group.thy
author paulson <lp15@cam.ac.uk>
Wed Jun 06 14:25:53 2018 +0100 (12 months ago)
changeset 68399 0b71d08528f0
parent 68188 2af1f142f855
child 68443 43055b016688
permissions -rw-r--r--
resolution of name clashes in Algebra
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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 Complete_Lattice "HOL-Library.FuncSet"
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begin
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section \<open>Monoids and Groups\<close>
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subsection \<open>Definitions\<close>
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text \<open>
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  Definitions follow @{cite "Jacobson:1985"}.
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\<close>
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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 \<and> x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> 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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  \<comment> \<open>The set of invertible elements\<close>
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  where "Units G = {y. y \<in> carrier G \<and> (\<exists>x \<in> carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> 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 = rec_nat \<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 = rec_nat \<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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lemma int_pow_int: "x [^]\<^bsub>G\<^esub> (int n) = x [^]\<^bsub>G\<^esub> n"
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by(simp add: int_pow_def nat_pow_def)
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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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    \<comment> \<open>Must avoid premature use of \<open>hyp_subst_tac\<close>.\<close>
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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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lemma (in monoid) carrier_not_empty: "carrier G \<noteq> {}"
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by auto
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text \<open>Power\<close>
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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 \<open>Groups\<close>
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text \<open>
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  A group is a monoid all of whose elements are invertible.
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\<close>
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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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    "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y \<in> carrier G. y \<otimes> x = \<one> \<and> 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> \<and> 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 \<subseteq> Units G"
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    by (unfold Units_def) fast
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  show ?thesis
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    by standard (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:
paulson@14963
   301
    "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
ballarin@13936
   302
  shows "group G"
ballarin@13936
   303
  by (rule groupI) (auto intro: m_assoc l_inv_ex)
ballarin@13936
   304
ballarin@13936
   305
lemma (in group) Units_eq [simp]:
ballarin@13936
   306
  "Units G = carrier G"
ballarin@13936
   307
proof
wenzelm@67091
   308
  show "Units G \<subseteq> carrier G" by fast
ballarin@13936
   309
next
wenzelm@67091
   310
  show "carrier G \<subseteq> Units G" by (rule Units)
ballarin@13936
   311
qed
ballarin@13936
   312
ballarin@13936
   313
lemma (in group) inv_closed [intro, simp]:
ballarin@13936
   314
  "x \<in> carrier G ==> inv x \<in> carrier G"
ballarin@13936
   315
  using Units_inv_closed by simp
ballarin@13936
   316
ballarin@19981
   317
lemma (in group) l_inv_ex [simp]:
ballarin@19981
   318
  "x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
ballarin@19981
   319
  using Units_l_inv_ex by simp
ballarin@19981
   320
ballarin@19981
   321
lemma (in group) r_inv_ex [simp]:
ballarin@19981
   322
  "x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"
ballarin@19981
   323
  using Units_r_inv_ex by simp
ballarin@19981
   324
paulson@14963
   325
lemma (in group) l_inv [simp]:
ballarin@13936
   326
  "x \<in> carrier G ==> inv x \<otimes> x = \<one>"
lp15@68399
   327
  by simp
ballarin@13813
   328
ballarin@20318
   329
wenzelm@61382
   330
subsection \<open>Cancellation Laws and Basic Properties\<close>
ballarin@13813
   331
paulson@14963
   332
lemma (in group) r_inv [simp]:
ballarin@13813
   333
  "x \<in> carrier G ==> x \<otimes> inv x = \<one>"
lp15@68399
   334
  by simp
ballarin@13813
   335
lp15@68399
   336
lemma (in group) right_cancel [simp]:
ballarin@13813
   337
  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
ballarin@13813
   338
   (y \<otimes> x = z \<otimes> x) = (y = z)"
lp15@68399
   339
  by (metis inv_closed m_assoc r_inv r_one)
ballarin@13813
   340
ballarin@13854
   341
lemma (in group) inv_one [simp]:
ballarin@13854
   342
  "inv \<one> = \<one>"
ballarin@13854
   343
proof -
ballarin@27698
   344
  have "inv \<one> = \<one> \<otimes> (inv \<one>)" by (simp del: r_inv Units_r_inv)
paulson@14963
   345
  moreover have "... = \<one>" by simp
ballarin@13854
   346
  finally show ?thesis .
ballarin@13854
   347
qed
ballarin@13854
   348
ballarin@13813
   349
lemma (in group) inv_inv [simp]:
ballarin@13813
   350
  "x \<in> carrier G ==> inv (inv x) = x"
ballarin@13936
   351
  using Units_inv_inv by simp
ballarin@13936
   352
ballarin@13936
   353
lemma (in group) inv_inj:
ballarin@13936
   354
  "inj_on (m_inv G) (carrier G)"
ballarin@13936
   355
  using inv_inj_on_Units by simp
ballarin@13813
   356
ballarin@13854
   357
lemma (in group) inv_mult_group:
ballarin@13813
   358
  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"
ballarin@13813
   359
proof -
wenzelm@14693
   360
  assume G: "x \<in> carrier G"  "y \<in> carrier G"
ballarin@13813
   361
  then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)"
wenzelm@44472
   362
    by (simp add: m_assoc) (simp add: m_assoc [symmetric])
ballarin@27698
   363
  with G show ?thesis by (simp del: l_inv Units_l_inv)
ballarin@13813
   364
qed
ballarin@13813
   365
ballarin@13940
   366
lemma (in group) inv_comm:
ballarin@13940
   367
  "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
wenzelm@14693
   368
  by (rule Units_inv_comm) auto
ballarin@13940
   369
paulson@13944
   370
lemma (in group) inv_equality:
paulson@13943
   371
     "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"
lp15@68399
   372
  using inv_unique r_inv by blast
paulson@13943
   373
ballarin@57271
   374
(* Contributed by Joachim Breitner *)
ballarin@57271
   375
lemma (in group) inv_solve_left:
ballarin@57271
   376
  "\<lbrakk> a \<in> carrier G; b \<in> carrier G; c \<in> carrier G \<rbrakk> \<Longrightarrow> a = inv b \<otimes> c \<longleftrightarrow> c = b \<otimes> a"
ballarin@57271
   377
  by (metis inv_equality l_inv_ex l_one m_assoc r_inv)
ballarin@57271
   378
lemma (in group) inv_solve_right:
ballarin@57271
   379
  "\<lbrakk> a \<in> carrier G; b \<in> carrier G; c \<in> carrier G \<rbrakk> \<Longrightarrow> a = b \<otimes> inv c \<longleftrightarrow> b = a \<otimes> c"
ballarin@57271
   380
  by (metis inv_equality l_inv_ex l_one m_assoc r_inv)
ballarin@57271
   381
wenzelm@61382
   382
text \<open>Power\<close>
ballarin@13936
   383
ballarin@13936
   384
lemma (in group) int_pow_def2:
nipkow@67341
   385
  "a [^] (z::int) = (if z < 0 then inv (a [^] (nat (-z))) else a [^] (nat z))"
ballarin@13936
   386
  by (simp add: int_pow_def nat_pow_def Let_def)
ballarin@13936
   387
ballarin@13936
   388
lemma (in group) int_pow_0 [simp]:
nipkow@67341
   389
  "x [^] (0::int) = \<one>"
ballarin@13936
   390
  by (simp add: int_pow_def2)
ballarin@13936
   391
ballarin@13936
   392
lemma (in group) int_pow_one [simp]:
nipkow@67341
   393
  "\<one> [^] (z::int) = \<one>"
ballarin@13936
   394
  by (simp add: int_pow_def2)
ballarin@13936
   395
ballarin@57271
   396
(* The following are contributed by Joachim Breitner *)
ballarin@20318
   397
ballarin@57271
   398
lemma (in group) int_pow_closed [intro, simp]:
nipkow@67341
   399
  "x \<in> carrier G ==> x [^] (i::int) \<in> carrier G"
ballarin@57271
   400
  by (simp add: int_pow_def2)
ballarin@57271
   401
ballarin@57271
   402
lemma (in group) int_pow_1 [simp]:
nipkow@67341
   403
  "x \<in> carrier G \<Longrightarrow> x [^] (1::int) = x"
ballarin@57271
   404
  by (simp add: int_pow_def2)
ballarin@57271
   405
ballarin@57271
   406
lemma (in group) int_pow_neg:
nipkow@67341
   407
  "x \<in> carrier G \<Longrightarrow> x [^] (-i::int) = inv (x [^] i)"
ballarin@57271
   408
  by (simp add: int_pow_def2)
ballarin@57271
   409
ballarin@57271
   410
lemma (in group) int_pow_mult:
nipkow@67341
   411
  "x \<in> carrier G \<Longrightarrow> x [^] (i + j::int) = x [^] i \<otimes> x [^] j"
ballarin@57271
   412
proof -
ballarin@57271
   413
  have [simp]: "-i - j = -j - i" by simp
wenzelm@67613
   414
  assume "x \<in> carrier G" then
ballarin@57271
   415
  show ?thesis
ballarin@57271
   416
    by (auto simp add: int_pow_def2 inv_solve_left inv_solve_right nat_add_distrib [symmetric] nat_pow_mult )
ballarin@57271
   417
qed
ballarin@57271
   418
Andreas@61628
   419
lemma (in group) int_pow_diff:
nipkow@67341
   420
  "x \<in> carrier G \<Longrightarrow> x [^] (n - m :: int) = x [^] n \<otimes> inv (x [^] m)"
Andreas@61628
   421
by(simp only: diff_conv_add_uminus int_pow_mult int_pow_neg)
Andreas@61628
   422
Andreas@61628
   423
lemma (in group) inj_on_multc: "c \<in> carrier G \<Longrightarrow> inj_on (\<lambda>x. x \<otimes> c) (carrier G)"
Andreas@61628
   424
by(simp add: inj_on_def)
Andreas@61628
   425
Andreas@61628
   426
lemma (in group) inj_on_cmult: "c \<in> carrier G \<Longrightarrow> inj_on (\<lambda>x. c \<otimes> x) (carrier G)"
Andreas@61628
   427
by(simp add: inj_on_def)
Andreas@61628
   428
wenzelm@61382
   429
subsection \<open>Subgroups\<close>
ballarin@13813
   430
ballarin@19783
   431
locale subgroup =
ballarin@19783
   432
  fixes H and G (structure)
paulson@14963
   433
  assumes subset: "H \<subseteq> carrier G"
paulson@14963
   434
    and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H"
ballarin@20318
   435
    and one_closed [simp]: "\<one> \<in> H"
paulson@14963
   436
    and m_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> inv x \<in> H"
ballarin@13813
   437
ballarin@20318
   438
lemma (in subgroup) is_subgroup:
wenzelm@26199
   439
  "subgroup H G" by (rule subgroup_axioms)
ballarin@20318
   440
ballarin@13813
   441
declare (in subgroup) group.intro [intro]
ballarin@13949
   442
paulson@14963
   443
lemma (in subgroup) mem_carrier [simp]:
paulson@14963
   444
  "x \<in> H \<Longrightarrow> x \<in> carrier G"
paulson@14963
   445
  using subset by blast
ballarin@13813
   446
paulson@14963
   447
lemma subgroup_imp_subset:
paulson@14963
   448
  "subgroup H G \<Longrightarrow> H \<subseteq> carrier G"
paulson@14963
   449
  by (rule subgroup.subset)
paulson@14963
   450
paulson@14963
   451
lemma (in subgroup) subgroup_is_group [intro]:
ballarin@27611
   452
  assumes "group G"
ballarin@27611
   453
  shows "group (G\<lparr>carrier := H\<rparr>)"
ballarin@27611
   454
proof -
ballarin@29237
   455
  interpret group G by fact
ballarin@27611
   456
  show ?thesis
ballarin@27698
   457
    apply (rule monoid.group_l_invI)
ballarin@27698
   458
    apply (unfold_locales) [1]
ballarin@27698
   459
    apply (auto intro: m_assoc l_inv mem_carrier)
ballarin@27698
   460
    done
ballarin@27611
   461
qed
ballarin@13813
   462
wenzelm@61382
   463
text \<open>
ballarin@13813
   464
  Since @{term H} is nonempty, it contains some element @{term x}.  Since
wenzelm@63167
   465
  it is closed under inverse, it contains \<open>inv x\<close>.  Since
wenzelm@63167
   466
  it is closed under product, it contains \<open>x \<otimes> inv x = \<one>\<close>.
wenzelm@61382
   467
\<close>
ballarin@13813
   468
ballarin@13813
   469
lemma (in group) one_in_subset:
ballarin@13813
   470
  "[| 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
   471
   ==> \<one> \<in> H"
wenzelm@44472
   472
by force
ballarin@13813
   473
wenzelm@61382
   474
text \<open>A characterization of subgroups: closed, non-empty subset.\<close>
ballarin@13813
   475
ballarin@13813
   476
lemma (in group) subgroupI:
ballarin@13813
   477
  assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"
paulson@14963
   478
    and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H"
paulson@14963
   479
    and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H"
ballarin@13813
   480
  shows "subgroup H G"
ballarin@27714
   481
proof (simp add: subgroup_def assms)
ballarin@27714
   482
  show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: assms)
ballarin@13813
   483
qed
ballarin@13813
   484
ballarin@13936
   485
declare monoid.one_closed [iff] group.inv_closed [simp]
ballarin@13936
   486
  monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]
ballarin@13813
   487
ballarin@13813
   488
lemma subgroup_nonempty:
wenzelm@67091
   489
  "\<not> subgroup {} G"
ballarin@13813
   490
  by (blast dest: subgroup.one_closed)
ballarin@13813
   491
ballarin@13813
   492
lemma (in subgroup) finite_imp_card_positive:
ballarin@13813
   493
  "finite (carrier G) ==> 0 < card H"
ballarin@13813
   494
proof (rule classical)
wenzelm@67091
   495
  assume "finite (carrier G)" and a: "\<not> 0 < card H"
paulson@14963
   496
  then have "finite H" by (blast intro: finite_subset [OF subset])
wenzelm@41528
   497
  with is_subgroup a have "subgroup {} G" by simp
ballarin@13813
   498
  with subgroup_nonempty show ?thesis by contradiction
ballarin@13813
   499
qed
ballarin@13813
   500
ballarin@13936
   501
(*
ballarin@13936
   502
lemma (in monoid) Units_subgroup:
ballarin@13936
   503
  "subgroup (Units G) G"
ballarin@13936
   504
*)
ballarin@13936
   505
ballarin@20318
   506
wenzelm@61382
   507
subsection \<open>Direct Products\<close>
ballarin@13813
   508
wenzelm@35848
   509
definition
wenzelm@35848
   510
  DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid" (infixr "\<times>\<times>" 80) where
wenzelm@35848
   511
  "G \<times>\<times> H =
wenzelm@35848
   512
    \<lparr>carrier = carrier G \<times> carrier H,
wenzelm@35848
   513
     mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')),
wenzelm@35848
   514
     one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"
ballarin@13813
   515
paulson@14963
   516
lemma DirProd_monoid:
ballarin@27611
   517
  assumes "monoid G" and "monoid H"
paulson@14963
   518
  shows "monoid (G \<times>\<times> H)"
paulson@14963
   519
proof -
wenzelm@30729
   520
  interpret G: monoid G by fact
wenzelm@30729
   521
  interpret H: monoid H by fact
ballarin@27714
   522
  from assms
paulson@14963
   523
  show ?thesis by (unfold monoid_def DirProd_def, auto) 
paulson@14963
   524
qed
ballarin@13813
   525
ballarin@13813
   526
wenzelm@61382
   527
text\<open>Does not use the previous result because it's easier just to use auto.\<close>
paulson@14963
   528
lemma DirProd_group:
ballarin@27611
   529
  assumes "group G" and "group H"
paulson@14963
   530
  shows "group (G \<times>\<times> H)"
ballarin@27611
   531
proof -
wenzelm@30729
   532
  interpret G: group G by fact
wenzelm@30729
   533
  interpret H: group H by fact
ballarin@27611
   534
  show ?thesis by (rule groupI)
paulson@14963
   535
     (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
paulson@14963
   536
           simp add: DirProd_def)
ballarin@27611
   537
qed
ballarin@13813
   538
paulson@14963
   539
lemma carrier_DirProd [simp]:
paulson@14963
   540
     "carrier (G \<times>\<times> H) = carrier G \<times> carrier H"
paulson@14963
   541
  by (simp add: DirProd_def)
paulson@13944
   542
paulson@14963
   543
lemma one_DirProd [simp]:
paulson@14963
   544
     "\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)"
paulson@14963
   545
  by (simp add: DirProd_def)
paulson@13944
   546
paulson@14963
   547
lemma mult_DirProd [simp]:
paulson@14963
   548
     "(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
   549
  by (simp add: DirProd_def)
paulson@13944
   550
paulson@14963
   551
lemma inv_DirProd [simp]:
ballarin@27611
   552
  assumes "group G" and "group H"
paulson@13944
   553
  assumes g: "g \<in> carrier G"
paulson@13944
   554
      and h: "h \<in> carrier H"
paulson@14963
   555
  shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"
ballarin@27611
   556
proof -
wenzelm@30729
   557
  interpret G: group G by fact
wenzelm@30729
   558
  interpret H: group H by fact
wenzelm@30729
   559
  interpret Prod: group "G \<times>\<times> H"
ballarin@27714
   560
    by (auto intro: DirProd_group group.intro group.axioms assms)
paulson@14963
   561
  show ?thesis by (simp add: Prod.inv_equality g h)
paulson@14963
   562
qed
ballarin@27698
   563
paulson@14963
   564
wenzelm@61382
   565
subsection \<open>Homomorphisms and Isomorphisms\<close>
ballarin@13813
   566
wenzelm@35847
   567
definition
wenzelm@35847
   568
  hom :: "_ => _ => ('a => 'b) set" where
wenzelm@35848
   569
  "hom G H =
wenzelm@67091
   570
    {h. h \<in> carrier G \<rightarrow> carrier H \<and>
wenzelm@14693
   571
      (\<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
   572
paulson@14761
   573
lemma (in group) hom_compose:
nipkow@31754
   574
  "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"
nipkow@44890
   575
by (fastforce simp add: hom_def compose_def)
paulson@13943
   576
wenzelm@35848
   577
definition
wenzelm@35848
   578
  iso :: "_ => _ => ('a => 'b) set" (infixr "\<cong>" 60)
wenzelm@67091
   579
  where "G \<cong> H = {h. h \<in> hom G H \<and> bij_betw h (carrier G) (carrier H)}"
paulson@14761
   580
wenzelm@67091
   581
lemma iso_refl: "(\<lambda>x. x) \<in> G \<cong> G"
nipkow@31727
   582
by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)
paulson@14761
   583
paulson@14761
   584
lemma (in group) iso_sym:
nipkow@33057
   585
     "h \<in> G \<cong> H \<Longrightarrow> inv_into (carrier G) h \<in> H \<cong> G"
nipkow@33057
   586
apply (simp add: iso_def bij_betw_inv_into) 
nipkow@33057
   587
apply (subgoal_tac "inv_into (carrier G) h \<in> carrier H \<rightarrow> carrier G") 
nipkow@33057
   588
 prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_inv_into]) 
nipkow@33057
   589
apply (simp add: hom_def bij_betw_def inv_into_f_eq f_inv_into_f Pi_def)
paulson@14761
   590
done
paulson@14761
   591
paulson@14761
   592
lemma (in group) iso_trans: 
paulson@14803
   593
     "[|h \<in> G \<cong> H; i \<in> H \<cong> I|] ==> (compose (carrier G) i h) \<in> G \<cong> I"
paulson@14761
   594
by (auto simp add: iso_def hom_compose bij_betw_compose)
paulson@14761
   595
paulson@14963
   596
lemma DirProd_commute_iso:
paulson@14963
   597
  shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)"
nipkow@31754
   598
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)
paulson@14761
   599
paulson@14963
   600
lemma DirProd_assoc_iso:
paulson@14963
   601
  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
   602
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)
paulson@14761
   603
paulson@14761
   604
wenzelm@61382
   605
text\<open>Basis for homomorphism proofs: we assume two groups @{term G} and
wenzelm@61382
   606
  @{term H}, with a homomorphism @{term h} between them\<close>
ballarin@61565
   607
locale group_hom = G?: group G + H?: group H for G (structure) and H (structure) +
ballarin@29237
   608
  fixes h
ballarin@13813
   609
  assumes homh: "h \<in> hom G H"
ballarin@29240
   610
ballarin@29240
   611
lemma (in group_hom) hom_mult [simp]:
ballarin@29240
   612
  "[| 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
   613
proof -
ballarin@29240
   614
  assume "x \<in> carrier G" "y \<in> carrier G"
ballarin@29240
   615
  with homh [unfolded hom_def] show ?thesis by simp
ballarin@29240
   616
qed
ballarin@29240
   617
ballarin@29240
   618
lemma (in group_hom) hom_closed [simp]:
ballarin@29240
   619
  "x \<in> carrier G ==> h x \<in> carrier H"
ballarin@29240
   620
proof -
ballarin@29240
   621
  assume "x \<in> carrier G"
nipkow@31754
   622
  with homh [unfolded hom_def] show ?thesis by auto
ballarin@29240
   623
qed
ballarin@13813
   624
ballarin@13813
   625
lemma (in group_hom) one_closed [simp]:
ballarin@13813
   626
  "h \<one> \<in> carrier H"
ballarin@13813
   627
  by simp
ballarin@13813
   628
ballarin@13813
   629
lemma (in group_hom) hom_one [simp]:
wenzelm@14693
   630
  "h \<one> = \<one>\<^bsub>H\<^esub>"
ballarin@13813
   631
proof -
ballarin@15076
   632
  have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>"
ballarin@13813
   633
    by (simp add: hom_mult [symmetric] del: hom_mult)
ballarin@13813
   634
  then show ?thesis by (simp del: r_one)
ballarin@13813
   635
qed
ballarin@13813
   636
ballarin@13813
   637
lemma (in group_hom) inv_closed [simp]:
ballarin@13813
   638
  "x \<in> carrier G ==> h (inv x) \<in> carrier H"
ballarin@13813
   639
  by simp
ballarin@13813
   640
ballarin@13813
   641
lemma (in group_hom) hom_inv [simp]:
wenzelm@14693
   642
  "x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"
ballarin@13813
   643
proof -
ballarin@13813
   644
  assume x: "x \<in> carrier G"
wenzelm@14693
   645
  then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>"
paulson@14963
   646
    by (simp add: hom_mult [symmetric] del: hom_mult)
wenzelm@14693
   647
  also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)"
paulson@14963
   648
    by (simp add: hom_mult [symmetric] del: hom_mult)
wenzelm@14693
   649
  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
   650
  with x show ?thesis by (simp del: H.r_inv H.Units_r_inv)
ballarin@13813
   651
qed
ballarin@13813
   652
ballarin@57271
   653
(* Contributed by Joachim Breitner *)
ballarin@57271
   654
lemma (in group) int_pow_is_hom:
nipkow@67399
   655
  "x \<in> carrier G \<Longrightarrow> (([^]) x) \<in> hom \<lparr> carrier = UNIV, mult = (+), one = 0::int \<rparr> G "
ballarin@57271
   656
  unfolding hom_def by (simp add: int_pow_mult)
ballarin@57271
   657
ballarin@20318
   658
wenzelm@61382
   659
subsection \<open>Commutative Structures\<close>
ballarin@13936
   660
wenzelm@61382
   661
text \<open>
ballarin@13936
   662
  Naming convention: multiplicative structures that are commutative
ballarin@13936
   663
  are called \emph{commutative}, additive structures are called
ballarin@13936
   664
  \emph{Abelian}.
wenzelm@61382
   665
\<close>
ballarin@13813
   666
paulson@14963
   667
locale comm_monoid = monoid +
paulson@14963
   668
  assumes m_comm: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x"
ballarin@13813
   669
paulson@14963
   670
lemma (in comm_monoid) m_lcomm:
paulson@14963
   671
  "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
ballarin@13813
   672
   x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
ballarin@13813
   673
proof -
wenzelm@14693
   674
  assume xyz: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
ballarin@13813
   675
  from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)
ballarin@13813
   676
  also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)
ballarin@13813
   677
  also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)
ballarin@13813
   678
  finally show ?thesis .
ballarin@13813
   679
qed
ballarin@13813
   680
paulson@14963
   681
lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm
ballarin@13813
   682
ballarin@13936
   683
lemma comm_monoidI:
ballarin@19783
   684
  fixes G (structure)
ballarin@13936
   685
  assumes m_closed:
wenzelm@14693
   686
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
wenzelm@14693
   687
    and one_closed: "\<one> \<in> carrier G"
ballarin@13936
   688
    and m_assoc:
ballarin@13936
   689
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
wenzelm@14693
   690
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
wenzelm@14693
   691
    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
ballarin@13936
   692
    and m_comm:
wenzelm@14693
   693
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
ballarin@13936
   694
  shows "comm_monoid G"
ballarin@13936
   695
  using l_one
paulson@14963
   696
    by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro 
ballarin@27714
   697
             intro: assms simp: m_closed one_closed m_comm)
ballarin@13817
   698
ballarin@13936
   699
lemma (in monoid) monoid_comm_monoidI:
ballarin@13936
   700
  assumes m_comm:
wenzelm@14693
   701
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
ballarin@13936
   702
  shows "comm_monoid G"
ballarin@13936
   703
  by (rule comm_monoidI) (auto intro: m_assoc m_comm)
paulson@14963
   704
wenzelm@14693
   705
(*lemma (in comm_monoid) r_one [simp]:
ballarin@13817
   706
  "x \<in> carrier G ==> x \<otimes> \<one> = x"
ballarin@13817
   707
proof -
ballarin@13817
   708
  assume G: "x \<in> carrier G"
ballarin@13817
   709
  then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)
ballarin@13817
   710
  also from G have "... = x" by simp
ballarin@13817
   711
  finally show ?thesis .
wenzelm@14693
   712
qed*)
paulson@14963
   713
ballarin@13936
   714
lemma (in comm_monoid) nat_pow_distr:
ballarin@13936
   715
  "[| x \<in> carrier G; y \<in> carrier G |] ==>
nipkow@67341
   716
  (x \<otimes> y) [^] (n::nat) = x [^] n \<otimes> y [^] n"
ballarin@13936
   717
  by (induct n) (simp, simp add: m_ac)
ballarin@13936
   718
ballarin@13936
   719
locale comm_group = comm_monoid + group
ballarin@13936
   720
ballarin@13936
   721
lemma (in group) group_comm_groupI:
ballarin@13936
   722
  assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>
wenzelm@14693
   723
      x \<otimes> y = y \<otimes> x"
ballarin@13936
   724
  shows "comm_group G"
wenzelm@61169
   725
  by standard (simp_all add: m_comm)
ballarin@13817
   726
ballarin@13936
   727
lemma comm_groupI:
ballarin@19783
   728
  fixes G (structure)
ballarin@13936
   729
  assumes m_closed:
wenzelm@14693
   730
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
wenzelm@14693
   731
    and one_closed: "\<one> \<in> carrier G"
ballarin@13936
   732
    and m_assoc:
ballarin@13936
   733
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
wenzelm@14693
   734
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
ballarin@13936
   735
    and m_comm:
wenzelm@14693
   736
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
wenzelm@14693
   737
    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
paulson@14963
   738
    and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
ballarin@13936
   739
  shows "comm_group G"
ballarin@27714
   740
  by (fast intro: group.group_comm_groupI groupI assms)
ballarin@13936
   741
ballarin@13936
   742
lemma (in comm_group) inv_mult:
ballarin@13854
   743
  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"
ballarin@13936
   744
  by (simp add: m_ac inv_mult_group)
ballarin@13854
   745
ballarin@20318
   746
wenzelm@61382
   747
subsection \<open>The Lattice of Subgroups of a Group\<close>
ballarin@14751
   748
wenzelm@61382
   749
text_raw \<open>\label{sec:subgroup-lattice}\<close>
ballarin@14751
   750
ballarin@14751
   751
theorem (in group) subgroups_partial_order:
nipkow@67399
   752
  "partial_order \<lparr>carrier = {H. subgroup H G}, eq = (=), le = (\<subseteq>)\<rparr>"
wenzelm@61169
   753
  by standard simp_all
ballarin@14751
   754
ballarin@14751
   755
lemma (in group) subgroup_self:
ballarin@14751
   756
  "subgroup (carrier G) G"
ballarin@14751
   757
  by (rule subgroupI) auto
ballarin@14751
   758
ballarin@14751
   759
lemma (in group) subgroup_imp_group:
wenzelm@55926
   760
  "subgroup H G ==> group (G\<lparr>carrier := H\<rparr>)"
wenzelm@26199
   761
  by (erule subgroup.subgroup_is_group) (rule group_axioms)
ballarin@14751
   762
ballarin@14751
   763
lemma (in group) is_monoid [intro, simp]:
ballarin@14751
   764
  "monoid G"
paulson@14963
   765
  by (auto intro: monoid.intro m_assoc) 
ballarin@14751
   766
ballarin@14751
   767
lemma (in group) subgroup_inv_equality:
wenzelm@55926
   768
  "[| subgroup H G; x \<in> H |] ==> m_inv (G \<lparr>carrier := H\<rparr>) x = inv x"
ballarin@14751
   769
apply (rule_tac inv_equality [THEN sym])
paulson@14761
   770
  apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)
paulson@14761
   771
 apply (rule subsetD [OF subgroup.subset], assumption+)
paulson@14761
   772
apply (rule subsetD [OF subgroup.subset], assumption)
paulson@14761
   773
apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+)
ballarin@14751
   774
done
ballarin@14751
   775
ballarin@14751
   776
theorem (in group) subgroups_Inter:
wenzelm@67091
   777
  assumes subgr: "(\<And>H. H \<in> A \<Longrightarrow> subgroup H G)"
wenzelm@67091
   778
    and not_empty: "A \<noteq> {}"
ballarin@14751
   779
  shows "subgroup (\<Inter>A) G"
ballarin@14751
   780
proof (rule subgroupI)
ballarin@14751
   781
  from subgr [THEN subgroup.subset] and not_empty
ballarin@14751
   782
  show "\<Inter>A \<subseteq> carrier G" by blast
ballarin@14751
   783
next
ballarin@14751
   784
  from subgr [THEN subgroup.one_closed]
wenzelm@67091
   785
  show "\<Inter>A \<noteq> {}" by blast
ballarin@14751
   786
next
ballarin@14751
   787
  fix x assume "x \<in> \<Inter>A"
ballarin@14751
   788
  with subgr [THEN subgroup.m_inv_closed]
ballarin@14751
   789
  show "inv x \<in> \<Inter>A" by blast
ballarin@14751
   790
next
ballarin@14751
   791
  fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"
ballarin@14751
   792
  with subgr [THEN subgroup.m_closed]
ballarin@14751
   793
  show "x \<otimes> y \<in> \<Inter>A" by blast
ballarin@14751
   794
qed
ballarin@14751
   795
ballarin@66579
   796
theorem (in group) subgroups_complete_lattice:
nipkow@67399
   797
  "complete_lattice \<lparr>carrier = {H. subgroup H G}, eq = (=), le = (\<subseteq>)\<rparr>"
ballarin@66579
   798
    (is "complete_lattice ?L")
ballarin@66579
   799
proof (rule partial_order.complete_lattice_criterion1)
ballarin@66579
   800
  show "partial_order ?L" by (rule subgroups_partial_order)
ballarin@66579
   801
next
ballarin@66579
   802
  have "greatest ?L (carrier G) (carrier ?L)"
ballarin@66579
   803
    by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)
ballarin@66579
   804
  then show "\<exists>G. greatest ?L G (carrier ?L)" ..
ballarin@66579
   805
next
ballarin@66579
   806
  fix A
wenzelm@67091
   807
  assume L: "A \<subseteq> carrier ?L" and non_empty: "A \<noteq> {}"
ballarin@66579
   808
  then have Int_subgroup: "subgroup (\<Inter>A) G"
ballarin@66579
   809
    by (fastforce intro: subgroups_Inter)
ballarin@66579
   810
  have "greatest ?L (\<Inter>A) (Lower ?L A)" (is "greatest _ ?Int _")
ballarin@66579
   811
  proof (rule greatest_LowerI)
ballarin@66579
   812
    fix H
ballarin@66579
   813
    assume H: "H \<in> A"
ballarin@66579
   814
    with L have subgroupH: "subgroup H G" by auto
ballarin@66579
   815
    from subgroupH have groupH: "group (G \<lparr>carrier := H\<rparr>)" (is "group ?H")
ballarin@66579
   816
      by (rule subgroup_imp_group)
ballarin@66579
   817
    from groupH have monoidH: "monoid ?H"
ballarin@66579
   818
      by (rule group.is_monoid)
ballarin@66579
   819
    from H have Int_subset: "?Int \<subseteq> H" by fastforce
ballarin@66579
   820
    then show "le ?L ?Int H" by simp
ballarin@66579
   821
  next
ballarin@66579
   822
    fix H
ballarin@66579
   823
    assume H: "H \<in> Lower ?L A"
ballarin@66579
   824
    with L Int_subgroup show "le ?L H ?Int"
ballarin@66579
   825
      by (fastforce simp: Lower_def intro: Inter_greatest)
ballarin@66579
   826
  next
ballarin@66579
   827
    show "A \<subseteq> carrier ?L" by (rule L)
ballarin@66579
   828
  next
ballarin@66579
   829
    show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
ballarin@66579
   830
  qed
ballarin@66579
   831
  then show "\<exists>I. greatest ?L I (Lower ?L A)" ..
ballarin@66579
   832
qed
ballarin@66579
   833
ballarin@13813
   834
end