src/HOL/Algebra/Group.thy
author paulson
Wed May 19 11:30:18 2004 +0200 (2004-05-19)
changeset 14761 28b5eb4a867f
parent 14751 0d7850e27fed
child 14803 f7557773cc87
permissions -rw-r--r--
more results about isomorphisms
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(*
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  Title:  HOL/Algebra/Group.thy
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  Id:     $Id$
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  Author: Clemens Ballarin, started 4 February 2003
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Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.
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*)
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header {* Groups *}
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theory Group = FuncSet + Lattice:
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section {* From Magmas to Groups *}
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text {*
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  Definitions follow \cite{Jacobson:1985}; with the exception of
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  \emph{magma} which, following Bourbaki, is a set together with a
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  binary, closed operation.
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*}
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subsection {* Definitions *}
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record 'a semigroup = "'a partial_object" +
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  mult :: "['a, 'a] => 'a" (infixl "\<otimes>\<index>" 70)
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record 'a monoid = "'a semigroup" +
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  one :: 'a ("\<one>\<index>")
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constdefs (structure G)
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  m_inv :: "_ => 'a => 'a" ("inv\<index> _" [81] 80)
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  "inv x == (THE y. y \<in> carrier G & x \<otimes> y = \<one> & y \<otimes> x = \<one>)"
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  Units :: "_ => 'a set"
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  "Units G == {y. y \<in> carrier G & (EX x : carrier G. x \<otimes> y = \<one> & y \<otimes> x = \<one>)}"
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consts
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  pow :: "[('a, 'm) monoid_scheme, 'a, 'b::number] => 'a" (infixr "'(^')\<index>" 75)
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defs (overloaded)
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  nat_pow_def: "pow G a n == nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n"
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  int_pow_def: "pow G a z ==
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    let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a)
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    in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z)"
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locale magma = struct G +
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  assumes m_closed [intro, simp]:
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    "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
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locale semigroup = magma +
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  assumes m_assoc:
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    "[| 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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locale monoid = semigroup +
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  assumes one_closed [intro, simp]: "\<one> \<in> carrier G"
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    and l_one [simp]: "x \<in> carrier G ==> \<one> \<otimes> x = x"
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    and r_one [simp]: "x \<in> carrier G ==> x \<otimes> \<one> = x"
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lemma monoidI:
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  includes struct G
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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 magma.intro semigroup_axioms.intro
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    semigroup.intro monoid_axioms.intro
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    intro: prems)
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lemma (in monoid) Units_closed [dest]:
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  "x \<in> Units G ==> x \<in> carrier G"
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  by (unfold Units_def) fast
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lemma (in monoid) inv_unique:
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  assumes eq: "y \<otimes> x = \<one>"  "x \<otimes> y' = \<one>"
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    and G: "x \<in> carrier G"  "y \<in> carrier G"  "y' \<in> carrier G"
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  shows "y = y'"
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proof -
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  from G eq have "y = y \<otimes> (x \<otimes> y')" by simp
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  also from G have "... = (y \<otimes> x) \<otimes> y'" by (simp add: m_assoc)
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  also from G eq have "... = y'" by simp
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  finally show ?thesis .
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qed
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lemma (in monoid) Units_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:
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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:
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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)
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  with G show "y = z" by (simp add: Units_l_inv)
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next
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  assume eq: "y = z"
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    and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"
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  then show "x \<otimes> y = x \<otimes> z" by simp
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qed
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lemma (in monoid) Units_inv_inv [simp]:
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  "x \<in> Units G ==> inv (inv x) = x"
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proof -
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  assume x: "x \<in> Units G"
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  then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x"
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    by (simp add: Units_l_inv Units_r_inv)
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  with x show ?thesis by (simp add: Units_closed)
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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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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"
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  by (rule group.intro [OF prems]) 
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theorem groupI:
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  includes struct G
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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 ==> EX y : 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 ==> EX y : 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 "EX y : 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
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    by (fast intro!: group.intro magma.intro semigroup_axioms.intro
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      semigroup.intro monoid_axioms.intro group_axioms.intro
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      carrier_subset_Units intro: prems r_one)
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qed
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lemma (in monoid) monoid_groupI:
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  assumes l_inv_ex:
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    "!!x. x \<in> carrier G ==> EX y : carrier G. y \<otimes> x = \<one>"
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  shows "group G"
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  by (rule groupI) (auto intro: m_assoc l_inv_ex)
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lemma (in group) Units_eq [simp]:
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  "Units G = carrier G"
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proof
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  show "Units G <= carrier G" by fast
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next
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  show "carrier G <= Units G" by (rule Units)
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qed
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lemma (in group) inv_closed [intro, simp]:
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  "x \<in> carrier G ==> inv x \<in> carrier G"
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  using Units_inv_closed by simp
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lemma (in group) l_inv:
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  "x \<in> carrier G ==> inv x \<otimes> x = \<one>"
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  using Units_l_inv by simp
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subsection {* Cancellation Laws and Basic Properties *}
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lemma (in group) l_cancel [simp]:
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  "[| 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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  using Units_l_inv by simp
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lemma (in group) r_inv:
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  "x \<in> carrier G ==> x \<otimes> inv x = \<one>"
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proof -
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  assume x: "x \<in> carrier G"
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  then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>"
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    by (simp add: m_assoc [symmetric] l_inv)
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  with x show ?thesis by (simp del: r_one)
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qed
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lemma (in group) r_cancel [simp]:
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  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
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   (y \<otimes> x = z \<otimes> x) = (y = z)"
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proof
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  assume eq: "y \<otimes> x = z \<otimes> x"
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    and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
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  then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)"
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    by (simp add: m_assoc [symmetric])
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  with G show "y = z" by (simp add: r_inv)
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next
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  assume eq: "y = z"
wenzelm@14693
   315
    and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
ballarin@13813
   316
  then show "y \<otimes> x = z \<otimes> x" by simp
ballarin@13813
   317
qed
ballarin@13813
   318
ballarin@13854
   319
lemma (in group) inv_one [simp]:
ballarin@13854
   320
  "inv \<one> = \<one>"
ballarin@13854
   321
proof -
ballarin@13854
   322
  have "inv \<one> = \<one> \<otimes> (inv \<one>)" by simp
ballarin@13854
   323
  moreover have "... = \<one>" by (simp add: r_inv)
ballarin@13854
   324
  finally show ?thesis .
ballarin@13854
   325
qed
ballarin@13854
   326
ballarin@13813
   327
lemma (in group) inv_inv [simp]:
ballarin@13813
   328
  "x \<in> carrier G ==> inv (inv x) = x"
ballarin@13936
   329
  using Units_inv_inv by simp
ballarin@13936
   330
ballarin@13936
   331
lemma (in group) inv_inj:
ballarin@13936
   332
  "inj_on (m_inv G) (carrier G)"
ballarin@13936
   333
  using inv_inj_on_Units by simp
ballarin@13813
   334
ballarin@13854
   335
lemma (in group) inv_mult_group:
ballarin@13813
   336
  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"
ballarin@13813
   337
proof -
wenzelm@14693
   338
  assume G: "x \<in> carrier G"  "y \<in> carrier G"
ballarin@13813
   339
  then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)"
ballarin@13813
   340
    by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric] l_inv)
ballarin@13813
   341
  with G show ?thesis by simp
ballarin@13813
   342
qed
ballarin@13813
   343
ballarin@13940
   344
lemma (in group) inv_comm:
ballarin@13940
   345
  "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
wenzelm@14693
   346
  by (rule Units_inv_comm) auto
ballarin@13940
   347
paulson@13944
   348
lemma (in group) inv_equality:
paulson@13943
   349
     "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"
paulson@13943
   350
apply (simp add: m_inv_def)
paulson@13943
   351
apply (rule the_equality)
wenzelm@14693
   352
 apply (simp add: inv_comm [of y x])
wenzelm@14693
   353
apply (rule r_cancel [THEN iffD1], auto)
paulson@13943
   354
done
paulson@13943
   355
ballarin@13936
   356
text {* Power *}
ballarin@13936
   357
ballarin@13936
   358
lemma (in group) int_pow_def2:
ballarin@13936
   359
  "a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))"
ballarin@13936
   360
  by (simp add: int_pow_def nat_pow_def Let_def)
ballarin@13936
   361
ballarin@13936
   362
lemma (in group) int_pow_0 [simp]:
ballarin@13936
   363
  "x (^) (0::int) = \<one>"
ballarin@13936
   364
  by (simp add: int_pow_def2)
ballarin@13936
   365
ballarin@13936
   366
lemma (in group) int_pow_one [simp]:
ballarin@13936
   367
  "\<one> (^) (z::int) = \<one>"
ballarin@13936
   368
  by (simp add: int_pow_def2)
ballarin@13936
   369
ballarin@13813
   370
subsection {* Substructures *}
ballarin@13813
   371
ballarin@13813
   372
locale submagma = var H + struct G +
ballarin@13813
   373
  assumes subset [intro, simp]: "H \<subseteq> carrier G"
ballarin@13813
   374
    and m_closed [intro, simp]: "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"
ballarin@13813
   375
ballarin@13813
   376
declare (in submagma) magma.intro [intro] semigroup.intro [intro]
ballarin@13936
   377
  semigroup_axioms.intro [intro]
ballarin@13813
   378
ballarin@13813
   379
lemma submagma_imp_subset:
ballarin@13813
   380
  "submagma H G ==> H \<subseteq> carrier G"
ballarin@13813
   381
  by (rule submagma.subset)
ballarin@13813
   382
ballarin@13813
   383
lemma (in submagma) subsetD [dest, simp]:
ballarin@13813
   384
  "x \<in> H ==> x \<in> carrier G"
ballarin@13813
   385
  using subset by blast
ballarin@13813
   386
ballarin@13813
   387
lemma (in submagma) magmaI [intro]:
ballarin@13813
   388
  includes magma G
ballarin@13813
   389
  shows "magma (G(| carrier := H |))"
ballarin@13813
   390
  by rule simp
ballarin@13813
   391
ballarin@13813
   392
lemma (in submagma) semigroup_axiomsI [intro]:
ballarin@13813
   393
  includes semigroup G
ballarin@13813
   394
  shows "semigroup_axioms (G(| carrier := H |))"
ballarin@13813
   395
    by rule (simp add: m_assoc)
ballarin@13813
   396
ballarin@13813
   397
lemma (in submagma) semigroupI [intro]:
ballarin@13813
   398
  includes semigroup G
ballarin@13813
   399
  shows "semigroup (G(| carrier := H |))"
ballarin@13813
   400
  using prems by fast
ballarin@13813
   401
ballarin@14551
   402
ballarin@13813
   403
locale subgroup = submagma H G +
ballarin@13813
   404
  assumes one_closed [intro, simp]: "\<one> \<in> H"
ballarin@13813
   405
    and m_inv_closed [intro, simp]: "x \<in> H ==> inv x \<in> H"
ballarin@13813
   406
ballarin@13813
   407
declare (in subgroup) group.intro [intro]
ballarin@13949
   408
ballarin@13813
   409
lemma (in subgroup) group_axiomsI [intro]:
ballarin@13813
   410
  includes group G
ballarin@13813
   411
  shows "group_axioms (G(| carrier := H |))"
ballarin@14254
   412
  by (rule group_axioms.intro) (auto intro: l_inv r_inv simp add: Units_def)
ballarin@13813
   413
ballarin@13813
   414
lemma (in subgroup) groupI [intro]:
ballarin@13813
   415
  includes group G
ballarin@13813
   416
  shows "group (G(| carrier := H |))"
ballarin@13936
   417
  by (rule groupI) (auto intro: m_assoc l_inv)
ballarin@13813
   418
ballarin@13813
   419
text {*
ballarin@13813
   420
  Since @{term H} is nonempty, it contains some element @{term x}.  Since
ballarin@13813
   421
  it is closed under inverse, it contains @{text "inv x"}.  Since
ballarin@13813
   422
  it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}.
ballarin@13813
   423
*}
ballarin@13813
   424
ballarin@13813
   425
lemma (in group) one_in_subset:
ballarin@13813
   426
  "[| 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
   427
   ==> \<one> \<in> H"
ballarin@13813
   428
by (force simp add: l_inv)
ballarin@13813
   429
ballarin@13813
   430
text {* A characterization of subgroups: closed, non-empty subset. *}
ballarin@13813
   431
ballarin@13813
   432
lemma (in group) subgroupI:
ballarin@13813
   433
  assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"
ballarin@13813
   434
    and inv: "!!a. a \<in> H ==> inv a \<in> H"
ballarin@13813
   435
    and mult: "!!a b. [|a \<in> H; b \<in> H|] ==> a \<otimes> b \<in> H"
ballarin@13813
   436
  shows "subgroup H G"
ballarin@14254
   437
proof (rule subgroup.intro)
ballarin@14254
   438
  from subset and mult show "submagma H G" by (rule submagma.intro)
ballarin@13813
   439
next
ballarin@13813
   440
  have "\<one> \<in> H" by (rule one_in_subset) (auto simp only: prems)
ballarin@13813
   441
  with inv show "subgroup_axioms H G"
ballarin@13813
   442
    by (intro subgroup_axioms.intro) simp_all
ballarin@13813
   443
qed
ballarin@13813
   444
ballarin@13813
   445
text {*
ballarin@13813
   446
  Repeat facts of submagmas for subgroups.  Necessary???
ballarin@13813
   447
*}
ballarin@13813
   448
ballarin@13813
   449
lemma (in subgroup) subset:
ballarin@13813
   450
  "H \<subseteq> carrier G"
ballarin@13813
   451
  ..
ballarin@13813
   452
ballarin@13813
   453
lemma (in subgroup) m_closed:
ballarin@13813
   454
  "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"
ballarin@13813
   455
  ..
ballarin@13813
   456
ballarin@13813
   457
declare magma.m_closed [simp]
ballarin@13813
   458
ballarin@13936
   459
declare monoid.one_closed [iff] group.inv_closed [simp]
ballarin@13936
   460
  monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]
ballarin@13813
   461
ballarin@13813
   462
lemma subgroup_nonempty:
ballarin@13813
   463
  "~ subgroup {} G"
ballarin@13813
   464
  by (blast dest: subgroup.one_closed)
ballarin@13813
   465
ballarin@13813
   466
lemma (in subgroup) finite_imp_card_positive:
ballarin@13813
   467
  "finite (carrier G) ==> 0 < card H"
ballarin@13813
   468
proof (rule classical)
ballarin@14254
   469
  have sub: "subgroup H G" using prems by (rule subgroup.intro)
ballarin@13813
   470
  assume fin: "finite (carrier G)"
ballarin@13813
   471
    and zero: "~ 0 < card H"
ballarin@13813
   472
  then have "finite H" by (blast intro: finite_subset dest: subset)
ballarin@13813
   473
  with zero sub have "subgroup {} G" by simp
ballarin@13813
   474
  with subgroup_nonempty show ?thesis by contradiction
ballarin@13813
   475
qed
ballarin@13813
   476
ballarin@13936
   477
(*
ballarin@13936
   478
lemma (in monoid) Units_subgroup:
ballarin@13936
   479
  "subgroup (Units G) G"
ballarin@13936
   480
*)
ballarin@13936
   481
ballarin@13813
   482
subsection {* Direct Products *}
ballarin@13813
   483
wenzelm@14651
   484
constdefs (structure G and H)
wenzelm@14651
   485
  DirProdSemigroup :: "_ => _ => ('a \<times> 'b) semigroup"  (infixr "\<times>\<^sub>s" 80)
ballarin@13817
   486
  "G \<times>\<^sub>s H == (| carrier = carrier G \<times> carrier H,
wenzelm@14693
   487
    mult = (%(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')) |)"
ballarin@13817
   488
wenzelm@14651
   489
  DirProdGroup :: "_ => _ => ('a \<times> 'b) monoid"  (infixr "\<times>\<^sub>g" 80)
wenzelm@14693
   490
  "G \<times>\<^sub>g H == semigroup.extend (G \<times>\<^sub>s H) (| one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>) |)"
ballarin@13813
   491
ballarin@13817
   492
lemma DirProdSemigroup_magma:
ballarin@13813
   493
  includes magma G + magma H
ballarin@13817
   494
  shows "magma (G \<times>\<^sub>s H)"
ballarin@14254
   495
  by (rule magma.intro) (auto simp add: DirProdSemigroup_def)
ballarin@13813
   496
ballarin@13817
   497
lemma DirProdSemigroup_semigroup_axioms:
ballarin@13813
   498
  includes semigroup G + semigroup H
ballarin@13817
   499
  shows "semigroup_axioms (G \<times>\<^sub>s H)"
ballarin@14254
   500
  by (rule semigroup_axioms.intro)
ballarin@14254
   501
    (auto simp add: DirProdSemigroup_def G.m_assoc H.m_assoc)
ballarin@13813
   502
ballarin@13817
   503
lemma DirProdSemigroup_semigroup:
ballarin@13813
   504
  includes semigroup G + semigroup H
ballarin@13817
   505
  shows "semigroup (G \<times>\<^sub>s H)"
ballarin@13813
   506
  using prems
ballarin@13813
   507
  by (fast intro: semigroup.intro
ballarin@13817
   508
    DirProdSemigroup_magma DirProdSemigroup_semigroup_axioms)
ballarin@13813
   509
ballarin@13813
   510
lemma DirProdGroup_magma:
ballarin@13813
   511
  includes magma G + magma H
ballarin@13813
   512
  shows "magma (G \<times>\<^sub>g H)"
ballarin@14254
   513
  by (rule magma.intro)
wenzelm@14651
   514
    (auto simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
ballarin@13813
   515
ballarin@13813
   516
lemma DirProdGroup_semigroup_axioms:
ballarin@13813
   517
  includes semigroup G + semigroup H
ballarin@13813
   518
  shows "semigroup_axioms (G \<times>\<^sub>g H)"
ballarin@14254
   519
  by (rule semigroup_axioms.intro)
wenzelm@14651
   520
    (auto simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs
ballarin@13817
   521
      G.m_assoc H.m_assoc)
ballarin@13813
   522
ballarin@13813
   523
lemma DirProdGroup_semigroup:
ballarin@13813
   524
  includes semigroup G + semigroup H
ballarin@13813
   525
  shows "semigroup (G \<times>\<^sub>g H)"
ballarin@13813
   526
  using prems
ballarin@13813
   527
  by (fast intro: semigroup.intro
ballarin@13813
   528
    DirProdGroup_magma DirProdGroup_semigroup_axioms)
ballarin@13813
   529
wenzelm@14651
   530
text {* \dots\ and further lemmas for group \dots *}
ballarin@13813
   531
ballarin@13817
   532
lemma DirProdGroup_group:
ballarin@13813
   533
  includes group G + group H
ballarin@13813
   534
  shows "group (G \<times>\<^sub>g H)"
ballarin@13936
   535
  by (rule groupI)
ballarin@13936
   536
    (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
wenzelm@14651
   537
      simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
ballarin@13813
   538
paulson@13944
   539
lemma carrier_DirProdGroup [simp]:
paulson@13944
   540
     "carrier (G \<times>\<^sub>g H) = carrier G \<times> carrier H"
wenzelm@14651
   541
  by (simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
paulson@13944
   542
paulson@13944
   543
lemma one_DirProdGroup [simp]:
wenzelm@14693
   544
     "\<one>\<^bsub>(G \<times>\<^sub>g H)\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)"
wenzelm@14651
   545
  by (simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
paulson@13944
   546
paulson@13944
   547
lemma mult_DirProdGroup [simp]:
wenzelm@14693
   548
     "(g, h) \<otimes>\<^bsub>(G \<times>\<^sub>g H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')"
wenzelm@14651
   549
  by (simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
paulson@13944
   550
paulson@13944
   551
lemma inv_DirProdGroup [simp]:
paulson@13944
   552
  includes group G + group H
paulson@13944
   553
  assumes g: "g \<in> carrier G"
paulson@13944
   554
      and h: "h \<in> carrier H"
wenzelm@14693
   555
  shows "m_inv (G \<times>\<^sub>g H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"
paulson@13944
   556
  apply (rule group.inv_equality [OF DirProdGroup_group])
paulson@13944
   557
  apply (simp_all add: prems group_def group.l_inv)
paulson@13944
   558
  done
paulson@13944
   559
paulson@14761
   560
subsection {* Isomorphisms *}
ballarin@13813
   561
wenzelm@14651
   562
constdefs (structure G and H)
wenzelm@14651
   563
  hom :: "_ => _ => ('a => 'b) set"
ballarin@13813
   564
  "hom G H ==
ballarin@13813
   565
    {h. h \<in> carrier G -> carrier H &
wenzelm@14693
   566
      (\<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
   567
ballarin@13813
   568
lemma (in semigroup) hom:
paulson@14761
   569
     "semigroup (| carrier = hom G G, mult = op o |)"
ballarin@14254
   570
proof (rule semigroup.intro)
ballarin@13813
   571
  show "magma (| carrier = hom G G, mult = op o |)"
ballarin@14254
   572
    by (rule magma.intro) (simp add: Pi_def hom_def)
ballarin@13813
   573
next
ballarin@13813
   574
  show "semigroup_axioms (| carrier = hom G G, mult = op o |)"
ballarin@14254
   575
    by (rule semigroup_axioms.intro) (simp add: o_assoc)
ballarin@13813
   576
qed
ballarin@13813
   577
ballarin@13813
   578
lemma hom_mult:
wenzelm@14693
   579
  "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |]
wenzelm@14693
   580
   ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
wenzelm@14693
   581
  by (simp add: hom_def)
ballarin@13813
   582
ballarin@13813
   583
lemma hom_closed:
ballarin@13813
   584
  "[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"
ballarin@13813
   585
  by (auto simp add: hom_def funcset_mem)
ballarin@13813
   586
paulson@14761
   587
lemma (in group) hom_compose:
paulson@14761
   588
     "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"
paulson@14761
   589
apply (auto simp add: hom_def funcset_compose) 
paulson@14761
   590
apply (simp add: compose_def funcset_mem)
paulson@13943
   591
done
paulson@13943
   592
paulson@14761
   593
paulson@14761
   594
subsection {* Isomorphisms *}
paulson@14761
   595
paulson@14761
   596
constdefs (structure G and H)
paulson@14761
   597
  iso :: "_ => _ => ('a => 'b) set"
paulson@14761
   598
  "iso G H == {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}"
paulson@14761
   599
paulson@14761
   600
lemma iso_refl: "(%x. x) \<in> iso G G"
paulson@14761
   601
by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
paulson@14761
   602
paulson@14761
   603
lemma (in group) iso_sym:
paulson@14761
   604
     "h \<in> iso G H \<Longrightarrow> Inv (carrier G) h \<in> iso H G"
paulson@14761
   605
apply (simp add: iso_def bij_betw_Inv) 
paulson@14761
   606
apply (subgoal_tac "Inv (carrier G) h \<in> carrier H \<rightarrow> carrier G") 
paulson@14761
   607
 prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_Inv]) 
paulson@14761
   608
apply (simp add: hom_def bij_betw_def Inv_f_eq funcset_mem f_Inv_f) 
paulson@14761
   609
done
paulson@14761
   610
paulson@14761
   611
lemma (in group) iso_trans: 
paulson@14761
   612
     "[|h \<in> iso G H; i \<in> iso H I|] ==> (compose (carrier G) i h) \<in> iso G I"
paulson@14761
   613
by (auto simp add: iso_def hom_compose bij_betw_compose)
paulson@14761
   614
paulson@14761
   615
lemma DirProdGroup_commute_iso:
paulson@14761
   616
  shows "(%(x,y). (y,x)) \<in> iso (G \<times>\<^sub>g H) (H \<times>\<^sub>g G)"
paulson@14761
   617
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
paulson@14761
   618
paulson@14761
   619
lemma DirProdGroup_assoc_iso:
paulson@14761
   620
  shows "(%(x,y,z). (x,(y,z))) \<in> iso (G \<times>\<^sub>g H \<times>\<^sub>g I) (G \<times>\<^sub>g (H \<times>\<^sub>g I))"
paulson@14761
   621
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
paulson@14761
   622
paulson@14761
   623
ballarin@13813
   624
locale group_hom = group G + group H + var h +
ballarin@13813
   625
  assumes homh: "h \<in> hom G H"
ballarin@13813
   626
  notes hom_mult [simp] = hom_mult [OF homh]
ballarin@13813
   627
    and hom_closed [simp] = hom_closed [OF homh]
ballarin@13813
   628
ballarin@13813
   629
lemma (in group_hom) one_closed [simp]:
ballarin@13813
   630
  "h \<one> \<in> carrier H"
ballarin@13813
   631
  by simp
ballarin@13813
   632
ballarin@13813
   633
lemma (in group_hom) hom_one [simp]:
wenzelm@14693
   634
  "h \<one> = \<one>\<^bsub>H\<^esub>"
ballarin@13813
   635
proof -
wenzelm@14693
   636
  have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^sub>2 h \<one>"
ballarin@13813
   637
    by (simp add: hom_mult [symmetric] del: hom_mult)
ballarin@13813
   638
  then show ?thesis by (simp del: r_one)
ballarin@13813
   639
qed
ballarin@13813
   640
ballarin@13813
   641
lemma (in group_hom) inv_closed [simp]:
ballarin@13813
   642
  "x \<in> carrier G ==> h (inv x) \<in> carrier H"
ballarin@13813
   643
  by simp
ballarin@13813
   644
ballarin@13813
   645
lemma (in group_hom) hom_inv [simp]:
wenzelm@14693
   646
  "x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"
ballarin@13813
   647
proof -
ballarin@13813
   648
  assume x: "x \<in> carrier G"
wenzelm@14693
   649
  then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>"
ballarin@13813
   650
    by (simp add: hom_mult [symmetric] G.r_inv del: hom_mult)
wenzelm@14693
   651
  also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)"
ballarin@13813
   652
    by (simp add: hom_mult [symmetric] H.r_inv del: hom_mult)
wenzelm@14693
   653
  finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" .
ballarin@13813
   654
  with x show ?thesis by simp
ballarin@13813
   655
qed
ballarin@13813
   656
ballarin@13949
   657
subsection {* Commutative Structures *}
ballarin@13936
   658
ballarin@13936
   659
text {*
ballarin@13936
   660
  Naming convention: multiplicative structures that are commutative
ballarin@13936
   661
  are called \emph{commutative}, additive structures are called
ballarin@13936
   662
  \emph{Abelian}.
ballarin@13936
   663
*}
ballarin@13813
   664
ballarin@13813
   665
subsection {* Definition *}
ballarin@13813
   666
ballarin@13936
   667
locale comm_semigroup = semigroup +
ballarin@13813
   668
  assumes m_comm: "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
ballarin@13813
   669
ballarin@13936
   670
lemma (in comm_semigroup) m_lcomm:
ballarin@13813
   671
  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
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
ballarin@13936
   681
lemmas (in comm_semigroup) m_ac = m_assoc m_comm m_lcomm
ballarin@13936
   682
ballarin@13936
   683
locale comm_monoid = comm_semigroup + monoid
ballarin@13813
   684
ballarin@13936
   685
lemma comm_monoidI:
wenzelm@14693
   686
  includes struct G
ballarin@13936
   687
  assumes m_closed:
wenzelm@14693
   688
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
wenzelm@14693
   689
    and one_closed: "\<one> \<in> carrier G"
ballarin@13936
   690
    and m_assoc:
ballarin@13936
   691
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
wenzelm@14693
   692
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
wenzelm@14693
   693
    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
ballarin@13936
   694
    and m_comm:
wenzelm@14693
   695
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
ballarin@13936
   696
  shows "comm_monoid G"
ballarin@13936
   697
  using l_one
ballarin@13936
   698
  by (auto intro!: comm_monoid.intro magma.intro semigroup_axioms.intro
ballarin@13936
   699
    comm_semigroup_axioms.intro monoid_axioms.intro
ballarin@13936
   700
    intro: prems simp: m_closed one_closed m_comm)
ballarin@13817
   701
ballarin@13936
   702
lemma (in monoid) monoid_comm_monoidI:
ballarin@13936
   703
  assumes m_comm:
wenzelm@14693
   704
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
ballarin@13936
   705
  shows "comm_monoid G"
ballarin@13936
   706
  by (rule comm_monoidI) (auto intro: m_assoc m_comm)
wenzelm@14693
   707
(*lemma (in comm_monoid) r_one [simp]:
ballarin@13817
   708
  "x \<in> carrier G ==> x \<otimes> \<one> = x"
ballarin@13817
   709
proof -
ballarin@13817
   710
  assume G: "x \<in> carrier G"
ballarin@13817
   711
  then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)
ballarin@13817
   712
  also from G have "... = x" by simp
ballarin@13817
   713
  finally show ?thesis .
wenzelm@14693
   714
qed*)
ballarin@13936
   715
lemma (in comm_monoid) nat_pow_distr:
ballarin@13936
   716
  "[| x \<in> carrier G; y \<in> carrier G |] ==>
ballarin@13936
   717
  (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"
ballarin@13936
   718
  by (induct n) (simp, simp add: m_ac)
ballarin@13936
   719
ballarin@13936
   720
locale comm_group = comm_monoid + group
ballarin@13936
   721
ballarin@13936
   722
lemma (in group) group_comm_groupI:
ballarin@13936
   723
  assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>
wenzelm@14693
   724
      x \<otimes> y = y \<otimes> x"
ballarin@13936
   725
  shows "comm_group G"
ballarin@13936
   726
  by (fast intro: comm_group.intro comm_semigroup_axioms.intro
paulson@14761
   727
                  is_group prems)
ballarin@13817
   728
ballarin@13936
   729
lemma comm_groupI:
wenzelm@14693
   730
  includes struct G
ballarin@13936
   731
  assumes m_closed:
wenzelm@14693
   732
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
wenzelm@14693
   733
    and one_closed: "\<one> \<in> carrier G"
ballarin@13936
   734
    and m_assoc:
ballarin@13936
   735
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
wenzelm@14693
   736
      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
ballarin@13936
   737
    and m_comm:
wenzelm@14693
   738
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
wenzelm@14693
   739
    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
wenzelm@14693
   740
    and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. y \<otimes> x = \<one>"
ballarin@13936
   741
  shows "comm_group G"
ballarin@13936
   742
  by (fast intro: group.group_comm_groupI groupI prems)
ballarin@13936
   743
ballarin@13936
   744
lemma (in comm_group) inv_mult:
ballarin@13854
   745
  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"
ballarin@13936
   746
  by (simp add: m_ac inv_mult_group)
ballarin@13854
   747
ballarin@14751
   748
subsection {* Lattice of subgroups of a group *}
ballarin@14751
   749
ballarin@14751
   750
text_raw {* \label{sec:subgroup-lattice} *}
ballarin@14751
   751
ballarin@14751
   752
theorem (in group) subgroups_partial_order:
ballarin@14751
   753
  "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
ballarin@14751
   754
  by (rule partial_order.intro) simp_all
ballarin@14751
   755
ballarin@14751
   756
lemma (in group) subgroup_self:
ballarin@14751
   757
  "subgroup (carrier G) G"
ballarin@14751
   758
  by (rule subgroupI) auto
ballarin@14751
   759
ballarin@14751
   760
lemma (in group) subgroup_imp_group:
ballarin@14751
   761
  "subgroup H G ==> group (G(| carrier := H |))"
ballarin@14751
   762
  using subgroup.groupI [OF _ group.intro] .
ballarin@14751
   763
ballarin@14751
   764
lemma (in group) is_monoid [intro, simp]:
ballarin@14751
   765
  "monoid G"
ballarin@14751
   766
  by (rule monoid.intro)
ballarin@14751
   767
ballarin@14751
   768
lemma (in group) subgroup_inv_equality:
ballarin@14751
   769
  "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"
ballarin@14751
   770
apply (rule_tac inv_equality [THEN sym])
paulson@14761
   771
  apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)
paulson@14761
   772
 apply (rule subsetD [OF subgroup.subset], assumption+)
paulson@14761
   773
apply (rule subsetD [OF subgroup.subset], assumption)
paulson@14761
   774
apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+)
ballarin@14751
   775
done
ballarin@14751
   776
ballarin@14751
   777
theorem (in group) subgroups_Inter:
ballarin@14751
   778
  assumes subgr: "(!!H. H \<in> A ==> subgroup H G)"
ballarin@14751
   779
    and not_empty: "A ~= {}"
ballarin@14751
   780
  shows "subgroup (\<Inter>A) G"
ballarin@14751
   781
proof (rule subgroupI)
ballarin@14751
   782
  from subgr [THEN subgroup.subset] and not_empty
ballarin@14751
   783
  show "\<Inter>A \<subseteq> carrier G" by blast
ballarin@14751
   784
next
ballarin@14751
   785
  from subgr [THEN subgroup.one_closed]
ballarin@14751
   786
  show "\<Inter>A ~= {}" by blast
ballarin@14751
   787
next
ballarin@14751
   788
  fix x assume "x \<in> \<Inter>A"
ballarin@14751
   789
  with subgr [THEN subgroup.m_inv_closed]
ballarin@14751
   790
  show "inv x \<in> \<Inter>A" by blast
ballarin@14751
   791
next
ballarin@14751
   792
  fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"
ballarin@14751
   793
  with subgr [THEN subgroup.m_closed]
ballarin@14751
   794
  show "x \<otimes> y \<in> \<Inter>A" by blast
ballarin@14751
   795
qed
ballarin@14751
   796
ballarin@14751
   797
theorem (in group) subgroups_complete_lattice:
ballarin@14751
   798
  "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
ballarin@14751
   799
    (is "complete_lattice ?L")
ballarin@14751
   800
proof (rule partial_order.complete_lattice_criterion1)
ballarin@14751
   801
  show "partial_order ?L" by (rule subgroups_partial_order)
ballarin@14751
   802
next
ballarin@14751
   803
  have "greatest ?L (carrier G) (carrier ?L)"
ballarin@14751
   804
    by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)
ballarin@14751
   805
  then show "EX G. greatest ?L G (carrier ?L)" ..
ballarin@14751
   806
next
ballarin@14751
   807
  fix A
ballarin@14751
   808
  assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"
ballarin@14751
   809
  then have Int_subgroup: "subgroup (\<Inter>A) G"
ballarin@14751
   810
    by (fastsimp intro: subgroups_Inter)
ballarin@14751
   811
  have "greatest ?L (\<Inter>A) (Lower ?L A)"
ballarin@14751
   812
    (is "greatest ?L ?Int _")
ballarin@14751
   813
  proof (rule greatest_LowerI)
ballarin@14751
   814
    fix H
ballarin@14751
   815
    assume H: "H \<in> A"
ballarin@14751
   816
    with L have subgroupH: "subgroup H G" by auto
ballarin@14751
   817
    from subgroupH have submagmaH: "submagma H G" by (rule subgroup.axioms)
ballarin@14751
   818
    from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
ballarin@14751
   819
      by (rule subgroup_imp_group)
ballarin@14751
   820
    from groupH have monoidH: "monoid ?H"
ballarin@14751
   821
      by (rule group.is_monoid)
ballarin@14751
   822
    from H have Int_subset: "?Int \<subseteq> H" by fastsimp
ballarin@14751
   823
    then show "le ?L ?Int H" by simp
ballarin@14751
   824
  next
ballarin@14751
   825
    fix H
ballarin@14751
   826
    assume H: "H \<in> Lower ?L A"
ballarin@14751
   827
    with L Int_subgroup show "le ?L H ?Int" by (fastsimp intro: Inter_greatest)
ballarin@14751
   828
  next
ballarin@14751
   829
    show "A \<subseteq> carrier ?L" by (rule L)
ballarin@14751
   830
  next
ballarin@14751
   831
    show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
ballarin@14751
   832
  qed
ballarin@14751
   833
  then show "EX I. greatest ?L I (Lower ?L A)" ..
ballarin@14751
   834
qed
ballarin@14751
   835
ballarin@13813
   836
end