src/HOL/Algebra/Group.thy
author ballarin
Wed Apr 30 18:32:06 2003 +0200 (2003-04-30)
changeset 13940 c67798653056
parent 13936 d3671b878828
child 13943 83d842ccd4aa
permissions -rw-r--r--
HOL-Algebra: New polynomial development added.
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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 {* Algebraic Structures up to Commutative Groups *}
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theory Group = FuncSet:
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axclass number < type
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instance nat :: number ..
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instance int :: number ..
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section {* From Magmas to Groups *}
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text {*
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  Definitions follow Jacobson, Basic Algebra I, Freeman, 1985; with
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  the exception of \emph{magma} which, following Bourbaki, is a set
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  together with a binary, closed operation.
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*}
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subsection {* Definitions *}
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record 'a semigroup =
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  carrier :: "'a set"
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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
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  m_inv :: "[('a, 'm) monoid_scheme, 'a] => 'a" ("inv\<index> _" [81] 80)
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  "m_inv G x == (THE y. y \<in> carrier G &
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                  mult G x y = one G & mult G y x = one G)"
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  Units :: "('a, 'm) monoid_scheme => 'a set"
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  "Units G == {y. y \<in> carrier G &
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                  (EX x : carrier G. mult G x y = one G & mult G y x = one G)}"
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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 G) (%u b. mult G b a) n"
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  int_pow_def: "pow G a z ==
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    let p = nat_rec (one G) (%u b. mult G b a)
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    in if neg z then m_inv G (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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  assumes m_closed:
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      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
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    and one_closed: "one G \<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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      mult G (mult G x y) z = mult G x (mult G y z)"
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    and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
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    and r_one: "!!x. x \<in> carrier G ==> mult G x (one G) = 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)
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  apply auto
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  apply (rule theI2, fast)
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   apply (fast intro: inv_unique)
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  apply 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)
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  apply auto
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  apply (rule theI2, fast)
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   apply (fast intro: inv_unique)
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  apply 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)
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  apply auto
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  apply (rule theI2, fast)
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   apply (fast intro: inv_unique)
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  apply 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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theorem groupI:
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  assumes m_closed [simp]:
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      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
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    and one_closed [simp]: "one G \<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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      mult G (mult G x y) z = mult G x (mult G y z)"
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    and l_one [simp]: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
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    and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G"
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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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    (mult G x y = mult G x z) = (y = z)"
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  proof
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    fix x y z
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    assume eq: "mult G x y = mult G x 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: "mult G x_inv x = one G" by fast
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    from G eq xG have "mult G (mult G x_inv x) y = mult G (mult G x_inv x) 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 "mult G x y = mult G x z" by simp
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  qed
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  have r_one:
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    "!!x. x \<in> carrier G ==> mult G x (one G) = 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: "mult G x_inv x = one G" by fast
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    from x xG have "mult G x_inv (mult G x (one G)) = mult G x_inv x"
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      by (simp add: m_assoc [symmetric] l_inv)
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    with x xG show "mult G x (one G) = 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. mult G y x = one G &
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      mult G x y = one G"
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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: "mult G y x = one G" by fast
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    from x y have "mult G y (mult G x y) = mult G y (one G)"
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      by (simp add: m_assoc [symmetric] l_inv r_one)
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    with x y have r_inv: "mult G x y = one G"
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      by simp
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    from x y show "EX y : carrier G. mult G y x = one G &
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      mult G x y = one G"
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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. mult G y x = one G"
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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"
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    and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
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   325
  then show "y \<otimes> x = z \<otimes> x" by simp
ballarin@13813
   326
qed
ballarin@13813
   327
ballarin@13854
   328
lemma (in group) inv_one [simp]:
ballarin@13854
   329
  "inv \<one> = \<one>"
ballarin@13854
   330
proof -
ballarin@13854
   331
  have "inv \<one> = \<one> \<otimes> (inv \<one>)" by simp
ballarin@13854
   332
  moreover have "... = \<one>" by (simp add: r_inv)
ballarin@13854
   333
  finally show ?thesis .
ballarin@13854
   334
qed
ballarin@13854
   335
ballarin@13813
   336
lemma (in group) inv_inv [simp]:
ballarin@13813
   337
  "x \<in> carrier G ==> inv (inv x) = x"
ballarin@13936
   338
  using Units_inv_inv by simp
ballarin@13936
   339
ballarin@13936
   340
lemma (in group) inv_inj:
ballarin@13936
   341
  "inj_on (m_inv G) (carrier G)"
ballarin@13936
   342
  using inv_inj_on_Units by simp
ballarin@13813
   343
ballarin@13854
   344
lemma (in group) inv_mult_group:
ballarin@13813
   345
  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"
ballarin@13813
   346
proof -
ballarin@13813
   347
  assume G: "x \<in> carrier G" "y \<in> carrier G"
ballarin@13813
   348
  then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)"
ballarin@13813
   349
    by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric] l_inv)
ballarin@13813
   350
  with G show ?thesis by simp
ballarin@13813
   351
qed
ballarin@13813
   352
ballarin@13940
   353
lemma (in group) inv_comm:
ballarin@13940
   354
  "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
ballarin@13940
   355
  by (rule Units_inv_comm) auto                          
ballarin@13940
   356
ballarin@13936
   357
text {* Power *}
ballarin@13936
   358
ballarin@13936
   359
lemma (in group) int_pow_def2:
ballarin@13936
   360
  "a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))"
ballarin@13936
   361
  by (simp add: int_pow_def nat_pow_def Let_def)
ballarin@13936
   362
ballarin@13936
   363
lemma (in group) int_pow_0 [simp]:
ballarin@13936
   364
  "x (^) (0::int) = \<one>"
ballarin@13936
   365
  by (simp add: int_pow_def2)
ballarin@13936
   366
ballarin@13936
   367
lemma (in group) int_pow_one [simp]:
ballarin@13936
   368
  "\<one> (^) (z::int) = \<one>"
ballarin@13936
   369
  by (simp add: int_pow_def2)
ballarin@13936
   370
ballarin@13813
   371
subsection {* Substructures *}
ballarin@13813
   372
ballarin@13813
   373
locale submagma = var H + struct G +
ballarin@13813
   374
  assumes subset [intro, simp]: "H \<subseteq> carrier G"
ballarin@13813
   375
    and m_closed [intro, simp]: "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"
ballarin@13813
   376
ballarin@13813
   377
declare (in submagma) magma.intro [intro] semigroup.intro [intro]
ballarin@13936
   378
  semigroup_axioms.intro [intro]
ballarin@13813
   379
(*
ballarin@13813
   380
alternative definition of submagma
ballarin@13813
   381
ballarin@13813
   382
locale submagma = var H + struct G +
ballarin@13813
   383
  assumes subset [intro, simp]: "carrier H \<subseteq> carrier G"
ballarin@13813
   384
    and m_equal [simp]: "mult H = mult G"
ballarin@13813
   385
    and m_closed [intro, simp]:
ballarin@13813
   386
      "[| x \<in> carrier H; y \<in> carrier H |] ==> x \<otimes> y \<in> carrier H"
ballarin@13813
   387
*)
ballarin@13813
   388
ballarin@13813
   389
lemma submagma_imp_subset:
ballarin@13813
   390
  "submagma H G ==> H \<subseteq> carrier G"
ballarin@13813
   391
  by (rule submagma.subset)
ballarin@13813
   392
ballarin@13813
   393
lemma (in submagma) subsetD [dest, simp]:
ballarin@13813
   394
  "x \<in> H ==> x \<in> carrier G"
ballarin@13813
   395
  using subset by blast
ballarin@13813
   396
ballarin@13813
   397
lemma (in submagma) magmaI [intro]:
ballarin@13813
   398
  includes magma G
ballarin@13813
   399
  shows "magma (G(| carrier := H |))"
ballarin@13813
   400
  by rule simp
ballarin@13813
   401
ballarin@13813
   402
lemma (in submagma) semigroup_axiomsI [intro]:
ballarin@13813
   403
  includes semigroup G
ballarin@13813
   404
  shows "semigroup_axioms (G(| carrier := H |))"
ballarin@13813
   405
    by rule (simp add: m_assoc)
ballarin@13813
   406
ballarin@13813
   407
lemma (in submagma) semigroupI [intro]:
ballarin@13813
   408
  includes semigroup G
ballarin@13813
   409
  shows "semigroup (G(| carrier := H |))"
ballarin@13813
   410
  using prems by fast
ballarin@13813
   411
ballarin@13813
   412
locale subgroup = submagma H G +
ballarin@13813
   413
  assumes one_closed [intro, simp]: "\<one> \<in> H"
ballarin@13813
   414
    and m_inv_closed [intro, simp]: "x \<in> H ==> inv x \<in> H"
ballarin@13813
   415
ballarin@13813
   416
declare (in subgroup) group.intro [intro]
ballarin@13936
   417
(*
ballarin@13817
   418
lemma (in subgroup) l_oneI [intro]:
ballarin@13817
   419
  includes l_one G
ballarin@13817
   420
  shows "l_one (G(| carrier := H |))"
ballarin@13817
   421
  by rule simp_all
ballarin@13936
   422
*)
ballarin@13813
   423
lemma (in subgroup) group_axiomsI [intro]:
ballarin@13813
   424
  includes group G
ballarin@13813
   425
  shows "group_axioms (G(| carrier := H |))"
ballarin@13936
   426
  by rule (auto intro: l_inv r_inv simp add: Units_def)
ballarin@13813
   427
ballarin@13813
   428
lemma (in subgroup) groupI [intro]:
ballarin@13813
   429
  includes group G
ballarin@13813
   430
  shows "group (G(| carrier := H |))"
ballarin@13936
   431
  by (rule groupI) (auto intro: m_assoc l_inv)
ballarin@13813
   432
ballarin@13813
   433
text {*
ballarin@13813
   434
  Since @{term H} is nonempty, it contains some element @{term x}.  Since
ballarin@13813
   435
  it is closed under inverse, it contains @{text "inv x"}.  Since
ballarin@13813
   436
  it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}.
ballarin@13813
   437
*}
ballarin@13813
   438
ballarin@13813
   439
lemma (in group) one_in_subset:
ballarin@13813
   440
  "[| 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
   441
   ==> \<one> \<in> H"
ballarin@13813
   442
by (force simp add: l_inv)
ballarin@13813
   443
ballarin@13813
   444
text {* A characterization of subgroups: closed, non-empty subset. *}
ballarin@13813
   445
ballarin@13813
   446
lemma (in group) subgroupI:
ballarin@13813
   447
  assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"
ballarin@13813
   448
    and inv: "!!a. a \<in> H ==> inv a \<in> H"
ballarin@13813
   449
    and mult: "!!a b. [|a \<in> H; b \<in> H|] ==> a \<otimes> b \<in> H"
ballarin@13813
   450
  shows "subgroup H G"
ballarin@13813
   451
proof
ballarin@13813
   452
  from subset and mult show "submagma H G" ..
ballarin@13813
   453
next
ballarin@13813
   454
  have "\<one> \<in> H" by (rule one_in_subset) (auto simp only: prems)
ballarin@13813
   455
  with inv show "subgroup_axioms H G"
ballarin@13813
   456
    by (intro subgroup_axioms.intro) simp_all
ballarin@13813
   457
qed
ballarin@13813
   458
ballarin@13813
   459
text {*
ballarin@13813
   460
  Repeat facts of submagmas for subgroups.  Necessary???
ballarin@13813
   461
*}
ballarin@13813
   462
ballarin@13813
   463
lemma (in subgroup) subset:
ballarin@13813
   464
  "H \<subseteq> carrier G"
ballarin@13813
   465
  ..
ballarin@13813
   466
ballarin@13813
   467
lemma (in subgroup) m_closed:
ballarin@13813
   468
  "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"
ballarin@13813
   469
  ..
ballarin@13813
   470
ballarin@13813
   471
declare magma.m_closed [simp]
ballarin@13813
   472
ballarin@13936
   473
declare monoid.one_closed [iff] group.inv_closed [simp]
ballarin@13936
   474
  monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]
ballarin@13813
   475
ballarin@13813
   476
lemma subgroup_nonempty:
ballarin@13813
   477
  "~ subgroup {} G"
ballarin@13813
   478
  by (blast dest: subgroup.one_closed)
ballarin@13813
   479
ballarin@13813
   480
lemma (in subgroup) finite_imp_card_positive:
ballarin@13813
   481
  "finite (carrier G) ==> 0 < card H"
ballarin@13813
   482
proof (rule classical)
ballarin@13813
   483
  have sub: "subgroup H G" using prems ..
ballarin@13813
   484
  assume fin: "finite (carrier G)"
ballarin@13813
   485
    and zero: "~ 0 < card H"
ballarin@13813
   486
  then have "finite H" by (blast intro: finite_subset dest: subset)
ballarin@13813
   487
  with zero sub have "subgroup {} G" by simp
ballarin@13813
   488
  with subgroup_nonempty show ?thesis by contradiction
ballarin@13813
   489
qed
ballarin@13813
   490
ballarin@13936
   491
(*
ballarin@13936
   492
lemma (in monoid) Units_subgroup:
ballarin@13936
   493
  "subgroup (Units G) G"
ballarin@13936
   494
*)
ballarin@13936
   495
ballarin@13813
   496
subsection {* Direct Products *}
ballarin@13813
   497
ballarin@13813
   498
constdefs
ballarin@13817
   499
  DirProdSemigroup ::
ballarin@13854
   500
    "[('a, 'm) semigroup_scheme, ('b, 'n) semigroup_scheme]
ballarin@13817
   501
    => ('a \<times> 'b) semigroup"
ballarin@13817
   502
    (infixr "\<times>\<^sub>s" 80)
ballarin@13817
   503
  "G \<times>\<^sub>s H == (| carrier = carrier G \<times> carrier H,
ballarin@13817
   504
    mult = (%(xg, xh) (yg, yh). (mult G xg yg, mult H xh yh)) |)"
ballarin@13817
   505
ballarin@13936
   506
  DirProdGroup ::
ballarin@13854
   507
    "[('a, 'm) monoid_scheme, ('b, 'n) monoid_scheme] => ('a \<times> 'b) monoid"
ballarin@13936
   508
    (infixr "\<times>\<^sub>g" 80)
ballarin@13936
   509
  "G \<times>\<^sub>g H == (| carrier = carrier (G \<times>\<^sub>s H),
ballarin@13817
   510
    mult = mult (G \<times>\<^sub>s H),
ballarin@13817
   511
    one = (one G, one H) |)"
ballarin@13936
   512
(*
ballarin@13813
   513
  DirProdGroup ::
ballarin@13854
   514
    "[('a, 'm) group_scheme, ('b, 'n) group_scheme] => ('a \<times> 'b) group"
ballarin@13813
   515
    (infixr "\<times>\<^sub>g" 80)
ballarin@13813
   516
  "G \<times>\<^sub>g H == (| carrier = carrier (G \<times>\<^sub>m H),
ballarin@13813
   517
    mult = mult (G \<times>\<^sub>m H),
ballarin@13817
   518
    one = one (G \<times>\<^sub>m H),
ballarin@13813
   519
    m_inv = (%(g, h). (m_inv G g, m_inv H h)) |)"
ballarin@13936
   520
*)
ballarin@13813
   521
ballarin@13817
   522
lemma DirProdSemigroup_magma:
ballarin@13813
   523
  includes magma G + magma H
ballarin@13817
   524
  shows "magma (G \<times>\<^sub>s H)"
ballarin@13817
   525
  by rule (auto simp add: DirProdSemigroup_def)
ballarin@13813
   526
ballarin@13817
   527
lemma DirProdSemigroup_semigroup_axioms:
ballarin@13813
   528
  includes semigroup G + semigroup H
ballarin@13817
   529
  shows "semigroup_axioms (G \<times>\<^sub>s H)"
ballarin@13817
   530
  by rule (auto simp add: DirProdSemigroup_def G.m_assoc H.m_assoc)
ballarin@13813
   531
ballarin@13817
   532
lemma DirProdSemigroup_semigroup:
ballarin@13813
   533
  includes semigroup G + semigroup H
ballarin@13817
   534
  shows "semigroup (G \<times>\<^sub>s H)"
ballarin@13813
   535
  using prems
ballarin@13813
   536
  by (fast intro: semigroup.intro
ballarin@13817
   537
    DirProdSemigroup_magma DirProdSemigroup_semigroup_axioms)
ballarin@13813
   538
ballarin@13813
   539
lemma DirProdGroup_magma:
ballarin@13813
   540
  includes magma G + magma H
ballarin@13813
   541
  shows "magma (G \<times>\<^sub>g H)"
ballarin@13817
   542
  by rule
ballarin@13936
   543
    (auto simp add: DirProdGroup_def DirProdSemigroup_def)
ballarin@13813
   544
ballarin@13813
   545
lemma DirProdGroup_semigroup_axioms:
ballarin@13813
   546
  includes semigroup G + semigroup H
ballarin@13813
   547
  shows "semigroup_axioms (G \<times>\<^sub>g H)"
ballarin@13813
   548
  by rule
ballarin@13936
   549
    (auto simp add: DirProdGroup_def DirProdSemigroup_def
ballarin@13817
   550
      G.m_assoc H.m_assoc)
ballarin@13813
   551
ballarin@13813
   552
lemma DirProdGroup_semigroup:
ballarin@13813
   553
  includes semigroup G + semigroup H
ballarin@13813
   554
  shows "semigroup (G \<times>\<^sub>g H)"
ballarin@13813
   555
  using prems
ballarin@13813
   556
  by (fast intro: semigroup.intro
ballarin@13813
   557
    DirProdGroup_magma DirProdGroup_semigroup_axioms)
ballarin@13813
   558
ballarin@13813
   559
(* ... and further lemmas for group ... *)
ballarin@13813
   560
ballarin@13817
   561
lemma DirProdGroup_group:
ballarin@13813
   562
  includes group G + group H
ballarin@13813
   563
  shows "group (G \<times>\<^sub>g H)"
ballarin@13936
   564
  by (rule groupI)
ballarin@13936
   565
    (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
ballarin@13936
   566
      simp add: DirProdGroup_def DirProdSemigroup_def)
ballarin@13813
   567
ballarin@13813
   568
subsection {* Homomorphisms *}
ballarin@13813
   569
ballarin@13813
   570
constdefs
ballarin@13817
   571
  hom :: "[('a, 'c) semigroup_scheme, ('b, 'd) semigroup_scheme]
ballarin@13817
   572
    => ('a => 'b)set"
ballarin@13813
   573
  "hom G H ==
ballarin@13813
   574
    {h. h \<in> carrier G -> carrier H &
ballarin@13813
   575
      (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (mult G x y) = mult H (h x) (h y))}"
ballarin@13813
   576
ballarin@13813
   577
lemma (in semigroup) hom:
ballarin@13813
   578
  includes semigroup G
ballarin@13813
   579
  shows "semigroup (| carrier = hom G G, mult = op o |)"
ballarin@13813
   580
proof
ballarin@13813
   581
  show "magma (| carrier = hom G G, mult = op o |)"
ballarin@13813
   582
    by rule (simp add: Pi_def hom_def)
ballarin@13813
   583
next
ballarin@13813
   584
  show "semigroup_axioms (| carrier = hom G G, mult = op o |)"
ballarin@13813
   585
    by rule (simp add: o_assoc)
ballarin@13813
   586
qed
ballarin@13813
   587
ballarin@13813
   588
lemma hom_mult:
ballarin@13813
   589
  "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |] 
ballarin@13813
   590
   ==> h (mult G x y) = mult H (h x) (h y)"
ballarin@13813
   591
  by (simp add: hom_def) 
ballarin@13813
   592
ballarin@13813
   593
lemma hom_closed:
ballarin@13813
   594
  "[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"
ballarin@13813
   595
  by (auto simp add: hom_def funcset_mem)
ballarin@13813
   596
ballarin@13813
   597
locale group_hom = group G + group H + var h +
ballarin@13813
   598
  assumes homh: "h \<in> hom G H"
ballarin@13813
   599
  notes hom_mult [simp] = hom_mult [OF homh]
ballarin@13813
   600
    and hom_closed [simp] = hom_closed [OF homh]
ballarin@13813
   601
ballarin@13813
   602
lemma (in group_hom) one_closed [simp]:
ballarin@13813
   603
  "h \<one> \<in> carrier H"
ballarin@13813
   604
  by simp
ballarin@13813
   605
ballarin@13813
   606
lemma (in group_hom) hom_one [simp]:
ballarin@13813
   607
  "h \<one> = \<one>\<^sub>2"
ballarin@13813
   608
proof -
ballarin@13813
   609
  have "h \<one> \<otimes>\<^sub>2 \<one>\<^sub>2 = h \<one> \<otimes>\<^sub>2 h \<one>"
ballarin@13813
   610
    by (simp add: hom_mult [symmetric] del: hom_mult)
ballarin@13813
   611
  then show ?thesis by (simp del: r_one)
ballarin@13813
   612
qed
ballarin@13813
   613
ballarin@13813
   614
lemma (in group_hom) inv_closed [simp]:
ballarin@13813
   615
  "x \<in> carrier G ==> h (inv x) \<in> carrier H"
ballarin@13813
   616
  by simp
ballarin@13813
   617
ballarin@13813
   618
lemma (in group_hom) hom_inv [simp]:
ballarin@13813
   619
  "x \<in> carrier G ==> h (inv x) = inv\<^sub>2 (h x)"
ballarin@13813
   620
proof -
ballarin@13813
   621
  assume x: "x \<in> carrier G"
ballarin@13813
   622
  then have "h x \<otimes>\<^sub>2 h (inv x) = \<one>\<^sub>2"
ballarin@13813
   623
    by (simp add: hom_mult [symmetric] G.r_inv del: hom_mult)
ballarin@13813
   624
  also from x have "... = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)"
ballarin@13813
   625
    by (simp add: hom_mult [symmetric] H.r_inv del: hom_mult)
ballarin@13813
   626
  finally have "h x \<otimes>\<^sub>2 h (inv x) = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)" .
ballarin@13813
   627
  with x show ?thesis by simp
ballarin@13813
   628
qed
ballarin@13813
   629
ballarin@13936
   630
section {* Commutative Structures *}
ballarin@13936
   631
ballarin@13936
   632
text {*
ballarin@13936
   633
  Naming convention: multiplicative structures that are commutative
ballarin@13936
   634
  are called \emph{commutative}, additive structures are called
ballarin@13936
   635
  \emph{Abelian}.
ballarin@13936
   636
*}
ballarin@13813
   637
ballarin@13813
   638
subsection {* Definition *}
ballarin@13813
   639
ballarin@13936
   640
locale comm_semigroup = semigroup +
ballarin@13813
   641
  assumes m_comm: "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
ballarin@13813
   642
ballarin@13936
   643
lemma (in comm_semigroup) m_lcomm:
ballarin@13813
   644
  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
ballarin@13813
   645
   x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
ballarin@13813
   646
proof -
ballarin@13813
   647
  assume xyz: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
ballarin@13813
   648
  from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)
ballarin@13813
   649
  also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)
ballarin@13813
   650
  also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)
ballarin@13813
   651
  finally show ?thesis .
ballarin@13813
   652
qed
ballarin@13813
   653
ballarin@13936
   654
lemmas (in comm_semigroup) m_ac = m_assoc m_comm m_lcomm
ballarin@13936
   655
ballarin@13936
   656
locale comm_monoid = comm_semigroup + monoid
ballarin@13813
   657
ballarin@13936
   658
lemma comm_monoidI:
ballarin@13936
   659
  assumes m_closed:
ballarin@13936
   660
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
ballarin@13936
   661
    and one_closed: "one G \<in> carrier G"
ballarin@13936
   662
    and m_assoc:
ballarin@13936
   663
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
ballarin@13936
   664
      mult G (mult G x y) z = mult G x (mult G y z)"
ballarin@13936
   665
    and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
ballarin@13936
   666
    and m_comm:
ballarin@13936
   667
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x"
ballarin@13936
   668
  shows "comm_monoid G"
ballarin@13936
   669
  using l_one
ballarin@13936
   670
  by (auto intro!: comm_monoid.intro magma.intro semigroup_axioms.intro
ballarin@13936
   671
    comm_semigroup_axioms.intro monoid_axioms.intro
ballarin@13936
   672
    intro: prems simp: m_closed one_closed m_comm)
ballarin@13817
   673
ballarin@13936
   674
lemma (in monoid) monoid_comm_monoidI:
ballarin@13936
   675
  assumes m_comm:
ballarin@13936
   676
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x"
ballarin@13936
   677
  shows "comm_monoid G"
ballarin@13936
   678
  by (rule comm_monoidI) (auto intro: m_assoc m_comm)
ballarin@13936
   679
(*
ballarin@13936
   680
lemma (in comm_monoid) r_one [simp]:
ballarin@13817
   681
  "x \<in> carrier G ==> x \<otimes> \<one> = x"
ballarin@13817
   682
proof -
ballarin@13817
   683
  assume G: "x \<in> carrier G"
ballarin@13817
   684
  then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)
ballarin@13817
   685
  also from G have "... = x" by simp
ballarin@13817
   686
  finally show ?thesis .
ballarin@13817
   687
qed
ballarin@13936
   688
*)
ballarin@13817
   689
ballarin@13936
   690
lemma (in comm_monoid) nat_pow_distr:
ballarin@13936
   691
  "[| x \<in> carrier G; y \<in> carrier G |] ==>
ballarin@13936
   692
  (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"
ballarin@13936
   693
  by (induct n) (simp, simp add: m_ac)
ballarin@13936
   694
ballarin@13936
   695
locale comm_group = comm_monoid + group
ballarin@13936
   696
ballarin@13936
   697
lemma (in group) group_comm_groupI:
ballarin@13936
   698
  assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>
ballarin@13936
   699
      mult G x y = mult G y x"
ballarin@13936
   700
  shows "comm_group G"
ballarin@13936
   701
  by (fast intro: comm_group.intro comm_semigroup_axioms.intro
ballarin@13936
   702
    group.axioms prems)
ballarin@13817
   703
ballarin@13936
   704
lemma comm_groupI:
ballarin@13936
   705
  assumes m_closed:
ballarin@13936
   706
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
ballarin@13936
   707
    and one_closed: "one G \<in> carrier G"
ballarin@13936
   708
    and m_assoc:
ballarin@13936
   709
      "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
ballarin@13936
   710
      mult G (mult G x y) z = mult G x (mult G y z)"
ballarin@13936
   711
    and m_comm:
ballarin@13936
   712
      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x"
ballarin@13936
   713
    and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
ballarin@13936
   714
    and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G"
ballarin@13936
   715
  shows "comm_group G"
ballarin@13936
   716
  by (fast intro: group.group_comm_groupI groupI prems)
ballarin@13936
   717
ballarin@13936
   718
lemma (in comm_group) inv_mult:
ballarin@13854
   719
  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"
ballarin@13936
   720
  by (simp add: m_ac inv_mult_group)
ballarin@13854
   721
ballarin@13813
   722
end