(*  Title:      HOL/AxClasses/Group.thy

    ID:         $Id$

    Author:     Markus Wenzel, TU Muenchen

*)

theory Group = Main:

subsection {* Monoids and Groups *}

consts

  times :: "'a => 'a => 'a"    (infixl "[*]" 70)

  inverse :: "'a => 'a"

  one :: 'a

axclass monoid < "term"

  assoc:      "(x [*] y) [*] z = x [*] (y [*] z)"

  left_unit:  "one [*] x = x"

  right_unit: "x [*] one = x"

axclass semigroup < "term"

  assoc: "(x [*] y) [*] z = x [*] (y [*] z)"

axclass group < semigroup

  left_unit:    "one [*] x = x"

  left_inverse: "inverse x [*] x = one"

axclass agroup < group

  commute: "x [*] y = y [*] x"

subsection {* Abstract reasoning *}

theorem group_right_inverse: "x [*] inverse x = (one::'a::group)"

proof -

  have "x [*] inverse x = one [*] (x [*] inverse x)"

    by (simp only: group.left_unit)

  also have "... = one [*] x [*] inverse x"

    by (simp only: semigroup.assoc)

  also have "... = inverse (inverse x) [*] inverse x [*] x [*] inverse x"

    by (simp only: group.left_inverse)

  also have "... = inverse (inverse x) [*] (inverse x [*] x) [*] inverse x"

    by (simp only: semigroup.assoc)

  also have "... = inverse (inverse x) [*] one [*] inverse x"

    by (simp only: group.left_inverse)

  also have "... = inverse (inverse x) [*] (one [*] inverse x)"

    by (simp only: semigroup.assoc)

  also have "... = inverse (inverse x) [*] inverse x"

    by (simp only: group.left_unit)

  also have "... = one"

    by (simp only: group.left_inverse)

  finally show ?thesis .

qed

theorem group_right_unit: "x [*] one = (x::'a::group)"

proof -

  have "x [*] one = x [*] (inverse x [*] x)"

    by (simp only: group.left_inverse)

  also have "... = x [*] inverse x [*] x"

    by (simp only: semigroup.assoc)

  also have "... = one [*] x"

    by (simp only: group_right_inverse)

  also have "... = x"

    by (simp only: group.left_unit)

  finally show ?thesis .

qed

subsection {* Abstract instantiation *}

instance monoid < semigroup

proof intro_classes

  fix x y z :: "'a::monoid"

  show "x [*] y [*] z = x [*] (y [*] z)"

    by (rule monoid.assoc)

qed

instance group < monoid

proof intro_classes

  fix x y z :: "'a::group"

  show "x [*] y [*] z = x [*] (y [*] z)"

    by (rule semigroup.assoc)

  show "one [*] x = x"

    by (rule group.left_unit)

  show "x [*] one = x"

    by (rule group_right_unit)

qed

subsection {* Concrete instantiation *}

defs (overloaded)

  times_bool_def:   "x [*] y == x ~= (y::bool)"

  inverse_bool_def: "inverse x == x::bool"

  unit_bool_def:    "one == False"

instance bool :: agroup

proof (intro_classes,

    unfold times_bool_def inverse_bool_def unit_bool_def)

  fix x y z

  show "((x ~= y) ~= z) = (x ~= (y ~= z))" by blast

  show "(False ~= x) = x" by blast

  show "(x ~= x) = False" by blast

  show "(x ~= y) = (y ~= x)" by blast

qed

subsection {* Lifting and Functors *}

defs (overloaded)

  times_prod_def: "p [*] q == (fst p [*] fst q, snd p [*] snd q)"

instance * :: (semigroup, semigroup) semigroup

proof (intro_classes, unfold times_prod_def)

  fix p q r :: "'a::semigroup * 'b::semigroup"

  show

    "(fst (fst p [*] fst q, snd p [*] snd q) [*] fst r,

      snd (fst p [*] fst q, snd p [*] snd q) [*] snd r) =

       (fst p [*] fst (fst q [*] fst r, snd q [*] snd r),

        snd p [*] snd (fst q [*] fst r, snd q [*] snd r))"

    by (simp add: semigroup.assoc)

qed

end