1.1 --- a/src/HOL/AxClasses/Group.thy	Wed Mar 04 11:05:02 2009 +0100
     1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.3 @@ -1,124 +0,0 @@
     1.4 -(*  Title:      HOL/AxClasses/Group.thy
     1.5 -    ID:         $Id$
     1.6 -    Author:     Markus Wenzel, TU Muenchen
     1.7 -*)
     1.8 -
     1.9 -theory Group imports Main begin
    1.10 -
    1.11 -subsection {* Monoids and Groups *}
    1.12 -
    1.13 -consts
    1.14 -  times :: "'a => 'a => 'a"    (infixl "[*]" 70)
    1.15 -  invers :: "'a => 'a"
    1.16 -  one :: 'a
    1.17 -
    1.18 -
    1.19 -axclass monoid < type
    1.20 -  assoc:      "(x [*] y) [*] z = x [*] (y [*] z)"
    1.21 -  left_unit:  "one [*] x = x"
    1.22 -  right_unit: "x [*] one = x"
    1.23 -
    1.24 -axclass semigroup < type
    1.25 -  assoc: "(x [*] y) [*] z = x [*] (y [*] z)"
    1.26 -
    1.27 -axclass group < semigroup
    1.28 -  left_unit:    "one [*] x = x"
    1.29 -  left_inverse: "invers x [*] x = one"
    1.30 -
    1.31 -axclass agroup < group
    1.32 -  commute: "x [*] y = y [*] x"
    1.33 -
    1.34 -
    1.35 -subsection {* Abstract reasoning *}
    1.36 -
    1.37 -theorem group_right_inverse: "x [*] invers x = (one::'a::group)"
    1.38 -proof -
    1.39 -  have "x [*] invers x = one [*] (x [*] invers x)"
    1.40 -    by (simp only: group_class.left_unit)
    1.41 -  also have "... = one [*] x [*] invers x"
    1.42 -    by (simp only: semigroup_class.assoc)
    1.43 -  also have "... = invers (invers x) [*] invers x [*] x [*] invers x"
    1.44 -    by (simp only: group_class.left_inverse)
    1.45 -  also have "... = invers (invers x) [*] (invers x [*] x) [*] invers x"
    1.46 -    by (simp only: semigroup_class.assoc)
    1.47 -  also have "... = invers (invers x) [*] one [*] invers x"
    1.48 -    by (simp only: group_class.left_inverse)
    1.49 -  also have "... = invers (invers x) [*] (one [*] invers x)"
    1.50 -    by (simp only: semigroup_class.assoc)
    1.51 -  also have "... = invers (invers x) [*] invers x"
    1.52 -    by (simp only: group_class.left_unit)
    1.53 -  also have "... = one"
    1.54 -    by (simp only: group_class.left_inverse)
    1.55 -  finally show ?thesis .
    1.56 -qed
    1.57 -
    1.58 -theorem group_right_unit: "x [*] one = (x::'a::group)"
    1.59 -proof -
    1.60 -  have "x [*] one = x [*] (invers x [*] x)"
    1.61 -    by (simp only: group_class.left_inverse)
    1.62 -  also have "... = x [*] invers x [*] x"
    1.63 -    by (simp only: semigroup_class.assoc)
    1.64 -  also have "... = one [*] x"
    1.65 -    by (simp only: group_right_inverse)
    1.66 -  also have "... = x"
    1.67 -    by (simp only: group_class.left_unit)
    1.68 -  finally show ?thesis .
    1.69 -qed
    1.70 -
    1.71 -
    1.72 -subsection {* Abstract instantiation *}
    1.73 -
    1.74 -instance monoid < semigroup
    1.75 -proof intro_classes
    1.76 -  fix x y z :: "'a::monoid"
    1.77 -  show "x [*] y [*] z = x [*] (y [*] z)"
    1.78 -    by (rule monoid_class.assoc)
    1.79 -qed
    1.80 -
    1.81 -instance group < monoid
    1.82 -proof intro_classes
    1.83 -  fix x y z :: "'a::group"
    1.84 -  show "x [*] y [*] z = x [*] (y [*] z)"
    1.85 -    by (rule semigroup_class.assoc)
    1.86 -  show "one [*] x = x"
    1.87 -    by (rule group_class.left_unit)
    1.88 -  show "x [*] one = x"
    1.89 -    by (rule group_right_unit)
    1.90 -qed
    1.91 -
    1.92 -
    1.93 -subsection {* Concrete instantiation *}
    1.94 -
    1.95 -defs (overloaded)
    1.96 -  times_bool_def:   "x [*] y == x ~= (y::bool)"
    1.97 -  inverse_bool_def: "invers x == x::bool"
    1.98 -  unit_bool_def:    "one == False"
    1.99 -
   1.100 -instance bool :: agroup
   1.101 -proof (intro_classes,
   1.102 -    unfold times_bool_def inverse_bool_def unit_bool_def)
   1.103 -  fix x y z
   1.104 -  show "((x ~= y) ~= z) = (x ~= (y ~= z))" by blast
   1.105 -  show "(False ~= x) = x" by blast
   1.106 -  show "(x ~= x) = False" by blast
   1.107 -  show "(x ~= y) = (y ~= x)" by blast
   1.108 -qed
   1.109 -
   1.110 -
   1.111 -subsection {* Lifting and Functors *}
   1.112 -
   1.113 -defs (overloaded)
   1.114 -  times_prod_def: "p [*] q == (fst p [*] fst q, snd p [*] snd q)"
   1.115 -
   1.116 -instance * :: (semigroup, semigroup) semigroup
   1.117 -proof (intro_classes, unfold times_prod_def)
   1.118 -  fix p q r :: "'a::semigroup * 'b::semigroup"
   1.119 -  show
   1.120 -    "(fst (fst p [*] fst q, snd p [*] snd q) [*] fst r,
   1.121 -      snd (fst p [*] fst q, snd p [*] snd q) [*] snd r) =
   1.122 -       (fst p [*] fst (fst q [*] fst r, snd q [*] snd r),
   1.123 -        snd p [*] snd (fst q [*] fst r, snd q [*] snd r))"
   1.124 -    by (simp add: semigroup_class.assoc)
   1.125 -qed
   1.126 -
   1.127 -end