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(* Title: HOL/AxClasses/Group.thy
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ID: $Id$
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Author: Markus Wenzel, TU Muenchen
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License: GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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theory Group = Main:
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subsection {* Monoids and Groups *}
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consts
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times :: "'a => 'a => 'a" (infixl "[*]" 70)
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invers :: "'a => 'a"
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one :: 'a
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axclass monoid < "term"
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assoc: "(x [*] y) [*] z = x [*] (y [*] z)"
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left_unit: "one [*] x = x"
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right_unit: "x [*] one = x"
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axclass semigroup < "term"
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assoc: "(x [*] y) [*] z = x [*] (y [*] z)"
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axclass group < semigroup
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left_unit: "one [*] x = x"
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left_inverse: "invers x [*] x = one"
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axclass agroup < group
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commute: "x [*] y = y [*] x"
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subsection {* Abstract reasoning *}
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theorem group_right_inverse: "x [*] invers x = (one::'a::group)"
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proof -
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have "x [*] invers x = one [*] (x [*] invers x)"
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by (simp only: group.left_unit)
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also have "... = one [*] x [*] invers x"
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by (simp only: semigroup.assoc)
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also have "... = invers (invers x) [*] invers x [*] x [*] invers x"
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by (simp only: group.left_inverse)
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also have "... = invers (invers x) [*] (invers x [*] x) [*] invers x"
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by (simp only: semigroup.assoc)
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also have "... = invers (invers x) [*] one [*] invers x"
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by (simp only: group.left_inverse)
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also have "... = invers (invers x) [*] (one [*] invers x)"
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by (simp only: semigroup.assoc)
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also have "... = invers (invers x) [*] invers x"
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by (simp only: group.left_unit)
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also have "... = one"
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by (simp only: group.left_inverse)
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finally show ?thesis .
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qed
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theorem group_right_unit: "x [*] one = (x::'a::group)"
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proof -
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have "x [*] one = x [*] (invers x [*] x)"
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by (simp only: group.left_inverse)
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also have "... = x [*] invers x [*] x"
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by (simp only: semigroup.assoc)
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also have "... = one [*] x"
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by (simp only: group_right_inverse)
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also have "... = x"
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by (simp only: group.left_unit)
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finally show ?thesis .
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qed
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subsection {* Abstract instantiation *}
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instance monoid < semigroup
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proof intro_classes
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fix x y z :: "'a::monoid"
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show "x [*] y [*] z = x [*] (y [*] z)"
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by (rule monoid.assoc)
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qed
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instance group < monoid
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proof intro_classes
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fix x y z :: "'a::group"
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show "x [*] y [*] z = x [*] (y [*] z)"
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by (rule semigroup.assoc)
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show "one [*] x = x"
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by (rule group.left_unit)
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show "x [*] one = x"
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by (rule group_right_unit)
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qed
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subsection {* Concrete instantiation *}
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defs (overloaded)
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times_bool_def: "x [*] y == x ~= (y::bool)"
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inverse_bool_def: "invers x == x::bool"
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unit_bool_def: "one == False"
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instance bool :: agroup
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proof (intro_classes,
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unfold times_bool_def inverse_bool_def unit_bool_def)
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fix x y z
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show "((x ~= y) ~= z) = (x ~= (y ~= z))" by blast
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show "(False ~= x) = x" by blast
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show "(x ~= x) = False" by blast
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show "(x ~= y) = (y ~= x)" by blast
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qed
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subsection {* Lifting and Functors *}
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defs (overloaded)
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times_prod_def: "p [*] q == (fst p [*] fst q, snd p [*] snd q)"
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instance * :: (semigroup, semigroup) semigroup
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proof (intro_classes, unfold times_prod_def)
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fix p q r :: "'a::semigroup * 'b::semigroup"
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show
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"(fst (fst p [*] fst q, snd p [*] snd q) [*] fst r,
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snd (fst p [*] fst q, snd p [*] snd q) [*] snd r) =
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(fst p [*] fst (fst q [*] fst r, snd q [*] snd r),
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snd p [*] snd (fst q [*] fst r, snd q [*] snd r))"
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by (simp add: semigroup.assoc)
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qed
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end
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