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(* Title: HOL/AxClasses/Tutorial/Group.thy
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ID: $Id$
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Author: Markus Wenzel, TU Muenchen
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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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inverse :: "'a => 'a"
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one :: 'a;
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axclass
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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
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semigroup < "term"
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assoc: "(x [*] y) [*] z = x [*] (y [*] z)";
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axclass
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group < semigroup
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left_unit: "one [*] x = x"
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left_inverse: "inverse x [*] x = one";
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axclass
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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 [*] inverse x = (one::'a::group)";
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proof -;
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have "x [*] inverse x = one [*] (x [*] inverse x)";
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by (simp only: group.left_unit);
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also; have "... = one [*] x [*] inverse x";
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by (simp only: semigroup.assoc);
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also; have "... = inverse (inverse x) [*] inverse x [*] x [*] inverse x";
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by (simp only: group.left_inverse);
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also; have "... = inverse (inverse x) [*] (inverse x [*] x) [*] inverse x";
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by (simp only: semigroup.assoc);
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also; have "... = inverse (inverse x) [*] one [*] inverse x";
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by (simp only: group.left_inverse);
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also; have "... = inverse (inverse x) [*] (one [*] inverse x)";
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by (simp only: semigroup.assoc);
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also; have "... = inverse (inverse x) [*] inverse 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 [*] (inverse x [*] x)";
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by (simp only: group.left_inverse);
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also; have "... = x [*] inverse 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
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times_bool_def: "x [*] y == x ~= (y::bool)"
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inverse_bool_def: "inverse 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
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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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