| author | haftmann |
| Fri, 17 Jun 2005 16:12:49 +0200 | |
| changeset 16417 | 9bc16273c2d4 |
| parent 14981 | e73f8140af78 |
| child 17274 | 746bb4c56800 |
| permissions | -rw-r--r-- |
| 10134 | 1 |
(* Title: HOL/AxClasses/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 imports Main begin |
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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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12338
de0f4a63baa5
renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents:
11072
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changeset
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axclass monoid < type |
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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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12338
de0f4a63baa5
renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents:
11072
diff
changeset
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axclass semigroup < type |
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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 |