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(* Title: HOL/ex/MonoidGroup.thy
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ID: $Id$
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Author: Markus Wenzel
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License: GPL (GNU GENERAL PUBLIC LICENSE)
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Monoids and Groups as predicates over record schemes.
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*)
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theory MonoidGroup = Main:
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record 'a monoid_sig =
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times :: "['a, 'a] => 'a"
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one :: 'a
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record 'a group_sig = "'a monoid_sig" +
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inv :: "'a => 'a"
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constdefs
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monoid :: "(| times :: ['a, 'a] => 'a, one :: 'a, ... :: 'b::more |) => bool"
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"monoid M == ALL x y z.
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times M (times M x y) z = times M x (times M y z) &
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times M (one M) x = x & times M x (one M) = x"
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group :: "(| times :: ['a, 'a] => 'a, one :: 'a, inv :: 'a => 'a, ... :: 'b::more |) => bool"
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"group G == monoid G & (ALL x. times G (inv G x) x = one G)"
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reverse :: "(| times :: ['a, 'a] => 'a, one :: 'a, ... :: 'b::more |) =>
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(| times :: ['a, 'a] => 'a, one :: 'a, ... :: 'b |)"
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"reverse M == M (| times := %x y. times M y x |)"
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end
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