(* Title: HOL/ex/MonoidGroup.thy
ID: $Id$
Author: Markus Wenzel
Copyright 1996 TU Muenchen
Monoids and Groups as predicates over record schemes.
*)
MonoidGroup = HOL + Record +
record 'a monoid_sig =
times :: "['a, 'a] => 'a"
one :: 'a
record 'a group_sig = 'a monoid_sig +
inv :: "'a => 'a"
constdefs
monoid :: "(| times :: ['a, 'a] => 'a, one :: 'a, ... :: 'more |) => bool"
"monoid M == ALL x y z.
times M (times M x y) z = times M x (times M y z) &
times M (one M) x & times M x (one M) = x"
group :: "(| times :: ['a, 'a] => 'a, one :: 'a, inv :: 'a => 'a, ... :: 'more |) => bool"
"group G == monoid G & (ALL x. times G (inv G x) x = one G)"
reverse :: "(| times :: ['a, 'a] => 'a, one :: 'a, ... :: 'more |) =>
(| times :: ['a, 'a] => 'a, one :: 'a, ... :: 'more |)"
"reverse M == M (| times := %x y. times M y x |)"
end