header {* Monoids and Groups as predicates over record schemes *}
theory MonoidGroup imports Main begin
record 'a monoid_sig =
times :: "'a => 'a => 'a"
one :: 'a
record 'a group_sig = "'a monoid_sig" +
inv :: "'a => 'a"
definition
monoid :: "(| times :: 'a => 'a => 'a, one :: 'a, ... :: 'b |) => bool" where
"monoid M = (∀x y z.
times M (times M x y) z = times M x (times M y z) ∧
times M (one M) x = x ∧ times M x (one M) = x)"
definition
group :: "(| times :: 'a => 'a => 'a, one :: 'a, inv :: 'a => 'a, ... :: 'b |) => bool" where
"group G = (monoid G ∧ (∀x. times G (inv G x) x = one G))"
definition
reverse :: "(| times :: 'a => 'a => 'a, one :: 'a, ... :: 'b |) =>
(| times :: 'a => 'a => 'a, one :: 'a, ... :: 'b |)" where
"reverse M = M (| times := λx y. times M y x |)"
end