src/HOL/ex/Group.thy

+(*  Title:      HOL/Integ/Group.thy
+    ID:         $Id$
+    Author:     Tobias Nipkow
+    Copyright   1996 TU Muenchen
+
+A little bit of group theory leading up to rings. Hence groups are additive.
+*)
+
+Group = Set +
+
+(* 0 already used in Nat *)
+axclass  zero < term
+consts   zero :: "'a::zero"
+
+(* additive semigroups *)
+
+axclass  add_semigroup < plus
+  plus_assoc   "(x + y) + z = x + (y + z)"
+
+
+(* additive monoids *)
+
+axclass  add_monoid < add_semigroup, zero
+  zeroL    "zero + x = x"
+  zeroR    "x + zero = x"
+
+(* additive groups *)
+(*
+The inverse is the binary `-'. Groups with unary and binary inverse are
+interdefinable: x-y := x+(zero-y) and -x := zero-x. The law left_inv is
+simply the translation of (-x)+x = zero. This characterizes groups already,
+provided we only allow (zero-x). Law minus_inv `defines' the general x-y in
+terms of the specific zero-y.
+*)
+axclass  add_group < add_monoid, minus
+  left_inv  "(zero-x)+x = zero"
+  minus_inv "x-y = x + (zero-y)"
+
+(* additive abelian groups *)
+
+axclass  add_agroup < add_group
+  plus_commute  "x + y = y + x"
+
+end