# Theory More_Group

theory More_Group
imports Ring
```(*  Title:      HOL/Algebra/More_Group.thy
*)

section ‹More on groups›

theory More_Group
imports
Ring
begin

text ‹
Show that the units in any monoid give rise to a group.

The file Residues.thy provides some infrastructure to use
facts about the unit group within the ring locale.
›

definition units_of :: "('a, 'b) monoid_scheme => 'a monoid" where
"units_of G == (| carrier = Units G,
Group.monoid.mult = Group.monoid.mult G,
one  = one G |)"

lemma (in monoid) units_group: "group(units_of G)"
apply (unfold units_of_def)
apply (rule groupI)
apply auto
apply (subst m_assoc)
apply auto
apply (rule_tac x = "inv x" in bexI)
apply auto
done

lemma (in comm_monoid) units_comm_group: "comm_group(units_of G)"
apply (rule group.group_comm_groupI)
apply (rule units_group)
apply (insert comm_monoid_axioms)
apply (unfold units_of_def Units_def comm_monoid_def comm_monoid_axioms_def)
apply auto
done

lemma units_of_carrier: "carrier (units_of G) = Units G"
unfolding units_of_def by auto

lemma units_of_mult: "mult(units_of G) = mult G"
unfolding units_of_def by auto

lemma units_of_one: "one(units_of G) = one G"
unfolding units_of_def by auto

lemma (in monoid) units_of_inv: "x : Units G ==> m_inv (units_of G) x = m_inv G x"
apply (rule sym)
apply (subst m_inv_def)
apply (rule the1_equality)
apply (rule ex_ex1I)
apply (subst (asm) Units_def)
apply auto
apply (erule inv_unique)
apply auto
apply (rule Units_closed)
apply (simp_all only: units_of_carrier [symmetric])
apply (insert units_group)
apply auto
apply (subst units_of_mult [symmetric])
apply (subst units_of_one [symmetric])
apply (erule group.r_inv, assumption)
apply (subst units_of_mult [symmetric])
apply (subst units_of_one [symmetric])
apply (erule group.l_inv, assumption)
done

lemma (in group) inj_on_const_mult: "a: (carrier G) ==> inj_on (%x. a ⊗ x) (carrier G)"
unfolding inj_on_def by auto

lemma (in group) surj_const_mult: "a : (carrier G) ==> (%x. a ⊗ x) ` (carrier G) = (carrier G)"
apply (auto simp add: image_def)
apply (rule_tac x = "(m_inv G a) ⊗ x" in bexI)
apply auto
(* auto should get this. I suppose we need "comm_monoid_simprules"
for ac_simps rewriting. *)
apply (subst m_assoc [symmetric])
apply auto
done

lemma (in group) l_cancel_one [simp]:
"x : carrier G ⟹ a : carrier G ⟹ (x ⊗ a = x) = (a = one G)"
apply auto
apply (subst l_cancel [symmetric])
prefer 4
apply (erule ssubst)
apply auto
done

lemma (in group) r_cancel_one [simp]: "x : carrier G ⟹ a : carrier G ⟹
(a ⊗ x = x) = (a = one G)"
apply auto
apply (subst r_cancel [symmetric])
prefer 4
apply (erule ssubst)
apply auto
done

(* Is there a better way to do this? *)
lemma (in group) l_cancel_one' [simp]: "x : carrier G ⟹ a : carrier G ⟹
(x = x ⊗ a) = (a = one G)"
apply (subst eq_commute)
apply simp
done

lemma (in group) r_cancel_one' [simp]: "x : carrier G ⟹ a : carrier G ⟹
(x = a ⊗ x) = (a = one G)"
apply (subst eq_commute)
apply simp
done

(* This should be generalized to arbitrary groups, not just commutative
ones, using Lagrange's theorem. *)

lemma (in comm_group) power_order_eq_one:
assumes fin [simp]: "finite (carrier G)"
and a [simp]: "a : carrier G"
shows "a (^) card(carrier G) = one G"
proof -
have "(⨂x∈carrier G. x) = (⨂x∈carrier G. a ⊗ x)"
by (subst (2) finprod_reindex [symmetric],
auto simp add: Pi_def inj_on_const_mult surj_const_mult)
also have "… = (⨂x∈carrier G. a) ⊗ (⨂x∈carrier G. x)"
by (auto simp add: finprod_multf Pi_def)
also have "(⨂x∈carrier G. a) = a (^) card(carrier G)"
by (auto simp add: finprod_const)
finally show ?thesis
(* uses the preceeding lemma *)
by auto
qed

end
```