Theory More_Group

theory More_Group
imports Ring
(*  Title:      HOL/Algebra/More_Group.thy
    Author:     Jeremy Avigad
*)

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