Theory More_Finite_Product

(*  Title:      HOL/Algebra/More_Finite_Product.thy
    Author:     Jeremy Avigad
*)

section ‹More on finite products›

theory More_Finite_Product
imports
  More_Group
begin

lemma (in comm_monoid) finprod_UN_disjoint:
  "finite I ⟹ (ALL i:I. finite (A i)) ⟶ (ALL i:I. ALL j:I. i ~= j ⟶
     (A i) Int (A j) = {}) ⟶
      (ALL i:I. ALL x: (A i). g x : carrier G) ⟶
        finprod G g (UNION I A) = finprod G (%i. finprod G g (A i)) I"
  apply (induct set: finite)
  apply force
  apply clarsimp
  apply (subst finprod_Un_disjoint)
  apply blast
  apply (erule finite_UN_I)
  apply blast
  apply (fastforce)
  apply (auto intro!: funcsetI finprod_closed)
  done

lemma (in comm_monoid) finprod_Union_disjoint:
  "[| finite C; (ALL A:C. finite A & (ALL x:A. f x : carrier G));
      (ALL A:C. ALL B:C. A ~= B --> A Int B = {}) |]
   ==> finprod G f (⋃C) = finprod G (finprod G f) C"
  apply (frule finprod_UN_disjoint [of C id f])
  apply auto
  done

lemma (in comm_monoid) finprod_one:
    "finite A ⟹ (⋀x. x:A ⟹ f x = 𝟭) ⟹ finprod G f A = 𝟭"
  by (induct set: finite) auto


(* need better simplification rules for rings *)
(* the next one holds more generally for abelian groups *)

lemma (in cring) sum_zero_eq_neg: "x : carrier R ⟹ y : carrier R ⟹ x ⊕ y = 𝟬 ⟹ x = ⊖ y"
  by (metis minus_equality)

lemma (in domain) square_eq_one:
  fixes x
  assumes [simp]: "x : carrier R"
    and "x ⊗ x = 𝟭"
  shows "x = 𝟭 | x = ⊖𝟭"
proof -
  have "(x ⊕ 𝟭) ⊗ (x ⊕ ⊖ 𝟭) = x ⊗ x ⊕ ⊖ 𝟭"
    by (simp add: ring_simprules)
  also from ‹x ⊗ x = 𝟭› have "… = 𝟬"
    by (simp add: ring_simprules)
  finally have "(x ⊕ 𝟭) ⊗ (x ⊕ ⊖ 𝟭) = 𝟬" .
  then have "(x ⊕ 𝟭) = 𝟬 | (x ⊕ ⊖ 𝟭) = 𝟬"
    by (intro integral, auto)
  then show ?thesis
    apply auto
    apply (erule notE)
    apply (rule sum_zero_eq_neg)
    apply auto
    apply (subgoal_tac "x = ⊖ (⊖ 𝟭)")
    apply (simp add: ring_simprules)
    apply (rule sum_zero_eq_neg)
    apply auto
    done
qed

lemma (in Ring.domain) inv_eq_self: "x : Units R ⟹ x = inv x ⟹ x = 𝟭 ∨ x = ⊖𝟭"
  by (metis Units_closed Units_l_inv square_eq_one)


text ‹
  The following translates theorems about groups to the facts about
  the units of a ring. (The list should be expanded as more things are
  needed.)
›

lemma (in ring) finite_ring_finite_units [intro]: "finite (carrier R) ⟹ finite (Units R)"
  by (rule finite_subset) auto

lemma (in monoid) units_of_pow:
  fixes n :: nat
  shows "x ∈ Units G ⟹ x (^)⇘units_of G⇙ n = x (^)⇘G⇙ n"
  apply (induct n)
  apply (auto simp add: units_group group.is_monoid
    monoid.nat_pow_0 monoid.nat_pow_Suc units_of_one units_of_mult)
  done

lemma (in cring) units_power_order_eq_one: "finite (Units R) ⟹ a : Units R
    ⟹ a (^) card(Units R) = 𝟭"
  apply (subst units_of_carrier [symmetric])
  apply (subst units_of_one [symmetric])
  apply (subst units_of_pow [symmetric])
  apply assumption
  apply (rule comm_group.power_order_eq_one)
  apply (rule units_comm_group)
  apply (unfold units_of_def, auto)
  done

end