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Datei: Ring.thy Sprache: Isabelle

Original von: Isabelle^©

(*  Title:      HOL/Algebra/Ring.thy
    Author:     Clemens Ballarin, started 9 December 1996

With contributions by Martin Baillon.
*)

theory Ring
imports FiniteProduct
begin

section \<open>The Algebraic Hierarchy of Rings\<close>

subsection \<open>Abelian Groups\<close>

record 'a ring = "'a monoid" +
  zero :: 'a ("\\")
  add :: "['a, 'a] \ 'a" (infixl "\\" 65)

abbreviation
  add_monoid :: "('a, 'm) ring_scheme \ ('a, 'm) monoid_scheme"
  where "add_monoid R \ \ carrier = carrier R, mult = add R, one = zero R, \ = (undefined :: 'm) \"

text \<open>Derived operations.\<close>

definition
  a_inv :: "[('a, 'm) ring_scheme, 'a ] \ 'a" ("\\ _" [81] 80)
  where "a_inv R = m_inv (add_monoid R)"

definition
  a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" ("(_ \\ _)" [65,66] 65)
  where "x \\<^bsub>R\<^esub> y = x \\<^bsub>R\<^esub> (\\<^bsub>R\<^esub> y)"

definition
  add_pow :: "[_, ('b :: semiring_1), 'a] \ 'a" ("[_] \\ _" [81, 81] 80)
  where "add_pow R k a = pow (add_monoid R) a k"

locale abelian_monoid =
  fixes G (structure)
  assumes a_comm_monoid:
     "comm_monoid (add_monoid G)"

definition
  finsum :: "[('b, 'm) ring_scheme, 'a \ 'b, 'a set] \ 'b" where
  "finsum G = finprod (add_monoid G)"

syntax
  "_finsum" :: "index \ idt \ 'a set \ 'b \ 'b"
      ("(3\__\_. _)" [1000, 0, 51, 10] 10)
translations
  "\\<^bsub>G\<^esub>i\A. b" \ "CONST finsum G (\i. b) A"
  \<comment> \<open>Beware of argument permutation!\<close>

locale abelian_group = abelian_monoid +
  assumes a_comm_group:
     "comm_group (add_monoid G)"

subsection \<open>Basic Properties\<close>

lemma abelian_monoidI:
  fixes R (structure)
  assumes "\x y. \ x \ carrier R; y \ carrier R \ \ x \ y \ carrier R"
      and "\ \ carrier R"
      and "\x y z. \ x \ carrier R; y \ carrier R; z \ carrier R \ \ (x \ y) \ z = x \ (y \ z)"
      and "\x. x \ carrier R \ \ \ x = x"
      and "\x y. \ x \ carrier R; y \ carrier R \ \ x \ y = y \ x"
  shows "abelian_monoid R"
  by (auto intro!: abelian_monoid.intro comm_monoidI intro: assms)

lemma abelian_monoidE:
  fixes R (structure)
  assumes "abelian_monoid R"
  shows "\x y. \ x \ carrier R; y \ carrier R \ \ x \ y \ carrier R"
    and "\ \ carrier R"
    and "\x y z. \ x \ carrier R; y \ carrier R; z \ carrier R \ \ (x \ y) \ z = x \ (y \ z)"
    and "\x. x \ carrier R \ \ \ x = x"
    and "\x y. \ x \ carrier R; y \ carrier R \ \ x \ y = y \ x"
  using assms unfolding abelian_monoid_def comm_monoid_def comm_monoid_axioms_def monoid_def by auto

lemma abelian_groupI:
  fixes R (structure)
  assumes "\x y. \ x \ carrier R; y \ carrier R \ \ x \ y \ carrier R"
      and "\ \ carrier R"
      and "\x y z. \ x \ carrier R; y \ carrier R; z \ carrier R \ \ (x \ y) \ z = x \ (y \ z)"
      and "\x y. \ x \ carrier R; y \ carrier R \ \ x \ y = y \ x"
      and "\x. x \ carrier R \ \ \ x = x"
      and "\x. x \ carrier R \ \y \ carrier R. y \ x = \"
  shows "abelian_group R"
  by (auto intro!: abelian_group.intro abelian_monoidI
      abelian_group_axioms.intro comm_monoidI comm_groupI
    intro: assms)

lemma abelian_groupE:
  fixes R (structure)
  assumes "abelian_group R"
  shows "\x y. \ x \ carrier R; y \ carrier R \ \ x \ y \ carrier R"
    and "\ \ carrier R"
    and "\x y z. \ x \ carrier R; y \ carrier R; z \ carrier R \ \ (x \ y) \ z = x \ (y \ z)"
    and "\x y. \ x \ carrier R; y \ carrier R \ \ x \ y = y \ x"
    and "\x. x \ carrier R \ \ \ x = x"
    and "\x. x \ carrier R \ \y \ carrier R. y \ x = \"
  using abelian_group.a_comm_group assms comm_groupE by fastforce+

lemma (in abelian_monoid) a_monoid:
  "monoid (add_monoid G)"
by (rule comm_monoid.axioms, rule a_comm_monoid)

lemma (in abelian_group) a_group:
  "group (add_monoid G)"
  by (simp add: group_def a_monoid)
    (simp add: comm_group.axioms group.axioms a_comm_group)

lemmas monoid_record_simps = partial_object.simps monoid.simps

text \<open>Transfer facts from multiplicative structures via interpretation.\<close>

sublocale abelian_monoid <
       add: monoid "(add_monoid G)"
  rewrites "carrier (add_monoid G) = carrier G"
       and "mult (add_monoid G) = add G"
       and "one (add_monoid G) = zero G"
       and "(\a k. pow (add_monoid G) a k) = (\a k. add_pow G k a)"
  by (rule a_monoid) (auto simp add: add_pow_def)

context abelian_monoid
begin

lemmas a_closed = add.m_closed
lemmas zero_closed = add.one_closed
lemmas a_assoc = add.m_assoc
lemmas l_zero = add.l_one
lemmas r_zero = add.r_one
lemmas minus_unique = add.inv_unique

end

sublocale abelian_monoid <
  add: comm_monoid "(add_monoid G)"
  rewrites "carrier (add_monoid G) = carrier G"
       and "mult (add_monoid G) = add G"
       and "one (add_monoid G) = zero G"
       and "finprod (add_monoid G) = finsum G"
       and "pow (add_monoid G) = (\a k. add_pow G k a)"
  by (rule a_comm_monoid) (auto simp: finsum_def add_pow_def)

context abelian_monoid begin

lemmas a_comm = add.m_comm
lemmas a_lcomm = add.m_lcomm
lemmas a_ac = a_assoc a_comm a_lcomm

lemmas finsum_empty = add.finprod_empty
lemmas finsum_insert = add.finprod_insert
lemmas finsum_zero = add.finprod_one
lemmas finsum_closed = add.finprod_closed
lemmas finsum_Un_Int = add.finprod_Un_Int
lemmas finsum_Un_disjoint = add.finprod_Un_disjoint
lemmas finsum_addf = add.finprod_multf
lemmas finsum_cong' = add.finprod_cong'
lemmas finsum_0 = add.finprod_0
lemmas finsum_Suc = add.finprod_Suc
lemmas finsum_Suc2 = add.finprod_Suc2
lemmas finsum_infinite = add.finprod_infinite

lemmas finsum_cong = add.finprod_cong
text \<open>Usually, if this rule causes a failed congruence proof error,
   the reason is that the premise \<open>g \<in> B \<rightarrow> carrier G\<close> cannot be shown.
   Adding @{thm [source] Pi_def} to the simpset is often useful.\<close>

lemmas finsum_reindex = add.finprod_reindex

(* The following would be wrong.  Needed is the equivalent of [^] for addition,
  or indeed the canonical embedding from Nat into the monoid.

lemma finsum_const:
  assumes fin [simp]: "finite A"
      and a [simp]: "a : carrier G"
    shows "finsum G (%x. a) A = a [^] card A"
  using fin apply induct
  apply force
  apply (subst finsum_insert)
  apply auto
  apply (force simp add: Pi_def)
  apply (subst m_comm)
  apply auto
done
*)

lemmas finsum_singleton = add.finprod_singleton

end

sublocale abelian_group <
        add: group "(add_monoid G)"
  rewrites "carrier (add_monoid G) = carrier G"
       and "mult (add_monoid G) = add G"
       and "one (add_monoid G) = zero G"
       and "m_inv (add_monoid G) = a_inv G"
       and "pow (add_monoid G) = (\a k. add_pow G k a)"
  by (rule a_group) (auto simp: m_inv_def a_inv_def add_pow_def)

context abelian_group
begin

lemmas a_inv_closed = add.inv_closed

lemma minus_closed [intro, simp]:
  "[| x \ carrier G; y \ carrier G |] ==> x \ y \ carrier G"
  by (simp add: a_minus_def)

lemmas l_neg = add.l_inv [simp del]
lemmas r_neg = add.r_inv [simp del]
lemmas minus_minus = add.inv_inv
lemmas a_inv_inj = add.inv_inj
lemmas minus_equality = add.inv_equality

end

sublocale abelian_group <
   add: comm_group "(add_monoid G)"
  rewrites "carrier (add_monoid G) = carrier G"
       and "mult (add_monoid G) = add G"
       and "one (add_monoid G) = zero G"
       and "m_inv (add_monoid G) = a_inv G"
       and "finprod (add_monoid G) = finsum G"
       and "pow (add_monoid G) = (\a k. add_pow G k a)"
  by (rule a_comm_group) (auto simp: m_inv_def a_inv_def finsum_def add_pow_def)

lemmas (in abelian_group) minus_add = add.inv_mult

text \<open>Derive an \<open>abelian_group\<close> from a \<open>comm_group\<close>\<close>

lemma comm_group_abelian_groupI:
  fixes G (structure)
  assumes cg: "comm_group (add_monoid G)"
  shows "abelian_group G"
proof -
  interpret comm_group "(add_monoid G)"
    by (rule cg)
  show "abelian_group G" ..
qed

subsection \<open>Rings: Basic Definitions\<close>

locale semiring = abelian_monoid (* for add *) R + monoid (* for mult *) R for R (structure) +
  assumes l_distr: "\ x \ carrier R; y \ carrier R; z \ carrier R \ \ (x \ y) \ z = x \ z \ y \ z"
      and r_distr: "\ x \ carrier R; y \ carrier R; z \ carrier R \ \ z \ (x \ y) = z \ x \ z \ y"
      and l_null[simp]: "x \ carrier R \ \ \ x = \"
      and r_null[simp]: "x \ carrier R \ x \ \ = \"

locale ring = abelian_group (* for add *) R + monoid (* for mult *) R for R (structure) +
  assumes "\ x \ carrier R; y \ carrier R; z \ carrier R \ \ (x \ y) \ z = x \ z \ y \ z"
      and "\ x \ carrier R; y \ carrier R; z \ carrier R \ \ z \ (x \ y) = z \ x \ z \ y"

locale cring = ring + comm_monoid (* for mult *) R

locale "domain" = cring +
  assumes one_not_zero [simp]: "\ \ \"
      and integral: "\ a \ b = \; a \ carrier R; b \ carrier R \ \ a = \ \ b = \"

locale field = "domain" +
  assumes field_Units: "Units R = carrier R - {\}"

subsection \<open>Rings\<close>

lemma ringI:
  fixes R (structure)
  assumes "abelian_group R"
      and "monoid R"
      and "\x y z. \ x \ carrier R; y \ carrier R; z \ carrier R \ \ (x \ y) \ z = x \ z \ y \ z"
      and "\x y z. \ x \ carrier R; y \ carrier R; z \ carrier R \ \ z \ (x \ y) = z \ x \ z \ y"
  shows "ring R"
  by (auto intro: ring.intro
    abelian_group.axioms ring_axioms.intro assms)

lemma ringE:
  fixes R (structure)
  assumes "ring R"
  shows "abelian_group R"
    and "monoid R"
    and "\x y z. \ x \ carrier R; y \ carrier R; z \ carrier R \ \ (x \ y) \ z = x \ z \ y \ z"
    and "\x y z. \ x \ carrier R; y \ carrier R; z \ carrier R \ \ z \ (x \ y) = z \ x \ z \ y"
  using assms unfolding ring_def ring_axioms_def by auto

context ring begin

lemma is_abelian_group: "abelian_group R" ..

lemma is_monoid: "monoid R"
  by (auto intro!: monoidI m_assoc)

lemma is_ring: "ring R"
  by (rule ring_axioms)

end
thm monoid_record_simps
lemmas ring_record_simps = monoid_record_simps ring.simps

lemma cringI:
  fixes R (structure)
  assumes abelian_group: "abelian_group R"
    and comm_monoid: "comm_monoid R"
    and l_distr: "\x y z. \ x \ carrier R; y \ carrier R; z \ carrier R \ \
                            (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
  shows "cring R"
proof (intro cring.intro ring.intro)
  show "ring_axioms R"
    \<comment> \<open>Right-distributivity follows from left-distributivity and
          commutativity.\<close>
  proof (rule ring_axioms.intro)
    fix x y z
    assume R: "x \ carrier R" "y \ carrier R" "z \ carrier R"
    note [simp] = comm_monoid.axioms [OF comm_monoid]
      abelian_group.axioms [OF abelian_group]
      abelian_monoid.a_closed

    from R have "z \ (x \ y) = (x \ y) \ z"
      by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
    also from R have "... = x \ z \ y \ z" by (simp add: l_distr)
    also from R have "... = z \ x \ z \ y"
      by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
    finally show "z \ (x \ y) = z \ x \ z \ y" .
  qed (rule l_distr)
qed (auto intro: cring.intro
  abelian_group.axioms comm_monoid.axioms ring_axioms.intro assms)

lemma cringE:
  fixes R (structure)
  assumes "cring R"
  shows "comm_monoid R"
    and "\x y z. \ x \ carrier R; y \ carrier R; z \ carrier R \ \ (x \ y) \ z = x \ z \ y \ z"
  using assms cring_def by auto (simp add: assms cring.axioms(1) ringE(3))

lemma (in cring) is_cring:
  "cring R" by (rule cring_axioms)

lemma (in ring) minus_zero [simp]: "\ \ = \"
  by (simp add: a_inv_def)

subsubsection \<open>Normaliser for Rings\<close>

lemma (in abelian_group) r_neg1:
  "\ x \ carrier G; y \ carrier G \ \ (\ x) \ (x \ y) = y"
proof -
  assume G: "x \ carrier G" "y \ carrier G"
  then have "(\ x \ x) \ y = y"
    by (simp only: l_neg l_zero)
  with G show ?thesis by (simp add: a_ac)
qed

lemma (in abelian_group) r_neg2:
  "\ x \ carrier G; y \ carrier G \ \ x \ ((\ x) \ y) = y"
proof -
  assume G: "x \ carrier G" "y \ carrier G"
  then have "(x \ \ x) \ y = y"
    by (simp only: r_neg l_zero)
  with G show ?thesis
    by (simp add: a_ac)
qed

context ring begin

text \<open>
  The following proofs are from Jacobson, Basic Algebra I, pp.~88--89.
\<close>

sublocale semiring
proof -
  note [simp] = ring_axioms[unfolded ring_def ring_axioms_def]
  show "semiring R"
  proof (unfold_locales)
    fix x
    assume R: "x \ carrier R"
    then have "\ \ x \ \ \ x = (\ \ \) \ x"
      by (simp del: l_zero r_zero)
    also from R have "... = \ \ x \ \" by simp
    finally have "\ \ x \ \ \ x = \ \ x \ \" .
    with R show "\ \ x = \" by (simp del: r_zero)
    from R have "x \ \ \ x \ \ = x \ (\ \ \)"
      by (simp del: l_zero r_zero)
    also from R have "... = x \ \ \ \" by simp
    finally have "x \ \ \ x \ \ = x \ \ \ \" .
    with R show "x \ \ = \" by (simp del: r_zero)
  qed auto
qed

lemma l_minus:
  "\ x \ carrier R; y \ carrier R \ \ (\ x) \ y = \ (x \ y)"
proof -
  assume R: "x \ carrier R" "y \ carrier R"
  then have "(\ x) \ y \ x \ y = (\ x \ x) \ y" by (simp add: l_distr)
  also from R have "... = \" by (simp add: l_neg)
  finally have "(\ x) \ y \ x \ y = \" .
  with R have "(\ x) \ y \ x \ y \ \ (x \ y) = \ \ \ (x \ y)" by simp
  with R show ?thesis by (simp add: a_assoc r_neg)
qed

lemma r_minus:
  "\ x \ carrier R; y \ carrier R \ \ x \ (\ y) = \ (x \ y)"
proof -
  assume R: "x \ carrier R" "y \ carrier R"
  then have "x \ (\ y) \ x \ y = x \ (\ y \ y)" by (simp add: r_distr)
  also from R have "... = \" by (simp add: l_neg)
  finally have "x \ (\ y) \ x \ y = \" .
  with R have "x \ (\ y) \ x \ y \ \ (x \ y) = \ \ \ (x \ y)" by simp
  with R show ?thesis by (simp add: a_assoc r_neg )
qed

end

lemma (in abelian_group) minus_eq: "x \ y = x \ (\ y)"
  by (rule a_minus_def)

text \<open>Setup algebra method:
  compute distributive normal form in locale contexts\<close>

ML_file \<open>ringsimp.ML\<close>

attribute_setup algebra = \<open>
  Scan.lift ((Args.add >> K true || Args.del >> K false) --| Args.colon || Scan.succeed true)
    -- Scan.lift Args.name -- Scan.repeat Args.term
    >> (fn ((b, n), ts) => if b then Ringsimp.add_struct (n, ts) else Ringsimp.del_struct (n, ts))
\<close> "theorems controlling algebra method"

method_setup algebra = \<open>
  Scan.succeed (SIMPLE_METHOD' o Ringsimp.algebra_tac)
\<close> "normalisation of algebraic structure"

lemmas (in semiring) semiring_simprules
  [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
  a_closed zero_closed  m_closed one_closed
  a_assoc l_zero  a_comm m_assoc l_one l_distr r_zero
  a_lcomm r_distr l_null r_null

lemmas (in ring) ring_simprules
  [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
  a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
  a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq
  r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
  a_lcomm r_distr l_null r_null l_minus r_minus

lemmas (in cring)
  [algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
  _

lemmas (in cring) cring_simprules
  [algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
  a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
  a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq
  r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
  a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus

lemma (in semiring) nat_pow_zero:
  "(n::nat) \ 0 \ \ [^] n = \"
  by (induct n) simp_all

context semiring begin

lemma one_zeroD:
  assumes onezero: "\ = \"
  shows "carrier R = {\}"
proof (rule, rule)
  fix x
  assume xcarr: "x \ carrier R"
  from xcarr have "x = x \ \" by simp
  with onezero have "x = x \ \" by simp
  with xcarr have "x = \" by simp
  then show "x \ {\}" by fast
qed fast

lemma one_zeroI:
  assumes carrzero: "carrier R = {\}"
  shows "\ = \"
proof -
  from one_closed and carrzero
      show "\ = \" by simp
qed

lemma carrier_one_zero: "(carrier R = {\}) = (\ = \)"
  using one_zeroD by blast

lemma carrier_one_not_zero: "(carrier R \ {\}) = (\ \ \)"
  by (simp add: carrier_one_zero)

end

text \<open>Two examples for use of method algebra\<close>

lemma
  fixes R (structure) and S (structure)
  assumes "ring R" "cring S"
  assumes RS: "a \ carrier R" "b \ carrier R" "c \ carrier S" "d \ carrier S"
  shows "a \ (\ (a \ (\ b))) = b \ c \\<^bsub>S\<^esub> d = d \\<^bsub>S\<^esub> c"
proof -
  interpret ring R by fact
  interpret cring S by fact
  from RS show ?thesis by algebra
qed

lemma
  fixes R (structure)
  assumes "ring R"
  assumes R: "a \ carrier R" "b \ carrier R"
  shows "a \ (a \ b) = b"
proof -
  interpret ring R by fact
  from R show ?thesis by algebra
qed

subsubsection \<open>Sums over Finite Sets\<close>

lemma (in semiring) finsum_ldistr:
  "\ finite A; a \ carrier R; f: A \ carrier R \ \
    (\<Oplus> i \<in> A. (f i)) \<otimes> a = (\<Oplus> i \<in> A. ((f i) \<otimes> a))"
proof (induct set: finite)
  case empty then show ?case by simp
next
  case (insert x F) then show ?case by (simp add: Pi_def l_distr)
qed

lemma (in semiring) finsum_rdistr:
  "\ finite A; a \ carrier R; f: A \ carrier R \ \
   a \<otimes> (\<Oplus> i \<in> A. (f i)) = (\<Oplus> i \<in> A. (a \<otimes> (f i)))"
proof (induct set: finite)
  case empty then show ?case by simp
next
  case (insert x F) then show ?case by (simp add: Pi_def r_distr)
qed

text \<open>A quick detour\<close>

lemma add_pow_int_ge: "(k :: int) \ 0 \ [ k ] \\<^bsub>R\<^esub> a = [ nat k ] \\<^bsub>R\<^esub> a" \<^marker>\contributor \Paulo Emílio de Vilhena\\
  by (simp add: add_pow_def int_pow_def nat_pow_def)

lemma add_pow_int_lt: "(k :: int) < 0 \ [ k ] \\<^bsub>R\<^esub> a = \\<^bsub>R\<^esub> ([ nat (- k) ] \\<^bsub>R\<^esub> a)" \<^marker>\contributor \Paulo Emílio de Vilhena\\
  by (simp add: int_pow_def nat_pow_def a_inv_def add_pow_def)

corollary (in semiring) add_pow_ldistr: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  assumes "a \ carrier R" "b \ carrier R"
  shows "([(k :: nat)] \ a) \ b = [k] \ (a \ b)"
proof -
  have "([k] \ a) \ b = (\ i \ {..< k}. a) \ b"
    using add.finprod_const[OF assms(1), of "{..] by simp
  also have " ... = (\ i \ {..< k}. (a \ b))"
    using finsum_ldistr[of "{.. b "\x. a"] assms by simp
  also have " ... = [k] \ (a \ b)"
    using add.finprod_const[of "a \ b" "{..
  finally show ?thesis .
qed

corollary (in semiring) add_pow_rdistr: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  assumes "a \ carrier R" "b \ carrier R"
  shows "a \ ([(k :: nat)] \ b) = [k] \ (a \ b)"
proof -
  have "a \ ([k] \ b) = a \ (\ i \ {..< k}. b)"
    using add.finprod_const[OF assms(2), of "{..] by simp
  also have " ... = (\ i \ {..< k}. (a \ b))"
    using finsum_rdistr[of "{.. a "\x. b"] assms by simp
  also have " ... = [k] \ (a \ b)"
    using add.finprod_const[of "a \ b" "{..
  finally show ?thesis .
qed

(* For integers, we need the uniqueness of the additive inverse *)
lemma (in ring) add_pow_ldistr_int: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  assumes "a \ carrier R" "b \ carrier R"
  shows "([(k :: int)] \ a) \ b = [k] \ (a \ b)"
proof (cases "k \ 0")
  case True thus ?thesis
    using add_pow_int_ge[of k R] add_pow_ldistr[OF assms] by auto
next
  case False thus ?thesis
    using add_pow_int_lt[of k R a] add_pow_int_lt[of k R "a \ b"]
          add_pow_ldistr[OF assms, of "nat (- k)"] assms l_minus by auto
qed

lemma (in ring) add_pow_rdistr_int: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  assumes "a \ carrier R" "b \ carrier R"
  shows "a \ ([(k :: int)] \ b) = [k] \ (a \ b)"
proof (cases "k \ 0")
  case True thus ?thesis
    using add_pow_int_ge[of k R] add_pow_rdistr[OF assms] by auto
next
  case False thus ?thesis
    using add_pow_int_lt[of k R b] add_pow_int_lt[of k R "a \ b"]
          add_pow_rdistr[OF assms, of "nat (- k)"] assms r_minus by auto
qed

subsection \<open>Integral Domains\<close>

context "domain" begin

lemma zero_not_one [simp]: "\ \ \"
  by (rule not_sym) simp

lemma integral_iff: (* not by default a simp rule! *)
  "\ a \ carrier R; b \ carrier R \ \ (a \ b = \) = (a = \ \ b = \)"
proof
  assume "a \ carrier R" "b \ carrier R" "a \ b = \"
  then show "a = \ \ b = \" by (simp add: integral)
next
  assume "a \ carrier R" "b \ carrier R" "a = \ \ b = \"
  then show "a \ b = \" by auto
qed

lemma m_lcancel:
  assumes prem: "a \ \"
    and R: "a \ carrier R" "b \ carrier R" "c \ carrier R"
  shows "(a \ b = a \ c) = (b = c)"
proof
  assume eq: "a \ b = a \ c"
  with R have "a \ (b \ c) = \" by algebra
  with R have "a = \ \ (b \ c) = \" by (simp add: integral_iff)
  with prem and R have "b \ c = \" by auto
  with R have "b = b \ (b \ c)" by algebra
  also from R have "b \ (b \ c) = c" by algebra
  finally show "b = c" .
next
  assume "b = c" then show "a \ b = a \ c" by simp
qed

lemma m_rcancel:
  assumes prem: "a \ \"
    and R: "a \ carrier R" "b \ carrier R" "c \ carrier R"
  shows conc: "(b \ a = c \ a) = (b = c)"
proof -
  from prem and R have "(a \ b = a \ c) = (b = c)" by (rule m_lcancel)
  with R show ?thesis by algebra
qed

end

subsection \<open>Fields\<close>

text \<open>Field would not need to be derived from domain, the properties
  for domain follow from the assumptions of field\<close>

lemma (in field) is_ring: "ring R"
  using ring_axioms .

lemma fieldE :
  fixes R (structure)
  assumes "field R"
  shows "cring R"
    and one_not_zero : "\ \ \"
    and integral: "\a b. \ a \ b = \; a \ carrier R; b \ carrier R \ \ a = \ \ b = \"
  and field_Units: "Units R = carrier R - {\}"
  using assms unfolding field_def field_axioms_def domain_def domain_axioms_def by simp_all

lemma (in cring) cring_fieldI:
  assumes field_Units: "Units R = carrier R - {\}"
  shows "field R"
proof
  from field_Units have "\ \ Units R" by fast
  moreover have "\ \ Units R" by fast
  ultimately show "\ \ \" by force
next
  fix a b
  assume acarr: "a \ carrier R"
    and bcarr: "b \ carrier R"
    and ab: "a \ b = \"
  show "a = \ \ b = \"
  proof (cases "a = \", simp)
    assume "a \ \"
    with field_Units and acarr have aUnit: "a \ Units R" by fast
    from bcarr have "b = \ \ b" by algebra
    also from aUnit acarr have "... = (inv a \ a) \ b" by simp
    also from acarr bcarr aUnit[THEN Units_inv_closed]
    have "... = (inv a) \ (a \ b)" by algebra
    also from ab and acarr bcarr aUnit have "... = (inv a) \ \" by simp
    also from aUnit[THEN Units_inv_closed] have "... = \" by algebra
    finally have "b = \" .
    then show "a = \ \ b = \" by simp
  qed
qed (rule field_Units)

text \<open>Another variant to show that something is a field\<close>
lemma (in cring) cring_fieldI2:
  assumes notzero: "\ \ \"
    and invex: "\a. \a \ carrier R; a \ \\ \ \b\carrier R. a \ b = \"
  shows "field R"
proof -
  have *: "carrier R - {\} \ {y \ carrier R. \x\carrier R. x \ y = \ \ y \ x = \}"
  proof (clarsimp)
    fix x
    assume xcarr: "x \ carrier R" and "x \ \"
    obtain y where ycarr: "y \ carrier R" and xy: "x \ y = \"
      using \<open>x \<noteq> \<zero>\<close> invex xcarr by blast
    with ycarr and xy show "\y\carrier R. y \ x = \ \ x \ y = \"
      using m_comm xcarr by fastforce
  qed
  show ?thesis
    apply (rule cring_fieldI, simp add: Units_def)
    using *
    using group_l_invI notzero set_diff_eq by auto
qed

subsection \<open>Morphisms\<close>

definition
  ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
  where "ring_hom R S =
    {h. h \<in> carrier R \<rightarrow> carrier S \<and>
      (\<forall>x y. x \<in> carrier R \<and> y \<in> carrier R \<longrightarrow>
        h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y \<and> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y) \<and>
      h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}"

lemma ring_hom_memI:
  fixes R (structure) and S (structure)
  assumes "\x. x \ carrier R \ h x \ carrier S"
      and "\x y. \ x \ carrier R; y \ carrier R \ \ h (x \ y) = h x \\<^bsub>S\<^esub> h y"
      and "\x y. \ x \ carrier R; y \ carrier R \ \ h (x \ y) = h x \\<^bsub>S\<^esub> h y"
      and "h \ = \\<^bsub>S\<^esub>"
  shows "h \ ring_hom R S"
  by (auto simp add: ring_hom_def assms Pi_def)

lemma ring_hom_memE:
  fixes R (structure) and S (structure)
  assumes "h \ ring_hom R S"
  shows "\x. x \ carrier R \ h x \ carrier S"
    and "\x y. \ x \ carrier R; y \ carrier R \ \ h (x \ y) = h x \\<^bsub>S\<^esub> h y"
    and "\x y. \ x \ carrier R; y \ carrier R \ \ h (x \ y) = h x \\<^bsub>S\<^esub> h y"
    and "h \ = \\<^bsub>S\<^esub>"
  using assms unfolding ring_hom_def by auto

lemma ring_hom_closed:
  "\ h \ ring_hom R S; x \ carrier R \ \ h x \ carrier S"
  by (auto simp add: ring_hom_def funcset_mem)

lemma ring_hom_mult:
  fixes R (structure) and S (structure)
  shows "\ h \ ring_hom R S; x \ carrier R; y \ carrier R \ \ h (x \ y) = h x \\<^bsub>S\<^esub> h y"
    by (simp add: ring_hom_def)

lemma ring_hom_add:
  fixes R (structure) and S (structure)
  shows "\ h \ ring_hom R S; x \ carrier R; y \ carrier R \ \ h (x \ y) = h x \\<^bsub>S\<^esub> h y"
    by (simp add: ring_hom_def)

lemma ring_hom_one:
  fixes R (structure) and S (structure)
  shows "h \ ring_hom R S \ h \ = \\<^bsub>S\<^esub>"
  by (simp add: ring_hom_def)

lemma ring_hom_zero:
  fixes R (structure) and S (structure)
  assumes "h \ ring_hom R S" "ring R" "ring S"
  shows "h \ = \\<^bsub>S\<^esub>"
proof -
  have "h \ = h \ \\<^bsub>S\<^esub> h \"
    using ring_hom_add[OF assms(1), of \<zero> \<zero>] assms(2)
    by (simp add: ring.ring_simprules(2) ring.ring_simprules(15))
  thus ?thesis
    by (metis abelian_group.l_neg assms ring.is_abelian_group ring.ring_simprules(18) ring.ring_simprules(2) ring_hom_closed)
qed

locale ring_hom_cring =
  R?: cring R + S?: cring S for R (structure) and S (structure) + fixes h
  assumes homh [simp, intro]: "h \ ring_hom R S"
  notes hom_closed [simp, intro] = ring_hom_closed [OF homh]
    and hom_mult [simp] = ring_hom_mult [OF homh]
    and hom_add [simp] = ring_hom_add [OF homh]
    and hom_one [simp] = ring_hom_one [OF homh]

lemma (in ring_hom_cring) hom_zero [simp]: "h \ = \\<^bsub>S\<^esub>"
proof -
  have "h \ \\<^bsub>S\<^esub> h \ = h \ \\<^bsub>S\<^esub> \\<^bsub>S\<^esub>"
    by (simp add: hom_add [symmetric] del: hom_add)
  then show ?thesis by (simp del: S.r_zero)
qed

lemma (in ring_hom_cring) hom_a_inv [simp]:
  "x \ carrier R \ h (\ x) = \\<^bsub>S\<^esub> h x"
proof -
  assume R: "x \ carrier R"
  then have "h x \\<^bsub>S\<^esub> h (\ x) = h x \\<^bsub>S\<^esub> (\\<^bsub>S\<^esub> h x)"
    by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add)
  with R show ?thesis by simp
qed

lemma (in ring_hom_cring) hom_finsum [simp]:
  assumes "f: A \ carrier R"
  shows "h (\ i \ A. f i) = (\\<^bsub>S\<^esub> i \ A. (h o f) i)"
  using assms by (induct A rule: infinite_finite_induct, auto simp: Pi_def)

lemma (in ring_hom_cring) hom_finprod:
  assumes "f: A \ carrier R"
  shows "h (\ i \ A. f i) = (\\<^bsub>S\<^esub> i \ A. (h o f) i)"
  using assms by (induct A rule: infinite_finite_induct, auto simp: Pi_def)

declare ring_hom_cring.hom_finprod [simp]

lemma id_ring_hom [simp]: "id \ ring_hom R R"
  by (auto intro!: ring_hom_memI)

lemma ring_hom_trans: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  "\ f \ ring_hom R S; g \ ring_hom S T \ \ g \ f \ ring_hom R T"
  by (rule ring_hom_memI) (auto simp add: ring_hom_closed ring_hom_mult ring_hom_add ring_hom_one)

subsection\<open>Jeremy Avigad's \<open>More_Finite_Product\<close> material\<close>

(* 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 = \ \ x = \ y"
  by (metis minus_equality)

lemma (in domain) square_eq_one:
  fixes x
  assumes [simp]: "x \ carrier R"
    and "x \ x = \"
  shows "x = \ \ x = \\"
proof -
  have "(x \ \) \ (x \ \ \) = x \ x \ \ \"
    by (simp add: ring_simprules)
  also from \<open>x \<otimes> x = \<one>\<close> have "\<dots> = \<zero>"
    by (simp add: ring_simprules)
  finally have "(x \ \) \ (x \ \ \) = \" .
  then have "(x \ \) = \ \ (x \ \ \) = \"
    by (intro integral) auto
  then show ?thesis
    by (metis add.inv_closed add.inv_solve_right assms(1) l_zero one_closed zero_closed)
qed

lemma (in domain) inv_eq_self: "x \ Units R \ x = inv x \ x = \ \ x = \\"
  by (metis Units_closed Units_l_inv square_eq_one)

text \<open>
  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.)
\<close>

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 [^]\<^bsub>units_of G\<^esub> n = x [^]\<^bsub>G\<^esub> 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) = \"
  by (metis comm_group.power_order_eq_one units_comm_group units_of_carrier units_of_one units_of_pow)

subsection\<open>Jeremy Avigad's \<open>More_Ring\<close> material\<close>

lemma (in cring) field_intro2:
  assumes "\\<^bsub>R\<^esub> \ \\<^bsub>R\<^esub>" and un: "\x. x \ carrier R - {\\<^bsub>R\<^esub>} \ x \ Units R"
  shows "field R"
proof unfold_locales
  show "\ \ \" using assms by auto
  show "\a \ b = \; a \ carrier R;
            b \<in> carrier R\<rbrakk>
           \<Longrightarrow> a = \<zero> \<or> b = \<zero>" for a b
    by (metis un Units_l_cancel insert_Diff_single insert_iff r_null zero_closed)
qed (use assms in \<open>auto simp: Units_def\<close>)

lemma (in monoid) inv_char:
  assumes "x \ carrier G" "y \ carrier G" "x \ y = \" "y \ x = \"
  shows "inv x = y"
  using assms inv_unique' by auto

lemma (in comm_monoid) comm_inv_char: "x \ carrier G \ y \ carrier G \ x \ y = \ \ inv x = y"
  by (simp add: inv_char m_comm)

lemma (in ring) inv_neg_one [simp]: "inv (\ \) = \ \"
  by (simp add: inv_char local.ring_axioms ring.r_minus)

lemma (in monoid) inv_eq_imp_eq: "x \ Units G \ y \ Units G \ inv x = inv y \ x = y"
  by (metis Units_inv_inv)

lemma (in ring) Units_minus_one_closed [intro]: "\ \ \ Units R"
  by (simp add: Units_def) (metis add.l_inv_ex local.minus_minus minus_equality one_closed r_minus r_one)

lemma (in ring) inv_eq_neg_one_eq: "x \ Units R \ inv x = \ \ \ x = \ \"
  by (metis Units_inv_inv inv_neg_one)

lemma (in monoid) inv_eq_one_eq: "x \ Units G \ inv x = \ \ x = \"
  by (metis Units_inv_inv inv_one)

end

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