# Theory Ring

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

With contributions by Martin Baillon.
*)

theory Ring
imports FiniteProduct
begin

section ‹The Algebraic Hierarchy of Rings›

subsection ‹Abelian Groups›

record 'a ring = "'a monoid" +
zero :: 'a ("𝟬ı")
add :: "['a, 'a] ⇒ 'a" (infixl "⊕ı" 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 ‹Derived operations.›

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

definition
a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" ("(_ ⊖ı _)" [65,66] 65)
where "x ⊖⇘R⇙ y = x ⊕⇘R⇙ (⊖⇘R⇙ y)"

definition
add_pow :: "[_, ('b :: semiring_1), 'a] ⇒ 'a" ("[_] ⋅ı _" [81, 81] 80)

locale abelian_monoid =
fixes G (structure)
assumes a_comm_monoid:

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
"⨁⇘G⇙i∈A. b" ⇌ "CONST finsum G (λi. b) A"
― ‹Beware of argument permutation!›

locale abelian_group = abelian_monoid +
assumes a_comm_group:

subsection ‹Basic Properties›

lemma abelian_monoidI:
fixes R (structure)
assumes "⋀x y. ⟦ x ∈ carrier R; y ∈ carrier R ⟧ ⟹ x ⊕ y ∈ carrier R"
and "𝟬 ∈ 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 ⟹ 𝟬 ⊕ 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 "𝟬 ∈ 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 ⟹ 𝟬 ⊕ 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 "𝟬 ∈ 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 ⟹ 𝟬 ⊕ x = x"
and "⋀x. x ∈ carrier R ⟹ ∃y ∈ carrier R. y ⊕ x = 𝟬"
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 "𝟬 ∈ 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 ⟹ 𝟬 ⊕ x = x"
and "⋀x. x ∈ carrier R ⟹ ∃y ∈ carrier R. y ⊕ x = 𝟬"
using abelian_group.a_comm_group assms comm_groupE by fastforce+

lemma (in abelian_monoid) a_monoid:
by (rule comm_monoid.axioms, rule a_comm_monoid)

lemma (in abelian_group) a_group:

lemmas monoid_record_simps = partial_object.simps monoid.simps

text ‹Transfer facts from multiplicative structures via interpretation.›

sublocale abelian_monoid <
rewrites "carrier (add_monoid G) = carrier G"
and "one     (add_monoid G) = zero G"
and "(λa k. pow (add_monoid G) a k) = (λa k. add_pow G k a)"

context abelian_monoid
begin

end

sublocale abelian_monoid <
rewrites "carrier (add_monoid G) = carrier G"
and "one     (add_monoid G) = zero G"
and "finprod (add_monoid G) = finsum G"
by (rule a_comm_monoid) (auto simp: finsum_def add_pow_def)

context abelian_monoid begin

lemmas a_ac = a_assoc a_comm a_lcomm

text ‹Usually, if this rule causes a failed congruence proof error,
the reason is that the premise ‹g ∈ B → carrier G› cannot be shown.
Adding @{thm [source] Pi_def} to the simpset is often useful.›

(* 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 (subst m_comm)
apply auto
done
*)

end

sublocale abelian_group <
rewrites "carrier (add_monoid G) = carrier G"
and "one     (add_monoid G) = zero G"
and "m_inv   (add_monoid G) = a_inv G"
by (rule a_group) (auto simp: m_inv_def a_inv_def add_pow_def)

context abelian_group
begin

lemma minus_closed [intro, simp]:
"[| x ∈ carrier G; y ∈ carrier G |] ==> x ⊖ y ∈ carrier G"

lemmas l_neg = add.l_inv [simp del]
lemmas r_neg = add.r_inv [simp del]

end

sublocale abelian_group <
rewrites "carrier (add_monoid G) = carrier G"
and "one     (add_monoid G) = zero G"
and "m_inv   (add_monoid G) = a_inv G"
and "finprod (add_monoid G) = finsum G"
by (rule a_comm_group) (auto simp: m_inv_def a_inv_def finsum_def add_pow_def)

text ‹Derive an ‹abelian_group› from a ‹comm_group››

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

subsection ‹Rings: Basic Definitions›

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 ⟹ 𝟬 ⊗ x = 𝟬"
and r_null[simp]: "x ∈ carrier R ⟹ x ⊗ 𝟬 = 𝟬"

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]: "𝟭 ≠ 𝟬"
and integral: "⟦ a ⊗ b = 𝟬; a ∈ carrier R; b ∈ carrier R ⟧ ⟹ a = 𝟬 ∨ b = 𝟬"

locale field = "domain" +
assumes field_Units: "Units R = carrier R - {𝟬}"

subsection ‹Rings›

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 ⊕ y) ⊗ z = x ⊗ z ⊕ y ⊗ z"
shows "cring R"
proof (intro cring.intro ring.intro)
show "ring_axioms R"
― ‹Right-distributivity follows from left-distributivity and
commutativity.›
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]: "⊖ 𝟬 = 𝟬"

subsubsection ‹Normaliser for Rings›

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
qed

context ring begin

text ‹
The following proofs are from Jacobson, Basic Algebra I, pp.~88--89.
›

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 "𝟬 ⊗ x ⊕ 𝟬 ⊗ x = (𝟬 ⊕ 𝟬) ⊗ x"
by (simp del: l_zero r_zero)
also from R have "... = 𝟬 ⊗ x ⊕ 𝟬" by simp
finally have "𝟬 ⊗ x ⊕ 𝟬 ⊗ x = 𝟬 ⊗ x ⊕ 𝟬" .
with R show "𝟬 ⊗ x = 𝟬" by (simp del: r_zero)
from R have "x ⊗ 𝟬 ⊕ x ⊗ 𝟬 = x ⊗ (𝟬 ⊕ 𝟬)"
by (simp del: l_zero r_zero)
also from R have "... = x ⊗ 𝟬 ⊕ 𝟬" by simp
finally have "x ⊗ 𝟬 ⊕ x ⊗ 𝟬 = x ⊗ 𝟬 ⊕ 𝟬" .
with R show "x ⊗ 𝟬 = 𝟬" 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 "... = 𝟬" by (simp add: l_neg)
finally have "(⊖ x) ⊗ y ⊕ x ⊗ y = 𝟬" .
with R have "(⊖ x) ⊗ y ⊕ x ⊗ y ⊕ ⊖ (x ⊗ y) = 𝟬 ⊕ ⊖ (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 "... = 𝟬" by (simp add: l_neg)
finally have "x ⊗ (⊖ y) ⊕ x ⊗ y = 𝟬" .
with R have "x ⊗ (⊖ y) ⊕ x ⊗ y ⊕ ⊖ (x ⊗ y) = 𝟬 ⊕ ⊖ (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 ‹Setup algebra method:
compute distributive normal form in locale contexts›

ML_file "ringsimp.ML"

attribute_setup algebra = ‹
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))
› "theorems controlling algebra method"

method_setup algebra = ‹
Scan.succeed (SIMPLE_METHOD' o Ringsimp.algebra_tac)
› "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 ⟹ 𝟬 [^] n = 𝟬"
by (induct n) simp_all

context semiring begin

lemma one_zeroD:
assumes onezero: "𝟭 = 𝟬"
shows "carrier R = {𝟬}"
proof (rule, rule)
fix x
assume xcarr: "x ∈ carrier R"
from xcarr have "x = x ⊗ 𝟭" by simp
with onezero have "x = x ⊗ 𝟬" by simp
with xcarr have "x = 𝟬" by simp
then show "x ∈ {𝟬}" by fast
qed fast

lemma one_zeroI:
assumes carrzero: "carrier R = {𝟬}"
shows "𝟭 = 𝟬"
proof -
from one_closed and carrzero
show "𝟭 = 𝟬" by simp
qed

lemma carrier_one_zero: "(carrier R = {𝟬}) = (𝟭 = 𝟬)"
using one_zeroD by blast

lemma carrier_one_not_zero: "(carrier R ≠ {𝟬}) = (𝟭 ≠ 𝟬)"

end

text ‹Two examples for use of method algebra›

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 ⊗⇘S⇙ d = d ⊗⇘S⇙ 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 ‹Sums over Finite Sets›

lemma (in semiring) finsum_ldistr:
"⟦ finite A; a ∈ carrier R; f: A → carrier R ⟧ ⟹
(⨁ i ∈ A. (f i)) ⊗ a = (⨁ i ∈ A. ((f i) ⊗ 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 ⊗ (⨁ i ∈ A. (f i)) = (⨁ i ∈ A. (a ⊗ (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

(* ************************************************************************** *)
(* Contributed by Paulo E. de Vilhena.                                        *)

text ‹A quick detour›

lemma add_pow_int_ge: "(k :: int) ≥ 0 ⟹ [ k ] ⋅⇘R⇙ a = [ nat k ] ⋅⇘R⇙ a"

lemma add_pow_int_lt: "(k :: int) < 0 ⟹ [ k ] ⋅⇘R⇙ a = ⊖⇘R⇙ ([ nat (- k) ] ⋅⇘R⇙ a)"

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 "{..<k}"] by simp
also have " ... = (⨁ i ∈ {..< k}. (a ⊗ b))"
using finsum_ldistr[of "{..<k}" b "λx. a"] assms by simp
also have " ... = [k] ⋅ (a ⊗ b)"
using add.finprod_const[of "a ⊗ b" "{..<k}"] assms by simp
finally show ?thesis .
qed

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 "{..<k}"] by simp
also have " ... = (⨁ i ∈ {..< k}. (a ⊗ b))"
using finsum_rdistr[of "{..<k}" a "λx. b"] assms by simp
also have " ... = [k] ⋅ (a ⊗ b)"
using add.finprod_const[of "a ⊗ b" "{..<k}"] assms by simp
finally show ?thesis .
qed

(* For integers, we need the uniqueness of the additive inverse *)
assumes "a ∈ carrier R" "b ∈ carrier R"
shows "([(k :: int)] ⋅ a) ⊗ b = [k] ⋅ (a ⊗ b)"
proof (cases "k ≥ 0")
case True thus ?thesis
next
case False thus ?thesis
add_pow_ldistr[OF assms, of "nat (- k)"] assms l_minus by auto
qed

assumes "a ∈ carrier R" "b ∈ carrier R"
shows "a ⊗ ([(k :: int)] ⋅ b) = [k] ⋅ (a ⊗ b)"
proof (cases "k ≥ 0")
case True thus ?thesis
next
case False thus ?thesis
add_pow_rdistr[OF assms, of "nat (- k)"] assms r_minus by auto
qed

subsection ‹Integral Domains›

context "domain" begin

lemma zero_not_one [simp]: "𝟬 ≠ 𝟭"
by (rule not_sym) simp

lemma integral_iff: (* not by default a simp rule! *)
"⟦ a ∈ carrier R; b ∈ carrier R ⟧ ⟹ (a ⊗ b = 𝟬) = (a = 𝟬 ∨ b = 𝟬)"
proof
assume "a ∈ carrier R" "b ∈ carrier R" "a ⊗ b = 𝟬"
then show "a = 𝟬 ∨ b = 𝟬" by (simp add: integral)
next
assume "a ∈ carrier R" "b ∈ carrier R" "a = 𝟬 ∨ b = 𝟬"
then show "a ⊗ b = 𝟬" by auto
qed

lemma m_lcancel:
assumes prem: "a ≠ 𝟬"
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) = 𝟬" by algebra
with R have "a = 𝟬 ∨ (b ⊖ c) = 𝟬" by (simp add: integral_iff)
with prem and R have "b ⊖ c = 𝟬" 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 ≠ 𝟬"
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 ‹Fields›

text ‹Field would not need to be derived from domain, the properties
for domain follow from the assumptions of field›

lemma fieldE :
fixes R (structure)
assumes "field R"
shows "cring R"
and one_not_zero : "𝟭 ≠ 𝟬"
and integral: "⋀a b. ⟦ a ⊗ b = 𝟬; a ∈ carrier R; b ∈ carrier R ⟧ ⟹ a = 𝟬 ∨ b = 𝟬"
and field_Units: "Units R = carrier R - {𝟬}"
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 - {𝟬}"
shows "field R"
proof
from field_Units have "𝟬 ∉ Units R" by fast
moreover have "𝟭 ∈ Units R" by fast
ultimately show "𝟭 ≠ 𝟬" by force
next
fix a b
assume acarr: "a ∈ carrier R"
and bcarr: "b ∈ carrier R"
and ab: "a ⊗ b = 𝟬"
show "a = 𝟬 ∨ b = 𝟬"
proof (cases "a = 𝟬", simp)
assume "a ≠ 𝟬"
with field_Units and acarr have aUnit: "a ∈ Units R" by fast
from bcarr have "b = 𝟭 ⊗ 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) ⊗ 𝟬" by simp
also from aUnit[THEN Units_inv_closed] have "... = 𝟬" by algebra
finally have "b = 𝟬" .
then show "a = 𝟬 ∨ b = 𝟬" by simp
qed
qed (rule field_Units)

text ‹Another variant to show that something is a field›
lemma (in cring) cring_fieldI2:
assumes notzero: "𝟬 ≠ 𝟭"
and invex: "⋀a. ⟦a ∈ carrier R; a ≠ 𝟬⟧ ⟹ ∃b∈carrier R. a ⊗ b = 𝟭"
shows "field R"
proof -
have *: "carrier R - {𝟬} ⊆ {y ∈ carrier R. ∃x∈carrier R. x ⊗ y = 𝟭 ∧ y ⊗ x = 𝟭}"
proof (clarsimp)
fix x
assume xcarr: "x ∈ carrier R" and "x ≠ 𝟬"
obtain y where ycarr: "y ∈ carrier R" and xy: "x ⊗ y = 𝟭"
using ‹x ≠ 𝟬› invex xcarr by blast
with ycarr and xy show "∃y∈carrier R. y ⊗ x = 𝟭 ∧ x ⊗ y = 𝟭"
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 ‹Morphisms›

definition
ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
where "ring_hom R S =
{h. h ∈ carrier R → carrier S ∧
(∀x y. x ∈ carrier R ∧ y ∈ carrier R ⟶
h (x ⊗⇘R⇙ y) = h x ⊗⇘S⇙ h y ∧ h (x ⊕⇘R⇙ y) = h x ⊕⇘S⇙ h y) ∧
h 𝟭⇘R⇙ = 𝟭⇘S⇙}"

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 ⊗⇘S⇙ h y"
and "⋀x y. ⟦ x ∈ carrier R; y ∈ carrier R ⟧ ⟹ h (x ⊕ y) = h x ⊕⇘S⇙ h y"
and "h 𝟭 = 𝟭⇘S⇙"
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 ⊗⇘S⇙ h y"
and "⋀x y. ⟦ x ∈ carrier R; y ∈ carrier R ⟧ ⟹ h (x ⊕ y) = h x ⊕⇘S⇙ h y"
and "h 𝟭 = 𝟭⇘S⇙"
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 ⊗⇘S⇙ h y"

fixes R (structure) and S (structure)
shows "⟦ h ∈ ring_hom R S; x ∈ carrier R; y ∈ carrier R ⟧ ⟹ h (x ⊕ y) = h x ⊕⇘S⇙ h y"

lemma ring_hom_one:
fixes R (structure) and S (structure)
shows "h ∈ ring_hom R S ⟹ h 𝟭 = 𝟭⇘S⇙"

lemma ring_hom_zero:
fixes R (structure) and S (structure)
assumes "h ∈ ring_hom R S" "ring R" "ring S"
shows "h 𝟬 = 𝟬⇘S⇙"
proof -
have "h 𝟬 = h 𝟬 ⊕⇘S⇙ h 𝟬"
using ring_hom_add[OF assms(1), of 𝟬 𝟬] assms(2)
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_one [simp] = ring_hom_one [OF homh]

lemma (in ring_hom_cring) hom_zero [simp]: "h 𝟬 = 𝟬⇘S⇙"
proof -
have "h 𝟬 ⊕⇘S⇙ h 𝟬 = h 𝟬 ⊕⇘S⇙ 𝟬⇘S⇙"
then show ?thesis by (simp del: S.r_zero)
qed

lemma (in ring_hom_cring) hom_a_inv [simp]:
"x ∈ carrier R ⟹ h (⊖ x) = ⊖⇘S⇙ h x"
proof -
assume R: "x ∈ carrier R"
then have "h x ⊕⇘S⇙ h (⊖ x) = h x ⊕⇘S⇙ (⊖⇘S⇙ h x)"
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) = (⨁⇘S⇙ 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) = (⨂⇘S⇙ 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)

(* Next lemma contributed by Paulo EmÃ­lio de Vilhena. *)

lemma ring_hom_trans:
"⟦ f ∈ ring_hom R S; g ∈ ring_hom S T ⟧ ⟹ g ∘ f ∈ ring_hom R T"

(* 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"
also from ‹x ⊗ x = 𝟭› have "… = 𝟬"
finally have "(x ⊕ 𝟭) ⊗ (x ⊕ ⊖ 𝟭) = 𝟬" .
then have "(x ⊕ 𝟭) = 𝟬 ∨ (x ⊕ ⊖ 𝟭) = 𝟬"
by (intro integral) auto
then show ?thesis
qed

lemma (in 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 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"
by (metis comm_group.power_order_eq_one units_comm_group units_of_carrier units_of_one units_of_pow)

lemma (in cring) field_intro2:
assumes "𝟬⇘R⇙ ≠ 𝟭⇘R⇙" and un: "⋀x. x ∈ carrier R - {𝟬⇘R⇙} ⟹ x ∈ Units R"
shows "field R"
proof unfold_locales
show "𝟭 ≠ 𝟬" using assms by auto
show "⟦a ⊗ b = 𝟬; a ∈ carrier R;
b ∈ carrier R⟧
⟹ a = 𝟬 ∨ b = 𝟬" for a b
by (metis un Units_l_cancel insert_Diff_single insert_iff r_null zero_closed)
qed (use assms in ‹auto simp: Units_def›)

lemma (in monoid) inv_char:
assumes "x ∈ carrier G" "y ∈ carrier G" "x ⊗ y = 𝟭" "y ⊗ x = 𝟭"
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 = 𝟭 ⟹ inv x = y"

lemma (in ring) inv_neg_one [simp]: "inv (⊖ 𝟭) = ⊖ 𝟭"
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]: "⊖ 𝟭 ∈ Units R"