Theory Ring

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


theory Ring
imports FiniteProduct
begin

section {* The Algebraic Hierarchy of Rings *}

subsection {* Abelian Groups *}

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

text {* Derived operations. *}

definition
a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _" [81] 80)
where "a_inv R = m_inv (| carrier = carrier R, mult = add R, one = zero R |)"

definition
a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" (infixl "\<ominus>\<index>" 65)
where "[| x ∈ carrier R; y ∈ carrier R |] ==> x \<ominus>R y = x ⊕R (\<ominus>R y)"

locale abelian_monoid =
fixes G (structure)
assumes a_comm_monoid:
"comm_monoid (| carrier = carrier G, mult = add G, one = zero G |)"

definition
finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b" where
"finsum G = finprod (| carrier = carrier G, mult = add G, one = zero G |)"

syntax
"_finsum" :: "index => idt => 'a set => 'b => 'b"
("(3\<Oplus>__:_. _)" [1000, 0, 51, 10] 10)
syntax (xsymbols)
"_finsum" :: "index => idt => 'a set => 'b => 'b"
("(3\<Oplus>__∈_. _)" [1000, 0, 51, 10] 10)
syntax (HTML output)
"_finsum" :: "index => idt => 'a set => 'b => 'b"
("(3\<Oplus>__∈_. _)" [1000, 0, 51, 10] 10)
translations
"\<Oplus>\<index>i:A. b" == "CONST finsum \<struct>\<index> (%i. b) A"
-- {* Beware of argument permutation! *}


locale abelian_group = abelian_monoid +
assumes a_comm_group:
"comm_group (| carrier = carrier G, mult = add G, one = zero G |)"


subsection {* Basic Properties *}

lemma abelian_monoidI:
fixes R (structure)
assumes a_closed:
"!!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> x ⊕ y ∈ carrier R"
and zero_closed: "\<zero> ∈ carrier R"
and a_assoc:
"!!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==>
(x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)"

and l_zero: "!!x. x ∈ carrier R ==> \<zero> ⊕ x = x"
and a_comm:
"!!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_groupI:
fixes R (structure)
assumes a_closed:
"!!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> x ⊕ y ∈ carrier R"
and zero_closed: "zero R ∈ carrier R"
and a_assoc:
"!!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |] ==>
(x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)"

and a_comm:
"!!x y. [| x ∈ carrier R; y ∈ carrier R |] ==> x ⊕ y = y ⊕ x"
and l_zero: "!!x. x ∈ carrier R ==> \<zero> ⊕ x = x"
and l_inv_ex: "!!x. x ∈ carrier R ==> EX y : carrier R. y ⊕ x = \<zero>"
shows "abelian_group R"
by (auto intro!: abelian_group.intro abelian_monoidI
abelian_group_axioms.intro comm_monoidI comm_groupI
intro: assms)

lemma (in abelian_monoid) a_monoid:
"monoid (| carrier = carrier G, mult = add G, one = zero G |)"
by (rule comm_monoid.axioms, rule a_comm_monoid)

lemma (in abelian_group) a_group:
"group (| carrier = carrier G, mult = add G, one = zero 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 {* Transfer facts from multiplicative structures via interpretation. *}

sublocale abelian_monoid <
add!: monoid "(| carrier = carrier G, mult = add G, one = zero G |)"
where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
by (rule a_monoid) auto

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 "(| carrier = carrier G, mult = add G, one = zero G |)"
where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
and "finprod (| carrier = carrier G, mult = add G, one = zero G |) = finsum G"
by (rule a_comm_monoid) (auto simp: finsum_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_add = add.finprod_mult

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


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 "(| carrier = carrier G, mult = add G, one = zero G |)"
where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
and "m_inv (| carrier = carrier G, mult = add G, one = zero G |) = a_inv G"
by (rule a_group) (auto simp: m_inv_def a_inv_def)

context abelian_group begin

lemmas a_inv_closed = add.inv_closed

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

lemmas a_l_cancel = add.l_cancel
lemmas a_r_cancel = add.r_cancel
lemmas l_neg = add.l_inv [simp del]
lemmas r_neg = add.r_inv [simp del]
lemmas minus_zero = add.inv_one
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 "(| carrier = carrier G, mult = add G, one = zero G |)"
where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
and "m_inv (| carrier = carrier G, mult = add G, one = zero G |) = a_inv G"
and "finprod (| carrier = carrier G, mult = add G, one = zero G |) = finsum G"
by (rule a_comm_group) (auto simp: m_inv_def a_inv_def finsum_def)

lemmas (in abelian_group) minus_add = add.inv_mult

text {* Derive an @{text "abelian_group"} from a @{text "comm_group"} *}

lemma comm_group_abelian_groupI:
fixes G (structure)
assumes cg: "comm_group (|carrier = carrier G, mult = add G, one = zero G|)),"
shows "abelian_group G"
proof -
interpret comm_group "(|carrier = carrier G, mult = add G, one = zero G|)),"
by (rule cg)
show "abelian_group G" ..
qed


subsection {* Rings: Basic Definitions *}

locale ring = abelian_group R + monoid 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"


locale cring = ring + comm_monoid R

locale "domain" = cring +
assumes one_not_zero [simp]: "\<one> ~= \<zero>"
and integral: "[| a ⊗ b = \<zero>; a ∈ carrier R; b ∈ carrier R |] ==>
a = \<zero> | b = \<zero>"


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


subsection {* Rings *}

lemma ringI:
fixes R (structure)
assumes abelian_group: "abelian_group R"
and monoid: "monoid R"
and l_distr: "!!x y z. [| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |]
==> (x ⊕ y) ⊗ z = x ⊗ z ⊕ y ⊗ z"

and r_distr: "!!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)

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

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 (in cring) is_comm_monoid:
"comm_monoid R"
by (auto intro!: comm_monoidI m_assoc m_comm)
*)


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


subsubsection {* Normaliser for Rings *}

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

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

context ring begin

text {*
The following proofs are from Jacobson, Basic Algebra I, pp.~88--89.
*}


lemma l_null [simp]:
"x ∈ carrier R ==> \<zero> ⊗ x = \<zero>"
proof -
assume R: "x ∈ carrier R"
then have "\<zero> ⊗ x ⊕ \<zero> ⊗ x = (\<zero> ⊕ \<zero>) ⊗ x"
by (simp add: l_distr del: l_zero r_zero)
also from R have "... = \<zero> ⊗ x ⊕ \<zero>" by simp
finally have "\<zero> ⊗ x ⊕ \<zero> ⊗ x = \<zero> ⊗ x ⊕ \<zero>" .
with R show ?thesis by (simp del: r_zero)
qed

lemma r_null [simp]:
"x ∈ carrier R ==> x ⊗ \<zero> = \<zero>"
proof -
assume R: "x ∈ carrier R"
then have "x ⊗ \<zero> ⊕ x ⊗ \<zero> = x ⊗ (\<zero> ⊕ \<zero>)"
by (simp add: r_distr del: l_zero r_zero)
also from R have "... = x ⊗ \<zero> ⊕ \<zero>" by simp
finally have "x ⊗ \<zero> ⊕ x ⊗ \<zero> = x ⊗ \<zero> ⊕ \<zero>" .
with R show ?thesis by (simp del: r_zero)
qed

lemma l_minus:
"[| x ∈ carrier R; y ∈ carrier R |] ==> \<ominus> x ⊗ y = \<ominus> (x ⊗ y)"
proof -
assume R: "x ∈ carrier R" "y ∈ carrier R"
then have "(\<ominus> x) ⊗ y ⊕ x ⊗ y = (\<ominus> x ⊕ x) ⊗ y" by (simp add: l_distr)
also from R have "... = \<zero>" by (simp add: l_neg)
finally have "(\<ominus> x) ⊗ y ⊕ x ⊗ y = \<zero>" .
with R have "(\<ominus> x) ⊗ y ⊕ x ⊗ y ⊕ \<ominus> (x ⊗ y) = \<zero> ⊕ \<ominus> (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 ⊗ \<ominus> y = \<ominus> (x ⊗ y)"
proof -
assume R: "x ∈ carrier R" "y ∈ carrier R"
then have "x ⊗ (\<ominus> y) ⊕ x ⊗ y = x ⊗ (\<ominus> y ⊕ y)" by (simp add: r_distr)
also from R have "... = \<zero>" by (simp add: l_neg)
finally have "x ⊗ (\<ominus> y) ⊕ x ⊗ y = \<zero>" .
with R have "x ⊗ (\<ominus> y) ⊕ x ⊗ y ⊕ \<ominus> (x ⊗ y) = \<zero> ⊕ \<ominus> (x ⊗ y)" by simp
with R show ?thesis by (simp add: a_assoc r_neg )
qed

end

lemma (in abelian_group) minus_eq:
"[| x ∈ carrier G; y ∈ carrier G |] ==> x \<ominus> y = x ⊕ \<ominus> y"
by (simp only: a_minus_def)

text {* Setup algebra method:
compute distributive normal form in locale contexts *}


ML_file "ringsimp.ML"

setup Algebra.attrib_setup

method_setup algebra = {*
Scan.succeed (SIMPLE_METHOD' o Algebra.algebra_tac)
*}
"normalisation of algebraic structure"

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 cring) nat_pow_zero:
"(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>"
by (induct n) simp_all

context ring begin

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

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

lemma carrier_one_zero: "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
apply rule
apply (erule one_zeroI)
apply (erule one_zeroD)
done

lemma carrier_one_not_zero: "(carrier R ≠ {\<zero>}) = (\<one> ≠ \<zero>)"
by (simp add: carrier_one_zero)

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 ⊕ \<ominus> (a ⊕ \<ominus> 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 \<ominus> (a \<ominus> b) = b"
proof -
interpret ring R by fact
from R show ?thesis by algebra
qed


subsubsection {* Sums over Finite Sets *}

lemma (in ring) finsum_ldistr:
"[| finite A; a ∈ carrier R; f ∈ A -> carrier R |] ==>
finsum R f A ⊗ a = finsum R (%i. f i ⊗ a) 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 ring) finsum_rdistr:
"[| finite A; a ∈ carrier R; f ∈ A -> carrier R |] ==>
a ⊗ finsum R f A = finsum R (%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 r_distr)
qed


subsection {* Integral Domains *}

context "domain" begin

lemma zero_not_one [simp]:
"\<zero> ~= \<one>"
by (rule not_sym) simp

lemma integral_iff: (* not by default a simp rule! *)
"[| a ∈ carrier R; b ∈ carrier R |] ==> (a ⊗ b = \<zero>) = (a = \<zero> | b = \<zero>)"
proof
assume "a ∈ carrier R" "b ∈ carrier R" "a ⊗ b = \<zero>"
then show "a = \<zero> | b = \<zero>" by (simp add: integral)
next
assume "a ∈ carrier R" "b ∈ carrier R" "a = \<zero> | b = \<zero>"
then show "a ⊗ b = \<zero>" by auto
qed

lemma m_lcancel:
assumes prem: "a ~= \<zero>"
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 \<ominus> c) = \<zero>" by algebra
with R have "a = \<zero> | (b \<ominus> c) = \<zero>" by (simp add: integral_iff)
with prem and R have "b \<ominus> c = \<zero>" by auto
with R have "b = b \<ominus> (b \<ominus> c)" by algebra
also from R have "b \<ominus> (b \<ominus> 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 ~= \<zero>"
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 (in cring) cring_fieldI:
assumes field_Units: "Units R = carrier R - {\<zero>}"
shows "field R"
proof
from field_Units have "\<zero> ∉ Units R" by fast
moreover have "\<one> ∈ Units R" by fast
ultimately show "\<one> ≠ \<zero>" by force
next
fix a b
assume acarr: "a ∈ carrier R"
and bcarr: "b ∈ carrier R"
and ab: "a ⊗ b = \<zero>"
show "a = \<zero> ∨ b = \<zero>"
proof (cases "a = \<zero>", simp)
assume "a ≠ \<zero>"
with field_Units and acarr have aUnit: "a ∈ Units R" by fast
from bcarr have "b = \<one> ⊗ 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) ⊗ \<zero>" by simp
also from aUnit[THEN Units_inv_closed] have "... = \<zero>" by algebra
finally have "b = \<zero>" .
then show "a = \<zero> ∨ b = \<zero>" by simp
qed
qed (rule field_Units)

text {* Another variant to show that something is a field *}
lemma (in cring) cring_fieldI2:
assumes notzero: "\<zero> ≠ \<one>"
and invex: "!!a. [|a ∈ carrier R; a ≠ \<zero>|] ==> ∃b∈carrier R. a ⊗ b = \<one>"
shows "field R"
apply (rule cring_fieldI, simp add: Units_def)
apply (rule, clarsimp)
apply (simp add: notzero)
proof (clarsimp)
fix x
assume xcarr: "x ∈ carrier R"
and "x ≠ \<zero>"
then have "∃y∈carrier R. x ⊗ y = \<one>" by (rule invex)
then obtain y where ycarr: "y ∈ carrier R" and xy: "x ⊗ y = \<one>" by fast
from xy xcarr ycarr have "y ⊗ x = \<one>" by (simp add: m_comm)
with ycarr and xy show "∃y∈carrier R. y ⊗ x = \<one> ∧ x ⊗ y = \<one>" by fast
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 &
(ALL 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 \<one>R = \<one>S}"


lemma ring_hom_memI:
fixes R (structure) and S (structure)
assumes hom_closed: "!!x. x ∈ carrier R ==> h x ∈ carrier S"
and hom_mult: "!!x y. [| x ∈ carrier R; y ∈ carrier R |] ==>
h (x ⊗ y) = h x ⊗S h y"

and hom_add: "!!x y. [| x ∈ carrier R; y ∈ carrier R |] ==>
h (x ⊕ y) = h x ⊕S h y"

and hom_one: "h \<one> = \<one>S"
shows "h ∈ ring_hom R S"
by (auto simp add: ring_hom_def assms Pi_def)

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"

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 ⊕S 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 \<one> = \<one>S"
by (simp add: ring_hom_def)

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 \<zero> = \<zero>S"
proof -
have "h \<zero> ⊕S h \<zero> = h \<zero> ⊕S \<zero>S"
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 (\<ominus> x) = \<ominus>S h x"
proof -
assume R: "x ∈ carrier R"
then have "h x ⊕S h (\<ominus> x) = h x ⊕S (\<ominus>S 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]:
"[| finite A; f ∈ A -> carrier R |] ==>
h (finsum R f A) = finsum S (h o f) A"

proof (induct set: finite)
case empty then show ?case by simp
next
case insert then show ?case by (simp add: Pi_def)
qed

lemma (in ring_hom_cring) hom_finprod:
"[| finite A; f ∈ A -> carrier R |] ==>
h (finprod R f A) = finprod S (h o f) A"

proof (induct set: finite)
case empty then show ?case by simp
next
case insert then show ?case by (simp add: Pi_def)
qed

declare ring_hom_cring.hom_finprod [simp]

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

end