(* 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