src/HOL/Algebra/Ring.thy
 author nipkow Fri, 13 Nov 2009 14:14:04 +0100 changeset 33657 a4179bf442d1 parent 29237 e90d9d51106b child 35054 a5db9779b026 permissions -rw-r--r--
renamed lemmas "anti_sym" -> "antisym"
```
(*
Title:     The algebraic hierarchy of rings
Author:    Clemens Ballarin, started 9 December 1996
*)

theory Ring
imports FiniteProduct
uses ("ringsimp.ML") begin

section {* The Algebraic Hierarchy of Rings *}

subsection {* Abelian Groups *}

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

text {* Derived operations. *}

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

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

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

text {*
The following definition is redundant but simple to use.
*}

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 \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R"
and zero_closed: "\<zero> \<in> carrier R"
and a_assoc:
"!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==>
(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x"
and a_comm:
"!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> 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 \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R"
and zero_closed: "zero R \<in> carrier R"
and a_assoc:
"!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==>
(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
and a_comm:
"!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x"
and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x"
and l_inv_ex: "!!x. x \<in> carrier R ==> EX y : carrier R. y \<oplus> 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 |)"

lemmas monoid_record_simps = partial_object.simps monoid.simps

lemma (in abelian_monoid) a_closed [intro, simp]:
"\<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier G"
by (rule monoid.m_closed [OF a_monoid, simplified monoid_record_simps])

lemma (in abelian_monoid) zero_closed [intro, simp]:
"\<zero> \<in> carrier G"
by (rule monoid.one_closed [OF a_monoid, simplified monoid_record_simps])

lemma (in abelian_group) a_inv_closed [intro, simp]:
"x \<in> carrier G ==> \<ominus> x \<in> carrier G"
group.inv_closed [OF a_group, simplified monoid_record_simps])

lemma (in abelian_group) minus_closed [intro, simp]:
"[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y \<in> carrier G"

lemma (in abelian_group) a_l_cancel [simp]:
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
(x \<oplus> y = x \<oplus> z) = (y = z)"
by (rule group.l_cancel [OF a_group, simplified monoid_record_simps])

lemma (in abelian_group) a_r_cancel [simp]:
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
(y \<oplus> x = z \<oplus> x) = (y = z)"
by (rule group.r_cancel [OF a_group, simplified monoid_record_simps])

lemma (in abelian_monoid) a_assoc:
"\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
by (rule monoid.m_assoc [OF a_monoid, simplified monoid_record_simps])

lemma (in abelian_monoid) l_zero [simp]:
"x \<in> carrier G ==> \<zero> \<oplus> x = x"
by (rule monoid.l_one [OF a_monoid, simplified monoid_record_simps])

lemma (in abelian_group) l_neg:
"x \<in> carrier G ==> \<ominus> x \<oplus> x = \<zero>"
group.l_inv [OF a_group, simplified monoid_record_simps])

lemma (in abelian_monoid) a_comm:
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x"
by (rule comm_monoid.m_comm [OF a_comm_monoid,
simplified monoid_record_simps])

lemma (in abelian_monoid) a_lcomm:
"\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
by (rule comm_monoid.m_lcomm [OF a_comm_monoid,
simplified monoid_record_simps])

lemma (in abelian_monoid) r_zero [simp]:
"x \<in> carrier G ==> x \<oplus> \<zero> = x"
using monoid.r_one [OF a_monoid]
by simp

lemma (in abelian_group) r_neg:
"x \<in> carrier G ==> x \<oplus> (\<ominus> x) = \<zero>"
using group.r_inv [OF a_group]

lemma (in abelian_group) minus_zero [simp]:
"\<ominus> \<zero> = \<zero>"
group.inv_one [OF a_group, simplified monoid_record_simps])

lemma (in abelian_group) minus_minus [simp]:
"x \<in> carrier G ==> \<ominus> (\<ominus> x) = x"
using group.inv_inv [OF a_group, simplified monoid_record_simps]

lemma (in abelian_group) a_inv_inj:
"inj_on (a_inv G) (carrier G)"
using group.inv_inj [OF a_group, simplified monoid_record_simps]

"[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> (x \<oplus> y) = \<ominus> x \<oplus> \<ominus> y"
using comm_group.inv_mult [OF a_comm_group]

lemma (in abelian_group) minus_equality:
"[| x \<in> carrier G; y \<in> carrier G; y \<oplus> x = \<zero> |] ==> \<ominus> x = y"
using group.inv_equality [OF a_group]

lemma (in abelian_monoid) minus_unique:
"[| x \<in> carrier G; y \<in> carrier G; y' \<in> carrier G;
y \<oplus> x = \<zero>; x \<oplus> y' = \<zero> |] ==> y = y'"
using monoid.inv_unique [OF a_monoid]

lemmas (in abelian_monoid) a_ac = a_assoc a_comm a_lcomm

text {* Derive an @{text "abelian_group"} from a @{text "comm_group"} *}
lemma comm_group_abelian_groupI:
fixes G (structure)
assumes cg: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
shows "abelian_group G"
proof -
interpret comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
by (rule cg)
show "abelian_group G" ..
qed

subsection {* Sums over Finite Sets *}

text {*
This definition makes it easy to lift lemmas from @{term finprod}.
*}

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

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

context abelian_monoid begin

(*
lemmas finsum_empty [simp] =
comm_monoid.finprod_empty [OF a_comm_monoid, simplified]
is dangeous, because attributes (like simplified) are applied upon opening
the locale, simplified refers to the simpset at that time!!!

lemmas finsum_empty [simp] =
abelian_monoid.finprod_empty [OF a_abelian_monoid, folded finsum_def,
simplified monoid_record_simps]
makes the locale slow, because proofs are repeated for every
"lemma (in abelian_monoid)" command.
When lemma is used time in UnivPoly.thy from beginning to UP_cring goes down
from 110 secs to 60 secs.
*)

lemma finsum_empty [simp]:
"finsum G f {} = \<zero>"
by (rule comm_monoid.finprod_empty [OF a_comm_monoid,
folded finsum_def, simplified monoid_record_simps])

lemma finsum_insert [simp]:
"[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |]
==> finsum G f (insert a F) = f a \<oplus> finsum G f F"
by (rule comm_monoid.finprod_insert [OF a_comm_monoid,
folded finsum_def, simplified monoid_record_simps])

lemma finsum_zero [simp]:
"finite A ==> (\<Oplus>i\<in>A. \<zero>) = \<zero>"
by (rule comm_monoid.finprod_one [OF a_comm_monoid, folded finsum_def,
simplified monoid_record_simps])

lemma finsum_closed [simp]:
fixes A
assumes fin: "finite A" and f: "f \<in> A -> carrier G"
shows "finsum G f A \<in> carrier G"
apply (rule comm_monoid.finprod_closed [OF a_comm_monoid,
folded finsum_def, simplified monoid_record_simps])
apply (rule fin)
apply (rule f)
done

lemma finsum_Un_Int:
"[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==>
finsum G g (A Un B) \<oplus> finsum G g (A Int B) =
finsum G g A \<oplus> finsum G g B"
by (rule comm_monoid.finprod_Un_Int [OF a_comm_monoid,
folded finsum_def, simplified monoid_record_simps])

lemma finsum_Un_disjoint:
"[| finite A; finite B; A Int B = {};
g \<in> A -> carrier G; g \<in> B -> carrier G |]
==> finsum G g (A Un B) = finsum G g A \<oplus> finsum G g B"
by (rule comm_monoid.finprod_Un_disjoint [OF a_comm_monoid,
folded finsum_def, simplified monoid_record_simps])

"[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==>
finsum G (%x. f x \<oplus> g x) A = (finsum G f A \<oplus> finsum G g A)"
by (rule comm_monoid.finprod_multf [OF a_comm_monoid,
folded finsum_def, simplified monoid_record_simps])

lemma finsum_cong':
"[| A = B; g : B -> carrier G;
!!i. i : B ==> f i = g i |] ==> finsum G f A = finsum G g B"
by (rule comm_monoid.finprod_cong' [OF a_comm_monoid,
folded finsum_def, simplified monoid_record_simps]) auto

lemma finsum_0 [simp]:
"f : {0::nat} -> carrier G ==> finsum G f {..0} = f 0"
by (rule comm_monoid.finprod_0 [OF a_comm_monoid, folded finsum_def,
simplified monoid_record_simps])

lemma finsum_Suc [simp]:
"f : {..Suc n} -> carrier G ==>
finsum G f {..Suc n} = (f (Suc n) \<oplus> finsum G f {..n})"
by (rule comm_monoid.finprod_Suc [OF a_comm_monoid, folded finsum_def,
simplified monoid_record_simps])

lemma finsum_Suc2:
"f : {..Suc n} -> carrier G ==>
finsum G f {..Suc n} = (finsum G (%i. f (Suc i)) {..n} \<oplus> f 0)"
by (rule comm_monoid.finprod_Suc2 [OF a_comm_monoid, folded finsum_def,
simplified monoid_record_simps])

"[| f : {..n} -> carrier G; g : {..n} -> carrier G |] ==>
finsum G (%i. f i \<oplus> g i) {..n::nat} =
finsum G f {..n} \<oplus> finsum G g {..n}"
by (rule comm_monoid.finprod_mult [OF a_comm_monoid, folded finsum_def,
simplified monoid_record_simps])

lemma finsum_cong:
"[| A = B; f : B -> carrier G;
!!i. i : B =simp=> f i = g i |] ==> finsum G f A = finsum G g B"
by (rule comm_monoid.finprod_cong [OF a_comm_monoid, folded finsum_def,
simplified monoid_record_simps]) (auto simp add: simp_implies_def)

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

lemma finsum_reindex:
assumes fin: "finite A"
shows "f : (h ` A) \<rightarrow> carrier G \<Longrightarrow>
inj_on h A ==> finsum G f (h ` A) = finsum G (%x. f (h x)) A"
using fin apply induct
apply (auto simp add: finsum_insert Pi_def)
done

(* The following is 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
*)

(* By Jesus Aransay. *)

lemma finsum_singleton:
assumes i_in_A: "i \<in> A" and fin_A: "finite A" and f_Pi: "f \<in> A \<rightarrow> carrier G"
shows "(\<Oplus>j\<in>A. if i = j then f j else \<zero>) = f i"
using i_in_A finsum_insert [of "A - {i}" i "(\<lambda>j. if i = j then f j else \<zero>)"]
fin_A f_Pi finsum_zero [of "A - {i}"]
finsum_cong [of "A - {i}" "A - {i}" "(\<lambda>j. if i = j then f j else \<zero>)" "(\<lambda>i. \<zero>)"]
unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)

end

subsection {* Rings: Basic Definitions *}

locale ring = abelian_group R + monoid R for R (structure) +
assumes l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
and r_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"

locale cring = ring + comm_monoid R

locale "domain" = cring +
assumes one_not_zero [simp]: "\<one> ~= \<zero>"
and integral: "[| a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> 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 \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
and r_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
shows "ring R"
by (auto intro: ring.intro
abelian_group.axioms ring_axioms.intro assms)

lemma (in ring) is_abelian_group:
"abelian_group R"
..

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

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

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 \<in> carrier R; y \<in> carrier R; z \<in> 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"
-- {* Right-distributivity follows from left-distributivity and
commutativity. *}
proof (rule ring_axioms.intro)
fix x y z
assume R: "x \<in> carrier R" "y \<in> carrier R" "z \<in> carrier R"
note [simp] = comm_monoid.axioms [OF comm_monoid]
abelian_group.axioms [OF abelian_group]
abelian_monoid.a_closed

from R have "z \<otimes> (x \<oplus> y) = (x \<oplus> y) \<otimes> z"
by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
also from R have "... = x \<otimes> z \<oplus> y \<otimes> z" by (simp add: l_distr)
also from R have "... = z \<otimes> x \<oplus> z \<otimes> y"
by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
finally show "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> 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 \<in> carrier G; y \<in> carrier G |] ==> x \<oplus> (\<ominus> x \<oplus> y) = y"
proof -
assume G: "x \<in> carrier G" "y \<in> carrier G"
then have "(x \<oplus> \<ominus> x) \<oplus> y = y"
by (simp only: r_neg l_zero)
with G show ?thesis
qed

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

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

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

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

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

lemma (in ring) r_minus:
"[| x \<in> carrier R; y \<in> carrier R |] ==> x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
proof -
assume R: "x \<in> carrier R" "y \<in> carrier R"
then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr)
also from R have "... = \<zero>" by (simp add: l_neg r_null)
finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" .
with R have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
with R show ?thesis by (simp add: a_assoc r_neg )
qed

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

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

use "ringsimp.ML"

setup Algebra.setup

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

lemma (in ring) one_zeroD:
assumes onezero: "\<one> = \<zero>"
shows "carrier R = {\<zero>}"
proof (rule, rule)
fix x
assume xcarr: "x \<in> carrier R"
from xcarr
have "x = x \<otimes> \<one>" by simp
from this and onezero
have "x = x \<otimes> \<zero>" by simp
from this and xcarr
have "x = \<zero>" by simp
thus "x \<in> {\<zero>}" by fast
qed fast

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

lemma (in ring) carrier_one_zero:
shows "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
by (rule, erule one_zeroI, erule one_zeroD)

lemma (in ring) carrier_one_not_zero:
shows "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"

text {* Two examples for use of method algebra *}

lemma
fixes R (structure) and S (structure)
assumes "ring R" "cring S"
assumes RS: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier S" "d \<in> carrier S"
shows "a \<oplus> \<ominus> (a \<oplus> \<ominus> b) = b & c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c"
proof -
interpret ring R by fact
interpret cring S by fact
ML_val {* Algebra.print_structures @{context} *}
from RS show ?thesis by algebra
qed

lemma
fixes R (structure)
assumes "ring R"
assumes R: "a \<in> carrier R" "b \<in> 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 \<in> carrier R; f \<in> A -> carrier R |] ==>
finsum R f A \<otimes> a = finsum R (%i. f i \<otimes> 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 \<in> carrier R; f \<in> A -> carrier R |] ==>
a \<otimes> finsum R f A = finsum R (%i. a \<otimes> 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 *}

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

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

lemma (in "domain") m_lcancel:
assumes prem: "a ~= \<zero>"
and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
proof
assume eq: "a \<otimes> b = a \<otimes> c"
with R have "a \<otimes> (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 \<otimes> b = a \<otimes> c" by simp
qed

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

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 a: "\<zero> \<notin> Units R" by fast
have "\<one> \<in> Units R" by fast
from this and a
show "\<one> \<noteq> \<zero>" by force
next
fix a b
assume acarr: "a \<in> carrier R"
and bcarr: "b \<in> carrier R"
and ab: "a \<otimes> b = \<zero>"
show "a = \<zero> \<or> b = \<zero>"
proof (cases "a = \<zero>", simp)
assume "a \<noteq> \<zero>"
from this and field_Units and acarr
have aUnit: "a \<in> Units R" by fast
from bcarr
have "b = \<one> \<otimes> b" by algebra
also from aUnit acarr
have "... = (inv a \<otimes> a) \<otimes> b" by (simp add: Units_l_inv)
also from acarr bcarr aUnit[THEN Units_inv_closed]
have "... = (inv a) \<otimes> (a \<otimes> b)" by algebra
also from ab and acarr bcarr aUnit
have "... = (inv a) \<otimes> \<zero>" by simp
also from aUnit[THEN Units_inv_closed]
have "... = \<zero>" by algebra
finally
have "b = \<zero>" .
thus "a = \<zero> \<or> 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> \<noteq> \<one>"
and invex: "\<And>a. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>\<rbrakk> \<Longrightarrow> \<exists>b\<in>carrier R. a \<otimes> b = \<one>"
shows "field R"
apply (rule cring_fieldI, simp add: Units_def)
apply (rule, clarsimp)
proof (clarsimp)
fix x
assume xcarr: "x \<in> carrier R"
and "x \<noteq> \<zero>"
from this
have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex)
from this
obtain y
where ycarr: "y \<in> carrier R"
and xy: "x \<otimes> y = \<one>"
by fast
from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm)
from ycarr and this and xy
show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
qed

subsection {* Morphisms *}

constdefs (structure R S)
ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
"ring_hom R S == {h. h \<in> carrier R -> carrier S &
(ALL x y. x \<in> carrier R & y \<in> carrier R -->
h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y & h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y) &
h \<one> = \<one>\<^bsub>S\<^esub>}"

lemma ring_hom_memI:
fixes R (structure) and S (structure)
assumes hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
and hom_mult: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
and hom_add: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
and hom_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
shows "h \<in> ring_hom R S"
by (auto simp add: ring_hom_def assms Pi_def)

lemma ring_hom_closed:
"[| h \<in> ring_hom R S; x \<in> carrier R |] ==> h x \<in> carrier S"
by (auto simp add: ring_hom_def funcset_mem)

lemma ring_hom_mult:
fixes R (structure) and S (structure)
shows
"[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"

fixes R (structure) and S (structure)
shows
"[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"

lemma ring_hom_one:
fixes R (structure) and S (structure)
shows "h \<in> ring_hom R S ==> h \<one> = \<one>\<^bsub>S\<^esub>"

locale ring_hom_cring = R: cring R + S: cring S
for R (structure) and S (structure) +
fixes h
assumes homh [simp, intro]: "h \<in> 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 \<zero> = \<zero>\<^bsub>S\<^esub>"
proof -
have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>"
then show ?thesis by (simp del: S.r_zero)
qed

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

lemma (in ring_hom_cring) hom_finsum [simp]:
"[| finite A; f \<in> 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 \<in> 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 \<in> ring_hom R R"
by (auto intro!: ring_hom_memI)

end
```