(* Title: HOL/Algebra/Ring.thy
Author: Clemens Ballarin, started 9 December 1996
With contributions by Martin Baillon.
*)
theory Ring
imports FiniteProduct
begin
section \<open>The Algebraic Hierarchy of Rings\<close>
subsection \<open>Abelian Groups\<close>
record 'a ring = "'a monoid" +
zero :: 'a ("\<zero>\<index>")
add :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<oplus>\<index>" 65)
abbreviation
add_monoid :: "('a, 'm) ring_scheme \<Rightarrow> ('a, 'm) monoid_scheme"
where "add_monoid R \<equiv> \<lparr> carrier = carrier R, mult = add R, one = zero R, \<dots> = (undefined :: 'm) \<rparr>"
text \<open>Derived operations.\<close>
definition
a_inv :: "[('a, 'm) ring_scheme, 'a ] \<Rightarrow> 'a" ("\<ominus>\<index> _" [81] 80)
where "a_inv R = m_inv (add_monoid R)"
definition
a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" ("(_ \<ominus>\<index> _)" [65,66] 65)
where "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> (\<ominus>\<^bsub>R\<^esub> y)"
definition
add_pow :: "[_, ('b :: semiring_1), 'a] \<Rightarrow> 'a" ("[_] \<cdot>\<index> _" [81, 81] 80)
where "add_pow R k a = pow (add_monoid R) a k"
locale abelian_monoid =
fixes G (structure)
assumes a_comm_monoid:
"comm_monoid (add_monoid G)"
definition
finsum :: "[('b, 'm) ring_scheme, 'a \<Rightarrow> 'b, 'a set] \<Rightarrow> 'b" where
"finsum G = finprod (add_monoid G)"
syntax
"_finsum" :: "index \<Rightarrow> idt \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"
("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
translations
"\<Oplus>\<^bsub>G\<^esub>i\<in>A. b" \<rightleftharpoons> "CONST finsum G (\<lambda>i. b) A"
\<comment> \<open>Beware of argument permutation!\<close>
locale abelian_group = abelian_monoid +
assumes a_comm_group:
"comm_group (add_monoid G)"
subsection \<open>Basic Properties\<close>
lemma abelian_monoidI:
fixes R (structure)
assumes "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier R"
and "\<zero> \<in> carrier R"
and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero> \<oplus> x = x"
and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> 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 "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier R"
and "\<zero> \<in> carrier R"
and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero> \<oplus> x = x"
and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> 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 "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier R"
and "\<zero> \<in> carrier R"
and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x"
and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero> \<oplus> x = x"
and "\<And>x. x \<in> carrier R \<Longrightarrow> \<exists>y \<in> 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 abelian_groupE:
fixes R (structure)
assumes "abelian_group R"
shows "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier R"
and "\<zero> \<in> carrier R"
and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x"
and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero> \<oplus> x = x"
and "\<And>x. x \<in> carrier R \<Longrightarrow> \<exists>y \<in> carrier R. y \<oplus> x = \<zero>"
using abelian_group.a_comm_group assms comm_groupE by fastforce+
lemma (in abelian_monoid) a_monoid:
"monoid (add_monoid G)"
by (rule comm_monoid.axioms, rule a_comm_monoid)
lemma (in abelian_group) a_group:
"group (add_monoid G)"
by (simp add: group_def a_monoid)
(simp add: comm_group.axioms group.axioms a_comm_group)
lemmas monoid_record_simps = partial_object.simps monoid.simps
text \<open>Transfer facts from multiplicative structures via interpretation.\<close>
sublocale abelian_monoid <
add: monoid "(add_monoid G)"
rewrites "carrier (add_monoid G) = carrier G"
and "mult (add_monoid G) = add G"
and "one (add_monoid G) = zero G"
and "(\<lambda>a k. pow (add_monoid G) a k) = (\<lambda>a k. add_pow G k a)"
by (rule a_monoid) (auto simp add: add_pow_def)
context abelian_monoid
begin
lemmas a_closed = add.m_closed
lemmas zero_closed = add.one_closed
lemmas a_assoc = add.m_assoc
lemmas l_zero = add.l_one
lemmas r_zero = add.r_one
lemmas minus_unique = add.inv_unique
end
sublocale abelian_monoid <
add: comm_monoid "(add_monoid G)"
rewrites "carrier (add_monoid G) = carrier G"
and "mult (add_monoid G) = add G"
and "one (add_monoid G) = zero G"
and "finprod (add_monoid G) = finsum G"
and "pow (add_monoid G) = (\<lambda>a k. add_pow G k a)"
by (rule a_comm_monoid) (auto simp: finsum_def add_pow_def)
context abelian_monoid begin
lemmas a_comm = add.m_comm
lemmas a_lcomm = add.m_lcomm
lemmas a_ac = a_assoc a_comm a_lcomm
lemmas finsum_empty = add.finprod_empty
lemmas finsum_insert = add.finprod_insert
lemmas finsum_zero = add.finprod_one
lemmas finsum_closed = add.finprod_closed
lemmas finsum_Un_Int = add.finprod_Un_Int
lemmas finsum_Un_disjoint = add.finprod_Un_disjoint
lemmas finsum_addf = add.finprod_multf
lemmas finsum_cong' = add.finprod_cong'
lemmas finsum_0 = add.finprod_0
lemmas finsum_Suc = add.finprod_Suc
lemmas finsum_Suc2 = add.finprod_Suc2
lemmas finsum_infinite = add.finprod_infinite
lemmas finsum_cong = add.finprod_cong
text \<open>Usually, if this rule causes a failed congruence proof error,
the reason is that the premise \<open>g \<in> B \<rightarrow> carrier G\<close> cannot be shown.
Adding @{thm [source] Pi_def} to the simpset is often useful.\<close>
lemmas finsum_reindex = add.finprod_reindex
(* The following would be wrong. Needed is the equivalent of [^] for addition,
or indeed the canonical embedding from Nat into the monoid.
lemma finsum_const:
assumes fin [simp]: "finite A"
and a [simp]: "a : carrier G"
shows "finsum G (%x. a) A = a [^] card A"
using fin apply induct
apply force
apply (subst finsum_insert)
apply auto
apply (force simp add: Pi_def)
apply (subst m_comm)
apply auto
done
*)
lemmas finsum_singleton = add.finprod_singleton
end
sublocale abelian_group <
add: group "(add_monoid G)"
rewrites "carrier (add_monoid G) = carrier G"
and "mult (add_monoid G) = add G"
and "one (add_monoid G) = zero G"
and "m_inv (add_monoid G) = a_inv G"
and "pow (add_monoid G) = (\<lambda>a k. add_pow G k a)"
by (rule a_group) (auto simp: m_inv_def a_inv_def add_pow_def)
context abelian_group
begin
lemmas a_inv_closed = add.inv_closed
lemma minus_closed [intro, simp]:
"[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y \<in> carrier G"
by (simp add: a_minus_def)
lemmas l_neg = add.l_inv [simp del]
lemmas r_neg = add.r_inv [simp del]
lemmas minus_minus = add.inv_inv
lemmas a_inv_inj = add.inv_inj
lemmas minus_equality = add.inv_equality
end
sublocale abelian_group <
add: comm_group "(add_monoid G)"
rewrites "carrier (add_monoid G) = carrier G"
and "mult (add_monoid G) = add G"
and "one (add_monoid G) = zero G"
and "m_inv (add_monoid G) = a_inv G"
and "finprod (add_monoid G) = finsum G"
and "pow (add_monoid G) = (\<lambda>a k. add_pow G k a)"
by (rule a_comm_group) (auto simp: m_inv_def a_inv_def finsum_def add_pow_def)
lemmas (in abelian_group) minus_add = add.inv_mult
text \<open>Derive an \<open>abelian_group\<close> from a \<open>comm_group\<close>\<close>
lemma comm_group_abelian_groupI:
fixes G (structure)
assumes cg: "comm_group (add_monoid G)"
shows "abelian_group G"
proof -
interpret comm_group "(add_monoid G)"
by (rule cg)
show "abelian_group G" ..
qed
subsection \<open>Rings: Basic Definitions\<close>
locale semiring = abelian_monoid (* for add *) R + monoid (* for mult *) R for R (structure) +
assumes l_distr: "\<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
and r_distr: "\<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
and l_null[simp]: "x \<in> carrier R \<Longrightarrow> \<zero> \<otimes> x = \<zero>"
and r_null[simp]: "x \<in> carrier R \<Longrightarrow> x \<otimes> \<zero> = \<zero>"
locale ring = abelian_group (* for add *) R + monoid (* for mult *) R for R (structure) +
assumes "\<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
and "\<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
locale cring = ring + comm_monoid (* for mult *) R
locale "domain" = cring +
assumes one_not_zero [simp]: "\<one> \<noteq> \<zero>"
and integral: "\<lbrakk> a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R \<rbrakk> \<Longrightarrow> a = \<zero> \<or> b = \<zero>"
locale field = "domain" +
assumes field_Units: "Units R = carrier R - {\<zero>}"
subsection \<open>Rings\<close>
lemma ringI:
fixes R (structure)
assumes "abelian_group R"
and "monoid R"
and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> 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 ringE:
fixes R (structure)
assumes "ring R"
shows "abelian_group R"
and "monoid R"
and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> 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: "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow>
(x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
shows "cring R"
proof (intro cring.intro ring.intro)
show "ring_axioms R"
\<comment> \<open>Right-distributivity follows from left-distributivity and
commutativity.\<close>
proof (rule ring_axioms.intro)
fix x y z
assume R: "x \<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 cringE:
fixes R (structure)
assumes "cring R"
shows "comm_monoid R"
and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> 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]: "\<ominus> \<zero> = \<zero>"
by (simp add: a_inv_def)
subsubsection \<open>Normaliser for Rings\<close>
lemma (in abelian_group) r_neg1:
"\<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> (\<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
lemma (in abelian_group) r_neg2:
"\<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> 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
by (simp add: a_ac)
qed
context ring begin
text \<open>
The following proofs are from Jacobson, Basic Algebra I, pp.~88--89.
\<close>
sublocale semiring
proof -
note [simp] = ring_axioms[unfolded ring_def ring_axioms_def]
show "semiring R"
proof (unfold_locales)
fix x
assume R: "x \<in> carrier R"
then have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = (\<zero> \<oplus> \<zero>) \<otimes> x"
by (simp 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 "\<zero> \<otimes> x = \<zero>" by (simp del: r_zero)
from R have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> (\<zero> \<oplus> \<zero>)"
by (simp 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 "x \<otimes> \<zero> = \<zero>" by (simp del: r_zero)
qed auto
qed
lemma l_minus:
"\<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> (\<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)
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 r_minus:
"\<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> 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)
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
end
lemma (in abelian_group) minus_eq: "x \<ominus> y = x \<oplus> (\<ominus> y)"
by (rule a_minus_def)
text \<open>Setup algebra method:
compute distributive normal form in locale contexts\<close>
ML_file \<open>ringsimp.ML\<close>
attribute_setup algebra = \<open>
Scan.lift ((Args.add >> K true || Args.del >> K false) --| Args.colon || Scan.succeed true)
-- Scan.lift Args.name -- Scan.repeat Args.term
>> (fn ((b, n), ts) => if b then Ringsimp.add_struct (n, ts) else Ringsimp.del_struct (n, ts))
\<close> "theorems controlling algebra method"
method_setup algebra = \<open>
Scan.succeed (SIMPLE_METHOD' o Ringsimp.algebra_tac)
\<close> "normalisation of algebraic structure"
lemmas (in semiring) semiring_simprules
[algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
a_closed zero_closed m_closed one_closed
a_assoc l_zero a_comm m_assoc l_one l_distr r_zero
a_lcomm r_distr l_null r_null
lemmas (in ring) ring_simprules
[algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq
r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
a_lcomm r_distr l_null r_null l_minus r_minus
lemmas (in cring)
[algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
_
lemmas (in cring) cring_simprules
[algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq
r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus
lemma (in semiring) nat_pow_zero:
"(n::nat) \<noteq> 0 \<Longrightarrow> \<zero> [^] n = \<zero>"
by (induct n) simp_all
context semiring begin
lemma 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
with onezero have "x = x \<otimes> \<zero>" by simp
with xcarr have "x = \<zero>" by simp
then show "x \<in> {\<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>)"
using one_zeroD by blast
lemma carrier_one_not_zero: "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
by (simp add: carrier_one_zero)
end
text \<open>Two examples for use of method algebra\<close>
lemma
fixes R (structure) and S (structure)
assumes "ring R" "cring S"
assumes RS: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier S" "d \<in> carrier S"
shows "a \<oplus> (\<ominus> (a \<oplus> (\<ominus> b))) = b \<and> c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c"
proof -
interpret ring R by fact
interpret cring S by fact
from RS show ?thesis by algebra
qed
lemma
fixes R (structure)
assumes "ring R"
assumes R: "a \<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 \<open>Sums over Finite Sets\<close>
lemma (in semiring) finsum_ldistr:
"\<lbrakk> finite A; a \<in> carrier R; f: A \<rightarrow> carrier R \<rbrakk> \<Longrightarrow>
(\<Oplus> i \<in> A. (f i)) \<otimes> a = (\<Oplus> i \<in> A. ((f i) \<otimes> a))"
proof (induct set: finite)
case empty then show ?case by simp
next
case (insert x F) then show ?case by (simp add: Pi_def l_distr)
qed
lemma (in semiring) finsum_rdistr:
"\<lbrakk> finite A; a \<in> carrier R; f: A \<rightarrow> carrier R \<rbrakk> \<Longrightarrow>
a \<otimes> (\<Oplus> i \<in> A. (f i)) = (\<Oplus> i \<in> A. (a \<otimes> (f i)))"
proof (induct set: finite)
case empty then show ?case by simp
next
case (insert x F) then show ?case by (simp add: Pi_def r_distr)
qed
text \<open>A quick detour\<close>
lemma add_pow_int_ge: "(k :: int) \<ge> 0 \<Longrightarrow> [ k ] \<cdot>\<^bsub>R\<^esub> a = [ nat k ] \<cdot>\<^bsub>R\<^esub> a" \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
by (simp add: add_pow_def int_pow_def nat_pow_def)
lemma add_pow_int_lt: "(k :: int) < 0 \<Longrightarrow> [ k ] \<cdot>\<^bsub>R\<^esub> a = \<ominus>\<^bsub>R\<^esub> ([ nat (- k) ] \<cdot>\<^bsub>R\<^esub> a)" \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
by (simp add: int_pow_def nat_pow_def a_inv_def add_pow_def)
corollary (in semiring) add_pow_ldistr: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
assumes "a \<in> carrier R" "b \<in> carrier R"
shows "([(k :: nat)] \<cdot> a) \<otimes> b = [k] \<cdot> (a \<otimes> b)"
proof -
have "([k] \<cdot> a) \<otimes> b = (\<Oplus> i \<in> {..< k}. a) \<otimes> b"
using add.finprod_const[OF assms(1), of "{..<k}"] by simp
also have " ... = (\<Oplus> i \<in> {..< k}. (a \<otimes> b))"
using finsum_ldistr[of "{..<k}" b "\<lambda>x. a"] assms by simp
also have " ... = [k] \<cdot> (a \<otimes> b)"
using add.finprod_const[of "a \<otimes> b" "{..<k}"] assms by simp
finally show ?thesis .
qed
corollary (in semiring) add_pow_rdistr: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
assumes "a \<in> carrier R" "b \<in> carrier R"
shows "a \<otimes> ([(k :: nat)] \<cdot> b) = [k] \<cdot> (a \<otimes> b)"
proof -
have "a \<otimes> ([k] \<cdot> b) = a \<otimes> (\<Oplus> i \<in> {..< k}. b)"
using add.finprod_const[OF assms(2), of "{..<k}"] by simp
also have " ... = (\<Oplus> i \<in> {..< k}. (a \<otimes> b))"
using finsum_rdistr[of "{..<k}" a "\<lambda>x. b"] assms by simp
also have " ... = [k] \<cdot> (a \<otimes> b)"
using add.finprod_const[of "a \<otimes> b" "{..<k}"] assms by simp
finally show ?thesis .
qed
(* For integers, we need the uniqueness of the additive inverse *)
lemma (in ring) add_pow_ldistr_int: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
assumes "a \<in> carrier R" "b \<in> carrier R"
shows "([(k :: int)] \<cdot> a) \<otimes> b = [k] \<cdot> (a \<otimes> b)"
proof (cases "k \<ge> 0")
case True thus ?thesis
using add_pow_int_ge[of k R] add_pow_ldistr[OF assms] by auto
next
case False thus ?thesis
using add_pow_int_lt[of k R a] add_pow_int_lt[of k R "a \<otimes> b"]
add_pow_ldistr[OF assms, of "nat (- k)"] assms l_minus by auto
qed
lemma (in ring) add_pow_rdistr_int: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
assumes "a \<in> carrier R" "b \<in> carrier R"
shows "a \<otimes> ([(k :: int)] \<cdot> b) = [k] \<cdot> (a \<otimes> b)"
proof (cases "k \<ge> 0")
case True thus ?thesis
using add_pow_int_ge[of k R] add_pow_rdistr[OF assms] by auto
next
case False thus ?thesis
using add_pow_int_lt[of k R b] add_pow_int_lt[of k R "a \<otimes> b"]
add_pow_rdistr[OF assms, of "nat (- k)"] assms r_minus by auto
qed
subsection \<open>Integral Domains\<close>
context "domain" begin
lemma zero_not_one [simp]: "\<zero> \<noteq> \<one>"
by (rule not_sym) simp
lemma integral_iff: (* not by default a simp rule! *)
"\<lbrakk> a \<in> carrier R; b \<in> carrier R \<rbrakk> \<Longrightarrow> (a \<otimes> b = \<zero>) = (a = \<zero> \<or> b = \<zero>)"
proof
assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>"
then show "a = \<zero> \<or> b = \<zero>" by (simp add: integral)
next
assume "a \<in> carrier R" "b \<in> carrier R" "a = \<zero> \<or> b = \<zero>"
then show "a \<otimes> b = \<zero>" by auto
qed
lemma m_lcancel:
assumes prem: "a \<noteq> \<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> \<or> (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 m_rcancel:
assumes prem: "a \<noteq> \<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
end
subsection \<open>Fields\<close>
text \<open>Field would not need to be derived from domain, the properties
for domain follow from the assumptions of field\<close>
lemma (in field) is_ring: "ring R"
using ring_axioms .
lemma fieldE :
fixes R (structure)
assumes "field R"
shows "cring R"
and one_not_zero : "\<one> \<noteq> \<zero>"
and integral: "\<And>a b. \<lbrakk> a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R \<rbrakk> \<Longrightarrow> a = \<zero> \<or> b = \<zero>"
and field_Units: "Units R = carrier R - {\<zero>}"
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 - {\<zero>}"
shows "field R"
proof
from field_Units have "\<zero> \<notin> Units R" by fast
moreover have "\<one> \<in> Units R" by fast
ultimately 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>"
with 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
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>" .
then show "a = \<zero> \<or> b = \<zero>" by simp
qed
qed (rule field_Units)
text \<open>Another variant to show that something is a field\<close>
lemma (in cring) cring_fieldI2:
assumes notzero: "\<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"
proof -
have *: "carrier R - {\<zero>} \<subseteq> {y \<in> carrier R. \<exists>x\<in>carrier R. x \<otimes> y = \<one> \<and> y \<otimes> x = \<one>}"
proof (clarsimp)
fix x
assume xcarr: "x \<in> carrier R" and "x \<noteq> \<zero>"
obtain y where ycarr: "y \<in> carrier R" and xy: "x \<otimes> y = \<one>"
using \<open>x \<noteq> \<zero>\<close> invex xcarr by blast
with ycarr and xy show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>"
using m_comm xcarr by fastforce
qed
show ?thesis
apply (rule cring_fieldI, simp add: Units_def)
using *
using group_l_invI notzero set_diff_eq by auto
qed
subsection \<open>Morphisms\<close>
definition
ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
where "ring_hom R S =
{h. h \<in> carrier R \<rightarrow> carrier S \<and>
(\<forall>x y. x \<in> carrier R \<and> y \<in> carrier R \<longrightarrow>
h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y \<and> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y) \<and>
h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}"
lemma ring_hom_memI:
fixes R (structure) and S (structure)
assumes "\<And>x. x \<in> carrier R \<Longrightarrow> h x \<in> carrier S"
and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
and "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_memE:
fixes R (structure) and S (structure)
assumes "h \<in> ring_hom R S"
shows "\<And>x. x \<in> carrier R \<Longrightarrow> h x \<in> carrier S"
and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
and "\<And>x y. \<lbrakk> x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
and "h \<one> = \<one>\<^bsub>S\<^esub>"
using assms unfolding ring_hom_def by auto
lemma ring_hom_closed:
"\<lbrakk> h \<in> ring_hom R S; x \<in> carrier R \<rbrakk> \<Longrightarrow> 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 "\<lbrakk> h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
by (simp add: ring_hom_def)
lemma ring_hom_add:
fixes R (structure) and S (structure)
shows "\<lbrakk> h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R \<rbrakk> \<Longrightarrow> h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
by (simp add: ring_hom_def)
lemma ring_hom_one:
fixes R (structure) and S (structure)
shows "h \<in> ring_hom R S \<Longrightarrow> h \<one> = \<one>\<^bsub>S\<^esub>"
by (simp add: ring_hom_def)
lemma ring_hom_zero:
fixes R (structure) and S (structure)
assumes "h \<in> ring_hom R S" "ring R" "ring S"
shows "h \<zero> = \<zero>\<^bsub>S\<^esub>"
proof -
have "h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero>"
using ring_hom_add[OF assms(1), of \<zero> \<zero>] assms(2)
by (simp add: ring.ring_simprules(2) ring.ring_simprules(15))
thus ?thesis
by (metis abelian_group.l_neg assms ring.is_abelian_group ring.ring_simprules(18) ring.ring_simprules(2) ring_hom_closed)
qed
locale ring_hom_cring =
R?: cring R + S?: cring S for R (structure) and S (structure) + fixes h
assumes homh [simp, intro]: "h \<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_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>\<^bsub>S\<^esub>"
proof -
have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>"
by (simp add: hom_add [symmetric] del: hom_add)
then show ?thesis by (simp del: S.r_zero)
qed
lemma (in ring_hom_cring) hom_a_inv [simp]:
"x \<in> carrier R \<Longrightarrow> 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)"
by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add)
with R show ?thesis by simp
qed
lemma (in ring_hom_cring) hom_finsum [simp]:
assumes "f: A \<rightarrow> carrier R"
shows "h (\<Oplus> i \<in> A. f i) = (\<Oplus>\<^bsub>S\<^esub> i \<in> 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 \<rightarrow> carrier R"
shows "h (\<Otimes> i \<in> A. f i) = (\<Otimes>\<^bsub>S\<^esub> i \<in> 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 \<in> ring_hom R R"
by (auto intro!: ring_hom_memI)
lemma ring_hom_trans: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
"\<lbrakk> f \<in> ring_hom R S; g \<in> ring_hom S T \<rbrakk> \<Longrightarrow> g \<circ> f \<in> ring_hom R T"
by (rule ring_hom_memI) (auto simp add: ring_hom_closed ring_hom_mult ring_hom_add ring_hom_one)
subsection\<open>Jeremy Avigad's \<open>More_Finite_Product\<close> material\<close>
(* need better simplification rules for rings *)
(* the next one holds more generally for abelian groups *)
lemma (in cring) sum_zero_eq_neg: "x \<in> carrier R \<Longrightarrow> y \<in> carrier R \<Longrightarrow> x \<oplus> y = \<zero> \<Longrightarrow> x = \<ominus> y"
by (metis minus_equality)
lemma (in domain) square_eq_one:
fixes x
assumes [simp]: "x \<in> carrier R"
and "x \<otimes> x = \<one>"
shows "x = \<one> \<or> x = \<ominus>\<one>"
proof -
have "(x \<oplus> \<one>) \<otimes> (x \<oplus> \<ominus> \<one>) = x \<otimes> x \<oplus> \<ominus> \<one>"
by (simp add: ring_simprules)
also from \<open>x \<otimes> x = \<one>\<close> have "\<dots> = \<zero>"
by (simp add: ring_simprules)
finally have "(x \<oplus> \<one>) \<otimes> (x \<oplus> \<ominus> \<one>) = \<zero>" .
then have "(x \<oplus> \<one>) = \<zero> \<or> (x \<oplus> \<ominus> \<one>) = \<zero>"
by (intro integral) auto
then show ?thesis
by (metis add.inv_closed add.inv_solve_right assms(1) l_zero one_closed zero_closed)
qed
lemma (in domain) inv_eq_self: "x \<in> Units R \<Longrightarrow> x = inv x \<Longrightarrow> x = \<one> \<or> x = \<ominus>\<one>"
by (metis Units_closed Units_l_inv square_eq_one)
text \<open>
The following translates theorems about groups to the facts about
the units of a ring. (The list should be expanded as more things are
needed.)
\<close>
lemma (in ring) finite_ring_finite_units [intro]: "finite (carrier R) \<Longrightarrow> finite (Units R)"
by (rule finite_subset) auto
lemma (in monoid) units_of_pow:
fixes n :: nat
shows "x \<in> Units G \<Longrightarrow> x [^]\<^bsub>units_of G\<^esub> n = x [^]\<^bsub>G\<^esub> n"
apply (induct n)
apply (auto simp add: units_group group.is_monoid
monoid.nat_pow_0 monoid.nat_pow_Suc units_of_one units_of_mult)
done
lemma (in cring) units_power_order_eq_one:
"finite (Units R) \<Longrightarrow> a \<in> Units R \<Longrightarrow> a [^] card(Units R) = \<one>"
by (metis comm_group.power_order_eq_one units_comm_group units_of_carrier units_of_one units_of_pow)
subsection\<open>Jeremy Avigad's \<open>More_Ring\<close> material\<close>
lemma (in cring) field_intro2:
assumes "\<zero>\<^bsub>R\<^esub> \<noteq> \<one>\<^bsub>R\<^esub>" and un: "\<And>x. x \<in> carrier R - {\<zero>\<^bsub>R\<^esub>} \<Longrightarrow> x \<in> Units R"
shows "field R"
proof unfold_locales
show "\<one> \<noteq> \<zero>" using assms by auto
show "\<lbrakk>a \<otimes> b = \<zero>; a \<in> carrier R;
b \<in> carrier R\<rbrakk>
\<Longrightarrow> a = \<zero> \<or> b = \<zero>" for a b
by (metis un Units_l_cancel insert_Diff_single insert_iff r_null zero_closed)
qed (use assms in \<open>auto simp: Units_def\<close>)
lemma (in monoid) inv_char:
assumes "x \<in> carrier G" "y \<in> carrier G" "x \<otimes> y = \<one>" "y \<otimes> x = \<one>"
shows "inv x = y"
using assms inv_unique' by auto
lemma (in comm_monoid) comm_inv_char: "x \<in> carrier G \<Longrightarrow> y \<in> carrier G \<Longrightarrow> x \<otimes> y = \<one> \<Longrightarrow> inv x = y"
by (simp add: inv_char m_comm)
lemma (in ring) inv_neg_one [simp]: "inv (\<ominus> \<one>) = \<ominus> \<one>"
by (simp add: inv_char local.ring_axioms ring.r_minus)
lemma (in monoid) inv_eq_imp_eq: "x \<in> Units G \<Longrightarrow> y \<in> Units G \<Longrightarrow> inv x = inv y \<Longrightarrow> x = y"
by (metis Units_inv_inv)
lemma (in ring) Units_minus_one_closed [intro]: "\<ominus> \<one> \<in> Units R"
by (simp add: Units_def) (metis add.l_inv_ex local.minus_minus minus_equality one_closed r_minus r_one)
lemma (in ring) inv_eq_neg_one_eq: "x \<in> Units R \<Longrightarrow> inv x = \<ominus> \<one> \<longleftrightarrow> x = \<ominus> \<one>"
by (metis Units_inv_inv inv_neg_one)
lemma (in monoid) inv_eq_one_eq: "x \<in> Units G \<Longrightarrow> inv x = \<one> \<longleftrightarrow> x = \<one>"
by (metis Units_inv_inv inv_one)
end