(* Title: HOL/Algebra/IntRing.thy
Author: Stephan Hohe, TU Muenchen
Author: Clemens Ballarin
*)
theory IntRing
imports QuotRing Lattice Int "~~/src/HOL/Number_Theory/Primes"
begin
section \<open>The Ring of Integers\<close>
subsection \<open>Some properties of @{typ int}\<close>
lemma dvds_eq_abseq:
fixes k :: int
shows "l dvd k \<and> k dvd l \<longleftrightarrow> \<bar>l\<bar> = \<bar>k\<bar>"
apply rule
apply (simp add: zdvd_antisym_abs)
apply (simp add: dvd_if_abs_eq)
done
subsection \<open>@{text "\<Z>"}: The Set of Integers as Algebraic Structure\<close>
abbreviation int_ring :: "int ring" ("\<Z>")
where "int_ring \<equiv> \<lparr>carrier = UNIV, mult = op *, one = 1, zero = 0, add = op +\<rparr>"
lemma int_Zcarr [intro!, simp]: "k \<in> carrier \<Z>"
by simp
lemma int_is_cring: "cring \<Z>"
apply (rule cringI)
apply (rule abelian_groupI, simp_all)
defer 1
apply (rule comm_monoidI, simp_all)
apply (rule distrib_right)
apply (fast intro: left_minus)
done
(*
lemma int_is_domain:
"domain \<Z>"
apply (intro domain.intro domain_axioms.intro)
apply (rule int_is_cring)
apply (unfold int_ring_def, simp+)
done
*)
subsection \<open>Interpretations\<close>
text \<open>Since definitions of derived operations are global, their
interpretation needs to be done as early as possible --- that is,
with as few assumptions as possible.\<close>
interpretation int: monoid \<Z>
rewrites "carrier \<Z> = UNIV"
and "mult \<Z> x y = x * y"
and "one \<Z> = 1"
and "pow \<Z> x n = x^n"
proof -
-- "Specification"
show "monoid \<Z>" by standard auto
then interpret int: monoid \<Z> .
-- "Carrier"
show "carrier \<Z> = UNIV" by simp
-- "Operations"
{ fix x y show "mult \<Z> x y = x * y" by simp }
show "one \<Z> = 1" by simp
show "pow \<Z> x n = x^n" by (induct n) simp_all
qed
interpretation int: comm_monoid \<Z>
rewrites "finprod \<Z> f A = setprod f A"
proof -
-- "Specification"
show "comm_monoid \<Z>" by standard auto
then interpret int: comm_monoid \<Z> .
-- "Operations"
{ fix x y have "mult \<Z> x y = x * y" by simp }
note mult = this
have one: "one \<Z> = 1" by simp
show "finprod \<Z> f A = setprod f A"
by (induct A rule: infinite_finite_induct, auto)
qed
interpretation int: abelian_monoid \<Z>
rewrites int_carrier_eq: "carrier \<Z> = UNIV"
and int_zero_eq: "zero \<Z> = 0"
and int_add_eq: "add \<Z> x y = x + y"
and int_finsum_eq: "finsum \<Z> f A = setsum f A"
proof -
-- "Specification"
show "abelian_monoid \<Z>" by standard auto
then interpret int: abelian_monoid \<Z> .
-- "Carrier"
show "carrier \<Z> = UNIV" by simp
-- "Operations"
{ fix x y show "add \<Z> x y = x + y" by simp }
note add = this
show zero: "zero \<Z> = 0"
by simp
show "finsum \<Z> f A = setsum f A"
by (induct A rule: infinite_finite_induct, auto)
qed
interpretation int: abelian_group \<Z>
(* The equations from the interpretation of abelian_monoid need to be repeated.
Since the morphisms through which the abelian structures are interpreted are
not the identity, the equations of these interpretations are not inherited. *)
(* FIXME *)
rewrites "carrier \<Z> = UNIV"
and "zero \<Z> = 0"
and "add \<Z> x y = x + y"
and "finsum \<Z> f A = setsum f A"
and int_a_inv_eq: "a_inv \<Z> x = - x"
and int_a_minus_eq: "a_minus \<Z> x y = x - y"
proof -
-- "Specification"
show "abelian_group \<Z>"
proof (rule abelian_groupI)
fix x
assume "x \<in> carrier \<Z>"
then show "\<exists>y \<in> carrier \<Z>. y \<oplus>\<^bsub>\<Z>\<^esub> x = \<zero>\<^bsub>\<Z>\<^esub>"
by simp arith
qed auto
then interpret int: abelian_group \<Z> .
-- "Operations"
{ fix x y have "add \<Z> x y = x + y" by simp }
note add = this
have zero: "zero \<Z> = 0" by simp
{
fix x
have "add \<Z> (- x) x = zero \<Z>"
by (simp add: add zero)
then show "a_inv \<Z> x = - x"
by (simp add: int.minus_equality)
}
note a_inv = this
show "a_minus \<Z> x y = x - y"
by (simp add: int.minus_eq add a_inv)
qed (simp add: int_carrier_eq int_zero_eq int_add_eq int_finsum_eq)+
interpretation int: "domain" \<Z>
rewrites "carrier \<Z> = UNIV"
and "zero \<Z> = 0"
and "add \<Z> x y = x + y"
and "finsum \<Z> f A = setsum f A"
and "a_inv \<Z> x = - x"
and "a_minus \<Z> x y = x - y"
proof -
show "domain \<Z>"
by unfold_locales (auto simp: distrib_right distrib_left)
qed (simp add: int_carrier_eq int_zero_eq int_add_eq int_finsum_eq int_a_inv_eq int_a_minus_eq)+
text \<open>Removal of occurrences of @{term UNIV} in interpretation result
--- experimental.\<close>
lemma UNIV:
"x \<in> UNIV \<longleftrightarrow> True"
"A \<subseteq> UNIV \<longleftrightarrow> True"
"(\<forall>x \<in> UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
"(EX x : UNIV. P x) \<longleftrightarrow> (EX x. P x)"
"(True \<longrightarrow> Q) \<longleftrightarrow> Q"
"(True \<Longrightarrow> PROP R) \<equiv> PROP R"
by simp_all
interpretation int (* FIXME [unfolded UNIV] *) :
partial_order "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
rewrites "carrier \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> = UNIV"
and "le \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x \<le> y)"
and "lless \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x < y)"
proof -
show "partial_order \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
by standard simp_all
show "carrier \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> = UNIV"
by simp
show "le \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x \<le> y)"
by simp
show "lless \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x < y)"
by (simp add: lless_def) auto
qed
interpretation int (* FIXME [unfolded UNIV] *) :
lattice "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
rewrites "join \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = max x y"
and "meet \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = min x y"
proof -
let ?Z = "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
show "lattice ?Z"
apply unfold_locales
apply (simp add: least_def Upper_def)
apply arith
apply (simp add: greatest_def Lower_def)
apply arith
done
then interpret int: lattice "?Z" .
show "join ?Z x y = max x y"
apply (rule int.joinI)
apply (simp_all add: least_def Upper_def)
apply arith
done
show "meet ?Z x y = min x y"
apply (rule int.meetI)
apply (simp_all add: greatest_def Lower_def)
apply arith
done
qed
interpretation int (* [unfolded UNIV] *) :
total_order "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
by standard clarsimp
subsection \<open>Generated Ideals of @{text "\<Z>"}\<close>
lemma int_Idl: "Idl\<^bsub>\<Z>\<^esub> {a} = {x * a | x. True}"
apply (subst int.cgenideal_eq_genideal[symmetric]) apply simp
apply (simp add: cgenideal_def)
done
lemma multiples_principalideal: "principalideal {x * a | x. True } \<Z>"
by (metis UNIV_I int.cgenideal_eq_genideal int.cgenideal_is_principalideal int_Idl)
lemma prime_primeideal:
assumes prime: "prime p"
shows "primeideal (Idl\<^bsub>\<Z>\<^esub> {p}) \<Z>"
apply (rule primeidealI)
apply (rule int.genideal_ideal, simp)
apply (rule int_is_cring)
apply (simp add: int.cgenideal_eq_genideal[symmetric] cgenideal_def)
apply clarsimp defer 1
apply (simp add: int.cgenideal_eq_genideal[symmetric] cgenideal_def)
apply (elim exE)
proof -
fix a b x
assume "a * b = x * int p"
then have "p dvd a * b" by simp
then have "p dvd a \<or> p dvd b"
by (metis prime prime_dvd_mult_eq_int)
then show "(\<exists>x. a = x * int p) \<or> (\<exists>x. b = x * int p)"
by (metis dvd_def mult.commute)
next
assume "UNIV = {uu. EX x. uu = x * int p}"
then obtain x where "1 = x * int p" by best
then have "\<bar>int p * x\<bar> = 1" by (simp add: mult.commute)
then show False
by (metis abs_of_nat of_nat_1 of_nat_eq_iff abs_zmult_eq_1 one_not_prime_nat prime)
qed
subsection \<open>Ideals and Divisibility\<close>
lemma int_Idl_subset_ideal: "Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l} = (k \<in> Idl\<^bsub>\<Z>\<^esub> {l})"
by (rule int.Idl_subset_ideal') simp_all
lemma Idl_subset_eq_dvd: "Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l} \<longleftrightarrow> l dvd k"
apply (subst int_Idl_subset_ideal, subst int_Idl, simp)
apply (rule, clarify)
apply (simp add: dvd_def)
apply (simp add: dvd_def ac_simps)
done
lemma dvds_eq_Idl: "l dvd k \<and> k dvd l \<longleftrightarrow> Idl\<^bsub>\<Z>\<^esub> {k} = Idl\<^bsub>\<Z>\<^esub> {l}"
proof -
have a: "l dvd k \<longleftrightarrow> (Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l})"
by (rule Idl_subset_eq_dvd[symmetric])
have b: "k dvd l \<longleftrightarrow> (Idl\<^bsub>\<Z>\<^esub> {l} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {k})"
by (rule Idl_subset_eq_dvd[symmetric])
have "l dvd k \<and> k dvd l \<longleftrightarrow> Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l} \<and> Idl\<^bsub>\<Z>\<^esub> {l} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {k}"
by (subst a, subst b, simp)
also have "Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l} \<and> Idl\<^bsub>\<Z>\<^esub> {l} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {k} \<longleftrightarrow> Idl\<^bsub>\<Z>\<^esub> {k} = Idl\<^bsub>\<Z>\<^esub> {l}"
by blast
finally show ?thesis .
qed
lemma Idl_eq_abs: "Idl\<^bsub>\<Z>\<^esub> {k} = Idl\<^bsub>\<Z>\<^esub> {l} \<longleftrightarrow> \<bar>l\<bar> = \<bar>k\<bar>"
apply (subst dvds_eq_abseq[symmetric])
apply (rule dvds_eq_Idl[symmetric])
done
subsection \<open>Ideals and the Modulus\<close>
definition ZMod :: "int \<Rightarrow> int \<Rightarrow> int set"
where "ZMod k r = (Idl\<^bsub>\<Z>\<^esub> {k}) +>\<^bsub>\<Z>\<^esub> r"
lemmas ZMod_defs =
ZMod_def genideal_def
lemma rcos_zfact:
assumes kIl: "k \<in> ZMod l r"
shows "\<exists>x. k = x * l + r"
proof -
from kIl[unfolded ZMod_def] have "\<exists>xl\<in>Idl\<^bsub>\<Z>\<^esub> {l}. k = xl + r"
by (simp add: a_r_coset_defs)
then obtain xl where xl: "xl \<in> Idl\<^bsub>\<Z>\<^esub> {l}" and k: "k = xl + r"
by auto
from xl obtain x where "xl = x * l"
by (auto simp: int_Idl)
with k have "k = x * l + r"
by simp
then show "\<exists>x. k = x * l + r" ..
qed
lemma ZMod_imp_zmod:
assumes zmods: "ZMod m a = ZMod m b"
shows "a mod m = b mod m"
proof -
interpret ideal "Idl\<^bsub>\<Z>\<^esub> {m}" \<Z>
by (rule int.genideal_ideal) fast
from zmods have "b \<in> ZMod m a"
unfolding ZMod_def by (simp add: a_repr_independenceD)
then have "\<exists>x. b = x * m + a"
by (rule rcos_zfact)
then obtain x where "b = x * m + a"
by fast
then have "b mod m = (x * m + a) mod m"
by simp
also have "\<dots> = ((x * m) mod m) + (a mod m)"
by (simp add: mod_add_eq)
also have "\<dots> = a mod m"
by simp
finally have "b mod m = a mod m" .
then show "a mod m = b mod m" ..
qed
lemma ZMod_mod: "ZMod m a = ZMod m (a mod m)"
proof -
interpret ideal "Idl\<^bsub>\<Z>\<^esub> {m}" \<Z>
by (rule int.genideal_ideal) fast
show ?thesis
unfolding ZMod_def
apply (rule a_repr_independence'[symmetric])
apply (simp add: int_Idl a_r_coset_defs)
proof -
have "a = m * (a div m) + (a mod m)"
by (simp add: zmod_zdiv_equality)
then have "a = (a div m) * m + (a mod m)"
by simp
then show "\<exists>h. (\<exists>x. h = x * m) \<and> a = h + a mod m"
by fast
qed simp
qed
lemma zmod_imp_ZMod:
assumes modeq: "a mod m = b mod m"
shows "ZMod m a = ZMod m b"
proof -
have "ZMod m a = ZMod m (a mod m)"
by (rule ZMod_mod)
also have "\<dots> = ZMod m (b mod m)"
by (simp add: modeq[symmetric])
also have "\<dots> = ZMod m b"
by (rule ZMod_mod[symmetric])
finally show ?thesis .
qed
corollary ZMod_eq_mod: "ZMod m a = ZMod m b \<longleftrightarrow> a mod m = b mod m"
apply (rule iffI)
apply (erule ZMod_imp_zmod)
apply (erule zmod_imp_ZMod)
done
subsection \<open>Factorization\<close>
definition ZFact :: "int \<Rightarrow> int set ring"
where "ZFact k = \<Z> Quot (Idl\<^bsub>\<Z>\<^esub> {k})"
lemmas ZFact_defs = ZFact_def FactRing_def
lemma ZFact_is_cring: "cring (ZFact k)"
apply (unfold ZFact_def)
apply (rule ideal.quotient_is_cring)
apply (intro ring.genideal_ideal)
apply (simp add: cring.axioms[OF int_is_cring] ring.intro)
apply simp
apply (rule int_is_cring)
done
lemma ZFact_zero: "carrier (ZFact 0) = (\<Union>a. {{a}})"
apply (insert int.genideal_zero)
apply (simp add: ZFact_defs A_RCOSETS_defs r_coset_def)
done
lemma ZFact_one: "carrier (ZFact 1) = {UNIV}"
apply (simp only: ZFact_defs A_RCOSETS_defs r_coset_def ring_record_simps)
apply (subst int.genideal_one)
apply (rule, rule, clarsimp)
apply (rule, rule, clarsimp)
apply (rule, clarsimp, arith)
apply (rule, clarsimp)
apply (rule exI[of _ "0"], clarsimp)
done
lemma ZFact_prime_is_domain:
assumes pprime: "prime p"
shows "domain (ZFact p)"
apply (unfold ZFact_def)
apply (rule primeideal.quotient_is_domain)
apply (rule prime_primeideal[OF pprime])
done
end