(*
Title: HOL/Algebra/IntRing.thy
Id: $Id$
Author: Stephan Hohe, TU Muenchen
*)
theory IntRing
imports QuotRing Lattice Int Primes
begin
section {* The Ring of Integers *}
subsection {* Some properties of @{typ int} *}
lemma dvds_imp_abseq:
"\<lbrakk>l dvd k; k dvd l\<rbrakk> \<Longrightarrow> abs l = abs (k::int)"
apply (subst abs_split, rule conjI)
apply (clarsimp, subst abs_split, rule conjI)
apply (clarsimp)
apply (cases "k=0", simp)
apply (cases "l=0", simp)
apply (simp add: zdvd_anti_sym)
apply clarsimp
apply (cases "k=0", simp)
apply (simp add: zdvd_anti_sym)
apply (clarsimp, subst abs_split, rule conjI)
apply (clarsimp)
apply (cases "l=0", simp)
apply (simp add: zdvd_anti_sym)
apply (clarsimp)
apply (subgoal_tac "-l = -k", simp)
apply (intro zdvd_anti_sym, simp+)
done
lemma abseq_imp_dvd:
assumes a_lk: "abs l = abs (k::int)"
shows "l dvd k"
proof -
from a_lk
have "nat (abs l) = nat (abs k)" by simp
hence "nat (abs l) dvd nat (abs k)" by simp
hence "int (nat (abs l)) dvd k" by (subst int_dvd_iff)
hence "abs l dvd k" by simp
thus "l dvd k"
apply (unfold dvd_def, cases "l<0")
defer 1 apply clarsimp
proof (clarsimp)
fix k
assume l0: "l < 0"
have "- (l * k) = l * (-k)" by simp
thus "\<exists>ka. - (l * k) = l * ka" by fast
qed
qed
lemma dvds_eq_abseq:
"(l dvd k \<and> k dvd l) = (abs l = abs (k::int))"
apply rule
apply (simp add: dvds_imp_abseq)
apply (rule conjI)
apply (simp add: abseq_imp_dvd)+
done
subsection {* @{text "\<Z>"}: The Set of Integers as Algebraic Structure *}
constdefs
int_ring :: "int ring" ("\<Z>")
"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 add: int_ring_def)
lemma int_is_cring:
"cring \<Z>"
unfolding int_ring_def
apply (rule cringI)
apply (rule abelian_groupI, simp_all)
defer 1
apply (rule comm_monoidI, simp_all)
apply (rule zadd_zmult_distrib)
apply (fast intro: zadd_zminus_inverse2)
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 {* Interpretations *}
text {* 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. *}
interpretation int: monoid ["\<Z>"]
where "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>" proof qed (auto simp: int_ring_def)
then interpret int: monoid ["\<Z>"] .
-- "Carrier"
show "carrier \<Z> = UNIV" by (simp add: int_ring_def)
-- "Operations"
{ fix x y show "mult \<Z> x y = x * y" by (simp add: int_ring_def) }
note mult = this
show one: "one \<Z> = 1" by (simp add: int_ring_def)
show "pow \<Z> x n = x^n" by (induct n) (simp, simp add: int_ring_def)+
qed
interpretation int: comm_monoid ["\<Z>"]
where "finprod \<Z> f A = (if finite A then setprod f A else undefined)"
proof -
-- "Specification"
show "comm_monoid \<Z>" proof qed (auto simp: int_ring_def)
then interpret int: comm_monoid ["\<Z>"] .
-- "Operations"
{ fix x y have "mult \<Z> x y = x * y" by (simp add: int_ring_def) }
note mult = this
have one: "one \<Z> = 1" by (simp add: int_ring_def)
show "finprod \<Z> f A = (if finite A then setprod f A else undefined)"
proof (cases "finite A")
case True then show ?thesis proof induct
case empty show ?case by (simp add: one)
next
case insert then show ?case by (simp add: Pi_def mult)
qed
next
case False then show ?thesis by (simp add: finprod_def)
qed
qed
interpretation int: abelian_monoid ["\<Z>"]
where "zero \<Z> = 0"
and "add \<Z> x y = x + y"
and "finsum \<Z> f A = (if finite A then setsum f A else undefined)"
proof -
-- "Specification"
show "abelian_monoid \<Z>" proof qed (auto simp: int_ring_def)
then interpret int: abelian_monoid ["\<Z>"] .
-- "Operations"
{ fix x y show "add \<Z> x y = x + y" by (simp add: int_ring_def) }
note add = this
show zero: "zero \<Z> = 0" by (simp add: int_ring_def)
show "finsum \<Z> f A = (if finite A then setsum f A else undefined)"
proof (cases "finite A")
case True then show ?thesis proof induct
case empty show ?case by (simp add: zero)
next
case insert then show ?case by (simp add: Pi_def add)
qed
next
case False then show ?thesis by (simp add: finsum_def finprod_def)
qed
qed
interpretation int: abelian_group ["\<Z>"]
where "a_inv \<Z> x = - x"
and "a_minus \<Z> x y = x - y"
proof -
-- "Specification"
show "abelian_group \<Z>"
proof (rule abelian_groupI)
show "!!x. x \<in> carrier \<Z> ==> EX y : carrier \<Z>. y \<oplus>\<^bsub>\<Z>\<^esub> x = \<zero>\<^bsub>\<Z>\<^esub>"
by (simp add: int_ring_def) arith
qed (auto simp: int_ring_def)
then interpret int: abelian_group ["\<Z>"] .
-- "Operations"
{ fix x y have "add \<Z> x y = x + y" by (simp add: int_ring_def) }
note add = this
have zero: "zero \<Z> = 0" by (simp add: int_ring_def)
{ 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
interpretation int: "domain" ["\<Z>"]
proof qed (auto simp: int_ring_def left_distrib right_distrib)
text {* Removal of occurrences of @{term UNIV} in interpretation result
--- experimental. *}
lemma UNIV:
"x \<in> UNIV = True"
"A \<subseteq> UNIV = True"
"(ALL x : UNIV. P x) = (ALL x. P x)"
"(EX x : UNIV. P x) = (EX x. P x)"
"(True --> Q) = Q"
"(True ==> PROP R) == PROP R"
by simp_all
interpretation int (* FIXME [unfolded UNIV] *) :
partial_order ["(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"]
where "carrier (| carrier = UNIV::int set, eq = op =, le = op \<le> |) = UNIV"
and "le (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x \<le> y)"
and "lless (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x < y)"
proof -
show "partial_order (| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
proof qed simp_all
show "carrier (| carrier = UNIV::int set, eq = op =, le = op \<le> |) = UNIV"
by simp
show "le (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x \<le> y)"
by simp
show "lless (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x < y)"
by (simp add: lless_def) auto
qed
interpretation int (* FIXME [unfolded UNIV] *) :
lattice ["(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"]
where "join (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = max x y"
and "meet (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = min x y"
proof -
let ?Z = "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
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 ["(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"]
proof qed clarsimp
subsection {* Generated Ideals of @{text "\<Z>"} *}
lemma int_Idl:
"Idl\<^bsub>\<Z>\<^esub> {a} = {x * a | x. True}"
apply (subst int.cgenideal_eq_genideal[symmetric]) apply (simp add: int_ring_def)
apply (simp add: cgenideal_def int_ring_def)
done
lemma multiples_principalideal:
"principalideal {x * a | x. True } \<Z>"
apply (subst int_Idl[symmetric], rule principalidealI)
apply (rule int.genideal_ideal, simp)
apply fast
done
lemma prime_primeideal:
assumes prime: "prime (nat 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 (simp add: int_ring_def)
apply clarsimp defer 1
apply (simp add: int.cgenideal_eq_genideal[symmetric] cgenideal_def)
apply (simp add: int_ring_def)
apply (elim exE)
proof -
fix a b x
from prime
have ppos: "0 <= p" by (simp add: prime_def)
have unnat: "!!x. nat p dvd nat (abs x) ==> p dvd x"
proof -
fix x
assume "nat p dvd nat (abs x)"
hence "int (nat p) dvd x" by (simp add: int_dvd_iff[symmetric])
thus "p dvd x" by (simp add: ppos)
qed
assume "a * b = x * p"
hence "p dvd a * b" by simp
hence "nat p dvd nat (abs (a * b))"
apply (subst nat_dvd_iff, clarsimp)
apply (rule conjI, clarsimp, simp add: zabs_def)
proof (clarsimp)
assume a: " ~ 0 <= p"
from prime
have "0 < p" by (simp add: prime_def)
from a and this
have "False" by simp
thus "nat (abs (a * b)) = 0" ..
qed
hence "nat p dvd (nat (abs a) * nat (abs b))" by (simp add: nat_abs_mult_distrib)
hence "nat p dvd nat (abs a) | nat p dvd nat (abs b)" by (rule prime_dvd_mult[OF prime])
hence "p dvd a | p dvd b" by (fast intro: unnat)
thus "(EX x. a = x * p) | (EX x. b = x * p)"
proof
assume "p dvd a"
hence "EX x. a = p * x" by (simp add: dvd_def)
from this obtain x
where "a = p * x" by fast
hence "a = x * p" by simp
hence "EX x. a = x * p" by simp
thus "(EX x. a = x * p) | (EX x. b = x * p)" ..
next
assume "p dvd b"
hence "EX x. b = p * x" by (simp add: dvd_def)
from this obtain x
where "b = p * x" by fast
hence "b = x * p" by simp
hence "EX x. b = x * p" by simp
thus "(EX x. a = x * p) | (EX x. b = x * p)" ..
qed
next
assume "UNIV = {uu. EX x. uu = x * p}"
from this obtain x
where "1 = x * p" by fast
from this [symmetric]
have "p * x = 1" by (subst zmult_commute)
hence "\<bar>p * x\<bar> = 1" by simp
hence "\<bar>p\<bar> = 1" by (rule abs_zmult_eq_1)
from this and prime
show "False" by (simp add: prime_def)
qed
subsection {* Ideals and Divisibility *}
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+)
lemma Idl_subset_eq_dvd:
"(Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l}) = (l dvd k)"
apply (subst int_Idl_subset_ideal, subst int_Idl, simp)
apply (rule, clarify)
apply (simp add: dvd_def, clarify)
apply (simp add: int.m_comm)
done
lemma dvds_eq_Idl:
"(l dvd k \<and> k dvd l) = (Idl\<^bsub>\<Z>\<^esub> {k} = Idl\<^bsub>\<Z>\<^esub> {l})"
proof -
have a: "l dvd k = (Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l})" by (rule Idl_subset_eq_dvd[symmetric])
have b: "k dvd l = (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) = ((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})) = (Idl\<^bsub>\<Z>\<^esub> {k} = Idl\<^bsub>\<Z>\<^esub> {l})" by (rule, fast+)
finally
show ?thesis .
qed
lemma Idl_eq_abs:
"(Idl\<^bsub>\<Z>\<^esub> {k} = Idl\<^bsub>\<Z>\<^esub> {l}) = (abs l = abs k)"
apply (subst dvds_eq_abseq[symmetric])
apply (rule dvds_eq_Idl[symmetric])
done
subsection {* Ideals and the Modulus *}
constdefs
ZMod :: "int => int => int set"
"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 "EX 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 int_ring_def)
from this 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 (simp add: int_Idl, fast)
from k and this
have "k = x * l + r" by simp
thus "\<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)
from this
have "EX x. b = x * m + a" by (rule rcos_zfact)
from this obtain x
where "b = x * m + a"
by fast
hence "b mod m = (x * m + a) mod m" by simp
also
have "\<dots> = ((x * m) mod m) + (a mod m)" by (simp add: zmod_zadd1_eq)
also
have "\<dots> = a mod m" by simp
finally
have "b mod m = a mod m" .
thus "a mod m = b mod m" ..
qed
lemma ZMod_mod:
shows "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)
apply (simp add: int_ring_def)
proof -
have "a = m * (a div m) + (a mod m)" by (simp add: zmod_zdiv_equality)
hence "a = (a div m) * m + (a mod m)" by simp
thus "\<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:
shows "(ZMod m a = ZMod m b) = (a mod m = b mod m)"
by (rule, erule ZMod_imp_zmod, erule zmod_imp_ZMod)
subsection {* Factorization *}
constdefs
ZFact :: "int \<Rightarrow> int set ring"
"ZFact k == \<Z> Quot (Idl\<^bsub>\<Z>\<^esub> {k})"
lemmas ZFact_defs = ZFact_def FactRing_def
lemma ZFact_is_cring:
shows "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 int_ring_def ring_record_simps)
done
lemma ZFact_one:
"carrier (ZFact 1) = {UNIV}"
apply (simp only: ZFact_defs A_RCOSETS_defs r_coset_def int_ring_def ring_record_simps)
apply (subst int.genideal_one[unfolded int_ring_def, simplified ring_record_simps])
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 (nat p)"
shows "domain (ZFact p)"
apply (unfold ZFact_def)
apply (rule primeideal.quotient_is_domain)
apply (rule prime_primeideal[OF pprime])
done
end