(*  Title:      HOL/Algebra/IntRing.thy

     Author:     Stephan Hohe, TU Muenchen

     Author:     Clemens Ballarin

 *)

 theory IntRing

 imports QuotRing Lattice Int "~~/src/HOL/Old_Number_Theory/Primes"

 begin

 section {* The Ring of Integers *}

 subsection {* Some properties of @{typ int} *}

 lemma dvds_eq_abseq:

   "(l dvd k \<and> k dvd l) = (abs l = abs (k::int))"

 apply rule

  apply (simp add: zdvd_antisym_abs)

 apply (simp add: dvd_if_abs_eq)

 done

 subsection {* @{text "\<Z>"}: The Set of Integers as Algebraic Structure *}

 abbreviation

   int_ring :: "int ring" ("\<Z>") where

   "int_ring == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"

 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 {* 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>" by default 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 }

   note mult = this

   show one: "one \<Z> = 1" by simp

   show "pow \<Z> x n = x^n" by (induct n) simp_all

 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>" by default 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 = (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 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 = (if finite A then setsum f A else undefined)"

 proof -

   -- "Specification"

   show "abelian_monoid \<Z>" by default 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 = (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>

   (* 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 *)

   where "carrier \<Z> = UNIV"

     and "zero \<Z> = 0"

     and "add \<Z> x y = x + y"

     and "finsum \<Z> f A = (if finite A then setsum f A else undefined)"

     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)

     show "!!x. x \<in> carrier \<Z> ==> EX y : 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>

   where "carrier \<Z> = UNIV"

     and "zero \<Z> = 0"

     and "add \<Z> x y = x + y"

     and "finsum \<Z> f A = (if finite A then setsum f A else undefined)"

     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 {* 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> |)"

     by default 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> |)"

   by default 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

   apply (simp add: cgenideal_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 clarsimp defer 1

  apply (simp add: int.cgenideal_eq_genideal[symmetric] cgenideal_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))" using ppos by (simp add: nat_dvd_iff)

   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 best

   from this [symmetric]

       have "p * x = 1" by (subst mult_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)

 apply (simp add: dvd_def mult_ac)

 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 *}

 definition

   ZMod :: "int => int => 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 "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)

   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: mod_add_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)

   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 *}

 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:

   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 ring_record_simps)

 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 (nat p)"

   shows "domain (ZFact p)"

 apply (unfold ZFact_def)

 apply (rule primeideal.quotient_is_domain)

 apply (rule prime_primeideal[OF pprime])

 done

 end