Theory IntRing

(*  Title:      HOL/Algebra/IntRing.thy
    Author:     Stephan Hohe, TU Muenchen
    Author:     Clemens Ballarin
*)

theory IntRing
imports "HOL-Computational_Algebra.Primes" QuotRing Lattice 
begin

section ‹The Ring of Integers›

subsection ‹Some properties of \<^typ>‹int››

lemma dvds_eq_abseq:
  fixes k :: int
  shows "l dvd k ∧ k dvd l ⟷ ¦l¦ = ¦k¦"
  by (metis dvd_if_abs_eq lcm.commute lcm_proj1_iff_int)


subsection ‹‹𝒵›: The Set of Integers as Algebraic Structure›

abbreviation int_ring :: "int ring" ("𝒵")
  where "int_ring ≡ ⦇carrier = UNIV, mult = (*), one = 1, zero = 0, add = (+)⦈"

lemma int_Zcarr [intro!, simp]: "k ∈ carrier 𝒵"
  by simp

lemma int_is_cring: "cring 𝒵"
proof (rule cringI)
  show "abelian_group 𝒵"
    by (rule abelian_groupI) (auto intro: left_minus)
  show "Group.comm_monoid 𝒵"
    by (simp add: Group.monoid.intro monoid.monoid_comm_monoidI)
qed (auto simp: distrib_right)


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
  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 = prod 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 = prod 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 = sum 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 = sum 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 = sum 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 ∈ carrier 𝒵"
    then show "∃y ∈ carrier 𝒵. y ⊕⇘𝒵⇙ x = 𝟬⇘𝒵⇙"
      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 = sum 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 ‹Removal of occurrences of \<^term>‹UNIV› in interpretation result
  --- experimental.›

lemma UNIV:
  "x ∈ UNIV ⟷ True"
  "A ⊆ UNIV ⟷ True"
  "(∀x ∈ UNIV. P x) ⟷ (∀x. P x)"
  "(∃x ∈ UNIV. P x) ⟷ (∃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 = (=), le = (≤)⦈"
  rewrites "carrier ⦇carrier = UNIV::int set, eq = (=), le = (≤)⦈ = UNIV"
    and "le ⦇carrier = UNIV::int set, eq = (=), le = (≤)⦈ x y = (x ≤ y)"
    and "lless ⦇carrier = UNIV::int set, eq = (=), le = (≤)⦈ x y = (x < y)"
proof -
  show "partial_order ⦇carrier = UNIV::int set, eq = (=), le = (≤)⦈"
    by standard simp_all
  show "carrier ⦇carrier = UNIV::int set, eq = (=), le = (≤)⦈ = UNIV"
    by simp
  show "le ⦇carrier = UNIV::int set, eq = (=), le = (≤)⦈ x y = (x ≤ y)"
    by simp
  show "lless ⦇carrier = UNIV::int set, eq = (=), le = (≤)⦈ x y = (x < y)"
    by (simp add: lless_def) auto
qed

interpretation int (* FIXME [unfolded UNIV] *) :
  lattice "⦇carrier = UNIV::int set, eq = (=), le = (≤)⦈"
  rewrites "join ⦇carrier = UNIV::int set, eq = (=), le = (≤)⦈ x y = max x y"
    and "meet ⦇carrier = UNIV::int set, eq = (=), le = (≤)⦈ x y = min x y"
proof -
  let ?Z = "⦇carrier = UNIV::int set, eq = (=), le = (≤)⦈"
  show "lattice ?Z"
    apply unfold_locales
    apply (simp_all add: least_def Upper_def greatest_def Lower_def)
    apply arith+
    done
  then interpret int: lattice "?Z" .
  show "join ?Z x y = max x y"
    by (metis int.le_iff_meet iso_tuple_UNIV_I join_comm linear max.absorb_iff2 max_def)
  show "meet ?Z x y = min x y"
    using int.meet_le int.meet_left int.meet_right by auto
qed

interpretation int (* [unfolded UNIV] *) :
  total_order "⦇carrier = UNIV::int set, eq = (=), le = (≤)⦈"
  by standard clarsimp


subsection ‹Generated Ideals of ‹𝒵››

lemma int_Idl: "Idl⇘𝒵⇙ {a} = {x * a | x. True}"
  by (simp_all add: cgenideal_def int.cgenideal_eq_genideal[symmetric])

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: "Factorial_Ring.prime p"
  shows "primeideal (Idl⇘𝒵⇙ {p}) 𝒵"
proof (rule primeidealI)
  show "ideal (Idl⇘𝒵⇙ {p}) 𝒵"
    by (rule int.genideal_ideal, simp)
  show "cring 𝒵"
    by (rule int_is_cring)
  have False if "UNIV = {v::int. ∃x. v = x * p}"
  proof -
    from that
    obtain i where "1 = i * p"
      by (blast intro:  elim: )
    then show False
      using prime by (auto simp add: abs_mult zmult_eq_1_iff)
  qed
  then show "carrier 𝒵 ≠ Idl⇘𝒵⇙ {p}"
    by (auto simp add: int.cgenideal_eq_genideal[symmetric] cgenideal_def)
  have "p dvd a ∨ p dvd b" if "a * b = x * p" for a b x
    by (simp add: prime prime_dvd_multD that)
  then show "⋀a b. ⟦a ∈ carrier 𝒵; b ∈ carrier 𝒵; a ⊗⇘𝒵⇙ b ∈ Idl⇘𝒵⇙ {p}⟧
           ⟹ a ∈ Idl⇘𝒵⇙ {p} ∨ b ∈ Idl⇘𝒵⇙ {p}"
    by (auto simp add: int.cgenideal_eq_genideal[symmetric] cgenideal_def dvd_def mult.commute)
qed

subsection ‹Ideals and Divisibility›

lemma int_Idl_subset_ideal: "Idl⇘𝒵⇙ {k} ⊆ Idl⇘𝒵⇙ {l} = (k ∈ Idl⇘𝒵⇙ {l})"
  by (rule int.Idl_subset_ideal') simp_all

lemma Idl_subset_eq_dvd: "Idl⇘𝒵⇙ {k} ⊆ Idl⇘𝒵⇙ {l} ⟷ l dvd k"
  by (subst int_Idl_subset_ideal) (auto simp: dvd_def int_Idl)

lemma dvds_eq_Idl: "l dvd k ∧ k dvd l ⟷ Idl⇘𝒵⇙ {k} = Idl⇘𝒵⇙ {l}"
proof -
  have a: "l dvd k ⟷ (Idl⇘𝒵⇙ {k} ⊆ Idl⇘𝒵⇙ {l})"
    by (rule Idl_subset_eq_dvd[symmetric])
  have b: "k dvd l ⟷ (Idl⇘𝒵⇙ {l} ⊆ Idl⇘𝒵⇙ {k})"
    by (rule Idl_subset_eq_dvd[symmetric])

  have "l dvd k ∧ k dvd l ⟷ Idl⇘𝒵⇙ {k} ⊆ Idl⇘𝒵⇙ {l} ∧ Idl⇘𝒵⇙ {l} ⊆ Idl⇘𝒵⇙ {k}"
    by (subst a, subst b, simp)
  also have "Idl⇘𝒵⇙ {k} ⊆ Idl⇘𝒵⇙ {l} ∧ Idl⇘𝒵⇙ {l} ⊆ Idl⇘𝒵⇙ {k} ⟷ Idl⇘𝒵⇙ {k} = Idl⇘𝒵⇙ {l}"
    by blast
  finally show ?thesis .
qed

lemma Idl_eq_abs: "Idl⇘𝒵⇙ {k} = Idl⇘𝒵⇙ {l} ⟷ ¦l¦ = ¦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⇘𝒵⇙ {k}) +>⇘𝒵⇙ r"

lemmas ZMod_defs =
  ZMod_def genideal_def

lemma rcos_zfact:
  assumes kIl: "k ∈ ZMod l r"
  shows "∃x. k = x * l + r"
proof -
  from kIl[unfolded ZMod_def] have "∃xl∈Idl⇘𝒵⇙ {l}. k = xl + r"
    by (simp add: a_r_coset_defs)
  then obtain xl where xl: "xl ∈ Idl⇘𝒵⇙ {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 "∃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⇘𝒵⇙ {m}" 𝒵
    by (rule int.genideal_ideal) fast
  from zmods have "b ∈ ZMod m a"
    unfolding ZMod_def by (simp add: a_repr_independenceD)
  then have "∃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 "… = ((x * m) mod m) + (a mod m)"
    by (simp add: mod_add_eq)
  also have "… = 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⇘𝒵⇙ {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: mult_div_mod_eq [symmetric])
    then have "a = (a div m) * m + (a mod m)"
      by simp
    then show "∃h. (∃x. h = x * m) ∧ 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 "… = ZMod m (b mod m)"
    by (simp add: modeq[symmetric])
  also have "… = ZMod m b"
    by (rule ZMod_mod[symmetric])
  finally show ?thesis .
qed

corollary ZMod_eq_mod: "ZMod m a = ZMod m b ⟷ a mod m = b mod m"
  apply (rule iffI)
  apply (erule ZMod_imp_zmod)
  apply (erule zmod_imp_ZMod)
  done


subsection ‹Factorization›

definition ZFact :: "int ⇒ int set ring"
  where "ZFact k = 𝒵 Quot (Idl⇘𝒵⇙ {k})"

lemmas ZFact_defs = ZFact_def FactRing_def

lemma ZFact_is_cring: "cring (ZFact k)"
  by (simp add: ZFact_def ideal.quotient_is_cring int.cring_axioms int.genideal_ideal)

lemma ZFact_zero: "carrier (ZFact 0) = (⋃a. {{a}})"
  by (simp add: ZFact_defs A_RCOSETS_defs r_coset_def int.genideal_zero)

lemma ZFact_one: "carrier (ZFact 1) = {UNIV}"
  unfolding ZFact_defs A_RCOSETS_defs r_coset_def ring_record_simps int.genideal_one
proof
  have "⋀a b::int. ∃x. b = x + a"
    by presburger
  then show "(⋃a::int. {⋃h. {h + a}}) ⊆ {UNIV}"
    by force
  then show "{UNIV} ⊆ (⋃a::int. {⋃h. {h + a}})"
    by (metis (no_types, lifting) UNIV_I UN_I singletonD singletonI subset_iff)
qed

lemma ZFact_prime_is_domain:
  assumes pprime: "Factorial_Ring.prime p"
  shows "domain (ZFact p)"
  by (simp add: ZFact_def pprime prime_primeideal primeideal.quotient_is_domain)

end