import OAI.LinearAlgebra.CirculantHadamard.LocalDVR
import Mathlib.RingTheory.Localization.AtPrime.Basic
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Int.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.Ring.Int.Parity

namespace OAI

/-!
# The integers localized at the prime two

The ring is the actual localization of `ℤ` at the complement of `(2)`. Its
maximal ideal is generated by the image of two, and its nonzero generator makes
it a discrete valuation ring. The characteristic-zero and odd-unit interfaces
refer to this concrete localization, without assuming any valuation facts.
-/

noncomputable section

namespace CirculantHadamard.IntegerTwoLocal

/-- The prime ideal `(2)` in the integers. -/
def ideal : Ideal ℤ := Ideal.span {(2 : ℤ)}

instance ideal_isPrime : ideal.IsPrime := by
  apply (Ideal.span_singleton_prime (by norm_num : (2 : ℤ) ≠ 0)).mpr
  exact Nat.prime_iff_prime_int.mp Nat.prime_two

/-- The actual ring `ℤ_(2)`. -/
abbrev Ring := Localization.AtPrime ideal

/-- The localization has no zero divisors. -/
theorem isDomain : IsDomain Ring := inferInstance

/-- The complement-of-prime localization is local. -/
theorem isLocalRing : IsLocalRing Ring := inferInstance

/-- Integer coefficients embed faithfully in the concrete localization. -/
theorem algebraMap_injective : Function.Injective (algebraMap ℤ Ring) :=
  IsLocalization.injective Ring ideal.primeCompl_le_nonZeroDivisors

instance charZero : CharZero Ring :=
  charZero_of_injective_algebraMap algebraMap_injective

/-- Noetherianity is inherited from the integer ring by localization. -/
instance isNoetherianRing : IsNoetherianRing Ring :=
  IsLocalization.isNoetherianRing ideal.primeCompl Ring
    (inferInstance : IsNoetherianRing ℤ)

/-- The maximal ideal is exactly the principal ideal generated by two. -/
theorem maximalIdeal_eq_span :
    IsLocalRing.maximalIdeal Ring = Ideal.span {(2 : Ring)} := by
  calc
    IsLocalRing.maximalIdeal Ring = Ideal.map (algebraMap ℤ Ring) ideal :=
      (Localization.AtPrime.map_eq_maximalIdeal (I := ideal)).symm
    _ = Ideal.span {(2 : Ring)} := by
      simp only [ideal, Ideal.map_span, Set.image_singleton, map_ofNat]

theorem two_ne_zero : (2 : Ring) ≠ 0 := by norm_num

theorem two_not_isUnit : ¬ IsUnit (2 : Ring) := by
  rw [← mem_nonunits_iff, ← IsLocalRing.mem_maximalIdeal, maximalIdeal_eq_span]
  exact Ideal.mem_span_singleton_self _

/-- The concrete localization is a discrete valuation ring with uniformizer
(two); this conclusion is obtained from its proved principal maximal ideal. -/
instance isDiscreteValuationRing : IsDiscreteValuationRing Ring :=
  LocalDVR.isDiscreteValuationRing_of_maximalIdeal_eq_span two_ne_zero
    maximalIdeal_eq_span

@[simp] theorem mem_ideal_iff_even (z : ℤ) : z ∈ ideal ↔ Even z := by
  simp only [ideal, Ideal.mem_span_singleton, even_iff_two_dvd]

/-- Exactly the odd integers become units. -/
theorem isUnit_algebraMap_iff_odd (z : ℤ) :
    IsUnit (algebraMap ℤ Ring z) ↔ Odd z := by
  rw [IsLocalization.AtPrime.isUnit_to_map_iff Ring ideal z]
  change z ∉ ideal ↔ Odd z
  rw [mem_ideal_iff_even, Int.not_even_iff_odd]

theorem isUnit_intCast_of_odd (z : ℤ) (hz : Odd z) : IsUnit (z : Ring) := by
  simpa using (isUnit_algebraMap_iff_odd z).mpr hz

theorem isUnit_natCast_of_odd (u : ℕ) (hu : Odd u) : IsUnit (u : Ring) := by
  simpa only [Int.cast_natCast] using
    isUnit_intCast_of_odd (u : ℤ) ((Int.odd_coe_nat u).mpr hu)

end CirculantHadamard.IntegerTwoLocal

end

end OAI