Blame for lean/OAI/LinearAlgebra/CirculantHadamard/LocalComparison.lean

import OAI.LinearAlgebra.CirculantHadamard.LocalComparisonInduction
import OAI.LinearAlgebra.CirculantHadamard.CyclotomicFactorDivision
import OAI.LinearAlgebra.CirculantHadamard.CyclicEvaluationProjection
import OAI.LinearAlgebra.CirculantHadamard.RamifiedLocal

namespace OAI

universe uA

/-!
# Local character comparison in the actual cyclotomic quotient

Starting with a characteristic-zero DVR whose maximal ideal is generated by
the rational prime `p`, all ramified rings needed by the induction are
constructed as the actual cyclotomic quotients. No projection-divisibility,
comparison, or existence-of-valuation premise is taken from the caller.

The conclusion uses membership in the ideal generated by `root - 1` to state
residue one. This is independent of how a local-ring instance is installed and
can be transported directly to a later localization.
-/

noncomputable section

namespace CirculantHadamard.LocalComparison

open CyclicRing Polynomial

variable (A : Type uA) [CommRing A]

/-- The fixed primitive-order-`p` root used at every induction level. -/
def orderPRoot (p : ℕ) : RamifiedLocal.Extension A p 0 :=
  AdjoinRoot.root (cyclotomic (p ^ (0 + 1)) A)

theorem orderPRoot_pow (p : ℕ) : orderPRoot A p ^ p = 1 := by
  calc
    orderPRoot A p ^ p = orderPRoot A p ^ (p ^ (0 + 1)) :=
      congrArg (fun n : ℕ => orderPRoot A p ^ n) (by simp)
    _ = 1 := cyclotomicAdjoinRoot_pow (R := A) (p ^ (0 + 1))

theorem orderPRoot_pow_level (p e : ℕ) : orderPRoot A p ^ (p ^ (e + 1)) = 1 := by
  calc
    orderPRoot A p ^ (p ^ (e + 1)) = orderPRoot A p ^ (p * p ^ e) :=
      congrArg (fun n : ℕ => orderPRoot A p ^ n) (pow_succ' p e)
    _ = (orderPRoot A p ^ p) ^ (p ^ e) := pow_mul _ _ _
    _ = 1 := by rw [orderPRoot_pow, one_pow]

/-- Actual order-`p` evaluation of an element of the cyclic group ring of
order `p ^ (e + 1)`. -/
def orderPValue (p e : ℕ) [NeZero p] :
    Elem A (p ^ (e + 1)) →ₐ[A] RamifiedLocal.Extension A p 0 :=
  evaluateAt (p ^ (e + 1)) (orderPRoot A p) (orderPRoot_pow_level A p e)

@[simp] theorem orderPValue_scalar (p e : ℕ) [NeZero p] (a : A) :
    orderPValue A p e (scalar (p ^ (e + 1)) a) =
      algebraMap A (RamifiedLocal.Extension A p 0) a :=
  (orderPValue A p e).commutes a

theorem orderPValue_projection (p e : ℕ) [NeZero p]
    (F : Elem A (p ^ (e + 2))) :
    orderPValue A p e
        (cyclicProjection A (pow_dvd_pow p (Nat.le_succ (e + 1))) F) =
      orderPValue A p (e + 1) F :=
  evaluateAt_cyclicProjection _ _ _ _ F

/-- The conclusion of comparison, expressed by an actual unit and membership
in the actual ramified maximal ideal. -/
def OrderPCompared (p e : ℕ) [NeZero p] (F : Elem A (p ^ (e + 1))) : Prop :=
  ∃ u : (RamifiedLocal.Extension A p 0)ˣ,
    orderPValue A p e F = (u : RamifiedLocal.Extension A p 0) *
      algebraMap A (RamifiedLocal.Extension A p 0) (augmentation (p ^ (e + 1)) F) ∧
    (u : RamifiedLocal.Extension A p 0) - 1 ∈
      Ideal.span {RamifiedLocal.uniformizer A p 0}

section DVR

variable [IsDomain A] [CharZero A] [IsDiscreteValuationRing A]

/-- Character comparison with every ramified extension constructed from the
input DVR. Both factors are compared, with matching group/scalar exponents. -/
theorem local_character_comparison (p : ℕ) [NeZero p] (hp : p.Prime)
    (hmax : IsLocalRing.maximalIdeal A = Ideal.span {(p : A)})
    (e : ℕ) (b : Aˣ) (F K : Elem A (p ^ (e + 1)))
    (hFK : F * K = scalar (p ^ (e + 1)) ((p : A) ^ (e + 1) * (b : A))) :
    OrderPCompared A p e F ∧ OrderPCompared A p e K := by
  let : IsDomain (RamifiedLocal.Extension A p 0) :=
    RamifiedLocal.isDomain A hp 0 hmax
  let : IsLocalRing (RamifiedLocal.Extension A p 0) :=
    RamifiedLocal.isLocalRing A hp 0 hmax
  have hpA : Prime (p : A) :=
    ((IsDiscreteValuationRing.irreducible_iff_uniformizer (p : A)).mpr hmax).prime
  have hpS : algebraMap A (RamifiedLocal.Extension A p 0) (p : A) ≠ 0 := by
    intro h
    exact hpA.ne_zero (RamifiedLocal.algebraMap_injective A hp 0
      (by simpa only [map_zero] using h))
  have hroot : IsLocalRing.residue (RamifiedLocal.Extension A p 0) (orderPRoot A p) = 1 := by
    have hpi : RamifiedLocal.uniformizer A p 0 ∈
        IsLocalRing.maximalIdeal (RamifiedLocal.Extension A p 0) := by
      rw [RamifiedLocal.maximalIdeal_eq_span A hp 0 hmax]
      exact Ideal.subset_span (Set.mem_singleton _)
    have hz := (IsLocalRing.residue_eq_zero_iff _).mpr hpi
    simpa only [RamifiedLocal.uniformizer, orderPRoot, map_sub, map_one, sub_eq_zero] using hz
  have hc := comparison_induction p (algebraMap A (RamifiedLocal.Extension A p 0))
    hpA hpS (fun k => (orderPValue A p k).toRingHom)
    (fun k a => orderPValue_scalar A p k a)
    (by
      intro U
      change IsLocalRing.residue (RamifiedLocal.Extension A p 0) (orderPValue A p 0 U) =
        IsLocalRing.residue (RamifiedLocal.Extension A p 0)
          (algebraMap A (RamifiedLocal.Extension A p 0) (augmentation (p ^ 1) U))
      simp only [orderPValue, evaluateAt_apply,
        augmentation_apply, map_sum, map_mul, map_pow, hroot, one_pow, mul_one])
    (fun k U => orderPValue_projection A p k U)
    (by
      intro k c U V hUV
      let : IsDomain (RamifiedLocal.Extension A p (k + 1)) :=
        RamifiedLocal.isDomain A hp (k + 1) hmax
      let : IsLocalRing (RamifiedLocal.Extension A p (k + 1)) :=
        RamifiedLocal.isLocalRing A hp (k + 1) hmax
      let : IsDiscreteValuationRing (RamifiedLocal.Extension A p (k + 1)) :=
        RamifiedLocal.isDiscreteValuationRing A hp (k + 1) hmax
      have hpR : (p : RamifiedLocal.Extension A p (k + 1)) ≠ 0 := by
        intro h
        apply hpA.ne_zero
        apply RamifiedLocal.algebraMap_injective A hp (k + 1)
        simpa only [map_natCast, map_zero] using h
      exact projected_factor_divisible hp hpR
        (Nat.mul_pos (Nat.sub_pos_of_lt hp.one_lt) (pow_pos hp.pos (k + 1)))
        (RamifiedLocal.addVal_prime A hp (k + 1) hmax) c U V hUV)
    e b F K hFK
  have toCompared {U : Elem A (p ^ (e + 1))}
      (hU : ResidueOneAssociated (IsLocalRing.residue (RamifiedLocal.Extension A p 0))
        (orderPValue A p e U)
        (algebraMap A (RamifiedLocal.Extension A p 0) (augmentation (p ^ (e + 1)) U))) :
      OrderPCompared A p e U := by
    obtain ⟨u, hu, hres⟩ := hU
    refine ⟨u, hu, ?_⟩
    rw [← RamifiedLocal.maximalIdeal_eq_span A hp 0 hmax]
    apply (IsLocalRing.residue_eq_zero_iff _).mp
    rw [map_sub, hres, map_one, sub_self]
  exact ⟨toCompared hc.1, toCompared hc.2⟩

omit [IsDiscreteValuationRing A] in
/-- The augmentation denominators used in the corresponding ratio statement
are nonzero in the coefficient DVR. -/
theorem local_character_comparison_denominators [IsDiscreteValuationRing A] (p : ℕ) (hp : p.Prime)
    (e : ℕ) (b : Aˣ) (F K : Elem A (p ^ (e + 1)))
    (hFK : F * K = scalar (p ^ (e + 1)) ((p : A) ^ (e + 1) * (b : A))) :
    augmentation (p ^ (e + 1)) F ≠ 0 ∧ augmentation (p ^ (e + 1)) K ≠ 0 :=
  augmentation_ne_zero_of_scalar_product (Nat.cast_ne_zero.mpr hp.ne_zero) b F K hFK

end DVR

end CirculantHadamard.LocalComparison

end

end OAI