Definitions/Def_AutomorphicForm_ProductionPinsGeneral.lean

import Definitions.Def_AutomorphicForm_ProductionPinsCompact
import Definitions.Def_AutomorphicForm_ArithCuspRealization
import Definitions.Def_AutomorphicForm_SiegelCovering

set_option autoImplicit false

open IsDedekindDomain NumberField MeasureTheory Matrix
open NumberField.AdelicHaar NumberField.AdelicLevel NumberField.AdelicBox
open AutomorphicForm AutomorphicForm.WindowedSiegel AutomorphicForm.SiegelCovering
open NumberField.SiegelVolume

noncomputable section

namespace AutomorphicForm

variable (F : Type) [Field F] [NumberField F]

open scoped nonZeroDivisors in

noncomputable def finIdeleExponentAt (v : HeightOneSpectrum (𝓞 F))
    (δ : (FiniteAdeleRing (𝓞 F) F)ˣ) : ℤ :=
  -WithZero.log (Valued.v ((δ : FiniteAdeleRing (𝓞 F) F) v))

private theorem valued_finIdele_ne_zero (v : HeightOneSpectrum (𝓞 F))
    (δ : (FiniteAdeleRing (𝓞 F) F)ˣ) :
    Valued.v ((δ : FiniteAdeleRing (𝓞 F) F) v) ≠ 0 := by
  rw [ne_eq, map_eq_zero]
  intro h
  have : ((δ * δ⁻¹ : (FiniteAdeleRing (𝓞 F) F)ˣ) : FiniteAdeleRing (𝓞 F) F) v = 1 := by
    rw [mul_inv_cancel, Units.val_one, coe_one_apply]
  rw [Units.val_mul, coe_mul_apply, h, zero_mul] at this
  exact zero_ne_one this

@[simp] theorem finIdeleExponentAt_one (v : HeightOneSpectrum (𝓞 F)) :
    finIdeleExponentAt F v 1 = 0 := by
  unfold finIdeleExponentAt
  rw [Units.val_one, coe_one_apply, map_one, WithZero.log_one, neg_zero]

theorem finIdeleExponentAt_mul (v : HeightOneSpectrum (𝓞 F))
    (δ₁ δ₂ : (FiniteAdeleRing (𝓞 F) F)ˣ) :
    finIdeleExponentAt F v (δ₁ * δ₂) = finIdeleExponentAt F v δ₁ + finIdeleExponentAt F v δ₂ := by
  unfold finIdeleExponentAt
  rw [Units.val_mul, coe_mul_apply, map_mul,
    WithZero.log_mul (valued_finIdele_ne_zero F v δ₁) (valued_finIdele_ne_zero F v δ₂), neg_add]

private theorem finite_support_finIdeleExponentAt (δ : (FiniteAdeleRing (𝓞 F) F)ˣ) :
    {v | finIdeleExponentAt F v δ ≠ 0}.Finite := by
  have hδ : ∀ᶠ v in Filter.cofinite, (δ : FiniteAdeleRing (𝓞 F) F) v
      ∈ v.adicCompletionIntegers F := (δ : FiniteAdeleRing (𝓞 F) F).2
  have hδi : ∀ᶠ v in Filter.cofinite, ((δ⁻¹ : (FiniteAdeleRing (𝓞 F) F)ˣ) :
      FiniteAdeleRing (𝓞 F) F) v ∈ v.adicCompletionIntegers F :=
    ((δ⁻¹ : (FiniteAdeleRing (𝓞 F) F)ˣ) : FiniteAdeleRing (𝓞 F) F).2
  refine Set.Finite.subset (Filter.eventually_cofinite.mp (hδ.and hδi)) ?_
  intro v hv hgood
  apply hv

  have hprod : Valued.v ((δ : FiniteAdeleRing (𝓞 F) F) v) *
      Valued.v (((δ⁻¹ : (FiniteAdeleRing (𝓞 F) F)ˣ) : FiniteAdeleRing (𝓞 F) F) v) = 1 := by
    rw [← map_mul, ← coe_mul_apply, ← Units.val_mul, mul_inv_cancel,
      Units.val_one, coe_one_apply, map_one]
  have heq : Valued.v ((δ : FiniteAdeleRing (𝓞 F) F) v) = 1 :=
    le_antisymm hgood.1 (le_of_eq_of_le hprod.symm (mul_le_of_le_one_right' hgood.2))
  simp [finIdeleExponentAt, heq]

open scoped nonZeroDivisors in

noncomputable def finAssocFracIdeal (δ : (FiniteAdeleRing (𝓞 F) F)ˣ) :
    FractionalIdeal (𝓞 F)⁰ F :=
  ∏ᶠ v : HeightOneSpectrum (𝓞 F), (v.asIdeal : FractionalIdeal (𝓞 F)⁰ F) ^ finIdeleExponentAt F v δ

open scoped nonZeroDivisors in
theorem finAssocFracIdeal_ne_zero (δ : (FiniteAdeleRing (𝓞 F) F)ˣ) :
    finAssocFracIdeal F δ ≠ 0 := by
  classical
  unfold finAssocFracIdeal
  rw [finprod_eq_prod_of_mulSupport_subset _
    (s := (finite_support_finIdeleExponentAt F δ).toFinset)
    (fun v hv => by
      simp only [Set.Finite.coe_toFinset, Set.mem_setOf_eq]
      intro h
      exact hv (by simp only [h, zpow_zero]))]
  exact Finset.prod_ne_zero_iff.mpr fun v _ =>
    zpow_ne_zero _ (FractionalIdeal.coeIdeal_ne_zero.mpr v.ne_bot)

open scoped nonZeroDivisors in
theorem count_finAssocFracIdeal (v : HeightOneSpectrum (𝓞 F))
    (δ : (FiniteAdeleRing (𝓞 F) F)ˣ) :
    FractionalIdeal.count F v (finAssocFracIdeal F δ) = finIdeleExponentAt F v δ :=
  FractionalIdeal.count_finprod F v _ (Filter.eventually_cofinite.mpr
    (by simpa using finite_support_finIdeleExponentAt F δ))

open scoped nonZeroDivisors in
theorem finAssocFracIdeal_mul (δ₁ δ₂ : (FiniteAdeleRing (𝓞 F) F)ˣ) :
    finAssocFracIdeal F (δ₁ * δ₂) = finAssocFracIdeal F δ₁ * finAssocFracIdeal F δ₂ := by
  have h0 : finAssocFracIdeal F (δ₁ * δ₂) ≠ 0 := finAssocFracIdeal_ne_zero F _
  have h12 : finAssocFracIdeal F δ₁ * finAssocFracIdeal F δ₂ ≠ 0 :=
    mul_ne_zero (finAssocFracIdeal_ne_zero F _) (finAssocFracIdeal_ne_zero F _)
  rw [← FractionalIdeal.finprod_heightOneSpectrum_factorization' (R := 𝓞 F) (K := F) h0,
    ← FractionalIdeal.finprod_heightOneSpectrum_factorization' (R := 𝓞 F) (K := F) h12]
  refine finprod_congr fun v => ?_
  rw [count_finAssocFracIdeal, finIdeleExponentAt_mul,
    FractionalIdeal.count_mul F v
      (finAssocFracIdeal_ne_zero F δ₁) (finAssocFracIdeal_ne_zero F δ₂),
    count_finAssocFracIdeal, count_finAssocFracIdeal]

open scoped nonZeroDivisors Classical in
theorem finIdeleExponentAt_localUnit_uniformizer (v w : HeightOneSpectrum (𝓞 F)) :
    finIdeleExponentAt F w (localUnit (𝓞 F) F v (uniformizerUnit F v)) =
      if w = v then 1 else 0 := by
  unfold finIdeleExponentAt
  by_cases hw : w = v
  · subst hw
    rw [if_pos rfl, localUnit_apply_self, valued_uniformizerUnit, WithZero.log_exp]
    ring
  · rw [if_neg hw, localUnit_apply_of_ne (𝓞 F) F v _ hw, map_one, WithZero.log_one, neg_zero]

open scoped nonZeroDivisors in

noncomputable def contentHomFin : (FiniteAdeleRing (𝓞 F) F)ˣ →* ClassGroup (𝓞 F) where
  toFun δ := ClassGroup.mk F (Units.mk0 (finAssocFracIdeal F δ) (finAssocFracIdeal_ne_zero F δ))
  map_one' := by
    rw [← map_one (ClassGroup.mk (R := 𝓞 F) (K := F))]
    congr 1
    refine Units.ext ?_
    simp only [Units.val_mk0, Units.val_one, finAssocFracIdeal]
    exact finprod_eq_one_of_forall_eq_one fun v => by rw [finIdeleExponentAt_one, zpow_zero]
  map_mul' δ₁ δ₂ := by
    rw [← map_mul]
    congr 1
    exact Units.ext (by simpa using finAssocFracIdeal_mul F δ₁ δ₂)

open scoped nonZeroDivisors in
theorem contentHomFin_apply (δ : (FiniteAdeleRing (𝓞 F) F)ˣ) :
    contentHomFin F δ =
      ClassGroup.mk F (Units.mk0 (finAssocFracIdeal F δ) (finAssocFracIdeal_ne_zero F δ)) := rfl

open scoped nonZeroDivisors Classical in

theorem contentHomFin_surjective : Function.Surjective (contentHomFin F) := by
  classical
  intro c
  obtain ⟨I, hI⟩ := ClassGroup.mk0_surjective c
  have hI0 : ((I : Ideal (𝓞 F)) : FractionalIdeal (𝓞 F)⁰ F) ≠ 0 :=
    FractionalIdeal.coeIdeal_ne_zero.mpr (nonZeroDivisors.coe_ne_zero I)
  have hfin : {v : HeightOneSpectrum (𝓞 F) |
      FractionalIdeal.count F v ((I : Ideal (𝓞 F)) : FractionalIdeal (𝓞 F)⁰ F) ≠ 0}.Finite :=
    Filter.eventually_cofinite.mp (FractionalIdeal.finite_factors _)
  set s : Finset (HeightOneSpectrum (𝓞 F)) := hfin.toFinset with hs
  refine ⟨∏ v ∈ s, (localUnit (𝓞 F) F v (uniformizerUnit F v)) ^
      (FractionalIdeal.count F v ((I : Ideal (𝓞 F)) : FractionalIdeal (𝓞 F)⁰ F)).toNat, ?_⟩
  show ClassGroup.mk F _ = c
  rw [← hI, ← ClassGroup.mk_mk0 F]
  refine congrArg (ClassGroup.mk F) (Units.ext ?_)
  show finAssocFracIdeal F _ = ((FractionalIdeal.mk0 F I : (FractionalIdeal (𝓞 F)⁰ F)ˣ) :
    FractionalIdeal (𝓞 F)⁰ F)
  rw [FractionalIdeal.coe_mk0]
  conv_rhs =>
    rw [← FractionalIdeal.finprod_heightOneSpectrum_factorization' (R := 𝓞 F) (K := F) hI0]
  rw [← FractionalIdeal.finprod_heightOneSpectrum_factorization' (R := 𝓞 F) (K := F)
      (finAssocFracIdeal_ne_zero F _)]
  refine finprod_congr fun w => congrArg _ ?_
  rw [count_finAssocFracIdeal]

  have hprod : finIdeleExponentAt F w (∏ v ∈ s, (localUnit (𝓞 F) F v (uniformizerUnit F v)) ^
        (FractionalIdeal.count F v ((I : Ideal (𝓞 F)) : FractionalIdeal (𝓞 F)⁰ F)).toNat)
      = ∑ v ∈ s, (FractionalIdeal.count F v
          ((I : Ideal (𝓞 F)) : FractionalIdeal (𝓞 F)⁰ F)).toNat *
            finIdeleExponentAt F w (localUnit (𝓞 F) F v (uniformizerUnit F v)) := by
    induction s using Finset.induction_on with
    | empty => simp [finIdeleExponentAt_one]
    | insert v t hvt ih =>
        rw [Finset.prod_insert hvt, finIdeleExponentAt_mul, ih, Finset.sum_insert hvt]
        congr 1
        induction (FractionalIdeal.count F v _).toNat with
        | zero => simp [finIdeleExponentAt_one]
        | succ n ihn => rw [pow_succ, finIdeleExponentAt_mul, ihn]; push_cast; ring
  rw [hprod]
  simp_rw [finIdeleExponentAt_localUnit_uniformizer, mul_ite, mul_one, mul_zero]
  rw [Finset.sum_ite_eq s w]
  by_cases hws : w ∈ s
  · rw [if_pos hws, Int.toNat_of_nonneg (FractionalIdeal.count_coe_nonneg F w _)]
  · rw [if_neg hws]
    simp only [hs, Set.Finite.mem_toFinset, Set.mem_setOf_eq, not_not] at hws
    exact hws.symm

def classSq : ClassGroup (𝓞 F) →* ClassGroup (𝓞 F) := powMonoidHom 2

omit [NumberField F] in
@[simp] theorem classSq_apply (C : ClassGroup (𝓞 F)) : classSq F C = C ^ 2 := rfl

open scoped Classical in

noncomputable def classRepFinIdele (C : ClassGroup (𝓞 F) ⧸ (classSq F).range) :
    (FiniteAdeleRing (𝓞 F) F)ˣ :=
  if C = 1 then 1
  else Function.surjInv
    ((QuotientGroup.mk'_surjective (classSq F).range).comp (contentHomFin_surjective F)) C

theorem classRepFinIdele_spec (C : ClassGroup (𝓞 F) ⧸ (classSq F).range) :
    QuotientGroup.mk' (classSq F).range (contentHomFin F (classRepFinIdele F C)) = C := by
  classical
  unfold classRepFinIdele
  split_ifs with hC
  · simp [hC]
  · exact Function.surjInv_eq
      ((QuotientGroup.mk'_surjective (classSq F).range).comp (contentHomFin_surjective F)) C

@[simp] theorem classRepFinIdele_one : classRepFinIdele F 1 = 1 := by
  classical
  unfold classRepFinIdele
  rw [if_pos rfl]

noncomputable def finIdeleDiag : (FiniteAdeleRing (𝓞 F) F)ˣ →* AdelicGL2 (𝓞 F) F :=
  diagOne.comp (Units.map (finIncl (𝓞 F) F))

theorem glArch_finIdeleDiag (δ : (FiniteAdeleRing (𝓞 F) F)ˣ) :
    glArch (𝓞 F) F (finIdeleDiag F δ) = 1 := by
  refine Matrix.GeneralLinearGroup.ext fun i j => ?_

  fin_cases i <;> fin_cases j <;> rfl

theorem injective_finIdeleDiag : Function.Injective (finIdeleDiag F) := by
  intro δ₁ δ₂ h
  have h00 := congrArg (fun g : AdelicGL2 (𝓞 F) F =>
    ((g : Matrix (Fin 2) (Fin 2) (AdeleRing (𝓞 F) F)) 0 0).2) h
  simp only [finIdeleDiag, MonoidHom.comp_apply, diagOne_coe_apply, Matrix.diagonal_apply_eq,
    Matrix.cons_val_zero, Units.coe_map, finIncl_apply_snd] at h00
  exact Units.ext h00

noncomputable def classRepEmbedding :
    (ClassGroup (𝓞 F) ⧸ (classSq F).range) ↪ AdelicGL2 (𝓞 F) F :=
  ⟨fun C => finIdeleDiag F (classRepFinIdele F C),
    (injective_finIdeleDiag F).comp (Function.LeftInverse.injective
      (g := fun δ => QuotientGroup.mk' (classSq F).range (contentHomFin F δ))
      (classRepFinIdele_spec F))⟩

theorem classRepEmbedding_apply (C : ClassGroup (𝓞 F) ⧸ (classSq F).range) :
    classRepEmbedding F C = finIdeleDiag F (classRepFinIdele F C) := rfl

theorem glArch_classRepEmbedding (C : ClassGroup (𝓞 F) ⧸ (classSq F).range) :
    glArch (𝓞 F) F (classRepEmbedding F C) = 1 :=
  glArch_finIdeleDiag F _

@[simp] theorem classRepEmbedding_one :
    classRepEmbedding F 1 = 1 := by
  rw [classRepEmbedding_apply, classRepFinIdele_one, map_one]

noncomputable def classRepTranslates : Finset (AdelicGL2 (𝓞 F) F) :=
  haveI := Fintype.ofFinite (ClassGroup (𝓞 F) ⧸ (classSq F).range)
  Finset.univ.map (classRepEmbedding F)

theorem card_classRepTranslates :
    (classRepTranslates F).card = Nat.card (ClassGroup (𝓞 F) ⧸ (classSq F).range) := by
  letI := Fintype.ofFinite (ClassGroup (𝓞 F) ⧸ (classSq F).range)
  rw [classRepTranslates, Finset.card_map, Finset.card_univ, Nat.card_eq_fintype_card]

theorem one_mem_classRepTranslates : (1 : AdelicGL2 (𝓞 F) F) ∈ classRepTranslates F := by
  letI := Fintype.ofFinite (ClassGroup (𝓞 F) ⧸ (classSq F).range)
  rw [classRepTranslates, Finset.mem_map]
  exact ⟨1, Finset.mem_univ _, classRepEmbedding_one F⟩

noncomputable def classRepSiegelSet (c u d₁ d₂ : ℝ) : Set (AdelicGL2 (𝓞 F) F) :=
  ⋃ x ∈ classRepTranslates F, (· * x) '' centreCutSiegelSet F c u d₁ d₂

theorem surjective_classSq_of_odd (h : Odd (classNumber F)) :
    Function.Surjective (classSq F) := by
  intro C
  obtain ⟨k, hk⟩ := h
  refine ⟨C ^ (k + 1), ?_⟩
  have hcard : C ^ classNumber F = 1 := pow_card_eq_one
  simp only [classSq, powMonoidHom_apply, ← pow_mul]
  have h2 : (k + 1) * 2 = classNumber F + 1 := by omega
  rw [h2, pow_succ, hcard, one_mul]

theorem subsingleton_quotient_classSq_of_odd (h : Odd (classNumber F)) :
    Subsingleton (ClassGroup (𝓞 F) ⧸ (classSq F).range) := by
  have hr : (classSq F).range = ⊤ := MonoidHom.range_eq_top.mpr (surjective_classSq_of_odd F h)
  rw [hr]
  exact QuotientGroup.subsingleton_quotient_top

theorem classRepTranslates_eq_singleton_one_of_odd (h : Odd (classNumber F)) :
    classRepTranslates F = {1} := by
  letI := Fintype.ofFinite (ClassGroup (𝓞 F) ⧸ (classSq F).range)
  haveI := subsingleton_quotient_classSq_of_odd F h
  rw [classRepTranslates]
  have huniv : (Finset.univ : Finset (ClassGroup (𝓞 F) ⧸ (classSq F).range)) = {1} := by
    rw [Finset.eq_singleton_iff_unique_mem]
    exact ⟨Finset.mem_univ _, fun x _ => Subsingleton.elim x 1⟩
  rw [huniv, Finset.map_singleton, classRepEmbedding_one]

theorem classRepSiegelSet_eq_of_odd (h : Odd (classNumber F)) (c u d₁ d₂ : ℝ) :
    classRepSiegelSet F c u d₁ d₂ = centreCutSiegelSet F c u d₁ d₂ := by
  unfold classRepSiegelSet
  rw [classRepTranslates_eq_singleton_one_of_odd F h]
  ext g
  simp only [Finset.mem_singleton, Set.mem_iUnion, exists_prop, exists_eq_left, Set.mem_image,
    mul_one]
  exact ⟨fun ⟨x, hx, hxg⟩ => hxg ▸ hx, fun hg => ⟨g, hg, rfl⟩⟩

def productionPinsGeneralOf (c u d₁ d₂ : ℝ) (_hd : d₁ < d₂) : CarrierPins F :=
  productionPinsOf F
    (classRepSiegelSet F c u d₁ d₂)
    (fun N => levelOne (𝓞 F) F N ⊓ finiteAdelicGL2Subgroup F)
    (fun v => heckeGen (𝓞 F) F v)
    (adelicBox F)

def productionPinsGeneral : CarrierPins F :=
  productionPinsGeneralOf F (1/2 : ℝ) 1 (1/2) 2 (by norm_num)

@[simp] theorem productionPinsGeneralOf_D (c u d₁ d₂ : ℝ) (hd : d₁ < d₂) :
    (productionPinsGeneralOf F c u d₁ d₂ hd).D = classRepSiegelSet F c u d₁ d₂ := rfl

@[simp] theorem productionPinsGeneralOf_μ (c u d₁ d₂ : ℝ) (hd : d₁ < d₂) :
    (productionPinsGeneralOf F c u d₁ d₂ hd).μ = adelicGLHaar (Fin 2) (𝓞 F) F := rfl

@[simp] theorem productionPinsGeneralOf_U (c u d₁ d₂ : ℝ) (hd : d₁ < d₂) (N : Ideal (𝓞 F)) :
    (productionPinsGeneralOf F c u d₁ d₂ hd).U N =
      levelOne (𝓞 F) F N ⊓ finiteAdelicGL2Subgroup F := rfl

@[simp] theorem productionPinsGeneral_D :
    (productionPinsGeneral F).D = classRepSiegelSet F (1/2 : ℝ) 1 (1/2) 2 := rfl

theorem productionPinsGeneral_eq_compact_of_odd (h : Odd (classNumber F)) :
    productionPinsGeneral F = productionPinsCompact F := by
  unfold productionPinsGeneral productionPinsGeneralOf productionPinsCompact
  rw [classRepSiegelSet_eq_of_odd F h]

theorem centreCutSiegelSet_subset_classRepSiegelSet (c u d₁ d₂ : ℝ) :
    centreCutSiegelSet F c u d₁ d₂ ⊆ classRepSiegelSet F c u d₁ d₂ := by
  intro g hg
  rw [classRepSiegelSet, Set.mem_iUnion]
  exact ⟨1, Set.mem_iUnion.mpr ⟨one_mem_classRepTranslates F, g, hg, mul_one g⟩⟩

theorem adelicGLHaar_mul_right_centreCutSiegelSet_lt_top
    {c : ℝ} (hc : 0 < c) (u : ℝ) {d₁ : ℝ} (hd₁ : 0 < d₁) (d₂ : ℝ) (x : AdelicGL2 (𝓞 F) F) :
    (letI := glBorel (Fin 2) (𝓞 F) F;
      adelicGLHaar (Fin 2) (𝓞 F) F ((· * x) '' centreCutSiegelSet F c u d₁ d₂) < ⊤) := by
  letI := glBorel (Fin 2) (𝓞 F) F
  haveI := borelSpace_glBorel (Fin 2) (𝓞 F) F
  haveI := isHaarMeasure_adelicGLHaar (Fin 2) (𝓞 F) F

  have himg : (· * x) '' centreCutSiegelSet F c u d₁ d₂
      = (· * x⁻¹) ⁻¹' centreCutSiegelSet F c u d₁ d₂ := by
    ext g; simp [Set.mem_preimage]
  rw [himg, ← Measure.map_apply (measurable_mul_const x⁻¹)
    (measurableSet_centreCutSiegelSet c u d₁ d₂)]

  haveI : (Measure.map (· * x⁻¹) (adelicGLHaar (Fin 2) (𝓞 F) F)).IsMulLeftInvariant :=
    isMulLeftInvariant_map_mul_right _
  haveI : IsFiniteMeasureOnCompacts (Measure.map (· * x⁻¹) (adelicGLHaar (Fin 2) (𝓞 F) F)) :=
    Measure.IsFiniteMeasureOnCompacts.map _ (Homeomorph.mulRight x⁻¹)
  exact measure_centreCutSiegelSet_lt_top _ hc u hd₁ d₂

theorem productionPinsGeneral_μ_D_pos_lt_top :
    (letI := (productionPinsGeneral F).mS;
      0 < (productionPinsGeneral F).μ (productionPinsGeneral F).D ∧
        (productionPinsGeneral F).μ (productionPinsGeneral F).D < ⊤) := by
  letI := glBorel (Fin 2) (𝓞 F) F
  have hcompact := productionPinsCompact_μ_D_pos_lt_top F
  constructor
  · refine lt_of_lt_of_le hcompact.1 (measure_mono ?_)
    exact centreCutSiegelSet_subset_classRepSiegelSet F (1/2) 1 (1/2) 2
  · calc (productionPinsGeneral F).μ (productionPinsGeneral F).D
        = adelicGLHaar (Fin 2) (𝓞 F) F (classRepSiegelSet F (1/2) 1 (1/2) 2) := rfl
      _ ≤ ∑ x ∈ classRepTranslates F,
            adelicGLHaar (Fin 2) (𝓞 F) F ((· * x) '' centreCutSiegelSet F (1/2) 1 (1/2) 2) := by
          unfold classRepSiegelSet
          exact measure_biUnion_finset_le _ _
      _ < ⊤ := ENNReal.sum_lt_top.mpr fun x _ =>
          adelicGLHaar_mul_right_centreCutSiegelSet_lt_top F (by norm_num) 1 (by norm_num) 2 x

theorem not_ae_zero_restrict_of_continuous_of_mem_interior
    {G : Type*} [TopologicalSpace G] [MeasurableSpace G] [OpensMeasurableSpace G]
    {μ : Measure G} [μ.IsOpenPosMeasure]
    {φ : G → ℂ} (hφ : Continuous φ) {g₀ : G} (hg₀ : φ g₀ ≠ 0)
    {C : Set G} (hC : g₀ ∈ interior C) :
    ¬ (φ =ᵐ[μ.restrict C] 0) := by
  intro hae
  set U : Set G := interior C ∩ {g | φ g ≠ 0} with hU
  have hUopen : IsOpen U := isOpen_interior.inter (isOpen_ne.preimage hφ)
  have hUne : U.Nonempty := ⟨g₀, hC, hg₀⟩
  have hUpos : 0 < μ U := hUopen.measure_pos μ hUne
  have hUsub : U ⊆ C := (Set.inter_subset_left).trans interior_subset
  have hres : 0 < (μ.restrict C) U := by
    rw [Measure.restrict_apply hUopen.measurableSet, Set.inter_eq_self_of_subset_left hUsub]
    exact hUpos
  have hzero : U ⊆ {g | ¬ (φ g = (0 : G → ℂ) g)} := fun g hg => by
    simpa using hg.2
  exact absurd (measure_mono_null hzero hae) hres.ne'

def IsGenuineCuspRealizationAt (pins : CarrierPins F) (Φ : HeckeEigensystem F ℂ)
    (R : SmoothCuspRealizationAt F pins Φ) : Prop :=
  Continuous R.toFun

def IsGenuineCuspRealizable (pins : CarrierPins F) (Φ : HeckeEigensystem F ℂ) : Prop :=
  ∃ R : SmoothCuspRealizationAt F pins Φ, IsGenuineCuspRealizationAt F pins Φ R

def IsArithGenuineCuspRealizable (pins : CarrierPins F) (Φ : HeckeEigensystem F ℂ) : Prop :=
  IsGenuineCuspRealizable F pins Φ.toRawCentral

def IsArithGenuineCuspRealizableVia (pins : CarrierPins F) {R : Type*} [CommRing R]
    (ι : R →+* ℂ) (Φ : HeckeEigensystem F R) : Prop :=
  IsArithGenuineCuspRealizable F pins (Φ.map ι)

def genuineCuspNotionOf
    (pins : ∀ (F : Type) [Field F] [NumberField F], CarrierPins F) :
    CuspidalityNotion ℂ where
  IsCusp := fun F _i1 _i2 Φ => @IsArithGenuineCuspRealizable F _i1 _i2 (pins F) Φ

theorem isGenuineCuspRealizable_iff (pins : CarrierPins F) (Φ : HeckeEigensystem F ℂ) :
    IsGenuineCuspRealizable F pins Φ ↔
      ∃ R : SmoothCuspRealizationAt F pins Φ, Continuous R.toFun := Iff.rfl

theorem IsGenuineCuspRealizable.isSmoothCuspRealizable {pins : CarrierPins F}
    {Φ : HeckeEigensystem F ℂ} (h : IsGenuineCuspRealizable F pins Φ) :
    IsSmoothCuspRealizable F pins Φ :=
  ⟨h.choose⟩

theorem IsArithGenuineCuspRealizable.isArithCuspRealizable {pins : CarrierPins F}
    {Φ : HeckeEigensystem F ℂ} (h : IsArithGenuineCuspRealizable F pins Φ) :
    IsArithCuspRealizable F pins Φ :=
  h.isSmoothCuspRealizable

end AutomorphicForm

end

section Battery
open AutomorphicForm
#check @finIdeleExponentAt
#check @finAssocFracIdeal
#check @contentHomFin
#check @contentHomFin_apply
#check @contentHomFin_surjective
#check @classSq
#check @classRepFinIdele
#check @classRepFinIdele_spec
#check @finIdeleDiag
#check @classRepEmbedding
#check @classRepEmbedding_apply
#check @classRepSiegelSet
#check @card_classRepTranslates
#check @productionPinsGeneralOf
#check @productionPinsGeneral
#check @IsGenuineCuspRealizationAt
#check @IsGenuineCuspRealizable
#check @IsArithGenuineCuspRealizable
#check @IsArithGenuineCuspRealizableVia
#check @genuineCuspNotionOf
#print axioms AutomorphicForm.finAssocFracIdeal_ne_zero
#print axioms AutomorphicForm.count_finAssocFracIdeal
#print axioms AutomorphicForm.finAssocFracIdeal_mul
#print axioms AutomorphicForm.finIdeleExponentAt_localUnit_uniformizer
#print axioms AutomorphicForm.contentHomFin_surjective
#print axioms AutomorphicForm.classRepFinIdele_spec
#print axioms AutomorphicForm.glArch_finIdeleDiag
#print axioms AutomorphicForm.injective_finIdeleDiag
#print axioms AutomorphicForm.glArch_classRepEmbedding
#print axioms AutomorphicForm.classRepEmbedding_one
#print axioms AutomorphicForm.productionPinsGeneral_eq_compact_of_odd
#print axioms AutomorphicForm.IsGenuineCuspRealizable.isSmoothCuspRealizable
#print axioms AutomorphicForm.classRepSiegelSet_eq_of_odd
#print axioms AutomorphicForm.card_classRepTranslates
#print axioms AutomorphicForm.one_mem_classRepTranslates
#print axioms AutomorphicForm.classRepTranslates_eq_singleton_one_of_odd
#print axioms AutomorphicForm.not_ae_zero_restrict_of_continuous_of_mem_interior
#print axioms AutomorphicForm.productionPinsGeneral_μ_D_pos_lt_top
end Battery