File: Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean

full_name: _aux_Mathlib_NumberTheory_ClassNumber_AdmissibleAbsoluteValue___unexpand_EuclideanDomain_r_1
code: local infixl:50 " ≺ " => EuclideanDomain.r
goal: Lean.PrettyPrinter.Unexpander

full_name: «_aux_Mathlib_NumberTheory_ClassNumber_AdmissibleAbsoluteValue___macroRules_term_≺__1»
code: local infixl:50 " ≺ " => EuclideanDomain.r
goal: Lean.Macro

full_name: AbsoluteValue.IsAdmissible.card
code: /-- An absolute value `R → ℤ` is admissible if it respects the Euclidean domain
structure and a large enough set of elements in `R^n` will contain a pair of
elements whose remainders are pointwise close together. -/
structure IsAdmissible extends IsEuclidean abv where
  protected card : ℝ → ℕ
  /-- For all `ε > 0` and finite families `A`, we can partition the remainders of `A` mod `b`
  into `abv.card ε` sets, such that all elements in each part of remainders are close together. -/
  exists_partition' :
    ∀ (n : ℕ) {ε : ℝ} (_ : 0 < ε) {b : R} (_ : b ≠ 0) (A : Fin n → R),
      ∃ t : Fin n → Fin (card ε), ∀ i₀ i₁, t i₀ = t i₁ → (abv (A i₁ % b - A i₀ % b) : ℝ) < abv b • ε
goal: ℝ → ℕ

full_name: AbsoluteValue.IsAdmissible.exists_partition'
code: /-- An absolute value `R → ℤ` is admissible if it respects the Euclidean domain
structure and a large enough set of elements in `R^n` will contain a pair of
elements whose remainders are pointwise close together. -/
structure IsAdmissible extends IsEuclidean abv where
  protected card : ℝ → ℕ
  /-- For all `ε > 0` and finite families `A`, we can partition the remainders of `A` mod `b`
  into `abv.card ε` sets, such that all elements in each part of remainders are close together. -/
  exists_partition' :
    ∀ (n : ℕ) {ε : ℝ} (_ : 0 < ε) {b : R} (_ : b ≠ 0) (A : Fin n → R),
      ∃ t : Fin n → Fin (card ε), ∀ i₀ i₁, t i₀ = t i₁ → (abv (A i₁ % b - A i₀ % b) : ℝ) < abv b • ε
goal: 0 < ε →
  ∀ {b : R}, b ≠ 0 → ∀ (A : Fin n → R), ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ → ↑(abv (A i₁ % b - A i₀ % b)) < abv b • ε

full_name: AbsoluteValue.IsAdmissible
code: /-- An absolute value `R → ℤ` is admissible if it respects the Euclidean domain
structure and a large enough set of elements in `R^n` will contain a pair of
elements whose remainders are pointwise close together. -/
structure IsAdmissible extends IsEuclidean abv where
  protected card : ℝ → ℕ
  /-- For all `ε > 0` and finite families `A`, we can partition the remainders of `A` mod `b`
  into `abv.card ε` sets, such that all elements in each part of remainders are close together. -/
  exists_partition' :
    ∀ (n : ℕ) {ε : ℝ} (_ : 0 < ε) {b : R} (_ : b ≠ 0) (A : Fin n → R),
      ∃ t : Fin n → Fin (card ε), ∀ i₀ i₁, t i₀ = t i₁ → (abv (A i₁ % b - A i₀ % b) : ℝ) < abv b • ε
goal: Type

full_name: AbsoluteValue.IsAdmissible.exists_partition
code: /-- For all `ε > 0` and finite families `A`, we can partition the remainders of `A` mod `b`
into `abv.card ε` sets, such that all elements in each part of remainders are close together. -/
theorem exists_partition {ι : Type*} [Finite ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0)
    (A : ι → R) (h : abv.IsAdmissible) : ∃ t : ι → Fin (h.card ε),
      ∀ i₀ i₁, t i₀ = t i₁ → (abv (A i₁ % b - A i₀ % b) : ℝ) < abv b • ε 
goal: ∃ t, ∀ (i₀ i₁ : ι), t i₀ = t i₁ → ↑(abv (A i₁ % b - A i₀ % b)) < abv b • ε

full_name: AbsoluteValue.IsAdmissible.exists_approx_aux
code: /-- Any large enough family of vectors in `R^n` has a pair of elements
whose remainders are close together, pointwise. -/
theorem exists_approx_aux (n : ℕ) (h : abv.IsAdmissible) :
    ∀ {ε : ℝ} (_hε : 0 < ε) {b : R} (_hb : b ≠ 0) (A : Fin (h.card ε ^ n).succ → Fin n → R),
      ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ k, (abv (A i₁ k % b - A i₀ k % b) : ℝ) < abv b • ε 
goal: ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : Fin n), ↑(abv (A i₁ k % b - A i₀ k % b)) < abv b • ε

full_name: AbsoluteValue.IsAdmissible.exists_approx
code: /-- Any large enough family of vectors in `R^ι` has a pair of elements
whose remainders are close together, pointwise. -/
theorem exists_approx {ι : Type*} [Fintype ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0)
    (h : abv.IsAdmissible) (A : Fin (h.card ε ^ Fintype.card ι).succ → ι → R) :
    ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ k, (abv (A i₁ k % b - A i₀ k % b) : ℝ) < abv b • ε 
goal: ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : ι), ↑(abv (A i₁ k % b - A i₀ k % b)) < abv b • ε