l3lab/ntp-mathlib huggingface dataset:
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      (1 : ℂ) / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) := by
  let f := fun x : ℝ => cexp (-π * a * (x : ℂ) ^ 2)
  have h1 : 0 < (↑π * a).re := by
    rw [ofReal_mul_re]
    exact mul_pos pi_pos ha
  have h2 : 0 < (↑π / a).re := by
    rw [div_eq_mul_inv, ofReal_mul_re, inv_re]
    refine' mul_pos pi_pos (div_pos ha <| normSq_pos.mpr _)
    contrapose! ha
    

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℂ
f : ℝ → ℂ := fun x => cexp (-↑π * a * ↑x ^ 2)
h1 : 0 < (↑π * a).re
ha : a = 0
⊢ a.re ≤ 0

Next tactic:
rw [ha, zero_re]
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      (1 : ℂ) / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) := by
  let f := fun x : ℝ => cexp (-π * a * (x : ℂ) ^ 2)
  have h1 : 0 < (↑π * a).re := by
    rw [ofReal_mul_re]
    exact mul_pos pi_pos ha
  have h2 : 0 < (↑π / a).re := by
    rw [div_eq_mul_inv, ofReal_mul_re, inv_re]
    refine' mul_pos pi_pos (div_pos ha <| normSq_pos.mpr _)
    contrapose! ha
    rw [ha, zero_re]
  

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℂ
ha : 0 < a.re
f : ℝ → ℂ := fun x => cexp (-↑π * a * ↑x ^ 2)
h1 : 0 < (↑π * a).re
h2 : 0 < (↑π / a).re
⊢ ∑' (n : ℤ), cexp (-↑π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), cexp (-↑π / a * ↑n ^ 2)

Next tactic:
have f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    convert (isLittleO_exp_neg_mul_sq_cocompact h1 (-2)).isBigO using 2
    dsimp only
    congr 1
    ring
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      (1 : ℂ) / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) := by
  let f := fun x : ℝ => cexp (-π * a * (x : ℂ) ^ 2)
  have h1 : 0 < (↑π * a).re := by
    rw [ofReal_mul_re]
    exact mul_pos pi_pos ha
  have h2 : 0 < (↑π / a).re := by
    rw [div_eq_mul_inv, ofReal_mul_re, inv_re]
    refine' mul_pos pi_pos (div_pos ha <| normSq_pos.mpr _)
    contrapose! ha
    rw [ha, zero_re]
  have f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℂ
ha : 0 < a.re
f : ℝ → ℂ := fun x => cexp (-↑π * a * ↑x ^ 2)
h1 : 0 < (↑π * a).re
h2 : 0 < (↑π / a).re
⊢ f =O[cocompact ℝ] fun x => |x| ^ (-2)

Next tactic:
convert (isLittleO_exp_neg_mul_sq_cocompact h1 (-2)).isBigO using 2
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      (1 : ℂ) / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) := by
  let f := fun x : ℝ => cexp (-π * a * (x : ℂ) ^ 2)
  have h1 : 0 < (↑π * a).re := by
    rw [ofReal_mul_re]
    exact mul_pos pi_pos ha
  have h2 : 0 < (↑π / a).re := by
    rw [div_eq_mul_inv, ofReal_mul_re, inv_re]
    refine' mul_pos pi_pos (div_pos ha <| normSq_pos.mpr _)
    contrapose! ha
    rw [ha, zero_re]
  have f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    convert (isLittleO_exp_neg_mul_sq_cocompact h1 (-2)).isBigO using 2
    

Proof state:
case h.e'_7.h
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℂ
ha : 0 < a.re
f : ℝ → ℂ := fun x => cexp (-↑π * a * ↑x ^ 2)
h1 : 0 < (↑π * a).re
h2 : 0 < (↑π / a).re
x✝ : ℝ
⊢ f x✝ = cexp (-(↑π * a) * ↑x✝ ^ 2)

Next tactic:
dsimp only
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      (1 : ℂ) / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) := by
  let f := fun x : ℝ => cexp (-π * a * (x : ℂ) ^ 2)
  have h1 : 0 < (↑π * a).re := by
    rw [ofReal_mul_re]
    exact mul_pos pi_pos ha
  have h2 : 0 < (↑π / a).re := by
    rw [div_eq_mul_inv, ofReal_mul_re, inv_re]
    refine' mul_pos pi_pos (div_pos ha <| normSq_pos.mpr _)
    contrapose! ha
    rw [ha, zero_re]
  have f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    convert (isLittleO_exp_neg_mul_sq_cocompact h1 (-2)).isBigO using 2
    dsimp only
    

Proof state:
case h.e'_7.h
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℂ
ha : 0 < a.re
f : ℝ → ℂ := fun x => cexp (-↑π * a * ↑x ^ 2)
h1 : 0 < (↑π * a).re
h2 : 0 < (↑π / a).re
x✝ : ℝ
⊢ cexp (-↑π * a * ↑x✝ ^ 2) = cexp (-(↑π * a) * ↑x✝ ^ 2)

Next tactic:
congr 1
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      (1 : ℂ) / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) := by
  let f := fun x : ℝ => cexp (-π * a * (x : ℂ) ^ 2)
  have h1 : 0 < (↑π * a).re := by
    rw [ofReal_mul_re]
    exact mul_pos pi_pos ha
  have h2 : 0 < (↑π / a).re := by
    rw [div_eq_mul_inv, ofReal_mul_re, inv_re]
    refine' mul_pos pi_pos (div_pos ha <| normSq_pos.mpr _)
    contrapose! ha
    rw [ha, zero_re]
  have f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    convert (isLittleO_exp_neg_mul_sq_cocompact h1 (-2)).isBigO using 2
    dsimp only
    congr 1
    

Proof state:
case h.e'_7.h.e_z
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℂ
ha : 0 < a.re
f : ℝ → ℂ := fun x => cexp (-↑π * a * ↑x ^ 2)
h1 : 0 < (↑π * a).re
h2 : 0 < (↑π / a).re
x✝ : ℝ
⊢ -↑π * a * ↑x✝ ^ 2 = -(↑π * a) * ↑x✝ ^ 2

Next tactic:
ring
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      (1 : ℂ) / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) := by
  let f := fun x : ℝ => cexp (-π * a * (x : ℂ) ^ 2)
  have h1 : 0 < (↑π * a).re := by
    rw [ofReal_mul_re]
    exact mul_pos pi_pos ha
  have h2 : 0 < (↑π / a).re := by
    rw [div_eq_mul_inv, ofReal_mul_re, inv_re]
    refine' mul_pos pi_pos (div_pos ha <| normSq_pos.mpr _)
    contrapose! ha
    rw [ha, zero_re]
  have f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    convert (isLittleO_exp_neg_mul_sq_cocompact h1 (-2)).isBigO using 2
    dsimp only
    congr 1
    ring
  

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℂ
ha : 0 < a.re
f : ℝ → ℂ := fun x => cexp (-↑π * a * ↑x ^ 2)
h1 : 0 < (↑π * a).re
h2 : 0 < (↑π / a).re
f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2)
⊢ ∑' (n : ℤ), cexp (-↑π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), cexp (-↑π / a * ↑n ^ 2)

Next tactic:
have Ff_bd : 𝓕 f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    rw [fourier_transform_gaussian_pi ha]
    convert (isLittleO_exp_neg_mul_sq_cocompact h2 (-2)).isBigO.const_mul_left
      ((1 : ℂ) / a ^ (1 / 2 : ℂ)) using 2
    congr 1
    ring_nf
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      (1 : ℂ) / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) := by
  let f := fun x : ℝ => cexp (-π * a * (x : ℂ) ^ 2)
  have h1 : 0 < (↑π * a).re := by
    rw [ofReal_mul_re]
    exact mul_pos pi_pos ha
  have h2 : 0 < (↑π / a).re := by
    rw [div_eq_mul_inv, ofReal_mul_re, inv_re]
    refine' mul_pos pi_pos (div_pos ha <| normSq_pos.mpr _)
    contrapose! ha
    rw [ha, zero_re]
  have f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    convert (isLittleO_exp_neg_mul_sq_cocompact h1 (-2)).isBigO using 2
    dsimp only
    congr 1
    ring
  have Ff_bd : 𝓕 f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℂ
ha : 0 < a.re
f : ℝ → ℂ := fun x => cexp (-↑π * a * ↑x ^ 2)
h1 : 0 < (↑π * a).re
h2 : 0 < (↑π / a).re
f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2)
⊢ 𝓕 f =O[cocompact ℝ] fun x => |x| ^ (-2)

Next tactic:
rw [fourier_transform_gaussian_pi ha]
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      (1 : ℂ) / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) := by
  let f := fun x : ℝ => cexp (-π * a * (x : ℂ) ^ 2)
  have h1 : 0 < (↑π * a).re := by
    rw [ofReal_mul_re]
    exact mul_pos pi_pos ha
  have h2 : 0 < (↑π / a).re := by
    rw [div_eq_mul_inv, ofReal_mul_re, inv_re]
    refine' mul_pos pi_pos (div_pos ha <| normSq_pos.mpr _)
    contrapose! ha
    rw [ha, zero_re]
  have f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    convert (isLittleO_exp_neg_mul_sq_cocompact h1 (-2)).isBigO using 2
    dsimp only
    congr 1
    ring
  have Ff_bd : 𝓕 f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    rw [fourier_transform_gaussian_pi ha]
    

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℂ
ha : 0 < a.re
f : ℝ → ℂ := fun x => cexp (-↑π * a * ↑x ^ 2)
h1 : 0 < (↑π * a).re
h2 : 0 < (↑π / a).re
f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2)
⊢ (fun t => 1 / a ^ (1 / 2) * cexp (-↑π / a * ↑t ^ 2)) =O[cocompact ℝ] fun x => |x| ^ (-2)

Next tactic:
convert (isLittleO_exp_neg_mul_sq_cocompact h2 (-2)).isBigO.const_mul_left
      ((1 : ℂ) / a ^ (1 / 2 : ℂ)) using 2
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      (1 : ℂ) / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) := by
  let f := fun x : ℝ => cexp (-π * a * (x : ℂ) ^ 2)
  have h1 : 0 < (↑π * a).re := by
    rw [ofReal_mul_re]
    exact mul_pos pi_pos ha
  have h2 : 0 < (↑π / a).re := by
    rw [div_eq_mul_inv, ofReal_mul_re, inv_re]
    refine' mul_pos pi_pos (div_pos ha <| normSq_pos.mpr _)
    contrapose! ha
    rw [ha, zero_re]
  have f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    convert (isLittleO_exp_neg_mul_sq_cocompact h1 (-2)).isBigO using 2
    dsimp only
    congr 1
    ring
  have Ff_bd : 𝓕 f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    rw [fourier_transform_gaussian_pi ha]
    convert (isLittleO_exp_neg_mul_sq_cocompact h2 (-2)).isBigO.const_mul_left
      ((1 : ℂ) / a ^ (1 / 2 : ℂ)) using 2
    

Proof state:
case h.e'_7.h
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℂ
ha : 0 < a.re
f : ℝ → ℂ := fun x => cexp (-↑π * a * ↑x ^ 2)
h1 : 0 < (↑π * a).re
h2 : 0 < (↑π / a).re
f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2)
x✝ : ℝ
⊢ 1 / a ^ (1 / 2) * cexp (-↑π / a * ↑x✝ ^ 2) = 1 / a ^ (1 / 2) * cexp (-(↑π / a) * ↑x✝ ^ 2)

Next tactic:
congr 1
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      (1 : ℂ) / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) := by
  let f := fun x : ℝ => cexp (-π * a * (x : ℂ) ^ 2)
  have h1 : 0 < (↑π * a).re := by
    rw [ofReal_mul_re]
    exact mul_pos pi_pos ha
  have h2 : 0 < (↑π / a).re := by
    rw [div_eq_mul_inv, ofReal_mul_re, inv_re]
    refine' mul_pos pi_pos (div_pos ha <| normSq_pos.mpr _)
    contrapose! ha
    rw [ha, zero_re]
  have f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    convert (isLittleO_exp_neg_mul_sq_cocompact h1 (-2)).isBigO using 2
    dsimp only
    congr 1
    ring
  have Ff_bd : 𝓕 f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    rw [fourier_transform_gaussian_pi ha]
    convert (isLittleO_exp_neg_mul_sq_cocompact h2 (-2)).isBigO.const_mul_left
      ((1 : ℂ) / a ^ (1 / 2 : ℂ)) using 2
    congr 1
    

Proof state:
case h.e'_7.h.e_a
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℂ
ha : 0 < a.re
f : ℝ → ℂ := fun x => cexp (-↑π * a * ↑x ^ 2)
h1 : 0 < (↑π * a).re
h2 : 0 < (↑π / a).re
f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2)
x✝ : ℝ
⊢ cexp (-↑π / a * ↑x✝ ^ 2) = cexp (-(↑π / a) * ↑x✝ ^ 2)

Next tactic:
ring_nf
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      (1 : ℂ) / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) := by
  let f := fun x : ℝ => cexp (-π * a * (x : ℂ) ^ 2)
  have h1 : 0 < (↑π * a).re := by
    rw [ofReal_mul_re]
    exact mul_pos pi_pos ha
  have h2 : 0 < (↑π / a).re := by
    rw [div_eq_mul_inv, ofReal_mul_re, inv_re]
    refine' mul_pos pi_pos (div_pos ha <| normSq_pos.mpr _)
    contrapose! ha
    rw [ha, zero_re]
  have f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    convert (isLittleO_exp_neg_mul_sq_cocompact h1 (-2)).isBigO using 2
    dsimp only
    congr 1
    ring
  have Ff_bd : 𝓕 f =O[cocompact ℝ] fun x => |x| ^ (-2 : ℝ) := by
    rw [fourier_transform_gaussian_pi ha]
    convert (isLittleO_exp_neg_mul_sq_cocompact h2 (-2)).isBigO.const_mul_left
      ((1 : ℂ) / a ^ (1 / 2 : ℂ)) using 2
    congr 1
    ring_nf
  

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℂ
ha : 0 < a.re
f : ℝ → ℂ := fun x => cexp (-↑π * a * ↑x ^ 2)
h1 : 0 < (↑π * a).re
h2 : 0 < (↑π / a).re
f_bd : f =O[cocompact ℝ] fun x => |x| ^ (-2)
Ff_bd : 𝓕 f =O[cocompact ℝ] fun x => |x| ^ (-2)
⊢ ∑' (n : ℤ), cexp (-↑π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), cexp (-↑π / a * ↑n ^ 2)

Next tactic:
simpa only [fourier_transform_gaussian_pi ha, tsum_mul_left, Function.comp] using
    Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay
      (Complex.continuous_exp.comp (continuous_const.mul (continuous_ofReal.pow 2)) : Continuous f)
      one_lt_two f_bd Ff_bd
theorem Real.tsum_exp_neg_mul_int_sq {a : ℝ} (ha : 0 < a) :
    (∑' n : ℤ, exp (-π * a * (n : ℝ) ^ 2)) =
      (1 : ℝ) / a ^ (1 / 2 : ℝ) * (∑' n : ℤ, exp (-π / a * (n : ℝ) ^ 2)) := by
  

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
ha : 0 < a
⊢ ∑' (n : ℤ), rexp (-π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), rexp (-π / a * ↑n ^ 2)

Next tactic:
simpa only [← ofReal_inj, ofReal_mul, ofReal_tsum, ofReal_exp, ofReal_div, ofReal_pow,
    ofReal_int_cast, ofReal_neg, ofReal_cpow ha.le, ofReal_ofNat, ofReal_one] using
    Complex.tsum_exp_neg_mul_int_sq (by rwa [ofReal_re] : 0 < (a : ℂ).re)
theorem Real.tsum_exp_neg_mul_int_sq {a : ℝ} (ha : 0 < a) :
    (∑' n : ℤ, exp (-π * a * (n : ℝ) ^ 2)) =
      (1 : ℝ) / a ^ (1 / 2 : ℝ) * (∑' n : ℤ, exp (-π / a * (n : ℝ) ^ 2)) := by
  simpa only [← ofReal_inj, ofReal_mul, ofReal_tsum, ofReal_exp, ofReal_div, ofReal_pow,
    ofReal_int_cast, ofReal_neg, ofReal_cpow ha.le, ofReal_ofNat, ofReal_one] using
    Complex.tsum_exp_neg_mul_int_sq (by 

Proof state:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
ha : 0 < a
⊢ 0 < (↑a).re

Next tactic:
rwa [ofReal_re]
theorem exists_seq_separating (α : Type*) {p : Set α → Prop} {s₀} (hp : p s₀) (t : Set α)
    [HasCountableSeparatingOn α p t] :
    ∃ S : ℕ → Set α, (∀ n, p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ n, x ∈ S n ↔ y ∈ S n) → x = y := by
  

Proof state:
α : Type u_1
p : Set α → Prop
s₀ : Set α
hp : p s₀
t : Set α
inst✝ : HasCountableSeparatingOn α p t
⊢ ∃ S, (∀ (n : ℕ), p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ (n : ℕ), x ∈ S n ↔ y ∈ S n) → x = y

Next tactic:
rcases exists_nonempty_countable_separating α hp t with ⟨S, hSne, hSc, hS⟩
theorem exists_seq_separating (α : Type*) {p : Set α → Prop} {s₀} (hp : p s₀) (t : Set α)
    [HasCountableSeparatingOn α p t] :
    ∃ S : ℕ → Set α, (∀ n, p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ n, x ∈ S n ↔ y ∈ S n) → x = y := by
  rcases exists_nonempty_countable_separating α hp t with ⟨S, hSne, hSc, hS⟩
  

Proof state:
case intro.intro.intro
α : Type u_1
p : Set α → Prop
s₀ : Set α
hp : p s₀
t : Set α
inst✝ : HasCountableSeparatingOn α p t
S : Set (Set α)
hSne : Set.Nonempty S
hSc : Set.Countable S
hS : (∀ s ∈ S, p s) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y
⊢ ∃ S, (∀ (n : ℕ), p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ (n : ℕ), x ∈ S n ↔ y ∈ S n) → x = y

Next tactic:
rcases hSc.exists_eq_range hSne with ⟨S, rfl⟩
theorem exists_seq_separating (α : Type*) {p : Set α → Prop} {s₀} (hp : p s₀) (t : Set α)
    [HasCountableSeparatingOn α p t] :
    ∃ S : ℕ → Set α, (∀ n, p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ n, x ∈ S n ↔ y ∈ S n) → x = y := by
  rcases exists_nonempty_countable_separating α hp t with ⟨S, hSne, hSc, hS⟩
  rcases hSc.exists_eq_range hSne with ⟨S, rfl⟩
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
p : Set α → Prop
s₀ : Set α
hp : p s₀
t : Set α
inst✝ : HasCountableSeparatingOn α p t
S : ℕ → Set α
hSne : Set.Nonempty (range S)
hSc : Set.Countable (range S)
hS : (∀ s ∈ range S, p s) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ s ∈ range S, x ∈ s ↔ y ∈ s) → x = y
⊢ ∃ S, (∀ (n : ℕ), p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ (n : ℕ), x ∈ S n ↔ y ∈ S n) → x = y

Next tactic:
use S
theorem exists_seq_separating (α : Type*) {p : Set α → Prop} {s₀} (hp : p s₀) (t : Set α)
    [HasCountableSeparatingOn α p t] :
    ∃ S : ℕ → Set α, (∀ n, p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ n, x ∈ S n ↔ y ∈ S n) → x = y := by
  rcases exists_nonempty_countable_separating α hp t with ⟨S, hSne, hSc, hS⟩
  rcases hSc.exists_eq_range hSne with ⟨S, rfl⟩
  use S
  

Proof state:
case h
α : Type u_1
p : Set α → Prop
s₀ : Set α
hp : p s₀
t : Set α
inst✝ : HasCountableSeparatingOn α p t
S : ℕ → Set α
hSne : Set.Nonempty (range S)
hSc : Set.Countable (range S)
hS : (∀ s ∈ range S, p s) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ s ∈ range S, x ∈ s ↔ y ∈ s) → x = y
⊢ (∀ (n : ℕ), p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ (n : ℕ), x ∈ S n ↔ y ∈ S n) → x = y

Next tactic:
simpa only [forall_range_iff] using hS
theorem HasCountableSeparatingOn.of_subtype {α : Type*} {p : Set α → Prop} {t : Set α}
    {q : Set t → Prop} [h : HasCountableSeparatingOn t q univ]
    (hpq : ∀ U, q U → ∃ V, p V ∧ (↑) ⁻¹' V = U) : HasCountableSeparatingOn α p t := by
  

Proof state:
α : Type u_1
p : Set α → Prop
t : Set α
q : Set ↑t → Prop
h : HasCountableSeparatingOn (↑t) q univ
hpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U
⊢ HasCountableSeparatingOn α p t

Next tactic:
rcases h.1 with ⟨S, hSc, hSq, hS⟩
theorem HasCountableSeparatingOn.of_subtype {α : Type*} {p : Set α → Prop} {t : Set α}
    {q : Set t → Prop} [h : HasCountableSeparatingOn t q univ]
    (hpq : ∀ U, q U → ∃ V, p V ∧ (↑) ⁻¹' V = U) : HasCountableSeparatingOn α p t := by
  rcases h.1 with ⟨S, hSc, hSq, hS⟩
  

Proof state:
case intro.intro.intro
α : Type u_1
p : Set α → Prop
t : Set α
q : Set ↑t → Prop
h : HasCountableSeparatingOn (↑t) q univ
hpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U
S : Set (Set ↑t)
hSc : Set.Countable S
hSq : ∀ s ∈ S, q s
hS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y
⊢ HasCountableSeparatingOn α p t

Next tactic:
choose! V hpV hV using fun s hs ↦ hpq s (hSq s hs)
theorem HasCountableSeparatingOn.of_subtype {α : Type*} {p : Set α → Prop} {t : Set α}
    {q : Set t → Prop} [h : HasCountableSeparatingOn t q univ]
    (hpq : ∀ U, q U → ∃ V, p V ∧ (↑) ⁻¹' V = U) : HasCountableSeparatingOn α p t := by
  rcases h.1 with ⟨S, hSc, hSq, hS⟩
  choose! V hpV hV using fun s hs ↦ hpq s (hSq s hs)
  

Proof state:
case intro.intro.intro
α : Type u_1
p : Set α → Prop
t : Set α
q : Set ↑t → Prop
h : HasCountableSeparatingOn (↑t) q univ
hpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U
S : Set (Set ↑t)
hSc : Set.Countable S
hSq : ∀ s ∈ S, q s
hS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y
V : Set ↑t → Set α
hpV : ∀ s ∈ S, p (V s)
hV : ∀ s ∈ S, Subtype.val ⁻¹' V s = s
⊢ HasCountableSeparatingOn α p t

Next tactic:
refine ⟨⟨V '' S, hSc.image _, ball_image_iff.2 hpV, fun x hx y hy h ↦ ?_⟩⟩
theorem HasCountableSeparatingOn.of_subtype {α : Type*} {p : Set α → Prop} {t : Set α}
    {q : Set t → Prop} [h : HasCountableSeparatingOn t q univ]
    (hpq : ∀ U, q U → ∃ V, p V ∧ (↑) ⁻¹' V = U) : HasCountableSeparatingOn α p t := by
  rcases h.1 with ⟨S, hSc, hSq, hS⟩
  choose! V hpV hV using fun s hs ↦ hpq s (hSq s hs)
  refine ⟨⟨V '' S, hSc.image _, ball_image_iff.2 hpV, fun x hx y hy h ↦ ?_⟩⟩
  

Proof state:
case intro.intro.intro
α : Type u_1
p : Set α → Prop
t : Set α
q : Set ↑t → Prop
h✝ : HasCountableSeparatingOn (↑t) q univ
hpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U
S : Set (Set ↑t)
hSc : Set.Countable S
hSq : ∀ s ∈ S, q s
hS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y
V : Set ↑t → Set α
hpV : ∀ s ∈ S, p (V s)
hV : ∀ s ∈ S, Subtype.val ⁻¹' V s = s
x : α
hx : x ∈ t
y : α
hy : y ∈ t
h : ∀ s ∈ V '' S, x ∈ s ↔ y ∈ s
⊢ x = y

Next tactic:
refine congr_arg Subtype.val (hS ⟨x, hx⟩ trivial ⟨y, hy⟩ trivial fun U hU ↦ ?_)
theorem HasCountableSeparatingOn.of_subtype {α : Type*} {p : Set α → Prop} {t : Set α}
    {q : Set t → Prop} [h : HasCountableSeparatingOn t q univ]
    (hpq : ∀ U, q U → ∃ V, p V ∧ (↑) ⁻¹' V = U) : HasCountableSeparatingOn α p t := by
  rcases h.1 with ⟨S, hSc, hSq, hS⟩
  choose! V hpV hV using fun s hs ↦ hpq s (hSq s hs)
  refine ⟨⟨V '' S, hSc.image _, ball_image_iff.2 hpV, fun x hx y hy h ↦ ?_⟩⟩
  refine congr_arg Subtype.val (hS ⟨x, hx⟩ trivial ⟨y, hy⟩ trivial fun U hU ↦ ?_)
  

Proof state:
case intro.intro.intro
α : Type u_1
p : Set α → Prop
t : Set α
q : Set ↑t → Prop
h✝ : HasCountableSeparatingOn (↑t) q univ
hpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U
S : Set (Set ↑t)
hSc : Set.Countable S
hSq : ∀ s ∈ S, q s
hS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y
V : Set ↑t → Set α
hpV : ∀ s ∈ S, p (V s)
hV : ∀ s ∈ S, Subtype.val ⁻¹' V s = s
x : α
hx : x ∈ t
y : α
hy : y ∈ t
h : ∀ s ∈ V '' S, x ∈ s ↔ y ∈ s
U : Set ↑t
hU : U ∈ S
⊢ { val := x, property := hx } ∈ U ↔ { val := y, property := hy } ∈ U

Next tactic:
rw [← hV U hU]
theorem HasCountableSeparatingOn.of_subtype {α : Type*} {p : Set α → Prop} {t : Set α}
    {q : Set t → Prop} [h : HasCountableSeparatingOn t q univ]
    (hpq : ∀ U, q U → ∃ V, p V ∧ (↑) ⁻¹' V = U) : HasCountableSeparatingOn α p t := by
  rcases h.1 with ⟨S, hSc, hSq, hS⟩
  choose! V hpV hV using fun s hs ↦ hpq s (hSq s hs)
  refine ⟨⟨V '' S, hSc.image _, ball_image_iff.2 hpV, fun x hx y hy h ↦ ?_⟩⟩
  refine congr_arg Subtype.val (hS ⟨x, hx⟩ trivial ⟨y, hy⟩ trivial fun U hU ↦ ?_)
  rw [← hV U hU]
  

Proof state:
case intro.intro.intro
α : Type u_1
p : Set α → Prop
t : Set α
q : Set ↑t → Prop
h✝ : HasCountableSeparatingOn (↑t) q univ
hpq : ∀ (U : Set ↑t), q U → ∃ V, p V ∧ Subtype.val ⁻¹' V = U
S : Set (Set ↑t)
hSc : Set.Countable S
hSq : ∀ s ∈ S, q s
hS : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y
V : Set ↑t → Set α
hpV : ∀ s ∈ S, p (V s)
hV : ∀ s ∈ S, Subtype.val ⁻¹' V s = s
x : α
hx : x ∈ t
y : α
hy : y ∈ t
h : ∀ s ∈ V '' S, x ∈ s ↔ y ∈ s
U : Set ↑t
hU : U ∈ S
⊢ { val := x, property := hx } ∈ Subtype.val ⁻¹' V U ↔ { val := y, property := hy } ∈ Subtype.val ⁻¹' V U

Next tactic:
exact h _ (mem_image_of_mem _ hU)
theorem exists_subset_subsingleton_mem_of_forall_separating (p : Set α → Prop)
    {s : Set α} [h : HasCountableSeparatingOn α p s] (hs : s ∈ l)
    (hl : ∀ U, p U → U ∈ l ∨ Uᶜ ∈ l) : ∃ t, t ⊆ s ∧ t.Subsingleton ∧ t ∈ l := by
  

Proof state:
α : Type u_1
β : Sort ?u.3736
l : Filter α
inst✝ : CountableInterFilter l
f g : α → β
p : Set α → Prop
s : Set α
h : HasCountableSeparatingOn α p s
hs : s ∈ l
hl : ∀ (U : Set α), p U → U ∈ l ∨ Uᶜ ∈ l
⊢ ∃ t ⊆ s, Set.Subsingleton t ∧ t ∈ l

Next tactic:
rcases h.1 with ⟨S, hSc, hSp, hS⟩
theorem exists_subset_subsingleton_mem_of_forall_separating (p : Set α → Prop)
    {s : Set α} [h : HasCountableSeparatingOn α p s] (hs : s ∈ l)
    (hl : ∀ U, p U → U ∈ l ∨ Uᶜ ∈ l) : ∃ t, t ⊆ s ∧ t.Subsingleton ∧ t ∈ l := by
  rcases h.1 with ⟨S, hSc, hSp, hS⟩
  

Proof state:
case intro.intro.intro
α : Type u_1
β : Sort ?u.3736
l : Filter α
inst✝ : CountableInterFilter l
f g : α → β
p : Set α → Prop
s : Set α
h : HasCountableSeparatingOn α p s
hs : s ∈ l
hl : ∀ (U : Set α), p U → U ∈ l ∨ Uᶜ ∈ l
S : Set (Set α)
hSc : Set.Countable S
hSp : ∀ s ∈ S, p s
hS : ∀ x ∈ s, ∀ y ∈ s, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y
⊢ ∃ t ⊆ s, Set.Subsingleton t ∧ t ∈ l

Next tactic:
refine ⟨s ∩ ⋂₀ (S ∩ l.sets) ∩ ⋂ (U ∈ S) (_ : Uᶜ ∈ l), Uᶜ, ?_, ?_, ?_⟩
theorem exists_subset_subsingleton_mem_of_forall_separating (p : Set α → Prop)
    {s : Set α} [h : HasCountableSeparatingOn α p s] (hs : s ∈ l)
    (hl : ∀ U, p U → U ∈ l ∨ Uᶜ ∈ l) : ∃ t, t ⊆ s ∧ t.Subsingleton ∧ t ∈ l := by
  rcases h.1 with ⟨S, hSc, hSp, hS⟩
  refine ⟨s ∩ ⋂₀ (S ∩ l.sets) ∩ ⋂ (U ∈ S) (_ : Uᶜ ∈ l), Uᶜ, ?_, ?_, ?_⟩
  · 

Proof state:
case intro.intro.intro.refine_1
α : Type u_1
β : Sort ?u.3736
l : Filter α
inst✝ : CountableInterFilter l
f g : α → β
p : Set α → Prop
s : Set α
h : HasCountableSeparatingOn α p s
hs : s ∈ l
hl : ∀ (U : Set α), p U → U ∈ l ∨ Uᶜ ∈ l
S : Set (Set α)
hSc : Set.Countable S
hSp : ∀ s ∈ S, p s
hS : ∀ x ∈ s, ∀ y ∈ s, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y
⊢ s ∩ ⋂₀ (S ∩ l.sets) ∩ ⋂ U ∈ S, ⋂ (_ : Uᶜ ∈ l), Uᶜ ⊆ s

Next tactic:
exact fun _ h ↦ h.1.1