Module: Mathlib.NumberTheory.WellApproximable

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p ↦ Nat.Prime p
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p ↦ ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n ↦ approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∧ B ↑p =ᵐ[μ] ∅ → E =ᵐ[μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∨ A ↑p =ᵐ[μ] univ
⊢ (∀ᵐ (x : 𝕊) ∂μ, x ∉ E) ∨ ∀ᵐ (x : 𝕊) ∂μ, x ∈ E

Tactic:
have hB : ∀ p : Nat.Primes, B p =ᵐ[μ] (∅ : Set 𝕊) ∨ B p =ᵐ[μ] univ := by
    rintro ⟨p, hp⟩
    let x := u ⟨p, hp⟩
    let f : 𝕊 → 𝕊 := fun y => p • y + x
    suffices
      f '' B p ⊆ blimsup (fun n => approxAddOrderOf 𝕊 n (p * δ n)) atTop fun n => 0 < n ∧ p∣∣n by
      apply (ergodic_nsmul_add x hp.one_lt).ae_empty_or_univ_of_image_ae_le
        (hB₀ p).nullMeasurableSet
      apply (LE.le.eventuallyLE this).congr EventuallyEq.rfl
      exact blimsup_thickening_mul_ae_eq μ (fun n => 0 < n ∧ p∣∣n) (fun n => {y | addOrderOf y = n})
        (Nat.cast_pos.mpr hp.pos) _ hδ
    refine (sSupHom.setImage f).apply_blimsup_le.trans (mono_blimsup ?_)
    rintro n ⟨hn, h_div, h_ndiv⟩
    have h_cop : (addOrderOf x).Coprime (n / p) := by
      obtain ⟨q, rfl⟩ := h_div
      rw [hu₀, Subtype.coe_mk, hp.coprime_iff_not_dvd, q.mul_div_cancel_left hp.pos]
      exact fun contra => h_ndiv (mul_dvd_mul_left p contra)
    replace h_div : n / p * p = n := Nat.div_mul_cancel h_div
    have hf : f = (fun y => x + y) ∘ fun y => p • y := by
      ext; simp [f, add_comm x]
    simp_rw [Function.comp_apply]
    rw [sSupHom.setImage_toFun, hf, image_comp]
    have := @monotone_image 𝕊 𝕊 fun y => x + y
    specialize this (approxAddOrderOf.image_nsmul_subset (δ n) (n / p) hp.pos)
    simp only [h_div] at this ⊢
    refine this.trans ?_
    convert! approxAddOrderOf.vadd_subset_of_coprime (p * δ n) h_cop
    rw [hu₀, Subtype.coe_mk, mul_comm p, h_div]

Goal state after:
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p ↦ Nat.Prime p
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p ↦ ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n ↦ approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∧ B ↑p =ᵐ[μ] ∅ → E =ᵐ[μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∨ A ↑p =ᵐ[μ] univ
hB : ∀ (p : Nat.Primes), B ↑p =ᵐ[μ] ∅ ∨ B ↑p =ᵐ[μ] univ
⊢ (∀ᵐ (x : 𝕊) ∂μ, x ∉ E) ∨ ∀ᵐ (x : 𝕊) ∂μ, x ∈ E