-- Mathlib/NumberTheory/WellApproximable.lean
theorem addWellApproximable_ae_empty_or_univ (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) :
    (∀ᵐ x, ¬addWellApproximable 𝕊 δ x) ∨ ∀ᵐ x, addWellApproximable 𝕊 δ x

Proof state:
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
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 =ᶠ[ae μ] ∅ ∧ B ↑p =ᶠ[ae μ] ∅ → E =ᶠ[ae μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᶠ[ae μ] ∅ ∨ A ↑p =ᶠ[ae μ] univ
hB : ∀ (p : Nat.Primes), B ↑p =ᶠ[ae μ] ∅ ∨ B ↑p =ᶠ[ae μ] univ
p : Nat.Primes
⊢ u p +ᵥ C ↑p = C ↑p

Tactic:
let e := (AddAction.toPerm (u p) : Equiv.Perm 𝕊).toOrderIsoSet

Result:
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
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 =ᶠ[ae μ] ∅ ∧ B ↑p =ᶠ[ae μ] ∅ → E =ᶠ[ae μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᶠ[ae μ] ∅ ∨ A ↑p =ᶠ[ae μ] univ
hB : ∀ (p : Nat.Primes), B ↑p =ᶠ[ae μ] ∅ ∨ B ↑p =ᶠ[ae μ] univ
p : Nat.Primes
e : Set 𝕊 ≃o Set 𝕊 := Equiv.toOrderIsoSet (AddAction.toPerm (u p))
⊢ u p +ᵥ C ↑p = C ↑p

Proof state:
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
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
⊢ (∀ᵐ (x : 𝕊) ∂μ, ¬E x) ∨ ∀ᵐ (x : 𝕊) ∂μ, E x

Tactic:
have hE₂ : ∀ p : Nat.Primes, A p =ᵐ[μ] (∅ : Set 𝕊) ∧ B p =ᵐ[μ] (∅ : Set 𝕊) → E =ᵐ[μ] C p := by
  rintro p ⟨hA, hB⟩
  rw [hE₁ p]
  exact union_ae_eq_right_of_ae_eq_empty ((union_ae_eq_right_of_ae_eq_empty hA).trans hB)

Result:
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
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 =ᶠ[ae μ] ∅ ∧ B ↑p =ᶠ[ae μ] ∅ → E =ᶠ[ae μ] C ↑p
⊢ (∀ᵐ (x : 𝕊) ∂μ, ¬E x) ∨ ∀ᵐ (x : 𝕊) ∂μ, E x

Proof state:
case h.e'_2.h.e'_3
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this✝ : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
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 =ᶠ[ae μ] ∅ ∧ B ↑p =ᶠ[ae μ] ∅ → E =ᶠ[ae μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᶠ[ae μ] ∅ ∨ A ↑p =ᶠ[ae μ] univ
p : ℕ
hp : Nat.Prime p
x : 𝕊 := u ⟨p, hp⟩
f : 𝕊 → 𝕊 := fun y => p • y + x
n : ℕ
hn : 0 < n
h_ndiv : ¬p * p ∣ n
h_cop : (addOrderOf x).Coprime (n / p)
h_div : n / p * p = n
hf : f = (fun y => x + y) ∘ fun y => p • y
this :
  (fun y => x + y) '' ((fun y => p • y) '' approxAddOrderOf 𝕊 n (δ n)) ≤
    (fun y => x + y) '' approxAddOrderOf 𝕊 (n / p) (↑p * δ n)
⊢ n = addOrderOf x * (n / p)

Tactic:
rw [hu₀, Subtype.coe_mk, mul_comm p, h_div]

Result:
no goals

Proof state:
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
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)
⊢ (∀ᵐ (x : 𝕊) ∂μ, ¬E x) ∨ ∀ᵐ (x : 𝕊) ∂μ, E x

Tactic:
set A : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p∤n

Result:
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
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
⊢ (∀ᵐ (x : 𝕊) ∂μ, ¬E x) ∨ ∀ᵐ (x : 𝕊) ∂μ, E x

Proof state:
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this✝ : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
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 =ᶠ[ae μ] ∅ ∧ B ↑p =ᶠ[ae μ] ∅ → E =ᶠ[ae μ] C ↑p
p : ℕ
hp : Nat.Prime p
f : 𝕊 → 𝕊 := fun y => p • y
this : f '' A p ⊆ blimsup (fun n => approxAddOrderOf 𝕊 n (↑p * δ n)) atTop fun n => 0 < n ∧ ¬p ∣ n
⊢ (blimsup (fun n => approxAddOrderOf 𝕊 n (↑p * δ n)) atTop fun n => 0 < n ∧ ¬p ∣ n) =ᶠ[ae volume] A p

Tactic:
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

Result:
no goals