-- Module: Mathlib.Combinatorics.Additive.ApproximateSubgroup
Goal:
G : Type u_1
inst : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
obtain :
  ∀ (F₁ : Finset G) (hF₁ : (↑(#F₁) : ℝ) ≤ K) (hAF₁ : HPow.hPow (β := ℕ) A 2 ⊆ (↑F₁ : Set G) • A),
    CovBySMul G (K ^ (m - 1) * L ^ (n - 1)) (A ^ m ∩ B ^ n) (HPow.hPow (β := ℕ) A 2 ∩ HPow.hPow (β := ℕ) B 2)
obtain₁ :
  ∀ (F₁ : Finset G) (hF₁ : (↑(#F₁) : ℝ) ≤ K) (hAF₁ : HPow.hPow (β := ℕ) A 2 ⊆ (↑F₁ : Set G) • A) (F₂ : Finset G)
    (hF₂ : (↑(#F₂) : ℝ) ≤ L) (hBF₂ : HPow.hPow (β := ℕ) B 2 ⊆ (↑F₂ : Set G) • B),
    CovBySMul G (K ^ (m - 1) * L ^ (n - 1)) (A ^ m ∩ B ^ n) (HPow.hPow (β := ℕ) A 2 ∩ HPow.hPow (β := ℕ) B 2)
obtain₂ :
  ∀ (F₁ : Finset G) (hF₁ : (↑(#F₁) : ℝ) ≤ K) (hAF₁ : HPow.hPow (β := ℕ) A 2 ⊆ (↑F₁ : Set G) • A) (F₂ : Finset G)
    (hF₂ : (↑(#F₂) : ℝ) ≤ L) (hBF₂ : HPow.hPow (β := ℕ) B 2 ⊆ (↑F₂ : Set G) • B) (this : 1 ≤ K),
    CovBySMul G (K ^ (m - 1) * L ^ (n - 1)) (A ^ m ∩ B ^ n) (HPow.hPow (β := ℕ) A 2 ∩ HPow.hPow (β := ℕ) B 2)
obtain₃ :
  ∀ (F₁ : Finset G) (hF₁ : (↑(#F₁) : ℝ) ≤ K) (hAF₁ : HPow.hPow (β := ℕ) A 2 ⊆ (↑F₁ : Set G) • A) (F₂ : Finset G)
    (hF₂ : (↑(#F₂) : ℝ) ≤ L) (hBF₂ : HPow.hPow (β := ℕ) B 2 ⊆ (↑F₂ : Set G) • B) (this : 1 ≤ K) (f : (a b : G) → G)
    (hf : ∀ (a b : G), a • A ∩ b • B ⊆ f a b • (A⁻¹ * A ∩ (B⁻¹ * B))),
    CovBySMul G (K ^ (m - 1) * L ^ (n - 1)) (A ^ m ∩ B ^ n) (HPow.hPow (β := ℕ) A 2 ∩ HPow.hPow (β := ℕ) B 2)
refine_1 :
  ∀ (F₁ : Finset G) (hF₁ : (↑(#F₁) : ℝ) ≤ K) (hAF₁ : HPow.hPow (β := ℕ) A 2 ⊆ (↑F₁ : Set G) • A) (F₂ : Finset G)
    (hF₂ : (↑(#F₂) : ℝ) ≤ L) (hBF₂ : HPow.hPow (β := ℕ) B 2 ⊆ (↑F₂ : Set G) • B) (this : 1 ≤ K) (f : (a b : G) → G)
    (hf : ∀ (a b : G), a • A ∩ b • B ⊆ f a b • (A⁻¹ * A ∩ (B⁻¹ * B))),
    (↑(#(Finset.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1)))) : ℝ) ≤ K ^ (m - 1) * L ^ (n - 1)
refine_2 refine_2₁ :
  ∀ (F₁ : Finset G) (hF₁ : (↑(#F₁) : ℝ) ≤ K) (hAF₁ : HPow.hPow (β := ℕ) A 2 ⊆ (↑F₁ : Set G) • A) (F₂ : Finset G)
    (hF₂ : (↑(#F₂) : ℝ) ≤ L) (hBF₂ : HPow.hPow (β := ℕ) B 2 ⊆ (↑F₂ : Set G) • B) (this : 1 ≤ K) (f : (a b : G) → G)
    (hf : ∀ (a b : G), a • A ∩ b • B ⊆ f a b • (A⁻¹ * A ∩ (B⁻¹ * B))),
    A ^ m ∩ B ^ n ⊆
      (↑(Finset.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1))) : Set G) • (HPow.hPow (β := ℕ) A 2 ∩ HPow.hPow (β := ℕ) B 2)
⊢ CovBySMul G (K ^ (m - 1) * L ^ (n - 1)) (A ^ m ∩ B ^ n) (HPow.hPow (β := ℕ) A 2 ∩ HPow.hPow (β := ℕ) B 2)

Tactic:
classical
  obtain ⟨F₁, hF₁, hAF₁⟩ := hA.sq_covBySMul
  obtain ⟨F₂, hF₂, hBF₂⟩ := hB.sq_covBySMul
  have := hA.one_le
  choose f hf using exists_smul_inter_smul_subset_smul_inv_mul_inter_inv_mul A B
  refine ⟨.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1)), ?_, ?_⟩
  · calc
      (#(.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1))) : ℝ)
      _ ≤ #(F₁ ^ (m - 1)) * #(F₂ ^ (n - 1)) := mod_cast Finset.card_image₂_le ..
      _ ≤ #F₁ ^ (m - 1) * #F₂ ^ (n - 1) := by gcongr <;> exact mod_cast Finset.card_pow_le
      _ ≤ K ^ (m - 1) * L ^ (n - 1) := by gcongr
  · calc
      A ^ m ∩ B ^ n ⊆ (F₁ ^ (m - 1) * A) ∩ (F₂ ^ (n - 1) * B) := by
        gcongr <;> apply pow_subset_pow_mul_of_sq_subset_mul <;> norm_cast <;> lia
      _ = ⋃ (a ∈ F₁ ^ (m - 1)) (b ∈ F₂ ^ (n - 1)), a • A ∩ b • B := by
        simp_rw [← smul_eq_mul, ← iUnion_smul_set, iUnion₂_inter_iUnion₂]; norm_cast
      _ ⊆ ⋃ (a ∈ F₁ ^ (m - 1)) (b ∈ F₂ ^ (n - 1)), f a b • (A⁻¹ * A ∩ (B⁻¹ * B)) := by
        gcongr; exact hf ..
      _ = (Finset.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1))) * (A ^ 2 ∩ B ^ 2) := by
        simp_rw [hA.inv_eq_self, hB.inv_eq_self, ← sq]
        rw [Finset.coe_image₂, ← smul_eq_mul, ← iUnion_smul_set, biUnion_image2]
        simp_rw [Finset.mem_coe]

Premises used: HMul.hMul, Finset.card_image₂_le, Set.exists_smul_inter_smul_subset_smul_inv_mul_inter_inv_mul, HSub.hSub, Inter.inter, Inv.inv, HPow.hPow, IsApproximateSubgroup.sq_covBySMul, OfNat.ofNat