private theorem density_sub_eps_le_sum_density_div_card [Nonempty α]
    (hPα : P.parts.card * 16 ^ P.parts.card ≤ card α) (hPε : ↑100 ≤ ↑4 ^ P.parts.card * ε ^ 5)
    {hU : U ∈ P.parts} {hV : V ∈ P.parts} {A B : Finset (Finset α)}
    (hA : A ⊆ (chunk hP G ε hU).parts) (hB : B ⊆ (chunk hP G ε hV).parts) :
    (G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (∑ ab in A.product B, (G.edgeDensity ab.1 ab.2 : ℝ)) / (A.card * B.card) := by
  have : ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / ↑50 ≤
      (↑1 - ε ^ 5 / 50) * G.edgeDensity (A.biUnion id) (B.biUnion id) := by
    rw [sub_mul, one_mul, sub_le_sub_iff_left]
    refine' mul_le_of_le_one_right (by sz_positivity) _
    exact mod_cast G.edgeDensity_le_one _ _
  refine' this.trans _
  conv_rhs => -- Porting note: LHS and RHS need separate treatment to get the desired form
    simp only [SimpleGraph.edgeDensity_def, sum_div, Rat.cast_div, div_div]
  conv_lhs =>
    rw [SimpleGraph.edgeDensity_def, SimpleGraph.interedges, ← sup_eq_biUnion, ← sup_eq_biUnion,
      Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl), ofSubset_parts,
      ofSubset_parts]
    simp only [cast_sum, sum_div, mul_sum, Rat.cast_sum, Rat.cast_div,
      mul_div_left_comm ((1 : ℝ) - _)]
  push_cast
  apply sum_le_sum
  simp only [and_imp, Prod.forall, mem_product]
  rintro x y hx hy
  rw [mul_mul_mul_comm, mul_comm (x.card : ℝ), mul_comm (y.card : ℝ), le_div_iff, mul_assoc]
  · refine' mul_le_of_le_one_right (cast_nonneg _) _
    rw [div_mul_eq_mul_div, ← mul_assoc, mul_assoc]
    refine' div_le_one_of_le _ (by positivity)
    refine' (mul_le_mul_of_nonneg_right (one_sub_le_m_div_m_add_one_sq hPα hPε) _).trans _
    · exact mod_cast _root_.zero_le _
    rw [sq, mul_mul_mul_comm, mul_comm ((m : ℝ) / _), mul_comm ((m : ℝ) / _)]
    refine' mul_le_mul _ _ _ (cast_nonneg _)
    apply le_sum_card_subset_chunk_parts hA hx
    apply le_sum_card_subset_chunk_parts hB hy
    positivity
  refine' mul_pos (mul_pos _ _) (mul_pos _ _) <;> rw [cast_pos, Finset.card_pos]
  

Proof state:
case h.refine'_1
α : Type u_1
inst✝¹ : Fintype α
P : Finpartition univ
hP : IsEquipartition P
G : SimpleGraph α
ε : ℝ
U : Finset α
hU✝ : U ∈ P.parts
V : Finset α
𝒜 : Finset (Finset α)
s : Finset α
inst✝ : Nonempty α
hPα : Finset.card P.parts * 16 ^ Finset.card P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ Finset.card P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ Finset.Nonempty A
case h.refine'_2
α : Type u_1
inst✝¹ : Fintype α
P : Finpartition univ
hP : IsEquipartition P
G : SimpleGraph α
ε : ℝ
U : Finset α
hU✝ : U ∈ P.parts
V : Finset α
𝒜 : Finset (Finset α)
s : Finset α
inst✝ : Nonempty α
hPα : Finset.card P.parts * 16 ^ Finset.card P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ Finset.card P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ Finset.Nonempty x
case h.refine'_3
α : Type u_1
inst✝¹ : Fintype α
P : Finpartition univ
hP : IsEquipartition P
G : SimpleGraph α
ε : ℝ
U : Finset α
hU✝ : U ∈ P.parts
V : Finset α
𝒜 : Finset (Finset α)
s : Finset α
inst✝ : Nonempty α
hPα : Finset.card P.parts * 16 ^ Finset.card P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ Finset.card P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ Finset.Nonempty B
case h.refine'_4
α : Type u_1
inst✝¹ : Fintype α
P : Finpartition univ
hP : IsEquipartition P
G : SimpleGraph α
ε : ℝ
U : Finset α
hU✝ : U ∈ P.parts
V : Finset α
𝒜 : Finset (Finset α)
s : Finset α
inst✝ : Nonempty α
hPα : Finset.card P.parts * 16 ^ Finset.card P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ Finset.card P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ Finset.Nonempty y

Next tactic:
exacts [⟨_, hx⟩, nonempty_of_mem_parts _ (hA hx), ⟨_, hy⟩, nonempty_of_mem_parts _ (hB hy)]