/-- Lower bound on the edge densities between non-uniform parts of `SzemerediRegularity.increment`.
-/
theorem edgeDensity_chunk_not_uniform [Nonempty α] (hPα : P.parts.card * 16 ^ P.parts.card ≤ card α)
    (hPε : ↑100 ≤ ↑4 ^ P.parts.card * ε ^ 5) (hε₁ : ε ≤ 1) {hU : U ∈ P.parts} {hV : V ∈ P.parts}
    (hUVne : U ≠ V) (hUV : ¬G.IsUniform ε U V) :
    (G.edgeDensity U V : ℝ) ^ 2 - ε ^ 5 / ↑25 + ε ^ 4 / ↑3 ≤
    (∑ ab in (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts,
      (G.edgeDensity ab.1 ab.2 : ℝ) ^ 2) / ↑16 ^ P.parts.card :=
  calc
    ↑(G.edgeDensity U V) ^ 2 - ε ^ 5 / 25 + ε ^ 4 / ↑3 ≤ ↑(G.edgeDensity U V) ^ 2 - ε ^ 5 / ↑25 +
        (star hP G ε hU V).card * (star hP G ε hV U).card / ↑16 ^ P.parts.card *
          (↑9 / ↑16) * ε ^ 2 := by
      apply add_le_add_left
      have Ul : 4 / 5 * ε ≤ (star hP G ε hU V).card / _ :=
        eps_le_card_star_div hPα hPε hε₁ hU hV hUVne hUV
      have Vl : 4 / 5 * ε ≤ (star hP G ε hV U).card / _ :=
        eps_le_card_star_div hPα hPε hε₁ hV hU hUVne.symm fun h => hUV h.symm
      rw [show (16 : ℝ) = ↑4 ^ 2 by norm_num, pow_right_comm, sq ((4 : ℝ) ^ _), ←
        _root_.div_mul_div_comm, mul_assoc]
      have : 0 < ε := by sz_positivity
      have UVl := mul_le_mul Ul Vl (by positivity) ?_
      swap
      · -- This seems faster than `exact div_nonneg (by positivity) (by positivity)` and *much*
        -- (tens of seconds) faster than `positivity` on its own.
        apply div_nonneg <;> positivity
      refine' le_trans _ (mul_le_mul_of_nonneg_right UVl _)
      · norm_num
        nlinarith
      · norm_num
        positivity
    _ ≤ (∑ ab in (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts,
        (G.edgeDensity ab.1 ab.2 : ℝ) ^ 2) / ↑16 ^ P.parts.card := by
      have t : (star hP G ε hU V).product (star hP G ε hV U) ⊆
          (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts :=
        product_subset_product star_subset_chunk star_subset_chunk
      have hε : 0 ≤ ε := by sz_positivity
      have sp : ∀ (a b : Finset (Finset α)), a.product b = a ×ˢ b := fun a b => rfl
      have := add_div_le_sum_sq_div_card t (fun x => (G.edgeDensity x.1 x.2 : ℝ))
        ((G.edgeDensity U V : ℝ) ^ 2 - ε ^ 5 / ↑25) (show 0 ≤ 3 / 4 * ε by linarith) ?_ ?_
      · simp_rw [sp, card_product, card_chunk (m_pos hPα).ne', ← mul_pow, cast_pow, mul_pow,
          div_pow, ← mul_assoc] at this
        norm_num at this
        

Proof state:
case 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
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
hUVne : U ≠ V
hUV : ¬SimpleGraph.IsUniform G ε U V
t :
  Finset.product (star hP G ε hU V) (star hP G ε hV U) ⊆ Finset.product (chunk hP G ε hU).parts (chunk hP G ε hV).parts
hε : 0 ≤ ε
sp : ∀ (a b : Finset (Finset α)), Finset.product a b = a ×ˢ b
this :
  ↑(SimpleGraph.edgeDensity G U V) ^ 2 - ε ^ 5 / 25 +
      ↑(Finset.card (star hP G ε hU V)) * ↑(Finset.card (star hP G ε hV U)) / 16 ^ Finset.card P.parts * (9 / 16) *
        ε ^ 2 ≤
    (∑ x in (chunk hP G ε hU).parts ×ˢ (chunk hP G ε hV).parts, ↑(SimpleGraph.edgeDensity G x.1 x.2) ^ 2) /
      16 ^ Finset.card P.parts
⊢ ↑(SimpleGraph.edgeDensity G U V) ^ 2 - ε ^ 5 / 25 +
      ↑(Finset.card (star hP G ε hU V)) * ↑(Finset.card (star hP G ε hV U)) / 16 ^ Finset.card P.parts * (9 / 16) *
        ε ^ 2 ≤
    (∑ ab in Finset.product (chunk hP G ε hU).parts (chunk hP G ε hV).parts,
        ↑(SimpleGraph.edgeDensity G ab.1 ab.2) ^ 2) /
      16 ^ Finset.card P.parts

Next tactic:
exact this