private theorem one_sub_eps_mul_card_nonuniformWitness_le_card_star (hV : V ∈ P.parts)
    (hUV : U ≠ V) (hunif : ¬G.IsUniform ε U V) (hPε : ↑100 ≤ ↑4 ^ P.parts.card * ε ^ 5)
    (hε₁ : ε ≤ 1) :
    (1 - ε / 10) * (G.nonuniformWitness ε U V).card ≤ ((star hP G ε hU V).biUnion id).card := by
  have hP₁ : 0 < P.parts.card := Finset.card_pos.2 ⟨_, hU⟩
  have : (↑2 ^ P.parts.card : ℝ) * m / (U.card * ε) ≤ ε / 10 := by
    rw [← div_div, div_le_iff']
    swap
    sz_positivity
    refine' le_of_mul_le_mul_left _ (pow_pos zero_lt_two P.parts.card)
    calc
      ↑2 ^ P.parts.card * ((↑2 ^ P.parts.card * m : ℝ) / U.card) =
          ((2 : ℝ) * 2) ^ P.parts.card * m / U.card := by
        rw [mul_pow, ← mul_div_assoc, mul_assoc]
      _ = ↑4 ^ P.parts.card * m / U.card := by norm_num
      _ ≤ 1 := (div_le_one_of_le (pow_mul_m_le_card_part hP hU) (cast_nonneg _))
      _ ≤ ↑2 ^ P.parts.card * ε ^ 2 / 10 := by
        refine' (one_le_sq_iff <| by positivity).1 _
        rw [div_pow, mul_pow, pow_right_comm, ← pow_mul ε,
          one_le_div (sq_pos_of_ne_zero (10 : ℝ) <| by norm_num)]
        calc
          (↑10 ^ 2) = 100 := by norm_num
          _ ≤ ↑4 ^ P.parts.card * ε ^ 5 := hPε
          _ ≤ ↑4 ^ P.parts.card * ε ^ 4 :=
            (mul_le_mul_of_nonneg_left (pow_le_pow_of_le_one (by sz_positivity) hε₁ <| le_succ _)
              (by positivity))
          _ = (↑2 ^ 2) ^ P.parts.card * ε ^ (2 * 2) := by norm_num
      _ = ↑2 ^ P.parts.card * (ε * (ε / 10)) := by rw [mul_div_assoc, sq, mul_div_assoc]
  

Proof state:
α : Type u_1
inst✝ : 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
hV : V ∈ P.parts
hUV : U ≠ V
hunif : ¬SimpleGraph.IsUniform G ε U V
hPε : 100 ≤ 4 ^ Finset.card P.parts * ε ^ 5
hε₁ : ε ≤ 1
hP₁ : 0 < Finset.card P.parts
this : 2 ^ Finset.card P.parts * ↑m / (↑(Finset.card U) * ε) ≤ ε / 10
⊢ (1 - ε / 10) * ↑(Finset.card (SimpleGraph.nonuniformWitness G ε U V)) ≤
    ↑(Finset.card (Finset.biUnion (star hP G ε hU V) id))

Next tactic:
calc
    (↑1 - ε / 10) * (G.nonuniformWitness ε U V).card ≤
        (↑1 - ↑2 ^ P.parts.card * m / (U.card * ε)) * (G.nonuniformWitness ε U V).card :=
      mul_le_mul_of_nonneg_right (sub_le_sub_left this _) (cast_nonneg _)
    _ = (G.nonuniformWitness ε U V).card -
        ↑2 ^ P.parts.card * m / (U.card * ε) * (G.nonuniformWitness ε U V).card := by
      rw [sub_mul, one_mul]
    _ ≤ (G.nonuniformWitness ε U V).card - ↑2 ^ (P.parts.card - 1) * m := by
      refine' sub_le_sub_left _ _
      have : (2 : ℝ) ^ P.parts.card = ↑2 ^ (P.parts.card - 1) * 2 := by
        rw [← _root_.pow_succ', tsub_add_cancel_of_le (succ_le_iff.2 hP₁)]
      rw [← mul_div_right_comm, this, mul_right_comm _ (2 : ℝ), mul_assoc, le_div_iff]
      refine' mul_le_mul_of_nonneg_left _ (by positivity)
      exact (G.le_card_nonuniformWitness hunif).trans
        (le_mul_of_one_le_left (cast_nonneg _) one_le_two)
      have := Finset.card_pos.mpr (P.nonempty_of_mem_parts hU)
      sz_positivity
    _ ≤ ((star hP G ε hU V).biUnion id).card := by
      rw [sub_le_comm, ←
        cast_sub (card_le_of_subset <| biUnion_star_subset_nonuniformWitness hP G ε hU V), ←
        card_sdiff (biUnion_star_subset_nonuniformWitness hP G ε hU V)]
      exact mod_cast card_nonuniformWitness_sdiff_biUnion_star hV hUV hunif