l3lab/ntp-mathlib huggingface dataset:
private theorem eps_le_card_star_div [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)
    (hUV : U ≠ V) (hunif : ¬G.IsUniform ε U V) :
    ↑4 / ↑5 * ε ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
  have hm : (0 : ℝ) ≤ 1 - (↑m)⁻¹ := sub_nonneg_of_le (inv_le_one <| one_le_m_coe hPα)
  have hε : 0 ≤ 1 - ε / 10 :=
    sub_nonneg_of_le (div_le_one_of_le (hε₁.trans <| by norm_num) <| by norm_num)
  calc
    4 / 5 * ε = (1 - 1 / 10) * (1 - 9⁻¹) * ε := by norm_num
    _ ≤ (1 - ε / 10) * (1 - (↑m)⁻¹) * ((G.nonuniformWitness ε U V).card / U.card) :=
      (mul_le_mul (mul_le_mul (sub_le_sub_left (div_le_div_of_le_of_nonneg hε₁ <| by norm_num) _)
        (sub_le_sub_left (inv_le_inv_of_le (by norm_num) <|
          mod_cast (show 9 ≤ 100 by norm_num).trans
            (hundred_le_m hPα hPε hε₁)) _) (by norm_num) hε)
        ((le_div_iff' <| (@cast_pos ℝ _ _ _).2 (P.nonempty_of_mem_parts hU).card_pos).2 <|
          G.le_card_nonuniformWitness hunif) (by 

Proof state:
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
hUV : U ≠ V
hunif : ¬SimpleGraph.IsUniform G ε U V
hm : 0 ≤ 1 - (↑m)⁻¹
hε : 0 ≤ 1 - ε / 10
⊢ 0 ≤ ε

Next tactic:
sz_positivity
private theorem eps_le_card_star_div [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)
    (hUV : U ≠ V) (hunif : ¬G.IsUniform ε U V) :
    ↑4 / ↑5 * ε ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
  have hm : (0 : ℝ) ≤ 1 - (↑m)⁻¹ := sub_nonneg_of_le (inv_le_one <| one_le_m_coe hPα)
  have hε : 0 ≤ 1 - ε / 10 :=
    sub_nonneg_of_le (div_le_one_of_le (hε₁.trans <| by norm_num) <| by norm_num)
  calc
    4 / 5 * ε = (1 - 1 / 10) * (1 - 9⁻¹) * ε := by norm_num
    _ ≤ (1 - ε / 10) * (1 - (↑m)⁻¹) * ((G.nonuniformWitness ε U V).card / U.card) :=
      (mul_le_mul (mul_le_mul (sub_le_sub_left (div_le_div_of_le_of_nonneg hε₁ <| by norm_num) _)
        (sub_le_sub_left (inv_le_inv_of_le (by norm_num) <|
          mod_cast (show 9 ≤ 100 by norm_num).trans
            (hundred_le_m hPα hPε hε₁)) _) (by norm_num) hε)
        ((le_div_iff' <| (@cast_pos ℝ _ _ _).2 (P.nonempty_of_mem_parts hU).card_pos).2 <|
          G.le_card_nonuniformWitness hunif) (by sz_positivity) (by 

Proof state:
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
hUV : U ≠ V
hunif : ¬SimpleGraph.IsUniform G ε U V
hm : 0 ≤ 1 - (↑m)⁻¹
hε : 0 ≤ 1 - ε / 10
⊢ 0 ≤ (1 - ε / 10) * (1 - (↑m)⁻¹)

Next tactic:
positivity
private theorem eps_le_card_star_div [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)
    (hUV : U ≠ V) (hunif : ¬G.IsUniform ε U V) :
    ↑4 / ↑5 * ε ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
  have hm : (0 : ℝ) ≤ 1 - (↑m)⁻¹ := sub_nonneg_of_le (inv_le_one <| one_le_m_coe hPα)
  have hε : 0 ≤ 1 - ε / 10 :=
    sub_nonneg_of_le (div_le_one_of_le (hε₁.trans <| by norm_num) <| by norm_num)
  calc
    4 / 5 * ε = (1 - 1 / 10) * (1 - 9⁻¹) * ε := by norm_num
    _ ≤ (1 - ε / 10) * (1 - (↑m)⁻¹) * ((G.nonuniformWitness ε U V).card / U.card) :=
      (mul_le_mul (mul_le_mul (sub_le_sub_left (div_le_div_of_le_of_nonneg hε₁ <| by norm_num) _)
        (sub_le_sub_left (inv_le_inv_of_le (by norm_num) <|
          mod_cast (show 9 ≤ 100 by norm_num).trans
            (hundred_le_m hPα hPε hε₁)) _) (by norm_num) hε)
        ((le_div_iff' <| (@cast_pos ℝ _ _ _).2 (P.nonempty_of_mem_parts hU).card_pos).2 <|
          G.le_card_nonuniformWitness hunif) (by sz_positivity) (by positivity))
    _ = (1 - ε / 10) * (G.nonuniformWitness ε U V).card * ((1 - (↑m)⁻¹) / U.card) := by
      

Proof state:
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
hUV : U ≠ V
hunif : ¬SimpleGraph.IsUniform G ε U V
hm : 0 ≤ 1 - (↑m)⁻¹
hε : 0 ≤ 1 - ε / 10
⊢ (1 - ε / 10) * (1 - (↑m)⁻¹) * (↑(Finset.card (SimpleGraph.nonuniformWitness G ε U V)) / ↑(Finset.card U)) =
    (1 - ε / 10) * ↑(Finset.card (SimpleGraph.nonuniformWitness G ε U V)) * ((1 - (↑m)⁻¹) / ↑(Finset.card U))

Next tactic:
rw [mul_assoc, mul_assoc, mul_div_left_comm]
private theorem eps_le_card_star_div [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)
    (hUV : U ≠ V) (hunif : ¬G.IsUniform ε U V) :
    ↑4 / ↑5 * ε ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
  have hm : (0 : ℝ) ≤ 1 - (↑m)⁻¹ := sub_nonneg_of_le (inv_le_one <| one_le_m_coe hPα)
  have hε : 0 ≤ 1 - ε / 10 :=
    sub_nonneg_of_le (div_le_one_of_le (hε₁.trans <| by norm_num) <| by norm_num)
  calc
    4 / 5 * ε = (1 - 1 / 10) * (1 - 9⁻¹) * ε := by norm_num
    _ ≤ (1 - ε / 10) * (1 - (↑m)⁻¹) * ((G.nonuniformWitness ε U V).card / U.card) :=
      (mul_le_mul (mul_le_mul (sub_le_sub_left (div_le_div_of_le_of_nonneg hε₁ <| by norm_num) _)
        (sub_le_sub_left (inv_le_inv_of_le (by norm_num) <|
          mod_cast (show 9 ≤ 100 by norm_num).trans
            (hundred_le_m hPα hPε hε₁)) _) (by norm_num) hε)
        ((le_div_iff' <| (@cast_pos ℝ _ _ _).2 (P.nonempty_of_mem_parts hU).card_pos).2 <|
          G.le_card_nonuniformWitness hunif) (by sz_positivity) (by positivity))
    _ = (1 - ε / 10) * (G.nonuniformWitness ε U V).card * ((1 - (↑m)⁻¹) / U.card) := by
      rw [mul_assoc, mul_assoc, mul_div_left_comm]
    _ ≤ ((star hP G ε hU V).biUnion id).card * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right
        (one_sub_eps_mul_card_nonuniformWitness_le_card_star hV hUV hunif hPε hε₁) (by 

Proof state:
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
hUV : U ≠ V
hunif : ¬SimpleGraph.IsUniform G ε U V
hm : 0 ≤ 1 - (↑m)⁻¹
hε : 0 ≤ 1 - ε / 10
⊢ 0 ≤ (1 - (↑m)⁻¹) / ↑(Finset.card U)

Next tactic:
positivity
private theorem eps_le_card_star_div [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)
    (hUV : U ≠ V) (hunif : ¬G.IsUniform ε U V) :
    ↑4 / ↑5 * ε ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
  have hm : (0 : ℝ) ≤ 1 - (↑m)⁻¹ := sub_nonneg_of_le (inv_le_one <| one_le_m_coe hPα)
  have hε : 0 ≤ 1 - ε / 10 :=
    sub_nonneg_of_le (div_le_one_of_le (hε₁.trans <| by norm_num) <| by norm_num)
  calc
    4 / 5 * ε = (1 - 1 / 10) * (1 - 9⁻¹) * ε := by norm_num
    _ ≤ (1 - ε / 10) * (1 - (↑m)⁻¹) * ((G.nonuniformWitness ε U V).card / U.card) :=
      (mul_le_mul (mul_le_mul (sub_le_sub_left (div_le_div_of_le_of_nonneg hε₁ <| by norm_num) _)
        (sub_le_sub_left (inv_le_inv_of_le (by norm_num) <|
          mod_cast (show 9 ≤ 100 by norm_num).trans
            (hundred_le_m hPα hPε hε₁)) _) (by norm_num) hε)
        ((le_div_iff' <| (@cast_pos ℝ _ _ _).2 (P.nonempty_of_mem_parts hU).card_pos).2 <|
          G.le_card_nonuniformWitness hunif) (by sz_positivity) (by positivity))
    _ = (1 - ε / 10) * (G.nonuniformWitness ε U V).card * ((1 - (↑m)⁻¹) / U.card) := by
      rw [mul_assoc, mul_assoc, mul_div_left_comm]
    _ ≤ ((star hP G ε hU V).biUnion id).card * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right
        (one_sub_eps_mul_card_nonuniformWitness_le_card_star hV hUV hunif hPε hε₁) (by positivity))
    _ ≤ (star hP G ε hU V).card * (m + 1) * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right card_biUnion_star_le_m_add_one_card_star_mul (by 

Proof state:
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
hUV : U ≠ V
hunif : ¬SimpleGraph.IsUniform G ε U V
hm : 0 ≤ 1 - (↑m)⁻¹
hε : 0 ≤ 1 - ε / 10
⊢ 0 ≤ (1 - (↑m)⁻¹) / ↑(Finset.card U)

Next tactic:
positivity
private theorem eps_le_card_star_div [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)
    (hUV : U ≠ V) (hunif : ¬G.IsUniform ε U V) :
    ↑4 / ↑5 * ε ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
  have hm : (0 : ℝ) ≤ 1 - (↑m)⁻¹ := sub_nonneg_of_le (inv_le_one <| one_le_m_coe hPα)
  have hε : 0 ≤ 1 - ε / 10 :=
    sub_nonneg_of_le (div_le_one_of_le (hε₁.trans <| by norm_num) <| by norm_num)
  calc
    4 / 5 * ε = (1 - 1 / 10) * (1 - 9⁻¹) * ε := by norm_num
    _ ≤ (1 - ε / 10) * (1 - (↑m)⁻¹) * ((G.nonuniformWitness ε U V).card / U.card) :=
      (mul_le_mul (mul_le_mul (sub_le_sub_left (div_le_div_of_le_of_nonneg hε₁ <| by norm_num) _)
        (sub_le_sub_left (inv_le_inv_of_le (by norm_num) <|
          mod_cast (show 9 ≤ 100 by norm_num).trans
            (hundred_le_m hPα hPε hε₁)) _) (by norm_num) hε)
        ((le_div_iff' <| (@cast_pos ℝ _ _ _).2 (P.nonempty_of_mem_parts hU).card_pos).2 <|
          G.le_card_nonuniformWitness hunif) (by sz_positivity) (by positivity))
    _ = (1 - ε / 10) * (G.nonuniformWitness ε U V).card * ((1 - (↑m)⁻¹) / U.card) := by
      rw [mul_assoc, mul_assoc, mul_div_left_comm]
    _ ≤ ((star hP G ε hU V).biUnion id).card * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right
        (one_sub_eps_mul_card_nonuniformWitness_le_card_star hV hUV hunif hPε hε₁) (by positivity))
    _ ≤ (star hP G ε hU V).card * (m + 1) * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right card_biUnion_star_le_m_add_one_card_star_mul (by positivity))
    _ ≤ (star hP G ε hU V).card * (m + ↑1) * ((↑1 - (↑m)⁻¹) / (↑4 ^ P.parts.card * m)) :=
      (mul_le_mul_of_nonneg_left (div_le_div_of_le_left hm (by 

Proof state:
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
hUV : U ≠ V
hunif : ¬SimpleGraph.IsUniform G ε U V
hm : 0 ≤ 1 - (↑m)⁻¹
hε : 0 ≤ 1 - ε / 10
⊢ 0 < 4 ^ Finset.card P.parts * ↑m

Next tactic:
sz_positivity
private theorem eps_le_card_star_div [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)
    (hUV : U ≠ V) (hunif : ¬G.IsUniform ε U V) :
    ↑4 / ↑5 * ε ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
  have hm : (0 : ℝ) ≤ 1 - (↑m)⁻¹ := sub_nonneg_of_le (inv_le_one <| one_le_m_coe hPα)
  have hε : 0 ≤ 1 - ε / 10 :=
    sub_nonneg_of_le (div_le_one_of_le (hε₁.trans <| by norm_num) <| by norm_num)
  calc
    4 / 5 * ε = (1 - 1 / 10) * (1 - 9⁻¹) * ε := by norm_num
    _ ≤ (1 - ε / 10) * (1 - (↑m)⁻¹) * ((G.nonuniformWitness ε U V).card / U.card) :=
      (mul_le_mul (mul_le_mul (sub_le_sub_left (div_le_div_of_le_of_nonneg hε₁ <| by norm_num) _)
        (sub_le_sub_left (inv_le_inv_of_le (by norm_num) <|
          mod_cast (show 9 ≤ 100 by norm_num).trans
            (hundred_le_m hPα hPε hε₁)) _) (by norm_num) hε)
        ((le_div_iff' <| (@cast_pos ℝ _ _ _).2 (P.nonempty_of_mem_parts hU).card_pos).2 <|
          G.le_card_nonuniformWitness hunif) (by sz_positivity) (by positivity))
    _ = (1 - ε / 10) * (G.nonuniformWitness ε U V).card * ((1 - (↑m)⁻¹) / U.card) := by
      rw [mul_assoc, mul_assoc, mul_div_left_comm]
    _ ≤ ((star hP G ε hU V).biUnion id).card * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right
        (one_sub_eps_mul_card_nonuniformWitness_le_card_star hV hUV hunif hPε hε₁) (by positivity))
    _ ≤ (star hP G ε hU V).card * (m + 1) * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right card_biUnion_star_le_m_add_one_card_star_mul (by positivity))
    _ ≤ (star hP G ε hU V).card * (m + ↑1) * ((↑1 - (↑m)⁻¹) / (↑4 ^ P.parts.card * m)) :=
      (mul_le_mul_of_nonneg_left (div_le_div_of_le_left hm (by sz_positivity) <|
        pow_mul_m_le_card_part hP hU) (by 

Proof state:
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
hUV : U ≠ V
hunif : ¬SimpleGraph.IsUniform G ε U V
hm : 0 ≤ 1 - (↑m)⁻¹
hε : 0 ≤ 1 - ε / 10
⊢ 0 ≤ ↑(Finset.card (star hP G ε hU V)) * (↑m + 1)

Next tactic:
positivity
private theorem eps_le_card_star_div [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)
    (hUV : U ≠ V) (hunif : ¬G.IsUniform ε U V) :
    ↑4 / ↑5 * ε ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
  have hm : (0 : ℝ) ≤ 1 - (↑m)⁻¹ := sub_nonneg_of_le (inv_le_one <| one_le_m_coe hPα)
  have hε : 0 ≤ 1 - ε / 10 :=
    sub_nonneg_of_le (div_le_one_of_le (hε₁.trans <| by norm_num) <| by norm_num)
  calc
    4 / 5 * ε = (1 - 1 / 10) * (1 - 9⁻¹) * ε := by norm_num
    _ ≤ (1 - ε / 10) * (1 - (↑m)⁻¹) * ((G.nonuniformWitness ε U V).card / U.card) :=
      (mul_le_mul (mul_le_mul (sub_le_sub_left (div_le_div_of_le_of_nonneg hε₁ <| by norm_num) _)
        (sub_le_sub_left (inv_le_inv_of_le (by norm_num) <|
          mod_cast (show 9 ≤ 100 by norm_num).trans
            (hundred_le_m hPα hPε hε₁)) _) (by norm_num) hε)
        ((le_div_iff' <| (@cast_pos ℝ _ _ _).2 (P.nonempty_of_mem_parts hU).card_pos).2 <|
          G.le_card_nonuniformWitness hunif) (by sz_positivity) (by positivity))
    _ = (1 - ε / 10) * (G.nonuniformWitness ε U V).card * ((1 - (↑m)⁻¹) / U.card) := by
      rw [mul_assoc, mul_assoc, mul_div_left_comm]
    _ ≤ ((star hP G ε hU V).biUnion id).card * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right
        (one_sub_eps_mul_card_nonuniformWitness_le_card_star hV hUV hunif hPε hε₁) (by positivity))
    _ ≤ (star hP G ε hU V).card * (m + 1) * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right card_biUnion_star_le_m_add_one_card_star_mul (by positivity))
    _ ≤ (star hP G ε hU V).card * (m + ↑1) * ((↑1 - (↑m)⁻¹) / (↑4 ^ P.parts.card * m)) :=
      (mul_le_mul_of_nonneg_left (div_le_div_of_le_left hm (by sz_positivity) <|
        pow_mul_m_le_card_part hP hU) (by positivity))
    _ ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
      

Proof state:
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
hUV : U ≠ V
hunif : ¬SimpleGraph.IsUniform G ε U V
hm : 0 ≤ 1 - (↑m)⁻¹
hε : 0 ≤ 1 - ε / 10
⊢ ↑(Finset.card (star hP G ε hU V)) * (↑m + 1) * ((1 - (↑m)⁻¹) / (4 ^ Finset.card P.parts * ↑m)) ≤
    ↑(Finset.card (star hP G ε hU V)) / 4 ^ Finset.card P.parts

Next tactic:
rw [mul_assoc, mul_comm ((4 : ℝ) ^ P.parts.card), ← div_div, ← mul_div_assoc, ← mul_comm_div]
private theorem eps_le_card_star_div [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)
    (hUV : U ≠ V) (hunif : ¬G.IsUniform ε U V) :
    ↑4 / ↑5 * ε ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
  have hm : (0 : ℝ) ≤ 1 - (↑m)⁻¹ := sub_nonneg_of_le (inv_le_one <| one_le_m_coe hPα)
  have hε : 0 ≤ 1 - ε / 10 :=
    sub_nonneg_of_le (div_le_one_of_le (hε₁.trans <| by norm_num) <| by norm_num)
  calc
    4 / 5 * ε = (1 - 1 / 10) * (1 - 9⁻¹) * ε := by norm_num
    _ ≤ (1 - ε / 10) * (1 - (↑m)⁻¹) * ((G.nonuniformWitness ε U V).card / U.card) :=
      (mul_le_mul (mul_le_mul (sub_le_sub_left (div_le_div_of_le_of_nonneg hε₁ <| by norm_num) _)
        (sub_le_sub_left (inv_le_inv_of_le (by norm_num) <|
          mod_cast (show 9 ≤ 100 by norm_num).trans
            (hundred_le_m hPα hPε hε₁)) _) (by norm_num) hε)
        ((le_div_iff' <| (@cast_pos ℝ _ _ _).2 (P.nonempty_of_mem_parts hU).card_pos).2 <|
          G.le_card_nonuniformWitness hunif) (by sz_positivity) (by positivity))
    _ = (1 - ε / 10) * (G.nonuniformWitness ε U V).card * ((1 - (↑m)⁻¹) / U.card) := by
      rw [mul_assoc, mul_assoc, mul_div_left_comm]
    _ ≤ ((star hP G ε hU V).biUnion id).card * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right
        (one_sub_eps_mul_card_nonuniformWitness_le_card_star hV hUV hunif hPε hε₁) (by positivity))
    _ ≤ (star hP G ε hU V).card * (m + 1) * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right card_biUnion_star_le_m_add_one_card_star_mul (by positivity))
    _ ≤ (star hP G ε hU V).card * (m + ↑1) * ((↑1 - (↑m)⁻¹) / (↑4 ^ P.parts.card * m)) :=
      (mul_le_mul_of_nonneg_left (div_le_div_of_le_left hm (by sz_positivity) <|
        pow_mul_m_le_card_part hP hU) (by positivity))
    _ ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
      rw [mul_assoc, mul_comm ((4 : ℝ) ^ P.parts.card), ← div_div, ← mul_div_assoc, ← mul_comm_div]
      

Proof state:
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
hUV : U ≠ V
hunif : ¬SimpleGraph.IsUniform G ε U V
hm : 0 ≤ 1 - (↑m)⁻¹
hε : 0 ≤ 1 - ε / 10
⊢ ↑(Finset.card (star hP G ε hU V)) / 4 ^ Finset.card P.parts * ((↑m + 1) * ((1 - (↑m)⁻¹) / ↑m)) ≤
    ↑(Finset.card (star hP G ε hU V)) / 4 ^ Finset.card P.parts

Next tactic:
refine' mul_le_of_le_one_right (by positivity) _
private theorem eps_le_card_star_div [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)
    (hUV : U ≠ V) (hunif : ¬G.IsUniform ε U V) :
    ↑4 / ↑5 * ε ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
  have hm : (0 : ℝ) ≤ 1 - (↑m)⁻¹ := sub_nonneg_of_le (inv_le_one <| one_le_m_coe hPα)
  have hε : 0 ≤ 1 - ε / 10 :=
    sub_nonneg_of_le (div_le_one_of_le (hε₁.trans <| by norm_num) <| by norm_num)
  calc
    4 / 5 * ε = (1 - 1 / 10) * (1 - 9⁻¹) * ε := by norm_num
    _ ≤ (1 - ε / 10) * (1 - (↑m)⁻¹) * ((G.nonuniformWitness ε U V).card / U.card) :=
      (mul_le_mul (mul_le_mul (sub_le_sub_left (div_le_div_of_le_of_nonneg hε₁ <| by norm_num) _)
        (sub_le_sub_left (inv_le_inv_of_le (by norm_num) <|
          mod_cast (show 9 ≤ 100 by norm_num).trans
            (hundred_le_m hPα hPε hε₁)) _) (by norm_num) hε)
        ((le_div_iff' <| (@cast_pos ℝ _ _ _).2 (P.nonempty_of_mem_parts hU).card_pos).2 <|
          G.le_card_nonuniformWitness hunif) (by sz_positivity) (by positivity))
    _ = (1 - ε / 10) * (G.nonuniformWitness ε U V).card * ((1 - (↑m)⁻¹) / U.card) := by
      rw [mul_assoc, mul_assoc, mul_div_left_comm]
    _ ≤ ((star hP G ε hU V).biUnion id).card * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right
        (one_sub_eps_mul_card_nonuniformWitness_le_card_star hV hUV hunif hPε hε₁) (by positivity))
    _ ≤ (star hP G ε hU V).card * (m + 1) * ((1 - (↑m)⁻¹) / U.card) :=
      (mul_le_mul_of_nonneg_right card_biUnion_star_le_m_add_one_card_star_mul (by positivity))
    _ ≤ (star hP G ε hU V).card * (m + ↑1) * ((↑1 - (↑m)⁻¹) / (↑4 ^ P.parts.card * m)) :=
      (mul_le_mul_of_nonneg_left (div_le_div_of_le_left hm (by sz_positivity) <|
        pow_mul_m_le_card_part hP hU) (by positivity))
    _ ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
      rw [mul_assoc, mul_comm ((4 : ℝ) ^ P.parts.card), ← div_div, ← mul_div_assoc, ← mul_comm_div]
      refine' mul_le_of_le_one_right (by 

Proof state:
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
hUV : U ≠ V
hunif : ¬SimpleGraph.IsUniform G ε U V
hm : 0 ≤ 1 - (↑m)⁻¹
hε : 0 ≤ 1 - ε / 10
⊢ 0 ≤ ↑(Finset.card (star hP G ε hU V)) / 4 ^ Finset.card P.parts

Next tactic:
positivity