l3lab/ntp-mathlib huggingface dataset:
private theorem sum_density_div_card_le_density_add_eps [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} {A B : Finset (Finset α)}
    (hA : A ⊆ (chunk hP G ε hU).parts) (hB : B ⊆ (chunk hP G ε hV).parts) :
    (∑ ab in A.product B, G.edgeDensity ab.1 ab.2 : ℝ) / (A.card * B.card) ≤
    G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
  have : (↑1 + ε ^ 5 / ↑49) * G.edgeDensity (A.biUnion id) (B.biUnion id) ≤
      G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
    rw [add_mul, one_mul, add_le_add_iff_left]
    refine' mul_le_of_le_one_right (by sz_positivity) _
    exact mod_cast G.edgeDensity_le_one _ _
  refine' le_trans _ this
  conv_lhs => -- Porting note: LHS and RHS need separate treatment to get the desired form
    simp only [SimpleGraph.edgeDensity, edgeDensity, sum_div, Rat.cast_div, div_div]
  conv_rhs =>
    rw [SimpleGraph.edgeDensity, edgeDensity, ← sup_eq_biUnion, ← sup_eq_biUnion,
      Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl)]
    simp only [cast_sum, mul_sum, sum_div, 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, show A.product B = A ×ˢ B by rfl]
  intro x y hx hy
  rw [mul_mul_mul_comm, mul_comm (x.card : ℝ), mul_comm (y.card : ℝ), div_le_iff, mul_assoc]
  · refine' le_mul_of_one_le_right (cast_nonneg _) _
    rw [div_mul_eq_mul_div, one_le_div]
    refine' le_trans _ (mul_le_mul_of_nonneg_right (m_add_one_div_m_le_one_add hPα hPε hε₁) _)
    · rw [sq, mul_mul_mul_comm, mul_comm (_ / (m : ℝ)), mul_comm (_ / (m : ℝ))]
      exact mul_le_mul (sum_card_subset_chunk_parts_le (by sz_positivity) hA hx)
        (sum_card_subset_chunk_parts_le (by sz_positivity) hB hy) (by positivity) (by positivity)
    · exact mod_cast _root_.zero_le _
    rw [← cast_mul, cast_pos]
    apply mul_pos <;> rw [Finset.card_pos, sup_eq_biUnion, biUnion_nonempty]
    · 

Proof state:
case h.ha
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
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  (1 + ε ^ 5 / 49) * ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) ≤
    ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ ∃ x ∈ A, Finset.Nonempty (id x)

Next tactic:
exact ⟨_, hx, nonempty_of_mem_parts _ (hA hx)⟩
private theorem sum_density_div_card_le_density_add_eps [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} {A B : Finset (Finset α)}
    (hA : A ⊆ (chunk hP G ε hU).parts) (hB : B ⊆ (chunk hP G ε hV).parts) :
    (∑ ab in A.product B, G.edgeDensity ab.1 ab.2 : ℝ) / (A.card * B.card) ≤
    G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
  have : (↑1 + ε ^ 5 / ↑49) * G.edgeDensity (A.biUnion id) (B.biUnion id) ≤
      G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
    rw [add_mul, one_mul, add_le_add_iff_left]
    refine' mul_le_of_le_one_right (by sz_positivity) _
    exact mod_cast G.edgeDensity_le_one _ _
  refine' le_trans _ this
  conv_lhs => -- Porting note: LHS and RHS need separate treatment to get the desired form
    simp only [SimpleGraph.edgeDensity, edgeDensity, sum_div, Rat.cast_div, div_div]
  conv_rhs =>
    rw [SimpleGraph.edgeDensity, edgeDensity, ← sup_eq_biUnion, ← sup_eq_biUnion,
      Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl)]
    simp only [cast_sum, mul_sum, sum_div, 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, show A.product B = A ×ˢ B by rfl]
  intro x y hx hy
  rw [mul_mul_mul_comm, mul_comm (x.card : ℝ), mul_comm (y.card : ℝ), div_le_iff, mul_assoc]
  · refine' le_mul_of_one_le_right (cast_nonneg _) _
    rw [div_mul_eq_mul_div, one_le_div]
    refine' le_trans _ (mul_le_mul_of_nonneg_right (m_add_one_div_m_le_one_add hPα hPε hε₁) _)
    · rw [sq, mul_mul_mul_comm, mul_comm (_ / (m : ℝ)), mul_comm (_ / (m : ℝ))]
      exact mul_le_mul (sum_card_subset_chunk_parts_le (by sz_positivity) hA hx)
        (sum_card_subset_chunk_parts_le (by sz_positivity) hB hy) (by positivity) (by positivity)
    · exact mod_cast _root_.zero_le _
    rw [← cast_mul, cast_pos]
    apply mul_pos <;> rw [Finset.card_pos, sup_eq_biUnion, biUnion_nonempty]
    · exact ⟨_, hx, nonempty_of_mem_parts _ (hA hx)⟩
    · 

Proof state:
case h.hb
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
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  (1 + ε ^ 5 / 49) * ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) ≤
    ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ ∃ x ∈ B, Finset.Nonempty (id x)

Next tactic:
exact ⟨_, hy, nonempty_of_mem_parts _ (hB hy)⟩
private theorem sum_density_div_card_le_density_add_eps [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} {A B : Finset (Finset α)}
    (hA : A ⊆ (chunk hP G ε hU).parts) (hB : B ⊆ (chunk hP G ε hV).parts) :
    (∑ ab in A.product B, G.edgeDensity ab.1 ab.2 : ℝ) / (A.card * B.card) ≤
    G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
  have : (↑1 + ε ^ 5 / ↑49) * G.edgeDensity (A.biUnion id) (B.biUnion id) ≤
      G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
    rw [add_mul, one_mul, add_le_add_iff_left]
    refine' mul_le_of_le_one_right (by sz_positivity) _
    exact mod_cast G.edgeDensity_le_one _ _
  refine' le_trans _ this
  conv_lhs => -- Porting note: LHS and RHS need separate treatment to get the desired form
    simp only [SimpleGraph.edgeDensity, edgeDensity, sum_div, Rat.cast_div, div_div]
  conv_rhs =>
    rw [SimpleGraph.edgeDensity, edgeDensity, ← sup_eq_biUnion, ← sup_eq_biUnion,
      Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl)]
    simp only [cast_sum, mul_sum, sum_div, 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, show A.product B = A ×ˢ B by rfl]
  intro x y hx hy
  rw [mul_mul_mul_comm, mul_comm (x.card : ℝ), mul_comm (y.card : ℝ), div_le_iff, mul_assoc]
  · refine' le_mul_of_one_le_right (cast_nonneg _) _
    rw [div_mul_eq_mul_div, one_le_div]
    refine' le_trans _ (mul_le_mul_of_nonneg_right (m_add_one_div_m_le_one_add hPα hPε hε₁) _)
    · rw [sq, mul_mul_mul_comm, mul_comm (_ / (m : ℝ)), mul_comm (_ / (m : ℝ))]
      exact mul_le_mul (sum_card_subset_chunk_parts_le (by sz_positivity) hA hx)
        (sum_card_subset_chunk_parts_le (by sz_positivity) hB hy) (by positivity) (by positivity)
    · exact mod_cast _root_.zero_le _
    rw [← cast_mul, cast_pos]
    apply mul_pos <;> rw [Finset.card_pos, sup_eq_biUnion, biUnion_nonempty]
    · exact ⟨_, hx, nonempty_of_mem_parts _ (hA hx)⟩
    · exact ⟨_, hy, nonempty_of_mem_parts _ (hB hy)⟩
  

Proof state:
case h
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
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  (1 + ε ^ 5 / 49) * ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) ≤
    ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(Finset.card A) * ↑(Finset.card x) * (↑(Finset.card B) * ↑(Finset.card y))

Next tactic:
refine' mul_pos (mul_pos _ _) (mul_pos _ _)
private theorem sum_density_div_card_le_density_add_eps [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} {A B : Finset (Finset α)}
    (hA : A ⊆ (chunk hP G ε hU).parts) (hB : B ⊆ (chunk hP G ε hV).parts) :
    (∑ ab in A.product B, G.edgeDensity ab.1 ab.2 : ℝ) / (A.card * B.card) ≤
    G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
  have : (↑1 + ε ^ 5 / ↑49) * G.edgeDensity (A.biUnion id) (B.biUnion id) ≤
      G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
    rw [add_mul, one_mul, add_le_add_iff_left]
    refine' mul_le_of_le_one_right (by sz_positivity) _
    exact mod_cast G.edgeDensity_le_one _ _
  refine' le_trans _ this
  conv_lhs => -- Porting note: LHS and RHS need separate treatment to get the desired form
    simp only [SimpleGraph.edgeDensity, edgeDensity, sum_div, Rat.cast_div, div_div]
  conv_rhs =>
    rw [SimpleGraph.edgeDensity, edgeDensity, ← sup_eq_biUnion, ← sup_eq_biUnion,
      Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl)]
    simp only [cast_sum, mul_sum, sum_div, 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, show A.product B = A ×ˢ B by rfl]
  intro x y hx hy
  rw [mul_mul_mul_comm, mul_comm (x.card : ℝ), mul_comm (y.card : ℝ), div_le_iff, mul_assoc]
  · refine' le_mul_of_one_le_right (cast_nonneg _) _
    rw [div_mul_eq_mul_div, one_le_div]
    refine' le_trans _ (mul_le_mul_of_nonneg_right (m_add_one_div_m_le_one_add hPα hPε hε₁) _)
    · rw [sq, mul_mul_mul_comm, mul_comm (_ / (m : ℝ)), mul_comm (_ / (m : ℝ))]
      exact mul_le_mul (sum_card_subset_chunk_parts_le (by sz_positivity) hA hx)
        (sum_card_subset_chunk_parts_le (by sz_positivity) hB hy) (by positivity) (by positivity)
    · exact mod_cast _root_.zero_le _
    rw [← cast_mul, cast_pos]
    apply mul_pos <;> rw [Finset.card_pos, sup_eq_biUnion, biUnion_nonempty]
    · exact ⟨_, hx, nonempty_of_mem_parts _ (hA hx)⟩
    · exact ⟨_, hy, nonempty_of_mem_parts _ (hB hy)⟩
  refine' mul_pos (mul_pos _ _) (mul_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
hε₁ : ε ≤ 1
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 :
  (1 + ε ^ 5 / 49) * ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) ≤
    ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(Finset.card A)

Next tactic:
rw [cast_pos, Finset.card_pos]
private theorem sum_density_div_card_le_density_add_eps [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} {A B : Finset (Finset α)}
    (hA : A ⊆ (chunk hP G ε hU).parts) (hB : B ⊆ (chunk hP G ε hV).parts) :
    (∑ ab in A.product B, G.edgeDensity ab.1 ab.2 : ℝ) / (A.card * B.card) ≤
    G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
  have : (↑1 + ε ^ 5 / ↑49) * G.edgeDensity (A.biUnion id) (B.biUnion id) ≤
      G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
    rw [add_mul, one_mul, add_le_add_iff_left]
    refine' mul_le_of_le_one_right (by sz_positivity) _
    exact mod_cast G.edgeDensity_le_one _ _
  refine' le_trans _ this
  conv_lhs => -- Porting note: LHS and RHS need separate treatment to get the desired form
    simp only [SimpleGraph.edgeDensity, edgeDensity, sum_div, Rat.cast_div, div_div]
  conv_rhs =>
    rw [SimpleGraph.edgeDensity, edgeDensity, ← sup_eq_biUnion, ← sup_eq_biUnion,
      Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl)]
    simp only [cast_sum, mul_sum, sum_div, 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, show A.product B = A ×ˢ B by rfl]
  intro x y hx hy
  rw [mul_mul_mul_comm, mul_comm (x.card : ℝ), mul_comm (y.card : ℝ), div_le_iff, mul_assoc]
  · refine' le_mul_of_one_le_right (cast_nonneg _) _
    rw [div_mul_eq_mul_div, one_le_div]
    refine' le_trans _ (mul_le_mul_of_nonneg_right (m_add_one_div_m_le_one_add hPα hPε hε₁) _)
    · rw [sq, mul_mul_mul_comm, mul_comm (_ / (m : ℝ)), mul_comm (_ / (m : ℝ))]
      exact mul_le_mul (sum_card_subset_chunk_parts_le (by sz_positivity) hA hx)
        (sum_card_subset_chunk_parts_le (by sz_positivity) hB hy) (by positivity) (by positivity)
    · exact mod_cast _root_.zero_le _
    rw [← cast_mul, cast_pos]
    apply mul_pos <;> rw [Finset.card_pos, sup_eq_biUnion, biUnion_nonempty]
    · exact ⟨_, hx, nonempty_of_mem_parts _ (hA hx)⟩
    · exact ⟨_, hy, nonempty_of_mem_parts _ (hB hy)⟩
  refine' mul_pos (mul_pos _ _) (mul_pos _ _) <;> 

Proof state:
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
hε₁ : ε ≤ 1
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 :
  (1 + ε ^ 5 / 49) * ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) ≤
    ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(Finset.card x)

Next tactic:
rw [cast_pos, Finset.card_pos]
private theorem sum_density_div_card_le_density_add_eps [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} {A B : Finset (Finset α)}
    (hA : A ⊆ (chunk hP G ε hU).parts) (hB : B ⊆ (chunk hP G ε hV).parts) :
    (∑ ab in A.product B, G.edgeDensity ab.1 ab.2 : ℝ) / (A.card * B.card) ≤
    G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
  have : (↑1 + ε ^ 5 / ↑49) * G.edgeDensity (A.biUnion id) (B.biUnion id) ≤
      G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
    rw [add_mul, one_mul, add_le_add_iff_left]
    refine' mul_le_of_le_one_right (by sz_positivity) _
    exact mod_cast G.edgeDensity_le_one _ _
  refine' le_trans _ this
  conv_lhs => -- Porting note: LHS and RHS need separate treatment to get the desired form
    simp only [SimpleGraph.edgeDensity, edgeDensity, sum_div, Rat.cast_div, div_div]
  conv_rhs =>
    rw [SimpleGraph.edgeDensity, edgeDensity, ← sup_eq_biUnion, ← sup_eq_biUnion,
      Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl)]
    simp only [cast_sum, mul_sum, sum_div, 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, show A.product B = A ×ˢ B by rfl]
  intro x y hx hy
  rw [mul_mul_mul_comm, mul_comm (x.card : ℝ), mul_comm (y.card : ℝ), div_le_iff, mul_assoc]
  · refine' le_mul_of_one_le_right (cast_nonneg _) _
    rw [div_mul_eq_mul_div, one_le_div]
    refine' le_trans _ (mul_le_mul_of_nonneg_right (m_add_one_div_m_le_one_add hPα hPε hε₁) _)
    · rw [sq, mul_mul_mul_comm, mul_comm (_ / (m : ℝ)), mul_comm (_ / (m : ℝ))]
      exact mul_le_mul (sum_card_subset_chunk_parts_le (by sz_positivity) hA hx)
        (sum_card_subset_chunk_parts_le (by sz_positivity) hB hy) (by positivity) (by positivity)
    · exact mod_cast _root_.zero_le _
    rw [← cast_mul, cast_pos]
    apply mul_pos <;> rw [Finset.card_pos, sup_eq_biUnion, biUnion_nonempty]
    · exact ⟨_, hx, nonempty_of_mem_parts _ (hA hx)⟩
    · exact ⟨_, hy, nonempty_of_mem_parts _ (hB hy)⟩
  refine' mul_pos (mul_pos _ _) (mul_pos _ _) <;> 

Proof state:
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
hε₁ : ε ≤ 1
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 :
  (1 + ε ^ 5 / 49) * ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) ≤
    ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(Finset.card B)

Next tactic:
rw [cast_pos, Finset.card_pos]
private theorem sum_density_div_card_le_density_add_eps [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} {A B : Finset (Finset α)}
    (hA : A ⊆ (chunk hP G ε hU).parts) (hB : B ⊆ (chunk hP G ε hV).parts) :
    (∑ ab in A.product B, G.edgeDensity ab.1 ab.2 : ℝ) / (A.card * B.card) ≤
    G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
  have : (↑1 + ε ^ 5 / ↑49) * G.edgeDensity (A.biUnion id) (B.biUnion id) ≤
      G.edgeDensity (A.biUnion id) (B.biUnion id) + ε ^ 5 / 49 := by
    rw [add_mul, one_mul, add_le_add_iff_left]
    refine' mul_le_of_le_one_right (by sz_positivity) _
    exact mod_cast G.edgeDensity_le_one _ _
  refine' le_trans _ this
  conv_lhs => -- Porting note: LHS and RHS need separate treatment to get the desired form
    simp only [SimpleGraph.edgeDensity, edgeDensity, sum_div, Rat.cast_div, div_div]
  conv_rhs =>
    rw [SimpleGraph.edgeDensity, edgeDensity, ← sup_eq_biUnion, ← sup_eq_biUnion,
      Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl)]
    simp only [cast_sum, mul_sum, sum_div, 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, show A.product B = A ×ˢ B by rfl]
  intro x y hx hy
  rw [mul_mul_mul_comm, mul_comm (x.card : ℝ), mul_comm (y.card : ℝ), div_le_iff, mul_assoc]
  · refine' le_mul_of_one_le_right (cast_nonneg _) _
    rw [div_mul_eq_mul_div, one_le_div]
    refine' le_trans _ (mul_le_mul_of_nonneg_right (m_add_one_div_m_le_one_add hPα hPε hε₁) _)
    · rw [sq, mul_mul_mul_comm, mul_comm (_ / (m : ℝ)), mul_comm (_ / (m : ℝ))]
      exact mul_le_mul (sum_card_subset_chunk_parts_le (by sz_positivity) hA hx)
        (sum_card_subset_chunk_parts_le (by sz_positivity) hB hy) (by positivity) (by positivity)
    · exact mod_cast _root_.zero_le _
    rw [← cast_mul, cast_pos]
    apply mul_pos <;> rw [Finset.card_pos, sup_eq_biUnion, biUnion_nonempty]
    · exact ⟨_, hx, nonempty_of_mem_parts _ (hA hx)⟩
    · exact ⟨_, hy, nonempty_of_mem_parts _ (hB hy)⟩
  refine' mul_pos (mul_pos _ _) (mul_pos _ _) <;> 

Proof state:
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
hε₁ : ε ≤ 1
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 :
  (1 + ε ^ 5 / 49) * ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) ≤
    ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(Finset.card y)

Next tactic:
rw [cast_pos, Finset.card_pos]
private theorem average_density_near_total_density [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} {A B : Finset (Finset α)}
    (hA : A ⊆ (chunk hP G ε hU).parts) (hB : B ⊆ (chunk hP G ε hV).parts) :
    |(∑ ab in A.product B, G.edgeDensity ab.1 ab.2 : ℝ) / (A.card * B.card) -
      G.edgeDensity (A.biUnion id) (B.biUnion id)| ≤ ε ^ 5 / 49 := 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
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
⊢ |(∑ ab in Finset.product A B, ↑(SimpleGraph.edgeDensity G ab.1 ab.2)) / (↑(Finset.card A) * ↑(Finset.card B)) -
        ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))| ≤
    ε ^ 5 / 49

Next tactic:
rw [abs_sub_le_iff]
private theorem average_density_near_total_density [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} {A B : Finset (Finset α)}
    (hA : A ⊆ (chunk hP G ε hU).parts) (hB : B ⊆ (chunk hP G ε hV).parts) :
    |(∑ ab in A.product B, G.edgeDensity ab.1 ab.2 : ℝ) / (A.card * B.card) -
      G.edgeDensity (A.biUnion id) (B.biUnion id)| ≤ ε ^ 5 / 49 := by
  rw [abs_sub_le_iff]
  

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
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
⊢ (∑ ab in Finset.product A B, ↑(SimpleGraph.edgeDensity G ab.1 ab.2)) / (↑(Finset.card A) * ↑(Finset.card B)) -
        ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) ≤
      ε ^ 5 / 49 ∧
    ↑(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) -
        (∑ ab in Finset.product A B, ↑(SimpleGraph.edgeDensity G ab.1 ab.2)) / (↑(Finset.card A) * ↑(Finset.card B)) ≤
      ε ^ 5 / 49

Next tactic:
constructor