-- Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean
theorem density_sub_eps_le_sum_density_div_card [Nonempty α]
    (hPα : #P.parts * 16 ^ #P.parts ≤ card α) (hPε : ↑100 ≤ ↑4 ^ #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) :
    (G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (∑ ab ∈ A.product B, (G.edgeDensity ab.1 ab.2 : ℝ)) / (#A * #B)

Proof state:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #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 :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
⊢ (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) ≤
    ∑ x ∈ A.product B, ↑↑(#(G.interedges x.1 x.2)) / (↑(↑(#x.1) * ↑(#x.2)) * (↑(#A) * ↑(#B)))

Tactic:
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) - _)]

Result:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #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 :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
⊢ ∑ x ∈ A ×ˢ B, ↑↑(#(interedges G.Adj x.1 x.2)) * ((1 - ε ^ 5 / 50) / ↑(↑(#(A.sup id)) * ↑(#(B.sup id)))) ≤
    ∑ x ∈ A.product B, ↑↑(#(G.interedges x.1 x.2)) / (↑(↑(#x.1) * ↑(#x.2)) * (↑(#A) * ↑(#B)))

Proof state:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #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
⊢ ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) ≤ 1

Tactic:
exact mod_cast G.edgeDensity_le_one _ _

Result:
no goals

Proof state:
case h
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #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 :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ ↑(#(interedges G.Adj x y)) *
      ((1 - ε ^ 5 / 50) / (↑(#(A.sup id)) * ↑(#(B.sup id))) * (↑(#A) * ↑(#x) * (↑(#B) * ↑(#y)))) ≤
    ↑(#(G.interedges x y))

Tactic:
refine mul_le_of_le_one_right (cast_nonneg _) ?_

Result:
case h
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #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 :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ (1 - ε ^ 5 / 50) / (↑(#(A.sup id)) * ↑(#(B.sup id))) * (↑(#A) * ↑(#x) * (↑(#B) * ↑(#y))) ≤ 1

Proof state:
case h.refine_2.refine_2
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #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 :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ ↑(#B) * ↑(#y) * (↑m / (↑m + 1)) ≤ ↑(#(B.sup id))

Tactic:
apply le_sum_card_subset_chunk_parts hB hy

Result:
no goals

Proof state:
case h.refine_2
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #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 :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ (↑m / (↑m + 1)) ^ 2 * (↑(#A) * ↑(#x) * (↑(#B) * ↑(#y))) ≤ ↑(#(A.sup id)) * ↑(#(B.sup id))

Tactic:
rw [sq, mul_mul_mul_comm, mul_comm ((m : ℝ) / _), mul_comm ((m : ℝ) / _)]

Result:
case h.refine_2
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #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 :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ ↑(#A) * ↑(#x) * (↑m / (↑m + 1)) * (↑(#B) * ↑(#y) * (↑m / (↑m + 1))) ≤ ↑(#(A.sup id)) * ↑(#(B.sup id))

Proof state:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #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 :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
⊢ ∑ x ∈ A ×ˢ B, ↑↑(#(interedges G.Adj x.1 x.2)) * ((1 - ε ^ 5 / 50) / ↑(↑(#(A.sup id)) * ↑(#(B.sup id)))) ≤
    ∑ x ∈ A.product B, ↑↑(#(G.interedges x.1 x.2)) / (↑(↑(#x.1) * ↑(#x.2)) * (↑(#A) * ↑(#B)))

Tactic:
push_cast

Result:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #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 :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
⊢ ∑ x ∈ A ×ˢ B, ↑(#(interedges G.Adj x.1 x.2)) * ((1 - ε ^ 5 / 50) / (↑(#(A.sup id)) * ↑(#(B.sup id)))) ≤
    ∑ x ∈ A.product B, ↑(#(G.interedges x.1 x.2)) / (↑(#x.1) * ↑(#x.2) * (↑(#A) * ↑(#B)))