l3lab/ntp-mathlib huggingface dataset:
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    

Proof state:
case intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
⊢ lineCount L p = pointCount P l

Next tactic:
have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
⊢ ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2

Next tactic:
rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
⊢ ∑ y : L, ∑ x : P, lineCount L (x, y).1 = ∑ x : P, ∑ y : L, pointCount P (x, y).2

Next tactic:
simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
    

Proof state:
case intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ lineCount L p = pointCount P l

Next tactic:
have step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · obtain ⟨l, hl⟩ := hf1.2 p
        exact ⟨l, Finset.mem_univ l, hl.symm⟩
      all_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; exact fun p => Iff.rfl
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
    have step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2

Next tactic:
rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
    have step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∑ c : P, ∑ a in Set.toFinset {l | c ∈ l}, lineCount L (c, a).1 = ∑ i in s, pointCount P i.2
case h
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∀ (p : P × L), p ∈ s ↔ p.1 ∈ Finset.univ ∧ p.2 ∈ Set.toFinset {l | p.1 ∈ l}

Next tactic:
rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
    have step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∑ c : P, ∑ a in Set.toFinset {l | c ∈ l}, lineCount L (c, a).1 =
    ∑ c : L, ∑ a in Set.toFinset {p | p ∈ c}, pointCount P (a, c).2
case h
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∀ (p : P × L), p ∈ s ↔ p.2 ∈ Finset.univ ∧ p.1 ∈ Set.toFinset {p_1 | p_1 ∈ p.2}
case h
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∀ (p : P × L), p ∈ s ↔ p.1 ∈ Finset.univ ∧ p.2 ∈ Set.toFinset {l | p.1 ∈ l}

Next tactic:
refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
    have step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · 

Proof state:
case refine'_1
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l✝ : L
hpl : p ∉ l✝
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
l : L
hl : l ∈ Finset.univ
⊢ ∑ a in Set.toFinset {p | p ∈ l}, pointCount P (a, l).2 =
    ∑ a in Set.toFinset {l_1 | (fun l x => f l) l hl ∈ l_1}, lineCount L ((fun l x => f l) l hl, a).1

Next tactic:
simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
    have step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        

Proof state:
case refine'_1
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l✝ : L
hpl : p ∉ l✝
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
l : L
hl : l ∈ Finset.univ
⊢ Nat.card ↑{p | p ∈ l} • pointCount P l = Nat.card ↑{l_1 | f l ∈ l_1} • lineCount L (f l)

Next tactic:
change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
    have step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        

Proof state:
case refine'_1
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l✝ : L
hpl : p ∉ l✝
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
l : L
hl : l ∈ Finset.univ
⊢ pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)

Next tactic:
rw [hf2]
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
    have step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · 

Proof state:
case refine'_2
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p✝ : P
l : L
hpl : p✝ ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
p : P
x✝ : p ∈ Finset.univ
⊢ ∃ a, ∃ (ha : a ∈ Finset.univ), p = (fun l x => f l) a ha

Next tactic:
obtain ⟨l, hl⟩ := hf1.2 p
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
    have step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · obtain ⟨l, hl⟩ := hf1.2 p
        

Proof state:
case refine'_2.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p✝ : P
l✝ : L
hpl : p✝ ∉ l✝
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
p : P
x✝ : p ∈ Finset.univ
l : L
hl : f l = p
⊢ ∃ a, ∃ (ha : a ∈ Finset.univ), p = (fun l x => f l) a ha

Next tactic:
exact ⟨l, Finset.mem_univ l, hl.symm⟩
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
    have step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · obtain ⟨l, hl⟩ := hf1.2 p
        exact ⟨l, Finset.mem_univ l, hl.symm⟩
      

Proof state:
case h
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∀ (p : P × L), p ∈ s ↔ p.2 ∈ Finset.univ ∧ p.1 ∈ Set.toFinset {p_1 | p_1 ∈ p.2}
case h
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∀ (p : P × L), p ∈ s ↔ p.1 ∈ Finset.univ ∧ p.2 ∈ Set.toFinset {l | p.1 ∈ l}

Next tactic:
all_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; exact fun p => Iff.rfl
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
    have step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · obtain ⟨l, hl⟩ := hf1.2 p
        exact ⟨l, Finset.mem_univ l, hl.symm⟩
      all_goals 

Proof state:
case h
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∀ (p : P × L), p ∈ s ↔ p.2 ∈ Finset.univ ∧ p.1 ∈ Set.toFinset {p_1 | p_1 ∈ p.2}

Next tactic:
simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]