-- Mathlib/Combinatorics/Configuration.lean
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

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) := {i | i.1 ∈ i.2}.toFinset
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
step2 : ∑ i ∈ s, lineCount L i.1 = ∑ i ∈ s, pointCount P i.2
⊢ ∑ i ∈ sᶜ, lineCount L i.1 = ∑ i ∈ sᶜ, pointCount P i.2

Tactic:
rwa [← s.sum_add_sum_compl, ← s.sum_add_sum_compl, step2, add_left_cancel_iff] at step1

Result:
no goals

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) := {i | i.1 ∈ i.2}.toFinset
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
step2 : ∑ i ∈ s, lineCount L i.1 = ∑ i ∈ s, pointCount P i.2
step3 : ∑ i ∈ {i | i.1 ∈ i.2}ᶜ.toFinset, lineCount L i.1 = ∑ i ∈ {i | i.1 ∈ i.2}ᶜ.toFinset, pointCount P i.2
i : P × L
hi : i ∈ {i | i.1 ∈ i.2}ᶜ.toFinset
⊢ i.1 ∉ i.2

Tactic:
exact Set.mem_toFinset.mp hi

Result:
no goals

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) := {i | i.1 ∈ i.2}.toFinset
⊢ ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2

Tactic:
rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]

Result:
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) := {i | i.1 ∈ i.2}.toFinset
⊢ ∑ y : L, ∑ x : P, lineCount L (x, y).1 = ∑ x : P, ∑ y : L, pointCount P (x, y).2

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) := {i | i.1 ∈ i.2}.toFinset
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∑ c : P, ∑ a ∈ {l | c ∈ l}.toFinset, lineCount L (c, a).1 = ∑ i ∈ s, pointCount P i.2

Tactic:
rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }, eq_comm]

Result:
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) := {i | i.1 ∈ i.2}.toFinset
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∑ c : L, ∑ a ∈ {p | p ∈ c}.toFinset, pointCount P (a, c).2 = ∑ c : P, ∑ a ∈ {l | c ∈ l}.toFinset, lineCount L (c, a).1

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) := {i | i.1 ∈ i.2}.toFinset
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∀ (p : P × L), p ∈ s ↔ p.2 ∈ univ ∧ p.1 ∈ {p_1 | p_1 ∈ p.2}.toFinset

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) := {i | i.1 ∈ i.2}.toFinset
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
l : L
x✝ : l ∈ univ
⊢ ∑ a ∈ {p | p ∈ l}.toFinset, pointCount P (a, l).2 = ∑ a ∈ {l_1 | f l ∈ l_1}.toFinset, lineCount L (f l, a).1

Tactic:
simp_rw [hf2, sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]

Result:
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) := {i | i.1 ∈ i.2}.toFinset
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
l : L
x✝ : l ∈ univ
⊢ Nat.card ↑{p | p ∈ l} • lineCount L (f l) = Nat.card ↑{l_1 | f l ∈ l_1} • lineCount L (f l)

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) := {i | i.1 ∈ i.2}.toFinset
⊢ lineCount L p = pointCount P l

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]

Result:
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) := {i | i.1 ∈ i.2}.toFinset
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ lineCount L p = pointCount 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) := {i | i.1 ∈ i.2}.toFinset
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∀ (i : L), i ∈ univ ↔ f i ∈ univ

Tactic:
simp

Result:
no goals

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) := {i | i.1 ∈ i.2}.toFinset
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
step2 : ∑ i ∈ s, lineCount L i.1 = ∑ i ∈ s, pointCount P i.2
⊢ lineCount L p = pointCount P l

Tactic:
have step3 : ∑ i ∈ sᶜ, lineCount L i.1 = ∑ i ∈ sᶜ, pointCount P i.2 := by
  rwa [← s.sum_add_sum_compl, ← s.sum_add_sum_compl, step2, add_left_cancel_iff] at step1

Result:
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) := {i | i.1 ∈ i.2}.toFinset
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
step2 : ∑ i ∈ s, lineCount L i.1 = ∑ i ∈ s, pointCount P i.2
step3 : ∑ i ∈ sᶜ, lineCount L i.1 = ∑ i ∈ sᶜ, pointCount P i.2
⊢ lineCount L p = pointCount P l