awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
obtain :
  ∀ (f : L → P) (hf1 : Function.Bijective f) (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)),
    lineCount L p = pointCount P l
obtain₁ :
  ∀ (f : L → P) (hf1 : Function.Bijective f) (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)),
    lineCount L p = pointCount P l
obtain₂ :
  ∀ (f : L → P) (hf1 : Function.Bijective f) (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l))
    (step1 : Eq (α := ℕ) (∑ i : P × L, lineCount L i.1) (∑ i : P × L, pointCount P i.2)), lineCount L p = pointCount P l
obtain₃ :
  ∀ (f : L → P) (hf1 : Function.Bijective f) (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)),
    let s : Finset (P × L) := Set.toFinset {i : P × L | i.1 ∈ i.2};
    ∀ (step1 : Eq (α := ℕ) (∑ i : P × L, lineCount L i.1) (∑ i : P × L, pointCount P i.2))
      (step2 : Eq (α := ℕ) (∑ i ∈ s, lineCount L i.1) (∑ i ∈ s, pointCount P i.2)), lineCount L p = pointCount P l
⊢ lineCount L p = pointCount P l
have obtain₄ : ∀ (f : L → P) (hf1 : Function.Bijective f) (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)),
  let s : Finset (P × L) := Set.toFinset {i : P × L | i.1 ∈ i.2};
  ∀ (step1 : Eq (α := ℕ) (∑ i : P × L, lineCount L i.1) (∑ i : P × L, pointCount P i.2))
    (step2 : Eq (α := ℕ) (∑ i ∈ s, lineCount L i.1) (∑ i ∈ s, pointCount P i.2))
    (step3 : Eq (α := ℕ) (∑ i ∈ sᶜ, lineCount L i.1) (∑ i ∈ sᶜ, pointCount P i.2)), lineCount L p = pointCount P l
Tactic: 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 ∈ s, lineCount L i.1 = ∑ i ∈ s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      on_goal 1 =>
        rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }, eq_comm]
        · refine sum_bijective _ hf1 (by simp) fun l _ ↦ ?_
          simp_rw [hf2, sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
          change pointCount P l • _ = lineCount L (f l) • _
          rw [hf2]
      all_goals simp_rw [s, Finset.mem_univ, true_and, Set.mem_toFinset]; exact fun p => Iff.rfl
    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
    rw [← Set.toFinset_compl] at step3
    exact
      ((Finset.sum_eq_sum_iff_of_le fun i hi =>
              HasLines.pointCount_le_lineCount (by exact Set.mem_toFinset.mp hi)).mp
          step3.symm (p, l) (Set.mem_toFinset.mpr hpl)).symm
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂
have obtain : ∀ (f : L → P), Function.Bijective f → ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)), ∃ p ∈ l₁, p ∈ l₂
Tactic: obtain ⟨f, _, hf2⟩ := HasLines.exists_bijective_of_card_eq h
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
⊢ ∃ p ∈ l₁, p ∈ l₂
have this : Nontrivial L
Tactic: haveI : Nontrivial L := ⟨⟨l₁, l₂, hl⟩⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
this : Nontrivial L
⊢ ∃ p ∈ l₁, p ∈ l₂
have this₁ : Nontrivial P
Tactic: haveI := Fintype.one_lt_card_iff_nontrivial.mp ((congr_arg _ h).mpr Fintype.one_lt_card)
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
this_1 : Nontrivial L
this : Nontrivial P
⊢ ∃ p ∈ l₁, p ∈ l₂
have h₁ : ∀ (p : P), 0 < lineCount L p
Tactic: have h₁ : ∀ p : P, 0 < lineCount L p := fun p =>
        Exists.elim (exists_ne p) fun q hq =>
          (congr_arg _ Nat.card_eq_fintype_card).mpr
            (Fintype.card_pos_iff.mpr ⟨⟨mkLine hq, (mkLine_ax hq).2⟩⟩)
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
this_1 : Nontrivial L
this : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
⊢ ∃ p ∈ l₁, p ∈ l₂
have h₂ : ∀ (l : L), 0 < pointCount P l
Tactic: have h₂ : ∀ l : L, 0 < pointCount P l := fun l => (congr_arg _ (hf2 l)).mpr (h₁ (f l))
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
this_1 : Nontrivial L
this : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
⊢ ∃ p ∈ l₁, p ∈ l₂
have obtain : ∀ (p : P) (hl₁ : p ∈ l₁), ∃ p ∈ l₁, p ∈ l₂
Tactic: obtain ⟨p, hl₁⟩ := Fintype.card_pos_iff.mp ((congr_arg _ Nat.card_eq_fintype_card).mp (h₂ l₁))
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
this_1 : Nontrivial L
this : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
⊢ ∃ p ∈ l₁, p ∈ l₂
have pos : ∀ (hl₂ : p ∈ l₂), ∃ p ∈ l₁, p ∈ l₂
Tactic: by_cases hl₂ : p ∈ l₂
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
this_1 : Nontrivial L
this : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
pos : ∀ (hl₂ : p ∈ l₂), ∃ p ∈ l₁, p ∈ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂
have neg : ∀ (hl₂ : p ∉ l₂), ∃ p ∈ l₁, p ∈ l₂
Tactic: by_cases hl₂ : p ∈ l₂
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
this_1 : Nontrivial L
this : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂
have key' : Fintype.card.{u_1} { q : P // q ∈ l₂ } = Fintype.card.{u_2} { l : L // p ∈ l }
Tactic: have key' : Fintype.card { q : P // q ∈ l₂ } = Fintype.card { l : L // p ∈ l } :=
        ((HasLines.lineCount_eq_pointCount h hl₂).trans Nat.card_eq_fintype_card).symm.trans
          Nat.card_eq_fintype_card
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
this_1 : Nontrivial L
this : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
key' : Fintype.card.{u_1} { q : P // q ∈ l₂ } = Fintype.card.{u_2} { l : L // p ∈ l }
⊢ ∃ p ∈ l₁, p ∈ l₂
have this₁ : ∀ (q : { q : P // q ∈ l₂ }), p ≠ (↑q : P)
Tactic: have : ∀ q : { q // q ∈ l₂ }, p ≠ q := fun q hq => hl₂ ((congr_arg (· ∈ l₂) hq).mpr q.2)
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f_1 : L → P
left : Function.Bijective f_1
hf2 : ∀ (l : L), pointCount P l = lineCount L (f_1 l)
this_1 : Nontrivial L
this_2 : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
key' : Fintype.card.{u_1} { q : P // q ∈ l₂ } = Fintype.card.{u_2} { l : L // p ∈ l }
this : ∀ (q : { q : P // q ∈ l₂ }), p ≠ (↑q : P)
f : { q : P // q ∈ l₂ } → { l : L // p ∈ l } :=
  fun (q : { q : P // q ∈ l₂ }) ↦ Subtype.mk (p := fun (l : L) ↦ p ∈ l) (@mkLine _ _ inst _ _ _ ⋯) ⋯
⊢ ∃ p ∈ l₁, p ∈ l₂
have hf : Function.Injective f
Tactic: have hf : Function.Injective f := fun q₁ q₂ hq =>
        Subtype.ext ((eq_or_eq q₁.2 q₂.2 (mkLine_ax (this q₁)).2
            ((congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mpr (mkLine_ax (this q₂)).2)).resolve_right
            fun h => (congr_arg (p ∉ ·) h).mp hl₂ (mkLine_ax (this q₁)).1)
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (hypothesis)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f_1 : L → P
left : Function.Bijective f_1
hf2 : ∀ (l : L), pointCount P l = lineCount L (f_1 l)
this_1 : Nontrivial L
this_2 : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
key' : Fintype.card.{u_1} { q : P // q ∈ l₂ } = Fintype.card.{u_2} { l : L // p ∈ l }
this : ∀ (q : { q : P // q ∈ l₂ }), p ≠ (↑q : P)
f : { q : P // q ∈ l₂ } → { l : L // p ∈ l } :=
  fun (q : { q : P // q ∈ l₂ }) ↦ Subtype.mk (p := fun (l : L) ↦ p ∈ l) (@mkLine _ _ inst _ _ _ ⋯) ⋯
hf : Function.Injective f
⊢ ∃ p ∈ l₁, p ∈ l₂
have key'₁ : Function.Surjective f
Tactic: have key' := ((Fintype.bijective_iff_injective_and_card f).mpr ⟨hf, key'⟩).2
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f_1 : L → P
left : Function.Bijective f_1
hf2 : ∀ (l : L), pointCount P l = lineCount L (f_1 l)
this_1 : Nontrivial L
this_2 : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
key'_1 : Fintype.card.{u_1} { q : P // q ∈ l₂ } = Fintype.card.{u_2} { l : L // p ∈ l }
this : ∀ (q : { q : P // q ∈ l₂ }), p ≠ (↑q : P)
f : { q : P // q ∈ l₂ } → { l : L // p ∈ l } :=
  fun (q : { q : P // q ∈ l₂ }) ↦ Subtype.mk (p := fun (l : L) ↦ p ∈ l) (@mkLine _ _ inst _ _ _ ⋯) ⋯
hf : Function.Injective f
key' : Function.Surjective f
⊢ ∃ p ∈ l₁, p ∈ l₂
have neg : ∀ (q : { q : P // q ∈ l₂ }) (hq : f q = Subtype.mk (p := fun (l : L) ↦ p ∈ l) l₁ hl₁), ∃ p ∈ l₁, p ∈ l₂
Tactic: obtain ⟨q, hq⟩ := key' ⟨l₁, hl₁⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂
have obtain : ∀ (f : L → P), Function.Bijective f → ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)), ∃ p ∈ l₁, p ∈ l₂
Tactic: classical
      obtain ⟨f, _, hf2⟩ := HasLines.exists_bijective_of_card_eq h
      haveI : Nontrivial L := ⟨⟨l₁, l₂, hl⟩⟩
      haveI := Fintype.one_lt_card_iff_nontrivial.mp ((congr_arg _ h).mpr Fintype.one_lt_card)
      have h₁ : ∀ p : P, 0 < lineCount L p := fun p =>
        Exists.elim (exists_ne p) fun q hq =>
          (congr_arg _ Nat.card_eq_fintype_card).mpr
            (Fintype.card_pos_iff.mpr ⟨⟨mkLine hq, (mkLine_ax hq).2⟩⟩)
      have h₂ : ∀ l : L, 0 < pointCount P l := fun l => (congr_arg _ (hf2 l)).mpr (h₁ (f l))
      obtain ⟨p, hl₁⟩ := Fintype.card_pos_iff.mp ((congr_arg _ Nat.card_eq_fintype_card).mp (h₂ l₁))
      by_cases hl₂ : p ∈ l₂
      · exact ⟨p, hl₁, hl₂⟩
      have key' : Fintype.card { q : P // q ∈ l₂ } = Fintype.card { l : L // p ∈ l } :=
        ((HasLines.lineCount_eq_pointCount h hl₂).trans Nat.card_eq_fintype_card).symm.trans
          Nat.card_eq_fintype_card
      have : ∀ q : { q // q ∈ l₂ }, p ≠ q := fun q hq => hl₂ ((congr_arg (· ∈ l₂) hq).mpr q.2)
      let f : { q : P // q ∈ l₂ } → { l : L // p ∈ l } := fun q =>
        ⟨mkLine (this q), (mkLine_ax (this q)).1⟩
      have hf : Function.Injective f := fun q₁ q₂ hq =>
        Subtype.ext ((eq_or_eq q₁.2 q₂.2 (mkLine_ax (this q₁)).2
            ((congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mpr (mkLine_ax (this q₂)).2)).resolve_right
            fun h => (congr_arg (p ∉ ·) h).mp hl₂ (mkLine_ax (this q₁)).1)
      have key' := ((Fintype.bijective_iff_injective_and_card f).mpr ⟨hf, key'⟩).2
      obtain ⟨q, hq⟩ := key' ⟨l₁, hl₁⟩
      exact ⟨q, (congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mp (mkLine_ax (this q)).2, q.2⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
obtain : ∀ (f : L → P), Function.Bijective f → ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)), ∃ p ∈ l₁, p ∈ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂
have obtain₁ : ∀ (f : L → P),
  Function.Bijective f → ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)) (this : Nontrivial L), ∃ p ∈ l₁, p ∈ l₂
Tactic: classical
      obtain ⟨f, _, hf2⟩ := HasLines.exists_bijective_of_card_eq h
      haveI : Nontrivial L := ⟨⟨l₁, l₂, hl⟩⟩
      haveI := Fintype.one_lt_card_iff_nontrivial.mp ((congr_arg _ h).mpr Fintype.one_lt_card)
      have h₁ : ∀ p : P, 0 < lineCount L p := fun p =>
        Exists.elim (exists_ne p) fun q hq =>
          (congr_arg _ Nat.card_eq_fintype_card).mpr
            (Fintype.card_pos_iff.mpr ⟨⟨mkLine hq, (mkLine_ax hq).2⟩⟩)
      have h₂ : ∀ l : L, 0 < pointCount P l := fun l => (congr_arg _ (hf2 l)).mpr (h₁ (f l))
      obtain ⟨p, hl₁⟩ := Fintype.card_pos_iff.mp ((congr_arg _ Nat.card_eq_fintype_card).mp (h₂ l₁))
      by_cases hl₂ : p ∈ l₂
      · exact ⟨p, hl₁, hl₂⟩
      have key' : Fintype.card { q : P // q ∈ l₂ } = Fintype.card { l : L // p ∈ l } :=
        ((HasLines.lineCount_eq_pointCount h hl₂).trans Nat.card_eq_fintype_card).symm.trans
          Nat.card_eq_fintype_card
      have : ∀ q : { q // q ∈ l₂ }, p ≠ q := fun q hq => hl₂ ((congr_arg (· ∈ l₂) hq).mpr q.2)
      let f : { q : P // q ∈ l₂ } → { l : L // p ∈ l } := fun q =>
        ⟨mkLine (this q), (mkLine_ax (this q)).1⟩
      have hf : Function.Injective f := fun q₁ q₂ hq =>
        Subtype.ext ((eq_or_eq q₁.2 q₂.2 (mkLine_ax (this q₁)).2
            ((congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mpr (mkLine_ax (this q₂)).2)).resolve_right
            fun h => (congr_arg (p ∉ ·) h).mp hl₂ (mkLine_ax (this q₁)).1)
      have key' := ((Fintype.bijective_iff_injective_and_card f).mpr ⟨hf, key'⟩).2
      obtain ⟨q, hq⟩ := key' ⟨l₁, hl₁⟩
      exact ⟨q, (congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mp (mkLine_ax (this q)).2, q.2⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
obtain : ∀ (f : L → P), Function.Bijective f → ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)), ∃ p ∈ l₁, p ∈ l₂
obtain₁ :
  ∀ (f : L → P),
    Function.Bijective f →
      ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)) (this : Nontrivial L), ∃ p ∈ l₁, p ∈ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂
have obtain₂ : ∀ (f : L → P),
  Function.Bijective f →
    ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)) (this : Nontrivial L) (this : Nontrivial P),
      ∃ p ∈ l₁, p ∈ l₂
Tactic: classical
      obtain ⟨f, _, hf2⟩ := HasLines.exists_bijective_of_card_eq h
      haveI : Nontrivial L := ⟨⟨l₁, l₂, hl⟩⟩
      haveI := Fintype.one_lt_card_iff_nontrivial.mp ((congr_arg _ h).mpr Fintype.one_lt_card)
      have h₁ : ∀ p : P, 0 < lineCount L p := fun p =>
        Exists.elim (exists_ne p) fun q hq =>
          (congr_arg _ Nat.card_eq_fintype_card).mpr
            (Fintype.card_pos_iff.mpr ⟨⟨mkLine hq, (mkLine_ax hq).2⟩⟩)
      have h₂ : ∀ l : L, 0 < pointCount P l := fun l => (congr_arg _ (hf2 l)).mpr (h₁ (f l))
      obtain ⟨p, hl₁⟩ := Fintype.card_pos_iff.mp ((congr_arg _ Nat.card_eq_fintype_card).mp (h₂ l₁))
      by_cases hl₂ : p ∈ l₂
      · exact ⟨p, hl₁, hl₂⟩
      have key' : Fintype.card { q : P // q ∈ l₂ } = Fintype.card { l : L // p ∈ l } :=
        ((HasLines.lineCount_eq_pointCount h hl₂).trans Nat.card_eq_fintype_card).symm.trans
          Nat.card_eq_fintype_card
      have : ∀ q : { q // q ∈ l₂ }, p ≠ q := fun q hq => hl₂ ((congr_arg (· ∈ l₂) hq).mpr q.2)
      let f : { q : P // q ∈ l₂ } → { l : L // p ∈ l } := fun q =>
        ⟨mkLine (this q), (mkLine_ax (this q)).1⟩
      have hf : Function.Injective f := fun q₁ q₂ hq =>
        Subtype.ext ((eq_or_eq q₁.2 q₂.2 (mkLine_ax (this q₁)).2
            ((congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mpr (mkLine_ax (this q₂)).2)).resolve_right
            fun h => (congr_arg (p ∉ ·) h).mp hl₂ (mkLine_ax (this q₁)).1)
      have key' := ((Fintype.bijective_iff_injective_and_card f).mpr ⟨hf, key'⟩).2
      obtain ⟨q, hq⟩ := key' ⟨l₁, hl₁⟩
      exact ⟨q, (congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mp (mkLine_ax (this q)).2, q.2⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
obtain : ∀ (f : L → P), Function.Bijective f → ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)), ∃ p ∈ l₁, p ∈ l₂
obtain₁ :
  ∀ (f : L → P),
    Function.Bijective f →
      ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)) (this : Nontrivial L), ∃ p ∈ l₁, p ∈ l₂
obtain₂ :
  ∀ (f : L → P),
    Function.Bijective f →
      ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)) (this : Nontrivial L) (this : Nontrivial P),
        ∃ p ∈ l₁, p ∈ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂
have obtain₃ : ∀ (f : L → P),
  Function.Bijective f →
    ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)) (this : Nontrivial L) (this : Nontrivial P)
      (h₁ : ∀ (p : P), 0 < lineCount L p), ∃ p ∈ l₁, p ∈ l₂
Tactic: classical
      obtain ⟨f, _, hf2⟩ := HasLines.exists_bijective_of_card_eq h
      haveI : Nontrivial L := ⟨⟨l₁, l₂, hl⟩⟩
      haveI := Fintype.one_lt_card_iff_nontrivial.mp ((congr_arg _ h).mpr Fintype.one_lt_card)
      have h₁ : ∀ p : P, 0 < lineCount L p := fun p =>
        Exists.elim (exists_ne p) fun q hq =>
          (congr_arg _ Nat.card_eq_fintype_card).mpr
            (Fintype.card_pos_iff.mpr ⟨⟨mkLine hq, (mkLine_ax hq).2⟩⟩)
      have h₂ : ∀ l : L, 0 < pointCount P l := fun l => (congr_arg _ (hf2 l)).mpr (h₁ (f l))
      obtain ⟨p, hl₁⟩ := Fintype.card_pos_iff.mp ((congr_arg _ Nat.card_eq_fintype_card).mp (h₂ l₁))
      by_cases hl₂ : p ∈ l₂
      · exact ⟨p, hl₁, hl₂⟩
      have key' : Fintype.card { q : P // q ∈ l₂ } = Fintype.card { l : L // p ∈ l } :=
        ((HasLines.lineCount_eq_pointCount h hl₂).trans Nat.card_eq_fintype_card).symm.trans
          Nat.card_eq_fintype_card
      have : ∀ q : { q // q ∈ l₂ }, p ≠ q := fun q hq => hl₂ ((congr_arg (· ∈ l₂) hq).mpr q.2)
      let f : { q : P // q ∈ l₂ } → { l : L // p ∈ l } := fun q =>
        ⟨mkLine (this q), (mkLine_ax (this q)).1⟩
      have hf : Function.Injective f := fun q₁ q₂ hq =>
        Subtype.ext ((eq_or_eq q₁.2 q₂.2 (mkLine_ax (this q₁)).2
            ((congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mpr (mkLine_ax (this q₂)).2)).resolve_right
            fun h => (congr_arg (p ∉ ·) h).mp hl₂ (mkLine_ax (this q₁)).1)
      have key' := ((Fintype.bijective_iff_injective_and_card f).mpr ⟨hf, key'⟩).2
      obtain ⟨q, hq⟩ := key' ⟨l₁, hl₁⟩
      exact ⟨q, (congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mp (mkLine_ax (this q)).2, q.2⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Configuration (goal)
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
obtain : ∀ (f : L → P), Function.Bijective f → ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)), ∃ p ∈ l₁, p ∈ l₂
obtain₁ :
  ∀ (f : L → P),
    Function.Bijective f →
      ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)) (this : Nontrivial L), ∃ p ∈ l₁, p ∈ l₂
obtain₂ :
  ∀ (f : L → P),
    Function.Bijective f →
      ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)) (this : Nontrivial L) (this : Nontrivial P),
        ∃ p ∈ l₁, p ∈ l₂
obtain₃ :
  ∀ (f : L → P),
    Function.Bijective f →
      ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)) (this : Nontrivial L) (this : Nontrivial P)
        (h₁ : ∀ (p : P), 0 < lineCount L p), ∃ p ∈ l₁, p ∈ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂
have obtain₄ : ∀ (f : L → P),
  Function.Bijective f →
    ∀ (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)) (this : Nontrivial L) (this : Nontrivial P)
      (h₁ : ∀ (p : P), 0 < lineCount L p) (h₂ : ∀ (l : L), 0 < pointCount P l), ∃ p ∈ l₁, p ∈ l₂
Tactic: classical
      obtain ⟨f, _, hf2⟩ := HasLines.exists_bijective_of_card_eq h
      haveI : Nontrivial L := ⟨⟨l₁, l₂, hl⟩⟩
      haveI := Fintype.one_lt_card_iff_nontrivial.mp ((congr_arg _ h).mpr Fintype.one_lt_card)
      have h₁ : ∀ p : P, 0 < lineCount L p := fun p =>
        Exists.elim (exists_ne p) fun q hq =>
          (congr_arg _ Nat.card_eq_fintype_card).mpr
            (Fintype.card_pos_iff.mpr ⟨⟨mkLine hq, (mkLine_ax hq).2⟩⟩)
      have h₂ : ∀ l : L, 0 < pointCount P l := fun l => (congr_arg _ (hf2 l)).mpr (h₁ (f l))
      obtain ⟨p, hl₁⟩ := Fintype.card_pos_iff.mp ((congr_arg _ Nat.card_eq_fintype_card).mp (h₂ l₁))
      by_cases hl₂ : p ∈ l₂
      · exact ⟨p, hl₁, hl₂⟩
      have key' : Fintype.card { q : P // q ∈ l₂ } = Fintype.card { l : L // p ∈ l } :=
        ((HasLines.lineCount_eq_pointCount h hl₂).trans Nat.card_eq_fintype_card).symm.trans
          Nat.card_eq_fintype_card
      have : ∀ q : { q // q ∈ l₂ }, p ≠ q := fun q hq => hl₂ ((congr_arg (· ∈ l₂) hq).mpr q.2)
      let f : { q : P // q ∈ l₂ } → { l : L // p ∈ l } := fun q =>
        ⟨mkLine (this q), (mkLine_ax (this q)).1⟩
      have hf : Function.Injective f := fun q₁ q₂ hq =>
        Subtype.ext ((eq_or_eq q₁.2 q₂.2 (mkLine_ax (this q₁)).2
            ((congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mpr (mkLine_ax (this q₂)).2)).resolve_right
            fun h => (congr_arg (p ∉ ·) h).mp hl₂ (mkLine_ax (this q₁)).1)
      have key' := ((Fintype.bijective_iff_injective_and_card f).mpr ⟨hf, key'⟩).2
      obtain ⟨q, hq⟩ := key' ⟨l₁, hl₁⟩
      exact ⟨q, (congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mp (mkLine_ax (this q)).2, q.2⟩