-- Module: Mathlib.Combinatorics.Configuration
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : Nondegenerate P L
inst_2 : Fintype P
inst_3 : Fintype L
h : Fintype.card L ≤ Fintype.card P
let let₁ let₂ : ∃ (f : L → P), Function.Injective f ∧ ∀ (l : L), f l ∉ l
let₃ let₄ :
  let t : L → Finset P := fun (l : L) ↦ Set.toFinset {p : P | p ∉ l};
  ∀ (s : Finset L), #s ≤ #(s.biUnion t)
pos :
  let t : L → Finset P := fun (l : L) ↦ Set.toFinset {p : P | p ∉ l};
  ∀ (s : Finset L) (hs₀ : #s = 0), #s ≤ #(s.biUnion t)
neg neg₁ :
  let t : L → Finset P := fun (l : L) ↦ Set.toFinset {p : P | p ∉ l};
  ∀ (s : Finset L) (hs₀ : ¬#s = 0), #s ≤ #(s.biUnion t)
pos₁ :
  let t : L → Finset P := fun (l : L) ↦ Set.toFinset {p : P | p ∉ l};
  ∀ (s : Finset L) (hs₀ : ¬#s = 0) (hs₁ : #s = 1), #s ≤ #(s.biUnion t)
neg₂ neg₃ :
  let t : L → Finset P := fun (l : L) ↦ Set.toFinset {p : P | p ∉ l};
  ∀ (s : Finset L) (hs₀ : ¬#s = 0) (hs₁ : ¬#s = 1), #s ≤ #(s.biUnion t)
neg₄ :
  let t : L → Finset P := fun (l : L) ↦ Set.toFinset {p : P | p ∉ l};
  ∀ (s : Finset L) (hs₀ : ¬#s = 0) (hs₁ : ¬#s = 1), #(s.biUnion t)ᶜ ≤ #sᶜ
neg₅ :
  let t : L → Finset P := fun (l : L) ↦ Set.toFinset {p : P | p ∉ l};
  ∀ (s : Finset L) (hs₀ : ¬#s = 0) (hs₁ : ¬#s = 1) (hs₂ : #(s.biUnion t)ᶜ ≤ 1), #(s.biUnion t)ᶜ ≤ #sᶜ
pos₂ :
  let t : L → Finset P := fun (l : L) ↦ Set.toFinset {p : P | p ∉ l};
  ∀ (s : Finset L) (hs₀ : ¬#s = 0) (hs₁ : ¬#s = 1) (hs₂ : #(s.biUnion t)ᶜ ≤ 1) (hs₃ : #sᶜ = 0), #(s.biUnion t)ᶜ ≤ #sᶜ
neg₆ neg₇ :
  let t : L → Finset P := fun (l : L) ↦ Set.toFinset {p : P | p ∉ l};
  ∀ (s : Finset L) (hs₀ : ¬#s = 0) (hs₁ : ¬#s = 1) (hs₂ : #(s.biUnion t)ᶜ ≤ 1) (hs₃ : ¬#sᶜ = 0), #(s.biUnion t)ᶜ ≤ #sᶜ
⊢ ∃ (f : L → P), Function.Injective f ∧ ∀ (l : L), f l ∉ l

Tactic:
classical
    let t : L → Finset P := fun l => Set.toFinset { p | p ∉ l }
    suffices ∀ s : Finset L, #s ≤ (s.biUnion t).card by
      -- Hall's marriage theorem
      obtain ⟨f, hf1, hf2⟩ := (Finset.all_card_le_biUnion_card_iff_exists_injective t).mp this
      exact ⟨f, hf1, fun l => Set.mem_toFinset.mp (hf2 l)⟩
    intro s
    by_cases hs₀ : #s = 0
    -- If `s = ∅`, then `#s = 0 ≤ #(s.bUnion t)`
    · simp_rw [hs₀, zero_le]
    by_cases hs₁ : #s = 1
    -- If `s = {l}`, then pick a point `p ∉ l`
    · obtain ⟨l, rfl⟩ := Finset.card_eq_one.mp hs₁
      obtain ⟨p, hl⟩ := exists_point (P := P) l
      rw [Finset.card_singleton, Finset.singleton_biUnion, Nat.one_le_iff_ne_zero]
      exact Finset.card_ne_zero_of_mem (Set.mem_toFinset.mpr hl)
    suffices #(s.biUnion t)ᶜ ≤ #sᶜ by
      -- Rephrase in terms of complements (uses `h`)
      rw [Finset.card_compl, Finset.card_compl, tsub_le_iff_left] at this
      replace := h.trans this
      rwa [← add_tsub_assoc_of_le s.card_le_univ, le_tsub_iff_left (le_add_left s.card_le_univ),
        add_le_add_iff_right] at this
    have hs₂ : #(s.biUnion t)ᶜ ≤ 1 := by
      -- At most one line through two points of `s`
      refine Finset.card_le_one_iff.mpr @fun p₁ p₂ hp₁ hp₂ => ?_
      simp_rw [t, Finset.mem_compl, Finset.mem_biUnion, not_exists, not_and,
        Set.mem_toFinset, Set.mem_setOf_eq, Classical.not_not] at hp₁ hp₂
      obtain ⟨l₁, l₂, hl₁, hl₂, hl₃⟩ :=
        Finset.one_lt_card_iff.mp (Nat.one_lt_iff_ne_zero_and_ne_one.mpr ⟨hs₀, hs₁⟩)
      exact (eq_or_eq (hp₁ l₁ hl₁) (hp₂ l₁ hl₁) (hp₁ l₂ hl₂) (hp₂ l₂ hl₂)).resolve_right hl₃
    by_cases hs₃ : #sᶜ = 0
    · rw [hs₃, Nat.le_zero]
      rw [Finset.card_compl, tsub_eq_zero_iff_le, (Finset.card_le_univ _).ge_iff_eq', eq_comm,
        Finset.card_eq_iff_eq_univ] at hs₃ ⊢
      rw [hs₃]
      rw [Finset.eq_univ_iff_forall] at hs₃ ⊢
      exact fun p =>
        Exists.elim (exists_line p) -- If `s = univ`, then show `s.bUnion t = univ`
        fun l hl => Finset.mem_biUnion.mpr ⟨l, Finset.mem_univ l, Set.mem_toFinset.mpr hl⟩
    · exact hs₂.trans (Nat.one_le_iff_ne_zero.mpr hs₃)

Premises used: Iff.mpr, zero_le, Finset.mem_univ, Finset.eq_univ_iff_forall, Finset.card_eq_iff_eq_univ, Finset.card_compl, Nat.le_zero, Finset.singleton_biUnion, tsub_eq_zero_iff_le, eq_comm, LE.le.ge_iff_eq', Finset.card_le_univ, And.intro, Exists.intro, Nat.one_le_iff_ne_zero, Set.mem_toFinset, Configuration.Nondegenerate.exists_line, Finset.mem_biUnion, rfl, Exists.elim, Finset.card_singleton