awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
t : ι → Finset α
ι' : Finset ι
rw rw₁ rw₂ : Finite (↑{f : ↥ι' → α | Injective f ∧ ∀ (x : ↥ι'), f x ∈ t (↑x : ι)} : Type (max u v))
rw₃ : Finite (↑{f : ↥ι' → α | Injective f ∧ ∀ (x : ↥ι'), f x ∈ t (↑x : ι)} : Type (max u v))
rw₄ :
  let g : (↑(hallMatchingsOn t ι') : Type (max u v)) → ↥ι' → ↥(ι'.biUnion t) :=
    fun (f : (↑(hallMatchingsOn t ι') : Type (max u v))) (i : ↥ι') ↦
    Subtype.mk (p := fun (x : α) ↦ x ∈ ι'.biUnion t) ((↑f : ↥ι' → α) i) ⋯;
  Injective g
rw₅ :
  let g : (↑(hallMatchingsOn t ι') : Type (max u v)) → ↥ι' → ↥(ι'.biUnion t) :=
    fun (f : (↑(hallMatchingsOn t ι') : Type (max u v))) (i : ↥ι') ↦
    Subtype.mk (p := fun (x : α) ↦ x ∈ ι'.biUnion t) ((↑f : ↥ι' → α) i) ⋯;
  ∀ ⦃f f' : (↑(hallMatchingsOn t ι') : Type (max u v))⦄ (h : g f = g f'), f = f'
⊢ Finite (↑(hallMatchingsOn t ι') : Type (max u v))
have a : let g : (↑(hallMatchingsOn t ι') : Type (max u v)) → ↥ι' → ↥(ι'.biUnion t) :=
  fun (f : (↑(hallMatchingsOn t ι') : Type (max u v))) (i : ↥ι') ↦
  Subtype.mk (p := fun (x : α) ↦ x ∈ ι'.biUnion t) ((↑f : ↥ι' → α) i) sorry;
∀ ⦃f f' : (↑(hallMatchingsOn t ι') : Type (max u v))⦄ (h : g f = g f') (a : ↥ι'), (↑f : ↥ι' → α) a = (↑f' : ↥ι' → α) a
Tactic: classical
    rw [hallMatchingsOn]
    let g : hallMatchingsOn t ι' → ι' → ι'.biUnion t := by
      rintro f i
      refine ⟨f.val i, ?_⟩
      rw [mem_biUnion]
      exact ⟨i, i.property, f.property.2 i⟩
    apply Finite.of_injective g
    intro f f' h
    ext a
    rw [funext_iff] at h
    simpa [g] using h a
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
t : ι → Finset α
ι' : Finset ι
rw rw₁ rw₂ : Finite (↑{f : ↥ι' → α | Injective f ∧ ∀ (x : ↥ι'), f x ∈ t (↑x : ι)} : Type (max u v))
rw₃ : Finite (↑{f : ↥ι' → α | Injective f ∧ ∀ (x : ↥ι'), f x ∈ t (↑x : ι)} : Type (max u v))
rw₄ :
  let g : (↑(hallMatchingsOn t ι') : Type (max u v)) → ↥ι' → ↥(ι'.biUnion t) :=
    fun (f : (↑(hallMatchingsOn t ι') : Type (max u v))) (i : ↥ι') ↦
    Subtype.mk (p := fun (x : α) ↦ x ∈ ι'.biUnion t) ((↑f : ↥ι' → α) i) ⋯;
  Injective g
rw₅ :
  let g : (↑(hallMatchingsOn t ι') : Type (max u v)) → ↥ι' → ↥(ι'.biUnion t) :=
    fun (f : (↑(hallMatchingsOn t ι') : Type (max u v))) (i : ↥ι') ↦
    Subtype.mk (p := fun (x : α) ↦ x ∈ ι'.biUnion t) ((↑f : ↥ι' → α) i) ⋯;
  ∀ ⦃f f' : (↑(hallMatchingsOn t ι') : Type (max u v))⦄ (h : g f = g f'), f = f'
a :
  let g : (↑(hallMatchingsOn t ι') : Type (max u v)) → ↥ι' → ↥(ι'.biUnion t) :=
    fun (f : (↑(hallMatchingsOn t ι') : Type (max u v))) (i : ↥ι') ↦
    Subtype.mk (p := fun (x : α) ↦ x ∈ ι'.biUnion t) ((↑f : ↥ι' → α) i) ⋯;
  ∀ ⦃f f' : (↑(hallMatchingsOn t ι') : Type (max u v))⦄ (h : g f = g f') (a : ↥ι'), (↑f : ↥ι' → α) a = (↑f' : ↥ι' → α) a
⊢ Finite (↑(hallMatchingsOn t ι') : Type (max u v))
have a₁ : let g : (↑(hallMatchingsOn t ι') : Type (max u v)) → ↥ι' → ↥(ι'.biUnion t) :=
  fun (f : (↑(hallMatchingsOn t ι') : Type (max u v))) (i : ↥ι') ↦
  Subtype.mk (p := fun (x : α) ↦ x ∈ ι'.biUnion t) ((↑f : ↥ι' → α) i) sorry;
∀ ⦃f f' : (↑(hallMatchingsOn t ι') : Type (max u v))⦄ (h : ∀ (x : ↥ι'), g f x = g f' x) (a : ↥ι'),
  (↑f : ↥ι' → α) a = (↑f' : ↥ι' → α) a
Tactic: classical
    rw [hallMatchingsOn]
    let g : hallMatchingsOn t ι' → ι' → ι'.biUnion t := by
      rintro f i
      refine ⟨f.val i, ?_⟩
      rw [mem_biUnion]
      exact ⟨i, i.property, f.property.2 i⟩
    apply Finite.of_injective g
    intro f f' h
    ext a
    rw [funext_iff] at h
    simpa [g] using h a
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
⊢ (∀ (s : Finset ι), #s ≤ #(s.biUnion t)) ↔ ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
have mp : (∀ (s : Finset ι), #s ≤ #(s.biUnion t)) → ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
Tactic: constructor
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
mp : (∀ (s : Finset ι), #s ≤ #(s.biUnion t)) → ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
⊢ (∀ (s : Finset ι), #s ≤ #(s.biUnion t)) ↔ ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
have mpr : (∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x) → ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (hypothesis)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
⊢ ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
have this : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
Tactic: haveI : ∀ ι' : (Finset ι)ᵒᵖ, Nonempty ((hallMatchingsFunctor t).obj ι') := fun ι' =>
      hallMatchingsOn.nonempty t h ι'.unop
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (hypothesis)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
⊢ ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
have this₁ : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
Tactic: haveI : ∀ ι' : (Finset ι)ᵒᵖ, Finite ((hallMatchingsFunctor t).obj ι') := by
        intro ι'
        rw [hallMatchingsFunctor]
        infer_instance
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this_1 : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
⊢ ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
have mp : ∀ (u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j) (hu : u ∈ (hallMatchingsFunctor t).sections),
  ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
Tactic: obtain ⟨u, hu⟩ := nonempty_sections_of_finite_inverse_system (hallMatchingsFunctor t)
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this_1 : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
⊢ ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
have mp : ι → α
Tactic: refine ⟨?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this_1 : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
mp : ι → α
⊢ ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
have mp₁ : Injective.{u + 1, v + 1} (β := α) fun (i : ι) ↦
  (↑(u (Opposite.op {i})) : ↥(Opposite.unop (Opposite.op {i})) → α)
    (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) (Opposite.unop (Opposite.op {i})) x) i sorry)
Tactic: refine ⟨?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this_1 : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
mp : ι → α
mp₁ :
  Injective.{u + 1, v + 1} (β := α) fun (i : ι) ↦
    (↑(u (Opposite.op {i})) : ↥(Opposite.unop (Opposite.op {i})) → α)
      (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) (Opposite.unop (Opposite.op {i})) x) i ⋯)
⊢ ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
have mp₂ : ∀ (x : ι),
  Membership.mem (α := α) (t x)
    ((↑(u (Opposite.op {x})) : ↥(Opposite.unop (Opposite.op {x})) → α)
        (Subtype.mk (p := fun (x_1 : ι) ↦ Membership.mem (γ := Finset ι) (Opposite.unop (Opposite.op {x})) x_1) x sorry) :
      α)
Tactic: refine ⟨?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (hypothesis)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this_1 : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
i i' : ι
⊢ Eq.{v + 1} (α := α)
      ((fun (i : ι) ↦
          ((↑(u (Opposite.op {i})) : ↥(Opposite.unop (Opposite.op {i})) → α)
              (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) (Opposite.unop (Opposite.op {i})) x) i ⋯) :
            α))
        i)
      ((fun (i : ι) ↦
          ((↑(u (Opposite.op {i})) : ↥(Opposite.unop (Opposite.op {i})) → α)
              (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) (Opposite.unop (Opposite.op {i})) x) i ⋯) :
            α))
        i') →
    i = i'
have subi : Subset (α := Finset ι) {i} {i, i'}
Tactic: have subi : ({i} : Finset ι) ⊆ {i, i'} := by simp
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (hypothesis)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this_1 : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
i i' : ι
subi : Subset (α := Finset ι) {i} {i, i'}
⊢ Eq.{v + 1} (α := α)
      ((fun (i : ι) ↦
          ((↑(u (Opposite.op {i})) : ↥(Opposite.unop (Opposite.op {i})) → α)
              (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) (Opposite.unop (Opposite.op {i})) x) i ⋯) :
            α))
        i)
      ((fun (i : ι) ↦
          ((↑(u (Opposite.op {i})) : ↥(Opposite.unop (Opposite.op {i})) → α)
              (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) (Opposite.unop (Opposite.op {i})) x) i ⋯) :
            α))
        i') →
    i = i'
have subi' : Subset (α := Finset ι) {i'} {i, i'}
Tactic: have subi' : ({i'} : Finset ι) ⊆ {i, i'} := by simp
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (hypothesis)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this_1 : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
i i' : ι
subi : Subset (α := Finset ι) {i} {i, i'}
subi' : Subset (α := Finset ι) {i'} {i, i'}
⊢ Eq.{v + 1} (α := α)
      ((fun (i : ι) ↦
          ((↑(u (Opposite.op {i})) : ↥(Opposite.unop (Opposite.op {i})) → α)
              (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) (Opposite.unop (Opposite.op {i})) x) i ⋯) :
            α))
        i)
      ((fun (i : ι) ↦
          ((↑(u (Opposite.op {i})) : ↥(Opposite.unop (Opposite.op {i})) → α)
              (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) (Opposite.unop (Opposite.op {i})) x) i ⋯) :
            α))
        i') →
    i = i'
have subi₁ : LE.le (α := Finset ι) {i} {i, i'}
Tactic: rw [← Finset.le_iff_subset] at subi subi'
Cleared hypotheses: subi, subi'
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (hypothesis)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this_1 : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
i i' : ι
subi : Subset (α := Finset ι) {i} {i, i'}
subi' : Subset (α := Finset ι) {i'} {i, i'}
subi₁ : LE.le (α := Finset ι) {i} {i, i'}
⊢ Eq.{v + 1} (α := α)
      ((fun (i : ι) ↦
          ((↑(u (Opposite.op {i})) : ↥(Opposite.unop (Opposite.op {i})) → α)
              (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) (Opposite.unop (Opposite.op {i})) x) i ⋯) :
            α))
        i)
      ((fun (i : ι) ↦
          ((↑(u (Opposite.op {i})) : ↥(Opposite.unop (Opposite.op {i})) → α)
              (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) (Opposite.unop (Opposite.op {i})) x) i ⋯) :
            α))
        i') →
    i = i'
have subi'₁ : LE.le (α := Finset ι) {i'} {i, i'}
Tactic: rw [← Finset.le_iff_subset] at subi subi'
Cleared hypotheses: subi, subi'
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this_1 : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
i i' : ι
subi : LE.le (α := Finset ι) {i} {i, i'}
subi' : LE.le (α := Finset ι) {i'} {i, i'}
⊢ Eq.{v + 1} (α := α)
      ((↑(u (Opposite.op {i})) : ↥{i} → α) (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) {i} x) i ⋯) :
        α)
      ((↑(u (Opposite.op {i'})) : ↥{i'} → α)
          (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) {i'} x) i' ⋯) :
        α) →
    i = i'
have mp : Eq.{v + 1} (α := α)
    ((↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((hallMatchingsFunctor t).map (homOfLE subi).op) :
              (a : (fun (X : Type (max v u)) ↦ X) ((hallMatchingsFunctor t).obj (Opposite.op {i, i'}))) →
                (fun (X : Type (max v u)) ↦ X) ((hallMatchingsFunctor t).obj (Opposite.op {i})))
            (u (Opposite.op {i, i'}))) :
        ↥{i} → α)
      (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) {i} x) i sorry))
    ((↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((hallMatchingsFunctor t).map (homOfLE subi').op) :
              (a : (fun (X : Type (max v u)) ↦ X) ((hallMatchingsFunctor t).obj (Opposite.op {i, i'}))) →
                (fun (X : Type (max v u)) ↦ X) ((hallMatchingsFunctor t).obj (Opposite.op {i'})))
            (u (Opposite.op {i, i'}))) :
        ↥{i'} → α)
      (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) {i'} x) i' sorry)) →
  i = i'
Tactic: rw [← hu (CategoryTheory.homOfLE subi).op, ← hu (CategoryTheory.homOfLE subi').op]
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
⊢ ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
have mp : ∀ (this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')),
  ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
Tactic: classical
      haveI : ∀ ι' : (Finset ι)ᵒᵖ, Finite ((hallMatchingsFunctor t).obj ι') := by
        intro ι'
        rw [hallMatchingsFunctor]
        infer_instance
      -- Apply the compactness argument
      obtain ⟨u, hu⟩ := nonempty_sections_of_finite_inverse_system (hallMatchingsFunctor t)
      -- Interpret the resulting section of the inverse limit
      refine ⟨?_, ?_, ?_⟩
      · -- Build the matching function from the section
        exact fun i =>
          (u (Opposite.op ({i} : Finset ι))).val ⟨i, by simp only [mem_singleton]⟩
      · -- Show that it is injective
        intro i i'
        have subi : ({i} : Finset ι) ⊆ {i, i'} := by simp
        have subi' : ({i'} : Finset ι) ⊆ {i, i'} := by simp
        rw [← Finset.le_iff_subset] at subi subi'
        simp only
        rw [← hu (CategoryTheory.homOfLE subi).op, ← hu (CategoryTheory.homOfLE subi').op]
        let uii' := u (Opposite.op ({i, i'} : Finset ι))
        exact fun h => Subtype.mk_eq_mk.mp (uii'.property.1 h)
      · -- Show that it maps each index to the corresponding finite set
        intro i
        apply (u (Opposite.op ({i} : Finset ι))).property.2
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
mp mp₁ mp₂ :
  ∀ (this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')),
    ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
⊢ ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
have mp₃ : ∀ (this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι'))
  (u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j) (hu : u ∈ (hallMatchingsFunctor t).sections),
  ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
Tactic: classical
      haveI : ∀ ι' : (Finset ι)ᵒᵖ, Finite ((hallMatchingsFunctor t).obj ι') := by
        intro ι'
        rw [hallMatchingsFunctor]
        infer_instance
      -- Apply the compactness argument
      obtain ⟨u, hu⟩ := nonempty_sections_of_finite_inverse_system (hallMatchingsFunctor t)
      -- Interpret the resulting section of the inverse limit
      refine ⟨?_, ?_, ?_⟩
      · -- Build the matching function from the section
        exact fun i =>
          (u (Opposite.op ({i} : Finset ι))).val ⟨i, by simp only [mem_singleton]⟩
      · -- Show that it is injective
        intro i i'
        have subi : ({i} : Finset ι) ⊆ {i, i'} := by simp
        have subi' : ({i'} : Finset ι) ⊆ {i, i'} := by simp
        rw [← Finset.le_iff_subset] at subi subi'
        simp only
        rw [← hu (CategoryTheory.homOfLE subi).op, ← hu (CategoryTheory.homOfLE subi').op]
        let uii' := u (Opposite.op ({i, i'} : Finset ι))
        exact fun h => Subtype.mk_eq_mk.mp (uii'.property.1 h)
      · -- Show that it maps each index to the corresponding finite set
        intro i
        apply (u (Opposite.op ({i} : Finset ι))).property.2
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
mp mp₁ mp₂ :
  ∀ (this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')),
    ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
mp₃ :
  ∀ (this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι'))
    (u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j) (hu : u ∈ (hallMatchingsFunctor t).sections),
    ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
⊢ ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
have mp₄ : (this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')) →
  (u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j) → (hu : u ∈ (hallMatchingsFunctor t).sections) → ι → α
Tactic: classical
      haveI : ∀ ι' : (Finset ι)ᵒᵖ, Finite ((hallMatchingsFunctor t).obj ι') := by
        intro ι'
        rw [hallMatchingsFunctor]
        infer_instance
      -- Apply the compactness argument
      obtain ⟨u, hu⟩ := nonempty_sections_of_finite_inverse_system (hallMatchingsFunctor t)
      -- Interpret the resulting section of the inverse limit
      refine ⟨?_, ?_, ?_⟩
      · -- Build the matching function from the section
        exact fun i =>
          (u (Opposite.op ({i} : Finset ι))).val ⟨i, by simp only [mem_singleton]⟩
      · -- Show that it is injective
        intro i i'
        have subi : ({i} : Finset ι) ⊆ {i, i'} := by simp
        have subi' : ({i'} : Finset ι) ⊆ {i, i'} := by simp
        rw [← Finset.le_iff_subset] at subi subi'
        simp only
        rw [← hu (CategoryTheory.homOfLE subi).op, ← hu (CategoryTheory.homOfLE subi').op]
        let uii' := u (Opposite.op ({i, i'} : Finset ι))
        exact fun h => Subtype.mk_eq_mk.mp (uii'.property.1 h)
      · -- Show that it maps each index to the corresponding finite set
        intro i
        apply (u (Opposite.op ({i} : Finset ι))).property.2
-- Lean 4 proof step from Mathlib.Combinatorics.Hall.Basic (goal)
Goal:
ι : Type u
α : Type v
inst : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
mp mp₁ mp₂ :
  ∀ (this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')),
    ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
mp₃ :
  ∀ (this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι'))
    (u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j) (hu : u ∈ (hallMatchingsFunctor t).sections),
    ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
mp₄ :
  (this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')) →
    (u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j) → (hu : u ∈ (hallMatchingsFunctor t).sections) → ι → α
⊢ ∃ (f : ι → α), Injective f ∧ ∀ (x : ι), f x ∈ t x
have mp₅ : ∀ (this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι'))
  (u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j) (hu : u ∈ (hallMatchingsFunctor t).sections),
  Injective.{u + 1, v + 1} (β := α) fun (i : ι) ↦
    (↑(u (Opposite.op {i})) : ↥(Opposite.unop (Opposite.op {i})) → α)
      (Subtype.mk (p := fun (x : ι) ↦ Membership.mem (γ := Finset ι) (Opposite.unop (Opposite.op {i})) x) i sorry)
Tactic: classical
      haveI : ∀ ι' : (Finset ι)ᵒᵖ, Finite ((hallMatchingsFunctor t).obj ι') := by
        intro ι'
        rw [hallMatchingsFunctor]
        infer_instance
      -- Apply the compactness argument
      obtain ⟨u, hu⟩ := nonempty_sections_of_finite_inverse_system (hallMatchingsFunctor t)
      -- Interpret the resulting section of the inverse limit
      refine ⟨?_, ?_, ?_⟩
      · -- Build the matching function from the section
        exact fun i =>
          (u (Opposite.op ({i} : Finset ι))).val ⟨i, by simp only [mem_singleton]⟩
      · -- Show that it is injective
        intro i i'
        have subi : ({i} : Finset ι) ⊆ {i, i'} := by simp
        have subi' : ({i'} : Finset ι) ⊆ {i, i'} := by simp
        rw [← Finset.le_iff_subset] at subi subi'
        simp only
        rw [← hu (CategoryTheory.homOfLE subi).op, ← hu (CategoryTheory.homOfLE subi').op]
        let uii' := u (Opposite.op ({i, i'} : Finset ι))
        exact fun h => Subtype.mk_eq_mk.mp (uii'.property.1 h)
      · -- Show that it maps each index to the corresponding finite set
        intro i
        apply (u (Opposite.op ({i} : Finset ι))).property.2