awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
f : α → β
inst : DecidableEq β
y : β
rw : 0 < #({x ∈ s | f x = y}) ↔ y ∈ image f s
rw₁ : {x ∈ s | f x = y}.Nonempty ↔ y ∈ image f s
⊢ #({x ∈ s | f x = y}) ≠ 0 ↔ y ∈ image f s
have rw₂ : y ∈ image f s ↔ y ∈ image f s
Tactic: rw [← Nat.pos_iff_ne_zero, card_pos, fiber_nonempty_iff_mem_image]
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
p : α → Prop
inst : DecidablePred p
⊢ #(filter p s) = #s ↔ ∀ x ∈ s, p x
have rw : #s ≤ #(filter p s) ↔ ∀ x ∈ s, p x
Tactic: rw [← (card_filter_le s p).ge_iff_eq, eq_iff_card_le_of_subset (filter_subset p s),
    filter_eq_self]
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
p : α → Prop
inst : DecidablePred p
rw : #s ≤ #(filter p s) ↔ ∀ x ∈ s, p x
⊢ #(filter p s) = #s ↔ ∀ x ∈ s, p x
have rw₁ : filter p s = s ↔ ∀ x ∈ s, p x
Tactic: rw [← (card_filter_le s p).ge_iff_eq, eq_iff_card_le_of_subset (filter_subset p s),
    filter_eq_self]
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
p : α → Prop
inst : DecidablePred p
rw : #s ≤ #(filter p s) ↔ ∀ x ∈ s, p x
rw₁ : filter p s = s ↔ ∀ x ∈ s, p x
⊢ #(filter p s) = #s ↔ ∀ x ∈ s, p x
have rw₂ : (∀ x ∈ s, p x) ↔ ∀ x ∈ s, p x
Tactic: rw [← (card_filter_le s p).ge_iff_eq, eq_iff_card_le_of_subset (filter_subset p s),
    filter_eq_self]
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
p : α → Prop
inst : DecidablePred p
⊢ #(filter p s) = 0 ↔ ∀ x ∈ s, ¬p x
have rw : filter p s = ∅ ↔ ∀ x ∈ s, ¬p x
Tactic: rw [card_eq_zero, filter_eq_empty_iff]
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
p : α → Prop
inst : DecidablePred p
rw : filter p s = ∅ ↔ ∀ x ∈ s, ¬p x
⊢ #(filter p s) = 0 ↔ ∀ x ∈ s, ¬p x
have rw₁ : (∀ ⦃x : α⦄, x ∈ s → ¬p x) ↔ ∀ x ∈ s, ¬p x
Tactic: rw [card_eq_zero, filter_eq_empty_iff]
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
⊢ s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach
have h : ∀ (a : α), a ∈ s ↔ Membership.mem (γ := Finset α) (image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (range n).attach) a
Tactic: ext a
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
a : α
⊢ a ∈ s ↔ Membership.mem (γ := Finset α) (image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach) a
have h : a ∈ s ↔ ∃ (i : ℕ) (hi : i ∈ range n), f i sorry = a
Tactic: suffices _ : a ∈ s ↔ ∃ (i : _) (hi : i ∈ range n), f i (mem_range.1 hi) = a by
      simpa only [mem_image, mem_attach, true_and, Subtype.exists]
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
a : α
⊢ a ∈ s ↔ ∃ (i : ℕ) (hi : i ∈ range n), f i ⋯ = a
have h : a ∈ s → ∃ (i : ℕ) (hi : i ∈ range n), f i sorry = a
Tactic: constructor
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
a : α
h : a ∈ s → ∃ (i : ℕ) (hi : i ∈ range n), f i ⋯ = a
⊢ a ∈ s ↔ ∃ (i : ℕ) (hi : i ∈ range n), f i ⋯ = a
have h₁ : (∃ (i : ℕ) (hi : i ∈ range n), f i sorry = a) → a ∈ s
Tactic: constructor
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
a : α
ha : a ∈ s
⊢ ∃ (i : ℕ) (hi : i ∈ range n), f i ⋯ = a
have h : ∀ (i : ℕ) (hi : i < n) (ha : f i hi ∈ s), ∃ (i_1 : ℕ) (hi_1 : i_1 ∈ range n), f i_1 sorry = f i hi
Tactic: obtain ⟨i, hi, rfl⟩ := hf a ha
Cleared hypotheses: a, ha
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
i : ℕ
hi : i < n
ha : f i hi ∈ s
w : ℕ
⊢ ∃ (i_1 : ℕ) (hi_1 : i_1 ∈ range n), f i_1 ⋯ = f i hi
have w₁ : i ∈ range n
Tactic: use i, mem_range.2 hi
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
i : ℕ
hi : i < n
ha : f i hi ∈ s
w : ℕ
w₁ : i ∈ range n
⊢ ∃ (i_1 : ℕ) (hi_1 : i_1 ∈ range n), f i_1 ⋯ = f i hi
have h : f i sorry = f i hi
Tactic: use i, mem_range.2 hi
-- Lean 4 proof step from Mathlib.Data.Finset.Card (hypothesis)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
⊢ #s = n
have this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (range n).attach
Tactic: have : s = (range n).attach.image fun i => f i.1 (mem_range.1 i.2) := by
    ext a
    suffices _ : a ∈ s ↔ ∃ (i : _) (hi : i ∈ range n), f i (mem_range.1 hi) = a by
      simpa only [mem_image, mem_attach, true_and, Subtype.exists]
    constructor
    · intro ha; obtain ⟨i, hi, rfl⟩ := hf a ha; use i, mem_range.2 hi
    · rintro ⟨i, hi, rfl⟩; apply hf'
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach
⊢ #s = card.{u_1} (α := α) (image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach)
have rw : card.{u_1} (α := α) (image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (range n).attach) =
  card.{u_1} (α := α) (image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (range n).attach)
Tactic: rw [this]
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach
⊢ #s = n
have fun : card.{u_1} (α := α) (image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (range n).attach) = #(range n).attach
Tactic: calc
    #s = #((range n).attach.image fun i => f i.1 (mem_range.1 i.2)) := by rw [this]
    _ = #(range n).attach := ?_
    _ = #(range n) := card_attach
    _ = n := card_range n
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach
⊢ card.{u_1} (α := α) (image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach) = #(range n).attach
have H : Injective.{1, u_1 + 1} (β := α) fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry
Tactic: apply card_image_of_injective
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach
i : ℕ
hi : i ∈ range n
⊢ ∀ ⦃a₂ : ↥(range n)⦄,
    Eq.{u_1 + 1} (α := α) ((fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (Subtype.mk (p := fun (x : ℕ) ↦ x ∈ range n) i hi))
        ((fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) a₂) →
      Subtype.mk (p := fun (x : ℕ) ↦ x ∈ range n) i hi = a₂
have H : ∀ (j : ℕ) (hj : j ∈ range n),
  Eq.{u_1 + 1} (α := α) ((fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (Subtype.mk (p := fun (x : ℕ) ↦ x ∈ range n) i hi))
      ((fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (Subtype.mk (p := fun (x : ℕ) ↦ x ∈ range n) j hj)) →
    Subtype.mk (p := fun (x : ℕ) ↦ x ∈ range n) i hi = Subtype.mk (p := fun (x : ℕ) ↦ x ∈ range n) j hj
Tactic: ⟨j, hj⟩
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
⊢ #s = n
have have : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (range n).attach
Tactic: classical
  have : s = (range n).attach.image fun i => f i.1 (mem_range.1 i.2) := by
    ext a
    suffices _ : a ∈ s ↔ ∃ (i : _) (hi : i ∈ range n), f i (mem_range.1 hi) = a by
      simpa only [mem_image, mem_attach, true_and, Subtype.exists]
    constructor
    · intro ha; obtain ⟨i, hi, rfl⟩ := hf a ha; use i, mem_range.2 hi
    · rintro ⟨i, hi, rfl⟩; apply hf'
  calc
    #s = #((range n).attach.image fun i => f i.1 (mem_range.1 i.2)) := by rw [this]
    _ = #(range n).attach := ?_
    _ = #(range n) := card_attach
    _ = n := card_range n
  apply card_image_of_injective
  intro ⟨i, hi⟩ ⟨j, hj⟩ eq
  exact Subtype.ext <| f_inj i j (mem_range.1 hi) (mem_range.1 hj) eq
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
have : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach
⊢ #s = n
have have₁ : ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (range n).attach), #s = n
Tactic: classical
  have : s = (range n).attach.image fun i => f i.1 (mem_range.1 i.2) := by
    ext a
    suffices _ : a ∈ s ↔ ∃ (i : _) (hi : i ∈ range n), f i (mem_range.1 hi) = a by
      simpa only [mem_image, mem_attach, true_and, Subtype.exists]
    constructor
    · intro ha; obtain ⟨i, hi, rfl⟩ := hf a ha; use i, mem_range.2 hi
    · rintro ⟨i, hi, rfl⟩; apply hf'
  calc
    #s = #((range n).attach.image fun i => f i.1 (mem_range.1 i.2)) := by rw [this]
    _ = #(range n).attach := ?_
    _ = #(range n) := card_attach
    _ = n := card_range n
  apply card_image_of_injective
  intro ⟨i, hi⟩ ⟨j, hj⟩ eq
  exact Subtype.ext <| f_inj i j (mem_range.1 hi) (mem_range.1 hj) eq
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
have : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach
have₁ : ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach), #s = n
have₂ : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach
have₃ have₄ : ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach), #s = n
⊢ #s = n
have have₅ : ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (range n).attach),
  card.{u_1} (α := α) (image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (range n).attach) = #(range n).attach
Tactic: classical
  have : s = (range n).attach.image fun i => f i.1 (mem_range.1 i.2) := by
    ext a
    suffices _ : a ∈ s ↔ ∃ (i : _) (hi : i ∈ range n), f i (mem_range.1 hi) = a by
      simpa only [mem_image, mem_attach, true_and, Subtype.exists]
    constructor
    · intro ha; obtain ⟨i, hi, rfl⟩ := hf a ha; use i, mem_range.2 hi
    · rintro ⟨i, hi, rfl⟩; apply hf'
  calc
    #s = #((range n).attach.image fun i => f i.1 (mem_range.1 i.2)) := by rw [this]
    _ = #(range n).attach := ?_
    _ = #(range n) := card_attach
    _ = n := card_range n
  apply card_image_of_injective
  intro ⟨i, hi⟩ ⟨j, hj⟩ eq
  exact Subtype.ext <| f_inj i j (mem_range.1 hi) (mem_range.1 hj) eq
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
have : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach
have₁ : ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach), #s = n
have₂ : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach
have₃ have₄ : ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach), #s = n
have₅ :
  ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach),
    card.{u_1} (α := α) (image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach) = #(range n).attach
⊢ #s = n
have H : ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (range n).attach),
  Injective.{1, u_1 + 1} (β := α) fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry
Tactic: classical
  have : s = (range n).attach.image fun i => f i.1 (mem_range.1 i.2) := by
    ext a
    suffices _ : a ∈ s ↔ ∃ (i : _) (hi : i ∈ range n), f i (mem_range.1 hi) = a by
      simpa only [mem_image, mem_attach, true_and, Subtype.exists]
    constructor
    · intro ha; obtain ⟨i, hi, rfl⟩ := hf a ha; use i, mem_range.2 hi
    · rintro ⟨i, hi, rfl⟩; apply hf'
  calc
    #s = #((range n).attach.image fun i => f i.1 (mem_range.1 i.2)) := by rw [this]
    _ = #(range n).attach := ?_
    _ = #(range n) := card_attach
    _ = n := card_range n
  apply card_image_of_injective
  intro ⟨i, hi⟩ ⟨j, hj⟩ eq
  exact Subtype.ext <| f_inj i j (mem_range.1 hi) (mem_range.1 hj) eq
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
s : Finset α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ (i : ℕ) (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
have : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach
have₁ : ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach), #s = n
have₂ : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach
have₃ have₄ : ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach), #s = n
have₅ :
  ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach),
    card.{u_1} (α := α) (image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach) = #(range n).attach
H :
  ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯) (range n).attach),
    Injective.{1, u_1 + 1} (β := α) fun (i : ↥(range n)) ↦ f (↑i : ℕ) ⋯
⊢ #s = n
have H₁ : ∀ (this : s = image (fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (range n).attach) (i : ℕ) (hi : i ∈ range n) (j : ℕ)
  (hj : j ∈ range n)
  (eq :
    Eq.{u_1 + 1} (α := α) ((fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (Subtype.mk (p := fun (x : ℕ) ↦ x ∈ range n) i hi))
      ((fun (i : ↥(range n)) ↦ f (↑i : ℕ) sorry) (Subtype.mk (p := fun (x : ℕ) ↦ x ∈ range n) j hj))),
  Subtype.mk (p := fun (x : ℕ) ↦ x ∈ range n) i hi = Subtype.mk (p := fun (x : ℕ) ↦ x ∈ range n) j hj
Tactic: classical
  have : s = (range n).attach.image fun i => f i.1 (mem_range.1 i.2) := by
    ext a
    suffices _ : a ∈ s ↔ ∃ (i : _) (hi : i ∈ range n), f i (mem_range.1 hi) = a by
      simpa only [mem_image, mem_attach, true_and, Subtype.exists]
    constructor
    · intro ha; obtain ⟨i, hi, rfl⟩ := hf a ha; use i, mem_range.2 hi
    · rintro ⟨i, hi, rfl⟩; apply hf'
  calc
    #s = #((range n).attach.image fun i => f i.1 (mem_range.1 i.2)) := by rw [this]
    _ = #(range n).attach := ?_
    _ = #(range n) := card_attach
    _ = n := card_range n
  apply card_image_of_injective
  intro ⟨i, hi⟩ ⟨j, hj⟩ eq
  exact Subtype.ext <| f_inj i j (mem_range.1 hi) (mem_range.1 hj) eq
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ (a : α) (ha : a ∈ s), i a ha = b
⊢ #s = #t
have calc_1 : card.{u_2} (α := β) (image (fun (a : ↥s) ↦ i (↑a : α) sorry) s.attach) = #s.attach
Tactic: calc
    #s = #s.attach := card_attach.symm
    _ = #(s.attach.image fun a ↦ i a.1 a.2) := Eq.symm ?_
    _ = #t := ?_
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ (a : α) (ha : a ∈ s), i a ha = b
calc_1 : card.{u_2} (α := β) (image (fun (a : ↥s) ↦ i (↑a : α) ⋯) s.attach) = #s.attach
⊢ #s = #t
have calc_2 : card.{u_2} (α := β) (image (fun (a : ↥s) ↦ i (↑a : α) sorry) s.attach) = #t
Tactic: calc
    #s = #s.attach := card_attach.symm
    _ = #(s.attach.image fun a ↦ i a.1 a.2) := Eq.symm ?_
    _ = #t := ?_
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ (a : α) (ha : a ∈ s), i a ha = b
⊢ card.{u_2} (α := β) (image (fun (a : ↥s) ↦ i (↑a : α) ⋯) s.attach) = #s.attach
have calc_1 : Injective.{u_1 + 1, u_2 + 1} (β := β) fun (a : ↥s) ↦ i (↑a : α) sorry
Tactic: apply card_image_of_injective
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ (a : α) (ha : a ∈ s), i a ha = b
val : α
property : val ∈ s
⊢ ∀ ⦃a₂ : ↥s⦄,
    Eq.{u_2 + 1} (α := β) ((fun (a : ↥s) ↦ i (↑a : α) ⋯) (Subtype.mk (p := fun (x : α) ↦ x ∈ s) val property))
        ((fun (a : ↥s) ↦ i (↑a : α) ⋯) a₂) →
      Subtype.mk (p := fun (x : α) ↦ x ∈ s) val property = a₂
have calc_1 : ∀ (val_1 : α) (property_1 : val_1 ∈ s),
  Eq.{u_2 + 1} (α := β) ((fun (a : ↥s) ↦ i (↑a : α) sorry) (Subtype.mk (p := fun (x : α) ↦ x ∈ s) val property))
      ((fun (a : ↥s) ↦ i (↑a : α) sorry) (Subtype.mk (p := fun (x : α) ↦ x ∈ s) val_1 property_1)) →
    Subtype.mk (p := fun (x : α) ↦ x ∈ s) val property = Subtype.mk (p := fun (x : α) ↦ x ∈ s) val_1 property_1
Tactic: ⟨_, _⟩
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ (a : α) (ha : a ∈ s), i a ha = b
⊢ card.{u_2} (α := β) (image (fun (a : ↥s) ↦ i (↑a : α) ⋯) s.attach) = #t
have calc_2 : image (fun (a : ↥s) ↦ i (↑a : α) sorry) s.attach = t
Tactic: congr 1
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ (a : α) (ha : a ∈ s), i a ha = b
⊢ image (fun (a : ↥s) ↦ i (↑a : α) ⋯) s.attach = t
have calc_2 : ∀ (b : β), Membership.mem (γ := Finset β) (image (fun (a : ↥s) ↦ i (↑a : α) sorry) s.attach) b ↔ b ∈ t
Tactic: ext b
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ (a : α) (ha : a ∈ s), i a ha = b
b : β
⊢ Membership.mem (γ := Finset β) (image (fun (a : ↥s) ↦ i (↑a : α) ⋯) s.attach) b ↔ b ∈ t
have calc_2 : Membership.mem (γ := Finset β) (image (fun (a : ↥s) ↦ i (↑a : α) sorry) s.attach) b → b ∈ t
Tactic: constructor
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ (a : α) (ha : a ∈ s), i a ha = b
b : β
calc_2 : Membership.mem (γ := Finset β) (image (fun (a : ↥s) ↦ i (↑a : α) ⋯) s.attach) b → b ∈ t
⊢ Membership.mem (γ := Finset β) (image (fun (a : ↥s) ↦ i (↑a : α) ⋯) s.attach) b ↔ b ∈ t
have calc_2₁ : b ∈ t → Membership.mem (γ := Finset β) (image (fun (a : ↥s) ↦ i (↑a : α) sorry) s.attach) b
Tactic: constructor
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ (a : α) (ha : a ∈ s), i a ha = b
b : β
h : Membership.mem (γ := Finset β) (image (fun (a : ↥s) ↦ i (↑a : α) ⋯) s.attach) b
⊢ b ∈ t
have calc_2 : ∀ w ∈ s.attach,
  ∀ (h : Membership.mem (γ := Finset β) (image (fun (a : ↥s) ↦ i (↑a : α) sorry) s.attach) (i (↑w : α) sorry)), i (↑w : α) sorry ∈ t
Tactic: obtain ⟨_, _, rfl⟩ := mem_image.1 h
Cleared hypotheses: b, h
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ (a : α) (ha : a ∈ s), i a ha = b
b : β
h : b ∈ t
⊢ Membership.mem (γ := Finset β) (image (fun (a : ↥s) ↦ i (↑a : α) ⋯) s.attach) b
have calc_2 : ∀ (a : α) (ha : a ∈ s) (h : i a ha ∈ t),
  Membership.mem (γ := Finset β) (image (fun (a : ↥s) ↦ i (↑a : α) sorry) s.attach) (i a ha)
Tactic: obtain ⟨a, ha, rfl⟩ := i_surj b h
Cleared hypotheses: b, h
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ (a : α) (ha : a ∈ s), i a ha = b
⊢ #s = #t
have calc_1 : card.{u_2} (α := β) (image (fun (a : ↥s) ↦ i (↑a : α) sorry) s.attach) = #s.attach
Tactic: classical
  calc
    #s = #s.attach := card_attach.symm
    _ = #(s.attach.image fun a ↦ i a.1 a.2) := Eq.symm ?_
    _ = #t := ?_
  · apply card_image_of_injective
    intro ⟨_, _⟩ ⟨_, _⟩ h
    simpa using i_inj _ _ _ _ h
  · congr 1
    ext b
    constructor <;> intro h
    · obtain ⟨_, _, rfl⟩ := mem_image.1 h; apply hi
    · obtain ⟨a, ha, rfl⟩ := i_surj b h; exact mem_image.2 ⟨⟨a, ha⟩, by simp⟩
-- Lean 4 proof step from Mathlib.Data.Finset.Card (goal)
Goal:
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ (a : α) (ha : a ∈ s), i a ha = b
calc_1 : card.{u_2} (α := β) (image (fun (a : ↥s) ↦ i (↑a : α) ⋯) s.attach) = #s.attach
⊢ #s = #t
have calc_2 : card.{u_2} (α := β) (image (fun (a : ↥s) ↦ i (↑a : α) sorry) s.attach) = #t
Tactic: classical
  calc
    #s = #s.attach := card_attach.symm
    _ = #(s.attach.image fun a ↦ i a.1 a.2) := Eq.symm ?_
    _ = #t := ?_
  · apply card_image_of_injective
    intro ⟨_, _⟩ ⟨_, _⟩ h
    simpa using i_inj _ _ _ _ h
  · congr 1
    ext b
    constructor <;> intro h
    · obtain ⟨_, _, rfl⟩ := mem_image.1 h; apply hi
    · obtain ⟨a, ha, rfl⟩ := i_surj b h; exact mem_image.2 ⟨⟨a, ha⟩, by simp⟩