awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
p n k : ℕ
hp : Prime p
hkn : k ≤ p ^ n
hk0 : k ≠ 0
hdisj : Disjoint ({i ∈ Ico 1 n.succ | p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i}) ({i ∈ Ico 1 n.succ | p ^ i ∣ k})
⊢ LE.le (α := ℕ∞)
    (↑(#({i ∈ Ico 1 (log p (p ^ n)).succ | p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i}) +
          #({i ∈ Ico 1 (log p (p ^ n)).succ | p ^ i ∣ k})) :
      ℕ∞)
    (↑n : ℕ∞)
have a : #({i ∈ Ico 1 (log p (p ^ n)).succ | p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i}) +
    #({i ∈ Ico 1 (log p (p ^ n)).succ | p ^ i ∣ k}) ≤
  n
Tactic: apply WithTop.coe_mono
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
p n k : ℕ
hp : Prime p
hkn : k ≤ p ^ n
hk0 : k ≠ 0
hdisj : Disjoint ({i ∈ Ico 1 n.succ | p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i}) ({i ∈ Ico 1 n.succ | p ^ i ∣ k})
⊢ #({i ∈ Ico 1 (log p (p ^ n)).succ | p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i}) +
      #({i ∈ Ico 1 (log p (p ^ n)).succ | p ^ i ∣ k}) ≤
    n
have a : #({x ∈ Ico 1 n.succ | p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k}) ≤ n
Tactic: rw [log_pow hp.one_lt, ← card_union_of_disjoint hdisj, filter_union_right]
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (hypothesis)
Goal:
p n k : ℕ
hp : Prime p
hkn : k ≤ p ^ n
hk0 : k ≠ 0
hdisj : Disjoint ({i ∈ Ico 1 n.succ | p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i}) ({i ∈ Ico 1 n.succ | p ^ i ∣ k})
⊢ #({x ∈ Ico 1 n.succ | p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k}) ≤ n
have filter_le_Ico : #({x ∈ Ico 1 n.succ | p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k}) ≤ #(Ico 1 n.succ)
Tactic: have filter_le_Ico := (Ico 1 n.succ).card_filter_le
        fun x => p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
p n k : ℕ
hp : Prime p
hkn : k ≤ p ^ n
hk0 : k ≠ 0
hdisj : Disjoint ({i ∈ Ico 1 n.succ | p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i}) ({i ∈ Ico 1 n.succ | p ^ i ∣ k})
filter_le_Ico : #({x ∈ Ico 1 n.succ | p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k}) ≤ #(Ico 1 n.succ)
⊢ #({x ∈ Ico 1 n.succ | p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k}) ≤ n
have a : ∀ (filter_le_Ico : #({x ∈ Ico 1 n.succ | p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k}) ≤ n.succ - 1),
  #({x ∈ Ico 1 n.succ | p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k}) ≤ n
Tactic: rwa [card_Ico 1 n.succ] at filter_le_Ico
Cleared hypotheses: filter_le_Ico
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
p n k : ℕ
hp : Prime p
hkn : k ≤ p ^ n
hk0 : k ≠ 0
⊢ (↑n : ℕ∞) ≤ emultiplicity p ((p ^ n).choose k) + emultiplicity p k
have rw : emultiplicity p (p ^ n) ≤ emultiplicity p ((p ^ n).choose k) + emultiplicity p k
Tactic: rw [← hp.emultiplicity_pow_self]
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
p n k : ℕ
hp : Prime p
hkn : k ≤ p ^ n
hk0 : k ≠ 0
⊢ emultiplicity p ((p ^ n).choose k) = HSub.hSub (α := ℕ∞) (↑n : ℕ∞) (↑(multiplicity p k) : ℕ∞)
have rw : (↑(multiplicity p ((p ^ n).choose k)) : ℕ∞) =
  HAdd.hAdd (α := ℕ∞) (↑(multiplicity p ((p ^ n).choose k)) : ℕ∞) (↑(multiplicity p k) : ℕ∞) -
    (↑(multiplicity p k) : ℕ∞)
Tactic: rw [← emultiplicity_choose_prime_pow_add_emultiplicity hp hkn hk0,
    (finiteMultiplicity_iff.2 ⟨hp.ne_one, Nat.pos_of_ne_zero hk0⟩).emultiplicity_eq_multiplicity,
    (finiteMultiplicity_iff.2 ⟨hp.ne_one, choose_pos hkn⟩).emultiplicity_eq_multiplicity]
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
p n k : ℕ
hp : Prime p
hkn : k ≤ p ^ n
hk0 : k ≠ 0
⊢ (↑(multiplicity p ((p ^ n).choose k)) : ℕ∞) =
    HAdd.hAdd (α := ℕ∞) (↑(multiplicity p ((p ^ n).choose k)) : ℕ∞) (↑(multiplicity p k) : ℕ∞) -
      (↑(multiplicity p k) : ℕ∞)
have norm_cast : multiplicity p ((p ^ n).choose k) = multiplicity p ((p ^ n).choose k) + multiplicity p k - multiplicity p k
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
p n k : ℕ
hp : Prime p
hkn : k ≤ p ^ n
hk0 : k ≠ 0
⊢ multiplicity p ((p ^ n).choose k) = multiplicity p ((p ^ n).choose k) + multiplicity p k - multiplicity p k
have rw : multiplicity p ((p ^ n).choose k) = multiplicity p ((p ^ n).choose k)
Tactic: rw [Nat.add_sub_cancel_right]
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
p n k : ℕ
hp : Prime p
hk : k ≠ 0
hkp : k ≠ p ^ n
⊢ p ∣ (p ^ n).choose k
have inl : ∀ (hkp : p ^ n < k), p ∣ (p ^ n).choose k
Tactic: obtain hkp | hkp := hkp.symm.lt_or_gt
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
p n k : ℕ
hp : Prime p
hk : k ≠ 0
hkp : k ≠ p ^ n
inl : ∀ (hkp : p ^ n < k), p ∣ (p ^ n).choose k
⊢ p ∣ (p ^ n).choose k
have inr : ∀ (hkp : k < p ^ n), p ∣ (p ^ n).choose k
Tactic: obtain hkp | hkp := hkp.symm.lt_or_gt
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
p n k : ℕ
hp : Prime p
hk : k ≠ 0
hkp_1 : k ≠ p ^ n
hkp : k < p ^ n
⊢ p ∣ (p ^ n).choose k
have inr : ∀ (h : emultiplicity p ((p ^ n).choose k) = 0), p ^ n ∣ k
Tactic: refine emultiplicity_ne_zero.1 fun h => hkp.not_ge <| Nat.le_of_dvd hk.bot_lt ?_
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (hypothesis)
Goal:
p n k : ℕ
hp : Prime p
hk : k ≠ 0
hkp_1 : k ≠ p ^ n
hkp : k < p ^ n
h : emultiplicity p ((p ^ n).choose k) = 0
⊢ p ^ n ∣ k
have H : emultiplicity p ((p ^ n).choose k) + emultiplicity p k = (↑n : ℕ∞)
Tactic: have H := hp.emultiplicity_choose_prime_pow_add_emultiplicity hkp.le hk
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (hypothesis)
Goal:
p n k : ℕ
hp : Prime p
hk : k ≠ 0
hkp_1 : k ≠ p ^ n
hkp : k < p ^ n
h : emultiplicity p ((p ^ n).choose k) = 0
H : emultiplicity p ((p ^ n).choose k) + emultiplicity p k = (↑n : ℕ∞)
⊢ p ^ n ∣ k
have H₁ : p ^ n ∣ k ∧ ¬p ^ (n + 1) ∣ k
Tactic: rw [h, zero_add, emultiplicity_eq_coe] at H
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
p n k : ℕ
hp : Prime p
⊢ p ∣ (p ^ n).choose k ↔ k ≠ 0 ∧ k ≠ p ^ n
have refine_1 : ∀ (h : p ∣ (p ^ n).choose k), k ≠ 0
Tactic: refine ⟨fun h => ⟨?_, ?_⟩, fun h => dvd_choose_pow hp h.1 h.2⟩
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
p n k : ℕ
hp : Prime p
refine_1 : ∀ (h : p ∣ (p ^ n).choose k), k ≠ 0
⊢ p ∣ (p ^ n).choose k ↔ k ≠ 0 ∧ k ≠ p ^ n
have refine_2 : ∀ (h : p ∣ (p ^ n).choose k), k ≠ p ^ n
Tactic: refine ⟨fun h => ⟨?_, ?_⟩, fun h => dvd_choose_pow hp h.1 h.2⟩
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (hypothesis)
Goal:
⊢ ∀ {n : ℕ}, n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
have h2 : _root_.Prime (M := ℕ) 2
Tactic: have h2 := prime_two.prime
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
⊢ ∀ {n : ℕ}, n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
have refine_1 : Ne (α := ℕ) 0 0 → emultiplicity 2 0! < (↑0 : ℕ∞)
Tactic: refine binaryRec ?_ ?_
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
refine_1 : Ne (α := ℕ) 0 0 → emultiplicity 2 0! < (↑0 : ℕ∞)
⊢ ∀ {n : ℕ}, n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
have refine_2 : ∀ (b : Bool) (n : ℕ),
  (n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)) → bit b n ≠ 0 → emultiplicity 2 (bit b n)! < (↑(bit b n) : ℕ∞)
Tactic: refine binaryRec ?_ ?_
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
h : bit b n ≠ 0
⊢ emultiplicity 2 (bit b n)! < (↑(bit b n) : ℕ∞)
have pos : ∀ (hn : n = 0), emultiplicity 2 (bit b n)! < (↑(bit b n) : ℕ∞)
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
h : bit b n ≠ 0
pos : ∀ (hn : n = 0), emultiplicity 2 (bit b n)! < (↑(bit b n) : ℕ∞)
⊢ emultiplicity 2 (bit b n)! < (↑(bit b n) : ℕ∞)
have neg : ∀ (hn : ¬n = 0), emultiplicity 2 (bit b n)! < (↑(bit b n) : ℕ∞)
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
h : bit b n ≠ 0
hn : n = 0
⊢ emultiplicity 2 (bit b n)! < (↑(bit b n) : ℕ∞)
have pos : ∀ (ih : Ne (α := ℕ) 0 0 → emultiplicity 2 0! < (↑0 : ℕ∞)) (h : bit b 0 ≠ 0),
  emultiplicity 2 (bit b 0)! < (↑(bit b 0) : ℕ∞)
Tactic: subst hn
Cleared hypotheses: n, ih, h, hn
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (hypothesis)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
ih : Ne (α := ℕ) 0 0 → emultiplicity 2 0! < (↑0 : ℕ∞)
h : bit b 0 ≠ 0
⊢ emultiplicity 2 (bit b 0)! < (↑(bit b 0) : ℕ∞)
have h₁ : b = true
Tactic: simp only [ne_eq, bit_eq_zero_iff, true_and, Bool.not_eq_false] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
ih : Ne (α := ℕ) 0 0 → emultiplicity 2 0! < (↑0 : ℕ∞)
h : b = true
⊢ emultiplicity 2 (bit b 0)! < (↑(bit b 0) : ℕ∞)
have pos : emultiplicity (α := ℕ) 2 1 < (↑1 : ℕ∞)
Tactic: simp only [bit, h, cond_true, mul_zero, zero_add, factorial_one]
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
ih : Ne (α := ℕ) 0 0 → emultiplicity 2 0! < (↑0 : ℕ∞)
h : b = true
⊢ emultiplicity (α := ℕ) 2 1 < (↑1 : ℕ∞)
have pos : LT.lt (α := ℕ∞) 0 (↑1 : ℕ∞)
Tactic: rw [Prime.emultiplicity_one]
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
ih : Ne (α := ℕ) 0 0 → emultiplicity 2 0! < (↑0 : ℕ∞)
h : b = true
pos : LT.lt (α := ℕ∞) 0 (↑1 : ℕ∞)
⊢ emultiplicity (α := ℕ) 2 1 < (↑1 : ℕ∞)
have pos₁ : Prime 2
Tactic: rw [Prime.emultiplicity_one]
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
h : bit b n ≠ 0
hn : ¬n = 0
⊢ emultiplicity 2 (2 * n)! < (↑(2 * n) : ℕ∞)
have rw : emultiplicity 2 n ! + (↑n : ℕ∞) < (↑(2 * n) : ℕ∞)
Tactic: rw [prime_two.emultiplicity_factorial_mul]
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
h : bit b n ≠ 0
hn : ¬n = 0
⊢ emultiplicity 2 n ! + (↑n : ℕ∞) < (↑(2 * n) : ℕ∞)
have rw : emultiplicity 2 n ! + (↑n : ℕ∞) < (↑(n + n) : ℕ∞)
Tactic: rw [two_mul]
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
h : bit b n ≠ 0
hn : ¬n = 0
⊢ emultiplicity 2 n ! + (↑n : ℕ∞) < HAdd.hAdd (α := ℕ∞) (↑n : ℕ∞) (↑n : ℕ∞)
have apply : Ne (α := WithTop ℕ) (↑n : ℕ∞) ⊤
Tactic: apply WithTop.add_lt_add_right _ (ih hn)
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (hypothesis)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
h : bit b n ≠ 0
hn : ¬n = 0
⊢ emultiplicity 2 (bit b n)! < (↑(bit b n) : ℕ∞)
have this : emultiplicity 2 (2 * n)! < (↑(2 * n) : ℕ∞)
Tactic: have : emultiplicity 2 (2 * n)! < (2 * n : ℕ) := by
      rw [prime_two.emultiplicity_factorial_mul]
      rw [two_mul]
      push_cast
      apply WithTop.add_lt_add_right _ (ih hn)
      exact Ne.symm nofun
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
h : bit b n ≠ 0
hn : ¬n = 0
this : emultiplicity 2 (2 * n)! < (↑(2 * n) : ℕ∞)
⊢ emultiplicity 2 (bit b n)! < (↑(bit b n) : ℕ∞)
have neg : ∀ (h : bit false n ≠ 0), emultiplicity 2 (bit false n)! < (↑(bit false n) : ℕ∞)
Tactic: cases b
Cleared hypotheses: b, h
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
b : Bool
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
h : bit b n ≠ 0
hn : ¬n = 0
this : emultiplicity 2 (2 * n)! < (↑(2 * n) : ℕ∞)
neg : ∀ (h : bit false n ≠ 0), emultiplicity 2 (bit false n)! < (↑(bit false n) : ℕ∞)
⊢ emultiplicity 2 (bit b n)! < (↑(bit b n) : ℕ∞)
have neg₁ : ∀ (h : bit true n ≠ 0), emultiplicity 2 (bit true n)! < (↑(bit true n) : ℕ∞)
Tactic: cases b
Cleared hypotheses: b, h
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
hn : ¬n = 0
this : emultiplicity 2 (2 * n)! < (↑(2 * n) : ℕ∞)
h : bit true n ≠ 0
⊢ emultiplicity 2 (bit true n)! < (↑(bit true n) : ℕ∞)
have neg : emultiplicity 2 (2 * n + 1) + emultiplicity 2 (2 * n)! < HAdd.hAdd (α := ℕ∞) (↑(2 * n) : ℕ∞) 1
Tactic: suffices emultiplicity 2 (2 * n + 1) + emultiplicity 2 (2 * n)! < ↑(2 * n) + 1 by
        simpa [emultiplicity_mul, h2, prime_two, bit, factorial]
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
hn : ¬n = 0
this : emultiplicity 2 (2 * n)! < (↑(2 * n) : ℕ∞)
h : bit true n ≠ 0
⊢ emultiplicity 2 (2 * n + 1) + emultiplicity 2 (2 * n)! < HAdd.hAdd (α := ℕ∞) (↑(2 * n) : ℕ∞) 1
have neg : emultiplicity 2 (2 * n)! < HAdd.hAdd (α := ℕ∞) (↑(2 * n) : ℕ∞) 1
Tactic: rw [emultiplicity_eq_zero.2 (two_not_dvd_two_mul_add_one n), zero_add]
-- Lean 4 proof step from Mathlib.Data.Nat.Multiplicity (goal)
Goal:
h2 : _root_.Prime (M := ℕ) 2
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < (↑n : ℕ∞)
hn : ¬n = 0
this : emultiplicity 2 (2 * n)! < (↑(2 * n) : ℕ∞)
h : bit true n ≠ 0
⊢ emultiplicity 2 (2 * n)! < HAdd.hAdd (α := ℕ∞) (↑(2 * n) : ℕ∞) 1
have neg : (↑(2 * n) : ℕ∞) < HAdd.hAdd (α := ℕ∞) (↑(2 * n) : ℕ∞) 1
Tactic: refine this.trans ?_
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Finite
n : ℕ
hn : #(hf.toFinset (s := setOf p)) ≤ n
⊢ nth p n = 0
have hn₁ : (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).length ≤ n
Tactic: rw [nth, dif_pos hf, List.getD_eq_default]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Finite
n : ℕ
hn : n < #(hf.toFinset (s := setOf p))
⊢ nth p n =
    (hf.toFinset (s := setOf p).orderEmbOfFin ⋯ : (a : Fin #hf.toFinset) → ℕ)
      (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p)))
have rw : (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).getD n 0 =
  (hf.toFinset (s := setOf p).orderEmbOfFin sorry : (a : Fin #hf.toFinset) → ℕ)
    (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p)))
Tactic: rw [nth_eq_getD_sort hf, Finset.orderEmbOfFin_apply, List.getD_eq_getElem, Fin.getElem_fin]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Finite
n : ℕ
hn : n < #(hf.toFinset (s := setOf p))
rw :
  (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).getD n 0 =
    (hf.toFinset (s := setOf p).orderEmbOfFin ⋯ : (a : Fin #hf.toFinset) → ℕ)
      (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p)))
⊢ nth p n =
    (hf.toFinset (s := setOf p).orderEmbOfFin ⋯ : (a : Fin #hf.toFinset) → ℕ)
      (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p)))
have rw₁ : (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).getD n 0 =
  getElem (valid := fun (xs : List ℕ) (i : Fin #(hf.toFinset (s := setOf p))) ↦ (↑i : ℕ) < xs.length)
    (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b) (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p))) sorry
Tactic: rw [nth_eq_getD_sort hf, Finset.orderEmbOfFin_apply, List.getD_eq_getElem, Fin.getElem_fin]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Finite
n : ℕ
hn : n < #(hf.toFinset (s := setOf p))
rw :
  (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).getD n 0 =
    (hf.toFinset (s := setOf p).orderEmbOfFin ⋯ : (a : Fin #hf.toFinset) → ℕ)
      (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p)))
rw₁ :
  (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).getD n 0 =
    getElem (valid := fun (xs : List ℕ) (i : Fin #(hf.toFinset (s := setOf p))) ↦ (↑i : ℕ) < xs.length)
      (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b) (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p))) ⋯
⊢ nth p n =
    (hf.toFinset (s := setOf p).orderEmbOfFin ⋯ : (a : Fin #hf.toFinset) → ℕ)
      (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p)))
have rw₂ : getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b) n
    sorry =
  getElem (valid := fun (xs : List ℕ) (i : Fin #(hf.toFinset (s := setOf p))) ↦ (↑i : ℕ) < xs.length)
    (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b) (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p))) sorry
Tactic: rw [nth_eq_getD_sort hf, Finset.orderEmbOfFin_apply, List.getD_eq_getElem, Fin.getElem_fin]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Finite
n : ℕ
hn : n < #(hf.toFinset (s := setOf p))
rw :
  (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).getD n 0 =
    (hf.toFinset (s := setOf p).orderEmbOfFin ⋯ : (a : Fin #hf.toFinset) → ℕ)
      (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p)))
rw₁ :
  (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).getD n 0 =
    getElem (valid := fun (xs : List ℕ) (i : Fin #(hf.toFinset (s := setOf p))) ↦ (↑i : ℕ) < xs.length)
      (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b) (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p))) ⋯
rw₂ :
  getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b) n
      ⋯ =
    getElem (valid := fun (xs : List ℕ) (i : Fin #(hf.toFinset (s := setOf p))) ↦ (↑i : ℕ) < xs.length)
      (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b) (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p))) ⋯
⊢ nth p n =
    (hf.toFinset (s := setOf p).orderEmbOfFin ⋯ : (a : Fin #hf.toFinset) → ℕ)
      (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p)))
have hn₁ : n < (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).length
Tactic: rw [nth_eq_getD_sort hf, Finset.orderEmbOfFin_apply, List.getD_eq_getElem, Fin.getElem_fin]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Finite
n : ℕ
hn : n < #(hf.toFinset (s := setOf p))
rw :
  (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).getD n 0 =
    (hf.toFinset (s := setOf p).orderEmbOfFin ⋯ : (a : Fin #hf.toFinset) → ℕ)
      (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p)))
rw₁ :
  (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).getD n 0 =
    getElem (valid := fun (xs : List ℕ) (i : Fin #(hf.toFinset (s := setOf p))) ↦ (↑i : ℕ) < xs.length)
      (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b) (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p))) ⋯
rw₂ :
  getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b) n
      ⋯ =
    getElem (valid := fun (xs : List ℕ) (i : Fin #(hf.toFinset (s := setOf p))) ↦ (↑i : ℕ) < xs.length)
      (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b) (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p))) ⋯
hn₁ : n < (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).length
⊢ nth p n =
    (hf.toFinset (s := setOf p).orderEmbOfFin ⋯ : (a : Fin #hf.toFinset) → ℕ)
      (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p)))
have rw₃ : getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b) n
    sorry =
  getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (hf.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b)
    (↑(⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p))) : ℕ) sorry
Tactic: rw [nth_eq_getD_sort hf, Finset.orderEmbOfFin_apply, List.getD_eq_getElem, Fin.getElem_fin]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Finite
m : ℕ
hm : m < #(hf.toFinset (s := setOf p))
n : ℕ
hn : n < #(hf.toFinset (s := setOf p))
h : m < n
⊢ nth p m < nth p n
have simp : LT.lt (α := ℕ)
  ((hf.toFinset (s := setOf p).orderEmbOfFin sorry : (a : Fin #hf.toFinset) → ℕ)
      (⟨m, hm⟩ : Fin #(hf.toFinset (s := setOf p))) :
    ℕ)
  ((hf.toFinset (s := setOf p).orderEmbOfFin sorry : (a : Fin #hf.toFinset) → ℕ)
      (⟨n, hn⟩ : Fin #(hf.toFinset (s := setOf p))) :
    ℕ)
Tactic: simp only [nth_eq_orderEmbOfFin, *]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Finite
⊢ nth p '' Set.Iio #(hf.toFinset (s := setOf p)) =
    Set.range (ι := Fin #(hf.toFinset (s := setOf p)))
      (⇑(hf.toFinset (s := setOf p).orderEmbOfFin ⋯) : (a : Fin #hf.toFinset) → ℕ)
have h : ∀ (x : ℕ),
  x ∈ nth p '' Set.Iio #(hf.toFinset (s := setOf p)) ↔
    Membership.mem (γ := Set ℕ)
      (Set.range (ι := Fin #(hf.toFinset (s := setOf p)))
        (⇑(hf.toFinset (s := setOf p).orderEmbOfFin sorry) : (a : Fin #hf.toFinset) → ℕ))
      x
Tactic: ext x
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Finite
⊢ Set.range (ι := Fin #(hf.toFinset (s := setOf p)))
      (⇑(hf.toFinset (s := setOf p).orderEmbOfFin ⋯) : (a : Fin #hf.toFinset) → ℕ) =
    setOf p
have rw : (↑(hf.toFinset (s := setOf p)) : Set ℕ) = setOf p
Tactic: rw [range_orderEmbOfFin, Set.Finite.coe_toFinset]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Finite
rw : (↑(hf.toFinset (s := setOf p)) : Set ℕ) = setOf p
⊢ Set.range (ι := Fin #(hf.toFinset (s := setOf p)))
      (⇑(hf.toFinset (s := setOf p).orderEmbOfFin ⋯) : (a : Fin #hf.toFinset) → ℕ) =
    setOf p
have rw₁ : setOf p = setOf p
Tactic: rw [range_orderEmbOfFin, Set.Finite.coe_toFinset]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Finite
x : ℕ
h : p x
⊢ ∃ n < #(hf.toFinset (s := setOf p)), nth p n = x
have rwa : ∀ (h : Membership.mem (γ := Set ℕ) {y : ℕ | p y} x), ∃ n < #(hf.toFinset (s := setOf p)), nth p n = x
Tactic: rwa [← @Set.mem_setOf_eq _ _ p, ← image_nth_Iio_card hf] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Finite
x : ℕ
h : p x
rwa : ∀ (h : Membership.mem (γ := Set ℕ) {y : ℕ | p y} x), ∃ n < #(hf.toFinset (s := setOf p)), nth p n = x
⊢ ∃ n < #(hf.toFinset (s := setOf p)), nth p n = x
have rwa₁ : ∀ (h : x ∈ nth p '' Set.Iio #(hf.toFinset (s := setOf p))), ∃ n < #(hf.toFinset (s := setOf p)), nth p n = x
Tactic: rwa [← @Set.mem_setOf_eq _ _ p, ← image_nth_Iio_card hf] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Infinite
n : ℕ
⊢ nth p n = (↑((Subtype.orderIsoOfNat (setOf p) : (a : ℕ) → (↑(setOf p) : Type)) n) : ℕ)
have rw : Eq (α := ℕ)
  (if h : (setOf p).Finite then (h.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).getD n 0
  else (↑((Subtype.orderIsoOfNat (setOf p) : (a : ℕ) → (↑(setOf p) : Type)) n) : ℕ))
  (↑((Subtype.orderIsoOfNat (setOf p) : (a : ℕ) → (↑(setOf p) : Type)) n) : ℕ)
Tactic: rw [nth, dif_neg hf]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Infinite
n : ℕ
rw :
  Eq (α := ℕ)
    (if h : (setOf p).Finite then (h.toFinset (s := setOf p).sort fun (a b : ℕ) ↦ a ≤ b).getD n 0
    else (↑((Subtype.orderIsoOfNat (setOf p) : (a : ℕ) → (↑(setOf p) : Type)) n) : ℕ))
    (↑((Subtype.orderIsoOfNat (setOf p) : (a : ℕ) → (↑(setOf p) : Type)) n) : ℕ)
⊢ nth p n = (↑((Subtype.orderIsoOfNat (setOf p) : (a : ℕ) → (↑(setOf p) : Type)) n) : ℕ)
have rw₁ : Eq (α := ℕ) (↑((Subtype.orderIsoOfNat (setOf p) : (a : ℕ) → (↑(setOf p) : Type)) n) : ℕ)
  (↑((Subtype.orderIsoOfNat (setOf p) : (a : ℕ) → (↑(setOf p) : Type)) n) : ℕ)
Tactic: rw [nth, dif_neg hf]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Infinite
⊢ StrictMono (nth p)
have rw : StrictMono (α := ℕ) (β := ℕ)
  (Function.comp (β := (↑(setOf p) : Type)) (Subtype.val (p := fun (x : ℕ) ↦ x ∈ setOf p))
    (⇑(Subtype.orderIsoOfNat (setOf p)) : (a : ℕ) → (↑(setOf p) : Type)))
Tactic: rw [nth_eq_orderIsoOfNat hf]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (goal)
Goal:
p : ℕ → Prop
hf : (setOf p).Infinite
⊢ Set.range (nth p) = setOf p
have rw : Set.range (ι := ℕ)
    (Function.comp (β := (↑(setOf p) : Type)) (Subtype.val (p := fun (x : ℕ) ↦ x ∈ setOf p))
      (⇑(Subtype.orderIsoOfNat (setOf p)) : (a : ℕ) → (↑(setOf p) : Type))) =
  setOf p
Tactic: rw [nth_eq_orderIsoOfNat hf]
-- Lean 4 proof step from Mathlib.Data.Nat.Nth (hypothesis)
Goal:
p : ℕ → Prop
hf : (setOf p).Infinite
⊢ Set.range (ι := ℕ)
      (Function.comp (β := (↑(setOf p) : Type)) (Subtype.val (p := fun (x : ℕ) ↦ x ∈ setOf p))
        (⇑(Subtype.orderIsoOfNat (setOf p)) : (a : ℕ) → (↑(setOf p) : Type))) =
    setOf p
have this : Infinite (↑(setOf p) : Type)
Tactic: haveI := hf.to_subtype