l3lab/ntp-mathlib huggingface dataset:
theorem nth_of_card_le (hf : (setOf p).Finite) {n : ℕ} (hn : hf.toFinset.card ≤ n) : nth p n = 0 :=
  by rw [nth, dif_pos hf, List.getD_eq_default]; 

Proof state:
case hn
p : ℕ → Prop
hf : Set.Finite (setOf p)
n : ℕ
hn : card (Set.Finite.toFinset hf) ≤ n
⊢ List.length (sort (fun x x_1 => x ≤ x_1) (Set.Finite.toFinset hf)) ≤ n

Next tactic:
rwa [Finset.length_sort]
theorem nth_eq_orderEmbOfFin (hf : (setOf p).Finite) {n : ℕ} (hn : n < hf.toFinset.card) :
    nth p n = hf.toFinset.orderEmbOfFin rfl ⟨n, hn⟩ := by
  

Proof state:
p : ℕ → Prop
hf : Set.Finite (setOf p)
n : ℕ
hn : n < card (Set.Finite.toFinset hf)
⊢ nth p n =
    (orderEmbOfFin (Set.Finite.toFinset hf) (_ : card (Set.Finite.toFinset hf) = card (Set.Finite.toFinset hf)))
      { val := n, isLt := hn }

Next tactic:
rw [nth_eq_getD_sort hf, Finset.orderEmbOfFin_apply, List.getD_eq_get]
theorem nth_eq_orderEmbOfFin (hf : (setOf p).Finite) {n : ℕ} (hn : n < hf.toFinset.card) :
    nth p n = hf.toFinset.orderEmbOfFin rfl ⟨n, hn⟩ := by
  rw [nth_eq_getD_sort hf, Finset.orderEmbOfFin_apply, List.getD_eq_get]; 

Proof state:
p : ℕ → Prop
hf : Set.Finite (setOf p)
n : ℕ
hn : n < card (Set.Finite.toFinset hf)
⊢ List.get (sort (fun x x_1 => x ≤ x_1) (Set.Finite.toFinset hf)) { val := n, isLt := ?hn } =
    List.nthLe (sort (fun x x_1 => x ≤ x_1) (Set.Finite.toFinset hf)) ↑{ val := n, isLt := hn }
      (_ : ↑{ val := n, isLt := hn } < List.length (sort (fun x x_1 => x ≤ x_1) (Set.Finite.toFinset hf)))
case hn
p : ℕ → Prop
hf : Set.Finite (setOf p)
n : ℕ
hn : n < card (Set.Finite.toFinset hf)
⊢ n < List.length (sort (fun x x_1 => x ≤ x_1) (Set.Finite.toFinset hf))

Next tactic:
rfl
theorem nth_strictMonoOn (hf : (setOf p).Finite) :
    StrictMonoOn (nth p) (Set.Iio hf.toFinset.card) := by
  

Proof state:
p : ℕ → Prop
hf : Set.Finite (setOf p)
⊢ StrictMonoOn (nth p) (Set.Iio (card (Set.Finite.toFinset hf)))

Next tactic:
rintro m (hm : m < _) n (hn : n < _) h
theorem nth_strictMonoOn (hf : (setOf p).Finite) :
    StrictMonoOn (nth p) (Set.Iio hf.toFinset.card) := by
  rintro m (hm : m < _) n (hn : n < _) h
  

Proof state:
p : ℕ → Prop
hf : Set.Finite (setOf p)
m : ℕ
hm : m < card (Set.Finite.toFinset hf)
n : ℕ
hn : n < card (Set.Finite.toFinset hf)
h : m < n
⊢ nth p m < nth p n

Next tactic:
simp only [nth_eq_orderEmbOfFin, *]
theorem nth_strictMonoOn (hf : (setOf p).Finite) :
    StrictMonoOn (nth p) (Set.Iio hf.toFinset.card) := by
  rintro m (hm : m < _) n (hn : n < _) h
  simp only [nth_eq_orderEmbOfFin, *]
  

Proof state:
p : ℕ → Prop
hf : Set.Finite (setOf p)
m : ℕ
hm : m < card (Set.Finite.toFinset hf)
n : ℕ
hn : n < card (Set.Finite.toFinset hf)
h : m < n
⊢ (orderEmbOfFin (Set.Finite.toFinset (_ : Set.Finite (setOf p)))
        (_ :
          card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) =
            card (Set.Finite.toFinset (_ : Set.Finite (setOf p)))))
      { val := m, isLt := (_ : m < card (Set.Finite.toFinset hf)) } <
    (orderEmbOfFin (Set.Finite.toFinset (_ : Set.Finite (setOf p)))
        (_ :
          card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) =
            card (Set.Finite.toFinset (_ : Set.Finite (setOf p)))))
      { val := n, isLt := (_ : n < card (Set.Finite.toFinset hf)) }

Next tactic:
exact OrderEmbedding.strictMono _ h
theorem range_nth_of_finite (hf : (setOf p).Finite) : Set.range (nth p) = insert 0 (setOf p) := by
  

Proof state:
p : ℕ → Prop
hf : Set.Finite (setOf p)
⊢ Set.range (nth p) = insert 0 (setOf p)

Next tactic:
simpa only [← nth_eq_getD_sort hf, mem_sort, Set.Finite.mem_toFinset]
    using Set.range_list_getD (hf.toFinset.sort (· ≤ ·)) 0
@[simp]
theorem image_nth_Iio_card (hf : (setOf p).Finite) : nth p '' Set.Iio hf.toFinset.card = setOf p :=
  calc
    nth p '' Set.Iio hf.toFinset.card = Set.range (hf.toFinset.orderEmbOfFin rfl) := by
      

Proof state:
p : ℕ → Prop
hf : Set.Finite (setOf p)
⊢ nth p '' Set.Iio (card (Set.Finite.toFinset hf)) =
    Set.range
      ⇑(orderEmbOfFin (Set.Finite.toFinset hf) (_ : card (Set.Finite.toFinset hf) = card (Set.Finite.toFinset hf)))

Next tactic:
ext x
@[simp]
theorem image_nth_Iio_card (hf : (setOf p).Finite) : nth p '' Set.Iio hf.toFinset.card = setOf p :=
  calc
    nth p '' Set.Iio hf.toFinset.card = Set.range (hf.toFinset.orderEmbOfFin rfl) := by
      ext x
      

Proof state:
case h
p : ℕ → Prop
hf : Set.Finite (setOf p)
x : ℕ
⊢ x ∈ nth p '' Set.Iio (card (Set.Finite.toFinset hf)) ↔
    x ∈
      Set.range
        ⇑(orderEmbOfFin (Set.Finite.toFinset hf) (_ : card (Set.Finite.toFinset hf) = card (Set.Finite.toFinset hf)))

Next tactic:
simp only [Set.mem_image, Set.mem_range, Fin.exists_iff, ← nth_eq_orderEmbOfFin hf,
        Set.mem_Iio, exists_prop]
@[simp]
theorem image_nth_Iio_card (hf : (setOf p).Finite) : nth p '' Set.Iio hf.toFinset.card = setOf p :=
  calc
    nth p '' Set.Iio hf.toFinset.card = Set.range (hf.toFinset.orderEmbOfFin rfl) := by
      ext x
      simp only [Set.mem_image, Set.mem_range, Fin.exists_iff, ← nth_eq_orderEmbOfFin hf,
        Set.mem_Iio, exists_prop]
    _ = setOf p := by 

Proof state:
p : ℕ → Prop
hf : Set.Finite (setOf p)
⊢ Set.range
      ⇑(orderEmbOfFin (Set.Finite.toFinset hf) (_ : card (Set.Finite.toFinset hf) = card (Set.Finite.toFinset hf))) =
    setOf p

Next tactic:
rw [range_orderEmbOfFin, Set.Finite.coe_toFinset]
theorem exists_lt_card_finite_nth_eq (hf : (setOf p).Finite) {x} (h : p x) :
    ∃ n, n < hf.toFinset.card ∧ nth p n = x := by
  

Proof state:
p : ℕ → Prop
hf : Set.Finite (setOf p)
x : ℕ
h : p x
⊢ ∃ n < card (Set.Finite.toFinset hf), nth p n = x

Next tactic:
rwa [← @Set.mem_setOf_eq _ _ p, ← image_nth_Iio_card hf] at h
/-- When `s` is an infinite set, `nth` agrees with `Nat.Subtype.orderIsoOfNat`. -/
theorem nth_apply_eq_orderIsoOfNat (hf : (setOf p).Infinite) (n : ℕ) :
    nth p n = @Nat.Subtype.orderIsoOfNat (setOf p) hf.to_subtype n := by 

Proof state:
p : ℕ → Prop
hf : Set.Infinite (setOf p)
n : ℕ
⊢ nth p n = ↑((Subtype.orderIsoOfNat (setOf p)) n)

Next tactic:
rw [nth, dif_neg hf]
theorem nth_strictMono (hf : (setOf p).Infinite) : StrictMono (nth p) := by
  

Proof state:
p : ℕ → Prop
hf : Set.Infinite (setOf p)
⊢ StrictMono (nth p)

Next tactic:
rw [nth_eq_orderIsoOfNat hf]
theorem nth_strictMono (hf : (setOf p).Infinite) : StrictMono (nth p) := by
  rw [nth_eq_orderIsoOfNat hf]
  

Proof state:
p : ℕ → Prop
hf : Set.Infinite (setOf p)
⊢ StrictMono (Subtype.val ∘ ⇑(Subtype.orderIsoOfNat (setOf p)))

Next tactic:
exact (Subtype.strictMono_coe _).comp (OrderIso.strictMono _)
theorem range_nth_of_infinite (hf : (setOf p).Infinite) : Set.range (nth p) = setOf p := by
  

Proof state:
p : ℕ → Prop
hf : Set.Infinite (setOf p)
⊢ Set.range (nth p) = setOf p

Next tactic:
rw [nth_eq_orderIsoOfNat hf]
theorem range_nth_of_infinite (hf : (setOf p).Infinite) : Set.range (nth p) = setOf p := by
  rw [nth_eq_orderIsoOfNat hf]
  

Proof state:
p : ℕ → Prop
hf : Set.Infinite (setOf p)
⊢ Set.range (Subtype.val ∘ ⇑(Subtype.orderIsoOfNat (setOf p))) = setOf p

Next tactic:
haveI := hf.to_subtype
theorem range_nth_of_infinite (hf : (setOf p).Infinite) : Set.range (nth p) = setOf p := by
  rw [nth_eq_orderIsoOfNat hf]
  haveI := hf.to_subtype
  -- porting note: added `classical`; probably, Lean 3 found instance by unification
  

Proof state:
p : ℕ → Prop
hf : Set.Infinite (setOf p)
this : Infinite ↑(setOf p)
⊢ Set.range (Subtype.val ∘ ⇑(Subtype.orderIsoOfNat (setOf p))) = setOf p

Next tactic:
classical exact Nat.Subtype.coe_comp_ofNat_range
theorem range_nth_of_infinite (hf : (setOf p).Infinite) : Set.range (nth p) = setOf p := by
  rw [nth_eq_orderIsoOfNat hf]
  haveI := hf.to_subtype
  -- porting note: added `classical`; probably, Lean 3 found instance by unification
  classical 

Proof state:
p : ℕ → Prop
hf : Set.Infinite (setOf p)
this : Infinite ↑(setOf p)
⊢ Set.range (Subtype.val ∘ ⇑(Subtype.orderIsoOfNat (setOf p))) = setOf p

Next tactic:
exact Nat.Subtype.coe_comp_ofNat_range
theorem exists_lt_card_nth_eq {x} (h : p x) :
    ∃ n, (∀ hf : (setOf p).Finite, n < hf.toFinset.card) ∧ nth p n = x := by
  

Proof state:
p : ℕ → Prop
x : ℕ
h : p x
⊢ ∃ n, (∀ (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)) ∧ nth p n = x

Next tactic:
refine' (setOf p).finite_or_infinite.elim (fun hf => _) fun hf => _
theorem exists_lt_card_nth_eq {x} (h : p x) :
    ∃ n, (∀ hf : (setOf p).Finite, n < hf.toFinset.card) ∧ nth p n = x := by
  refine' (setOf p).finite_or_infinite.elim (fun hf => _) fun hf => _
  · 

Proof state:
case refine'_1
p : ℕ → Prop
x : ℕ
h : p x
hf : Set.Finite (setOf p)
⊢ ∃ n, (∀ (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)) ∧ nth p n = x

Next tactic:
rcases exists_lt_card_finite_nth_eq hf h with ⟨n, hn, hx⟩
theorem exists_lt_card_nth_eq {x} (h : p x) :
    ∃ n, (∀ hf : (setOf p).Finite, n < hf.toFinset.card) ∧ nth p n = x := by
  refine' (setOf p).finite_or_infinite.elim (fun hf => _) fun hf => _
  · rcases exists_lt_card_finite_nth_eq hf h with ⟨n, hn, hx⟩
    

Proof state:
case refine'_1.intro.intro
p : ℕ → Prop
x : ℕ
h : p x
hf : Set.Finite (setOf p)
n : ℕ
hn : n < card (Set.Finite.toFinset hf)
hx : nth p n = x
⊢ ∃ n, (∀ (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)) ∧ nth p n = x

Next tactic:
exact ⟨n, fun _ => hn, hx⟩
theorem exists_lt_card_nth_eq {x} (h : p x) :
    ∃ n, (∀ hf : (setOf p).Finite, n < hf.toFinset.card) ∧ nth p n = x := by
  refine' (setOf p).finite_or_infinite.elim (fun hf => _) fun hf => _
  · rcases exists_lt_card_finite_nth_eq hf h with ⟨n, hn, hx⟩
    exact ⟨n, fun _ => hn, hx⟩
  · 

Proof state:
case refine'_2
p : ℕ → Prop
x : ℕ
h : p x
hf : Set.Infinite (setOf p)
⊢ ∃ n, (∀ (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)) ∧ nth p n = x

Next tactic:
rw [← @Set.mem_setOf_eq _ _ p, ← range_nth_of_infinite hf] at h
theorem exists_lt_card_nth_eq {x} (h : p x) :
    ∃ n, (∀ hf : (setOf p).Finite, n < hf.toFinset.card) ∧ nth p n = x := by
  refine' (setOf p).finite_or_infinite.elim (fun hf => _) fun hf => _
  · rcases exists_lt_card_finite_nth_eq hf h with ⟨n, hn, hx⟩
    exact ⟨n, fun _ => hn, hx⟩
  · rw [← @Set.mem_setOf_eq _ _ p, ← range_nth_of_infinite hf] at h
    

Proof state:
case refine'_2
p : ℕ → Prop
x : ℕ
h : x ∈ Set.range (nth p)
hf : Set.Infinite (setOf p)
⊢ ∃ n, (∀ (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)) ∧ nth p n = x

Next tactic:
rcases h with ⟨n, hx⟩
theorem exists_lt_card_nth_eq {x} (h : p x) :
    ∃ n, (∀ hf : (setOf p).Finite, n < hf.toFinset.card) ∧ nth p n = x := by
  refine' (setOf p).finite_or_infinite.elim (fun hf => _) fun hf => _
  · rcases exists_lt_card_finite_nth_eq hf h with ⟨n, hn, hx⟩
    exact ⟨n, fun _ => hn, hx⟩
  · rw [← @Set.mem_setOf_eq _ _ p, ← range_nth_of_infinite hf] at h
    rcases h with ⟨n, hx⟩
    

Proof state:
case refine'_2.intro
p : ℕ → Prop
x : ℕ
hf : Set.Infinite (setOf p)
n : ℕ
hx : nth p n = x
⊢ ∃ n, (∀ (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)) ∧ nth p n = x

Next tactic:
exact ⟨n, fun hf' => absurd hf' hf, hx⟩
/-- An alternative recursive definition of `Nat.nth`: `Nat.nth s n` is the infimum of `x ∈ s` such
that `Nat.nth s k < x` for all `k < n`, if this set is nonempty. We do not assume that the set is
nonempty because we use the same "garbage value" `0` both for `sInf` on `ℕ` and for `Nat.nth s n`
for `n ≥ card s`. -/
theorem nth_eq_sInf (p : ℕ → Prop) (n : ℕ) : nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x} := by
  

Proof state:
p✝ p : ℕ → Prop
n : ℕ
⊢ nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x}

Next tactic:
by_cases hn : ∀ hf : (setOf p).Finite, n < hf.toFinset.card
/-- An alternative recursive definition of `Nat.nth`: `Nat.nth s n` is the infimum of `x ∈ s` such
that `Nat.nth s k < x` for all `k < n`, if this set is nonempty. We do not assume that the set is
nonempty because we use the same "garbage value" `0` both for `sInf` on `ℕ` and for `Nat.nth s n`
for `n ≥ card s`. -/
theorem nth_eq_sInf (p : ℕ → Prop) (n : ℕ) : nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x} := by
  by_cases hn : ∀ hf : (setOf p).Finite, n < hf.toFinset.card
  · 

Proof state:
case pos
p✝ p : ℕ → Prop
n : ℕ
hn : ∀ (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)
⊢ nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x}

Next tactic:
exact (isLeast_nth hn).csInf_eq.symm
/-- An alternative recursive definition of `Nat.nth`: `Nat.nth s n` is the infimum of `x ∈ s` such
that `Nat.nth s k < x` for all `k < n`, if this set is nonempty. We do not assume that the set is
nonempty because we use the same "garbage value" `0` both for `sInf` on `ℕ` and for `Nat.nth s n`
for `n ≥ card s`. -/
theorem nth_eq_sInf (p : ℕ → Prop) (n : ℕ) : nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x} := by
  by_cases hn : ∀ hf : (setOf p).Finite, n < hf.toFinset.card
  · exact (isLeast_nth hn).csInf_eq.symm
  · 

Proof state:
case neg
p✝ p : ℕ → Prop
n : ℕ
hn : ¬∀ (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)
⊢ nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x}

Next tactic:
push_neg at hn
/-- An alternative recursive definition of `Nat.nth`: `Nat.nth s n` is the infimum of `x ∈ s` such
that `Nat.nth s k < x` for all `k < n`, if this set is nonempty. We do not assume that the set is
nonempty because we use the same "garbage value" `0` both for `sInf` on `ℕ` and for `Nat.nth s n`
for `n ≥ card s`. -/
theorem nth_eq_sInf (p : ℕ → Prop) (n : ℕ) : nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x} := by
  by_cases hn : ∀ hf : (setOf p).Finite, n < hf.toFinset.card
  · exact (isLeast_nth hn).csInf_eq.symm
  · push_neg at hn
    

Proof state:
case neg
p✝ p : ℕ → Prop
n : ℕ
hn : ∃ (h : Set.Finite (setOf p)), card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) ≤ n
⊢ nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x}

Next tactic:
rcases hn with ⟨hf, hn⟩
/-- An alternative recursive definition of `Nat.nth`: `Nat.nth s n` is the infimum of `x ∈ s` such
that `Nat.nth s k < x` for all `k < n`, if this set is nonempty. We do not assume that the set is
nonempty because we use the same "garbage value" `0` both for `sInf` on `ℕ` and for `Nat.nth s n`
for `n ≥ card s`. -/
theorem nth_eq_sInf (p : ℕ → Prop) (n : ℕ) : nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x} := by
  by_cases hn : ∀ hf : (setOf p).Finite, n < hf.toFinset.card
  · exact (isLeast_nth hn).csInf_eq.symm
  · push_neg at hn
    rcases hn with ⟨hf, hn⟩
    

Proof state:
case neg.intro
p✝ p : ℕ → Prop
n : ℕ
hf : Set.Finite (setOf p)
hn : card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) ≤ n
⊢ nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x}

Next tactic:
rw [nth_of_card_le _ hn]
/-- An alternative recursive definition of `Nat.nth`: `Nat.nth s n` is the infimum of `x ∈ s` such
that `Nat.nth s k < x` for all `k < n`, if this set is nonempty. We do not assume that the set is
nonempty because we use the same "garbage value" `0` both for `sInf` on `ℕ` and for `Nat.nth s n`
for `n ≥ card s`. -/
theorem nth_eq_sInf (p : ℕ → Prop) (n : ℕ) : nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x} := by
  by_cases hn : ∀ hf : (setOf p).Finite, n < hf.toFinset.card
  · exact (isLeast_nth hn).csInf_eq.symm
  · push_neg at hn
    rcases hn with ⟨hf, hn⟩
    rw [nth_of_card_le _ hn]
    

Proof state:
case neg.intro
p✝ p : ℕ → Prop
n : ℕ
hf : Set.Finite (setOf p)
hn : card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) ≤ n
⊢ 0 = sInf {x | p x ∧ ∀ k < n, nth p k < x}

Next tactic:
refine' ((congr_arg sInf <| Set.eq_empty_of_forall_not_mem fun k hk => _).trans sInf_empty).symm
/-- An alternative recursive definition of `Nat.nth`: `Nat.nth s n` is the infimum of `x ∈ s` such
that `Nat.nth s k < x` for all `k < n`, if this set is nonempty. We do not assume that the set is
nonempty because we use the same "garbage value" `0` both for `sInf` on `ℕ` and for `Nat.nth s n`
for `n ≥ card s`. -/
theorem nth_eq_sInf (p : ℕ → Prop) (n : ℕ) : nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x} := by
  by_cases hn : ∀ hf : (setOf p).Finite, n < hf.toFinset.card
  · exact (isLeast_nth hn).csInf_eq.symm
  · push_neg at hn
    rcases hn with ⟨hf, hn⟩
    rw [nth_of_card_le _ hn]
    refine' ((congr_arg sInf <| Set.eq_empty_of_forall_not_mem fun k hk => _).trans sInf_empty).symm
    

Proof state:
case neg.intro
p✝ p : ℕ → Prop
n : ℕ
hf : Set.Finite (setOf p)
hn : card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) ≤ n
k : ℕ
hk : k ∈ {x | p x ∧ ∀ k < n, nth p k < x}
⊢ False

Next tactic:
rcases exists_lt_card_nth_eq hk.1 with ⟨k, hlt, rfl⟩
/-- An alternative recursive definition of `Nat.nth`: `Nat.nth s n` is the infimum of `x ∈ s` such
that `Nat.nth s k < x` for all `k < n`, if this set is nonempty. We do not assume that the set is
nonempty because we use the same "garbage value" `0` both for `sInf` on `ℕ` and for `Nat.nth s n`
for `n ≥ card s`. -/
theorem nth_eq_sInf (p : ℕ → Prop) (n : ℕ) : nth p n = sInf {x | p x ∧ ∀ k < n, nth p k < x} := by
  by_cases hn : ∀ hf : (setOf p).Finite, n < hf.toFinset.card
  · exact (isLeast_nth hn).csInf_eq.symm
  · push_neg at hn
    rcases hn with ⟨hf, hn⟩
    rw [nth_of_card_le _ hn]
    refine' ((congr_arg sInf <| Set.eq_empty_of_forall_not_mem fun k hk => _).trans sInf_empty).symm
    rcases exists_lt_card_nth_eq hk.1 with ⟨k, hlt, rfl⟩
    

Proof state:
case neg.intro.intro.intro
p✝ p : ℕ → Prop
n : ℕ
hf : Set.Finite (setOf p)
hn : card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) ≤ n
k : ℕ
hlt : ∀ (hf : Set.Finite (setOf p)), k < card (Set.Finite.toFinset hf)
hk : nth p k ∈ {x | p x ∧ ∀ k < n, nth p k < x}
⊢ False

Next tactic:
exact (hk.2 _ ((hlt hf).trans_le hn)).false
theorem nth_zero : nth p 0 = sInf (setOf p) := by 

Proof state:
p : ℕ → Prop
⊢ nth p 0 = sInf (setOf p)

Next tactic:
rw [nth_eq_sInf]
theorem nth_zero : nth p 0 = sInf (setOf p) := by rw [nth_eq_sInf]; 

Proof state:
p : ℕ → Prop
⊢ sInf {x | p x ∧ ∀ k < 0, nth p k < x} = sInf (setOf p)

Next tactic:
simp
@[simp]
theorem nth_zero_of_zero (h : p 0) : nth p 0 = 0 := by 

Proof state:
p : ℕ → Prop
h : p 0
⊢ nth p 0 = 0

Next tactic:
simp [nth_zero, h]
theorem nth_zero_of_exists [DecidablePred p] (h : ∃ n, p n) : nth p 0 = Nat.find h := by
  

Proof state:
p : ℕ → Prop
inst✝ : DecidablePred p
h : ∃ n, p n
⊢ nth p 0 = Nat.find h

Next tactic:
rw [nth_zero]
theorem nth_zero_of_exists [DecidablePred p] (h : ∃ n, p n) : nth p 0 = Nat.find h := by
  rw [nth_zero]; 

Proof state:
p : ℕ → Prop
inst✝ : DecidablePred p
h : ∃ n, p n
⊢ sInf (setOf p) = Nat.find h

Next tactic:
convert Nat.sInf_def h
theorem nth_eq_zero {n} :
    nth p n = 0 ↔ p 0 ∧ n = 0 ∨ ∃ hf : (setOf p).Finite, hf.toFinset.card ≤ n := by
  

Proof state:
p : ℕ → Prop
n : ℕ
⊢ nth p n = 0 ↔ p 0 ∧ n = 0 ∨ ∃ (hf : Set.Finite (setOf p)), card (Set.Finite.toFinset hf) ≤ n

Next tactic:
refine' ⟨fun h => _, _⟩
theorem nth_eq_zero {n} :
    nth p n = 0 ↔ p 0 ∧ n = 0 ∨ ∃ hf : (setOf p).Finite, hf.toFinset.card ≤ n := by
  refine' ⟨fun h => _, _⟩
  · 

Proof state:
case refine'_1
p : ℕ → Prop
n : ℕ
h : nth p n = 0
⊢ p 0 ∧ n = 0 ∨ ∃ (hf : Set.Finite (setOf p)), card (Set.Finite.toFinset hf) ≤ n

Next tactic:
simp only [or_iff_not_imp_right, not_exists, not_le]
theorem nth_eq_zero {n} :
    nth p n = 0 ↔ p 0 ∧ n = 0 ∨ ∃ hf : (setOf p).Finite, hf.toFinset.card ≤ n := by
  refine' ⟨fun h => _, _⟩
  · simp only [or_iff_not_imp_right, not_exists, not_le]
    

Proof state:
case refine'_1
p : ℕ → Prop
n : ℕ
h : nth p n = 0
⊢ (∀ (x : Set.Finite (setOf p)), n < card (Set.Finite.toFinset x)) → p 0 ∧ n = 0

Next tactic:
exact fun hn => ⟨h ▸ nth_mem _ hn, nonpos_iff_eq_zero.1 <| h ▸ le_nth hn⟩
theorem nth_eq_zero {n} :
    nth p n = 0 ↔ p 0 ∧ n = 0 ∨ ∃ hf : (setOf p).Finite, hf.toFinset.card ≤ n := by
  refine' ⟨fun h => _, _⟩
  · simp only [or_iff_not_imp_right, not_exists, not_le]
    exact fun hn => ⟨h ▸ nth_mem _ hn, nonpos_iff_eq_zero.1 <| h ▸ le_nth hn⟩
  · 

Proof state:
case refine'_2
p : ℕ → Prop
n : ℕ
⊢ (p 0 ∧ n = 0 ∨ ∃ (hf : Set.Finite (setOf p)), card (Set.Finite.toFinset hf) ≤ n) → nth p n = 0

Next tactic:
rintro (⟨h₀, rfl⟩ | ⟨hf, hle⟩)
theorem nth_eq_zero {n} :
    nth p n = 0 ↔ p 0 ∧ n = 0 ∨ ∃ hf : (setOf p).Finite, hf.toFinset.card ≤ n := by
  refine' ⟨fun h => _, _⟩
  · simp only [or_iff_not_imp_right, not_exists, not_le]
    exact fun hn => ⟨h ▸ nth_mem _ hn, nonpos_iff_eq_zero.1 <| h ▸ le_nth hn⟩
  · rintro (⟨h₀, rfl⟩ | ⟨hf, hle⟩)
    

Proof state:
case refine'_2.inl.intro
p : ℕ → Prop
h₀ : p 0
⊢ nth p 0 = 0
case refine'_2.inr.intro
p : ℕ → Prop
n : ℕ
hf : Set.Finite (setOf p)
hle : card (Set.Finite.toFinset hf) ≤ n
⊢ nth p n = 0

Next tactic:
exacts [nth_zero_of_zero h₀, nth_of_card_le hf hle]
theorem nth_eq_zero_mono (h₀ : ¬p 0) {a b : ℕ} (hab : a ≤ b) (ha : nth p a = 0) : nth p b = 0 := by
  

Proof state:
p : ℕ → Prop
h₀ : ¬p 0
a b : ℕ
hab : a ≤ b
ha : nth p a = 0
⊢ nth p b = 0

Next tactic:
simp only [nth_eq_zero, h₀, false_and_iff, false_or_iff] at ha ⊢
theorem nth_eq_zero_mono (h₀ : ¬p 0) {a b : ℕ} (hab : a ≤ b) (ha : nth p a = 0) : nth p b = 0 := by
  simp only [nth_eq_zero, h₀, false_and_iff, false_or_iff] at ha ⊢
  

Proof state:
p : ℕ → Prop
h₀ : ¬p 0
a b : ℕ
hab : a ≤ b
ha : ∃ (hf : Set.Finite (setOf p)), card (Set.Finite.toFinset hf) ≤ a
⊢ ∃ (hf : Set.Finite (setOf p)), card (Set.Finite.toFinset hf) ≤ b

Next tactic:
exact ha.imp fun hf hle => hle.trans hab
theorem le_nth_of_lt_nth_succ {k a : ℕ} (h : a < nth p (k + 1)) (ha : p a) : a ≤ nth p k := by
  

Proof state:
p : ℕ → Prop
k a : ℕ
h : a < nth p (k + 1)
ha : p a
⊢ a ≤ nth p k

Next tactic:
cases' (setOf p).finite_or_infinite with hf hf
theorem le_nth_of_lt_nth_succ {k a : ℕ} (h : a < nth p (k + 1)) (ha : p a) : a ≤ nth p k := by
  cases' (setOf p).finite_or_infinite with hf hf
  · 

Proof state:
case inl
p : ℕ → Prop
k a : ℕ
h : a < nth p (k + 1)
ha : p a
hf : Set.Finite (setOf p)
⊢ a ≤ nth p k

Next tactic:
rcases exists_lt_card_finite_nth_eq hf ha with ⟨n, hn, rfl⟩
theorem le_nth_of_lt_nth_succ {k a : ℕ} (h : a < nth p (k + 1)) (ha : p a) : a ≤ nth p k := by
  cases' (setOf p).finite_or_infinite with hf hf
  · rcases exists_lt_card_finite_nth_eq hf ha with ⟨n, hn, rfl⟩
    

Proof state:
case inl.intro.intro
p : ℕ → Prop
k : ℕ
hf : Set.Finite (setOf p)
n : ℕ
hn : n < card (Set.Finite.toFinset hf)
h : nth p n < nth p (k + 1)
ha : p (nth p n)
⊢ nth p n ≤ nth p k

Next tactic:
cases' lt_or_le (k + 1) hf.toFinset.card with hk hk
theorem le_nth_of_lt_nth_succ {k a : ℕ} (h : a < nth p (k + 1)) (ha : p a) : a ≤ nth p k := by
  cases' (setOf p).finite_or_infinite with hf hf
  · rcases exists_lt_card_finite_nth_eq hf ha with ⟨n, hn, rfl⟩
    cases' lt_or_le (k + 1) hf.toFinset.card with hk hk
    · 

Proof state:
case inl.intro.intro.inl
p : ℕ → Prop
k : ℕ
hf : Set.Finite (setOf p)
n : ℕ
hn : n < card (Set.Finite.toFinset hf)
h : nth p n < nth p (k + 1)
ha : p (nth p n)
hk : k + 1 < card (Set.Finite.toFinset hf)
⊢ nth p n ≤ nth p k

Next tactic:
rwa [(nth_strictMonoOn hf).lt_iff_lt hn hk, lt_succ_iff,
        ← (nth_strictMonoOn hf).le_iff_le hn (k.lt_succ_self.trans hk)] at h
theorem le_nth_of_lt_nth_succ {k a : ℕ} (h : a < nth p (k + 1)) (ha : p a) : a ≤ nth p k := by
  cases' (setOf p).finite_or_infinite with hf hf
  · rcases exists_lt_card_finite_nth_eq hf ha with ⟨n, hn, rfl⟩
    cases' lt_or_le (k + 1) hf.toFinset.card with hk hk
    · rwa [(nth_strictMonoOn hf).lt_iff_lt hn hk, lt_succ_iff,
        ← (nth_strictMonoOn hf).le_iff_le hn (k.lt_succ_self.trans hk)] at h
    · 

Proof state:
case inl.intro.intro.inr
p : ℕ → Prop
k : ℕ
hf : Set.Finite (setOf p)
n : ℕ
hn : n < card (Set.Finite.toFinset hf)
h : nth p n < nth p (k + 1)
ha : p (nth p n)
hk : card (Set.Finite.toFinset hf) ≤ k + 1
⊢ nth p n ≤ nth p k

Next tactic:
rw [nth_of_card_le _ hk] at h
theorem le_nth_of_lt_nth_succ {k a : ℕ} (h : a < nth p (k + 1)) (ha : p a) : a ≤ nth p k := by
  cases' (setOf p).finite_or_infinite with hf hf
  · rcases exists_lt_card_finite_nth_eq hf ha with ⟨n, hn, rfl⟩
    cases' lt_or_le (k + 1) hf.toFinset.card with hk hk
    · rwa [(nth_strictMonoOn hf).lt_iff_lt hn hk, lt_succ_iff,
        ← (nth_strictMonoOn hf).le_iff_le hn (k.lt_succ_self.trans hk)] at h
    · rw [nth_of_card_le _ hk] at h
      

Proof state:
case inl.intro.intro.inr
p : ℕ → Prop
k : ℕ
hf : Set.Finite (setOf p)
n : ℕ
hn : n < card (Set.Finite.toFinset hf)
h : nth p n < 0
ha : p (nth p n)
hk : card (Set.Finite.toFinset hf) ≤ k + 1
⊢ nth p n ≤ nth p k

Next tactic:
exact absurd h (zero_le _).not_lt