ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Cardinal.prod_eq_of_fintype
theorem prod_eq_of_fintype {α : Type u} [h : Fintype α] (f : α → Cardinal.{v}) :
    prod f = Cardinal.lift.{u} (∏ i, f i) 
Arguments: {α : Type u} [h : Fintype α] (f : α → Cardinal.{v})
Goal: Cardinal.prod f = Cardinal.lift.{u, v} (∏ i : α, f i)
-- Cardinal.lift_sInf
@[simp]
theorem lift_sInf (s : Set Cardinal) : lift.{u, v} (sInf s) = sInf (lift.{u, v} '' s) 
Arguments: (s : Set Cardinal.{v})
Goal: Cardinal.lift.{u, v} (sInf s) = sInf (Cardinal.lift.{u, v} '' s)
-- Cardinal.lift_iInf
@[simp]
theorem lift_iInf {ι} (f : ι → Cardinal) : lift.{u, v} (iInf f) = ⨅ i, lift.{u, v} (f i) 
Arguments: {ι : Sort u_1} (f : ι → Cardinal.{v})
Goal: Cardinal.lift.{u, v} (iInf f) = ⨅ i, Cardinal.lift.{u, v} (f i)
-- Cardinal.lift_down
theorem lift_down {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
    b ≤ lift.{v,u} a → ∃ a', lift.{v,u} a' = b 
Arguments: {a : Cardinal.{u}} {b : Cardinal.{max u v}}
Goal: b ≤ Cardinal.lift.{v, u} a → ∃ a', Cardinal.lift.{v, u} a' = b
-- Cardinal.le_lift_iff
theorem le_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
    b ≤ lift.{v, u} a ↔ ∃ a', lift.{v, u} a' = b ∧ a' ≤ a 
Arguments: {a : Cardinal.{u}} {b : Cardinal.{max u v}}
Goal: b ≤ Cardinal.lift.{v, u} a ↔ ∃ a', Cardinal.lift.{v, u} a' = b ∧ a' ≤ a
-- Cardinal.lt_lift_iff
theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
    b < lift.{v, u} a ↔ ∃ a', lift.{v, u} a' = b ∧ a' < a 
Arguments: {a : Cardinal.{u}} {b : Cardinal.{max u v}}
Goal: b < Cardinal.lift.{v, u} a ↔ ∃ a', Cardinal.lift.{v, u} a' = b ∧ a' < a
-- Cardinal.lift_succ
@[simp]
theorem lift_succ (a) : lift.{v, u} (succ a) = succ (lift.{v, u} a) 
Arguments: (a : Cardinal.{u})
Goal: Cardinal.lift.{v, u} (Order.succ a) = Order.succ (Cardinal.lift.{v, u} a)
-- Cardinal.lift_umax_eq
@[simp, nolint simpNF]
theorem lift_umax_eq {a : Cardinal.{u}} {b : Cardinal.{v}} :
    lift.{max v w} a = lift.{max u w} b ↔ lift.{v} a = lift.{u} b 
Arguments: {a : Cardinal.{u}} {b : Cardinal.{v}}
Goal: Cardinal.lift.{max v w, u} a = Cardinal.lift.{max u w, v} b ↔ Cardinal.lift.{v, u} a = Cardinal.lift.{u, v} b
-- Cardinal.lift_min
@[simp]
theorem lift_min {a b : Cardinal} : lift.{u, v} (min a b) = min (lift.{u, v} a) (lift.{u, v} b) 
Arguments: {a b : Cardinal.{v}}
Goal: Cardinal.lift.{u, v} (min a b) = min (Cardinal.lift.{u, v} a) (Cardinal.lift.{u, v} b)
-- Cardinal.lift_max
@[simp]
theorem lift_max {a b : Cardinal} : lift.{u, v} (max a b) = max (lift.{u, v} a) (lift.{u, v} b) 
Arguments: {a b : Cardinal.{v}}
Goal: Cardinal.lift.{u, v} (max a b) = max (Cardinal.lift.{u, v} a) (Cardinal.lift.{u, v} b)
-- Cardinal.lift_sSup
theorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) :
    lift.{u} (sSup s) = sSup (lift.{u} '' s) 
Arguments: {s : Set Cardinal.{u_1}} (hs : BddAbove s)
Goal: Cardinal.lift.{u, u_1} (sSup s) = sSup (Cardinal.lift.{u, u_1} '' s)
-- Cardinal.lift_iSup
theorem lift_iSup {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f)) :
    lift.{u} (iSup f) = ⨆ i, lift.{u} (f i) 
Arguments: {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (Set.range f))
Goal: Cardinal.lift.{u, w} (iSup f) = ⨆ i, Cardinal.lift.{u, w} (f i)
-- Cardinal.lift_iSup_le
theorem lift_iSup_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))
    (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (iSup f) ≤ t 
Arguments: {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal.{max u w}} (hf : BddAbove (Set.range f)) (w : ∀ (i : ι), Cardinal.lift.{u, w} (f i) ≤ t)
Goal: Cardinal.lift.{u, w} (iSup f) ≤ t
-- Cardinal.lift_iSup_le_iff
@[simp]
theorem lift_iSup_le_iff {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f))
    {t : Cardinal} : lift.{u} (iSup f) ≤ t ↔ ∀ i, lift.{u} (f i) ≤ t 
Arguments: {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (Set.range f)) {t : Cardinal.{max u w}}
Goal: Cardinal.lift.{u, w} (iSup f) ≤ t ↔ ∀ (i : ι), Cardinal.lift.{u, w} (f i) ≤ t
-- Cardinal.lift_iSup_le_lift_iSup
theorem lift_iSup_le_lift_iSup {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{w}}
    {f' : ι' → Cardinal.{w'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) {g : ι → ι'}
    (h : ∀ i, lift.{w'} (f i) ≤ lift.{w} (f' (g i))) : lift.{w'} (iSup f) ≤ lift.{w} (iSup f') 
Arguments: {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{w}} {f' : ι' → Cardinal.{w'}} (hf : BddAbove (Set.range f)) (hf' : BddAbove (Set.range f')) {g : ι → ι'} (h : ∀ (i : ι), Cardinal.lift.{w', w} (f i) ≤ Cardinal.lift.{w, w'} (f' (g i)))
Goal: Cardinal.lift.{w', w} (iSup f) ≤ Cardinal.lift.{w, w'} (iSup f')
-- Cardinal.lift_iSup_le_lift_iSup'
theorem lift_iSup_le_lift_iSup' {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{v}}
    {f' : ι' → Cardinal.{v'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) (g : ι → ι')
    (h : ∀ i, lift.{v'} (f i) ≤ lift.{v} (f' (g i))) : lift.{v'} (iSup f) ≤ lift.{v} (iSup f') 
Arguments: {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{v}} {f' : ι' → Cardinal.{v'}} (hf : BddAbove (Set.range f)) (hf' : BddAbove (Set.range f')) (g : ι → ι') (h : ∀ (i : ι), Cardinal.lift.{v', v} (f i) ≤ Cardinal.lift.{v, v'} (f' (g i)))
Goal: Cardinal.lift.{v', v} (iSup f) ≤ Cardinal.lift.{v, v'} (iSup f')
-- Cardinal.mk_nat
theorem mk_nat : #ℕ = ℵ₀ 
Goal: Cardinal.mk ℕ = Cardinal.aleph0
-- Cardinal.aleph0_ne_zero
theorem aleph0_ne_zero : ℵ₀ ≠ 0 
Goal: Cardinal.aleph0 ≠ 0
-- Cardinal.aleph0_pos
theorem aleph0_pos : 0 < ℵ₀ 
Goal: 0 < Cardinal.aleph0
-- Cardinal.lift_aleph0
@[simp]
theorem lift_aleph0 : lift ℵ₀ = ℵ₀ 
Goal: Cardinal.lift.{u_1, u_2} Cardinal.aleph0 = Cardinal.aleph0
-- Cardinal.aleph0_le_lift
@[simp]
theorem aleph0_le_lift {c : Cardinal.{u}} : ℵ₀ ≤ lift.{v} c ↔ ℵ₀ ≤ c 
Arguments: {c : Cardinal.{u}}
Goal: Cardinal.aleph0 ≤ Cardinal.lift.{v, u} c ↔ Cardinal.aleph0 ≤ c
-- Cardinal.lift_le_aleph0
@[simp]
theorem lift_le_aleph0 {c : Cardinal.{u}} : lift.{v} c ≤ ℵ₀ ↔ c ≤ ℵ₀ 
Arguments: {c : Cardinal.{u}}
Goal: Cardinal.lift.{v, u} c ≤ Cardinal.aleph0 ↔ c ≤ Cardinal.aleph0
-- Cardinal.aleph0_lt_lift
@[simp]
theorem aleph0_lt_lift {c : Cardinal.{u}} : ℵ₀ < lift.{v} c ↔ ℵ₀ < c 
Arguments: {c : Cardinal.{u}}
Goal: Cardinal.aleph0 < Cardinal.lift.{v, u} c ↔ Cardinal.aleph0 < c
-- Cardinal.lift_lt_aleph0
@[simp]
theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ 
Arguments: {c : Cardinal.{u}}
Goal: Cardinal.lift.{v, u} c < Cardinal.aleph0 ↔ c < Cardinal.aleph0
-- Cardinal.mk_fin
theorem mk_fin (n : ℕ) : #(Fin n) = n 
Arguments: (n : ℕ)
Goal: Cardinal.mk (Fin n) = ↑n
-- Cardinal.lift_natCast
@[simp]
theorem lift_natCast (n : ℕ) : lift.{u} (n : Cardinal.{v}) = n 
Arguments: (n : ℕ)
Goal: Cardinal.lift.{u, v} ↑n = ↑n
-- Cardinal.lift_ofNat
@[simp]
theorem lift_ofNat (n : ℕ) [n.AtLeastTwo] :
    lift.{u} (no_index (OfNat.ofNat n : Cardinal.{v})) = OfNat.ofNat n 
Arguments: (n : ℕ) [n.AtLeastTwo]
Goal: Cardinal.lift.{u, v} (OfNat.ofNat n) = OfNat.ofNat n
-- Cardinal.lift_eq_nat_iff
@[simp]
theorem lift_eq_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a = n ↔ a = n 
Arguments: {a : Cardinal.{u}} {n : ℕ}
Goal: Cardinal.lift.{v, u} a = ↑n ↔ a = ↑n
-- Cardinal.lift_eq_ofNat_iff
@[simp]
theorem lift_eq_ofNat_iff {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo] :
    lift.{v} a = (no_index (OfNat.ofNat n)) ↔ a = OfNat.ofNat n 
Arguments: {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo]
Goal: Cardinal.lift.{v, u} a = OfNat.ofNat n ↔ a = OfNat.ofNat n
-- Cardinal.nat_eq_lift_iff
@[simp]
theorem nat_eq_lift_iff {n : ℕ} {a : Cardinal.{u}} :
    (n : Cardinal) = lift.{v} a ↔ (n : Cardinal) = a 
Arguments: {n : ℕ} {a : Cardinal.{u}}
Goal: ↑n = Cardinal.lift.{v, u} a ↔ ↑n = a
-- Cardinal.zero_eq_lift_iff
@[simp]
theorem zero_eq_lift_iff {a : Cardinal.{u}} :
    (0 : Cardinal) = lift.{v} a ↔ 0 = a 
Arguments: {a : Cardinal.{u}}
Goal: 0 = Cardinal.lift.{v, u} a ↔ 0 = a
-- Cardinal.one_eq_lift_iff
@[simp]
theorem one_eq_lift_iff {a : Cardinal.{u}} :
    (1 : Cardinal) = lift.{v} a ↔ 1 = a 
Arguments: {a : Cardinal.{u}}
Goal: 1 = Cardinal.lift.{v, u} a ↔ 1 = a
-- Cardinal.ofNat_eq_lift_iff
@[simp]
theorem ofNat_eq_lift_iff {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo] :
    (no_index (OfNat.ofNat n : Cardinal)) = lift.{v} a ↔ (OfNat.ofNat n : Cardinal) = a 
Arguments: {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo]
Goal: OfNat.ofNat n = Cardinal.lift.{v, u} a ↔ OfNat.ofNat n = a
-- Cardinal.lift_le_nat_iff
@[simp]
theorem lift_le_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a ≤ n ↔ a ≤ n 
Arguments: {a : Cardinal.{u}} {n : ℕ}
Goal: Cardinal.lift.{v, u} a ≤ ↑n ↔ a ≤ ↑n
-- Cardinal.lift_le_one_iff
@[simp]
theorem lift_le_one_iff {a : Cardinal.{u}} :
    lift.{v} a ≤ 1 ↔ a ≤ 1 
Arguments: {a : Cardinal.{u}}
Goal: Cardinal.lift.{v, u} a ≤ 1 ↔ a ≤ 1
-- Cardinal.lift_le_ofNat_iff
@[simp]
theorem lift_le_ofNat_iff {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo] :
    lift.{v} a ≤ (no_index (OfNat.ofNat n)) ↔ a ≤ OfNat.ofNat n 
Arguments: {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo]
Goal: Cardinal.lift.{v, u} a ≤ OfNat.ofNat n ↔ a ≤ OfNat.ofNat n
-- Cardinal.nat_le_lift_iff
@[simp]
theorem nat_le_lift_iff {n : ℕ} {a : Cardinal.{u}} : n ≤ lift.{v} a ↔ n ≤ a 
Arguments: {n : ℕ} {a : Cardinal.{u}}
Goal: ↑n ≤ Cardinal.lift.{v, u} a ↔ ↑n ≤ a
-- Cardinal.one_le_lift_iff
@[simp]
theorem one_le_lift_iff {a : Cardinal.{u}} :
    (1 : Cardinal) ≤ lift.{v} a ↔ 1 ≤ a 
Arguments: {a : Cardinal.{u}}
Goal: 1 ≤ Cardinal.lift.{v, u} a ↔ 1 ≤ a
-- Cardinal.ofNat_le_lift_iff
@[simp]
theorem ofNat_le_lift_iff {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo] :
    (no_index (OfNat.ofNat n : Cardinal)) ≤ lift.{v} a ↔ (OfNat.ofNat n : Cardinal) ≤ a 
Arguments: {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo]
Goal: OfNat.ofNat n ≤ Cardinal.lift.{v, u} a ↔ OfNat.ofNat n ≤ a
-- Cardinal.lift_lt_nat_iff
@[simp]
theorem lift_lt_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a < n ↔ a < n 
Arguments: {a : Cardinal.{u}} {n : ℕ}
Goal: Cardinal.lift.{v, u} a < ↑n ↔ a < ↑n
-- Cardinal.lift_lt_ofNat_iff
@[simp]
theorem lift_lt_ofNat_iff {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo] :
    lift.{v} a < (no_index (OfNat.ofNat n)) ↔ a < OfNat.ofNat n 
Arguments: {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo]
Goal: Cardinal.lift.{v, u} a < OfNat.ofNat n ↔ a < OfNat.ofNat n
-- Cardinal.nat_lt_lift_iff
@[simp]
theorem nat_lt_lift_iff {n : ℕ} {a : Cardinal.{u}} : n < lift.{v} a ↔ n < a 
Arguments: {n : ℕ} {a : Cardinal.{u}}
Goal: ↑n < Cardinal.lift.{v, u} a ↔ ↑n < a
-- Cardinal.zero_lt_lift_iff
@[simp]
theorem zero_lt_lift_iff {a : Cardinal.{u}} :
    (0 : Cardinal) < lift.{v} a ↔ 0 < a 
Arguments: {a : Cardinal.{u}}
Goal: 0 < Cardinal.lift.{v, u} a ↔ 0 < a
-- Cardinal.one_lt_lift_iff
@[simp]
theorem one_lt_lift_iff {a : Cardinal.{u}} :
    (1 : Cardinal) < lift.{v} a ↔ 1 < a 
Arguments: {a : Cardinal.{u}}
Goal: 1 < Cardinal.lift.{v, u} a ↔ 1 < a
-- Cardinal.ofNat_lt_lift_iff
@[simp]
theorem ofNat_lt_lift_iff {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo] :
    (no_index (OfNat.ofNat n : Cardinal)) < lift.{v} a ↔ (OfNat.ofNat n : Cardinal) < a 
Arguments: {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo]
Goal: OfNat.ofNat n < Cardinal.lift.{v, u} a ↔ OfNat.ofNat n < a
-- Cardinal.lift_mk_fin
theorem lift_mk_fin (n : ℕ) : lift #(Fin n) = n 
Arguments: (n : ℕ)
Goal: Cardinal.lift.{u_1, 0} (Cardinal.mk (Fin n)) = ↑n
-- Cardinal.mk_coe_finset
theorem mk_coe_finset {α : Type u} {s : Finset α} : #s = ↑(Finset.card s) 
Arguments: {α : Type u} {s : Finset α}
Goal: Cardinal.mk { x // x ∈ s } = ↑s.card
-- Cardinal.mk_finset_of_fintype
theorem mk_finset_of_fintype [Fintype α] : #(Finset α) = 2 ^ Fintype.card α 
Arguments: {α : Type u} [Fintype α]
Goal: Cardinal.mk (Finset α) = 2 ^ Fintype.card α
-- Cardinal.mk_finsupp_lift_of_fintype
@[simp]
theorem mk_finsupp_lift_of_fintype (α : Type u) (β : Type v) [Fintype α] [Zero β] :
    #(α →₀ β) = lift.{u} #β ^ Fintype.card α 
Arguments: (α : Type u) (β : Type v) [Fintype α] [Zero β]
Goal: Cardinal.mk (α →₀ β) = Cardinal.lift.{u, v} (Cardinal.mk β) ^ Fintype.card α
-- Cardinal.mk_finsupp_of_fintype
theorem mk_finsupp_of_fintype (α β : Type u) [Fintype α] [Zero β] :
    #(α →₀ β) = #β ^ Fintype.card α 
Arguments: (α β : Type u) [Fintype α] [Zero β]
Goal: Cardinal.mk (α →₀ β) = Cardinal.mk β ^ Fintype.card α
-- Cardinal.card_le_of_finset
theorem card_le_of_finset {α} (s : Finset α) : (s.card : Cardinal) ≤ #α 
Arguments: {α : Type u_1} (s : Finset α)
Goal: ↑s.card ≤ Cardinal.mk α
-- Cardinal.natCast_pow
@[norm_cast]
theorem natCast_pow {m n : ℕ} : (↑(m ^ n) : Cardinal) = (↑m : Cardinal) ^ (↑n : Cardinal) 
Arguments: {m n : ℕ}
Goal: ↑(m ^ n) = ↑m ^ ↑n
-- Cardinal.natCast_le
@[norm_cast]
theorem natCast_le {m n : ℕ} : (m : Cardinal) ≤ n ↔ m ≤ n 
Arguments: {m n : ℕ}
Goal: ↑m ≤ ↑n ↔ m ≤ n
-- Cardinal.natCast_lt
@[norm_cast]
theorem natCast_lt {m n : ℕ} : (m : Cardinal) < n ↔ m < n 
Arguments: {m n : ℕ}
Goal: ↑m < ↑n ↔ m < n
-- Cardinal.natCast_inj
theorem natCast_inj {m n : ℕ} : (m : Cardinal) = n ↔ m = n 
Arguments: {m n : ℕ}
Goal: ↑m = ↑n ↔ m = n
-- Cardinal.natCast_injective
theorem natCast_injective : Injective ((↑) : ℕ → Cardinal) 
Goal: Function.Injective Nat.cast
-- Cardinal.nat_succ
@[norm_cast]
theorem nat_succ (n : ℕ) : (n.succ : Cardinal) = succ ↑n 
Arguments: (n : ℕ)
Goal: ↑n.succ = Order.succ ↑n
-- Cardinal.succ_natCast
lemma succ_natCast (n : ℕ) : Order.succ (n : Cardinal) = n + 1 
Arguments: (n : ℕ)
Goal: Order.succ ↑n = ↑n + 1
-- Cardinal.natCast_add_one_le_iff
lemma natCast_add_one_le_iff {n : ℕ} {c : Cardinal} : n + 1 ≤ c ↔ n < c 
Arguments: {n : ℕ} {c : Cardinal.{u_1}}
Goal: ↑n + 1 ≤ c ↔ ↑n < c
-- Cardinal.two_le_iff_one_lt
lemma two_le_iff_one_lt {c : Cardinal} : 2 ≤ c ↔ 1 < c 
Arguments: {c : Cardinal.{u_1}}
Goal: 2 ≤ c ↔ 1 < c
-- Cardinal.succ_zero
@[simp]
theorem succ_zero : succ (0 : Cardinal) = 1 
Goal: Order.succ 0 = 1
-- Cardinal.exists_finset_le_card
theorem exists_finset_le_card (α : Type*) (n : ℕ) (h : n ≤ #α) :
    ∃ s : Finset α, n ≤ s.card 
Arguments: (α : Type u_1) (n : ℕ) (h : ↑n ≤ Cardinal.mk α)
Goal: ∃ s, n ≤ s.card
-- Cardinal.card_le_of
theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n) : #α ≤ n 
Arguments: {α : Type u} {n : ℕ} (H : ∀ (s : Finset α), s.card ≤ n)
Goal: Cardinal.mk α ≤ ↑n
-- Cardinal.cantor'
theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < b ^ a 
Arguments: (a : Cardinal.{u_1}) {b : Cardinal.{u_1}} (hb : 1 < b)
Goal: a < b ^ a
-- Cardinal.one_le_iff_pos
theorem one_le_iff_pos {c : Cardinal} : 1 ≤ c ↔ 0 < c 
Arguments: {c : Cardinal.{u_1}}
Goal: 1 ≤ c ↔ 0 < c
-- Cardinal.one_le_iff_ne_zero
theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 
Arguments: {c : Cardinal.{u_1}}
Goal: 1 ≤ c ↔ c ≠ 0
-- Cardinal.lt_one_iff_zero
@[simp]
theorem lt_one_iff_zero {c : Cardinal} : c < 1 ↔ c = 0 
Arguments: {c : Cardinal.{u_1}}
Goal: c < 1 ↔ c = 0
-- Cardinal.nat_lt_aleph0
theorem nat_lt_aleph0 (n : ℕ) : (n : Cardinal.{u}) < ℵ₀ 
Arguments: (n : ℕ)
Goal: ↑n < Cardinal.aleph0
-- Cardinal.one_lt_aleph0
@[simp]
theorem one_lt_aleph0 : 1 < ℵ₀ 
Goal: 1 < Cardinal.aleph0
-- Cardinal.one_le_aleph0
theorem one_le_aleph0 : 1 ≤ ℵ₀ 
Goal: 1 ≤ Cardinal.aleph0
-- Cardinal.lt_aleph0
theorem lt_aleph0 {c : Cardinal} : c < ℵ₀ ↔ ∃ n : ℕ, c = n 
Arguments: {c : Cardinal.{u_1}}
Goal: c < Cardinal.aleph0 ↔ ∃ n, c = ↑n
-- Cardinal.succ_eq_of_lt_aleph0
lemma succ_eq_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : Order.succ c = c + 1 
Arguments: {c : Cardinal.{u_1}} (h : c < Cardinal.aleph0)
Goal: Order.succ c = c + 1
-- Cardinal.aleph0_le
theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c 
Arguments: {c : Cardinal.{u_1}}
Goal: Cardinal.aleph0 ≤ c ↔ ∀ (n : ℕ), ↑n ≤ c
-- Cardinal.isSuccLimit_aleph0
theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ 
Goal: Order.IsSuccLimit Cardinal.aleph0
-- Cardinal.isLimit_aleph0
theorem isLimit_aleph0 : IsLimit ℵ₀ 
Goal: Cardinal.aleph0.IsLimit
-- Cardinal.not_isLimit_natCast
lemma not_isLimit_natCast : (n : ℕ) → ¬ IsLimit (n : Cardinal.{u})
  
Arguments: (n : ℕ)
Goal: ¬(↑n).IsLimit
-- Cardinal.IsLimit.aleph0_le
theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c 
Arguments: {c : Cardinal.{u_1}} (h : c.IsLimit)
Goal: Cardinal.aleph0 ≤ c
-- Cardinal.exists_eq_natCast_of_iSup_eq
lemma exists_eq_natCast_of_iSup_eq {ι : Type u} [Nonempty ι] (f : ι → Cardinal.{v})
    (hf : BddAbove (range f)) (n : ℕ) (h : ⨆ i, f i = n) : ∃ i, f i = n 
Arguments: {ι : Type u} [Nonempty ι] (f : ι → Cardinal.{v}) (hf : BddAbove (Set.range f)) (n : ℕ) (h : ⨆ i, f i = ↑n)
Goal: ∃ i, f i = ↑n
-- Cardinal.range_natCast
@[simp]
theorem range_natCast : range ((↑) : ℕ → Cardinal) = Iio ℵ₀ 
Goal: Set.range Nat.cast = Set.Iio Cardinal.aleph0
-- Cardinal.mk_eq_nat_iff
theorem mk_eq_nat_iff {α : Type u} {n : ℕ} : #α = n ↔ Nonempty (α ≃ Fin n) 
Arguments: {α : Type u} {n : ℕ}
Goal: Cardinal.mk α = ↑n ↔ Nonempty (α ≃ Fin n)
-- Cardinal.lt_aleph0_iff_finite
theorem lt_aleph0_iff_finite {α : Type u} : #α < ℵ₀ ↔ Finite α 
Arguments: {α : Type u}
Goal: Cardinal.mk α < Cardinal.aleph0 ↔ Finite α
-- Cardinal.lt_aleph0_iff_fintype
theorem lt_aleph0_iff_fintype {α : Type u} : #α < ℵ₀ ↔ Nonempty (Fintype α) 
Arguments: {α : Type u}
Goal: Cardinal.mk α < Cardinal.aleph0 ↔ Nonempty (Fintype α)
-- Cardinal.lt_aleph0_of_finite
theorem lt_aleph0_of_finite (α : Type u) [Finite α] : #α < ℵ₀ 
Arguments: (α : Type u) [Finite α]
Goal: Cardinal.mk α < Cardinal.aleph0
-- Cardinal.lt_aleph0_iff_set_finite
theorem lt_aleph0_iff_set_finite {S : Set α} : #S < ℵ₀ ↔ S.Finite 
Arguments: {α : Type u} {S : Set α}
Goal: Cardinal.mk ↑S < Cardinal.aleph0 ↔ S.Finite
-- Cardinal.lt_aleph0_iff_subtype_finite
@[simp]
theorem lt_aleph0_iff_subtype_finite {p : α → Prop} : #{ x // p x } < ℵ₀ ↔ { x | p x }.Finite 
Arguments: {α : Type u} {p : α → Prop}
Goal: Cardinal.mk { x // p x } < Cardinal.aleph0 ↔ {x | p x}.Finite
-- Cardinal.mk_le_aleph0_iff
theorem mk_le_aleph0_iff : #α ≤ ℵ₀ ↔ Countable α 
Arguments: {α : Type u}
Goal: Cardinal.mk α ≤ Cardinal.aleph0 ↔ Countable α
-- Cardinal.mk_le_aleph0
@[simp]
theorem mk_le_aleph0 [Countable α] : #α ≤ ℵ₀ 
Arguments: {α : Type u} [Countable α]
Goal: Cardinal.mk α ≤ Cardinal.aleph0
-- Cardinal.le_aleph0_iff_set_countable
theorem le_aleph0_iff_set_countable {s : Set α} : #s ≤ ℵ₀ ↔ s.Countable 
Arguments: {α : Type u} {s : Set α}
Goal: Cardinal.mk ↑s ≤ Cardinal.aleph0 ↔ s.Countable
-- Cardinal.le_aleph0_iff_subtype_countable
@[simp]
theorem le_aleph0_iff_subtype_countable {p : α → Prop} :
    #{ x // p x } ≤ ℵ₀ ↔ { x | p x }.Countable 
Arguments: {α : Type u} {p : α → Prop}
Goal: Cardinal.mk { x // p x } ≤ Cardinal.aleph0 ↔ {x | p x}.Countable
-- Cardinal.add_lt_aleph0
theorem add_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a + b < ℵ₀ 
Arguments: {a b : Cardinal.{u_1}} (ha : a < Cardinal.aleph0) (hb : b < Cardinal.aleph0)
Goal: a + b < Cardinal.aleph0
-- Cardinal.add_lt_aleph0_iff
theorem add_lt_aleph0_iff {a b : Cardinal} : a + b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ 
Arguments: {a b : Cardinal.{u_1}}
Goal: a + b < Cardinal.aleph0 ↔ a < Cardinal.aleph0 ∧ b < Cardinal.aleph0
-- Cardinal.aleph0_le_add_iff
theorem aleph0_le_add_iff {a b : Cardinal} : ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a ∨ ℵ₀ ≤ b 
Arguments: {a b : Cardinal.{u_1}}
Goal: Cardinal.aleph0 ≤ a + b ↔ Cardinal.aleph0 ≤ a ∨ Cardinal.aleph0 ≤ b
-- Cardinal.nsmul_lt_aleph0_iff
theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n = 0 ∨ a < ℵ₀ 
Arguments: {n : ℕ} {a : Cardinal.{u_1}}
Goal: n • a < Cardinal.aleph0 ↔ n = 0 ∨ a < Cardinal.aleph0
-- Cardinal.nsmul_lt_aleph0_iff_of_ne_zero
theorem nsmul_lt_aleph0_iff_of_ne_zero {n : ℕ} {a : Cardinal} (h : n ≠ 0) : n • a < ℵ₀ ↔ a < ℵ₀ 
Arguments: {n : ℕ} {a : Cardinal.{u_1}} (h : n ≠ 0)
Goal: n • a < Cardinal.aleph0 ↔ a < Cardinal.aleph0
-- Cardinal.mul_lt_aleph0
theorem mul_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a * b < ℵ₀ 
Arguments: {a b : Cardinal.{u_1}} (ha : a < Cardinal.aleph0) (hb : b < Cardinal.aleph0)
Goal: a * b < Cardinal.aleph0
-- Cardinal.mul_lt_aleph0_iff
theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0 ∨ a < ℵ₀ ∧ b < ℵ₀ 
Arguments: {a b : Cardinal.{u_1}}
Goal: a * b < Cardinal.aleph0 ↔ a = 0 ∨ b = 0 ∨ a < Cardinal.aleph0 ∧ b < Cardinal.aleph0
-- Cardinal.aleph0_le_mul_iff
theorem aleph0_le_mul_iff {a b : Cardinal} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (ℵ₀ ≤ a ∨ ℵ₀ ≤ b) 
Arguments: {a b : Cardinal.{u_1}}
Goal: Cardinal.aleph0 ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (Cardinal.aleph0 ≤ a ∨ Cardinal.aleph0 ≤ b)
-- Cardinal.aleph0_le_mul_iff'
theorem aleph0_le_mul_iff' {a b : Cardinal.{u}} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ ℵ₀ ≤ b ∨ ℵ₀ ≤ a ∧ b ≠ 0 
Arguments: {a b : Cardinal.{u}}
Goal: Cardinal.aleph0 ≤ a * b ↔ a ≠ 0 ∧ Cardinal.aleph0 ≤ b ∨ Cardinal.aleph0 ≤ a ∧ b ≠ 0
-- Cardinal.mul_lt_aleph0_iff_of_ne_zero
theorem mul_lt_aleph0_iff_of_ne_zero {a b : Cardinal} (ha : a ≠ 0) (hb : b ≠ 0) :
    a * b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ 
Arguments: {a b : Cardinal.{u_1}} (ha : a ≠ 0) (hb : b ≠ 0)
Goal: a * b < Cardinal.aleph0 ↔ a < Cardinal.aleph0 ∧ b < Cardinal.aleph0
-- Cardinal.power_lt_aleph0
theorem power_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a ^ b < ℵ₀ 
Arguments: {a b : Cardinal.{u_1}} (ha : a < Cardinal.aleph0) (hb : b < Cardinal.aleph0)
Goal: a ^ b < Cardinal.aleph0
-- Cardinal.eq_one_iff_unique
theorem eq_one_iff_unique {α : Type*} : #α = 1 ↔ Subsingleton α ∧ Nonempty α 
Arguments: {α : Type u_1}
Goal: Cardinal.mk α = 1 ↔ Subsingleton α ∧ Nonempty α
-- Cardinal.infinite_iff
theorem infinite_iff {α : Type u} : Infinite α ↔ ℵ₀ ≤ #α 
Arguments: {α : Type u}
Goal: Infinite α ↔ Cardinal.aleph0 ≤ Cardinal.mk α
-- Cardinal.aleph0_le_mk_iff
lemma aleph0_le_mk_iff : ℵ₀ ≤ #α ↔ Infinite α 
Arguments: {α : Type u}
Goal: Cardinal.aleph0 ≤ Cardinal.mk α ↔ Infinite α
-- Cardinal.mk_lt_aleph0_iff
lemma mk_lt_aleph0_iff : #α < ℵ₀ ↔ Finite α 
Arguments: {α : Type u}
Goal: Cardinal.mk α < Cardinal.aleph0 ↔ Finite α
-- Cardinal.aleph0_le_mk
@[simp]
theorem aleph0_le_mk (α : Type u) [Infinite α] : ℵ₀ ≤ #α 
Arguments: (α : Type u) [Infinite α]
Goal: Cardinal.aleph0 ≤ Cardinal.mk α
-- Cardinal.mk_eq_aleph0
@[simp]
theorem mk_eq_aleph0 (α : Type*) [Countable α] [Infinite α] : #α = ℵ₀ 
Arguments: (α : Type u_1) [Countable α] [Infinite α]
Goal: Cardinal.mk α = Cardinal.aleph0
-- Cardinal.denumerable_iff
theorem denumerable_iff {α : Type u} : Nonempty (Denumerable α) ↔ #α = ℵ₀ 
Arguments: {α : Type u}
Goal: Nonempty (Denumerable α) ↔ Cardinal.mk α = Cardinal.aleph0
-- Cardinal.mk_denumerable
theorem mk_denumerable (α : Type u) [Denumerable α] : #α = ℵ₀ 
Arguments: (α : Type u) [Denumerable α]
Goal: Cardinal.mk α = Cardinal.aleph0
-- Set.countable_infinite_iff_nonempty_denumerable
theorem _root_.Set.countable_infinite_iff_nonempty_denumerable {α : Type*} {s : Set α} :
    s.Countable ∧ s.Infinite ↔ Nonempty (Denumerable s) 
Arguments: {α : Type u_1} {s : Set α}
Goal: s.Countable ∧ s.Infinite ↔ Nonempty (Denumerable ↑s)
-- Cardinal.aleph0_add_aleph0
@[simp]
theorem aleph0_add_aleph0 : ℵ₀ + ℵ₀ = ℵ₀ 
Goal: Cardinal.aleph0 + Cardinal.aleph0 = Cardinal.aleph0
-- Cardinal.aleph0_mul_aleph0
theorem aleph0_mul_aleph0 : ℵ₀ * ℵ₀ = ℵ₀ 
Goal: Cardinal.aleph0 * Cardinal.aleph0 = Cardinal.aleph0
-- Cardinal.nat_mul_aleph0
@[simp]
theorem nat_mul_aleph0 {n : ℕ} (hn : n ≠ 0) : ↑n * ℵ₀ = ℵ₀ 
Arguments: {n : ℕ} (hn : n ≠ 0)
Goal: ↑n * Cardinal.aleph0 = Cardinal.aleph0
-- Cardinal.aleph0_mul_nat
@[simp]
theorem aleph0_mul_nat {n : ℕ} (hn : n ≠ 0) : ℵ₀ * n = ℵ₀ 
Arguments: {n : ℕ} (hn : n ≠ 0)
Goal: Cardinal.aleph0 * ↑n = Cardinal.aleph0
-- Cardinal.ofNat_mul_aleph0
@[simp]
theorem ofNat_mul_aleph0 {n : ℕ} [Nat.AtLeastTwo n] : no_index (OfNat.ofNat n) * ℵ₀ = ℵ₀ 
Arguments: {n : ℕ} [n.AtLeastTwo]
Goal: OfNat.ofNat n * Cardinal.aleph0 = Cardinal.aleph0
-- Cardinal.aleph0_mul_ofNat
@[simp]
theorem aleph0_mul_ofNat {n : ℕ} [Nat.AtLeastTwo n] : ℵ₀ * no_index (OfNat.ofNat n) = ℵ₀ 
Arguments: {n : ℕ} [n.AtLeastTwo]
Goal: Cardinal.aleph0 * OfNat.ofNat n = Cardinal.aleph0
-- Cardinal.add_le_aleph0
@[simp]
theorem add_le_aleph0 {c₁ c₂ : Cardinal} : c₁ + c₂ ≤ ℵ₀ ↔ c₁ ≤ ℵ₀ ∧ c₂ ≤ ℵ₀ 
Arguments: {c₁ c₂ : Cardinal.{u_1}}
Goal: c₁ + c₂ ≤ Cardinal.aleph0 ↔ c₁ ≤ Cardinal.aleph0 ∧ c₂ ≤ Cardinal.aleph0
-- Cardinal.aleph0_add_nat
@[simp]
theorem aleph0_add_nat (n : ℕ) : ℵ₀ + n = ℵ₀ 
Arguments: (n : ℕ)
Goal: Cardinal.aleph0 + ↑n = Cardinal.aleph0
-- Cardinal.nat_add_aleph0
@[simp]
theorem nat_add_aleph0 (n : ℕ) : ↑n + ℵ₀ = ℵ₀ 
Arguments: (n : ℕ)
Goal: ↑n + Cardinal.aleph0 = Cardinal.aleph0
-- Cardinal.ofNat_add_aleph0
@[simp]
theorem ofNat_add_aleph0 {n : ℕ} [Nat.AtLeastTwo n] : no_index (OfNat.ofNat n) + ℵ₀ = ℵ₀ 
Arguments: {n : ℕ} [n.AtLeastTwo]
Goal: OfNat.ofNat n + Cardinal.aleph0 = Cardinal.aleph0