ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Ctop.Realizer.mem_nhds
protected theorem mem_nhds [T : TopologicalSpace α] (F : Realizer α) {s : Set α} {a : α} :
    s ∈ 𝓝 a ↔ ∃ b, a ∈ F.F b ∧ F.F b ⊆ s 
Arguments: {α : Type u_1} [T : TopologicalSpace α] (F : Ctop.Realizer α) {s : Set α} {a : α}
Goal: s ∈ 𝓝 a ↔ ∃ b, a ∈ F.F.f b ∧ F.F.f b ⊆ s
-- Ctop.Realizer.isOpen_iff
theorem isOpen_iff [TopologicalSpace α] (F : Realizer α) {s : Set α} :
    IsOpen s ↔ ∀ a ∈ s, ∃ b, a ∈ F.F b ∧ F.F b ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] (F : Ctop.Realizer α) {s : Set α}
Goal: IsOpen s ↔ ∀ a ∈ s, ∃ b, a ∈ F.F.f b ∧ F.F.f b ⊆ s
-- Ctop.Realizer.isClosed_iff
theorem isClosed_iff [TopologicalSpace α] (F : Realizer α) {s : Set α} :
    IsClosed s ↔ ∀ a, (∀ b, a ∈ F.F b → ∃ z, z ∈ F.F b ∩ s) → a ∈ s 
Arguments: {α : Type u_1} [TopologicalSpace α] (F : Ctop.Realizer α) {s : Set α}
Goal: IsClosed s ↔ ∀ (a : α), (∀ (b : F.σ), a ∈ F.F.f b → ∃ z, z ∈ F.F.f b ∩ s) → a ∈ s
-- Ctop.Realizer.mem_interior_iff
theorem mem_interior_iff [TopologicalSpace α] (F : Realizer α) {s : Set α} {a : α} :
    a ∈ interior s ↔ ∃ b, a ∈ F.F b ∧ F.F b ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] (F : Ctop.Realizer α) {s : Set α} {a : α}
Goal: a ∈ interior s ↔ ∃ b, a ∈ F.F.f b ∧ F.F.f b ⊆ s
-- Ctop.Realizer.isOpen
protected theorem isOpen [TopologicalSpace α] (F : Realizer α) (s : F.σ) : IsOpen (F.F s) 
Arguments: {α : Type u_1} [TopologicalSpace α] (F : Ctop.Realizer α) (s : F.σ)
Goal: IsOpen (F.F.f s)
-- Ctop.Realizer.ext'
theorem ext' [T : TopologicalSpace α] {σ : Type*} {F : Ctop α σ}
    (H : ∀ a s, s ∈ 𝓝 a ↔ ∃ b, a ∈ F b ∧ F b ⊆ s) : F.toTopsp = T 
Arguments: {α : Type u_1} [T : TopologicalSpace α] {σ : Type u_5} {F : Ctop α σ} (H : ∀ (a : α) (s : Set α), s ∈ 𝓝 a ↔ ∃ b, a ∈ F.f b ∧ F.f b ⊆ s)
Goal: F.toTopsp = T
-- Ctop.Realizer.ext
theorem ext [T : TopologicalSpace α] {σ : Type*} {F : Ctop α σ} (H₁ : ∀ a, IsOpen (F a))
    (H₂ : ∀ a s, s ∈ 𝓝 a → ∃ b, a ∈ F b ∧ F b ⊆ s) : F.toTopsp = T 
Arguments: {α : Type u_1} [T : TopologicalSpace α] {σ : Type u_5} {F : Ctop α σ} (H₁ : ∀ (a : σ), IsOpen (F.f a)) (H₂ : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ b, a ∈ F.f b ∧ F.f b ⊆ s)
Goal: F.toTopsp = T
-- Ctop.Realizer.ofEquiv_σ
@[simp]
theorem ofEquiv_σ (F : Realizer α) (E : F.σ ≃ τ) : (F.ofEquiv E).σ = τ 
Arguments: {α : Type u_1} {τ : Type u_4} [TopologicalSpace α] (F : Ctop.Realizer α) (E : F.σ ≃ τ)
Goal: (F.ofEquiv E).σ = τ
-- Ctop.Realizer.ofEquiv_F
@[simp]
theorem ofEquiv_F (F : Realizer α) (E : F.σ ≃ τ) (s : τ) : (F.ofEquiv E).F s = F.F (E.symm s) 
Arguments: {α : Type u_1} {τ : Type u_4} [TopologicalSpace α] (F : Ctop.Realizer α) (E : F.σ ≃ τ) (s : τ)
Goal: (F.ofEquiv E).F.f s = F.F.f (E.symm s)
-- Ctop.Realizer.nhds_σ
@[simp]
theorem nhds_σ (F : Realizer α) (a : α) : (F.nhds a).σ = { s : F.σ // a ∈ F.F s } 
Arguments: {α : Type u_1} [TopologicalSpace α] (F : Ctop.Realizer α) (a : α)
Goal: (F.nhds a).σ = { s // a ∈ F.F.f s }
-- Ctop.Realizer.nhds_F
@[simp]
theorem nhds_F (F : Realizer α) (a : α) (s) : (F.nhds a).F s = F.F s.1 
Arguments: {α : Type u_1} [TopologicalSpace α] (F : Ctop.Realizer α) (a : α) (s : (F.nhds a).σ)
Goal: (F.nhds a).F.f s = F.F.f ↑s
-- Ctop.Realizer.tendsto_nhds_iff
theorem tendsto_nhds_iff {m : β → α} {f : Filter β} (F : f.Realizer) (R : Realizer α) {a : α} :
    Tendsto m f (𝓝 a) ↔ ∀ t, a ∈ R.F t → ∃ s, ∀ x ∈ F.F s, m x ∈ R.F t 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {m : β → α} {f : Filter β} (F : f.Realizer) (R : Ctop.Realizer α) {a : α}
Goal: Filter.Tendsto m f (𝓝 a) ↔ ∀ (t : R.σ), a ∈ R.F.f t → ∃ s, ∀ x ∈ F.F.f s, m x ∈ R.F.f t
-- LocallyFinite.Realizer.to_locallyFinite
theorem LocallyFinite.Realizer.to_locallyFinite [TopologicalSpace α] {F : Ctop.Realizer α}
    {f : β → Set α} (R : LocallyFinite.Realizer F f) : LocallyFinite f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {F : Ctop.Realizer α} {f : β → Set α} (R : LocallyFinite.Realizer F f)
Goal: LocallyFinite f
-- locallyFinite_iff_exists_realizer
theorem locallyFinite_iff_exists_realizer [TopologicalSpace α] (F : Ctop.Realizer α)
    {f : β → Set α} : LocallyFinite f ↔ Nonempty (LocallyFinite.Realizer F f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] (F : Ctop.Realizer α) {f : β → Set α}
Goal: LocallyFinite f ↔ Nonempty (LocallyFinite.Realizer F f)
-- ENat.some_eq_coe
@[simp] theorem some_eq_coe : (WithTop.some : ℕ → ℕ∞) = Nat.cast 
Goal: WithTop.some = Nat.cast
-- ENat.coe_zero
theorem coe_zero : ((0 : ℕ) : ℕ∞) = 0 
Goal: ↑0 = 0
-- ENat.coe_one
theorem coe_one : ((1 : ℕ) : ℕ∞) = 1 
Goal: ↑1 = 1
-- ENat.coe_add
theorem coe_add (m n : ℕ) : ↑(m + n) = (m + n : ℕ∞) 
Arguments: (m n : ℕ)
Goal: ↑(m + n) = ↑m + ↑n
-- ENat.coe_sub
@[simp, norm_cast]
theorem coe_sub (m n : ℕ) : ↑(m - n) = (m - n : ℕ∞) 
Arguments: (m n : ℕ)
Goal: ↑(m - n) = ↑m - ↑n
-- ENat.coe_mul
@[simp] lemma coe_mul (m n : ℕ) : ↑(m * n) = (m * n : ℕ∞) 
Arguments: (m n : ℕ)
Goal: ↑(m * n) = ↑m * ↑n
-- ENat.mul_top
@[simp] theorem mul_top (hm : m ≠ 0) : m * ⊤ = ⊤ 
Arguments: {m : ℕ∞} (hm : m ≠ 0)
Goal: m * ⊤ = ⊤
-- ENat.top_mul
@[simp] theorem top_mul (hm : m ≠ 0) : ⊤ * m = ⊤ 
Arguments: {m : ℕ∞} (hm : m ≠ 0)
Goal: ⊤ * m = ⊤
-- ENat.top_pow
theorem top_pow {n : ℕ} (n_pos : 0 < n) : (⊤ : ℕ∞) ^ n = ⊤ 
Arguments: {n : ℕ} (n_pos : 0 < n)
Goal: ⊤ ^ n = ⊤
-- ENat.coe_toNatHom
@[simp, norm_cast] lemma coe_toNatHom : toNatHom = toNat 
Goal: ⇑ENat.toNatHom = ENat.toNat
-- ENat.toNatHom_apply
lemma toNatHom_apply (n : ℕ) : toNatHom n = toNat n 
Arguments: (n : ℕ)
Goal: ENat.toNatHom ↑n = (↑n).toNat
-- ENat.toNat_coe
@[simp]
theorem toNat_coe (n : ℕ) : toNat n = n 
Arguments: (n : ℕ)
Goal: (↑n).toNat = n
-- ENat.toNat_ofNat
@[simp]
theorem toNat_ofNat (n : ℕ) [n.AtLeastTwo] : toNat (no_index (OfNat.ofNat n)) = n 
Arguments: (n : ℕ) [n.AtLeastTwo]
Goal: (OfNat.ofNat n).toNat = n
-- ENat.toNat_top
@[simp]
theorem toNat_top : toNat ⊤ = 0 
Goal: ⊤.toNat = 0
-- ENat.toNat_eq_zero
@[simp] theorem toNat_eq_zero : toNat n = 0 ↔ n = 0 ∨ n = ⊤ 
Arguments: {n : ℕ∞}
Goal: n.toNat = 0 ↔ n = 0 ∨ n = ⊤
-- ENat.recTopCoe_top
@[simp]
theorem recTopCoe_top {C : ℕ∞ → Sort*} (d : C ⊤) (f : ∀ a : ℕ, C a) :
    @recTopCoe C d f ⊤ = d 
Arguments: {C : ℕ∞ → Sort u_1} (d : C ⊤) (f : (a : ℕ) → C ↑a)
Goal: ENat.recTopCoe d f ⊤ = d
-- ENat.recTopCoe_coe
@[simp]
theorem recTopCoe_coe {C : ℕ∞ → Sort*} (d : C ⊤) (f : ∀ a : ℕ, C a) (x : ℕ) :
    @recTopCoe C d f ↑x = f x 
Arguments: {C : ℕ∞ → Sort u_1} (d : C ⊤) (f : (a : ℕ) → C ↑a) (x : ℕ)
Goal: ENat.recTopCoe d f ↑x = f x
-- ENat.recTopCoe_zero
@[simp]
theorem recTopCoe_zero {C : ℕ∞ → Sort*} (d : C ⊤) (f : ∀ a : ℕ, C a) : @recTopCoe C d f 0 = f 0 
Arguments: {C : ℕ∞ → Sort u_1} (d : C ⊤) (f : (a : ℕ) → C ↑a)
Goal: ENat.recTopCoe d f 0 = f 0
-- ENat.recTopCoe_one
@[simp]
theorem recTopCoe_one {C : ℕ∞ → Sort*} (d : C ⊤) (f : ∀ a : ℕ, C a) : @recTopCoe C d f 1 = f 1 
Arguments: {C : ℕ∞ → Sort u_1} (d : C ⊤) (f : (a : ℕ) → C ↑a)
Goal: ENat.recTopCoe d f 1 = f 1
-- ENat.recTopCoe_ofNat
@[simp]
theorem recTopCoe_ofNat {C : ℕ∞ → Sort*} (d : C ⊤) (f : ∀ a : ℕ, C a) (x : ℕ) [x.AtLeastTwo] :
    @recTopCoe C d f (no_index (OfNat.ofNat x)) = f (OfNat.ofNat x) 
Arguments: {C : ℕ∞ → Sort u_1} (d : C ⊤) (f : (a : ℕ) → C ↑a) (x : ℕ) [x.AtLeastTwo]
Goal: ENat.recTopCoe d f (OfNat.ofNat x) = f (OfNat.ofNat x)
-- ENat.top_ne_coe
@[simp]
theorem top_ne_coe (a : ℕ) : ⊤ ≠ (a : ℕ∞) 
Arguments: (a : ℕ)
Goal: ⊤ ≠ ↑a
-- ENat.top_ne_ofNat
@[simp]
theorem top_ne_ofNat (a : ℕ) [a.AtLeastTwo] : ⊤ ≠ (no_index (OfNat.ofNat a : ℕ∞)) 
Arguments: (a : ℕ) [a.AtLeastTwo]
Goal: ⊤ ≠ OfNat.ofNat a
-- ENat.top_ne_zero
@[simp] lemma top_ne_zero : (⊤ : ℕ∞) ≠ 0 
Goal: ⊤ ≠ 0
-- ENat.top_ne_one
@[simp] lemma top_ne_one : (⊤ : ℕ∞) ≠ 1 
Goal: ⊤ ≠ 1
-- ENat.coe_ne_top
@[simp]
theorem coe_ne_top (a : ℕ) : (a : ℕ∞) ≠ ⊤ 
Arguments: (a : ℕ)
Goal: ↑a ≠ ⊤
-- ENat.ofNat_ne_top
@[simp]
theorem ofNat_ne_top (a : ℕ) [a.AtLeastTwo] : (no_index (OfNat.ofNat a : ℕ∞)) ≠ ⊤ 
Arguments: (a : ℕ) [a.AtLeastTwo]
Goal: OfNat.ofNat a ≠ ⊤
-- ENat.zero_ne_top
@[simp] lemma zero_ne_top : 0 ≠ (⊤ : ℕ∞) 
Goal: 0 ≠ ⊤
-- ENat.one_ne_top
@[simp] lemma one_ne_top : 1 ≠ (⊤ : ℕ∞) 
Goal: 1 ≠ ⊤
-- ENat.top_sub_coe
@[simp]
theorem top_sub_coe (a : ℕ) : (⊤ : ℕ∞) - a = ⊤ 
Arguments: (a : ℕ)
Goal: ⊤ - ↑a = ⊤
-- ENat.top_sub_one
@[simp]
theorem top_sub_one : (⊤ : ℕ∞) - 1 = ⊤ 
Goal: ⊤ - 1 = ⊤
-- ENat.top_sub_ofNat
@[simp]
theorem top_sub_ofNat (a : ℕ) [a.AtLeastTwo] : (⊤ : ℕ∞) - (no_index (OfNat.ofNat a)) = ⊤ 
Arguments: (a : ℕ) [a.AtLeastTwo]
Goal: ⊤ - OfNat.ofNat a = ⊤
-- ENat.zero_lt_top
@[simp]
theorem zero_lt_top : (0 : ℕ∞) < ⊤ 
Goal: 0 < ⊤
-- ENat.sub_top
theorem sub_top (a : ℕ∞) : a - ⊤ = 0 
Arguments: (a : ℕ∞)
Goal: a - ⊤ = 0
-- ENat.coe_toNat_eq_self
@[simp]
theorem coe_toNat_eq_self : ENat.toNat n = n ↔ n ≠ ⊤ 
Arguments: {n : ℕ∞}
Goal: ↑n.toNat = n ↔ n ≠ ⊤
-- ENat.coe_toNat_le_self
theorem coe_toNat_le_self (n : ℕ∞) : ↑(toNat n) ≤ n 
Arguments: (n : ℕ∞)
Goal: ↑n.toNat ≤ n
-- ENat.toNat_add
theorem toNat_add {m n : ℕ∞} (hm : m ≠ ⊤) (hn : n ≠ ⊤) : toNat (m + n) = toNat m + toNat n 
Arguments: {m n : ℕ∞} (hm : m ≠ ⊤) (hn : n ≠ ⊤)
Goal: (m + n).toNat = m.toNat + n.toNat
-- ENat.toNat_sub
theorem toNat_sub {n : ℕ∞} (hn : n ≠ ⊤) (m : ℕ∞) : toNat (m - n) = toNat m - toNat n 
Arguments: {n : ℕ∞} (hn : n ≠ ⊤) (m : ℕ∞)
Goal: (m - n).toNat = m.toNat - n.toNat
-- ENat.toNat_eq_iff
theorem toNat_eq_iff {m : ℕ∞} {n : ℕ} (hn : n ≠ 0) : toNat m = n ↔ m = n 
Arguments: {m : ℕ∞} {n : ℕ} (hn : n ≠ 0)
Goal: m.toNat = n ↔ m = ↑n
-- ENat.toNat_le_of_le_coe
lemma toNat_le_of_le_coe {m : ℕ∞} {n : ℕ} (h : m ≤ n) : toNat m ≤ n 
Arguments: {m : ℕ∞} {n : ℕ} (h : m ≤ ↑n)
Goal: m.toNat ≤ n
-- ENat.toNat_le_toNat
@[gcongr]
lemma toNat_le_toNat {m n : ℕ∞} (h : m ≤ n) (hn : n ≠ ⊤) : toNat m ≤ toNat n 
Arguments: {m n : ℕ∞} (h : m ≤ n) (hn : n ≠ ⊤)
Goal: m.toNat ≤ n.toNat
-- ENat.succ_def
@[simp]
theorem succ_def (m : ℕ∞) : Order.succ m = m + 1 
Arguments: (m : ℕ∞)
Goal: Order.succ m = m + 1
-- ENat.add_one_le_of_lt
theorem add_one_le_of_lt (h : m < n) : m + 1 ≤ n 
Arguments: {m n : ℕ∞} (h : m < n)
Goal: m + 1 ≤ n
-- ENat.add_one_le_iff
theorem add_one_le_iff (hm : m ≠ ⊤) : m + 1 ≤ n ↔ m < n 
Arguments: {m n : ℕ∞} (hm : m ≠ ⊤)
Goal: m + 1 ≤ n ↔ m < n
-- ENat.one_le_iff_pos
theorem one_le_iff_pos : 1 ≤ n ↔ 0 < n 
Arguments: {n : ℕ∞}
Goal: 1 ≤ n ↔ 0 < n
-- ENat.one_le_iff_ne_zero
theorem one_le_iff_ne_zero : 1 ≤ n ↔ n ≠ 0 
Arguments: {n : ℕ∞}
Goal: 1 ≤ n ↔ n ≠ 0
-- ENat.lt_one_iff_eq_zero
lemma lt_one_iff_eq_zero : n < 1 ↔ n = 0 
Arguments: {n : ℕ∞}
Goal: n < 1 ↔ n = 0
-- ENat.le_of_lt_add_one
theorem le_of_lt_add_one (h : m < n + 1) : m ≤ n 
Arguments: {m n : ℕ∞} (h : m < n + 1)
Goal: m ≤ n
-- ENat.le_coe_iff
theorem le_coe_iff {n : ℕ∞} {k : ℕ} : n ≤ ↑k ↔ ∃ (n₀ : ℕ), n = n₀ ∧ n₀ ≤ k 
Arguments: {n : ℕ∞} {k : ℕ}
Goal: n ≤ ↑k ↔ ∃ n₀, n = ↑n₀ ∧ n₀ ≤ k
-- ENat.nat_induction
@[elab_as_elim]
theorem nat_induction {P : ℕ∞ → Prop} (a : ℕ∞) (h0 : P 0) (hsuc : ∀ n : ℕ, P n → P n.succ)
    (htop : (∀ n : ℕ, P n) → P ⊤) : P a 
Arguments: {P : ℕ∞ → Prop} (a : ℕ∞) (h0 : P 0) (hsuc : ∀ (n : ℕ), P ↑n → P ↑n.succ) (htop : (∀ (n : ℕ), P ↑n) → P ⊤)
Goal: P a
-- ZMod.eq_zero_iff_even
theorem eq_zero_iff_even {n : ℕ} : (n : ZMod 2) = 0 ↔ Even n 
Arguments: {n : ℕ}
Goal: ↑n = 0 ↔ Even n
-- ZMod.eq_one_iff_odd
theorem eq_one_iff_odd {n : ℕ} : (n : ZMod 2) = 1 ↔ Odd n 
Arguments: {n : ℕ}
Goal: ↑n = 1 ↔ Odd n
-- ZMod.ne_zero_iff_odd
theorem ne_zero_iff_odd {n : ℕ} : (n : ZMod 2) ≠ 0 ↔ Odd n 
Arguments: {n : ℕ}
Goal: ↑n ≠ 0 ↔ Odd n
-- ZMod.cast_descFactorial
theorem cast_descFactorial {n p : ℕ} (h : n ≤ p) :
    (descFactorial (p - 1) n : ZMod p) = (-1) ^ n * n ! 
Arguments: {n p : ℕ} (h : n ≤ p)
Goal: ↑((p - 1).descFactorial n) = (-1) ^ n * ↑n !
-- ZMod.eq_zero_iff_gcd_ne_one
theorem eq_zero_iff_gcd_ne_one {a : ℤ} {p : ℕ} [pp : Fact p.Prime] :
    (a : ZMod p) = 0 ↔ a.gcd p ≠ 1 
Arguments: {a : ℤ} {p : ℕ} [pp : Fact (Nat.Prime p)]
Goal: ↑a = 0 ↔ a.gcd ↑p ≠ 1
-- ZMod.ne_zero_of_gcd_eq_one
theorem ne_zero_of_gcd_eq_one {a : ℤ} {p : ℕ} (pp : p.Prime) (h : a.gcd p = 1) : (a : ZMod p) ≠ 0 
Arguments: {a : ℤ} {p : ℕ} (pp : Nat.Prime p) (h : a.gcd ↑p = 1)
Goal: ↑a ≠ 0
-- ZMod.eq_zero_of_gcd_ne_one
theorem eq_zero_of_gcd_ne_one {a : ℤ} {p : ℕ} (pp : p.Prime) (h : a.gcd p ≠ 1) : (a : ZMod p) = 0 
Arguments: {a : ℤ} {p : ℕ} (pp : Nat.Prime p) (h : a.gcd ↑p ≠ 1)
Goal: ↑a = 0
-- ZMod.smul_units_def
lemma ZMod.smul_units_def (z : ZMod 2) (au : Additive ℤˣ) :
    z • au = z.val • au 
Arguments: (z : ZMod 2) (au : Additive ℤˣ)
Goal: z • au = z.val • au
-- ZMod.natCast_smul_units
lemma ZMod.natCast_smul_units (n : ℕ) (au : Additive ℤˣ) : (n : ZMod 2) • au = n • au 
Arguments: (n : ℕ) (au : Additive ℤˣ)
Goal: ↑n • au = n • au
-- ofMul_uzpow
@[simp] lemma ofMul_uzpow (u : ℤˣ) (r : R) : Additive.ofMul (u ^ r) = r • Additive.ofMul u 
Arguments: {R : Type u_1} [CommSemiring R] [Module R (Additive ℤˣ)] (u : ℤˣ) (r : R)
Goal: Additive.ofMul (u ^ r) = r • Additive.ofMul u
-- toMul_uzpow
@[simp] lemma toMul_uzpow (u : Additive ℤˣ) (r : R) :
  Additive.toMul (r • u) = Additive.toMul u ^ r 
Arguments: {R : Type u_1} [CommSemiring R] [Module R (Additive ℤˣ)] (u : Additive ℤˣ) (r : R)
Goal: Additive.toMul (r • u) = Additive.toMul u ^ r
-- uzpow_natCast
@[norm_cast] lemma uzpow_natCast (u : ℤˣ) (n : ℕ) : u ^ (n : R) = u ^ n 
Arguments: {R : Type u_1} [CommSemiring R] [Module R (Additive ℤˣ)] (u : ℤˣ) (n : ℕ)
Goal: u ^ ↑n = u ^ n
-- uzpow_coe_nat
lemma uzpow_coe_nat (s : ℤˣ) (n : ℕ) [n.AtLeastTwo] :
    s ^ (no_index (OfNat.ofNat n : R)) = s ^ (no_index (OfNat.ofNat n : ℕ)) 
Arguments: {R : Type u_1} [CommSemiring R] [Module R (Additive ℤˣ)] (s : ℤˣ) (n : ℕ) [n.AtLeastTwo]
Goal: s ^ OfNat.ofNat n = s ^ OfNat.ofNat n
-- one_uzpow
@[simp] lemma one_uzpow (x : R) : (1 : ℤˣ) ^ x = 1 
Arguments: {R : Type u_1} [CommSemiring R] [Module R (Additive ℤˣ)] (x : R)
Goal: 1 ^ x = 1
-- mul_uzpow
lemma mul_uzpow (s₁ s₂ : ℤˣ) (x : R) : (s₁ * s₂) ^ x = s₁ ^ x * s₂ ^ x 
Arguments: {R : Type u_1} [CommSemiring R] [Module R (Additive ℤˣ)] (s₁ s₂ : ℤˣ) (x : R)
Goal: (s₁ * s₂) ^ x = s₁ ^ x * s₂ ^ x
-- uzpow_zero
@[simp] lemma uzpow_zero (s : ℤˣ) : (s ^ (0 : R) : ℤˣ) = (1 : ℤˣ) 
Arguments: {R : Type u_1} [CommSemiring R] [Module R (Additive ℤˣ)] (s : ℤˣ)
Goal: s ^ 0 = 1
-- uzpow_one
@[simp] lemma uzpow_one (s : ℤˣ) : (s ^ (1 : R) : ℤˣ) = s 
Arguments: {R : Type u_1} [CommSemiring R] [Module R (Additive ℤˣ)] (s : ℤˣ)
Goal: s ^ 1 = s
-- uzpow_mul
lemma uzpow_mul (s : ℤˣ) (x y : R) : s ^ (x * y) = (s ^ x) ^ y 
Arguments: {R : Type u_1} [CommSemiring R] [Module R (Additive ℤˣ)] (s : ℤˣ) (x y : R)
Goal: s ^ (x * y) = (s ^ x) ^ y
-- uzpow_add
lemma uzpow_add (s : ℤˣ) (x y : R) : s ^ (x + y) = s ^ x * s ^ y 
Arguments: {R : Type u_1} [CommSemiring R] [Module R (Additive ℤˣ)] (s : ℤˣ) (x y : R)
Goal: s ^ (x + y) = s ^ x * s ^ y
-- uzpow_sub
lemma uzpow_sub (s : ℤˣ) (x y : R) : s ^ (x - y) = s ^ x / s ^ y 
Arguments: {R : Type u_1} [CommRing R] [Module R (Additive ℤˣ)] (s : ℤˣ) (x y : R)
Goal: s ^ (x - y) = s ^ x / s ^ y
-- uzpow_neg
lemma uzpow_neg (s : ℤˣ) (x : R) : s ^ (-x) = (s ^ x)⁻¹ 
Arguments: {R : Type u_1} [CommRing R] [Module R (Additive ℤˣ)] (s : ℤˣ) (x : R)
Goal: s ^ (-x) = (s ^ x)⁻¹
-- uzpow_intCast
@[norm_cast] lemma uzpow_intCast (u : ℤˣ) (z : ℤ) : u ^ (z : R) = u ^ z 
Arguments: {R : Type u_1} [CommRing R] [Module R (Additive ℤˣ)] (u : ℤˣ) (z : ℤ)
Goal: u ^ ↑z = u ^ z
-- CFilter.inf_le_left
structure CFilter (α σ : Type*) [PartialOrder α] where
  f : σ → α
  pt : σ
  inf : σ → σ → σ
  inf_le_left : ∀ a b : σ, f (inf a b) ≤ f a
  inf_le_right : ∀ a b : σ, f (inf a b) ≤ f b
Arguments: {α : Type u_1} {σ : Type u_2} [PartialOrder α] (self : CFilter α σ) (a b : σ)
Goal: self.f (self.inf a b) ≤ self.f a
-- CFilter.inf_le_right
structure CFilter (α σ : Type*) [PartialOrder α] where
  f : σ → α
  pt : σ
  inf : σ → σ → σ
  inf_le_left : ∀ a b : σ, f (inf a b) ≤ f a
  inf_le_right : ∀ a b : σ, f (inf a b) ≤ f b
Arguments: {α : Type u_1} {σ : Type u_2} [PartialOrder α] (self : CFilter α σ) (a b : σ)
Goal: self.f (self.inf a b) ≤ self.f b
-- CFilter.coe_mk
theorem coe_mk (f pt inf h₁ h₂ a) : (@CFilter.mk α σ _ f pt inf h₁ h₂) a = f a 
Arguments: {α : Type u_1} {σ : Type u_3} [PartialOrder α] (f : σ → α) (pt : σ) (inf : σ → σ → σ) (h₁ : ∀ (a b : σ), f (inf a b) ≤ f a) (h₂ : ∀ (a b : σ), f (inf a b) ≤ f b) (a : σ)
Goal: { f := f, pt := pt, inf := inf, inf_le_left := h₁, inf_le_right := h₂ }.f a = f a
-- CFilter.ofEquiv_val
@[simp]
theorem ofEquiv_val (E : σ ≃ τ) (F : CFilter α σ) (a : τ) : F.ofEquiv E a = F (E.symm a) 
Arguments: {α : Type u_1} {σ : Type u_3} {τ : Type u_4} [PartialOrder α] (E : σ ≃ τ) (F : CFilter α σ) (a : τ)
Goal: (CFilter.ofEquiv E F).f a = F.f (E.symm a)
-- CFilter.mem_toFilter_sets
@[simp]
theorem mem_toFilter_sets (F : CFilter (Set α) σ) {a : Set α} : a ∈ F.toFilter ↔ ∃ b, F b ⊆ a 
Arguments: {α : Type u_1} {σ : Type u_3} (F : CFilter (Set α) σ) {a : Set α}
Goal: a ∈ F.toFilter ↔ ∃ b, F.f b ⊆ a
-- Filter.Realizer.eq
structure Filter.Realizer (f : Filter α) where
  σ : Type*
  F : CFilter (Set α) σ
  eq : F.toFilter = f
Arguments: {α : Type u_1} {f : Filter α} (self : f.Realizer)
Goal: self.F.toFilter = f
-- Filter.Realizer.mem_sets
theorem mem_sets {f : Filter α} (F : f.Realizer) {a : Set α} : a ∈ f ↔ ∃ b, F.F b ⊆ a 
Arguments: {α : Type u_1} {f : Filter α} (F : f.Realizer) {a : Set α}
Goal: a ∈ f ↔ ∃ b, F.F.f b ⊆ a
-- Filter.Realizer.ofEquiv_σ
@[simp]
theorem ofEquiv_σ {f : Filter α} (F : f.Realizer) (E : F.σ ≃ τ) : (F.ofEquiv E).σ = τ 
Arguments: {α : Type u_1} {τ : Type u_4} {f : Filter α} (F : f.Realizer) (E : F.σ ≃ τ)
Goal: (F.ofEquiv E).σ = τ
-- Filter.Realizer.ofEquiv_F
@[simp]
theorem ofEquiv_F {f : Filter α} (F : f.Realizer) (E : F.σ ≃ τ) (s : τ) :
    (F.ofEquiv E).F s = F.F (E.symm s) 
Arguments: {α : Type u_1} {τ : Type u_4} {f : Filter α} (F : f.Realizer) (E : F.σ ≃ τ) (s : τ)
Goal: (F.ofEquiv E).F.f s = F.F.f (E.symm s)
-- Filter.Realizer.principal_σ
@[simp]
theorem principal_σ (s : Set α) : (Realizer.principal s).σ = Unit 
Arguments: {α : Type u_1} (s : Set α)
Goal: (Filter.Realizer.principal s).σ = Unit
-- Filter.Realizer.principal_F
@[simp]
theorem principal_F (s : Set α) (u : Unit) : (Realizer.principal s).F u = s 
Arguments: {α : Type u_1} (s : Set α) (u : Unit)
Goal: (Filter.Realizer.principal s).F.f u = s
-- Filter.Realizer.top_σ
@[simp]
theorem top_σ : (@Realizer.top α).σ = Unit 
Arguments: {α : Type u_1}
Goal: Filter.Realizer.top.σ = Unit
-- Filter.Realizer.top_F
@[simp]
theorem top_F (u : Unit) : (@Realizer.top α).F u = univ 
Arguments: {α : Type u_1} (u : Unit)
Goal: Filter.Realizer.top.F.f u = Set.univ
-- Filter.Realizer.bot_σ
@[simp]
theorem bot_σ : (@Realizer.bot α).σ = Unit 
Arguments: {α : Type u_1}
Goal: Filter.Realizer.bot.σ = Unit
-- Filter.Realizer.bot_F
@[simp]
theorem bot_F (u : Unit) : (@Realizer.bot α).F u = ∅ 
Arguments: {α : Type u_1} (u : Unit)
Goal: Filter.Realizer.bot.F.f u = ∅
-- Filter.Realizer.map_σ
@[simp]
theorem map_σ (m : α → β) {f : Filter α} (F : f.Realizer) : (F.map m).σ = F.σ 
Arguments: {α : Type u_1} {β : Type u_2} (m : α → β) {f : Filter α} (F : f.Realizer)
Goal: (Filter.Realizer.map m F).σ = F.σ
-- Filter.Realizer.map_F
@[simp]
theorem map_F (m : α → β) {f : Filter α} (F : f.Realizer) (s) : (F.map m).F s = image m (F.F s) 
Arguments: {α : Type u_1} {β : Type u_2} (m : α → β) {f : Filter α} (F : f.Realizer) (s : (Filter.Realizer.map m F).σ)
Goal: (Filter.Realizer.map m F).F.f s = m '' F.F.f s
-- Filter.Realizer.le_iff
theorem le_iff {f g : Filter α} (F : f.Realizer) (G : g.Realizer) :
    f ≤ g ↔ ∀ b : G.σ, ∃ a : F.σ, F.F a ≤ G.F b 
Arguments: {α : Type u_1} {f g : Filter α} (F : f.Realizer) (G : g.Realizer)
Goal: f ≤ g ↔ ∀ (b : G.σ), ∃ a, F.F.f a ≤ G.F.f b
-- Filter.Realizer.tendsto_iff
theorem tendsto_iff (f : α → β) {l₁ : Filter α} {l₂ : Filter β} (L₁ : l₁.Realizer)
    (L₂ : l₂.Realizer) : Tendsto f l₁ l₂ ↔ ∀ b, ∃ a, ∀ x ∈ L₁.F a, f x ∈ L₂.F b 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) {l₁ : Filter α} {l₂ : Filter β} (L₁ : l₁.Realizer) (L₂ : l₂.Realizer)
Goal: Filter.Tendsto f l₁ l₂ ↔ ∀ (b : L₂.σ), ∃ a, ∀ x ∈ L₁.F.f a, f x ∈ L₂.F.f b
-- Filter.Realizer.ne_bot_iff
theorem ne_bot_iff {f : Filter α} (F : f.Realizer) : f ≠ ⊥ ↔ ∀ a : F.σ, (F.F a).Nonempty 
Arguments: {α : Type u_1} {f : Filter α} (F : f.Realizer)
Goal: f ≠ ⊥ ↔ ∀ (a : F.σ), (F.F.f a).Nonempty
-- ZMod.map_smul
theorem map_smul (f : F) (c : ZMod n) (x : M) : f (c • x) = c • f x 
Arguments: {n : ℕ} {M : Type u_1} {M₁ : Type u_2} {F : Type u_3} [AddCommGroup M] [AddCommGroup M₁] [FunLike F M M₁] [AddMonoidHomClass F M M₁] [Module (ZMod n) M] [Module (ZMod n) M₁] (f : F) (c : ZMod n) (x : M)
Goal: f (c • x) = c • f x
-- ZMod.smul_mem
theorem smul_mem (hx : x ∈ K) (c : ZMod n) : c • x ∈ K 
Arguments: {n : ℕ} {M : Type u_1} {S : Type u_4} [AddCommGroup M] [Module (ZMod n) M] [SetLike S M] [AddSubgroupClass S M] {x : M} {K : S} (hx : x ∈ K) (c : ZMod n)
Goal: c • x ∈ K
-- AddMonoidHom.toZModLinearMap_injective
theorem toZModLinearMap_injective : Function.Injective <| toZModLinearMap n (M := M) (M₁ := M₁) 
Arguments: (n : ℕ) {M : Type u_1} {M₁ : Type u_2} [AddCommGroup M] [AddCommGroup M₁] [Module (ZMod n) M] [Module (ZMod n) M₁]
Goal: Function.Injective (AddMonoidHom.toZModLinearMap n)
-- AddMonoidHom.coe_toZModLinearMap
@[simp]
theorem coe_toZModLinearMap (f : M →+ M₁) : ⇑(f.toZModLinearMap n) = f 
Arguments: (n : ℕ) {M : Type u_1} {M₁ : Type u_2} [AddCommGroup M] [AddCommGroup M₁] [Module (ZMod n) M] [Module (ZMod n) M₁] (f : M →+ M₁)
Goal: ⇑(AddMonoidHom.toZModLinearMap n f) = ⇑f
-- AddSubgroup.toZModSubmodule_symm
@[simp]
theorem toZModSubmodule_symm :
    ⇑((toZModSubmodule n).symm : _ ≃o AddSubgroup M) = Submodule.toAddSubgroup 
Arguments: (n : ℕ) {M : Type u_1} [AddCommGroup M] [Module (ZMod n) M]
Goal: ⇑(AddSubgroup.toZModSubmodule n).symm = Submodule.toAddSubgroup
-- AddSubgroup.coe_toZModSubmodule
@[simp]
theorem coe_toZModSubmodule (S : AddSubgroup M) : (toZModSubmodule n S : Set M) = S 
Arguments: (n : ℕ) {M : Type u_1} [AddCommGroup M] [Module (ZMod n) M] (S : AddSubgroup M)
Goal: ↑((AddSubgroup.toZModSubmodule n) S) = ↑S
-- AddSubgroup.toZModSubmodule_toAddSubgroup
@[simp]
theorem toZModSubmodule_toAddSubgroup (S : AddSubgroup M) :
    (toZModSubmodule n S).toAddSubgroup = S 
Arguments: (n : ℕ) {M : Type u_1} [AddCommGroup M] [Module (ZMod n) M] (S : AddSubgroup M)
Goal: ((AddSubgroup.toZModSubmodule n) S).toAddSubgroup = S
-- Submodule.toAddSubgroup_toZModSubmodule
@[simp]
theorem _root_.Submodule.toAddSubgroup_toZModSubmodule (S : Submodule (ZMod n) M) :
    toZModSubmodule n S.toAddSubgroup = S 
Arguments: (n : ℕ) {M : Type u_1} [AddCommGroup M] [Module (ZMod n) M] (S : Submodule (ZMod n) M)
Goal: (AddSubgroup.toZModSubmodule n) S.toAddSubgroup = S
-- Finset.Icc_add_Icc_subset
@[to_additive Icc_add_Icc_subset]
theorem Icc_mul_Icc_subset' [LocallyFiniteOrder α] (a b c d : α) :
    Icc a b * Icc c d ⊆ Icc (a * c) (b * d) 
Arguments: {α : Type u_1} [Add α] [Preorder α] [DecidableEq α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap HAdd.hAdd) LE.le] [LocallyFiniteOrder α] (a b c d : α)
Goal: Finset.Icc a b + Finset.Icc c d ⊆ Finset.Icc (a + c) (b + d)
-- Finset.Icc_mul_Icc_subset'
@[to_additive Icc_add_Icc_subset]
theorem Icc_mul_Icc_subset' [LocallyFiniteOrder α] (a b c d : α) :
    Icc a b * Icc c d ⊆ Icc (a * c) (b * d) 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [DecidableEq α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap HMul.hMul) LE.le] [LocallyFiniteOrder α] (a b c d : α)
Goal: Finset.Icc a b * Finset.Icc c d ⊆ Finset.Icc (a * c) (b * d)
-- Finset.Iic_mul_Iic_subset'
@[to_additive Iic_add_Iic_subset]
theorem Iic_mul_Iic_subset' [LocallyFiniteOrderBot α] (a b : α) : Iic a * Iic b ⊆ Iic (a * b) 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [DecidableEq α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap HMul.hMul) LE.le] [LocallyFiniteOrderBot α] (a b : α)
Goal: Finset.Iic a * Finset.Iic b ⊆ Finset.Iic (a * b)
-- Finset.Iic_add_Iic_subset
@[to_additive Iic_add_Iic_subset]
theorem Iic_mul_Iic_subset' [LocallyFiniteOrderBot α] (a b : α) : Iic a * Iic b ⊆ Iic (a * b) 
Arguments: {α : Type u_1} [Add α] [Preorder α] [DecidableEq α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap HAdd.hAdd) LE.le] [LocallyFiniteOrderBot α] (a b : α)
Goal: Finset.Iic a + Finset.Iic b ⊆ Finset.Iic (a + b)