ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- nhdsWithin_Iic_basis_Icc
theorem nhdsWithin_Iic_basis_Icc [NoMinOrder α] [DenselyOrdered α] {a : α} :
    (𝓝[≤] a).HasBasis (· < a) (Icc · a) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α] [DenselyOrdered α] {a : α}
Goal: (𝓝[≤] a).HasBasis (fun x => x < a) fun x => Set.Icc x a
-- nhds_eq_iInf_abs_sub
theorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α] (a : α)
Goal: 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}
-- orderTopology_of_nhds_abs
theorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]
    (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α 
Arguments: {α : Type u_4} [TopologicalSpace α] [LinearOrderedAddCommGroup α] (h_nhds : ∀ (a : α), 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r})
Goal: OrderTopology α
-- LinearOrderedAddCommGroup.tendsto_nhds
theorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :
    Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α] {f : β → α} {x : Filter β} {a : α}
Goal: Filter.Tendsto f x (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ (b : β) in x, |f b - a| < ε
-- eventually_abs_sub_lt
theorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α] (a : α) {ε : α} (hε : 0 < ε)
Goal: ∀ᶠ (x : α) in 𝓝 a, |x - a| < ε
-- Filter.Tendsto.add_atTop
theorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :
    Tendsto (fun x => f x + g x) l atTop 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α] {l : Filter β} {f g : β → α} {C : α} (hf : Filter.Tendsto f l (𝓝 C)) (hg : Filter.Tendsto g l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atTop
-- Filter.Tendsto.add_atBot
theorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :
    Tendsto (fun x => f x + g x) l atBot 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α] {l : Filter β} {f g : β → α} {C : α} (hf : Filter.Tendsto f l (𝓝 C)) (hg : Filter.Tendsto g l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atBot
-- Filter.Tendsto.atTop_add
theorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :
    Tendsto (fun x => f x + g x) l atTop 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α] {l : Filter β} {f g : β → α} {C : α} (hf : Filter.Tendsto f l Filter.atTop) (hg : Filter.Tendsto g l (𝓝 C))
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atTop
-- Filter.Tendsto.atBot_add
theorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :
    Tendsto (fun x => f x + g x) l atBot 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α] {l : Filter β} {f g : β → α} {C : α} (hf : Filter.Tendsto f l Filter.atBot) (hg : Filter.Tendsto g l (𝓝 C))
Goal: Filter.Tendsto (fun x => f x + g x) l Filter.atBot
-- nhds_basis_abs_sub_lt
theorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :
    (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α] [NoMaxOrder α] (a : α)
Goal: (𝓝 a).HasBasis (fun ε => 0 < ε) fun ε => {b | |b - a| < ε}
-- nhds_basis_Ioo_pos
theorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :
    (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α] [NoMaxOrder α] (a : α)
Goal: (𝓝 a).HasBasis (fun ε => 0 < ε) fun ε => Set.Ioo (a - ε) (a + ε)
-- nhds_basis_Icc_pos
theorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :
    (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α] [NoMaxOrder α] [DenselyOrdered α] (a : α)
Goal: (𝓝 a).HasBasis (fun x => 0 < x) fun ε => Set.Icc (a - ε) (a + ε)
-- nhds_basis_zero_abs_sub_lt
theorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :
    (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } 
Arguments: (α : Type u_1) [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α] [NoMaxOrder α]
Goal: (𝓝 0).HasBasis (fun ε => 0 < ε) fun ε => {b | |b| < ε}
-- nhds_basis_Ioo_pos_of_pos
theorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :
    (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrderedAddCommGroup α] [OrderTopology α] [NoMaxOrder α] {a : α} (ha : 0 < a)
Goal: (𝓝 a).HasBasis (fun ε => 0 < ε ∧ ε ≤ a) fun ε => Set.Ioo (a - ε) (a + ε)
-- Set.OrdConnected.mem_nhdsWithin_Ici
lemma mem_nhdsWithin_Ici [NoMaxOrder α] {S : Set α} (hS : OrdConnected S)
    {x y : α} (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) :
    S ∈ 𝓝[≥] x 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMaxOrder α] {S : Set α} (hS : S.OrdConnected) {x y : α} (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
Goal: S ∈ 𝓝[≥] x
-- Set.OrdConnected.mem_nhdsWithin_Ioi
lemma mem_nhdsWithin_Ioi [NoMaxOrder α] {S : Set α} (hS : OrdConnected S)
    {x y : α} (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) :
    S ∈ 𝓝[>] x 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMaxOrder α] {S : Set α} (hS : S.OrdConnected) {x y : α} (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
Goal: S ∈ 𝓝[>] x
-- Set.OrdConnected.mem_nhdsWithin_Iic
lemma mem_nhdsWithin_Iic [NoMinOrder α] {S : Set α} (hS : OrdConnected S)
    {x y : α} (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) :
    S ∈ 𝓝[≤] y 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] {S : Set α} (hS : S.OrdConnected) {x y : α} (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
Goal: S ∈ 𝓝[≤] y
-- Set.OrdConnected.mem_nhdsWithin_Iio
lemma mem_nhdsWithin_Iio [NoMinOrder α] {S : Set α} (hS : OrdConnected S)
    {x y : α} (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) :
    S ∈ 𝓝[<] y 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] [NoMinOrder α] {S : Set α} (hS : S.OrdConnected) {x y : α} (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
Goal: S ∈ 𝓝[<] y
-- ContinuousMap.nullhomotopic_of_constant
theorem nullhomotopic_of_constant (y : Y) : Nullhomotopic (ContinuousMap.const X y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (y : Y)
Goal: (ContinuousMap.const X y).Nullhomotopic
-- ContinuousMap.Nullhomotopic.comp_right
theorem Nullhomotopic.comp_right {f : C(X, Y)} (hf : f.Nullhomotopic) (g : C(Y, Z)) :
    (g.comp f).Nullhomotopic 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : f.Nullhomotopic) (g : C(Y, Z))
Goal: (g.comp f).Nullhomotopic
-- ContinuousMap.Nullhomotopic.comp_left
theorem Nullhomotopic.comp_left {f : C(Y, Z)} (hf : f.Nullhomotopic) (g : C(X, Y)) :
    (f.comp g).Nullhomotopic 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(Y, Z)} (hf : f.Nullhomotopic) (g : C(X, Y))
Goal: (f.comp g).Nullhomotopic
-- ContractibleSpace.hequiv_unit'
class ContractibleSpace (X : Type*) [TopologicalSpace X] : Prop where
  hequiv_unit' : Nonempty (X ≃ₕ Unit)
Arguments: {X : Type u_1} [TopologicalSpace X] [self : ContractibleSpace X]
Goal: Nonempty (X ≃ₕ Unit)
-- ContractibleSpace.hequiv_unit
theorem ContractibleSpace.hequiv_unit (X : Type*) [TopologicalSpace X] [ContractibleSpace X] :
    Nonempty (X ≃ₕ Unit) 
Arguments: (X : Type u_1) [TopologicalSpace X] [ContractibleSpace X]
Goal: Nonempty (X ≃ₕ Unit)
-- id_nullhomotopic
theorem id_nullhomotopic (X : Type*) [TopologicalSpace X] [ContractibleSpace X] :
    (ContinuousMap.id X).Nullhomotopic 
Arguments: (X : Type u_1) [TopologicalSpace X] [ContractibleSpace X]
Goal: (ContinuousMap.id X).Nullhomotopic
-- contractible_iff_id_nullhomotopic
theorem contractible_iff_id_nullhomotopic (Y : Type*) [TopologicalSpace Y] :
    ContractibleSpace Y ↔ (ContinuousMap.id Y).Nullhomotopic 
Arguments: (Y : Type u_1) [TopologicalSpace Y]
Goal: ContractibleSpace Y ↔ (ContinuousMap.id Y).Nullhomotopic
-- ContinuousMap.HomotopyEquiv.contractibleSpace
protected theorem ContinuousMap.HomotopyEquiv.contractibleSpace [ContractibleSpace Y] (e : X ≃ₕ Y) :
    ContractibleSpace X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [ContractibleSpace Y] (e : X ≃ₕ Y)
Goal: ContractibleSpace X
-- ContinuousMap.HomotopyEquiv.contractibleSpace_iff
protected theorem ContinuousMap.HomotopyEquiv.contractibleSpace_iff (e : X ≃ₕ Y) :
    ContractibleSpace X ↔ ContractibleSpace Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₕ Y)
Goal: ContractibleSpace X ↔ ContractibleSpace Y
-- Homeomorph.contractibleSpace
protected theorem Homeomorph.contractibleSpace [ContractibleSpace Y] (e : X ≃ₜ Y) :
    ContractibleSpace X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [ContractibleSpace Y] (e : X ≃ₜ Y)
Goal: ContractibleSpace X
-- Homeomorph.contractibleSpace_iff
protected theorem Homeomorph.contractibleSpace_iff (e : X ≃ₜ Y) :
    ContractibleSpace X ↔ ContractibleSpace Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y)
Goal: ContractibleSpace X ↔ ContractibleSpace Y
-- ContractibleSpace.hequiv
theorem hequiv [ContractibleSpace X] [ContractibleSpace Y] :
    Nonempty (X ≃ₕ Y) 
Arguments: (X : Type u_1) (Y : Type u_2) [TopologicalSpace X] [TopologicalSpace Y] [ContractibleSpace X] [ContractibleSpace Y]
Goal: Nonempty (X ≃ₕ Y)
-- ContinuousMap.HomotopyEquiv.right_inv
@[ext]
structure HomotopyEquiv (X : Type u) (Y : Type v) [TopologicalSpace X] [TopologicalSpace Y] where
  toFun : C(X, Y)
  invFun : C(Y, X)
  left_inv : (invFun.comp toFun).Homotopic (ContinuousMap.id X)
  right_inv : (toFun.comp invFun).Homotopic (ContinuousMap.id Y)
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (self : X ≃ₕ Y)
Goal: (self.toFun.comp self.invFun).Homotopic (ContinuousMap.id Y)
-- ContinuousMap.HomotopyEquiv.left_inv
@[ext]
structure HomotopyEquiv (X : Type u) (Y : Type v) [TopologicalSpace X] [TopologicalSpace Y] where
  toFun : C(X, Y)
  invFun : C(Y, X)
  left_inv : (invFun.comp toFun).Homotopic (ContinuousMap.id X)
  right_inv : (toFun.comp invFun).Homotopic (ContinuousMap.id Y)
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (self : X ≃ₕ Y)
Goal: (self.invFun.comp self.toFun).Homotopic (ContinuousMap.id X)
-- ContinuousMap.HomotopyEquiv.toFun_eq_coe
@[simp]
theorem toFun_eq_coe (h : HomotopyEquiv X Y) : (h.toFun : X → Y) = h 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₕ Y)
Goal: ⇑h.toFun = ↑h
-- ContinuousMap.HomotopyEquiv.continuous
@[continuity]
theorem continuous (h : HomotopyEquiv X Y) : Continuous h 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₕ Y)
Goal: Continuous ↑h
-- Homeomorph.coe_toHomotopyEquiv
@[simp]
theorem coe_toHomotopyEquiv (h : X ≃ₜ Y) : (h.toHomotopyEquiv : X → Y) = h 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: ↑h.toHomotopyEquiv = ⇑h
-- ContinuousMap.HomotopyEquiv.coe_invFun
@[simp]
theorem coe_invFun (h : HomotopyEquiv X Y) : (⇑h.invFun : Y → X) = ⇑h.symm 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₕ Y)
Goal: ⇑h.invFun = ↑h.symm
-- ContinuousMap.HomotopyEquiv.refl_apply
@[simps!]
def refl (X : Type u) [TopologicalSpace X] : X ≃ₕ X 
Arguments: (X : Type u) [TopologicalSpace X] (a : X)
Goal: ↑(ContinuousMap.HomotopyEquiv.refl X) a = a
-- ContinuousMap.HomotopyEquiv.refl_symm_apply
@[simps!]
def refl (X : Type u) [TopologicalSpace X] : X ≃ₕ X 
Arguments: (X : Type u) [TopologicalSpace X] (a : X)
Goal: ↑(ContinuousMap.HomotopyEquiv.refl X).symm a = a
-- ContinuousMap.HomotopyEquiv.trans_apply
@[simps!]
def trans (h₁ : X ≃ₕ Y) (h₂ : Y ≃ₕ Z) : X ≃ₕ Z 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (h₁ : X ≃ₕ Y) (h₂ : Y ≃ₕ Z)
Goal: ∀ (a : X), ↑(h₁.trans h₂) a = ↑h₂ (↑h₁ a)
-- ContinuousMap.HomotopyEquiv.trans_symm_apply
@[simps!]
def trans (h₁ : X ≃ₕ Y) (h₂ : Y ≃ₕ Z) : X ≃ₕ Z 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (h₁ : X ≃ₕ Y) (h₂ : Y ≃ₕ Z)
Goal: ∀ (a : Z), ↑(h₁.trans h₂).symm a = ↑h₁.symm (↑h₂.symm a)
-- ContinuousMap.HomotopyEquiv.symm_trans
theorem symm_trans (h₁ : X ≃ₕ Y) (h₂ : Y ≃ₕ Z) : (h₁.trans h₂).symm = h₂.symm.trans h₁.symm 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (h₁ : X ≃ₕ Y) (h₂ : Y ≃ₕ Z)
Goal: (h₁.trans h₂).symm = h₂.symm.trans h₁.symm
-- Homeomorph.refl_toHomotopyEquiv
@[simp]
theorem refl_toHomotopyEquiv (X : Type u) [TopologicalSpace X] :
    (Homeomorph.refl X).toHomotopyEquiv = HomotopyEquiv.refl X 
Arguments: (X : Type u) [TopologicalSpace X]
Goal: (Homeomorph.refl X).toHomotopyEquiv = ContinuousMap.HomotopyEquiv.refl X
-- Homeomorph.symm_toHomotopyEquiv
@[simp]
theorem symm_toHomotopyEquiv (h : X ≃ₜ Y) : h.symm.toHomotopyEquiv = h.toHomotopyEquiv.symm 
Arguments: {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: h.symm.toHomotopyEquiv = h.toHomotopyEquiv.symm
-- Homeomorph.trans_toHomotopyEquiv
@[simp]
theorem trans_toHomotopyEquiv (h₀ : X ≃ₜ Y) (h₁ : Y ≃ₜ Z) :
    (h₀.trans h₁).toHomotopyEquiv = h₀.toHomotopyEquiv.trans h₁.toHomotopyEquiv 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (h₀ : X ≃ₜ Y) (h₁ : Y ≃ₜ Z)
Goal: (h₀.trans h₁).toHomotopyEquiv = h₀.toHomotopyEquiv.trans h₁.toHomotopyEquiv
-- OrderTopology.topology_eq_generate_intervals
class OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
    topology_eq_generate_intervals : t = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }
Arguments: {α : Type u_1} [t : TopologicalSpace α] [Preorder α] [self : OrderTopology α]
Goal: t = TopologicalSpace.generateFrom {s | ∃ a, s = Set.Ioi a ∨ s = Set.Iio a}
-- isOpen_iff_generate_intervals
theorem isOpen_iff_generate_intervals [t : OrderTopology α] {s : Set α} :
    IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [t : OrderTopology α] {s : Set α}
Goal: IsOpen s ↔ TopologicalSpace.GenerateOpen {s | ∃ a, s = Set.Ioi a ∨ s = Set.Iio a} s
-- isOpen_lt'
theorem isOpen_lt' [OrderTopology α] (a : α) : IsOpen { b : α | a < b } 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α)
Goal: IsOpen {b | a < b}
-- isOpen_gt'
theorem isOpen_gt' [OrderTopology α] (a : α) : IsOpen { b : α | b < a } 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α)
Goal: IsOpen {b | b < a}
-- lt_mem_nhds
theorem lt_mem_nhds [OrderTopology α] {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTopology α] {a b : α} (h : a < b)
Goal: ∀ᶠ (x : α) in 𝓝 b, a < x
-- le_mem_nhds
theorem le_mem_nhds [OrderTopology α] {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTopology α] {a b : α} (h : a < b)
Goal: ∀ᶠ (x : α) in 𝓝 b, a ≤ x
-- gt_mem_nhds
theorem gt_mem_nhds [OrderTopology α] {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTopology α] {a b : α} (h : a < b)
Goal: ∀ᶠ (x : α) in 𝓝 a, x < b
-- ge_mem_nhds
theorem ge_mem_nhds [OrderTopology α] {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTopology α] {a b : α} (h : a < b)
Goal: ∀ᶠ (x : α) in 𝓝 a, x ≤ b
-- nhds_eq_order
theorem nhds_eq_order [OrderTopology α] (a : α) :
    𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α)
Goal: 𝓝 a = (⨅ b ∈ Set.Iio a, 𝓟 (Set.Ioi b)) ⊓ ⨅ b ∈ Set.Ioi a, 𝓟 (Set.Iio b)
-- tendsto_order
theorem tendsto_order [OrderTopology α] {f : β → α} {a : α} {x : Filter β} :
    Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [OrderTopology α] {f : β → α} {a : α} {x : Filter β}
Goal: Filter.Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ (b : β) in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ (b : β) in x, f b < a'
-- tendsto_of_tendsto_of_tendsto_of_le_of_le'
theorem tendsto_of_tendsto_of_tendsto_of_le_of_le' [OrderTopology α] {f g h : β → α} {b : Filter β}
    {a : α} (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)
    (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [OrderTopology α] {f g h : β → α} {b : Filter β} {a : α} (hg : Filter.Tendsto g b (𝓝 a)) (hh : Filter.Tendsto h b (𝓝 a)) (hgf : ∀ᶠ (b : β) in b, g b ≤ f b) (hfh : ∀ᶠ (b : β) in b, f b ≤ h b)
Goal: Filter.Tendsto f b (𝓝 a)
-- tendsto_of_tendsto_of_tendsto_of_le_of_le
theorem tendsto_of_tendsto_of_tendsto_of_le_of_le [OrderTopology α] {f g h : β → α} {b : Filter β}
    {a : α} (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :
    Tendsto f b (𝓝 a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [OrderTopology α] {f g h : β → α} {b : Filter β} {a : α} (hg : Filter.Tendsto g b (𝓝 a)) (hh : Filter.Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h)
Goal: Filter.Tendsto f b (𝓝 a)
-- nhds_order_unbounded
theorem nhds_order_unbounded [OrderTopology α] {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :
    𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a)
Goal: 𝓝 a = ⨅ l, ⨅ (_ : l < a), ⨅ u, ⨅ (_ : a < u), 𝓟 (Set.Ioo l u)
-- tendsto_order_unbounded
theorem tendsto_order_unbounded [OrderTopology α] {f : β → α} {a : α} {x : Filter β}
    (hu : ∃ u, a < u) (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :
    Tendsto f x (𝓝 a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder α] [OrderTopology α] {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u) (hl : ∃ l, l < a) (h : ∀ (l u : α), l < a → a < u → ∀ᶠ (b : β) in x, l < f b ∧ f b < u)
Goal: Filter.Tendsto f x (𝓝 a)
-- induced_topology_le_preorder
theorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]
    [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :
    induced f ‹TopologicalSpace β› ≤ Preorder.topology α 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : ∀ {x y : α}, f x < f y ↔ x < y)
Goal: TopologicalSpace.induced f inst✝ ≤ Preorder.topology α
-- induced_topology_eq_preorder
theorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]
    [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)
    (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)
    (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :
    induced f ‹TopologicalSpace β› = Preorder.topology α 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : ∀ {x y : α}, f x < f y ↔ x < y) (H₁ : ∀ {a : α} {b : β} {x : α}, b < f a → ¬b < f x → ∃ y < a, b ≤ f y) (H₂ : ∀ {a : α} {b : β} {x : α}, f a < b → ¬f x < b → ∃ y, a < y ∧ f y ≤ b)
Goal: TopologicalSpace.induced f inst✝ = Preorder.topology α
-- induced_orderTopology'
theorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]
    [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)
    (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :
    @OrderTopology _ (induced f ta) _ 
Arguments: {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β] [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y : α}, f x < f y ↔ x < y) (H₁ : ∀ {a : α} {x : β}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a : α} {x : β}, f a < x → ∃ b > a, f b ≤ x)
Goal: OrderTopology α
-- induced_orderTopology
theorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β]
    [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)
    (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ 
Arguments: {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β] [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y : α}, f x < f y ↔ x < y) (H : ∀ {x y : β}, x < y → ∃ a, x < f a ∧ f a < y)
Goal: OrderTopology α
-- StrictMono.induced_topology_eq_preorder
nonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]
    [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}
    (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : StrictMono f) (hc : (Set.range f).OrdConnected)
Goal: TopologicalSpace.induced f t = Preorder.topology α
-- StrictMono.embedding_of_ordConnected
theorem StrictMono.embedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]
    [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}
    (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : StrictMono f) (hc : (Set.range f).OrdConnected)
Goal: Embedding f
-- nhdsWithin_Ici_eq''
theorem nhdsWithin_Ici_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :
    𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α)
Goal: 𝓝[≥] a = (⨅ u, ⨅ (_ : a < u), 𝓟 (Set.Iio u)) ⊓ 𝓟 (Set.Ici a)
-- nhdsWithin_Iic_eq''
theorem nhdsWithin_Iic_eq'' [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :
    𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α)
Goal: 𝓝[≤] a = (⨅ l, ⨅ (_ : l < a), 𝓟 (Set.Ioi l)) ⊓ 𝓟 (Set.Iic a)
-- nhdsWithin_Ici_eq'
theorem nhdsWithin_Ici_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}
    (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α} (ha : ∃ u, a < u)
Goal: 𝓝[≥] a = ⨅ u, ⨅ (_ : a < u), 𝓟 (Set.Ico a u)
-- nhdsWithin_Iic_eq'
theorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}
    (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) 
Arguments: {α : Type u} [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α} (ha : ∃ l, l < a)
Goal: 𝓝[≤] a = ⨅ l, ⨅ (_ : l < a), 𝓟 (Set.Ioc l a)
-- nhdsWithin_Ici_basis'
theorem nhdsWithin_Ici_basis' [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}
    (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u 
Arguments: {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} (ha : ∃ u, a < u)
Goal: (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Set.Ico a u