ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LocallyConstant.indicator_apply_eq_if
@[to_additive]
theorem mulIndicator_apply_eq_if (hU : IsClopen U) :
    mulIndicator f hU a = if a ∈ U then f a else 1 
Arguments: {X : Type u_1} [TopologicalSpace X] {R : Type u_5} [Zero R] {U : Set X} (f : LocallyConstant X R) (a : X) (hU : IsClopen U)
Goal: (f.indicator hU) a = if a ∈ U then f a else 0
-- LocallyConstant.mulIndicator_of_mem
@[to_additive]
theorem mulIndicator_of_mem (hU : IsClopen U) (h : a ∈ U) : f.mulIndicator hU a = f a 
Arguments: {X : Type u_1} [TopologicalSpace X] {R : Type u_5} [One R] {U : Set X} (f : LocallyConstant X R) {a : X} (hU : IsClopen U) (h : a ∈ U)
Goal: (f.mulIndicator hU) a = f a
-- LocallyConstant.indicator_of_mem
@[to_additive]
theorem mulIndicator_of_mem (hU : IsClopen U) (h : a ∈ U) : f.mulIndicator hU a = f a 
Arguments: {X : Type u_1} [TopologicalSpace X] {R : Type u_5} [Zero R] {U : Set X} (f : LocallyConstant X R) {a : X} (hU : IsClopen U) (h : a ∈ U)
Goal: (f.indicator hU) a = f a
-- LocallyConstant.mulIndicator_of_not_mem
@[to_additive]
theorem mulIndicator_of_not_mem (hU : IsClopen U) (h : a ∉ U) : f.mulIndicator hU a = 1 
Arguments: {X : Type u_1} [TopologicalSpace X] {R : Type u_5} [One R] {U : Set X} (f : LocallyConstant X R) {a : X} (hU : IsClopen U) (h : a ∉ U)
Goal: (f.mulIndicator hU) a = 1
-- LocallyConstant.indicator_of_not_mem
@[to_additive]
theorem mulIndicator_of_not_mem (hU : IsClopen U) (h : a ∉ U) : f.mulIndicator hU a = 1 
Arguments: {X : Type u_1} [TopologicalSpace X] {R : Type u_5} [Zero R] {U : Set X} (f : LocallyConstant X R) {a : X} (hU : IsClopen U) (h : a ∉ U)
Goal: (f.indicator hU) a = 0
-- LocallyConstant.congrLeft_apply
@[simps]
def congrLeft [TopologicalSpace Y] (e : X ≃ₜ Y) : LocallyConstant X Z ≃ LocallyConstant Y Z 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (g : LocallyConstant X Z)
Goal: (LocallyConstant.congrLeft e) g = LocallyConstant.comap e.symm.toContinuousMap g
-- LocallyConstant.congrLeft_symm_apply
@[simps]
def congrLeft [TopologicalSpace Y] (e : X ≃ₜ Y) : LocallyConstant X Z ≃ LocallyConstant Y Z 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (g : LocallyConstant Y Z)
Goal: (LocallyConstant.congrLeft e).symm g = LocallyConstant.comap e.toContinuousMap g
-- LocallyConstant.congrRight_symm_apply
@[simps]
def congrRight (e : Y ≃ Z) : LocallyConstant X Y ≃ LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] (e : Y ≃ Z) (g : LocallyConstant X Z)
Goal: (LocallyConstant.congrRight e).symm g = LocallyConstant.map (⇑e.symm) g
-- LocallyConstant.congrRight_apply
@[simps]
def congrRight (e : Y ≃ Z) : LocallyConstant X Y ≃ LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] (e : Y ≃ Z) (g : LocallyConstant X Y)
Goal: (LocallyConstant.congrRight e) g = LocallyConstant.map (⇑e) g
-- LocallyConstant.piecewise_apply_left
@[simp]
lemma piecewise_apply_left {C₁ C₂ : Set X} (h₁ : IsClosed C₁) (h₂ : IsClosed C₂)
    (h : C₁ ∪ C₂ = Set.univ) (f : LocallyConstant C₁ Z) (g : LocallyConstant C₂ Z)
    (hfg : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f ⟨x, hx.1⟩ = g ⟨x, hx.2⟩)
    [DecidablePred (· ∈ C₁)] (x : X) (hx : x ∈ C₁) :
    piecewise h₁ h₂ h f g hfg x = f ⟨x, hx⟩ 
Arguments: {X : Type u_1} {Z : Type u_3} [TopologicalSpace X] {C₁ C₂ : Set X} (h₁ : IsClosed C₁) (h₂ : IsClosed C₂) (h : C₁ ∪ C₂ = Set.univ) (f : LocallyConstant (↑C₁) Z) (g : LocallyConstant (↑C₂) Z) (hfg : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f ⟨x, ⋯⟩ = g ⟨x, ⋯⟩) [DecidablePred fun x => x ∈ C₁] (x : X) (hx : x ∈ C₁)
Goal: (LocallyConstant.piecewise h₁ h₂ h f g hfg) x = f ⟨x, hx⟩
-- LocallyConstant.piecewise_apply_right
@[simp]
lemma piecewise_apply_right {C₁ C₂ : Set X} (h₁ : IsClosed C₁) (h₂ : IsClosed C₂)
    (h : C₁ ∪ C₂ = Set.univ) (f : LocallyConstant C₁ Z) (g : LocallyConstant C₂ Z)
    (hfg : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f ⟨x, hx.1⟩ = g ⟨x, hx.2⟩)
    [DecidablePred (· ∈ C₁)] (x : X) (hx : x ∈ C₂) :
    piecewise h₁ h₂ h f g hfg x = g ⟨x, hx⟩ 
Arguments: {X : Type u_1} {Z : Type u_3} [TopologicalSpace X] {C₁ C₂ : Set X} (h₁ : IsClosed C₁) (h₂ : IsClosed C₂) (h : C₁ ∪ C₂ = Set.univ) (f : LocallyConstant (↑C₁) Z) (g : LocallyConstant (↑C₂) Z) (hfg : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f ⟨x, ⋯⟩ = g ⟨x, ⋯⟩) [DecidablePred fun x => x ∈ C₁] (x : X) (hx : x ∈ C₂)
Goal: (LocallyConstant.piecewise h₁ h₂ h f g hfg) x = g ⟨x, hx⟩
-- LocallyConstant.piecewise'_apply_left
@[simp]
lemma piecewise'_apply_left {C₀ C₁ C₂ : Set X} (h₀ : C₀ ⊆ C₁ ∪ C₂) (h₁ : IsClosed C₁)
    (h₂ : IsClosed C₂) (f₁ : LocallyConstant C₁ Z) (f₂ : LocallyConstant C₂ Z)
    [DecidablePred (· ∈ C₁)] (hf : ∀ x (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, hx.1⟩ = f₂ ⟨x, hx.2⟩)
    (x : C₀) (hx : x.val ∈ C₁) :
    piecewise' h₀ h₁ h₂ f₁ f₂ hf x = f₁ ⟨x.val, hx⟩ 
Arguments: {X : Type u_1} {Z : Type u_3} [TopologicalSpace X] {C₀ C₁ C₂ : Set X} (h₀ : C₀ ⊆ C₁ ∪ C₂) (h₁ : IsClosed C₁) (h₂ : IsClosed C₂) (f₁ : LocallyConstant (↑C₁) Z) (f₂ : LocallyConstant (↑C₂) Z) [DecidablePred fun x => x ∈ C₁] (hf : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, ⋯⟩ = f₂ ⟨x, ⋯⟩) (x : ↑C₀) (hx : ↑x ∈ C₁)
Goal: (LocallyConstant.piecewise' h₀ h₁ h₂ f₁ f₂ hf) x = f₁ ⟨↑x, hx⟩
-- LocallyConstant.piecewise'_apply_right
@[simp]
lemma piecewise'_apply_right {C₀ C₁ C₂ : Set X} (h₀ : C₀ ⊆ C₁ ∪ C₂) (h₁ : IsClosed C₁)
    (h₂ : IsClosed C₂) (f₁ : LocallyConstant C₁ Z) (f₂ : LocallyConstant C₂ Z)
    [DecidablePred (· ∈ C₁)] (hf : ∀ x (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, hx.1⟩ = f₂ ⟨x, hx.2⟩)
    (x : C₀) (hx : x.val ∈ C₂) :
    piecewise' h₀ h₁ h₂ f₁ f₂ hf x = f₂ ⟨x.val, hx⟩ 
Arguments: {X : Type u_1} {Z : Type u_3} [TopologicalSpace X] {C₀ C₁ C₂ : Set X} (h₀ : C₀ ⊆ C₁ ∪ C₂) (h₁ : IsClosed C₁) (h₂ : IsClosed C₂) (f₁ : LocallyConstant (↑C₁) Z) (f₂ : LocallyConstant (↑C₂) Z) [DecidablePred fun x => x ∈ C₁] (hf : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, ⋯⟩ = f₂ ⟨x, ⋯⟩) (x : ↑C₀) (hx : ↑x ∈ C₂)
Goal: (LocallyConstant.piecewise' h₀ h₁ h₂ f₁ f₂ hf) x = f₂ ⟨↑x, hx⟩
-- continuous_abs
@[continuity]
theorem continuous_abs : Continuous (abs : G → G) 
Arguments: {G : Type u_2} [TopologicalSpace G] [LinearOrderedAddCommGroup G] [OrderTopology G]
Goal: Continuous abs
-- Filter.Tendsto.abs
protected theorem Filter.Tendsto.abs {a : G} (h : Tendsto f l (𝓝 a)) :
    Tendsto (fun x => |f x|) l (𝓝 |a|) 
Arguments: {α : Type u_1} {G : Type u_2} [TopologicalSpace G] [LinearOrderedAddCommGroup G] [OrderTopology G] {l : Filter α} {f : α → G} {a : G} (h : Filter.Tendsto f l (𝓝 a))
Goal: Filter.Tendsto (fun x => |f x|) l (𝓝 |a|)
-- tendsto_zero_iff_abs_tendsto_zero
theorem tendsto_zero_iff_abs_tendsto_zero (f : α → G) :
    Tendsto f l (𝓝 0) ↔ Tendsto (abs ∘ f) l (𝓝 0) 
Arguments: {α : Type u_1} {G : Type u_2} [TopologicalSpace G] [LinearOrderedAddCommGroup G] [OrderTopology G] {l : Filter α} (f : α → G)
Goal: Filter.Tendsto f l (𝓝 0) ↔ Filter.Tendsto (abs ∘ f) l (𝓝 0)
-- Continuous.abs
@[fun_prop]
protected theorem Continuous.abs (h : Continuous f) : Continuous fun x => |f x| 
Arguments: {α : Type u_1} {G : Type u_2} [TopologicalSpace G] [LinearOrderedAddCommGroup G] [OrderTopology G] {f : α → G} [TopologicalSpace α] (h : Continuous f)
Goal: Continuous fun x => |f x|
-- ContinuousAt.abs
@[fun_prop]
protected theorem ContinuousAt.abs (h : ContinuousAt f a) : ContinuousAt (fun x => |f x|) a 
Arguments: {α : Type u_1} {G : Type u_2} [TopologicalSpace G] [LinearOrderedAddCommGroup G] [OrderTopology G] {f : α → G} [TopologicalSpace α] {a : α} (h : ContinuousAt f a)
Goal: ContinuousAt (fun x => |f x|) a
-- ContinuousWithinAt.abs
protected theorem ContinuousWithinAt.abs (h : ContinuousWithinAt f s a) :
    ContinuousWithinAt (fun x => |f x|) s a 
Arguments: {α : Type u_1} {G : Type u_2} [TopologicalSpace G] [LinearOrderedAddCommGroup G] [OrderTopology G] {f : α → G} [TopologicalSpace α] {a : α} {s : Set α} (h : ContinuousWithinAt f s a)
Goal: ContinuousWithinAt (fun x => |f x|) s a
-- ContinuousOn.abs
@[fun_prop]
protected theorem ContinuousOn.abs (h : ContinuousOn f s) : ContinuousOn (fun x => |f x|) s 
Arguments: {α : Type u_1} {G : Type u_2} [TopologicalSpace G] [LinearOrderedAddCommGroup G] [OrderTopology G] {f : α → G} [TopologicalSpace α] {s : Set α} (h : ContinuousOn f s)
Goal: ContinuousOn (fun x => |f x|) s
-- tendsto_abs_nhdsWithin_zero
theorem tendsto_abs_nhdsWithin_zero : Tendsto (abs : G → G) (𝓝[≠] 0) (𝓝[>] 0) 
Arguments: {G : Type u_2} [TopologicalSpace G] [LinearOrderedAddCommGroup G] [OrderTopology G]
Goal: Filter.Tendsto abs (𝓝[≠] 0) (𝓝[>] 0)
-- tendsto_floor_atTop
theorem tendsto_floor_atTop : Tendsto (floor : α → ℤ) atTop atTop 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α]
Goal: Filter.Tendsto Int.floor Filter.atTop Filter.atTop
-- tendsto_floor_atBot
theorem tendsto_floor_atBot : Tendsto (floor : α → ℤ) atBot atBot 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α]
Goal: Filter.Tendsto Int.floor Filter.atBot Filter.atBot
-- tendsto_ceil_atTop
theorem tendsto_ceil_atTop : Tendsto (ceil : α → ℤ) atTop atTop 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α]
Goal: Filter.Tendsto Int.ceil Filter.atTop Filter.atTop
-- tendsto_ceil_atBot
theorem tendsto_ceil_atBot : Tendsto (ceil : α → ℤ) atBot atBot 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α]
Goal: Filter.Tendsto Int.ceil Filter.atBot Filter.atBot
-- continuousOn_floor
theorem continuousOn_floor (n : ℤ) :
    ContinuousOn (fun x => floor x : α → α) (Ico n (n + 1) : Set α) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] (n : ℤ)
Goal: ContinuousOn (fun x => ↑⌊x⌋) (Set.Ico (↑n) (↑n + 1))
-- continuousOn_ceil
theorem continuousOn_ceil (n : ℤ) :
    ContinuousOn (fun x => ceil x : α → α) (Ioc (n - 1) n : Set α) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] (n : ℤ)
Goal: ContinuousOn (fun x => ↑⌈x⌉) (Set.Ioc (↑n - 1) ↑n)
-- tendsto_floor_right_pure_floor
theorem tendsto_floor_right_pure_floor (x : α) : Tendsto (floor : α → ℤ) (𝓝[≥] x) (pure ⌊x⌋) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (x : α)
Goal: Filter.Tendsto Int.floor (𝓝[≥] x) (pure ⌊x⌋)
-- tendsto_floor_right_pure
theorem tendsto_floor_right_pure (n : ℤ) : Tendsto (floor : α → ℤ) (𝓝[≥] n) (pure n) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (n : ℤ)
Goal: Filter.Tendsto Int.floor (𝓝[≥] ↑n) (pure n)
-- tendsto_ceil_left_pure_ceil
theorem tendsto_ceil_left_pure_ceil (x : α) : Tendsto (ceil : α → ℤ) (𝓝[≤] x) (pure ⌈x⌉) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (x : α)
Goal: Filter.Tendsto Int.ceil (𝓝[≤] x) (pure ⌈x⌉)
-- tendsto_ceil_left_pure
theorem tendsto_ceil_left_pure (n : ℤ) : Tendsto (ceil : α → ℤ) (𝓝[≤] n) (pure n) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (n : ℤ)
Goal: Filter.Tendsto Int.ceil (𝓝[≤] ↑n) (pure n)
-- tendsto_floor_left_pure_ceil_sub_one
theorem tendsto_floor_left_pure_ceil_sub_one (x : α) :
    Tendsto (floor : α → ℤ) (𝓝[<] x) (pure (⌈x⌉ - 1)) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (x : α)
Goal: Filter.Tendsto Int.floor (𝓝[<] x) (pure (⌈x⌉ - 1))
-- tendsto_floor_left_pure_sub_one
theorem tendsto_floor_left_pure_sub_one (n : ℤ) :
    Tendsto (floor : α → ℤ) (𝓝[<] n) (pure (n - 1)) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (n : ℤ)
Goal: Filter.Tendsto Int.floor (𝓝[<] ↑n) (pure (n - 1))
-- tendsto_ceil_right_pure_floor_add_one
theorem tendsto_ceil_right_pure_floor_add_one (x : α) :
    Tendsto (ceil : α → ℤ) (𝓝[>] x) (pure (⌊x⌋ + 1)) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (x : α)
Goal: Filter.Tendsto Int.ceil (𝓝[>] x) (pure (⌊x⌋ + 1))
-- tendsto_ceil_right_pure_add_one
theorem tendsto_ceil_right_pure_add_one (n : ℤ) :
    Tendsto (ceil : α → ℤ) (𝓝[>] n) (pure (n + 1)) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (n : ℤ)
Goal: Filter.Tendsto Int.ceil (𝓝[>] ↑n) (pure (n + 1))
-- tendsto_floor_right
theorem tendsto_floor_right (n : ℤ) : Tendsto (fun x => floor x : α → α) (𝓝[≥] n) (𝓝[≥] n) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (n : ℤ)
Goal: Filter.Tendsto (fun x => ↑⌊x⌋) (𝓝[≥] ↑n) (𝓝[≥] ↑n)
-- tendsto_floor_right'
theorem tendsto_floor_right' (n : ℤ) : Tendsto (fun x => floor x : α → α) (𝓝[≥] n) (𝓝 n) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (n : ℤ)
Goal: Filter.Tendsto (fun x => ↑⌊x⌋) (𝓝[≥] ↑n) (𝓝 ↑n)
-- tendsto_ceil_left
theorem tendsto_ceil_left (n : ℤ) : Tendsto (fun x => ceil x : α → α) (𝓝[≤] n) (𝓝[≤] n) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (n : ℤ)
Goal: Filter.Tendsto (fun x => ↑⌈x⌉) (𝓝[≤] ↑n) (𝓝[≤] ↑n)
-- tendsto_ceil_left'
theorem tendsto_ceil_left' (n : ℤ) :
    Tendsto (fun x => ceil x : α → α) (𝓝[≤] n) (𝓝 n) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (n : ℤ)
Goal: Filter.Tendsto (fun x => ↑⌈x⌉) (𝓝[≤] ↑n) (𝓝 ↑n)
-- tendsto_floor_left
theorem tendsto_floor_left (n : ℤ) :
    Tendsto (fun x => floor x : α → α) (𝓝[<] n) (𝓝[≤] (n - 1)) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (n : ℤ)
Goal: Filter.Tendsto (fun x => ↑⌊x⌋) (𝓝[<] ↑n) (𝓝[≤] (↑n - 1))
-- tendsto_ceil_right
theorem tendsto_ceil_right (n : ℤ) :
    Tendsto (fun x => ceil x : α → α) (𝓝[>] n) (𝓝[≥] (n + 1)) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (n : ℤ)
Goal: Filter.Tendsto (fun x => ↑⌈x⌉) (𝓝[>] ↑n) (𝓝[≥] (↑n + 1))
-- tendsto_floor_left'
theorem tendsto_floor_left' (n : ℤ) :
    Tendsto (fun x => floor x : α → α) (𝓝[<] n) (𝓝 (n - 1)) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (n : ℤ)
Goal: Filter.Tendsto (fun x => ↑⌊x⌋) (𝓝[<] ↑n) (𝓝 (↑n - 1))
-- tendsto_ceil_right'
theorem tendsto_ceil_right' (n : ℤ) :
    Tendsto (fun x => ceil x : α → α) (𝓝[>] n) (𝓝 (n + 1)) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] (n : ℤ)
Goal: Filter.Tendsto (fun x => ↑⌈x⌉) (𝓝[>] ↑n) (𝓝 (↑n + 1))
-- continuousOn_fract
theorem continuousOn_fract [TopologicalAddGroup α] (n : ℤ) :
    ContinuousOn (fract : α → α) (Ico n (n + 1) : Set α) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [TopologicalAddGroup α] (n : ℤ)
Goal: ContinuousOn Int.fract (Set.Ico (↑n) (↑n + 1))
-- continuousAt_fract
theorem continuousAt_fract [OrderClosedTopology α] [TopologicalAddGroup α]
    {x : α} (h : x ≠ ⌊x⌋) : ContinuousAt fract x 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] [TopologicalAddGroup α] {x : α} (h : x ≠ ↑⌊x⌋)
Goal: ContinuousAt Int.fract x
-- tendsto_fract_left'
theorem tendsto_fract_left' [OrderClosedTopology α] [TopologicalAddGroup α] (n : ℤ) :
    Tendsto (fract : α → α) (𝓝[<] n) (𝓝 1) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] [TopologicalAddGroup α] (n : ℤ)
Goal: Filter.Tendsto Int.fract (𝓝[<] ↑n) (𝓝 1)
-- tendsto_fract_left
theorem tendsto_fract_left [OrderClosedTopology α] [TopologicalAddGroup α] (n : ℤ) :
    Tendsto (fract : α → α) (𝓝[<] n) (𝓝[<] 1) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] [TopologicalAddGroup α] (n : ℤ)
Goal: Filter.Tendsto Int.fract (𝓝[<] ↑n) (𝓝[<] 1)
-- tendsto_fract_right'
theorem tendsto_fract_right' [OrderClosedTopology α] [TopologicalAddGroup α] (n : ℤ) :
    Tendsto (fract : α → α) (𝓝[≥] n) (𝓝 0) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] [TopologicalAddGroup α] (n : ℤ)
Goal: Filter.Tendsto Int.fract (𝓝[≥] ↑n) (𝓝 0)
-- tendsto_fract_right
theorem tendsto_fract_right [OrderClosedTopology α] [TopologicalAddGroup α] (n : ℤ) :
    Tendsto (fract : α → α) (𝓝[≥] n) (𝓝[≥] 0) 
Arguments: {α : Type u_1} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderClosedTopology α] [TopologicalAddGroup α] (n : ℤ)
Goal: Filter.Tendsto Int.fract (𝓝[≥] ↑n) (𝓝[≥] 0)
-- ContinuousOn.comp_fract'
theorem ContinuousOn.comp_fract' {f : β → α → γ} (h : ContinuousOn (uncurry f) <| univ ×ˢ I)
    (hf : ∀ s, f s 0 = f s 1) : Continuous fun st : β × α => f st.1 (fract st.2) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ] {f : β → α → γ} (h : ContinuousOn (Function.uncurry f) (Set.univ ×ˢ Set.Icc 0 1)) (hf : ∀ (s : β), f s 0 = f s 1)
Goal: Continuous fun st => f st.1 (Int.fract st.2)
-- ContinuousOn.comp_fract
theorem ContinuousOn.comp_fract {s : β → α} {f : β → α → γ}
    (h : ContinuousOn (uncurry f) <| univ ×ˢ Icc 0 1) (hs : Continuous s)
    (hf : ∀ s, f s 0 = f s 1) : Continuous fun x : β => f x <| Int.fract (s x) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ] {s : β → α} {f : β → α → γ} (h : ContinuousOn (Function.uncurry f) (Set.univ ×ˢ Set.Icc 0 1)) (hs : Continuous s) (hf : ∀ (s : β), f s 0 = f s 1)
Goal: Continuous fun x => f x (Int.fract (s x))
-- ContinuousOn.comp_fract''
theorem ContinuousOn.comp_fract'' {f : α → β} (h : ContinuousOn f I) (hf : f 0 = f 1) :
    Continuous (f ∘ fract) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedRing α] [FloorRing α] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {f : α → β} (h : ContinuousOn f (Set.Icc 0 1)) (hf : f 0 = f 1)
Goal: Continuous (f ∘ Int.fract)
-- ZeroAtInftyContinuousMap.zero_at_infty'
structure ZeroAtInftyContinuousMap (α : Type u) (β : Type v) [TopologicalSpace α] [Zero β]
    [TopologicalSpace β] extends ContinuousMap α β : Type max u v where
    zero_at_infty' : Tendsto toFun (cocompact α) (𝓝 0)
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Zero β] [TopologicalSpace β] (self : α →C₀ β)
Goal: Filter.Tendsto self.toFun (Filter.cocompact α) (𝓝 0)
-- ZeroAtInftyContinuousMapClass.zero_at_infty
class ZeroAtInftyContinuousMapClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α]
    [Zero β] [TopologicalSpace β] [FunLike F α β] extends ContinuousMapClass F α β : Prop where
    zero_at_infty (f : F) : Tendsto f (cocompact α) (𝓝 0)
Arguments: {F : Type u_2} {α : outParam (Type u_3)} {β : outParam (Type u_4)} [TopologicalSpace α] [Zero β] [TopologicalSpace β] [FunLike F α β] [self : ZeroAtInftyContinuousMapClass F α β] (f : F)
Goal: Filter.Tendsto (⇑f) (Filter.cocompact α) (𝓝 0)
-- ZeroAtInftyContinuousMap.coe_toContinuousMap
@[simp]
theorem coe_toContinuousMap (f : C₀(α, β)) : (f.toContinuousMap : α → β) = f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (f : α →C₀ β)
Goal: ⇑f.toContinuousMap = ⇑f
-- ZeroAtInftyContinuousMap.ext
@[ext]
theorem ext {f g : C₀(α, β)} (h : ∀ x, f x = g x) : f = g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {f g : α →C₀ β} (h : ∀ (x : α), f x = g x)
Goal: f = g
-- ZeroAtInftyContinuousMap.coe_copy
@[simp]
theorem coe_copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (f : α →C₀ β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- ZeroAtInftyContinuousMap.copy_eq
theorem copy_eq (f : C₀(α, β)) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (f : α →C₀ β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- ZeroAtInftyContinuousMap.eq_of_empty
theorem eq_of_empty [IsEmpty α] (f g : C₀(α, β)) : f = g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] [IsEmpty α] (f g : α →C₀ β)
Goal: f = g
-- ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty_symm_apply
@[simps]
def ContinuousMap.liftZeroAtInfty [CompactSpace α] : C(α, β) ≃ C₀(α, β) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] [CompactSpace α] (f : α →C₀ β)
Goal: ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty.symm f = ↑f
-- ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty_apply_toFun
@[simps]
def ContinuousMap.liftZeroAtInfty [CompactSpace α] : C(α, β) ≃ C₀(α, β) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] [CompactSpace α] (f : C(α, β)) (a : α)
Goal: (ZeroAtInftyContinuousMap.ContinuousMap.liftZeroAtInfty f) a = f a
-- ZeroAtInftyContinuousMap.zeroAtInftyContinuousMapClass.ofCompact
lemma zeroAtInftyContinuousMapClass.ofCompact {G : Type*} [FunLike G α β]
    [ContinuousMapClass G α β] [CompactSpace α] : ZeroAtInftyContinuousMapClass G α β 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {G : Type u_2} [FunLike G α β] [ContinuousMapClass G α β] [CompactSpace α]
Goal: ZeroAtInftyContinuousMapClass G α β
-- ZeroAtInftyContinuousMap.coe_zero
@[simp]
theorem coe_zero [Zero β] : ⇑(0 : C₀(α, β)) = 0 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β]
Goal: ⇑0 = 0
-- ZeroAtInftyContinuousMap.zero_apply
theorem zero_apply [Zero β] : (0 : C₀(α, β)) x = 0 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] (x : α) [Zero β]
Goal: 0 x = 0
-- ZeroAtInftyContinuousMap.coe_mul
@[simp]
theorem coe_mul [MulZeroClass β] [ContinuousMul β] (f g : C₀(α, β)) : ⇑(f * g) = f * g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [MulZeroClass β] [ContinuousMul β] (f g : α →C₀ β)
Goal: ⇑(f * g) = ⇑f * ⇑g
-- ZeroAtInftyContinuousMap.mul_apply
theorem mul_apply [MulZeroClass β] [ContinuousMul β] (f g : C₀(α, β)) : (f * g) x = f x * g x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] (x : α) [MulZeroClass β] [ContinuousMul β] (f g : α →C₀ β)
Goal: (f * g) x = f x * g x
-- ZeroAtInftyContinuousMap.coe_add
@[simp]
theorem coe_add [AddZeroClass β] [ContinuousAdd β] (f g : C₀(α, β)) : ⇑(f + g) = f + g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [AddZeroClass β] [ContinuousAdd β] (f g : α →C₀ β)
Goal: ⇑(f + g) = ⇑f + ⇑g
-- ZeroAtInftyContinuousMap.add_apply
theorem add_apply [AddZeroClass β] [ContinuousAdd β] (f g : C₀(α, β)) : (f + g) x = f x + g x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] (x : α) [AddZeroClass β] [ContinuousAdd β] (f g : α →C₀ β)
Goal: (f + g) x = f x + g x
-- ZeroAtInftyContinuousMap.coe_smul
@[simp, norm_cast]
theorem coe_smul [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] (r : R)
    (f : C₀(α, β)) : ⇑(r • f) = r • ⇑f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {R : Type u_2} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] (r : R) (f : α →C₀ β)
Goal: ⇑(r • f) = r • ⇑f
-- ZeroAtInftyContinuousMap.smul_apply
theorem smul_apply [Zero β] {R : Type*} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β]
    (r : R) (f : C₀(α, β)) (x : α) : (r • f) x = r • f x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {R : Type u_2} [Zero R] [SMulWithZero R β] [ContinuousConstSMul R β] (r : R) (f : α →C₀ β) (x : α)
Goal: (r • f) x = r • f x
-- ZeroAtInftyContinuousMap.coe_neg
@[simp]
theorem coe_neg : ⇑(-f) = -f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [AddGroup β] [TopologicalAddGroup β] (f : α →C₀ β)
Goal: ⇑(-f) = -⇑f
-- ZeroAtInftyContinuousMap.neg_apply
theorem neg_apply : (-f) x = -f x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] (x : α) [AddGroup β] [TopologicalAddGroup β] (f : α →C₀ β)
Goal: (-f) x = -f x