File: Mathlib/Topology/Algebra/Order/Floor.lean

full_name: tendsto_floor_atTop
code: theorem tendsto_floor_atTop : Tendsto (floor : α → ℤ) atTop atTop 
goal: Filter.Tendsto Int.floor Filter.atTop Filter.atTop

full_name: tendsto_floor_atBot
code: theorem tendsto_floor_atBot : Tendsto (floor : α → ℤ) atBot atBot 
goal: Filter.Tendsto Int.floor Filter.atBot Filter.atBot

full_name: tendsto_ceil_atTop
code: theorem tendsto_ceil_atTop : Tendsto (ceil : α → ℤ) atTop atTop 
goal: Filter.Tendsto Int.ceil Filter.atTop Filter.atTop

full_name: tendsto_ceil_atBot
code: theorem tendsto_ceil_atBot : Tendsto (ceil : α → ℤ) atBot atBot 
goal: Filter.Tendsto Int.ceil Filter.atBot Filter.atBot

full_name: continuousOn_floor
code: theorem continuousOn_floor (n : ℤ) :
    ContinuousOn (fun x => floor x : α → α) (Ico n (n + 1) : Set α) 
goal: ContinuousOn (fun x => ↑⌊x⌋) (Set.Ico (↑n) (↑n + 1))

full_name: continuousOn_ceil
code: theorem continuousOn_ceil (n : ℤ) :
    ContinuousOn (fun x => ceil x : α → α) (Ioc (n - 1) n : Set α) 
goal: ContinuousOn (fun x => ↑⌈x⌉) (Set.Ioc (↑n - 1) ↑n)

full_name: tendsto_floor_right_pure_floor
code: theorem tendsto_floor_right_pure_floor (x : α) : Tendsto (floor : α → ℤ) (𝓝[≥] x) (pure ⌊x⌋) 
goal: Filter.Tendsto Int.floor (𝓝[≥] x) (pure ⌊x⌋)

full_name: tendsto_floor_right_pure
code: theorem tendsto_floor_right_pure (n : ℤ) : Tendsto (floor : α → ℤ) (𝓝[≥] n) (pure n) 
goal: Filter.Tendsto Int.floor (𝓝[≥] ↑n) (pure n)

full_name: tendsto_ceil_left_pure_ceil
code: theorem tendsto_ceil_left_pure_ceil (x : α) : Tendsto (ceil : α → ℤ) (𝓝[≤] x) (pure ⌈x⌉) 
goal: Filter.Tendsto Int.ceil (𝓝[≤] x) (pure ⌈x⌉)

full_name: tendsto_ceil_left_pure
code: theorem tendsto_ceil_left_pure (n : ℤ) : Tendsto (ceil : α → ℤ) (𝓝[≤] n) (pure n) 
goal: Filter.Tendsto Int.ceil (𝓝[≤] ↑n) (pure n)

full_name: tendsto_floor_left_pure_ceil_sub_one
code: theorem tendsto_floor_left_pure_ceil_sub_one (x : α) :
    Tendsto (floor : α → ℤ) (𝓝[<] x) (pure (⌈x⌉ - 1)) 
goal: Filter.Tendsto Int.floor (𝓝[<] x) (pure (⌈x⌉ - 1))

full_name: tendsto_floor_left_pure_sub_one
code: theorem tendsto_floor_left_pure_sub_one (n : ℤ) :
    Tendsto (floor : α → ℤ) (𝓝[<] n) (pure (n - 1)) 
goal: Filter.Tendsto Int.floor (𝓝[<] ↑n) (pure (n - 1))

full_name: tendsto_ceil_right_pure_floor_add_one
code: theorem tendsto_ceil_right_pure_floor_add_one (x : α) :
    Tendsto (ceil : α → ℤ) (𝓝[>] x) (pure (⌊x⌋ + 1)) 
goal: Filter.Tendsto Int.ceil (𝓝[>] x) (pure (⌊x⌋ + 1))

full_name: tendsto_ceil_right_pure_add_one
code: theorem tendsto_ceil_right_pure_add_one (n : ℤ) :
    Tendsto (ceil : α → ℤ) (𝓝[>] n) (pure (n + 1)) 
goal: Filter.Tendsto Int.ceil (𝓝[>] ↑n) (pure (n + 1))

full_name: tendsto_floor_right
code: theorem tendsto_floor_right (n : ℤ) : Tendsto (fun x => floor x : α → α) (𝓝[≥] n) (𝓝[≥] n) 
goal: Filter.Tendsto (fun x => ↑⌊x⌋) (𝓝[≥] ↑n) (𝓝[≥] ↑n)

full_name: tendsto_floor_right'
code: theorem tendsto_floor_right' (n : ℤ) : Tendsto (fun x => floor x : α → α) (𝓝[≥] n) (𝓝 n) 
goal: Filter.Tendsto (fun x => ↑⌊x⌋) (𝓝[≥] ↑n) (𝓝 ↑n)

full_name: tendsto_ceil_left
code: theorem tendsto_ceil_left (n : ℤ) : Tendsto (fun x => ceil x : α → α) (𝓝[≤] n) (𝓝[≤] n) 
goal: Filter.Tendsto (fun x => ↑⌈x⌉) (𝓝[≤] ↑n) (𝓝[≤] ↑n)

full_name: tendsto_ceil_left'
code: theorem tendsto_ceil_left' (n : ℤ) :
    Tendsto (fun x => ceil x : α → α) (𝓝[≤] n) (𝓝 n) 
goal: Filter.Tendsto (fun x => ↑⌈x⌉) (𝓝[≤] ↑n) (𝓝 ↑n)

full_name: tendsto_floor_left
code: theorem tendsto_floor_left (n : ℤ) :
    Tendsto (fun x => floor x : α → α) (𝓝[<] n) (𝓝[≤] (n - 1)) 
goal: Filter.Tendsto (fun x => ↑⌊x⌋) (𝓝[<] ↑n) (𝓝[≤] (↑n - 1))

full_name: tendsto_ceil_right
code: theorem tendsto_ceil_right (n : ℤ) :
    Tendsto (fun x => ceil x : α → α) (𝓝[>] n) (𝓝[≥] (n + 1)) 
goal: Filter.Tendsto (fun x => ↑⌈x⌉) (𝓝[>] ↑n) (𝓝[≥] (↑n + 1))

full_name: tendsto_floor_left'
code: theorem tendsto_floor_left' (n : ℤ) :
    Tendsto (fun x => floor x : α → α) (𝓝[<] n) (𝓝 (n - 1)) 
goal: Filter.Tendsto (fun x => ↑⌊x⌋) (𝓝[<] ↑n) (𝓝 (↑n - 1))

full_name: tendsto_ceil_right'
code: theorem tendsto_ceil_right' (n : ℤ) :
    Tendsto (fun x => ceil x : α → α) (𝓝[>] n) (𝓝 (n + 1)) 
goal: Filter.Tendsto (fun x => ↑⌈x⌉) (𝓝[>] ↑n) (𝓝 (↑n + 1))

full_name: continuousOn_fract
code: theorem continuousOn_fract [TopologicalAddGroup α] (n : ℤ) :
    ContinuousOn (fract : α → α) (Ico n (n + 1) : Set α) 
goal: ContinuousOn Int.fract (Set.Ico (↑n) (↑n + 1))

full_name: continuousAt_fract
code: theorem continuousAt_fract [OrderClosedTopology α] [TopologicalAddGroup α]
    {x : α} (h : x ≠ ⌊x⌋) : ContinuousAt fract x 
goal: ContinuousAt Int.fract x

full_name: tendsto_fract_left'
code: theorem tendsto_fract_left' [OrderClosedTopology α] [TopologicalAddGroup α] (n : ℤ) :
    Tendsto (fract : α → α) (𝓝[<] n) (𝓝 1) 
goal: Filter.Tendsto Int.fract (𝓝[<] ↑n) (𝓝 1)

full_name: tendsto_fract_left
code: theorem tendsto_fract_left [OrderClosedTopology α] [TopologicalAddGroup α] (n : ℤ) :
    Tendsto (fract : α → α) (𝓝[<] n) (𝓝[<] 1) 
goal: Filter.Tendsto Int.fract (𝓝[<] ↑n) (𝓝[<] 1)

full_name: tendsto_fract_right'
code: theorem tendsto_fract_right' [OrderClosedTopology α] [TopologicalAddGroup α] (n : ℤ) :
    Tendsto (fract : α → α) (𝓝[≥] n) (𝓝 0) 
goal: Filter.Tendsto Int.fract (𝓝[≥] ↑n) (𝓝 0)

full_name: tendsto_fract_right
code: theorem tendsto_fract_right [OrderClosedTopology α] [TopologicalAddGroup α] (n : ℤ) :
    Tendsto (fract : α → α) (𝓝[≥] n) (𝓝[≥] 0) 
goal: Filter.Tendsto Int.fract (𝓝[≥] ↑n) (𝓝[≥] 0)

full_name: _aux_Mathlib_Topology_Algebra_Order_Floor___macroRules_termI_1
code: local notation "I" => (Icc 0 1 : Set α)
goal: Lean.Macro

full_name: ContinuousOn.comp_fract'
code: /-- Do not use this, use `ContinuousOn.comp_fract` instead. -/
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) 
goal: Continuous fun st => f st.1 (Int.fract st.2)

full_name: ContinuousOn.comp_fract
code: 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) 
goal: Continuous fun x => f x (Int.fract (s x))

full_name: ContinuousOn.comp_fract''
code: /-- A special case of `ContinuousOn.comp_fract`. -/
theorem ContinuousOn.comp_fract'' {f : α → β} (h : ContinuousOn f I) (hf : f 0 = f 1) :
    Continuous (f ∘ fract) 
goal: Continuous (f ∘ Int.fract)