ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- PiNat.res_length
@[simp]
theorem res_length (x : ℕ → α) (n : ℕ) : (res x n).length = n 
Arguments: {α : Type u_2} (x : ℕ → α) (n : ℕ)
Goal: (PiNat.res x n).length = n
-- PiNat.res_eq_res
theorem res_eq_res {x y : ℕ → α} {n : ℕ} :
    res x n = res y n ↔ ∀ ⦃m⦄, m < n → x m = y m 
Arguments: {α : Type u_2} {x y : ℕ → α} {n : ℕ}
Goal: PiNat.res x n = PiNat.res y n ↔ ∀ ⦃m : ℕ⦄, m < n → x m = y m
-- PiNat.res_injective
theorem res_injective : Injective (@res α) 
Arguments: {α : Type u_2}
Goal: Function.Injective PiNat.res
-- PiNat.cylinder_eq_res
theorem cylinder_eq_res (x : ℕ → α) (n : ℕ) :
    cylinder x n = { y | res y n = res x n } 
Arguments: {α : Type u_2} (x : ℕ → α) (n : ℕ)
Goal: PiNat.cylinder x n = {y | PiNat.res y n = PiNat.res x n}
-- PiNat.dist_eq_of_ne
theorem dist_eq_of_ne {x y : ∀ n, E n} (h : x ≠ y) : dist x y = (1 / 2 : ℝ) ^ firstDiff x y 
Arguments: {E : ℕ → Type u_1} {x y : (n : ℕ) → E n} (h : x ≠ y)
Goal: dist x y = (1 / 2) ^ PiNat.firstDiff x y
-- PiNat.dist_self
protected theorem dist_self (x : ∀ n, E n) : dist x x = 0 
Arguments: {E : ℕ → Type u_1} (x : (n : ℕ) → E n)
Goal: dist x x = 0
-- PiNat.dist_comm
protected theorem dist_comm (x y : ∀ n, E n) : dist x y = dist y x 
Arguments: {E : ℕ → Type u_1} (x y : (n : ℕ) → E n)
Goal: dist x y = dist y x
-- PiNat.dist_nonneg
protected theorem dist_nonneg (x y : ∀ n, E n) : 0 ≤ dist x y 
Arguments: {E : ℕ → Type u_1} (x y : (n : ℕ) → E n)
Goal: 0 ≤ dist x y
-- PiNat.dist_triangle_nonarch
theorem dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y) (dist y z) 
Arguments: {E : ℕ → Type u_1} (x y z : (n : ℕ) → E n)
Goal: dist x z ≤ max (dist x y) (dist y z)
-- PiNat.dist_triangle
protected theorem dist_triangle (x y z : ∀ n, E n) : dist x z ≤ dist x y + dist y z 
Arguments: {E : ℕ → Type u_1} (x y z : (n : ℕ) → E n)
Goal: dist x z ≤ dist x y + dist y z
-- PiNat.eq_of_dist_eq_zero
protected theorem eq_of_dist_eq_zero (x y : ∀ n, E n) (hxy : dist x y = 0) : x = y 
Arguments: {E : ℕ → Type u_1} (x y : (n : ℕ) → E n) (hxy : dist x y = 0)
Goal: x = y
-- PiNat.mem_cylinder_iff_dist_le
theorem mem_cylinder_iff_dist_le {x y : ∀ n, E n} {n : ℕ} :
    y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n 
Arguments: {E : ℕ → Type u_1} {x y : (n : ℕ) → E n} {n : ℕ}
Goal: y ∈ PiNat.cylinder x n ↔ dist y x ≤ (1 / 2) ^ n
-- PiNat.apply_eq_of_dist_lt
theorem apply_eq_of_dist_lt {x y : ∀ n, E n} {n : ℕ} (h : dist x y < (1 / 2) ^ n) {i : ℕ}
    (hi : i ≤ n) : x i = y i 
Arguments: {E : ℕ → Type u_1} {x y : (n : ℕ) → E n} {n : ℕ} (h : dist x y < (1 / 2) ^ n) {i : ℕ} (hi : i ≤ n)
Goal: x i = y i
-- PiNat.lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder
theorem lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder {α : Type*}
    [PseudoMetricSpace α] {f : (∀ n, E n) → α} :
    (∀ x y : ∀ n, E n, dist (f x) (f y) ≤ dist x y) ↔
      ∀ x y n, y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n 
Arguments: {E : ℕ → Type u_1} {α : Type u_2} [PseudoMetricSpace α] {f : ((n : ℕ) → E n) → α}
Goal: (∀ (x y : (n : ℕ) → E n), dist (f x) (f y) ≤ dist x y) ↔
  ∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ PiNat.cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n
-- PiNat.isOpen_cylinder
theorem isOpen_cylinder (x : ∀ n, E n) (n : ℕ) : IsOpen (cylinder x n) 
Arguments: (E : ℕ → Type u_1) [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)] (x : (n : ℕ) → E n) (n : ℕ)
Goal: IsOpen (PiNat.cylinder x n)
-- PiNat.isTopologicalBasis_cylinders
theorem isTopologicalBasis_cylinders :
    IsTopologicalBasis { s : Set (∀ n, E n) | ∃ (x : ∀ n, E n) (n : ℕ), s = cylinder x n } 
Arguments: (E : ℕ → Type u_1) [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)]
Goal: TopologicalSpace.IsTopologicalBasis {s | ∃ x n, s = PiNat.cylinder x n}
-- PiNat.isOpen_iff_dist
theorem isOpen_iff_dist (s : Set (∀ n, E n)) :
    IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s 
Arguments: {E : ℕ → Type u_1} [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)] (s : Set ((n : ℕ) → E n))
Goal: IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ (y : (n : ℕ) → E n), dist x y < ε → y ∈ s
-- PiNat.completeSpace
protected theorem completeSpace : CompleteSpace (∀ n, E n) 
Arguments: {E : ℕ → Type u_1} [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)]
Goal: CompleteSpace ((n : ℕ) → E n)
-- PiNat.exists_disjoint_cylinder
theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
    (hx : x ∉ s) : ∃ n, Disjoint s (cylinder x n) 
Arguments: {E : ℕ → Type u_1} [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)] {s : Set ((n : ℕ) → E n)} (hs : IsClosed s) {x : (n : ℕ) → E n} (hx : x ∉ s)
Goal: ∃ n, Disjoint s (PiNat.cylinder x n)
-- PiNat.firstDiff_lt_shortestPrefixDiff
theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y < shortestPrefixDiff x s 
Arguments: {E : ℕ → Type u_1} [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)] {s : Set ((n : ℕ) → E n)} (hs : IsClosed s) {x y : (n : ℕ) → E n} (hx : x ∉ s) (hy : y ∈ s)
Goal: PiNat.firstDiff x y < PiNat.shortestPrefixDiff x s
-- PiNat.shortestPrefixDiff_pos
theorem shortestPrefixDiff_pos {s : Set (∀ n, E n)} (hs : IsClosed s) (hne : s.Nonempty)
    {x : ∀ n, E n} (hx : x ∉ s) : 0 < shortestPrefixDiff x s 
Arguments: {E : ℕ → Type u_1} [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)] {s : Set ((n : ℕ) → E n)} (hs : IsClosed s) (hne : s.Nonempty) {x : (n : ℕ) → E n} (hx : x ∉ s)
Goal: 0 < PiNat.shortestPrefixDiff x s
-- PiNat.firstDiff_le_longestPrefix
theorem firstDiff_le_longestPrefix {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
    (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y ≤ longestPrefix x s 
Arguments: {E : ℕ → Type u_1} [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)] {s : Set ((n : ℕ) → E n)} (hs : IsClosed s) {x y : (n : ℕ) → E n} (hx : x ∉ s) (hy : y ∈ s)
Goal: PiNat.firstDiff x y ≤ PiNat.longestPrefix x s
-- PiNat.inter_cylinder_longestPrefix_nonempty
theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty 
Arguments: {E : ℕ → Type u_1} [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)] {s : Set ((n : ℕ) → E n)} (hs : IsClosed s) (hne : s.Nonempty) (x : (n : ℕ) → E n)
Goal: (s ∩ PiNat.cylinder x (PiNat.longestPrefix x s)).Nonempty
-- PiNat.disjoint_cylinder_of_longestPrefix_lt
theorem disjoint_cylinder_of_longestPrefix_lt {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
    (hx : x ∉ s) {n : ℕ} (hn : longestPrefix x s < n) : Disjoint s (cylinder x n) 
Arguments: {E : ℕ → Type u_1} [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)] {s : Set ((n : ℕ) → E n)} (hs : IsClosed s) {x : (n : ℕ) → E n} (hx : x ∉ s) {n : ℕ} (hn : PiNat.longestPrefix x s < n)
Goal: Disjoint s (PiNat.cylinder x n)
-- PiNat.cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff
theorem cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff {x y : ∀ n, E n}
    {s : Set (∀ n, E n)} (hs : IsClosed s) (hne : s.Nonempty)
    (H : longestPrefix x s < firstDiff x y) (xs : x ∉ s) (ys : y ∉ s) :
    cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s) 
Arguments: {E : ℕ → Type u_1} [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)] {x y : (n : ℕ) → E n} {s : Set ((n : ℕ) → E n)} (hs : IsClosed s) (hne : s.Nonempty) (H : PiNat.longestPrefix x s < PiNat.firstDiff x y) (xs : x ∉ s) (ys : y ∉ s)
Goal: PiNat.cylinder x (PiNat.longestPrefix x s) = PiNat.cylinder y (PiNat.longestPrefix y s)
-- PiNat.exists_lipschitz_retraction_of_isClosed
theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) :
    ∃ f : (∀ n, E n) → ∀ n, E n, (∀ x ∈ s, f x = x) ∧ range f = s ∧ LipschitzWith 1 f 
Arguments: {E : ℕ → Type u_1} [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)] {s : Set ((n : ℕ) → E n)} (hs : IsClosed s) (hne : s.Nonempty)
Goal: ∃ f, (∀ x ∈ s, f x = x) ∧ Set.range f = s ∧ LipschitzWith 1 f
-- PiNat.exists_retraction_of_isClosed
theorem exists_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsClosed s) (hne : s.Nonempty) :
    ∃ f : (∀ n, E n) → ∀ n, E n, (∀ x ∈ s, f x = x) ∧ range f = s ∧ Continuous f 
Arguments: {E : ℕ → Type u_1} [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)] {s : Set ((n : ℕ) → E n)} (hs : IsClosed s) (hne : s.Nonempty)
Goal: ∃ f, (∀ x ∈ s, f x = x) ∧ Set.range f = s ∧ Continuous f
-- PiNat.exists_retraction_subtype_of_isClosed
theorem exists_retraction_subtype_of_isClosed {s : Set (∀ n, E n)} (hs : IsClosed s)
    (hne : s.Nonempty) :
    ∃ f : (∀ n, E n) → s, (∀ x : s, f x = x) ∧ Surjective f ∧ Continuous f 
Arguments: {E : ℕ → Type u_1} [(n : ℕ) → TopologicalSpace (E n)] [∀ (n : ℕ), DiscreteTopology (E n)] {s : Set ((n : ℕ) → E n)} (hs : IsClosed s) (hne : s.Nonempty)
Goal: ∃ f, (∀ (x : ↑s), f ↑x = x) ∧ Function.Surjective f ∧ Continuous f
-- exists_nat_nat_continuous_surjective_of_completeSpace
theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
    [CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
    ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f 
Arguments: (α : Type u_2) [MetricSpace α] [CompleteSpace α] [SecondCountableTopology α] [Nonempty α]
Goal: ∃ f, Continuous f ∧ Function.Surjective f
-- PiCountable.dist_eq_tsum
theorem dist_eq_tsum (x y : ∀ i, F i) :
    dist x y = ∑' i : ι, min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) 
Arguments: {ι : Type u_2} [Encodable ι] {F : ι → Type u_3} [(i : ι) → MetricSpace (F i)] (x y : (i : ι) → F i)
Goal: dist x y = ∑' (i : ι), min ((1 / 2) ^ Encodable.encode i) (dist (x i) (y i))
-- PiCountable.dist_summable
theorem dist_summable (x y : ∀ i, F i) :
    Summable fun i : ι => min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) 
Arguments: {ι : Type u_2} [Encodable ι] {F : ι → Type u_3} [(i : ι) → MetricSpace (F i)] (x y : (i : ι) → F i)
Goal: Summable fun i => min ((1 / 2) ^ Encodable.encode i) (dist (x i) (y i))
-- PiCountable.min_dist_le_dist_pi
theorem min_dist_le_dist_pi (x y : ∀ i, F i) (i : ι) :
    min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) ≤ dist x y 
Arguments: {ι : Type u_2} [Encodable ι] {F : ι → Type u_3} [(i : ι) → MetricSpace (F i)] (x y : (i : ι) → F i) (i : ι)
Goal: min ((1 / 2) ^ Encodable.encode i) (dist (x i) (y i)) ≤ dist x y
-- PiCountable.dist_le_dist_pi_of_dist_lt
theorem dist_le_dist_pi_of_dist_lt {x y : ∀ i, F i} {i : ι} (h : dist x y < (1 / 2) ^ encode i) :
    dist (x i) (y i) ≤ dist x y 
Arguments: {ι : Type u_2} [Encodable ι] {F : ι → Type u_3} [(i : ι) → MetricSpace (F i)] {x y : (i : ι) → F i} {i : ι} (h : dist x y < (1 / 2) ^ Encodable.encode i)
Goal: dist (x i) (y i) ≤ dist x y
-- ContinuousMap.continuous_toFun
structure ContinuousMap (α β : Type*) [TopologicalSpace α] [TopologicalSpace β] where
    protected toFun : α → β
    protected continuous_toFun : Continuous toFun := by continuity
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (self : C(α, β))
Goal: Continuous self.toFun
-- ContinuousMapClass.map_continuous
class ContinuousMapClass (F α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
    [FunLike F α β] : Prop where
    map_continuous (f : F) : Continuous f
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [self : ContinuousMapClass F α β] (f : F)
Goal: Continuous ⇑f
-- map_continuousAt
theorem map_continuousAt (f : F) (a : α) : ContinuousAt f a 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [ContinuousMapClass F α β] (f : F) (a : α)
Goal: ContinuousAt (⇑f) a
-- map_continuousWithinAt
theorem map_continuousWithinAt (f : F) (s : Set α) (a : α) : ContinuousWithinAt f s a 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [ContinuousMapClass F α β] (f : F) (s : Set α) (a : α)
Goal: ContinuousWithinAt (⇑f) s a
-- ContinuousMap.toFun_eq_coe
@[simp]
theorem toFun_eq_coe {f : C(α, β)} : f.toFun = (f : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : C(α, β)}
Goal: f.toFun = ⇑f
-- ContinuousMap.coe_coe
@[simp] -- Porting note: removed `norm_cast` attribute
protected theorem coe_coe {F : Type*} [FunLike F α β] [ContinuousMapClass F α β] (f : F) :
    ⇑(f : C(α, β)) = f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {F : Type u_5} [FunLike F α β] [ContinuousMapClass F α β] (f : F)
Goal: ⇑↑f = ⇑f
-- ContinuousMap.ext
@[ext]
theorem ext {f g : C(α, β)} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : C(α, β)} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- ContinuousMap.coe_copy
@[simp]
theorem coe_copy (f : C(α, β)) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- ContinuousMap.copy_eq
theorem copy_eq (f : C(α, β)) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- ContinuousMap.continuous
protected theorem continuous (f : C(α, β)) : Continuous f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β))
Goal: Continuous ⇑f
-- ContinuousMap.continuous_set_coe
@[continuity]
theorem continuous_set_coe (s : Set C(α, β)) (f : s) : Continuous (f : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (s : Set C(α, β)) (f : ↑s)
Goal: Continuous ⇑↑f
-- ContinuousMap.continuousAt
protected theorem continuousAt (f : C(α, β)) (x : α) : ContinuousAt f x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (x : α)
Goal: ContinuousAt (⇑f) x
-- ContinuousMap.congr_fun
protected theorem congr_fun {f g : C(α, β)} (H : f = g) (x : α) : f x = g x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : C(α, β)} (H : f = g) (x : α)
Goal: f x = g x
-- ContinuousMap.congr_arg
protected theorem congr_arg (f : C(α, β)) {x y : α} (h : x = y) : f x = f y 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) {x y : α} (h : x = y)
Goal: f x = f y
-- ContinuousMap.coe_injective
theorem coe_injective : @Function.Injective C(α, β) (α → β) (↑) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β]
Goal: Function.Injective DFunLike.coe
-- ContinuousMap.coe_mk
@[simp]
theorem coe_mk (f : α → β) (h : Continuous f) : ⇑(⟨f, h⟩ : C(α, β)) = f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (h : Continuous f)
Goal: ⇑{ toFun := f, continuous_toFun := h } = f
-- ContinuousMap.map_specializes
theorem map_specializes (f : C(α, β)) {x y : α} (h : x ⤳ y) : f x ⤳ f y 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) {x y : α} (h : x ⤳ y)
Goal: f x ⤳ f y
-- ContinuousMap.equivFnOfDiscrete_symm_apply_apply
@[simps]
def equivFnOfDiscrete [DiscreteTopology α] : C(α, β) ≃ (α → β) 
Arguments: (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [TopologicalSpace β] [DiscreteTopology α] (f : α → β)
Goal: ∀ (a : α), ((ContinuousMap.equivFnOfDiscrete α β).symm f) a = f a
-- ContinuousMap.equivFnOfDiscrete_apply
@[simps]
def equivFnOfDiscrete [DiscreteTopology α] : C(α, β) ≃ (α → β) 
Arguments: (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [TopologicalSpace β] [DiscreteTopology α] (f : C(α, β)) (a : α)
Goal: (ContinuousMap.equivFnOfDiscrete α β) f a = f a
-- ContinuousMap.coe_id
@[simp]
theorem coe_id : ⇑(ContinuousMap.id α) = id 
Arguments: (α : Type u_1) [TopologicalSpace α]
Goal: ⇑(ContinuousMap.id α) = id
-- ContinuousMap.coe_const
@[simp]
theorem coe_const (b : β) : ⇑(const α b) = Function.const α b 
Arguments: (α : Type u_1) {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (b : β)
Goal: ⇑(ContinuousMap.const α b) = Function.const α b
-- ContinuousMap.constPi_apply
@[simps (config := .asFn)]
def constPi : C(β, α → β) 
Arguments: (α : Type u_1) {β : Type u_2} [TopologicalSpace β]
Goal: ⇑(ContinuousMap.constPi α) = fun b => Function.const α b
-- ContinuousMap.id_apply
@[simp]
theorem id_apply (a : α) : ContinuousMap.id α a = a 
Arguments: {α : Type u_1} [TopologicalSpace α] (a : α)
Goal: (ContinuousMap.id α) a = a
-- ContinuousMap.const_apply
@[simp]
theorem const_apply (b : β) (a : α) : const α b a = b 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (b : β) (a : α)
Goal: (ContinuousMap.const α b) a = b
-- ContinuousMap.coe_comp
@[simp]
theorem coe_comp (f : C(β, γ)) (g : C(α, β)) : ⇑(comp f g) = f ∘ g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : C(β, γ)) (g : C(α, β))
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- ContinuousMap.comp_apply
@[simp]
theorem comp_apply (f : C(β, γ)) (g : C(α, β)) (a : α) : comp f g a = f (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : C(β, γ)) (g : C(α, β)) (a : α)
Goal: (f.comp g) a = f (g a)
-- ContinuousMap.comp_assoc
@[simp]
theorem comp_assoc (f : C(γ, δ)) (g : C(β, γ)) (h : C(α, β)) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (f : C(γ, δ)) (g : C(β, γ)) (h : C(α, β))
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- ContinuousMap.id_comp
@[simp]
theorem id_comp (f : C(α, β)) : (ContinuousMap.id _).comp f = f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β))
Goal: (ContinuousMap.id β).comp f = f
-- ContinuousMap.comp_id
@[simp]
theorem comp_id (f : C(α, β)) : f.comp (ContinuousMap.id _) = f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β))
Goal: f.comp (ContinuousMap.id α) = f
-- ContinuousMap.const_comp
@[simp]
theorem const_comp (c : γ) (f : C(α, β)) : (const β c).comp f = const α c 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (c : γ) (f : C(α, β))
Goal: (ContinuousMap.const β c).comp f = ContinuousMap.const α c
-- ContinuousMap.comp_const
@[simp]
theorem comp_const (f : C(β, γ)) (b : β) : f.comp (const α b) = const α (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : C(β, γ)) (b : β)
Goal: f.comp (ContinuousMap.const α b) = ContinuousMap.const α (f b)
-- ContinuousMap.cancel_right
@[simp]
theorem cancel_right {f₁ f₂ : C(β, γ)} {g : C(α, β)} (hg : Surjective g) :
    f₁.comp g = f₂.comp g ↔ f₁ = f₂ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f₁ f₂ : C(β, γ)} {g : C(α, β)} (hg : Function.Surjective ⇑g)
Goal: f₁.comp g = f₂.comp g ↔ f₁ = f₂
-- ContinuousMap.cancel_left
@[simp]
theorem cancel_left {f : C(β, γ)} {g₁ g₂ : C(α, β)} (hf : Injective f) :
    f.comp g₁ = f.comp g₂ ↔ g₁ = g₂ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : C(β, γ)} {g₁ g₂ : C(α, β)} (hf : Function.Injective ⇑f)
Goal: f.comp g₁ = f.comp g₂ ↔ g₁ = g₂
-- ContinuousMap.fst_apply
@[simps (config := .asFn)]
def fst : C(α × β, α) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β]
Goal: ⇑ContinuousMap.fst = Prod.fst
-- ContinuousMap.snd_apply
@[simps (config := .asFn)]
def snd : C(α × β, β) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β]
Goal: ⇑ContinuousMap.snd = Prod.snd
-- ContinuousMap.prodMap_apply
@[simps]
def prodMap (f : C(α₁, α₂)) (g : C(β₁, β₂)) : C(α₁ × β₁, α₂ × β₂) 
Arguments: {α₁ : Type u_5} {α₂ : Type u_6} {β₁ : Type u_7} {β₂ : Type u_8} [TopologicalSpace α₁] [TopologicalSpace α₂] [TopologicalSpace β₁] [TopologicalSpace β₂] (f : C(α₁, α₂)) (g : C(β₁, β₂))
Goal: ∀ (a : α₁ × β₁), (f.prodMap g) a = Prod.map (⇑f) (⇑g) a
-- ContinuousMap.prod_eval
@[simp]
theorem prod_eval (f : C(α, β₁)) (g : C(α, β₂)) (a : α) : (prodMk f g) a = (f a, g a) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β₁ : Type u_7} {β₂ : Type u_8} [TopologicalSpace β₁] [TopologicalSpace β₂] (f : C(α, β₁)) (g : C(α, β₂)) (a : α)
Goal: (f.prodMk g) a = (f a, g a)
-- ContinuousMap.prodSwap_apply
@[simps!]
def prodSwap : C(α × β, β × α) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (x : α × β)
Goal: ContinuousMap.prodSwap x = (x.2, x.1)
-- ContinuousMap.sigmaMk_apply
@[simps apply]
def sigmaMk (i : I) : C(X i, Σ i, X i) 
Arguments: {I : Type u_5} {X : I → Type u_7} [(i : I) → TopologicalSpace (X i)] (i : I) (snd : X i)
Goal: (ContinuousMap.sigmaMk i) snd = ⟨i, snd⟩
-- ContinuousMap.sigma_apply
@[simps]
def sigma (f : ∀ i, C(X i, A)) : C((Σ i, X i), A) 
Arguments: {I : Type u_5} {A : Type u_6} {X : I → Type u_7} [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : (i : I) → C(X i, A)) (ig : (i : I) × X i)
Goal: (ContinuousMap.sigma f) ig = (f ig.fst) ig.snd
-- ContinuousMap.sigmaEquiv_symm_apply
@[simps]
def sigmaEquiv : (∀ i, C(X i, A)) ≃ C((Σ i, X i), A) 
Arguments: {I : Type u_5} (A : Type u_6) (X : I → Type u_7) [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : C((i : I) × X i, A)) (i : I)
Goal: (ContinuousMap.sigmaEquiv A X).symm f i = f.comp (ContinuousMap.sigmaMk i)
-- ContinuousMap.sigmaEquiv_apply
@[simps]
def sigmaEquiv : (∀ i, C(X i, A)) ≃ C((Σ i, X i), A) 
Arguments: {I : Type u_5} (A : Type u_6) (X : I → Type u_7) [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : (i : I) → C(X i, A))
Goal: (ContinuousMap.sigmaEquiv A X) f = ContinuousMap.sigma f
-- ContinuousMap.pi_eval
@[simp]
theorem pi_eval (f : ∀ i, C(A, X i)) (a : A) : (pi f) a = fun i : I => (f i) a 
Arguments: {I : Type u_5} {A : Type u_6} {X : I → Type u_7} [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : (i : I) → C(A, X i)) (a : A)
Goal: (ContinuousMap.pi f) a = fun i => (f i) a
-- ContinuousMap.eval_apply
@[simps (config := .asFn)]
def eval (i : I) : C(∀ j, X j, X i) 
Arguments: {I : Type u_5} {X : I → Type u_7} [(i : I) → TopologicalSpace (X i)] (i : I)
Goal: ⇑(ContinuousMap.eval i) = Function.eval i
-- ContinuousMap.piEquiv_apply
@[simps]
def piEquiv : (∀ i, C(A, X i)) ≃ C(A, ∀ i, X i) 
Arguments: {I : Type u_5} (A : Type u_6) (X : I → Type u_7) [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : (i : I) → C(A, X i))
Goal: (ContinuousMap.piEquiv A X) f = ContinuousMap.pi f
-- ContinuousMap.piEquiv_symm_apply
@[simps]
def piEquiv : (∀ i, C(A, X i)) ≃ C(A, ∀ i, X i) 
Arguments: {I : Type u_5} (A : Type u_6) (X : I → Type u_7) [TopologicalSpace A] [(i : I) → TopologicalSpace (X i)] (f : C(A, (i : I) → X i)) (i : I)
Goal: (ContinuousMap.piEquiv A X).symm f i = (ContinuousMap.eval i).comp f
-- ContinuousMap.piMap_apply
@[simps!]
def piMap (f : ∀ i, C(X i, Y i)) : C((i : I) → X i, (i : I) → Y i) 
Arguments: {I : Type u_5} {X : I → Type u_7} {Y : I → Type u_8} [(i : I) → TopologicalSpace (X i)] [(i : I) → TopologicalSpace (Y i)] (f : (i : I) → C(X i, Y i)) (a : (i : I) → X i) (i : I)
Goal: (ContinuousMap.piMap f) a i = (f i) (a i)
-- ContinuousMap.coe_restrict
@[simp]
theorem coe_restrict (f : C(α, β)) : ⇑(f.restrict s) = f ∘ ((↑) : s → α) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (s : Set α) (f : C(α, β))
Goal: ⇑(ContinuousMap.restrict s f) = ⇑f ∘ Subtype.val
-- ContinuousMap.restrict_apply
@[simp]
theorem restrict_apply (f : C(α, β)) (s : Set α) (x : s) : f.restrict s x = f x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (s : Set α) (x : ↑s)
Goal: (ContinuousMap.restrict s f) x = f ↑x