ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsLocalHomeomorphOn.of_comp_right
theorem of_comp_right (hgf : IsLocalHomeomorphOn (g ∘ f) s) (hf : IsLocalHomeomorphOn f s) :
    IsLocalHomeomorphOn g (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {g : Y → Z} {f : X → Y} {s : Set X} (hgf : IsLocalHomeomorphOn (g ∘ f) s) (hf : IsLocalHomeomorphOn f s)
Goal: IsLocalHomeomorphOn g (f '' s)
-- IsLocalHomeomorphOn.map_nhds_eq
theorem map_nhds_eq (hf : IsLocalHomeomorphOn f s) {x : X} (hx : x ∈ s) : (𝓝 x).map f = 𝓝 (f x) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} (hf : IsLocalHomeomorphOn f s) {x : X} (hx : x ∈ s)
Goal: Filter.map f (𝓝 x) = 𝓝 (f x)
-- IsLocalHomeomorphOn.continuousAt
protected theorem continuousAt (hf : IsLocalHomeomorphOn f s) {x : X} (hx : x ∈ s) :
    ContinuousAt f x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} (hf : IsLocalHomeomorphOn f s) {x : X} (hx : x ∈ s)
Goal: ContinuousAt f x
-- IsLocalHomeomorphOn.continuousOn
protected theorem continuousOn (hf : IsLocalHomeomorphOn f s) : ContinuousOn f s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} (hf : IsLocalHomeomorphOn f s)
Goal: ContinuousOn f s
-- IsLocalHomeomorphOn.comp
protected theorem comp (hg : IsLocalHomeomorphOn g t) (hf : IsLocalHomeomorphOn f s)
    (h : Set.MapsTo f s t) : IsLocalHomeomorphOn (g ∘ f) s 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {g : Y → Z} {f : X → Y} {s : Set X} {t : Set Y} (hg : IsLocalHomeomorphOn g t) (hf : IsLocalHomeomorphOn f s) (h : Set.MapsTo f s t)
Goal: IsLocalHomeomorphOn (g ∘ f) s
-- Homeomorph.isLocalHomeomorph
theorem Homeomorph.isLocalHomeomorph (f : X ≃ₜ Y) : IsLocalHomeomorph f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X ≃ₜ Y)
Goal: IsLocalHomeomorph ⇑f
-- isLocalHomeomorph_iff_isLocalHomeomorphOn_univ
theorem isLocalHomeomorph_iff_isLocalHomeomorphOn_univ :
    IsLocalHomeomorph f ↔ IsLocalHomeomorphOn f Set.univ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: IsLocalHomeomorph f ↔ IsLocalHomeomorphOn f Set.univ
-- IsLocalHomeomorph.isLocalHomeomorphOn
protected theorem IsLocalHomeomorph.isLocalHomeomorphOn (hf : IsLocalHomeomorph f) :
    IsLocalHomeomorphOn f s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} (hf : IsLocalHomeomorph f)
Goal: IsLocalHomeomorphOn f s
-- isLocalHomeomorph_iff_openEmbedding_restrict
theorem isLocalHomeomorph_iff_openEmbedding_restrict {f : X → Y} :
    IsLocalHomeomorph f ↔ ∀ x : X, ∃ U ∈ 𝓝 x, OpenEmbedding (U.restrict f) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: IsLocalHomeomorph f ↔ ∀ (x : X), ∃ U ∈ 𝓝 x, OpenEmbedding (U.restrict f)
-- OpenEmbedding.isLocalHomeomorph
theorem OpenEmbedding.isLocalHomeomorph (hf : OpenEmbedding f) : IsLocalHomeomorph f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : OpenEmbedding f)
Goal: IsLocalHomeomorph f
-- IsLocalHomeomorph.mk
theorem mk (h : ∀ x : X, ∃ e : PartialHomeomorph X Y, x ∈ e.source ∧ Set.EqOn f e e.source) :
    IsLocalHomeomorph f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : ∀ (x : X), ∃ e, x ∈ e.source ∧ Set.EqOn f (↑e) e.source)
Goal: IsLocalHomeomorph f
-- IsLocalHomeomorph.Homeomorph.isLocalHomeomorph
lemma Homeomorph.isLocalHomeomorph (h : X ≃ₜ Y) : IsLocalHomeomorph h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: IsLocalHomeomorph ⇑h
-- IsLocalHomeomorph.isLocallyInjective
lemma isLocallyInjective (hf : IsLocalHomeomorph f) : IsLocallyInjective f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsLocalHomeomorph f)
Goal: IsLocallyInjective f
-- IsLocalHomeomorph.of_comp
theorem of_comp (hgf : IsLocalHomeomorph (g ∘ f)) (hg : IsLocalHomeomorph g)
    (cont : Continuous f) : IsLocalHomeomorph f 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {g : Y → Z} {f : X → Y} (hgf : IsLocalHomeomorph (g ∘ f)) (hg : IsLocalHomeomorph g) (cont : Continuous f)
Goal: IsLocalHomeomorph f
-- IsLocalHomeomorph.map_nhds_eq
theorem map_nhds_eq (hf : IsLocalHomeomorph f) (x : X) : (𝓝 x).map f = 𝓝 (f x) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsLocalHomeomorph f) (x : X)
Goal: Filter.map f (𝓝 x) = 𝓝 (f x)
-- IsLocalHomeomorph.continuous
protected theorem continuous (hf : IsLocalHomeomorph f) : Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsLocalHomeomorph f)
Goal: Continuous f
-- IsLocalHomeomorph.isOpenMap
protected theorem isOpenMap (hf : IsLocalHomeomorph f) : IsOpenMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsLocalHomeomorph f)
Goal: IsOpenMap f
-- IsLocalHomeomorph.comp
protected theorem comp (hg : IsLocalHomeomorph g) (hf : IsLocalHomeomorph f) :
    IsLocalHomeomorph (g ∘ f) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {g : Y → Z} {f : X → Y} (hg : IsLocalHomeomorph g) (hf : IsLocalHomeomorph f)
Goal: IsLocalHomeomorph (g ∘ f)
-- IsLocalHomeomorph.openEmbedding_of_injective
theorem openEmbedding_of_injective (hf : IsLocalHomeomorph f) (hi : f.Injective) :
    OpenEmbedding f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsLocalHomeomorph f) (hi : Function.Injective f)
Goal: OpenEmbedding f
-- IsLocalHomeomorph.openEmbedding_of_comp
theorem openEmbedding_of_comp (hf : IsLocalHomeomorph g) (hgf : OpenEmbedding (g ∘ f))
    (cont : Continuous f) : OpenEmbedding f 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {g : Y → Z} {f : X → Y} (hf : IsLocalHomeomorph g) (hgf : OpenEmbedding (g ∘ f)) (cont : Continuous f)
Goal: OpenEmbedding f
-- IsLocalHomeomorph.isTopologicalBasis
theorem isTopologicalBasis (hf : IsLocalHomeomorph f) : IsTopologicalBasis
    {U : Set X | ∃ V : Set Y, IsOpen V ∧ ∃ s : C(V,X), f ∘ s = (↑) ∧ Set.range s = U} 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsLocalHomeomorph f)
Goal: TopologicalSpace.IsTopologicalBasis {U | ∃ V, IsOpen V ∧ ∃ s, f ∘ ⇑s = Subtype.val ∧ Set.range ⇑s = U}
-- unitInterval.zero_mem
theorem zero_mem : (0 : ℝ) ∈ I 
Goal: 0 ∈ unitInterval
-- unitInterval.one_mem
theorem one_mem : (1 : ℝ) ∈ I 
Goal: 1 ∈ unitInterval
-- unitInterval.mul_mem
theorem mul_mem {x y : ℝ} (hx : x ∈ I) (hy : y ∈ I) : x * y ∈ I 
Arguments: {x y : ℝ} (hx : x ∈ unitInterval) (hy : y ∈ unitInterval)
Goal: x * y ∈ unitInterval
-- unitInterval.div_mem
theorem div_mem {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hxy : x ≤ y) : x / y ∈ I 
Arguments: {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hxy : x ≤ y)
Goal: x / y ∈ unitInterval
-- unitInterval.fract_mem
theorem fract_mem (x : ℝ) : fract x ∈ I 
Arguments: (x : ℝ)
Goal: Int.fract x ∈ unitInterval
-- unitInterval.mem_iff_one_sub_mem
theorem mem_iff_one_sub_mem {t : ℝ} : t ∈ I ↔ 1 - t ∈ I 
Arguments: {t : ℝ}
Goal: t ∈ unitInterval ↔ 1 - t ∈ unitInterval
-- unitInterval.univ_eq_Icc
lemma univ_eq_Icc : (univ : Set I) = Icc (0 : I) (1 : I) 
Goal: Set.univ = Set.Icc 0 1
-- unitInterval.coe_ne_zero
theorem coe_ne_zero {x : I} : (x : ℝ) ≠ 0 ↔ x ≠ 0 
Arguments: {x : ↑unitInterval}
Goal: ↑x ≠ 0 ↔ x ≠ 0
-- unitInterval.coe_ne_one
theorem coe_ne_one {x : I} : (x : ℝ) ≠ 1 ↔ x ≠ 1 
Arguments: {x : ↑unitInterval}
Goal: ↑x ≠ 1 ↔ x ≠ 1
-- unitInterval.mul_le_left
theorem mul_le_left {x y : I} : x * y ≤ x 
Arguments: {x y : ↑unitInterval}
Goal: x * y ≤ x
-- unitInterval.mul_le_right
theorem mul_le_right {x y : I} : x * y ≤ y 
Arguments: {x y : ↑unitInterval}
Goal: x * y ≤ y
-- unitInterval.symm_zero
@[simp]
theorem symm_zero : σ 0 = 1 
Goal: unitInterval.symm 0 = 1
-- unitInterval.symm_one
@[simp]
theorem symm_one : σ 1 = 0 
Goal: unitInterval.symm 1 = 0
-- unitInterval.symm_symm
@[simp]
theorem symm_symm (x : I) : σ (σ x) = x 
Arguments: (x : ↑unitInterval)
Goal: unitInterval.symm (unitInterval.symm x) = x
-- unitInterval.symm_involutive
theorem symm_involutive : Function.Involutive (symm : I → I) 
Goal: Function.Involutive unitInterval.symm
-- unitInterval.symm_bijective
theorem symm_bijective : Function.Bijective (symm : I → I) 
Goal: Function.Bijective unitInterval.symm
-- unitInterval.coe_symm_eq
@[simp]
theorem coe_symm_eq (x : I) : (σ x : ℝ) = 1 - x 
Arguments: (x : ↑unitInterval)
Goal: ↑(unitInterval.symm x) = 1 - ↑x
-- unitInterval.continuous_symm
@[continuity, fun_prop]
theorem continuous_symm : Continuous σ 
Goal: Continuous unitInterval.symm
-- unitInterval.symmHomeomorph_symm_apply
@[simps]
def symmHomeomorph : I ≃ₜ I 
Goal: ∀ (a : ↑unitInterval), unitInterval.symmHomeomorph.symm a = unitInterval.symm a
-- unitInterval.symmHomeomorph_apply
@[simps]
def symmHomeomorph : I ≃ₜ I 
Goal: ∀ (a : ↑unitInterval), unitInterval.symmHomeomorph a = unitInterval.symm a
-- unitInterval.strictAnti_symm
theorem strictAnti_symm : StrictAnti σ 
Goal: StrictAnti unitInterval.symm
-- unitInterval.half_le_symm_iff
theorem half_le_symm_iff (t : I) : 1 / 2 ≤ (σ t : ℝ) ↔ (t : ℝ) ≤ 1 / 2 
Arguments: (t : ↑unitInterval)
Goal: 1 / 2 ≤ ↑(unitInterval.symm t) ↔ ↑t ≤ 1 / 2
-- unitInterval.nonneg
theorem nonneg (x : I) : 0 ≤ (x : ℝ) 
Arguments: (x : ↑unitInterval)
Goal: 0 ≤ ↑x
-- unitInterval.one_minus_nonneg
theorem one_minus_nonneg (x : I) : 0 ≤ 1 - (x : ℝ) 
Arguments: (x : ↑unitInterval)
Goal: 0 ≤ 1 - ↑x
-- unitInterval.le_one
theorem le_one (x : I) : (x : ℝ) ≤ 1 
Arguments: (x : ↑unitInterval)
Goal: ↑x ≤ 1
-- unitInterval.one_minus_le_one
theorem one_minus_le_one (x : I) : 1 - (x : ℝ) ≤ 1 
Arguments: (x : ↑unitInterval)
Goal: 1 - ↑x ≤ 1
-- unitInterval.add_pos
theorem add_pos {t : I} {x : ℝ} (hx : 0 < x) : 0 < (x + t : ℝ) 
Arguments: {t : ↑unitInterval} {x : ℝ} (hx : 0 < x)
Goal: 0 < x + ↑t
-- unitInterval.nonneg'
theorem nonneg' {t : I} : 0 ≤ t 
Arguments: {t : ↑unitInterval}
Goal: 0 ≤ t
-- unitInterval.le_one'
theorem le_one' {t : I} : t ≤ 1 
Arguments: {t : ↑unitInterval}
Goal: t ≤ 1
-- unitInterval.mul_pos_mem_iff
theorem mul_pos_mem_iff {a t : ℝ} (ha : 0 < a) : a * t ∈ I ↔ t ∈ Set.Icc (0 : ℝ) (1 / a) 
Arguments: {a t : ℝ} (ha : 0 < a)
Goal: a * t ∈ unitInterval ↔ t ∈ Set.Icc 0 (1 / a)
-- unitInterval.two_mul_sub_one_mem_iff
theorem two_mul_sub_one_mem_iff {t : ℝ} : 2 * t - 1 ∈ I ↔ t ∈ Set.Icc (1 / 2 : ℝ) 1 
Arguments: {t : ℝ}
Goal: 2 * t - 1 ∈ unitInterval ↔ t ∈ Set.Icc (1 / 2) 1
-- Set.abs_projIcc_sub_projIcc
lemma _root_.Set.abs_projIcc_sub_projIcc : (|projIcc a b h c - projIcc a b h d| : α) ≤ |c - d| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b c d : α} (h : a ≤ b)
Goal: |↑(Set.projIcc a b h c) - ↑(Set.projIcc a b h d)| ≤ |c - d|
-- Set.Icc.addNSMul_zero
lemma addNSMul_zero : addNSMul h δ 0 = a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α} (h : a ≤ b) {δ : α}
Goal: ↑(Set.Icc.addNSMul h δ 0) = a
-- Set.Icc.addNSMul_eq_right
lemma addNSMul_eq_right [Archimedean α] (hδ : 0 < δ) :
    ∃ m, ∀ n ≥ m, addNSMul h δ n = b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α} (h : a ≤ b) {δ : α} [Archimedean α] (hδ : 0 < δ)
Goal: ∃ m, ∀ n ≥ m, ↑(Set.Icc.addNSMul h δ n) = b
-- Set.Icc.monotone_addNSMul
lemma monotone_addNSMul (hδ : 0 ≤ δ) : Monotone (addNSMul h δ) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α} (h : a ≤ b) {δ : α} (hδ : 0 ≤ δ)
Goal: Monotone (Set.Icc.addNSMul h δ)
-- Set.Icc.abs_sub_addNSMul_le
lemma abs_sub_addNSMul_le (hδ : 0 ≤ δ) {t : Icc a b} (n : ℕ)
    (ht : t ∈ Icc (addNSMul h δ n) (addNSMul h δ (n+1))) :
    (|t - addNSMul h δ n| : α) ≤ δ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α} (h : a ≤ b) {δ : α} (hδ : 0 ≤ δ) {t : ↑(Set.Icc a b)} (n : ℕ) (ht : t ∈ Set.Icc (Set.Icc.addNSMul h δ n) (Set.Icc.addNSMul h δ (n + 1)))
Goal: |↑t - ↑(Set.Icc.addNSMul h δ n)| ≤ δ
-- exists_monotone_Icc_subset_open_cover_Icc
lemma exists_monotone_Icc_subset_open_cover_Icc {ι} {a b : ℝ} (h : a ≤ b) {c : ι → Set (Icc a b)}
    (hc₁ : ∀ i, IsOpen (c i)) (hc₂ : univ ⊆ ⋃ i, c i) : ∃ t : ℕ → Icc a b, t 0 = a ∧
      Monotone t ∧ (∃ m, ∀ n ≥ m, t n = b) ∧ ∀ n, ∃ i, Icc (t n) (t (n + 1)) ⊆ c i 
Arguments: {ι : Sort u_1} {a b : ℝ} (h : a ≤ b) {c : ι → Set ↑(Set.Icc a b)} (hc₁ : ∀ (i : ι), IsOpen (c i)) (hc₂ : Set.univ ⊆ ⋃ i, c i)
Goal: ∃ t, ↑(t 0) = a ∧ Monotone t ∧ (∃ m, ∀ n ≥ m, ↑(t n) = b) ∧ ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ c i
-- exists_monotone_Icc_subset_open_cover_unitInterval
lemma exists_monotone_Icc_subset_open_cover_unitInterval {ι} {c : ι → Set I}
    (hc₁ : ∀ i, IsOpen (c i)) (hc₂ : univ ⊆ ⋃ i, c i) : ∃ t : ℕ → I, t 0 = 0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, t m = 1) ∧ ∀ n, ∃ i, Icc (t n) (t (n + 1)) ⊆ c i 
Arguments: {ι : Sort u_1} {c : ι → Set ↑unitInterval} (hc₁ : ∀ (i : ι), IsOpen (c i)) (hc₂ : Set.univ ⊆ ⋃ i, c i)
Goal: ∃ t, t 0 = 0 ∧ Monotone t ∧ (∃ n, ∀ m ≥ n, t m = 1) ∧ ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ c i
-- exists_monotone_Icc_subset_open_cover_unitInterval_prod_self
lemma exists_monotone_Icc_subset_open_cover_unitInterval_prod_self {ι} {c : ι → Set (I × I)}
    (hc₁ : ∀ i, IsOpen (c i)) (hc₂ : univ ⊆ ⋃ i, c i) :
    ∃ t : ℕ → I, t 0 = 0 ∧ Monotone t ∧ (∃ n, ∀ m ≥ n, t m = 1) ∧
      ∀ n m, ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i 
Arguments: {ι : Sort u_1} {c : ι → Set (↑unitInterval × ↑unitInterval)} (hc₁ : ∀ (i : ι), IsOpen (c i)) (hc₂ : Set.univ ⊆ ⋃ i, c i)
Goal: ∃ t,
  t 0 = 0 ∧
    Monotone t ∧
      (∃ n, ∀ m ≥ n, t m = 1) ∧ ∀ (n m : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ×ˢ Set.Icc (t m) (t (m + 1)) ⊆ c i
-- projIcc_eq_zero
@[simp]
theorem projIcc_eq_zero {x : ℝ} : projIcc (0 : ℝ) 1 zero_le_one x = 0 ↔ x ≤ 0 
Arguments: {x : ℝ}
Goal: Set.projIcc 0 1 ⋯ x = 0 ↔ x ≤ 0
-- projIcc_eq_one
@[simp]
theorem projIcc_eq_one {x : ℝ} : projIcc (0 : ℝ) 1 zero_le_one x = 1 ↔ 1 ≤ x 
Arguments: {x : ℝ}
Goal: Set.projIcc 0 1 ⋯ x = 1 ↔ 1 ≤ x
-- affineHomeomorph_image_I
theorem affineHomeomorph_image_I (a b : 𝕜) (h : 0 < a) :
    affineHomeomorph a b h.ne.symm '' Set.Icc 0 1 = Set.Icc b (a + b) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [TopologicalRing 𝕜] (a b : 𝕜) (h : 0 < a)
Goal: ⇑(affineHomeomorph a b ⋯) '' Set.Icc 0 1 = Set.Icc b (a + b)
-- iccHomeoI_apply_coe
@[simp]
theorem iccHomeoI_apply_coe (a b : 𝕜) (h : a < b) (x : Set.Icc a b) :
    ((iccHomeoI a b h) x : 𝕜) = (x - a) / (b - a) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [TopologicalRing 𝕜] (a b : 𝕜) (h : a < b) (x : ↑(Set.Icc a b))
Goal: ↑((iccHomeoI a b h) x) = (↑x - a) / (b - a)
-- iccHomeoI_symm_apply_coe
@[simp]
theorem iccHomeoI_symm_apply_coe (a b : 𝕜) (h : a < b) (x : Set.Icc (0 : 𝕜) (1 : 𝕜)) :
    ((iccHomeoI a b h).symm x : 𝕜) = (b - a) * x + a 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [TopologicalRing 𝕜] (a b : 𝕜) (h : a < b) (x : ↑(Set.Icc 0 1))
Goal: ↑((iccHomeoI a b h).symm x) = (b - a) * ↑x + a
-- rtendsto_nhds
theorem rtendsto_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
    RTendsto r l (𝓝 x) ↔ ∀ s, IsOpen s → x ∈ s → r.core s ∈ l 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {r : Rel Y X} {l : Filter Y} {x : X}
Goal: Filter.RTendsto r l (𝓝 x) ↔ ∀ (s : Set X), IsOpen s → x ∈ s → r.core s ∈ l
-- rtendsto'_nhds
theorem rtendsto'_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
    RTendsto' r l (𝓝 x) ↔ ∀ s, IsOpen s → x ∈ s → r.preimage s ∈ l 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {r : Rel Y X} {l : Filter Y} {x : X}
Goal: Filter.RTendsto' r l (𝓝 x) ↔ ∀ (s : Set X), IsOpen s → x ∈ s → r.preimage s ∈ l
-- ptendsto_nhds
theorem ptendsto_nhds {f : Y →. X} {l : Filter Y} {x : X} :
    PTendsto f l (𝓝 x) ↔ ∀ s, IsOpen s → x ∈ s → f.core s ∈ l 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f : Y →. X} {l : Filter Y} {x : X}
Goal: Filter.PTendsto f l (𝓝 x) ↔ ∀ (s : Set X), IsOpen s → x ∈ s → f.core s ∈ l
-- ptendsto'_nhds
theorem ptendsto'_nhds {f : Y →. X} {l : Filter Y} {x : X} :
    PTendsto' f l (𝓝 x) ↔ ∀ s, IsOpen s → x ∈ s → f.preimage s ∈ l 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f : Y →. X} {l : Filter Y} {x : X}
Goal: Filter.PTendsto' f l (𝓝 x) ↔ ∀ (s : Set X), IsOpen s → x ∈ s → f.preimage s ∈ l
-- open_dom_of_pcontinuous
theorem open_dom_of_pcontinuous {f : X →. Y} (h : PContinuous f) : IsOpen f.Dom 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X →. Y} (h : PContinuous f)
Goal: IsOpen f.Dom
-- pcontinuous_iff'
theorem pcontinuous_iff' {f : X →. Y} :
    PContinuous f ↔ ∀ {x y} (h : y ∈ f x), PTendsto' f (𝓝 x) (𝓝 y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X →. Y}
Goal: PContinuous f ↔ ∀ {x : X} {y : Y}, y ∈ f x → Filter.PTendsto' f (𝓝 x) (𝓝 y)
-- continuousWithinAt_iff_ptendsto_res
theorem continuousWithinAt_iff_ptendsto_res (f : X → Y) {x : X} {s : Set X} :
    ContinuousWithinAt f s x ↔ PTendsto (PFun.res f s) (𝓝 x) (𝓝 (f x)) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) {x : X} {s : Set X}
Goal: ContinuousWithinAt f s x ↔ Filter.PTendsto (PFun.res f s) (𝓝 x) (𝓝 (f x))
-- ContinuousMap.compactOpen_eq
theorem compactOpen_eq : @compactOpen X Y _ _ =
    .generateFrom (image2 (fun K U ↦ {f | MapsTo f K U}) {K | IsCompact K} {t | IsOpen t}) 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y]
Goal: ContinuousMap.compactOpen =
  TopologicalSpace.generateFrom (Set.image2 (fun K U => {f | Set.MapsTo (⇑f) K U}) {K | IsCompact K} {t | IsOpen t})
-- ContinuousMap.isOpen_setOf_mapsTo
theorem isOpen_setOf_mapsTo (hK : IsCompact K) (hU : IsOpen U) :
    IsOpen {f : C(X, Y) | MapsTo f K U} 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {K : Set X} {U : Set Y} (hK : IsCompact K) (hU : IsOpen U)
Goal: IsOpen {f | Set.MapsTo (⇑f) K U}
-- ContinuousMap.eventually_mapsTo
lemma eventually_mapsTo {f : C(X, Y)} (hK : IsCompact K) (hU : IsOpen U) (h : MapsTo f K U) :
    ∀ᶠ g : C(X, Y) in 𝓝 f, MapsTo g K U 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {K : Set X} {U : Set Y} {f : C(X, Y)} (hK : IsCompact K) (hU : IsOpen U) (h : Set.MapsTo (⇑f) K U)
Goal: ∀ᶠ (g : C(X, Y)) in 𝓝 f, Set.MapsTo (⇑g) K U
-- ContinuousMap.nhds_compactOpen
lemma nhds_compactOpen (f : C(X, Y)) :
    𝓝 f = ⨅ (K : Set X) (_ : IsCompact K) (U : Set Y) (_ : IsOpen U) (_ : MapsTo f K U),
      𝓟 {g : C(X, Y) | MapsTo g K U} 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (f : C(X, Y))
Goal: 𝓝 f = ⨅ K, ⨅ (_ : IsCompact K), ⨅ U, ⨅ (_ : IsOpen U), ⨅ (_ : Set.MapsTo (⇑f) K U), 𝓟 {g | Set.MapsTo (⇑g) K U}
-- ContinuousMap.tendsto_nhds_compactOpen
lemma tendsto_nhds_compactOpen {l : Filter α} {f : α → C(Y, Z)} {g : C(Y, Z)} :
    Tendsto f l (𝓝 g) ↔
      ∀ K, IsCompact K → ∀ U, IsOpen U → MapsTo g K U → ∀ᶠ a in l, MapsTo (f a) K U 
Arguments: {α : Type u_1} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace Y] [TopologicalSpace Z] {l : Filter α} {f : α → C(Y, Z)} {g : C(Y, Z)}
Goal: Filter.Tendsto f l (𝓝 g) ↔
  ∀ (K : Set Y), IsCompact K → ∀ (U : Set Z), IsOpen U → Set.MapsTo (⇑g) K U → ∀ᶠ (a : α) in l, Set.MapsTo (⇑(f a)) K U
-- ContinuousMap.continuous_compactOpen
lemma continuous_compactOpen {f : X → C(Y, Z)} :
    Continuous f ↔ ∀ K, IsCompact K → ∀ U, IsOpen U → IsOpen {x | MapsTo (f x) K U} 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → C(Y, Z)}
Goal: Continuous f ↔ ∀ (K : Set Y), IsCompact K → ∀ (U : Set Z), IsOpen U → IsOpen {x | Set.MapsTo (⇑(f x)) K U}
-- ContinuousMap.continuous_comp
theorem continuous_comp (g : C(Y, Z)) : Continuous (ContinuousMap.comp g : C(X, Y) → C(X, Z)) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (g : C(Y, Z))
Goal: Continuous g.comp
-- ContinuousMap.inducing_comp
theorem inducing_comp (g : C(Y, Z)) (hg : Inducing g) : Inducing (g.comp : C(X, Y) → C(X, Z)) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (g : C(Y, Z)) (hg : Inducing ⇑g)
Goal: Inducing g.comp
-- ContinuousMap.embedding_comp
theorem embedding_comp (g : C(Y, Z)) (hg : Embedding g) : Embedding (g.comp : C(X, Y) → C(X, Z)) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (g : C(Y, Z)) (hg : Embedding ⇑g)
Goal: Embedding g.comp
-- ContinuousMap.continuous_comp_left
theorem continuous_comp_left (f : C(X, Y)) : Continuous (fun g => g.comp f : C(Y, Z) → C(X, Z)) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : C(X, Y))
Goal: Continuous fun g => g.comp f
-- ContinuousMap.continuous_comp'
theorem continuous_comp' : Continuous fun x : C(X, Y) × C(Y, Z) => x.2.comp x.1 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z]
Goal: Continuous fun x => x.2.comp x.1
-- Filter.Tendsto.compCM
lemma _root_.Filter.Tendsto.compCM {α : Type*} {l : Filter α} {g : α → C(Y, Z)} {g₀ : C(Y, Z)}
    {f : α → C(X, Y)} {f₀ : C(X, Y)} (hg : Tendsto g l (𝓝 g₀)) (hf : Tendsto f l (𝓝 f₀)) :
    Tendsto (fun a ↦ (g a).comp (f a)) l (𝓝 (g₀.comp f₀)) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z] {α : Type u_6} {l : Filter α} {g : α → C(Y, Z)} {g₀ : C(Y, Z)} {f : α → C(X, Y)} {f₀ : C(X, Y)} (hg : Filter.Tendsto g l (𝓝 g₀)) (hf : Filter.Tendsto f l (𝓝 f₀))
Goal: Filter.Tendsto (fun a => (g a).comp (f a)) l (𝓝 (g₀.comp f₀))
-- ContinuousAt.compCM
nonrec lemma _root_.ContinuousAt.compCM (hg : ContinuousAt g a) (hf : ContinuousAt f a) :
    ContinuousAt (fun x ↦ (g x).comp (f x)) a 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z] {X' : Type u_6} [TopologicalSpace X'] {a : X'} {g : X' → C(Y, Z)} {f : X' → C(X, Y)} (hg : ContinuousAt g a) (hf : ContinuousAt f a)
Goal: ContinuousAt (fun x => (g x).comp (f x)) a
-- ContinuousWithinAt.compCM
nonrec lemma _root_.ContinuousWithinAt.compCM (hg : ContinuousWithinAt g s a)
    (hf : ContinuousWithinAt f s a) : ContinuousWithinAt (fun x ↦ (g x).comp (f x)) s a 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z] {X' : Type u_6} [TopologicalSpace X'] {a : X'} {g : X' → C(Y, Z)} {f : X' → C(X, Y)} {s : Set X'} (hg : ContinuousWithinAt g s a) (hf : ContinuousWithinAt f s a)
Goal: ContinuousWithinAt (fun x => (g x).comp (f x)) s a
-- ContinuousOn.compCM
lemma _root_.ContinuousOn.compCM (hg : ContinuousOn g s) (hf : ContinuousOn f s) :
    ContinuousOn (fun x ↦ (g x).comp (f x)) s 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z] {X' : Type u_6} [TopologicalSpace X'] {g : X' → C(Y, Z)} {f : X' → C(X, Y)} {s : Set X'} (hg : ContinuousOn g s) (hf : ContinuousOn f s)
Goal: ContinuousOn (fun x => (g x).comp (f x)) s
-- Continuous.compCM
lemma _root_.Continuous.compCM (hg : Continuous g) (hf : Continuous f) :
    Continuous fun x => (g x).comp (f x) 
Arguments: {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [LocallyCompactPair Y Z] {X' : Type u_6} [TopologicalSpace X'] {g : X' → C(Y, Z)} {f : X' → C(X, Y)} (hg : Continuous g) (hf : Continuous f)
Goal: Continuous fun x => (g x).comp (f x)