ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- OnePoint.nhdsWithin_coe_image
theorem nhdsWithin_coe_image (s : Set X) (x : X) :
    𝓝[(↑) '' s] (x : OnePoint X) = map (↑) (𝓝[s] x) 
Arguments: {X : Type u_1} [TopologicalSpace X] (s : Set X) (x : X)
Goal: 𝓝[OnePoint.some '' s] ↑x = Filter.map OnePoint.some (𝓝[s] x)
-- OnePoint.nhdsWithin_coe
theorem nhdsWithin_coe (s : Set (OnePoint X)) (x : X) : 𝓝[s] ↑x = map (↑) (𝓝[(↑) ⁻¹' s] x) 
Arguments: {X : Type u_1} [TopologicalSpace X] (s : Set (OnePoint X)) (x : X)
Goal: 𝓝[s] ↑x = Filter.map OnePoint.some (𝓝[OnePoint.some ⁻¹' s] x)
-- OnePoint.comap_coe_nhds
theorem comap_coe_nhds (x : X) : comap ((↑) : X → OnePoint X) (𝓝 x) = 𝓝 x 
Arguments: {X : Type u_1} [TopologicalSpace X] (x : X)
Goal: Filter.comap OnePoint.some (𝓝 ↑x) = 𝓝 x
-- OnePoint.nhdsWithin_compl_infty_eq
theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: 𝓝[≠] OnePoint.infty = Filter.map OnePoint.some (Filter.coclosedCompact X)
-- OnePoint.nhds_infty_eq
theorem nhds_infty_eq : 𝓝 (∞ : OnePoint X) = map (↑) (coclosedCompact X) ⊔ pure ∞ 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: 𝓝 OnePoint.infty = Filter.map OnePoint.some (Filter.coclosedCompact X) ⊔ pure OnePoint.infty
-- OnePoint.tendsto_coe_infty
theorem tendsto_coe_infty : Tendsto (↑) (coclosedCompact X) (𝓝 (∞ : OnePoint X)) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: Filter.Tendsto OnePoint.some (Filter.coclosedCompact X) (𝓝 OnePoint.infty)
-- OnePoint.hasBasis_nhds_infty
theorem hasBasis_nhds_infty :
    (𝓝 (∞ : OnePoint X)).HasBasis (fun s : Set X => IsClosed s ∧ IsCompact s) fun s =>
      (↑) '' sᶜ ∪ {∞} 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: (𝓝 OnePoint.infty).HasBasis (fun s => IsClosed s ∧ IsCompact s) fun s => OnePoint.some '' sᶜ ∪ {OnePoint.infty}
-- OnePoint.comap_coe_nhds_infty
@[simp]
theorem comap_coe_nhds_infty : comap ((↑) : X → OnePoint X) (𝓝 ∞) = coclosedCompact X 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: Filter.comap OnePoint.some (𝓝 OnePoint.infty) = Filter.coclosedCompact X
-- OnePoint.le_nhds_infty
theorem le_nhds_infty {f : Filter (OnePoint X)} :
    f ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → (↑) '' sᶜ ∪ {∞} ∈ f 
Arguments: {X : Type u_1} [TopologicalSpace X] {f : Filter (OnePoint X)}
Goal: f ≤ 𝓝 OnePoint.infty ↔ ∀ (s : Set X), IsClosed s → IsCompact s → OnePoint.some '' sᶜ ∪ {OnePoint.infty} ∈ f
-- OnePoint.ultrafilter_le_nhds_infty
theorem ultrafilter_le_nhds_infty {f : Ultrafilter (OnePoint X)} :
    (f : Filter (OnePoint X)) ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → (↑) '' s ∉ f 
Arguments: {X : Type u_1} [TopologicalSpace X] {f : Ultrafilter (OnePoint X)}
Goal: ↑f ≤ 𝓝 OnePoint.infty ↔ ∀ (s : Set X), IsClosed s → IsCompact s → OnePoint.some '' s ∉ f
-- OnePoint.tendsto_nhds_infty'
theorem tendsto_nhds_infty' {α : Type*} {f : OnePoint X → α} {l : Filter α} :
    Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ (↑)) (coclosedCompact X) l 
Arguments: {X : Type u_1} [TopologicalSpace X] {α : Type u_2} {f : OnePoint X → α} {l : Filter α}
Goal: Filter.Tendsto f (𝓝 OnePoint.infty) l ↔
  Filter.Tendsto f (pure OnePoint.infty) l ∧ Filter.Tendsto (f ∘ OnePoint.some) (Filter.coclosedCompact X) l
-- OnePoint.tendsto_nhds_infty
theorem tendsto_nhds_infty {α : Type*} {f : OnePoint X → α} {l : Filter α} :
    Tendsto f (𝓝 ∞) l ↔
      ∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {α : Type u_2} {f : OnePoint X → α} {l : Filter α}
Goal: Filter.Tendsto f (𝓝 OnePoint.infty) l ↔
  ∀ s ∈ l, f OnePoint.infty ∈ s ∧ ∃ t, IsClosed t ∧ IsCompact t ∧ Set.MapsTo (f ∘ OnePoint.some) tᶜ s
-- OnePoint.continuousAt_infty'
theorem continuousAt_infty' {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} :
    ContinuousAt f ∞ ↔ Tendsto (f ∘ (↑)) (coclosedCompact X) (𝓝 (f ∞)) 
Arguments: {X : Type u_1} [TopologicalSpace X] {Y : Type u_2} [TopologicalSpace Y] {f : OnePoint X → Y}
Goal: ContinuousAt f OnePoint.infty ↔ Filter.Tendsto (f ∘ OnePoint.some) (Filter.coclosedCompact X) (𝓝 (f OnePoint.infty))
-- OnePoint.continuousAt_infty
theorem continuousAt_infty {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} :
    ContinuousAt f ∞ ↔
      ∀ s ∈ 𝓝 (f ∞), ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s 
Arguments: {X : Type u_1} [TopologicalSpace X] {Y : Type u_2} [TopologicalSpace Y] {f : OnePoint X → Y}
Goal: ContinuousAt f OnePoint.infty ↔
  ∀ s ∈ 𝓝 (f OnePoint.infty), ∃ t, IsClosed t ∧ IsCompact t ∧ Set.MapsTo (f ∘ OnePoint.some) tᶜ s
-- OnePoint.continuousAt_coe
theorem continuousAt_coe {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X} :
    ContinuousAt f x ↔ ContinuousAt (f ∘ (↑)) x 
Arguments: {X : Type u_1} [TopologicalSpace X] {Y : Type u_2} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X}
Goal: ContinuousAt f ↑x ↔ ContinuousAt (f ∘ OnePoint.some) x
-- OnePoint.continuous_iff
lemma continuous_iff {Y : Type*} [TopologicalSpace Y] (f : OnePoint X → Y) : Continuous f ↔
    Tendsto (fun x : X ↦ f x) (coclosedCompact X) (𝓝 (f ∞)) ∧ Continuous (fun x : X ↦ f x) 
Arguments: {X : Type u_1} [TopologicalSpace X] {Y : Type u_2} [TopologicalSpace Y] (f : OnePoint X → Y)
Goal: Continuous f ↔
  Filter.Tendsto (fun x => f ↑x) (Filter.coclosedCompact X) (𝓝 (f OnePoint.infty)) ∧ Continuous fun x => f ↑x
-- OnePoint.continuous_iff_from_discrete
lemma continuous_iff_from_discrete {Y : Type*} [TopologicalSpace Y]
    [DiscreteTopology X] (f : OnePoint X → Y) :
    Continuous f ↔ Tendsto (fun x : X ↦ f x) cofinite (𝓝 (f ∞)) 
Arguments: {X : Type u_1} [TopologicalSpace X] {Y : Type u_2} [TopologicalSpace Y] [DiscreteTopology X] (f : OnePoint X → Y)
Goal: Continuous f ↔ Filter.Tendsto (fun x => f ↑x) Filter.cofinite (𝓝 (f OnePoint.infty))
-- OnePoint.continuous_iff_from_nat
lemma continuous_iff_from_nat {Y : Type*} [TopologicalSpace Y] (f : OnePoint ℕ → Y) :
    Continuous f ↔ Tendsto (fun x : ℕ ↦ f x) atTop (𝓝 (f ∞)) 
Arguments: {Y : Type u_2} [TopologicalSpace Y] (f : OnePoint ℕ → Y)
Goal: Continuous f ↔ Filter.Tendsto (fun x => f ↑x) Filter.atTop (𝓝 (f OnePoint.infty))
-- OnePoint.denseRange_coe
theorem denseRange_coe [NoncompactSpace X] : DenseRange ((↑) : X → OnePoint X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [NoncompactSpace X]
Goal: DenseRange OnePoint.some
-- OnePoint.denseEmbedding_coe
theorem denseEmbedding_coe [NoncompactSpace X] : DenseEmbedding ((↑) : X → OnePoint X) 
Arguments: {X : Type u_1} [TopologicalSpace X] [NoncompactSpace X]
Goal: DenseEmbedding OnePoint.some
-- OnePoint.specializes_coe
@[simp, norm_cast]
theorem specializes_coe {x y : X} : (x : OnePoint X) ⤳ y ↔ x ⤳ y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: ↑x ⤳ ↑y ↔ x ⤳ y
-- OnePoint.inseparable_coe
@[simp, norm_cast]
theorem inseparable_coe {x y : X} : Inseparable (x : OnePoint X) y ↔ Inseparable x y 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : X}
Goal: Inseparable ↑x ↑y ↔ Inseparable x y
-- OnePoint.not_specializes_infty_coe
theorem not_specializes_infty_coe {x : X} : ¬Specializes ∞ (x : OnePoint X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: ¬OnePoint.infty ⤳ ↑x
-- OnePoint.not_inseparable_infty_coe
theorem not_inseparable_infty_coe {x : X} : ¬Inseparable ∞ (x : OnePoint X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: ¬Inseparable OnePoint.infty ↑x
-- OnePoint.not_inseparable_coe_infty
theorem not_inseparable_coe_infty {x : X} : ¬Inseparable (x : OnePoint X) ∞ 
Arguments: {X : Type u_1} [TopologicalSpace X] {x : X}
Goal: ¬Inseparable (↑x) OnePoint.infty
-- OnePoint.inseparable_iff
theorem inseparable_iff {x y : OnePoint X} :
    Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' 
Arguments: {X : Type u_1} [TopologicalSpace X] {x y : OnePoint X}
Goal: Inseparable x y ↔ x = OnePoint.infty ∧ y = OnePoint.infty ∨ ∃ x', x = ↑x' ∧ ∃ y', y = ↑y' ∧ Inseparable x' y'
-- OnePoint.not_continuous_cofiniteTopology_of_symm
theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
    ¬Continuous (@CofiniteTopology.of (OnePoint X)).symm 
Arguments: {X : Type u_1} [TopologicalSpace X] [Infinite X] [DiscreteTopology X]
Goal: ¬Continuous ⇑CofiniteTopology.of.symm
-- Continuous.homeoOfEquivCompactToT2.t1_counterexample
theorem Continuous.homeoOfEquivCompactToT2.t1_counterexample :
    ∃ (α β : Type) (_ : TopologicalSpace α) (_ : TopologicalSpace β),
      CompactSpace α ∧ T1Space β ∧ ∃ f : α ≃ β, Continuous f ∧ ¬Continuous f.symm 
Goal: ∃ α β x x_1, CompactSpace α ∧ T1Space β ∧ ∃ f, Continuous ⇑f ∧ ¬Continuous ⇑f.symm
-- ContinuousMap.yonedaPresheaf_obj
@[simps]
def yonedaPresheaf : Cᵒᵖ ⥤ Type (max w w') 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (F : C ⥤ TopCat) (Y : Type w') [TopologicalSpace Y] (X : Cᵒᵖ)
Goal: (ContinuousMap.yonedaPresheaf F Y).obj X = C(↑(F.obj (Opposite.unop X)), Y)
-- ContinuousMap.yonedaPresheaf_map
@[simps]
def yonedaPresheaf : Cᵒᵖ ⥤ Type (max w w') 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (F : C ⥤ TopCat) (Y : Type w') [TopologicalSpace Y]
Goal: ∀ {X Y_1 : Cᵒᵖ} (f : X ⟶ Y_1) (g : C(↑(F.obj (Opposite.unop X)), Y)),
  (ContinuousMap.yonedaPresheaf F Y).map f g = g.comp (F.map f.unop)
-- ContinuousMap.yonedaPresheaf'_obj
@[simps]
def yonedaPresheaf' : TopCat.{w}ᵒᵖ ⥤ Type (max w w') 
Arguments: (Y : Type w') [TopologicalSpace Y] (X : TopCatᵒᵖ)
Goal: (ContinuousMap.yonedaPresheaf' Y).obj X = C(↑(Opposite.unop X), Y)
-- ContinuousMap.yonedaPresheaf'_map
@[simps]
def yonedaPresheaf' : TopCat.{w}ᵒᵖ ⥤ Type (max w w') 
Arguments: (Y : Type w') [TopologicalSpace Y]
Goal: ∀ {X Y_1 : TopCatᵒᵖ} (f : X ⟶ Y_1) (g : C(↑(Opposite.unop X), Y)),
  (ContinuousMap.yonedaPresheaf' Y).map f g = g.comp f.unop
-- ContinuousMap.comp_yonedaPresheaf'
theorem comp_yonedaPresheaf' : yonedaPresheaf F Y = F.op ⋙ yonedaPresheaf' Y 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (F : C ⥤ TopCat) (Y : Type w') [TopologicalSpace Y]
Goal: ContinuousMap.yonedaPresheaf F Y = F.op ⋙ ContinuousMap.yonedaPresheaf' Y
-- ContinuousMap.piComparison_fac
theorem piComparison_fac {α : Type} (X : α → TopCat) :
    piComparison (yonedaPresheaf'.{w, w'} Y) (fun x ↦ op (X x)) =
    (yonedaPresheaf' Y).map ((opCoproductIsoProduct X).inv ≫ (TopCat.sigmaIsoSigma X).inv.op) ≫
    (equivEquivIso (sigmaEquiv Y (fun x ↦ (X x).1))).inv ≫ (Types.productIso _).inv 
Arguments: (Y : Type w') [TopologicalSpace Y] {α : Type} (X : α → TopCat)
Goal: (CategoryTheory.Limits.piComparison (ContinuousMap.yonedaPresheaf' Y) fun x => Opposite.op (X x)) =
  (ContinuousMap.yonedaPresheaf' Y).map
      ((CategoryTheory.Limits.opCoproductIsoProduct X).inv ≫ (TopCat.sigmaIsoSigma X).inv.op) ≫
    (equivEquivIso (ContinuousMap.sigmaEquiv Y fun x => ↑(X x))).inv ≫
      (CategoryTheory.Limits.Types.productIso fun i => C(↑(X i), Y)).inv
-- continuous_right_toIcoMod
theorem continuous_right_toIcoMod : ContinuousWithinAt (toIcoMod hp a) (Ici x) x 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} (hp : 0 < p) (a x : 𝕜)
Goal: ContinuousWithinAt (toIcoMod hp a) (Set.Ici x) x
-- continuous_left_toIocMod
theorem continuous_left_toIocMod : ContinuousWithinAt (toIocMod hp a) (Iic x) x 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} (hp : 0 < p) (a x : 𝕜)
Goal: ContinuousWithinAt (toIocMod hp a) (Set.Iic x) x
-- toIcoMod_eventuallyEq_toIocMod
theorem toIcoMod_eventuallyEq_toIocMod : toIcoMod hp a =ᶠ[𝓝 x] toIocMod hp a 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} (hp : 0 < p) (a : 𝕜) {x : 𝕜} (hx : ↑x ≠ ↑a)
Goal: toIcoMod hp a =ᶠ[𝓝 x] toIocMod hp a
-- continuousAt_toIcoMod
theorem continuousAt_toIcoMod : ContinuousAt (toIcoMod hp a) x 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} (hp : 0 < p) (a : 𝕜) {x : 𝕜} (hx : ↑x ≠ ↑a)
Goal: ContinuousAt (toIcoMod hp a) x
-- continuousAt_toIocMod
theorem continuousAt_toIocMod : ContinuousAt (toIocMod hp a) x 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} (hp : 0 < p) (a : 𝕜) {x : 𝕜} (hx : ↑x ≠ ↑a)
Goal: ContinuousAt (toIocMod hp a) x
-- AddCircle.coe_nsmul
theorem coe_nsmul {n : ℕ} {x : 𝕜} : (↑(n • x) : AddCircle p) = n • (x : AddCircle p) 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] (p : 𝕜) {n : ℕ} {x : 𝕜}
Goal: ↑(n • x) = n • ↑x
-- AddCircle.coe_zsmul
theorem coe_zsmul {n : ℤ} {x : 𝕜} : (↑(n • x) : AddCircle p) = n • (x : AddCircle p) 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] (p : 𝕜) {n : ℤ} {x : 𝕜}
Goal: ↑(n • x) = n • ↑x
-- AddCircle.coe_add
theorem coe_add (x y : 𝕜) : (↑(x + y) : AddCircle p) = (x : AddCircle p) + (y : AddCircle p) 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] (p x y : 𝕜)
Goal: ↑(x + y) = ↑x + ↑y
-- AddCircle.coe_sub
theorem coe_sub (x y : 𝕜) : (↑(x - y) : AddCircle p) = (x : AddCircle p) - (y : AddCircle p) 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] (p x y : 𝕜)
Goal: ↑(x - y) = ↑x - ↑y
-- AddCircle.coe_neg
theorem coe_neg {x : 𝕜} : (↑(-x) : AddCircle p) = -(x : AddCircle p) 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] (p : 𝕜) {x : 𝕜}
Goal: ↑(-x) = -↑x
-- AddCircle.coe_eq_zero_iff
theorem coe_eq_zero_iff {x : 𝕜} : (x : AddCircle p) = 0 ↔ ∃ n : ℤ, n • p = x 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] (p : 𝕜) {x : 𝕜}
Goal: ↑x = 0 ↔ ∃ n, n • p = x
-- AddCircle.coe_eq_zero_of_pos_iff
theorem coe_eq_zero_of_pos_iff (hp : 0 < p) {x : 𝕜} (hx : 0 < x) :
    (x : AddCircle p) = 0 ↔ ∃ n : ℕ, n • p = x 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] (p : 𝕜) (hp : 0 < p) {x : 𝕜} (hx : 0 < x)
Goal: ↑x = 0 ↔ ∃ n, n • p = x
-- AddCircle.coe_period
theorem coe_period : (p : AddCircle p) = 0 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] (p : 𝕜)
Goal: ↑p = 0
-- AddCircle.coe_add_period
theorem coe_add_period (x : 𝕜) : ((x + p : 𝕜) : AddCircle p) = x 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] (p x : 𝕜)
Goal: ↑(x + p) = ↑x
-- AddCircle.continuous_mk'
@[continuity, nolint unusedArguments]
protected theorem continuous_mk' :
    Continuous (QuotientAddGroup.mk' (zmultiples p) : 𝕜 → AddCircle p) 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] (p : 𝕜)
Goal: Continuous ⇑(QuotientAddGroup.mk' (AddSubgroup.zmultiples p))
-- AddCircle.coe_eq_coe_iff_of_mem_Ico
theorem coe_eq_coe_iff_of_mem_Ico {x y : 𝕜} (hx : x ∈ Ico a (a + p)) (hy : y ∈ Ico a (a + p)) :
    (x : AddCircle p) = y ↔ x = y 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} [hp : Fact (0 < p)] {a : 𝕜} [Archimedean 𝕜] {x y : 𝕜} (hx : x ∈ Set.Ico a (a + p)) (hy : y ∈ Set.Ico a (a + p))
Goal: ↑x = ↑y ↔ x = y
-- AddCircle.liftIco_coe_apply
theorem liftIco_coe_apply {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Ico a (a + p)) :
    liftIco p a f ↑x = f x 
Arguments: {𝕜 : Type u_1} {B : Type u_2} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} [hp : Fact (0 < p)] {a : 𝕜} [Archimedean 𝕜] {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Set.Ico a (a + p))
Goal: AddCircle.liftIco p a f ↑x = f x
-- AddCircle.liftIoc_coe_apply
theorem liftIoc_coe_apply {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Ioc a (a + p)) :
    liftIoc p a f ↑x = f x 
Arguments: {𝕜 : Type u_1} {B : Type u_2} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} [hp : Fact (0 < p)] {a : 𝕜} [Archimedean 𝕜] {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Set.Ioc a (a + p))
Goal: AddCircle.liftIoc p a f ↑x = f x
-- AddCircle.eq_coe_Ico
lemma eq_coe_Ico (a : AddCircle p) : ∃ b, b ∈ Ico 0 p ∧ ↑b = a 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} [hp : Fact (0 < p)] [Archimedean 𝕜] (a : AddCircle p)
Goal: ∃ b ∈ Set.Ico 0 p, ↑b = a
-- AddCircle.coe_eq_zero_iff_of_mem_Ico
lemma coe_eq_zero_iff_of_mem_Ico (ha : a ∈ Ico 0 p) :
    (a : AddCircle p) = 0 ↔ a = 0 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} [hp : Fact (0 < p)] {a : 𝕜} [Archimedean 𝕜] (ha : a ∈ Set.Ico 0 p)
Goal: ↑a = 0 ↔ a = 0
-- AddCircle.continuous_equivIco_symm
@[continuity]
theorem continuous_equivIco_symm : Continuous (equivIco p a).symm 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] (a : 𝕜) [Archimedean 𝕜]
Goal: Continuous ⇑(AddCircle.equivIco p a).symm
-- AddCircle.continuous_equivIoc_symm
@[continuity]
theorem continuous_equivIoc_symm : Continuous (equivIoc p a).symm 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] (a : 𝕜) [Archimedean 𝕜]
Goal: Continuous ⇑(AddCircle.equivIoc p a).symm
-- AddCircle.continuousAt_equivIco
theorem continuousAt_equivIco : ContinuousAt (equivIco p a) x 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] (a : 𝕜) [Archimedean 𝕜] {x : AddCircle p} (hx : x ≠ ↑a)
Goal: ContinuousAt (⇑(AddCircle.equivIco p a)) x
-- AddCircle.continuousAt_equivIoc
theorem continuousAt_equivIoc : ContinuousAt (equivIoc p a) x 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] (a : 𝕜) [Archimedean 𝕜] {x : AddCircle p} (hx : x ≠ ↑a)
Goal: ContinuousAt (⇑(AddCircle.equivIoc p a)) x
-- AddCircle.partialHomeomorphCoe_target
@[simps] def partialHomeomorphCoe [DiscreteTopology (zmultiples p)] :
    PartialHomeomorph 𝕜 (AddCircle p) 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] (a : 𝕜) [Archimedean 𝕜] [DiscreteTopology ↥(AddSubgroup.zmultiples p)]
Goal: (AddCircle.partialHomeomorphCoe p a).target = {↑a}ᶜ
-- AddCircle.partialHomeomorphCoe_symm_apply
@[simps] def partialHomeomorphCoe [DiscreteTopology (zmultiples p)] :
    PartialHomeomorph 𝕜 (AddCircle p) 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] (a : 𝕜) [Archimedean 𝕜] [DiscreteTopology ↥(AddSubgroup.zmultiples p)] (x : AddCircle p)
Goal: ↑(AddCircle.partialHomeomorphCoe p a).symm x = ↑((AddCircle.equivIco p a) x)
-- AddCircle.partialHomeomorphCoe_source
@[simps] def partialHomeomorphCoe [DiscreteTopology (zmultiples p)] :
    PartialHomeomorph 𝕜 (AddCircle p) 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] (a : 𝕜) [Archimedean 𝕜] [DiscreteTopology ↥(AddSubgroup.zmultiples p)]
Goal: (AddCircle.partialHomeomorphCoe p a).source = Set.Ioo a (a + p)
-- AddCircle.partialHomeomorphCoe_apply
@[simps] def partialHomeomorphCoe [DiscreteTopology (zmultiples p)] :
    PartialHomeomorph 𝕜 (AddCircle p) 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] (a : 𝕜) [Archimedean 𝕜] [DiscreteTopology ↥(AddSubgroup.zmultiples p)]
Goal: ∀ (a_1 : 𝕜), ↑(AddCircle.partialHomeomorphCoe p a) a_1 = ↑a_1
-- AddCircle.isLocalHomeomorph_coe
lemma isLocalHomeomorph_coe [DiscreteTopology (zmultiples p)] [DenselyOrdered 𝕜] :
    IsLocalHomeomorph ((↑) : 𝕜 → AddCircle p) 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] [Archimedean 𝕜] [DiscreteTopology ↥(AddSubgroup.zmultiples p)] [DenselyOrdered 𝕜]
Goal: IsLocalHomeomorph QuotientAddGroup.mk
-- AddCircle.coe_image_Ico_eq
@[simp]
theorem coe_image_Ico_eq : ((↑) : 𝕜 → AddCircle p) '' Ico a (a + p) = univ 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] (a : 𝕜) [Archimedean 𝕜]
Goal: QuotientAddGroup.mk '' Set.Ico a (a + p) = Set.univ
-- AddCircle.coe_image_Ioc_eq
@[simp]
theorem coe_image_Ioc_eq : ((↑) : 𝕜 → AddCircle p) '' Ioc a (a + p) = univ 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] (a : 𝕜) [Archimedean 𝕜]
Goal: QuotientAddGroup.mk '' Set.Ioc a (a + p) = Set.univ
-- AddCircle.coe_image_Icc_eq
@[simp]
theorem coe_image_Icc_eq : ((↑) : 𝕜 → AddCircle p) '' Icc a (a + p) = univ 
Arguments: {𝕜 : Type u_1} [LinearOrderedAddCommGroup 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] (a : 𝕜) [Archimedean 𝕜]
Goal: QuotientAddGroup.mk '' Set.Icc a (a + p) = Set.univ
-- AddCircle.equivAddCircle_apply_mk
@[simp]
theorem equivAddCircle_apply_mk (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜) :
    equivAddCircle p q hp hq (x : 𝕜) = (x * (p⁻¹ * q) : 𝕜) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p q : 𝕜) (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜)
Goal: (AddCircle.equivAddCircle p q hp hq) ↑x = ↑(x * (p⁻¹ * q))
-- AddCircle.equivAddCircle_symm_apply_mk
@[simp]
theorem equivAddCircle_symm_apply_mk (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜) :
    (equivAddCircle p q hp hq).symm (x : 𝕜) = (x * (q⁻¹ * p) : 𝕜) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p q : 𝕜) (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜)
Goal: (AddCircle.equivAddCircle p q hp hq).symm ↑x = ↑(x * (q⁻¹ * p))
-- AddCircle.homeomorphAddCircle_apply_mk
@[simp]
theorem homeomorphAddCircle_apply_mk (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜) :
    homeomorphAddCircle p q hp hq (x : 𝕜) = (x * (p⁻¹ * q) : 𝕜) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p q : 𝕜) (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜)
Goal: (AddCircle.homeomorphAddCircle p q hp hq) ↑x = ↑(x * (p⁻¹ * q))
-- AddCircle.homeomorphAddCircle_symm_apply_mk
@[simp]
theorem homeomorphAddCircle_symm_apply_mk (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜) :
    (homeomorphAddCircle p q hp hq).symm (x : 𝕜) = (x * (q⁻¹ * p) : 𝕜) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p q : 𝕜) (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜)
Goal: (AddCircle.homeomorphAddCircle p q hp hq).symm ↑x = ↑(x * (q⁻¹ * p))
-- AddCircle.coe_equivIco_mk_apply
@[simp]
theorem coe_equivIco_mk_apply (x : 𝕜) :
    (equivIco p 0 <| QuotientAddGroup.mk x : 𝕜) = Int.fract (x / p) * p 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] (p : 𝕜) [hp : Fact (0 < p)] [FloorRing 𝕜] (x : 𝕜)
Goal: ↑((AddCircle.equivIco p 0) ↑x) = Int.fract (x / p) * p
-- AddCircle.addOrderOf_period_div
theorem addOrderOf_period_div {n : ℕ} (h : 0 < n) : addOrderOf ((p / n : 𝕜) : AddCircle p) = n 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {p : 𝕜} [hp : Fact (0 < p)] {n : ℕ} (h : 0 < n)
Goal: addOrderOf ↑(p / ↑n) = n
-- AddCircle.gcd_mul_addOrderOf_div_eq
theorem gcd_mul_addOrderOf_div_eq {n : ℕ} (m : ℕ) (hn : 0 < n) :
    m.gcd n * addOrderOf (↑(↑m / ↑n * p) : AddCircle p) = n 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] (p : 𝕜) [hp : Fact (0 < p)] {n : ℕ} (m : ℕ) (hn : 0 < n)
Goal: m.gcd n * addOrderOf ↑(↑m / ↑n * p) = n
-- AddCircle.addOrderOf_div_of_gcd_eq_one
theorem addOrderOf_div_of_gcd_eq_one {m n : ℕ} (hn : 0 < n) (h : m.gcd n = 1) :
    addOrderOf (↑(↑m / ↑n * p) : AddCircle p) = n 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {p : 𝕜} [hp : Fact (0 < p)] {m n : ℕ} (hn : 0 < n) (h : m.gcd n = 1)
Goal: addOrderOf ↑(↑m / ↑n * p) = n
-- AddCircle.addOrderOf_div_of_gcd_eq_one'
theorem addOrderOf_div_of_gcd_eq_one' {m : ℤ} {n : ℕ} (hn : 0 < n) (h : m.natAbs.gcd n = 1) :
    addOrderOf (↑(↑m / ↑n * p) : AddCircle p) = n 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {p : 𝕜} [hp : Fact (0 < p)] {m : ℤ} {n : ℕ} (hn : 0 < n) (h : m.natAbs.gcd n = 1)
Goal: addOrderOf ↑(↑m / ↑n * p) = n
-- AddCircle.addOrderOf_coe_rat
theorem addOrderOf_coe_rat {q : ℚ} : addOrderOf (↑(↑q * p) : AddCircle p) = q.den 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {p : 𝕜} [hp : Fact (0 < p)] {q : ℚ}
Goal: addOrderOf ↑(↑q * p) = q.den
-- AddCircle.addOrderOf_eq_pos_iff
theorem addOrderOf_eq_pos_iff {u : AddCircle p} {n : ℕ} (h : 0 < n) :
    addOrderOf u = n ↔ ∃ m < n, m.gcd n = 1 ∧ ↑(↑m / ↑n * p) = u 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} [hp : Fact (0 < p)] {u : AddCircle p} {n : ℕ} (h : 0 < n)
Goal: addOrderOf u = n ↔ ∃ m < n, m.gcd n = 1 ∧ ↑(↑m / ↑n * p) = u