ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- hasDerivAt_of_tendstoLocallyUniformlyOn
theorem hasDerivAt_of_tendstoLocallyUniformlyOn [NeBot l] {s : Set 𝕜} (hs : IsOpen s)
    (hf' : TendstoLocallyUniformlyOn f' g' l s)
    (hf : ∀ᶠ n in l, ∀ x ∈ s, HasDerivAt (f n) (f' n x) x)
    (hfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s) : HasDerivAt g (g' x) x 
Arguments: {ι : Type u_1} {l : Filter ι} {𝕜 : Type u_2} [RCLike 𝕜] {G : Type u_3} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → 𝕜 → G} {g : 𝕜 → G} {f' : ι → 𝕜 → G} {g' : 𝕜 → G} {x : 𝕜} [l.NeBot] {s : Set 𝕜} (hs : IsOpen s) (hf' : TendstoLocallyUniformlyOn f' g' l s) (hf : ∀ᶠ (n : ι) in l, ∀ x ∈ s, HasDerivAt (f n) (f' n x) x) (hfg : ∀ x ∈ s, Filter.Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s)
Goal: HasDerivAt g (g' x) x
-- hasDerivAt_of_tendsto_locally_uniformly_on'
theorem hasDerivAt_of_tendsto_locally_uniformly_on' [NeBot l] {s : Set 𝕜} (hs : IsOpen s)
    (hf' : TendstoLocallyUniformlyOn (deriv ∘ f) g' l s)
    (hf : ∀ᶠ n in l, DifferentiableOn 𝕜 (f n) s)
    (hfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s) : HasDerivAt g (g' x) x 
Arguments: {ι : Type u_1} {l : Filter ι} {𝕜 : Type u_2} [RCLike 𝕜] {G : Type u_3} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → 𝕜 → G} {g g' : 𝕜 → G} {x : 𝕜} [l.NeBot] {s : Set 𝕜} (hs : IsOpen s) (hf' : TendstoLocallyUniformlyOn (deriv ∘ f) g' l s) (hf : ∀ᶠ (n : ι) in l, DifferentiableOn 𝕜 (f n) s) (hfg : ∀ x ∈ s, Filter.Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s)
Goal: HasDerivAt g (g' x) x
-- hasDerivAt_of_tendstoUniformlyOn
theorem hasDerivAt_of_tendstoUniformlyOn [NeBot l] {s : Set 𝕜} (hs : IsOpen s)
    (hf' : TendstoUniformlyOn f' g' l s)
    (hf : ∀ᶠ n in l, ∀ x : 𝕜, x ∈ s → HasDerivAt (f n) (f' n x) x)
    (hfg : ∀ x : 𝕜, x ∈ s → Tendsto (fun n => f n x) l (𝓝 (g x))) :
    ∀ x : 𝕜, x ∈ s → HasDerivAt g (g' x) x 
Arguments: {ι : Type u_1} {l : Filter ι} {𝕜 : Type u_2} [RCLike 𝕜] {G : Type u_3} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → 𝕜 → G} {g : 𝕜 → G} {f' : ι → 𝕜 → G} {g' : 𝕜 → G} [l.NeBot] {s : Set 𝕜} (hs : IsOpen s) (hf' : TendstoUniformlyOn f' g' l s) (hf : ∀ᶠ (n : ι) in l, ∀ x ∈ s, HasDerivAt (f n) (f' n x) x) (hfg : ∀ x ∈ s, Filter.Tendsto (fun n => f n x) l (𝓝 (g x))) (x : 𝕜)
Goal: x ∈ s → HasDerivAt g (g' x) x
-- hasDerivAt_of_tendstoUniformly
theorem hasDerivAt_of_tendstoUniformly [NeBot l] (hf' : TendstoUniformly f' g' l)
    (hf : ∀ᶠ n in l, ∀ x : 𝕜, HasDerivAt (f n) (f' n x) x)
    (hfg : ∀ x : 𝕜, Tendsto (fun n => f n x) l (𝓝 (g x))) : ∀ x : 𝕜, HasDerivAt g (g' x) x 
Arguments: {ι : Type u_1} {l : Filter ι} {𝕜 : Type u_2} [RCLike 𝕜] {G : Type u_3} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → 𝕜 → G} {g : 𝕜 → G} {f' : ι → 𝕜 → G} {g' : 𝕜 → G} [l.NeBot] (hf' : TendstoUniformly f' g' l) (hf : ∀ᶠ (n : ι) in l, ∀ (x : 𝕜), HasDerivAt (f n) (f' n x) x) (hfg : ∀ (x : 𝕜), Filter.Tendsto (fun n => f n x) l (𝓝 (g x))) (x : 𝕜)
Goal: HasDerivAt g (g' x) x
-- Condensed.equalizerCondition
theorem equalizerCondition (X : Condensed A) : EqualizerCondition X.val 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_3, u_1} A] (X : Condensed A)
Goal: CategoryTheory.regularTopology.EqualizerCondition X.val
-- Condensed.equalizerCondition_profinite
theorem equalizerCondition_profinite (X : Sheaf (coherentTopology Profinite.{u}) A) :
    EqualizerCondition X.val 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] (X : CategoryTheory.Sheaf (CategoryTheory.coherentTopology Profinite) A)
Goal: CategoryTheory.regularTopology.EqualizerCondition X.val
-- HasStrictDerivAt.real_of_complex
theorem HasStrictDerivAt.real_of_complex (h : HasStrictDerivAt e e' z) :
    HasStrictDerivAt (fun x : ℝ => (e x).re) e'.re z 
Arguments: {e : ℂ → ℂ} {e' : ℂ} {z : ℝ} (h : HasStrictDerivAt e e' ↑z)
Goal: HasStrictDerivAt (fun x => (e ↑x).re) e'.re z
-- HasDerivAt.real_of_complex
theorem HasDerivAt.real_of_complex (h : HasDerivAt e e' z) :
    HasDerivAt (fun x : ℝ => (e x).re) e'.re z 
Arguments: {e : ℂ → ℂ} {e' : ℂ} {z : ℝ} (h : HasDerivAt e e' ↑z)
Goal: HasDerivAt (fun x => (e ↑x).re) e'.re z
-- ContDiffAt.real_of_complex
theorem ContDiffAt.real_of_complex {n : ℕ∞} (h : ContDiffAt ℂ n e z) :
    ContDiffAt ℝ n (fun x : ℝ => (e x).re) z 
Arguments: {e : ℂ → ℂ} {z : ℝ} {n : ℕ∞} (h : ContDiffAt ℂ n e ↑z)
Goal: ContDiffAt ℝ n (fun x => (e ↑x).re) z
-- ContDiff.real_of_complex
theorem ContDiff.real_of_complex {n : ℕ∞} (h : ContDiff ℂ n e) :
    ContDiff ℝ n fun x : ℝ => (e x).re 
Arguments: {e : ℂ → ℂ} {n : ℕ∞} (h : ContDiff ℂ n e)
Goal: ContDiff ℝ n fun x => (e ↑x).re
-- HasStrictDerivAt.complexToReal_fderiv'
theorem HasStrictDerivAt.complexToReal_fderiv' {f : ℂ → E} {x : ℂ} {f' : E}
    (h : HasStrictDerivAt f f' x) :
    HasStrictFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {x : ℂ} {f' : E} (h : HasStrictDerivAt f f' x)
Goal: HasStrictFDerivAt f (Complex.reCLM.smulRight f' + Complex.I • Complex.imCLM.smulRight f') x
-- HasDerivAt.complexToReal_fderiv'
theorem HasDerivAt.complexToReal_fderiv' {f : ℂ → E} {x : ℂ} {f' : E} (h : HasDerivAt f f' x) :
    HasFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {x : ℂ} {f' : E} (h : HasDerivAt f f' x)
Goal: HasFDerivAt f (Complex.reCLM.smulRight f' + Complex.I • Complex.imCLM.smulRight f') x
-- HasDerivWithinAt.complexToReal_fderiv'
theorem HasDerivWithinAt.complexToReal_fderiv' {f : ℂ → E} {s : Set ℂ} {x : ℂ} {f' : E}
    (h : HasDerivWithinAt f f' s x) :
    HasFDerivWithinAt f (reCLM.smulRight f' + I • imCLM.smulRight f') s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {s : Set ℂ} {x : ℂ} {f' : E} (h : HasDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt f (Complex.reCLM.smulRight f' + Complex.I • Complex.imCLM.smulRight f') s x
-- HasStrictDerivAt.complexToReal_fderiv
theorem HasStrictDerivAt.complexToReal_fderiv {f : ℂ → ℂ} {f' x : ℂ} (h : HasStrictDerivAt f f' x) :
    HasStrictFDerivAt f (f' • (1 : ℂ →L[ℝ] ℂ)) x 
Arguments: {f : ℂ → ℂ} {f' x : ℂ} (h : HasStrictDerivAt f f' x)
Goal: HasStrictFDerivAt f (f' • 1) x
-- HasDerivAt.complexToReal_fderiv
theorem HasDerivAt.complexToReal_fderiv {f : ℂ → ℂ} {f' x : ℂ} (h : HasDerivAt f f' x) :
    HasFDerivAt f (f' • (1 : ℂ →L[ℝ] ℂ)) x 
Arguments: {f : ℂ → ℂ} {f' x : ℂ} (h : HasDerivAt f f' x)
Goal: HasFDerivAt f (f' • 1) x
-- HasDerivWithinAt.complexToReal_fderiv
theorem HasDerivWithinAt.complexToReal_fderiv {f : ℂ → ℂ} {s : Set ℂ} {f' x : ℂ}
    (h : HasDerivWithinAt f f' s x) : HasFDerivWithinAt f (f' • (1 : ℂ →L[ℝ] ℂ)) s x 
Arguments: {f : ℂ → ℂ} {s : Set ℂ} {f' x : ℂ} (h : HasDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt f (f' • 1) s x
-- HasDerivAt.comp_ofReal
theorem HasDerivAt.comp_ofReal (hf : HasDerivAt e e' ↑z) : HasDerivAt (fun y : ℝ => e ↑y) e' z 
Arguments: {e : ℂ → ℂ} {e' : ℂ} {z : ℝ} (hf : HasDerivAt e e' ↑z)
Goal: HasDerivAt (fun y => e ↑y) e' z
-- HasDerivAt.ofReal_comp
theorem HasDerivAt.ofReal_comp {f : ℝ → ℝ} {u : ℝ} (hf : HasDerivAt f u z) :
    HasDerivAt (fun y : ℝ => ↑(f y) : ℝ → ℂ) u z 
Arguments: {z : ℝ} {f : ℝ → ℝ} {u : ℝ} (hf : HasDerivAt f u z)
Goal: HasDerivAt (fun y => ↑(f y)) (↑u) z
-- DifferentiableAt.conformalAt
theorem DifferentiableAt.conformalAt (h : DifferentiableAt ℂ f z) (hf' : deriv f z ≠ 0) :
    ConformalAt f z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {z : ℂ} {f : ℂ → E} (h : DifferentiableAt ℂ f z) (hf' : deriv f z ≠ 0)
Goal: ConformalAt f z
-- conformalAt_iff_differentiableAt_or_differentiableAt_comp_conj
theorem conformalAt_iff_differentiableAt_or_differentiableAt_comp_conj {f : ℂ → ℂ} {z : ℂ} :
    ConformalAt f z ↔
      (DifferentiableAt ℂ f z ∨ DifferentiableAt ℂ (f ∘ conj) (conj z)) ∧ fderiv ℝ f z ≠ 0 
Arguments: {f : ℂ → ℂ} {z : ℂ}
Goal: ConformalAt f z ↔
  (DifferentiableAt ℂ f z ∨ DifferentiableAt ℂ (f ∘ ⇑(starRingEnd ℂ)) ((starRingEnd ℂ) z)) ∧ fderiv ℝ f z ≠ 0
-- factorsThrough_of_pullbackCondition
theorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullback π π]
    [PreservesLimit (cospan π π) G]
    {a : C(G.obj Z, X)}
    (ha : a ∘ (G.map (pullback.fst _ _)) = a ∘ (G.map (pullback.snd π π))) :
    Function.FactorsThrough a (G.map π) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (G : C ⥤ TopCat) (X : Type w') [TopologicalSpace X] {Z B : C} {π : Z ⟶ B} [CategoryTheory.Limits.HasPullback π π] [CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan π π) G] {a : C(↑(G.obj Z), X)} (ha : ⇑a ∘ ⇑(G.map (CategoryTheory.Limits.pullback.fst π π)) = ⇑a ∘ ⇑(G.map (CategoryTheory.Limits.pullback.snd π π)))
Goal: Function.FactorsThrough ⇑a ⇑(G.map π)
-- equalizerCondition_yonedaPresheaf
theorem equalizerCondition_yonedaPresheaf
    [∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], PreservesLimit (cospan π π) G]
    (hq : ∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], QuotientMap (G.map π)) :
      EqualizerCondition (yonedaPresheaf G X) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (G : C ⥤ TopCat) (X : Type w') [TopologicalSpace X] [(Z B : C) →
    (π : Z ⟶ B) →
      [inst : CategoryTheory.EffectiveEpi π] →
        CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan π π) G] (hq : ∀ (Z B : C) (π : Z ⟶ B) [inst : CategoryTheory.EffectiveEpi π], QuotientMap ⇑(G.map π))
Goal: CategoryTheory.regularTopology.EqualizerCondition (ContinuousMap.yonedaPresheaf G X)
-- TopCat.toCondensedSet_val_map
@[simps! val_obj val_map]
noncomputable def TopCat.toCondensedSet (X : TopCat.{u+1}) : CondensedSet.{u} 
Arguments: (X : TopCat)
Goal: ∀ {X_1 Y : CompHausᵒᵖ} (f : X_1 ⟶ Y) (g : C(↑(compHausToTop.obj (Opposite.unop X_1)), ↑X)),
  X.toCondensedSet.val.map f g = g.comp f.unop
-- TopCat.toCondensedSet_val_obj
@[simps! val_obj val_map]
noncomputable def TopCat.toCondensedSet (X : TopCat.{u+1}) : CondensedSet.{u} 
Arguments: (X : TopCat)
Goal: ∀ (X_1 : CompHausᵒᵖ), X.toCondensedSet.val.obj X_1 = C(↑(Opposite.unop X_1).toTop, ↑X)
-- topCatToCondensedSet_obj
@[simps]
noncomputable def topCatToCondensedSet : TopCat.{u+1} ⥤ CondensedSet.{u} 
Arguments: (X : TopCat)
Goal: topCatToCondensedSet.obj X = X.toCondensedSet
-- topCatToCondensedSet_map_val_app
@[simps]
noncomputable def topCatToCondensedSet : TopCat.{u+1} ⥤ CondensedSet.{u} 
Goal: ∀ {X Y : TopCat} (f : X ⟶ Y) (x : CompHausᵒᵖ) (g : ((fun X => X.toCondensedSet) X).val.obj x),
  (topCatToCondensedSet.map f).val.app x g = ContinuousMap.comp f g
-- SSet.Quasicategory.hornFilling'
class Quasicategory (S : SSet) : Prop where
  hornFilling' : ∀ ⦃n : ℕ⦄ ⦃i : Fin (n+3)⦄ (σ₀ : Λ[n+2, i] ⟶ S)
    (_h0 : 0 < i) (_hn : i < Fin.last (n+2)),
      ∃ σ : Δ[n+2] ⟶ S, σ₀ = hornInclusion (n+2) i ≫ σ
Arguments: {S : SSet} [self : S.Quasicategory] ⦃n : ℕ⦄ ⦃i : Fin (n + 3)⦄ (σ₀ : SSet.horn (n + 2) i ⟶ S) (_h0 : 0 < i) (_hn : i < Fin.last (n + 2))
Goal: ∃ σ, σ₀ = CategoryTheory.CategoryStruct.comp (SSet.hornInclusion (n + 2) i) σ
-- SSet.Quasicategory.hornFilling
lemma Quasicategory.hornFilling {S : SSet} [Quasicategory S] ⦃n : ℕ⦄ ⦃i : Fin (n+1)⦄
    (h0 : 0 < i) (hn : i < Fin.last n)
    (σ₀ : Λ[n, i] ⟶ S) : ∃ σ : Δ[n] ⟶ S, σ₀ = hornInclusion n i ≫ σ 
Arguments: {S : SSet} [S.Quasicategory] ⦃n : ℕ⦄ ⦃i : Fin (n + 1)⦄ (h0 : 0 < i) (hn : i < Fin.last n) (σ₀ : SSet.horn n i ⟶ S)
Goal: ∃ σ, σ₀ = CategoryTheory.CategoryStruct.comp (SSet.hornInclusion n i) σ
-- SSet.quasicategory_of_filler
lemma quasicategory_of_filler (S : SSet)
    (filler : ∀ ⦃n : ℕ⦄ ⦃i : Fin (n+3)⦄ (σ₀ : Λ[n+2, i] ⟶ S)
      (_h0 : 0 < i) (_hn : i < Fin.last (n+2)),
      ∃ σ : S _[n+2], ∀ (j) (h : j ≠ i), S.δ j σ = σ₀.app _ (horn.face i j h)) :
    Quasicategory S 
Arguments: (S : SSet) (filler :
  ∀ ⦃n : ℕ⦄ ⦃i : Fin (n + 3)⦄ (σ₀ : SSet.horn (n + 2) i ⟶ S),
    0 < i →
      i < Fin.last (n + 2) →
        ∃ σ,
          ∀ (j : Fin (n + 3)) (h : j ≠ i),
            CategoryTheory.SimplicialObject.δ S j σ =
              σ₀.app (Opposite.op (SimplexCategory.mk (n + 1))) (SSet.horn.face i j h))
Goal: S.Quasicategory
-- SimplexCategory.toTopObj.ext
@[ext]
theorem toTopObj.ext {x : SimplexCategory} (f g : x.toTopObj) : (f : x → ℝ≥0) = g → f = g 
Arguments: {x : SimplexCategory} (f g : ↑x.toTopObj)
Goal: ↑f = ↑g → f = g
-- SimplexCategory.coe_toTopMap
@[simp]
theorem coe_toTopMap {x y : SimplexCategory} (f : x ⟶ y) (g : x.toTopObj) (i : y) :
    toTopMap f g i = ∑ j ∈ Finset.univ.filter (f · = i), g j 
Arguments: {x y : SimplexCategory} (f : x ⟶ y) (g : ↑x.toTopObj) (i : (CategoryTheory.forget SimplexCategory).obj y)
Goal: ↑(SimplexCategory.toTopMap f g) i = ∑ j ∈ Finset.filter (fun x_1 => f x_1 = i) Finset.univ, ↑g j
-- SimplexCategory.continuous_toTopMap
@[continuity]
theorem continuous_toTopMap {x y : SimplexCategory} (f : x ⟶ y) : Continuous (toTopMap f) 
Arguments: {x y : SimplexCategory} (f : x ⟶ y)
Goal: Continuous (SimplexCategory.toTopMap f)
-- SimplexCategory.toTop_map
@[simps obj map]
def toTop : SimplexCategory ⥤ TopCat 
Goal: ∀ {X Y : SimplexCategory} (f : X ⟶ Y),
  SimplexCategory.toTop.map f = { toFun := SimplexCategory.toTopMap f, continuous_toFun := ⋯ }
-- SimplexCategory.toTop_obj
@[simps obj map]
def toTop : SimplexCategory ⥤ TopCat 
Arguments: (x : SimplexCategory)
Goal: SimplexCategory.toTop.obj x = TopCat.of ↑x.toTopObj
-- simply_connected_def
@[mk_iff simply_connected_def]
class SimplyConnectedSpace (X : Type*) [TopologicalSpace X] : Prop where
  equiv_unit : Nonempty (FundamentalGroupoid X ≌ Discrete Unit)
Arguments: (X : Type u_1) [TopologicalSpace X]
Goal: SimplyConnectedSpace X ↔ Nonempty (FundamentalGroupoid X ≌ CategoryTheory.Discrete Unit)
-- SimplyConnectedSpace.equiv_unit
@[mk_iff simply_connected_def]
class SimplyConnectedSpace (X : Type*) [TopologicalSpace X] : Prop where
  equiv_unit : Nonempty (FundamentalGroupoid X ≌ Discrete Unit)
Arguments: {X : Type u_1} [TopologicalSpace X] [self : SimplyConnectedSpace X]
Goal: Nonempty (FundamentalGroupoid X ≌ CategoryTheory.Discrete Unit)
-- simply_connected_iff_unique_homotopic
theorem simply_connected_iff_unique_homotopic (X : Type*) [TopologicalSpace X] :
    SimplyConnectedSpace X ↔
      Nonempty X ∧ ∀ x y : X, Nonempty (Unique (Path.Homotopic.Quotient x y)) 
Arguments: (X : Type u_1) [TopologicalSpace X]
Goal: SimplyConnectedSpace X ↔ Nonempty X ∧ ∀ (x y : X), Nonempty (Unique (Path.Homotopic.Quotient x y))
-- SimplyConnectedSpace.paths_homotopic
theorem paths_homotopic {x y : X} (p₁ p₂ : Path x y) : Path.Homotopic p₁ p₂ 
Arguments: {X : Type u_1} [TopologicalSpace X] [SimplyConnectedSpace X] {x y : X} (p₁ p₂ : Path x y)
Goal: p₁.Homotopic p₂
-- simply_connected_iff_paths_homotopic
theorem simply_connected_iff_paths_homotopic {Y : Type*} [TopologicalSpace Y] :
    SimplyConnectedSpace Y ↔
      PathConnectedSpace Y ∧ ∀ x y : Y, Subsingleton (Path.Homotopic.Quotient x y) 
Arguments: {Y : Type u_1} [TopologicalSpace Y]
Goal: SimplyConnectedSpace Y ↔ PathConnectedSpace Y ∧ ∀ (x y : Y), Subsingleton (Path.Homotopic.Quotient x y)
-- simply_connected_iff_paths_homotopic'
theorem simply_connected_iff_paths_homotopic' {Y : Type*} [TopologicalSpace Y] :
    SimplyConnectedSpace Y ↔
      PathConnectedSpace Y ∧ ∀ {x y : Y} (p₁ p₂ : Path x y), Path.Homotopic p₁ p₂ 
Arguments: {Y : Type u_1} [TopologicalSpace Y]
Goal: SimplyConnectedSpace Y ↔ PathConnectedSpace Y ∧ ∀ {x y : Y} (p₁ p₂ : Path x y), p₁.Homotopic p₂
-- UpperHalfPlane.dist_eq
theorem dist_eq (z w : ℍ) : dist z w = 2 * arsinh (dist (z : ℂ) w / (2 * √(z.im * w.im))) 
Arguments: (z w : ℍ)
Goal: dist z w = 2 * Real.arsinh (dist ↑z ↑w / (2 * √(z.im * w.im)))
-- UpperHalfPlane.sinh_half_dist
theorem sinh_half_dist (z w : ℍ) :
    sinh (dist z w / 2) = dist (z : ℂ) w / (2 * √(z.im * w.im)) 
Arguments: (z w : ℍ)
Goal: Real.sinh (dist z w / 2) = dist ↑z ↑w / (2 * √(z.im * w.im))
-- UpperHalfPlane.cosh_half_dist
theorem cosh_half_dist (z w : ℍ) :
    cosh (dist z w / 2) = dist (z : ℂ) (conj (w : ℂ)) / (2 * √(z.im * w.im)) 
Arguments: (z w : ℍ)
Goal: Real.cosh (dist z w / 2) = dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.im * w.im))
-- UpperHalfPlane.tanh_half_dist
theorem tanh_half_dist (z w : ℍ) :
    tanh (dist z w / 2) = dist (z : ℂ) w / dist (z : ℂ) (conj ↑w) 
Arguments: (z w : ℍ)
Goal: Real.tanh (dist z w / 2) = dist ↑z ↑w / dist (↑z) ((starRingEnd ℂ) ↑w)
-- UpperHalfPlane.exp_half_dist
theorem exp_half_dist (z w : ℍ) :
    exp (dist z w / 2) = (dist (z : ℂ) w + dist (z : ℂ) (conj ↑w)) / (2 * √(z.im * w.im)) 
Arguments: (z w : ℍ)
Goal: rexp (dist z w / 2) = (dist ↑z ↑w + dist (↑z) ((starRingEnd ℂ) ↑w)) / (2 * √(z.im * w.im))
-- UpperHalfPlane.cosh_dist
theorem cosh_dist (z w : ℍ) : cosh (dist z w) = 1 + dist (z : ℂ) w ^ 2 / (2 * z.im * w.im) 
Arguments: (z w : ℍ)
Goal: Real.cosh (dist z w) = 1 + dist ↑z ↑w ^ 2 / (2 * z.im * w.im)
-- UpperHalfPlane.sinh_half_dist_add_dist
theorem sinh_half_dist_add_dist (a b c : ℍ) : sinh ((dist a b + dist b c) / 2) =
    (dist (a : ℂ) b * dist (c : ℂ) (conj ↑b) + dist (b : ℂ) c * dist (a : ℂ) (conj ↑b)) /
      (2 * √(a.im * c.im) * dist (b : ℂ) (conj ↑b)) 
Arguments: (a b c : ℍ)
Goal: Real.sinh ((dist a b + dist b c) / 2) =
  (dist ↑a ↑b * dist (↑c) ((starRingEnd ℂ) ↑b) + dist ↑b ↑c * dist (↑a) ((starRingEnd ℂ) ↑b)) /
    (2 * √(a.im * c.im) * dist (↑b) ((starRingEnd ℂ) ↑b))
-- UpperHalfPlane.dist_comm
protected theorem dist_comm (z w : ℍ) : dist z w = dist w z 
Arguments: (z w : ℍ)
Goal: dist z w = dist w z
-- UpperHalfPlane.dist_le_iff_le_sinh
theorem dist_le_iff_le_sinh :
    dist z w ≤ r ↔ dist (z : ℂ) w / (2 * √(z.im * w.im)) ≤ sinh (r / 2) 
Arguments: {z w : ℍ} {r : ℝ}
Goal: dist z w ≤ r ↔ dist ↑z ↑w / (2 * √(z.im * w.im)) ≤ Real.sinh (r / 2)
-- UpperHalfPlane.dist_eq_iff_eq_sinh
theorem dist_eq_iff_eq_sinh :
    dist z w = r ↔ dist (z : ℂ) w / (2 * √(z.im * w.im)) = sinh (r / 2) 
Arguments: {z w : ℍ} {r : ℝ}
Goal: dist z w = r ↔ dist ↑z ↑w / (2 * √(z.im * w.im)) = Real.sinh (r / 2)
-- UpperHalfPlane.dist_eq_iff_eq_sq_sinh
theorem dist_eq_iff_eq_sq_sinh (hr : 0 ≤ r) :
    dist z w = r ↔ dist (z : ℂ) w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2 
Arguments: {z w : ℍ} {r : ℝ} (hr : 0 ≤ r)
Goal: dist z w = r ↔ dist ↑z ↑w ^ 2 / (4 * z.im * w.im) = Real.sinh (r / 2) ^ 2
-- UpperHalfPlane.dist_triangle
protected theorem dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c 
Arguments: (a b c : ℍ)
Goal: dist a c ≤ dist a b + dist b c
-- UpperHalfPlane.dist_le_dist_coe_div_sqrt
theorem dist_le_dist_coe_div_sqrt (z w : ℍ) : dist z w ≤ dist (z : ℂ) w / √(z.im * w.im) 
Arguments: (z w : ℍ)
Goal: dist z w ≤ dist ↑z ↑w / √(z.im * w.im)
-- UpperHalfPlane.cosh_dist'
theorem cosh_dist' (z w : ℍ) :
    Real.cosh (dist z w) = ((z.re - w.re) ^ 2 + z.im ^ 2 + w.im ^ 2) / (2 * z.im * w.im) 
Arguments: (z w : ℍ)
Goal: Real.cosh (dist z w) = ((z.re - w.re) ^ 2 + z.im ^ 2 + w.im ^ 2) / (2 * z.im * w.im)
-- UpperHalfPlane.center_re
@[simp]
theorem center_re (z r) : (center z r).re = z.re 
Arguments: (z : ℍ) (r : ℝ)
Goal: (z.center r).re = z.re
-- UpperHalfPlane.center_im
@[simp]
theorem center_im (z r) : (center z r).im = z.im * Real.cosh r 
Arguments: (z : ℍ) (r : ℝ)
Goal: (z.center r).im = z.im * Real.cosh r
-- UpperHalfPlane.center_zero
@[simp]
theorem center_zero (z : ℍ) : center z 0 = z 
Arguments: (z : ℍ)
Goal: z.center 0 = z
-- UpperHalfPlane.dist_coe_center_sq
theorem dist_coe_center_sq (z w : ℍ) (r : ℝ) : dist (z : ℂ) (w.center r) ^ 2 =
    2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2 
Arguments: (z w : ℍ) (r : ℝ)
Goal: dist ↑z ↑(w.center r) ^ 2 = 2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2
-- UpperHalfPlane.dist_coe_center
theorem dist_coe_center (z w : ℍ) (r : ℝ) : dist (z : ℂ) (w.center r) =
    √(2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2) 
Arguments: (z w : ℍ) (r : ℝ)
Goal: dist ↑z ↑(w.center r) = √(2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2)
-- UpperHalfPlane.cmp_dist_eq_cmp_dist_coe_center
theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
    cmp (dist z w) r = cmp (dist (z : ℂ) (w.center r)) (w.im * Real.sinh r) 
Arguments: (z w : ℍ) (r : ℝ)
Goal: cmp (dist z w) r = cmp (dist ↑z ↑(w.center r)) (w.im * Real.sinh r)
-- UpperHalfPlane.dist_eq_iff_dist_coe_center_eq
theorem dist_eq_iff_dist_coe_center_eq :
    dist z w = r ↔ dist (z : ℂ) (w.center r) = w.im * Real.sinh r 
Arguments: {z w : ℍ} {r : ℝ}
Goal: dist z w = r ↔ dist ↑z ↑(w.center r) = w.im * Real.sinh r
-- UpperHalfPlane.dist_self_center
@[simp]
theorem dist_self_center (z : ℍ) (r : ℝ) :
    dist (z : ℂ) (z.center r) = z.im * (Real.cosh r - 1) 
Arguments: (z : ℍ) (r : ℝ)
Goal: dist ↑z ↑(z.center r) = z.im * (Real.cosh r - 1)
-- UpperHalfPlane.dist_center_dist
@[simp]
theorem dist_center_dist (z w : ℍ) :
    dist (z : ℂ) (w.center (dist z w)) = w.im * Real.sinh (dist z w) 
Arguments: (z w : ℍ)
Goal: dist ↑z ↑(w.center (dist z w)) = w.im * Real.sinh (dist z w)
-- UpperHalfPlane.dist_lt_iff_dist_coe_center_lt
theorem dist_lt_iff_dist_coe_center_lt :
    dist z w < r ↔ dist (z : ℂ) (w.center r) < w.im * Real.sinh r 
Arguments: {z w : ℍ} {r : ℝ}
Goal: dist z w < r ↔ dist ↑z ↑(w.center r) < w.im * Real.sinh r
-- UpperHalfPlane.lt_dist_iff_lt_dist_coe_center
theorem lt_dist_iff_lt_dist_coe_center :
    r < dist z w ↔ w.im * Real.sinh r < dist (z : ℂ) (w.center r) 
Arguments: {z w : ℍ} {r : ℝ}
Goal: r < dist z w ↔ w.im * Real.sinh r < dist ↑z ↑(w.center r)
-- UpperHalfPlane.dist_le_iff_dist_coe_center_le
theorem dist_le_iff_dist_coe_center_le :
    dist z w ≤ r ↔ dist (z : ℂ) (w.center r) ≤ w.im * Real.sinh r 
Arguments: {z w : ℍ} {r : ℝ}
Goal: dist z w ≤ r ↔ dist ↑z ↑(w.center r) ≤ w.im * Real.sinh r
-- UpperHalfPlane.le_dist_iff_le_dist_coe_center
theorem le_dist_iff_le_dist_coe_center :
    r < dist z w ↔ w.im * Real.sinh r < dist (z : ℂ) (w.center r) 
Arguments: {z w : ℍ} {r : ℝ}
Goal: r < dist z w ↔ w.im * Real.sinh r < dist ↑z ↑(w.center r)
-- UpperHalfPlane.dist_of_re_eq
nonrec theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log w.im) 
Arguments: {z w : ℍ} (h : z.re = w.re)
Goal: dist z w = dist (Real.log z.im) (Real.log w.im)
-- UpperHalfPlane.dist_log_im_le
theorem dist_log_im_le (z w : ℍ) : dist (log z.im) (log w.im) ≤ dist z w 
Arguments: (z w : ℍ)
Goal: dist (Real.log z.im) (Real.log w.im) ≤ dist z w
-- UpperHalfPlane.im_le_im_mul_exp_dist
theorem im_le_im_mul_exp_dist (z w : ℍ) : z.im ≤ w.im * Real.exp (dist z w) 
Arguments: (z w : ℍ)
Goal: z.im ≤ w.im * rexp (dist z w)
-- UpperHalfPlane.im_div_exp_dist_le
theorem im_div_exp_dist_le (z w : ℍ) : z.im / Real.exp (dist z w) ≤ w.im 
Arguments: (z w : ℍ)
Goal: z.im / rexp (dist z w) ≤ w.im
-- UpperHalfPlane.dist_coe_le
theorem dist_coe_le (z w : ℍ) : dist (z : ℂ) w ≤ w.im * (Real.exp (dist z w) - 1) 
Arguments: (z w : ℍ)
Goal: dist ↑z ↑w ≤ w.im * (rexp (dist z w) - 1)
-- UpperHalfPlane.le_dist_coe
theorem le_dist_coe (z w : ℍ) : w.im * (1 - Real.exp (-dist z w)) ≤ dist (z : ℂ) w 
Arguments: (z w : ℍ)
Goal: w.im * (1 - rexp (-dist z w)) ≤ dist ↑z ↑w
-- UpperHalfPlane.im_pos_of_dist_center_le
theorem im_pos_of_dist_center_le {z : ℍ} {r : ℝ} {w : ℂ}
    (h : dist w (center z r) ≤ z.im * Real.sinh r) : 0 < w.im 
Arguments: {z : ℍ} {r : ℝ} {w : ℂ} (h : dist w ↑(z.center r) ≤ z.im * Real.sinh r)
Goal: 0 < w.im
-- UpperHalfPlane.image_coe_closedBall
theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
    ((↑) : ℍ → ℂ) '' closedBall (α := ℍ) z r = closedBall ↑(z.center r) (z.im * Real.sinh r) 
Arguments: (z : ℍ) (r : ℝ)
Goal: UpperHalfPlane.coe '' Metric.closedBall z r = Metric.closedBall (↑(z.center r)) (z.im * Real.sinh r)
-- UpperHalfPlane.image_coe_ball
theorem image_coe_ball (z : ℍ) (r : ℝ) :
    ((↑) : ℍ → ℂ) '' ball (α := ℍ) z r = ball ↑(z.center r) (z.im * Real.sinh r) 
Arguments: (z : ℍ) (r : ℝ)
Goal: UpperHalfPlane.coe '' Metric.ball z r = Metric.ball (↑(z.center r)) (z.im * Real.sinh r)
-- UpperHalfPlane.image_coe_sphere
theorem image_coe_sphere (z : ℍ) (r : ℝ) :
    ((↑) : ℍ → ℂ) '' sphere (α := ℍ) z r = sphere ↑(z.center r) (z.im * Real.sinh r) 
Arguments: (z : ℍ) (r : ℝ)
Goal: UpperHalfPlane.coe '' Metric.sphere z r = Metric.sphere (↑(z.center r)) (z.im * Real.sinh r)
-- UpperHalfPlane.isometry_vertical_line
theorem isometry_vertical_line (a : ℝ) : Isometry fun y => mk ⟨a, exp y⟩ (exp_pos y) 
Arguments: (a : ℝ)
Goal: Isometry fun y => UpperHalfPlane.mk { re := a, im := rexp y } ⋯