ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
File: Mathlib/Analysis/Calculus/InverseFunctionTheorem/ContDiff.lean

full_name: ContDiffAt.toPartialHomeomorph
code: /-- Given a `ContDiff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible
derivative at `a`, returns a `PartialHomeomorph` with `to_fun = f` and `a ∈ source`. -/
def toPartialHomeomorph {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a)
    (hn : 1 ≤ n) : PartialHomeomorph E F 
goal: PartialHomeomorph E F

full_name: ContDiffAt.toPartialHomeomorph_coe
code: @[simp]
theorem toPartialHomeomorph_coe {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
    (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
    (hf.toPartialHomeomorph f hf' hn : E → F) = f 
goal: ↑(ContDiffAt.toPartialHomeomorph f hf hf' hn) = f

full_name: ContDiffAt.mem_toPartialHomeomorph_source
code: theorem mem_toPartialHomeomorph_source {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
    (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
    a ∈ (hf.toPartialHomeomorph f hf' hn).source 
goal: a ∈ (ContDiffAt.toPartialHomeomorph f hf hf' hn).source

full_name: ContDiffAt.image_mem_toPartialHomeomorph_target
code: theorem image_mem_toPartialHomeomorph_target {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
    (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
    f a ∈ (hf.toPartialHomeomorph f hf' hn).target 
goal: f a ∈ (ContDiffAt.toPartialHomeomorph f hf hf' hn).target

full_name: ContDiffAt.localInverse
code: /-- Given a `ContDiff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible derivative
at `a`, returns a function that is locally inverse to `f`. -/
def localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a)
    (hn : 1 ≤ n) : F → E 
goal: F → E

full_name: ContDiffAt.localInverse_apply_image
code: theorem localInverse_apply_image {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
    (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) : hf.localInverse hf' hn (f a) = a 
goal: hf.localInverse hf' hn (f a) = a

full_name: ContDiffAt.to_localInverse
code: /-- Given a `ContDiff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible derivative
at `a`, the inverse function (produced by `ContDiff.toPartialHomeomorph`) is
also `ContDiff`. -/
theorem to_localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
    (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
    ContDiffAt 𝕂 n (hf.localInverse hf' hn) (f a) 
goal: ContDiffAt 𝕂 n (hf.localInverse hf' hn) (f a)
File: Mathlib/Analysis/Hofer.lean

full_name: _aux_Mathlib_Analysis_Hofer___unexpand_Dist_dist_1
code: local notation "d" => dist
goal: Lean.PrettyPrinter.Unexpander

full_name: _aux_Mathlib_Analysis_Hofer___macroRules_termD_1
code: local notation "d" => dist
goal: Lean.Macro

full_name: hofer
code: theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
    {ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
    ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' 
goal: ∃ ε' > 0, ∃ x', ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ (y : X), d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x'
File: Mathlib/Analysis/Calculus/FDeriv/Extend.lean

full_name: hasFDerivWithinAt_closure_of_tendsto_fderiv
code: /-- If a function `f` is differentiable in a convex open set and continuous on its closure, and its
derivative converges to a limit `f'` at a point on the boundary, then `f` is differentiable there
with derivative `f'`. -/
theorem hasFDerivWithinAt_closure_of_tendsto_fderiv {f : E → F} {s : Set E} {x : E} {f' : E →L[ℝ] F}
    (f_diff : DifferentiableOn ℝ f s) (s_conv : Convex ℝ s) (s_open : IsOpen s)
    (f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y)
    (h : Tendsto (fun y => fderiv ℝ f y) (𝓝[s] x) (𝓝 f')) :
    HasFDerivWithinAt f f' (closure s) x 
goal: HasFDerivWithinAt f f' (closure s) x

full_name: hasDerivWithinAt_Ici_of_tendsto_deriv
code: /-- If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. -/
theorem hasDerivWithinAt_Ici_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a 
goal: HasDerivWithinAt f e (Set.Ici a) a

full_name: hasDerivWithinAt_Iic_of_tendsto_deriv
code: /-- If a function is differentiable on the left of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the left at `a`. -/
theorem hasDerivWithinAt_Iic_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ}
    {f : ℝ → E} (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a)
    (hs : s ∈ 𝓝[<] a) (f_lim' : Tendsto (fun x => deriv f x) (𝓝[<] a) (𝓝 e)) :
    HasDerivWithinAt f e (Iic a) a 
goal: HasDerivWithinAt f e (Set.Iic a) a

full_name: hasDerivAt_of_hasDerivAt_of_ne
code: /-- If a real function `f` has a derivative `g` everywhere but at a point, and `f` and `g` are
continuous at this point, then `g` is also the derivative of `f` at this point. -/
theorem hasDerivAt_of_hasDerivAt_of_ne {f g : ℝ → E} {x : ℝ}
    (f_diff : ∀ y ≠ x, HasDerivAt f (g y) y) (hf : ContinuousAt f x)
    (hg : ContinuousAt g x) : HasDerivAt f (g x) x 
goal: HasDerivAt f (g x) x

full_name: hasDerivAt_of_hasDerivAt_of_ne'
code: /-- If a real function `f` has a derivative `g` everywhere but at a point, and `f` and `g` are
continuous at this point, then `g` is the derivative of `f` everywhere. -/
theorem hasDerivAt_of_hasDerivAt_of_ne' {f g : ℝ → E} {x : ℝ}
    (f_diff : ∀ y ≠ x, HasDerivAt f (g y) y) (hf : ContinuousAt f x)
    (hg : ContinuousAt g x) (y : ℝ) : HasDerivAt f (g y) y 
goal: HasDerivAt f (g y) y
File: Mathlib/Analysis/Complex/Polynomial/UnitTrinomial.lean

full_name: Polynomial.IsUnitTrinomial.irreducible_of_coprime'
code: /-- A unit trinomial is irreducible if it has no complex roots in common with its mirror. -/
theorem irreducible_of_coprime' (hp : IsUnitTrinomial p)
    (h : ∀ z : ℂ, ¬(aeval z p = 0 ∧ aeval z (mirror p) = 0)) : Irreducible p 
goal: Irreducible p
File: Mathlib/Analysis/Complex/OperatorNorm.lean

full_name: Complex.det_conjLIE
code: /-- The determinant of `conjLIE`, as a linear map. -/
@[simp]
theorem det_conjLIE : LinearMap.det (conjLIE.toLinearEquiv : ℂ →ₗ[ℝ] ℂ) = -1 
goal: LinearMap.det ↑Complex.conjLIE.toLinearEquiv = -1

full_name: Complex.linearEquiv_det_conjLIE
code: /-- The determinant of `conjLIE`, as a linear equiv. -/
@[simp]
theorem linearEquiv_det_conjLIE : LinearEquiv.det conjLIE.toLinearEquiv = -1 
goal: LinearEquiv.det Complex.conjLIE.toLinearEquiv = -1

full_name: Complex.reCLM_norm
code: @[simp]
theorem reCLM_norm : ‖reCLM‖ = 1 
goal: ‖Complex.reCLM‖ = 1

full_name: Complex.reCLM_nnnorm
code: @[simp]
theorem reCLM_nnnorm : ‖reCLM‖₊ = 1 
goal: ‖Complex.reCLM‖₊ = 1

full_name: Complex.imCLM_norm
code: @[simp]
theorem imCLM_norm : ‖imCLM‖ = 1 
goal: ‖Complex.imCLM‖ = 1

full_name: Complex.imCLM_nnnorm
code: @[simp]
theorem imCLM_nnnorm : ‖imCLM‖₊ = 1 
goal: ‖Complex.imCLM‖₊ = 1

full_name: Complex.conjCLE_norm
code: @[simp]
theorem conjCLE_norm : ‖(conjCLE : ℂ →L[ℝ] ℂ)‖ = 1 
goal: ‖↑Complex.conjCLE‖ = 1

full_name: Complex.conjCLE_nnorm
code: @[simp]
theorem conjCLE_nnorm : ‖(conjCLE : ℂ →L[ℝ] ℂ)‖₊ = 1 
goal: ‖↑Complex.conjCLE‖₊ = 1

full_name: Complex.ofRealCLM_norm
code: @[simp]
theorem ofRealCLM_norm : ‖ofRealCLM‖ = 1 
goal: ‖Complex.ofRealCLM‖ = 1

full_name: Complex.ofRealCLM_nnnorm
code: @[simp]
theorem ofRealCLM_nnnorm : ‖ofRealCLM‖₊ = 1 
goal: ‖Complex.ofRealCLM‖₊ = 1
File: Mathlib/Analysis/Complex/Tietze.lean

full_name: TietzeExtension.of_tvs
code: theorem TietzeExtension.of_tvs (𝕜 : Type v) [NontriviallyNormedField 𝕜] {E : Type w}
    [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E]
    [T2Space E] [FiniteDimensional 𝕜 E] [CompleteSpace 𝕜] [TietzeExtension.{u, v} 𝕜] :
    TietzeExtension.{u, w} E 
goal: TietzeExtension E

full_name: Complex.instTietzeExtension
code: instance Complex.instTietzeExtension : TietzeExtension ℂ 
goal: TietzeExtension ℂ

full_name: RCLike.instTietzeExtension
code: instance (priority := 900) RCLike.instTietzeExtension {𝕜 : Type*} [RCLike 𝕜] :
    TietzeExtension 𝕜 
goal: TietzeExtension 𝕜

full_name: RCLike.instTietzeExtensionTVS
code: instance RCLike.instTietzeExtensionTVS {𝕜 : Type v} [RCLike 𝕜] {E : Type w}
    [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E]
    [ContinuousSMul 𝕜 E] [T2Space E] [FiniteDimensional 𝕜 E] :
    TietzeExtension.{u, w} E 
goal: TietzeExtension E

full_name: Set.instTietzeExtensionUnitBall
code: instance Set.instTietzeExtensionUnitBall {𝕜 : Type v} [RCLike 𝕜] {E : Type w}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] :
    TietzeExtension.{u, w} (Metric.ball (0 : E) 1) 
goal: TietzeExtension ↑(Metric.ball 0 1)

full_name: Set.instTietzeExtensionUnitClosedBall
code: instance Set.instTietzeExtensionUnitClosedBall {𝕜 : Type v} [RCLike 𝕜] {E : Type w}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] :
    TietzeExtension.{u, w} (Metric.closedBall (0 : E) 1) 
goal: TietzeExtension ↑(Metric.closedBall 0 1)

full_name: Metric.instTietzeExtensionBall
code: theorem Metric.instTietzeExtensionBall {𝕜 : Type v} [RCLike 𝕜] {E : Type w}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] {r : ℝ} (hr : 0 < r) :
    TietzeExtension.{u, w} (Metric.ball (0 : E) r) 
goal: TietzeExtension ↑(Metric.ball 0 r)

full_name: Metric.instTietzeExtensionClosedBall
code: theorem Metric.instTietzeExtensionClosedBall (𝕜 : Type v) [RCLike 𝕜] {E : Type w}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] (y : E) {r : ℝ} (hr : 0 < r) :
    TietzeExtension.{u, w} (Metric.closedBall y r) 
goal: TietzeExtension ↑(Metric.closedBall y r)

full_name: BoundedContinuousFunction.exists_norm_eq_restrict_eq
code: /-- **Tietze extension theorem** for real-valued bounded continuous maps, a version with a closed
embedding and bundled composition. If `e : C(X, Y)` is a closed embedding of a topological space
into a normal topological space and `f : X →ᵇ ℝ` is a bounded continuous function, then there exists
a bounded continuous function `g : Y →ᵇ ℝ` of the same norm such that `g ∘ e = f`. -/
theorem exists_norm_eq_restrict_eq (f : s →ᵇ E) :
    ∃ g : X →ᵇ E, ‖g‖ = ‖f‖ ∧ g.restrict s = f 
goal: ∃ g, ‖g‖ = ‖f‖ ∧ g.restrict s = f
File: Mathlib/Analysis/Complex/UpperHalfPlane/Manifold.lean

full_name: UpperHalfPlane.instChartedSpaceComplex
code: noncomputable instance : ChartedSpace ℂ ℍ 
goal: ChartedSpace ℂ ℍ

full_name: UpperHalfPlane.instSmoothManifoldWithCornersComplexModelWithCornersSelf
code: instance : SmoothManifoldWithCorners 𝓘(ℂ) ℍ 
goal: SmoothManifoldWithCorners 𝓘(ℂ, ℂ) ℍ

full_name: UpperHalfPlane.smooth_coe
code: /-- The inclusion map `ℍ → ℂ` is a smooth map of manifolds. -/
theorem smooth_coe : Smooth 𝓘(ℂ) 𝓘(ℂ) ((↑) : ℍ → ℂ) 
goal: Smooth 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) UpperHalfPlane.coe

full_name: UpperHalfPlane.mdifferentiable_coe
code: /-- The inclusion map `ℍ → ℂ` is a differentiable map of manifolds. -/
theorem mdifferentiable_coe : MDifferentiable 𝓘(ℂ) 𝓘(ℂ) ((↑) : ℍ → ℂ) 
goal: MDifferentiable 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) UpperHalfPlane.coe
File: Mathlib/Analysis/Complex/UpperHalfPlane/Exp.lean

full_name: UpperHalfPlane.abs_exp_two_pi_I_lt_one
code: theorem UpperHalfPlane.abs_exp_two_pi_I_lt_one (z : ℍ) :
    ‖(Complex.exp (2 * π * Complex.I * z))‖ < 1 
goal: ‖cexp (2 * ↑π * Complex.I * ↑z)‖ < 1
File: Mathlib/Analysis/Complex/UpperHalfPlane/FunctionsBoundedAtInfty.lean

full_name: UpperHalfPlane.atImInfty
code: /-- Filter for approaching `i∞`. -/
def atImInfty 
goal: Filter ℍ

full_name: UpperHalfPlane.atImInfty_basis
code: theorem atImInfty_basis : atImInfty.HasBasis (fun _ => True) fun i : ℝ => im ⁻¹' Set.Ici i 
goal: UpperHalfPlane.atImInfty.HasBasis (fun x => True) fun i => UpperHalfPlane.im ⁻¹' Set.Ici i

full_name: UpperHalfPlane.atImInfty_mem
code: theorem atImInfty_mem (S : Set ℍ) : S ∈ atImInfty ↔ ∃ A : ℝ, ∀ z : ℍ, A ≤ im z → z ∈ S 
goal: S ∈ UpperHalfPlane.atImInfty ↔ ∃ A, ∀ (z : ℍ), A ≤ z.im → z ∈ S

full_name: UpperHalfPlane.IsBoundedAtImInfty
code: /-- A function `f : ℍ → α` is bounded at infinity if it is bounded along `atImInfty`. -/
def IsBoundedAtImInfty {α : Type*} [Norm α] (f : ℍ → α) : Prop 
goal: Prop

full_name: UpperHalfPlane.IsZeroAtImInfty
code: /-- A function `f : ℍ → α` is zero at infinity it is zero along `atImInfty`. -/
def IsZeroAtImInfty {α : Type*} [Zero α] [TopologicalSpace α] (f : ℍ → α) : Prop 
goal: Prop

full_name: UpperHalfPlane.zero_form_isBoundedAtImInfty
code: theorem zero_form_isBoundedAtImInfty {α : Type*} [NormedField α] :
    IsBoundedAtImInfty (0 : ℍ → α) 
goal: UpperHalfPlane.IsBoundedAtImInfty 0

full_name: UpperHalfPlane.zeroAtImInftySubmodule
code: /-- Module of functions that are zero at infinity. -/
def zeroAtImInftySubmodule (α : Type*) [NormedField α] : Submodule α (ℍ → α) 
goal: Submodule α (ℍ → α)

full_name: UpperHalfPlane.boundedAtImInftySubalgebra
code: /-- Subalgebra of functions that are bounded at infinity. -/
def boundedAtImInftySubalgebra (α : Type*) [NormedField α] : Subalgebra α (ℍ → α) 
goal: Subalgebra α (ℍ → α)

full_name: UpperHalfPlane.IsBoundedAtImInfty.mul
code: nonrec theorem IsBoundedAtImInfty.mul {f g : ℍ → ℂ} (hf : IsBoundedAtImInfty f)
    (hg : IsBoundedAtImInfty g) : IsBoundedAtImInfty (f * g) 
goal: UpperHalfPlane.IsBoundedAtImInfty (f * g)

full_name: UpperHalfPlane.bounded_mem
code: theorem bounded_mem (f : ℍ → ℂ) :
    IsBoundedAtImInfty f ↔ ∃ M A : ℝ, ∀ z : ℍ, A ≤ im z → abs (f z) ≤ M 
goal: UpperHalfPlane.IsBoundedAtImInfty f ↔ ∃ M A, ∀ (z : ℍ), A ≤ z.im → Complex.abs (f z) ≤ M

full_name: UpperHalfPlane.zero_at_im_infty
code: theorem zero_at_im_infty (f : ℍ → ℂ) :
    IsZeroAtImInfty f ↔ ∀ ε : ℝ, 0 < ε → ∃ A : ℝ, ∀ z : ℍ, A ≤ im z → abs (f z) ≤ ε 
goal: UpperHalfPlane.IsZeroAtImInfty f ↔ ∀ (ε : ℝ), 0 < ε → ∃ A, ∀ (z : ℍ), A ≤ z.im → Complex.abs (f z) ≤ ε
File: Mathlib/Analysis/Complex/Asymptotics.lean

full_name: Complex.isTheta_ofReal
code: theorem isTheta_ofReal (f : α → ℝ) (l : Filter α) : (f · : α → ℂ) =Θ[l] f 
goal: (fun x => ↑(f x)) =Θ[l] f

full_name: Complex.isLittleO_ofReal_left
code: @[simp, norm_cast]
theorem isLittleO_ofReal_left {f : α → ℝ} {g : α → E} : (f · : α → ℂ) =o[l] g ↔ f =o[l] g 
goal: (fun x => ↑(f x)) =o[l] g ↔ f =o[l] g

full_name: Complex.isLittleO_ofReal_right
code: @[simp, norm_cast]
theorem isLittleO_ofReal_right {f : α → E} {g : α → ℝ} : f =o[l] (g · : α → ℂ) ↔ f =o[l] g 
goal: (f =o[l] fun x => ↑(g x)) ↔ f =o[l] g

full_name: Complex.isBigO_ofReal_left
code: @[simp, norm_cast]
theorem isBigO_ofReal_left {f : α → ℝ} {g : α → E} : (f · : α → ℂ) =O[l] g ↔ f =O[l] g 
goal: (fun x => ↑(f x)) =O[l] g ↔ f =O[l] g

full_name: Complex.isBigO_ofReal_right
code: @[simp, norm_cast]
theorem isBigO_ofReal_right {f : α → E} {g : α → ℝ} : f =O[l] (g · : α → ℂ) ↔ f =O[l] g 
goal: (f =O[l] fun x => ↑(g x)) ↔ f =O[l] g

full_name: Complex.isTheta_ofReal_left
code: @[simp, norm_cast]
theorem isTheta_ofReal_left {f : α → ℝ} {g : α → E} : (f · : α → ℂ) =Θ[l] g ↔ f =Θ[l] g 
goal: (fun x => ↑(f x)) =Θ[l] g ↔ f =Θ[l] g

full_name: Complex.isTheta_ofReal_right
code: @[simp, norm_cast]
theorem isTheta_ofReal_right {f : α → E} {g : α → ℝ} : f =Θ[l] (g · : α → ℂ) ↔ f =Θ[l] g 
goal: (f =Θ[l] fun x => ↑(g x)) ↔ f =Θ[l] g
File: Mathlib/Analysis/Complex/HalfPlane.lean

full_name: Complex.isOpen_re_lt_EReal
code: /-- An open left half-plane (with boundary real part given by an `EReal`) is an open set
in the complex plane. -/
lemma isOpen_re_lt_EReal (x : EReal) : IsOpen {z : ℂ | z.re < x} 
goal: IsOpen {z | ↑z.re < x}

full_name: Complex.isOpen_re_gt_EReal
code: /-- An open right half-plane (with boundary real part given by an `EReal`) is an open set
in the complex plane. -/
lemma isOpen_re_gt_EReal (x : EReal) : IsOpen {z : ℂ | x < z.re} 
goal: IsOpen {z | x < ↑z.re}

full_name: Complex.isOpen_im_lt_EReal
code: /-- An open lower half-plane (with boundary imaginary part given by an `EReal`) is an open set
in the complex plane. -/
lemma isOpen_im_lt_EReal (x : EReal) : IsOpen {z : ℂ | z.im < x} 
goal: IsOpen {z | ↑z.im < x}

full_name: Complex.isOpen_im_gt_EReal
code: /-- An open upper half-plane (with boundary imaginary part given by an `EReal`) is an open set
in the complex plane. -/
lemma isOpen_im_gt_EReal (x : EReal) : IsOpen {z : ℂ | x < z.im} 
goal: IsOpen {z | x < ↑z.im}
File: Mathlib/Analysis/Complex/Angle.lean

full_name: Complex.angle_eq_abs_arg
code: /-- The angle between two non-zero complex numbers is the absolute value of the argument of their
quotient.

Note that this does not hold when `x` or `y` is `0` as the LHS is `π / 2` while the RHS is `0`. -/
lemma angle_eq_abs_arg (hx : x ≠ 0) (hy : y ≠ 0) : angle x y = |(x / y).arg| 
goal: InnerProductGeometry.angle x y = |(x / y).arg|

full_name: Complex.angle_one_left
code: lemma angle_one_left (hy : y ≠ 0) : angle 1 y = |y.arg| 
goal: InnerProductGeometry.angle 1 y = |y.arg|

full_name: Complex.angle_one_right
code: lemma angle_one_right (hx : x ≠ 0) : angle x 1 = |x.arg| 
goal: InnerProductGeometry.angle x 1 = |x.arg|

full_name: Complex.angle_mul_left
code: @[simp] lemma angle_mul_left (ha : a ≠ 0) (x y : ℂ) : angle (a * x) (a * y) = angle x y 
goal: InnerProductGeometry.angle (a * x) (a * y) = InnerProductGeometry.angle x y

full_name: Complex.angle_mul_right
code: @[simp] lemma angle_mul_right (ha : a ≠ 0) (x y : ℂ) : angle (x * a) (y * a) = angle x y 
goal: InnerProductGeometry.angle (x * a) (y * a) = InnerProductGeometry.angle x y

full_name: Complex.angle_div_left_eq_angle_mul_right
code: lemma angle_div_left_eq_angle_mul_right (a x y : ℂ) : angle (x / a) y = angle x (y * a) 
goal: InnerProductGeometry.angle (x / a) y = InnerProductGeometry.angle x (y * a)

full_name: Complex.angle_div_right_eq_angle_mul_left
code: lemma angle_div_right_eq_angle_mul_left (a x y : ℂ) : angle x (y / a) = angle (x * a) y 
goal: InnerProductGeometry.angle x (y / a) = InnerProductGeometry.angle (x * a) y

full_name: Complex.angle_exp_exp
code: lemma angle_exp_exp (x y : ℝ) :
    angle (exp (x * I)) (exp (y * I)) = |toIocMod Real.two_pi_pos (-π) (x - y)| 
goal: InnerProductGeometry.angle (Complex.exp (↑x * Complex.I)) (Complex.exp (↑y * Complex.I)) =
  |toIocMod Real.two_pi_pos (-π) (x - y)|

full_name: Complex.angle_exp_one
code: lemma angle_exp_one (x : ℝ) : angle (exp (x * I)) 1 = |toIocMod Real.two_pi_pos (-π) x| 
goal: InnerProductGeometry.angle (Complex.exp (↑x * Complex.I)) 1 = |toIocMod Real.two_pi_pos (-π) x|

full_name: Complex.norm_sub_mem_Icc_angle
code: /-- Chord-length is a multiple of arc-length up to constants. -/
lemma norm_sub_mem_Icc_angle (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) :
    ‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y) 
goal: ‖x - y‖ ∈ Set.Icc (2 / π * InnerProductGeometry.angle x y) (InnerProductGeometry.angle x y)

full_name: Complex.norm_sub_le_angle
code: /-- Chord-length is always less than arc-length. -/
lemma norm_sub_le_angle (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) : ‖x - y‖ ≤ angle x y 
goal: ‖x - y‖ ≤ InnerProductGeometry.angle x y

full_name: Complex.mul_angle_le_norm_sub
code: /-- Chord-length is always greater than a multiple of arc-length. -/
lemma mul_angle_le_norm_sub (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) : 2 / π * angle x y ≤ ‖x - y‖ 
goal: 2 / π * InnerProductGeometry.angle x y ≤ ‖x - y‖

full_name: Complex.angle_le_mul_norm_sub
code: /-- Arc-length is always less than a multiple of chord-length. -/
lemma angle_le_mul_norm_sub (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) : angle x y ≤ π / 2 * ‖x - y‖ 
goal: InnerProductGeometry.angle x y ≤ π / 2 * ‖x - y‖
File: Mathlib/Analysis/Complex/Convex.lean

full_name: Complex.convexHull_reProdIm
code: /-- A version of `convexHull_prod` for `Set.reProdIm`. -/
lemma convexHull_reProdIm (s t : Set ℝ) :
    convexHull ℝ (s ×ℂ t) = convexHull ℝ s ×ℂ convexHull ℝ t 
goal: (convexHull ℝ) (s ×ℂ t) = (convexHull ℝ) s ×ℂ (convexHull ℝ) t

full_name: Complex.starConvex_slitPlane
code: /-- The slit plane is star-convex at a positive number. -/
lemma starConvex_slitPlane {z : ℂ} (hz : 0 < z) : StarConvex ℝ z slitPlane 
goal: StarConvex ℝ z Complex.slitPlane

full_name: Complex.starConvex_ofReal_slitPlane
code: /-- The slit plane is star-shaped at a positive real number. -/
lemma starConvex_ofReal_slitPlane {x : ℝ} (hx : 0 < x) : StarConvex ℝ ↑x slitPlane 
goal: StarConvex ℝ (↑x) Complex.slitPlane

full_name: Complex.starConvex_one_slitPlane
code: /-- The slit plane is star-shaped at `1`. -/
lemma starConvex_one_slitPlane : StarConvex ℝ 1 slitPlane 
goal: StarConvex ℝ 1 Complex.slitPlane
File: Mathlib/Analysis/Complex/Arg.lean

full_name: Complex.sameRay_iff
code: theorem sameRay_iff : SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg 
goal: SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg

full_name: Complex.sameRay_iff_arg_div_eq_zero
code: theorem sameRay_iff_arg_div_eq_zero : SameRay ℝ x y ↔ arg (x / y) = 0 
goal: SameRay ℝ x y ↔ (x / y).arg = 0

full_name: Complex.abs_add_eq_iff
code: theorem abs_add_eq_iff : abs (x + y) = abs x + abs y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg 
goal: Complex.abs (x + y) = Complex.abs x + Complex.abs y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg

full_name: Complex.abs_sub_eq_iff
code: theorem abs_sub_eq_iff : abs (x - y) = |abs x - abs y| ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg 
goal: Complex.abs (x - y) = |Complex.abs x - Complex.abs y| ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg

full_name: Complex.sameRay_of_arg_eq
code: theorem sameRay_of_arg_eq (h : x.arg = y.arg) : SameRay ℝ x y 
goal: SameRay ℝ x y

full_name: Complex.abs_add_eq
code: theorem abs_add_eq (h : x.arg = y.arg) : abs (x + y) = abs x + abs y 
goal: Complex.abs (x + y) = Complex.abs x + Complex.abs y

full_name: Complex.abs_sub_eq
code: theorem abs_sub_eq (h : x.arg = y.arg) : abs (x - y) = ‖abs x - abs y‖ 
goal: Complex.abs (x - y) = ‖Complex.abs x - Complex.abs y‖
File: Mathlib/Analysis/Complex/OpenMapping.lean

full_name: DiffContOnCl.ball_subset_image_closedBall
code: /-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. -/
theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
    (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
    ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r 
goal: Metric.ball (f z₀) (ε / 2) ⊆ f '' Metric.closedBall z₀ r

full_name: AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux
code: /-- A function `f : ℂ → ℂ` which is analytic at a point `z₀` is either constant in a neighborhood
of `z₀`, or behaves locally like an open function (in the sense that the image of every neighborhood
of `z₀` is a neighborhood of `f z₀`, as in `isOpenMap_iff_nhds_le`). For a function `f : E → ℂ`
the same result holds, see `AnalyticAt.eventually_constant_or_nhds_le_map_nhds`. -/
theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux (hf : AnalyticAt ℂ f z₀) :
    (∀ᶠ z in 𝓝 z₀, f z = f z₀) ∨ 𝓝 (f z₀) ≤ map f (𝓝 z₀) 
goal: (∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀) ∨ 𝓝 (f z₀) ≤ Filter.map f (𝓝 z₀)

full_name: AnalyticAt.eventually_constant_or_nhds_le_map_nhds
code: /-- The *open mapping theorem* for holomorphic functions, local version: is a function `g : E → ℂ`
is analytic at a point `z₀`, then either it is constant in a neighborhood of `z₀`, or it maps every
neighborhood of `z₀` to a neighborhood of `z₀`. For the particular case of a holomorphic function on
`ℂ`, see `AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux`. -/
theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : AnalyticAt ℂ g z₀) :
    (∀ᶠ z in 𝓝 z₀, g z = g z₀) ∨ 𝓝 (g z₀) ≤ map g (𝓝 z₀) 
goal: (∀ᶠ (z : E) in 𝓝 z₀, g z = g z₀) ∨ 𝓝 (g z₀) ≤ Filter.map g (𝓝 z₀)

full_name: AnalyticOn.is_constant_or_isOpen
code: /-- The *open mapping theorem* for holomorphic functions, global version: if a function `g : E → ℂ`
is analytic on a connected set `U`, then either it is constant on `U`, or it is open on `U` (in the
sense that it maps any open set contained in `U` to an open set in `ℂ`). -/
theorem AnalyticOn.is_constant_or_isOpen (hg : AnalyticOn ℂ g U) (hU : IsPreconnected U) :
    (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen s → IsOpen (g '' s) 
goal: (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen s → IsOpen (g '' s)