ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- abs_coe_circle
@[simp]
theorem abs_coe_circle (z : circle) : abs z = 1 
Arguments: (z : ↥circle)
Goal: Complex.abs ↑z = 1
-- mem_circle_iff_normSq
theorem mem_circle_iff_normSq {z : ℂ} : z ∈ circle ↔ normSq z = 1 
Arguments: {z : ℂ}
Goal: z ∈ circle ↔ Complex.normSq z = 1
-- normSq_eq_of_mem_circle
@[simp]
theorem normSq_eq_of_mem_circle (z : circle) : normSq z = 1 
Arguments: (z : ↥circle)
Goal: Complex.normSq ↑z = 1
-- ne_zero_of_mem_circle
theorem ne_zero_of_mem_circle (z : circle) : (z : ℂ) ≠ 0 
Arguments: (z : ↥circle)
Goal: ↑z ≠ 0
-- coe_inv_circle
@[simp]
theorem coe_inv_circle (z : circle) : ↑z⁻¹ = (z : ℂ)⁻¹ 
Arguments: (z : ↥circle)
Goal: ↑z⁻¹ = (↑z)⁻¹
-- coe_inv_circle_eq_conj
theorem coe_inv_circle_eq_conj (z : circle) : ↑z⁻¹ = conj (z : ℂ) 
Arguments: (z : ↥circle)
Goal: ↑z⁻¹ = (starRingEnd ℂ) ↑z
-- coe_div_circle
@[simp]
theorem coe_div_circle (z w : circle) : ↑(z / w) = (z : ℂ) / w 
Arguments: (z w : ↥circle)
Goal: ↑(z / w) = ↑z / ↑w
-- circle.toUnits_apply
@[simp]
theorem circle.toUnits_apply (z : circle) :
    circle.toUnits z = Units.mk0 ↑z (ne_zero_of_mem_circle z) 
Arguments: (z : ↥circle)
Goal: circle.toUnits z = Units.mk0 ↑z ⋯
-- circle.ofConjDivSelf_coe
@[simps]
def circle.ofConjDivSelf (z : ℂ) (hz : z ≠ 0) : circle 
Arguments: (z : ℂ) (hz : z ≠ 0)
Goal: ↑(circle.ofConjDivSelf z hz) = (starRingEnd ℂ) z / z
-- expMapCircle_apply
@[simp]
theorem expMapCircle_apply (t : ℝ) : ↑(expMapCircle t) = Complex.exp (t * Complex.I) 
Arguments: (t : ℝ)
Goal: ↑(expMapCircle t) = cexp (↑t * Complex.I)
-- expMapCircle_zero
@[simp]
theorem expMapCircle_zero : expMapCircle 0 = 1 
Goal: expMapCircle 0 = 1
-- expMapCircle_add
@[simp]
theorem expMapCircle_add (x y : ℝ) : expMapCircle (x + y) = expMapCircle x * expMapCircle y 
Arguments: (x y : ℝ)
Goal: expMapCircle (x + y) = expMapCircle x * expMapCircle y
-- expMapCircleHom_apply
@[simps]
def expMapCircleHom : ℝ →+ Additive circle 
Goal: ∀ (a : ℝ), expMapCircleHom a = (⇑Additive.ofMul ∘ ⇑expMapCircle) a
-- expMapCircle_sub
@[simp]
theorem expMapCircle_sub (x y : ℝ) : expMapCircle (x - y) = expMapCircle x / expMapCircle y 
Arguments: (x y : ℝ)
Goal: expMapCircle (x - y) = expMapCircle x / expMapCircle y
-- expMapCircle_neg
@[simp]
theorem expMapCircle_neg (x : ℝ) : expMapCircle (-x) = (expMapCircle x)⁻¹ 
Arguments: (x : ℝ)
Goal: expMapCircle (-x) = (expMapCircle x)⁻¹
-- norm_circle_smul
@[simp]
lemma norm_circle_smul {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℂ E]
    (u : circle) (v : E) :
    ‖u • v‖ = ‖v‖ 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℂ E] (u : ↥circle) (v : E)
Goal: ‖u • v‖ = ‖v‖
-- Complex.sameRay_iff
theorem sameRay_iff : SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg 
Arguments: {x y : ℂ}
Goal: SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg
-- Complex.sameRay_iff_arg_div_eq_zero
theorem sameRay_iff_arg_div_eq_zero : SameRay ℝ x y ↔ arg (x / y) = 0 
Arguments: {x y : ℂ}
Goal: SameRay ℝ x y ↔ (x / y).arg = 0
-- Complex.abs_add_eq_iff
theorem abs_add_eq_iff : abs (x + y) = abs x + abs y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg 
Arguments: {x y : ℂ}
Goal: Complex.abs (x + y) = Complex.abs x + Complex.abs y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg
-- Complex.abs_sub_eq_iff
theorem abs_sub_eq_iff : abs (x - y) = |abs x - abs y| ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg 
Arguments: {x y : ℂ}
Goal: Complex.abs (x - y) = |Complex.abs x - Complex.abs y| ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg
-- Complex.sameRay_of_arg_eq
theorem sameRay_of_arg_eq (h : x.arg = y.arg) : SameRay ℝ x y 
Arguments: {x y : ℂ} (h : x.arg = y.arg)
Goal: SameRay ℝ x y
-- Complex.abs_add_eq
theorem abs_add_eq (h : x.arg = y.arg) : abs (x + y) = abs x + abs y 
Arguments: {x y : ℂ} (h : x.arg = y.arg)
Goal: Complex.abs (x + y) = Complex.abs x + Complex.abs y
-- Complex.abs_sub_eq
theorem abs_sub_eq (h : x.arg = y.arg) : abs (x - y) = ‖abs x - abs y‖ 
Arguments: {x y : ℂ} (h : x.arg = y.arg)
Goal: Complex.abs (x - y) = ‖Complex.abs x - Complex.abs y‖
-- Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt
theorem analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt {f : ℂ → E} {c : ℂ}
    (hd : ∀ᶠ z in 𝓝[≠] c, DifferentiableAt ℂ f z) (hc : ContinuousAt f c) : AnalyticAt ℂ f c 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} {c : ℂ} (hd : ∀ᶠ (z : ℂ) in 𝓝[≠] c, DifferentiableAt ℂ f z) (hc : ContinuousAt f c)
Goal: AnalyticAt ℂ f c
-- Complex.differentiableOn_compl_singleton_and_continuousAt_iff
theorem differentiableOn_compl_singleton_and_continuousAt_iff {f : ℂ → E} {s : Set ℂ} {c : ℂ}
    (hs : s ∈ 𝓝 c) :
    DifferentiableOn ℂ f (s \ {c}) ∧ ContinuousAt f c ↔ DifferentiableOn ℂ f s 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hs : s ∈ 𝓝 c)
Goal: DifferentiableOn ℂ f (s \ {c}) ∧ ContinuousAt f c ↔ DifferentiableOn ℂ f s
-- Complex.differentiableOn_dslope
theorem differentiableOn_dslope {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s ∈ 𝓝 c) :
    DifferentiableOn ℂ (dslope f c) s ↔ DifferentiableOn ℂ f s 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s ∈ 𝓝 c)
Goal: DifferentiableOn ℂ (dslope f c) s ↔ DifferentiableOn ℂ f s
-- Complex.differentiableOn_update_limUnder_of_isLittleO
theorem differentiableOn_update_limUnder_of_isLittleO {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s ∈ 𝓝 c)
    (hd : DifferentiableOn ℂ f (s \ {c}))
    (ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹) :
    DifferentiableOn ℂ (update f c (limUnder (𝓝[≠] c) f)) s 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s ∈ 𝓝 c) (hd : DifferentiableOn ℂ f (s \ {c})) (ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹)
Goal: DifferentiableOn ℂ (Function.update f c (limUnder (𝓝[≠] c) f)) s
-- Complex.differentiableOn_update_limUnder_insert_of_isLittleO
theorem differentiableOn_update_limUnder_insert_of_isLittleO {f : ℂ → E} {s : Set ℂ} {c : ℂ}
    (hc : s ∈ 𝓝[≠] c) (hd : DifferentiableOn ℂ f s)
    (ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹) :
    DifferentiableOn ℂ (update f c (limUnder (𝓝[≠] c) f)) (insert c s) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s ∈ 𝓝[≠] c) (hd : DifferentiableOn ℂ f s) (ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹)
Goal: DifferentiableOn ℂ (Function.update f c (limUnder (𝓝[≠] c) f)) (insert c s)
-- Complex.differentiableOn_update_limUnder_of_bddAbove
theorem differentiableOn_update_limUnder_of_bddAbove {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s ∈ 𝓝 c)
    (hd : DifferentiableOn ℂ f (s \ {c})) (hb : BddAbove (norm ∘ f '' (s \ {c}))) :
    DifferentiableOn ℂ (update f c (limUnder (𝓝[≠] c) f)) s 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s ∈ 𝓝 c) (hd : DifferentiableOn ℂ f (s \ {c})) (hb : BddAbove (norm ∘ f '' (s \ {c})))
Goal: DifferentiableOn ℂ (Function.update f c (limUnder (𝓝[≠] c) f)) s
-- Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO
theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO {f : ℂ → E} {c : ℂ}
    (hd : ∀ᶠ z in 𝓝[≠] c, DifferentiableAt ℂ f z)
    (ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹) :
    Tendsto f (𝓝[≠] c) (𝓝 <| limUnder (𝓝[≠] c) f) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} {c : ℂ} (hd : ∀ᶠ (z : ℂ) in 𝓝[≠] c, DifferentiableAt ℂ f z) (ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹)
Goal: Filter.Tendsto f (𝓝[≠] c) (𝓝 (limUnder (𝓝[≠] c) f))
-- Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under
theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_bounded_under {f : ℂ → E} {c : ℂ}
    (hd : ∀ᶠ z in 𝓝[≠] c, DifferentiableAt ℂ f z)
    (hb : IsBoundedUnder (· ≤ ·) (𝓝[≠] c) fun z => ‖f z - f c‖) :
    Tendsto f (𝓝[≠] c) (𝓝 <| limUnder (𝓝[≠] c) f) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} {c : ℂ} (hd : ∀ᶠ (z : ℂ) in 𝓝[≠] c, DifferentiableAt ℂ f z) (hb : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) (𝓝[≠] c) fun z => ‖f z - f c‖)
Goal: Filter.Tendsto f (𝓝[≠] c) (𝓝 (limUnder (𝓝[≠] c) f))
-- Complex.two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable
theorem two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable {U : Set ℂ}
    (hU : IsOpen U) {c w₀ : ℂ} {R : ℝ} {f : ℂ → E} (hc : closedBall c R ⊆ U)
    (hf : DifferentiableOn ℂ f U) (hw₀ : w₀ ∈ ball c R) :
    ((2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = deriv f w₀ 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {U : Set ℂ} (hU : IsOpen U) {c w₀ : ℂ} {R : ℝ} {f : ℂ → E} (hc : Metric.closedBall c R ⊆ U) (hf : DifferentiableOn ℂ f U) (hw₀ : w₀ ∈ Metric.ball c R)
Goal: ((2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = deriv f w₀
-- DiffContOnCl.ball_subset_image_closedBall
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 
Arguments: {f : ℂ → ℂ} {z₀ : ℂ} {ε r : ℝ} (h : DiffContOnCl ℂ f (Metric.ball z₀ r)) (hr : 0 < r) (hf : ∀ z ∈ Metric.sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀)
Goal: Metric.ball (f z₀) (ε / 2) ⊆ f '' Metric.closedBall z₀ r
-- AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux
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) 
Arguments: {f : ℂ → ℂ} {z₀ : ℂ} (hf : AnalyticAt ℂ f z₀)
Goal: (∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀) ∨ 𝓝 (f z₀) ≤ Filter.map f (𝓝 z₀)
-- AnalyticAt.eventually_constant_or_nhds_le_map_nhds
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) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {g : E → ℂ} {z₀ : E} (hg : AnalyticAt ℂ g z₀)
Goal: (∀ᶠ (z : E) in 𝓝 z₀, g z = g z₀) ∨ 𝓝 (g z₀) ≤ Filter.map g (𝓝 z₀)
-- AnalyticOn.is_constant_or_isOpen
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) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {U : Set E} {g : E → ℂ} (hg : AnalyticOn ℂ g U) (hU : IsPreconnected U)
Goal: (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen s → IsOpen (g '' s)
-- Convex.taylor_approx_two_segment
theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
    (hw : x + v + w ∈ interior s) :
    (fun h : ℝ => f (x + h • v + h • w)
        - f (x + h • v) - h • f' x w - h ^ 2 • f'' v w - (h ^ 2 / 2) • f'' w w) =o[𝓝[>] 0]
      fun h => h ^ 2 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {s : Set E} (s_conv : Convex ℝ s) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s) (hx : HasFDerivWithinAt f' f'' (interior s) x) {v w : E} (hv : x + v ∈ interior s) (hw : x + v + w ∈ interior s)
Goal: (fun h => f (x + h • v + h • w) - f (x + h • v) - h • (f' x) w - h ^ 2 • (f'' v) w - (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]
  fun h => h ^ 2
-- Convex.isLittleO_alternate_sum_square
theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
    (h4w : x + (4 : ℝ) • w ∈ interior s) :
    (fun h : ℝ => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
        - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • f'' v w) =o[𝓝[>] 0]
      fun h => h ^ 2 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {s : Set E} (s_conv : Convex ℝ s) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s) (hx : HasFDerivWithinAt f' f'' (interior s) x) {v w : E} (h4v : x + 4 • v ∈ interior s) (h4w : x + 4 • w ∈ interior s)
Goal: (fun h =>
    f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -
      h ^ 2 • (f'' v) w) =o[𝓝[>] 0]
  fun h => h ^ 2
-- Convex.second_derivative_within_at_symmetric_of_mem_interior
theorem Convex.second_derivative_within_at_symmetric_of_mem_interior {v w : E}
    (h4v : x + (4 : ℝ) • v ∈ interior s) (h4w : x + (4 : ℝ) • w ∈ interior s) :
    f'' w v = f'' v w 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {s : Set E} (s_conv : Convex ℝ s) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s) (hx : HasFDerivWithinAt f' f'' (interior s) x) {v w : E} (h4v : x + 4 • v ∈ interior s) (h4w : x + 4 • w ∈ interior s)
Goal: (f'' w) v = (f'' v) w
-- Convex.second_derivative_within_at_symmetric
theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
    (hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
    (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
    (hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {s : Set E} (s_conv : Convex ℝ s) (hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s) (hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E)
Goal: (f'' v) w = (f'' w) v
-- second_derivative_symmetric_of_eventually
theorem second_derivative_symmetric_of_eventually {f : E → F} {f' : E → E →L[ℝ] F}
    {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hx : HasFDerivAt f' f'' x)
    (v w : E) : f'' v w = f'' w v 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {x : E} {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt f (f' y) y) (hx : HasFDerivAt f' f'' x) (v w : E)
Goal: (f'' v) w = (f'' w) v
-- second_derivative_symmetric
theorem second_derivative_symmetric {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
    (hf : ∀ y, HasFDerivAt f (f' y) y) (hx : HasFDerivAt f' f'' x) (v w : E) : f'' v w = f'' w v 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {x : E} {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ (y : E), HasFDerivAt f (f' y) y) (hx : HasFDerivAt f' f'' x) (v w : E)
Goal: (f'' v) w = (f'' w) v
-- Condensed.id_val
@[simp]
lemma id_val (X : Condensed.{u} C) : (𝟙 X : X ⟶ X).val = 𝟙 _ 
Arguments: {C : Type w} [CategoryTheory.Category.{v, w} C] (X : Condensed C)
Goal: (𝟙 X).val = 𝟙 X.val
-- Condensed.comp_val
@[simp]
lemma comp_val {X Y Z : Condensed.{u} C} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g).val = f.val ≫ g.val 
Arguments: {C : Type w} [CategoryTheory.Category.{v, w} C] {X Y Z : Condensed C} (f : X ⟶ Y) (g : Y ⟶ Z)
Goal: (f ≫ g).val = f.val ≫ g.val
-- Condensed.hom_ext
@[ext]
lemma hom_ext {X Y : Condensed.{u} C} (f g : X ⟶ Y) (h : ∀ S, f.val.app S = g.val.app S) :
    f = g 
Arguments: {C : Type w} [CategoryTheory.Category.{v, w} C] {X Y : Condensed C} (f g : X ⟶ Y) (h : ∀ (S : CompHausᵒᵖ), f.val.app S = g.val.app S)
Goal: f = g
-- CondensedSet.hom_naturality_apply
@[simp]
lemma hom_naturality_apply {X Y : CondensedSet.{u}} (f : X ⟶ Y)  {S T : CompHausᵒᵖ} (g : S ⟶ T)
    (x : X.val.obj S) : f.val.app T (X.val.map g x) = Y.val.map g (f.val.app S x) 
Arguments: {X Y : CondensedSet} (f : X ⟶ Y) {S T : CompHausᵒᵖ} (g : S ⟶ T) (x : X.val.obj S)
Goal: f.val.app T (X.val.map g x) = Y.val.map g (f.val.app S x)
-- Condensed.isSheafProfinite
lemma isSheafProfinite
    [∀ Y, HasLimitsOfShape (StructuredArrow Y profiniteToCompHaus.{u}.op) A] :
    Presheaf.IsSheaf (coherentTopology Profinite)
    (profiniteToCompHaus.op ⋙ X.val) 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] (X : Condensed A) [∀ (Y : CompHausᵒᵖ), CategoryTheory.Limits.HasLimitsOfShape (CategoryTheory.StructuredArrow Y profiniteToCompHaus.op) A]
Goal: CategoryTheory.Presheaf.IsSheaf (CategoryTheory.coherentTopology Profinite) (profiniteToCompHaus.op ⋙ X.val)
-- Condensed.isSheafStonean
lemma isSheafStonean
    [∀ Y, HasLimitsOfShape (StructuredArrow Y Stonean.toCompHaus.{u}.op) A] :
    Presheaf.IsSheaf (coherentTopology Stonean)
    (Stonean.toCompHaus.op ⋙ X.val) 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] (X : Condensed A) [∀ (Y : CompHausᵒᵖ), CategoryTheory.Limits.HasLimitsOfShape (CategoryTheory.StructuredArrow Y Stonean.toCompHaus.op) A]
Goal: CategoryTheory.Presheaf.IsSheaf (CategoryTheory.coherentTopology Stonean) (Stonean.toCompHaus.op ⋙ X.val)
-- LightCondensed.id_val
@[simp]
lemma id_val (X : LightCondensed.{u} C) : (𝟙 X : X ⟶ X).val = 𝟙 _ 
Arguments: {C : Type w} [CategoryTheory.Category.{v, w} C] (X : LightCondensed C)
Goal: (𝟙 X).val = 𝟙 X.val
-- LightCondensed.comp_val
@[simp]
lemma comp_val {X Y Z : LightCondensed.{u} C} (f : X ⟶ Y) (g : Y ⟶ Z) :
    (f ≫ g).val = f.val ≫ g.val 
Arguments: {C : Type w} [CategoryTheory.Category.{v, w} C] {X Y Z : LightCondensed C} (f : X ⟶ Y) (g : Y ⟶ Z)
Goal: (f ≫ g).val = f.val ≫ g.val
-- LightCondensed.hom_ext
@[ext]
lemma hom_ext {X Y : LightCondensed.{u} C} (f g : X ⟶ Y) (h : ∀ S, f.val.app S = g.val.app S) :
    f = g 
Arguments: {C : Type w} [CategoryTheory.Category.{v, w} C] {X Y : LightCondensed C} (f g : X ⟶ Y) (h : ∀ (S : LightProfiniteᵒᵖ), f.val.app S = g.val.app S)
Goal: f = g
-- LightCondSet.hom_naturality_apply
@[simp]
lemma hom_naturality_apply {X Y : LightCondSet.{u}} (f : X ⟶ Y)  {S T : LightProfiniteᵒᵖ}
    (g : S ⟶ T) (x : X.val.obj S) : f.val.app T (X.val.map g x) = Y.val.map g (f.val.app S x) 
Arguments: {X Y : LightCondSet} (f : X ⟶ Y) {S T : LightProfiniteᵒᵖ} (g : S ⟶ T) (x : X.val.obj S)
Goal: f.val.app T (X.val.map g x) = Y.val.map g (f.val.app S x)
-- Complex.schwarz_aux
theorem schwarz_aux {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball c R₁))
    (h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
    ‖dslope f c z‖ ≤ R₂ / R₁ 
Arguments: {R₁ R₂ : ℝ} {c z : ℂ} {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (Metric.ball c R₁)) (h_maps : Set.MapsTo f (Metric.ball c R₁) (Metric.ball (f c) R₂)) (hz : z ∈ Metric.ball c R₁)
Goal: ‖dslope f c z‖ ≤ R₂ / R₁
-- Complex.norm_dslope_le_div_of_mapsTo_ball
theorem norm_dslope_le_div_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
    (h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
    ‖dslope f c z‖ ≤ R₂ / R₁ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {R₁ R₂ : ℝ} {f : ℂ → E} {c z : ℂ} (hd : DifferentiableOn ℂ f (Metric.ball c R₁)) (h_maps : Set.MapsTo f (Metric.ball c R₁) (Metric.ball (f c) R₂)) (hz : z ∈ Metric.ball c R₁)
Goal: ‖dslope f c z‖ ≤ R₂ / R₁
-- Complex.affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div
theorem affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div [CompleteSpace E] [StrictConvexSpace ℝ E]
    (hd : DifferentiableOn ℂ f (ball c R₁)) (h_maps : Set.MapsTo f (ball c R₁) (ball (f c) R₂))
    (h_z₀ : z₀ ∈ ball c R₁) (h_eq : ‖dslope f c z₀‖ = R₂ / R₁) :
    Set.EqOn f (fun z => f c + (z - c) • dslope f c z₀) (ball c R₁) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {R₁ R₂ : ℝ} {f : ℂ → E} {c z₀ : ℂ} [CompleteSpace E] [StrictConvexSpace ℝ E] (hd : DifferentiableOn ℂ f (Metric.ball c R₁)) (h_maps : Set.MapsTo f (Metric.ball c R₁) (Metric.ball (f c) R₂)) (h_z₀ : z₀ ∈ Metric.ball c R₁) (h_eq : ‖dslope f c z₀‖ = R₂ / R₁)
Goal: Set.EqOn f (fun z => f c + (z - c) • dslope f c z₀) (Metric.ball c R₁)
-- Complex.affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div'
theorem affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div' [CompleteSpace E]
    [StrictConvexSpace ℝ E] (hd : DifferentiableOn ℂ f (ball c R₁))
    (h_maps : Set.MapsTo f (ball c R₁) (ball (f c) R₂))
    (h_z₀ : ∃ z₀ ∈ ball c R₁, ‖dslope f c z₀‖ = R₂ / R₁) :
    ∃ C : E, ‖C‖ = R₂ / R₁ ∧ Set.EqOn f (fun z => f c + (z - c) • C) (ball c R₁) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {R₁ R₂ : ℝ} {f : ℂ → E} {c : ℂ} [CompleteSpace E] [StrictConvexSpace ℝ E] (hd : DifferentiableOn ℂ f (Metric.ball c R₁)) (h_maps : Set.MapsTo f (Metric.ball c R₁) (Metric.ball (f c) R₂)) (h_z₀ : ∃ z₀ ∈ Metric.ball c R₁, ‖dslope f c z₀‖ = R₂ / R₁)
Goal: ∃ C, ‖C‖ = R₂ / R₁ ∧ Set.EqOn f (fun z => f c + (z - c) • C) (Metric.ball c R₁)
-- Complex.norm_deriv_le_div_of_mapsTo_ball
theorem norm_deriv_le_div_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
    (h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (h₀ : 0 < R₁) : ‖deriv f c‖ ≤ R₂ / R₁ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {R₁ R₂ : ℝ} {f : ℂ → E} {c : ℂ} (hd : DifferentiableOn ℂ f (Metric.ball c R₁)) (h_maps : Set.MapsTo f (Metric.ball c R₁) (Metric.ball (f c) R₂)) (h₀ : 0 < R₁)
Goal: ‖deriv f c‖ ≤ R₂ / R₁
-- Complex.dist_le_div_mul_dist_of_mapsTo_ball
theorem dist_le_div_mul_dist_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
    (h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
    dist (f z) (f c) ≤ R₂ / R₁ * dist z c 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {R₁ R₂ : ℝ} {f : ℂ → E} {c z : ℂ} (hd : DifferentiableOn ℂ f (Metric.ball c R₁)) (h_maps : Set.MapsTo f (Metric.ball c R₁) (Metric.ball (f c) R₂)) (hz : z ∈ Metric.ball c R₁)
Goal: dist (f z) (f c) ≤ R₂ / R₁ * dist z c
-- Complex.abs_deriv_le_div_of_mapsTo_ball
theorem abs_deriv_le_div_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
    (h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (h₀ : 0 < R₁) : abs (deriv f c) ≤ R₂ / R₁ 
Arguments: {f : ℂ → ℂ} {c : ℂ} {R₁ R₂ : ℝ} (hd : DifferentiableOn ℂ f (Metric.ball c R₁)) (h_maps : Set.MapsTo f (Metric.ball c R₁) (Metric.ball (f c) R₂)) (h₀ : 0 < R₁)
Goal: Complex.abs (deriv f c) ≤ R₂ / R₁
-- Complex.abs_deriv_le_one_of_mapsTo_ball
theorem abs_deriv_le_one_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R))
    (h_maps : MapsTo f (ball c R) (ball c R)) (hc : f c = c) (h₀ : 0 < R) : abs (deriv f c) ≤ 1 
Arguments: {f : ℂ → ℂ} {c : ℂ} {R : ℝ} (hd : DifferentiableOn ℂ f (Metric.ball c R)) (h_maps : Set.MapsTo f (Metric.ball c R) (Metric.ball c R)) (hc : f c = c) (h₀ : 0 < R)
Goal: Complex.abs (deriv f c) ≤ 1
-- Complex.dist_le_dist_of_mapsTo_ball_self
theorem dist_le_dist_of_mapsTo_ball_self (hd : DifferentiableOn ℂ f (ball c R))
    (h_maps : MapsTo f (ball c R) (ball c R)) (hc : f c = c) (hz : z ∈ ball c R) :
    dist (f z) c ≤ dist z c 
Arguments: {f : ℂ → ℂ} {c z : ℂ} {R : ℝ} (hd : DifferentiableOn ℂ f (Metric.ball c R)) (h_maps : Set.MapsTo f (Metric.ball c R) (Metric.ball c R)) (hc : f c = c) (hz : z ∈ Metric.ball c R)
Goal: dist (f z) c ≤ dist z c
-- Complex.abs_le_abs_of_mapsTo_ball_self
theorem abs_le_abs_of_mapsTo_ball_self (hd : DifferentiableOn ℂ f (ball 0 R))
    (h_maps : MapsTo f (ball 0 R) (ball 0 R)) (h₀ : f 0 = 0) (hz : abs z < R) :
    abs (f z) ≤ abs z 
Arguments: {f : ℂ → ℂ} {z : ℂ} {R : ℝ} (hd : DifferentiableOn ℂ f (Metric.ball 0 R)) (h_maps : Set.MapsTo f (Metric.ball 0 R) (Metric.ball 0 R)) (h₀ : f 0 = 0) (hz : Complex.abs z < R)
Goal: Complex.abs (f z) ≤ Complex.abs z
-- Complex.cderiv_eq_deriv
theorem cderiv_eq_deriv (hU : IsOpen U) (hf : DifferentiableOn ℂ f U) (hr : 0 < r)
    (hzr : closedBall z r ⊆ U) : cderiv r f z = deriv f z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {U : Set ℂ} {z : ℂ} {r : ℝ} {f : ℂ → E} (hU : IsOpen U) (hf : DifferentiableOn ℂ f U) (hr : 0 < r) (hzr : Metric.closedBall z r ⊆ U)
Goal: Complex.cderiv r f z = deriv f z
-- Complex.norm_cderiv_le
theorem norm_cderiv_le (hr : 0 < r) (hf : ∀ w ∈ sphere z r, ‖f w‖ ≤ M) :
    ‖cderiv r f z‖ ≤ M / r 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {z : ℂ} {M r : ℝ} {f : ℂ → E} (hr : 0 < r) (hf : ∀ w ∈ Metric.sphere z r, ‖f w‖ ≤ M)
Goal: ‖Complex.cderiv r f z‖ ≤ M / r