ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Complex.UnitDisc.coe_ne_one
theorem coe_ne_one (z : 𝔻) : (z : ℂ) ≠ 1 
Arguments: (z : Complex.UnitDisc)
Goal: ↑z ≠ 1
-- Complex.UnitDisc.coe_ne_neg_one
theorem coe_ne_neg_one (z : 𝔻) : (z : ℂ) ≠ -1 
Arguments: (z : Complex.UnitDisc)
Goal: ↑z ≠ -1
-- Complex.UnitDisc.one_add_coe_ne_zero
theorem one_add_coe_ne_zero (z : 𝔻) : (1 + z : ℂ) ≠ 0 
Arguments: (z : Complex.UnitDisc)
Goal: 1 + ↑z ≠ 0
-- Complex.UnitDisc.coe_mul
@[simp, norm_cast]
theorem coe_mul (z w : 𝔻) : ↑(z * w) = (z * w : ℂ) 
Arguments: (z w : Complex.UnitDisc)
Goal: ↑(z * w) = ↑z * ↑w
-- Complex.UnitDisc.coe_mk
@[simp]
theorem coe_mk (z : ℂ) (hz : abs z < 1) : (mk z hz : ℂ) = z 
Arguments: (z : ℂ) (hz : Complex.abs z < 1)
Goal: ↑(Complex.UnitDisc.mk z hz) = z
-- Complex.UnitDisc.mk_coe
@[simp]
theorem mk_coe (z : 𝔻) (hz : abs (z : ℂ) < 1 := z.abs_lt_one) : mk z hz = z 
Arguments: (z : Complex.UnitDisc) (hz : Complex.abs ↑z < 1 := ⋯)
Goal: Complex.UnitDisc.mk (↑z) hz = z
-- Complex.UnitDisc.mk_neg
@[simp]
theorem mk_neg (z : ℂ) (hz : abs (-z) < 1) : mk (-z) hz = -mk z (abs.map_neg z ▸ hz) 
Arguments: (z : ℂ) (hz : Complex.abs (-z) < 1)
Goal: Complex.UnitDisc.mk (-z) hz = -Complex.UnitDisc.mk z ⋯
-- Complex.UnitDisc.coe_zero
@[simp]
theorem coe_zero : ((0 : 𝔻) : ℂ) = 0 
Goal: ↑0 = 0
-- Complex.UnitDisc.coe_eq_zero
@[simp]
theorem coe_eq_zero {z : 𝔻} : (z : ℂ) = 0 ↔ z = 0 
Arguments: {z : Complex.UnitDisc}
Goal: ↑z = 0 ↔ z = 0
-- Complex.UnitDisc.coe_smul_circle
@[simp, norm_cast]
theorem coe_smul_circle (z : circle) (w : 𝔻) : ↑(z • w) = (z * w : ℂ) 
Arguments: (z : ↥circle) (w : Complex.UnitDisc)
Goal: ↑(z • w) = ↑z * ↑w
-- Complex.UnitDisc.coe_smul_closedBall
@[simp, norm_cast]
theorem coe_smul_closedBall (z : closedBall (0 : ℂ) 1) (w : 𝔻) : ↑(z • w) = (z * w : ℂ) 
Arguments: (z : ↑(Metric.closedBall 0 1)) (w : Complex.UnitDisc)
Goal: ↑(z • w) = ↑z * ↑w
-- Complex.UnitDisc.re_coe
@[simp, norm_cast]
theorem re_coe (z : 𝔻) : (z : ℂ).re = z.re 
Arguments: (z : Complex.UnitDisc)
Goal: (↑z).re = z.re
-- Complex.UnitDisc.im_coe
@[simp, norm_cast]
theorem im_coe (z : 𝔻) : (z : ℂ).im = z.im 
Arguments: (z : Complex.UnitDisc)
Goal: (↑z).im = z.im
-- Complex.UnitDisc.re_neg
@[simp]
theorem re_neg (z : 𝔻) : (-z).re = -z.re 
Arguments: (z : Complex.UnitDisc)
Goal: (-z).re = -z.re
-- Complex.UnitDisc.im_neg
@[simp]
theorem im_neg (z : 𝔻) : (-z).im = -z.im 
Arguments: (z : Complex.UnitDisc)
Goal: (-z).im = -z.im
-- Complex.UnitDisc.coe_conj
@[simp]
theorem coe_conj (z : 𝔻) : (z.conj : ℂ) = conj' ↑z 
Arguments: (z : Complex.UnitDisc)
Goal: ↑z.conj = conj' ↑z
-- Complex.UnitDisc.conj_zero
@[simp]
theorem conj_zero : conj 0 = 0 
Goal: Complex.UnitDisc.conj 0 = 0
-- Complex.UnitDisc.conj_conj
@[simp]
theorem conj_conj (z : 𝔻) : conj (conj z) = z 
Arguments: (z : Complex.UnitDisc)
Goal: z.conj.conj = z
-- Complex.UnitDisc.conj_neg
@[simp]
theorem conj_neg (z : 𝔻) : (-z).conj = -z.conj 
Arguments: (z : Complex.UnitDisc)
Goal: (-z).conj = -z.conj
-- Complex.UnitDisc.re_conj
@[simp]
theorem re_conj (z : 𝔻) : z.conj.re = z.re 
Arguments: (z : Complex.UnitDisc)
Goal: z.conj.re = z.re
-- Complex.UnitDisc.im_conj
@[simp]
theorem im_conj (z : 𝔻) : z.conj.im = -z.im 
Arguments: (z : Complex.UnitDisc)
Goal: z.conj.im = -z.im
-- Complex.UnitDisc.conj_mul
@[simp]
theorem conj_mul (z w : 𝔻) : (z * w).conj = z.conj * w.conj 
Arguments: (z w : Complex.UnitDisc)
Goal: (z * w).conj = z.conj * w.conj
-- Complex.isTheta_ofReal
theorem isTheta_ofReal (f : α → ℝ) (l : Filter α) : (f · : α → ℂ) =Θ[l] f 
Arguments: {α : Type u_1} (f : α → ℝ) (l : Filter α)
Goal: (fun x => ↑(f x)) =Θ[l] f
-- Complex.isLittleO_ofReal_left
@[simp, norm_cast]
theorem isLittleO_ofReal_left {f : α → ℝ} {g : α → E} : (f · : α → ℂ) =o[l] g ↔ f =o[l] g 
Arguments: {α : Type u_1} {E : Type u_2} [Norm E] {l : Filter α} {f : α → ℝ} {g : α → E}
Goal: (fun x => ↑(f x)) =o[l] g ↔ f =o[l] g
-- Complex.isLittleO_ofReal_right
@[simp, norm_cast]
theorem isLittleO_ofReal_right {f : α → E} {g : α → ℝ} : f =o[l] (g · : α → ℂ) ↔ f =o[l] g 
Arguments: {α : Type u_1} {E : Type u_2} [Norm E] {l : Filter α} {f : α → E} {g : α → ℝ}
Goal: (f =o[l] fun x => ↑(g x)) ↔ f =o[l] g
-- Complex.isBigO_ofReal_left
@[simp, norm_cast]
theorem isBigO_ofReal_left {f : α → ℝ} {g : α → E} : (f · : α → ℂ) =O[l] g ↔ f =O[l] g 
Arguments: {α : Type u_1} {E : Type u_2} [Norm E] {l : Filter α} {f : α → ℝ} {g : α → E}
Goal: (fun x => ↑(f x)) =O[l] g ↔ f =O[l] g
-- Complex.isBigO_ofReal_right
@[simp, norm_cast]
theorem isBigO_ofReal_right {f : α → E} {g : α → ℝ} : f =O[l] (g · : α → ℂ) ↔ f =O[l] g 
Arguments: {α : Type u_1} {E : Type u_2} [Norm E] {l : Filter α} {f : α → E} {g : α → ℝ}
Goal: (f =O[l] fun x => ↑(g x)) ↔ f =O[l] g
-- Complex.isTheta_ofReal_left
@[simp, norm_cast]
theorem isTheta_ofReal_left {f : α → ℝ} {g : α → E} : (f · : α → ℂ) =Θ[l] g ↔ f =Θ[l] g 
Arguments: {α : Type u_1} {E : Type u_2} [Norm E] {l : Filter α} {f : α → ℝ} {g : α → E}
Goal: (fun x => ↑(f x)) =Θ[l] g ↔ f =Θ[l] g
-- Complex.isTheta_ofReal_right
@[simp, norm_cast]
theorem isTheta_ofReal_right {f : α → E} {g : α → ℝ} : f =Θ[l] (g · : α → ℂ) ↔ f =Θ[l] g 
Arguments: {α : Type u_1} {E : Type u_2} [Norm E] {l : Filter α} {f : α → E} {g : α → ℝ}
Goal: (f =Θ[l] fun x => ↑(g x)) ↔ f =Θ[l] g
-- Complex.deriv_eq_smul_circleIntegral
theorem deriv_eq_smul_circleIntegral [CompleteSpace F] {R : ℝ} {c : ℂ} {f : ℂ → F} (hR : 0 < R)
    (hf : DiffContOnCl ℂ f (ball c R)) :
    deriv f c = (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - c) ^ (-2 : ℤ) • f z 
Arguments: {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F] {R : ℝ} {c : ℂ} {f : ℂ → F} (hR : 0 < R) (hf : DiffContOnCl ℂ f (Metric.ball c R))
Goal: deriv f c = (2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c) ^ (-2) • f z
-- Complex.norm_deriv_le_aux
theorem norm_deriv_le_aux [CompleteSpace F] {c : ℂ} {R C : ℝ} {f : ℂ → F} (hR : 0 < R)
    (hf : DiffContOnCl ℂ f (ball c R)) (hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) :
    ‖deriv f c‖ ≤ C / R 
Arguments: {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F] {c : ℂ} {R C : ℝ} {f : ℂ → F} (hR : 0 < R) (hf : DiffContOnCl ℂ f (Metric.ball c R)) (hC : ∀ z ∈ Metric.sphere c R, ‖f z‖ ≤ C)
Goal: ‖deriv f c‖ ≤ C / R
-- Complex.norm_deriv_le_of_forall_mem_sphere_norm_le
theorem norm_deriv_le_of_forall_mem_sphere_norm_le {c : ℂ} {R C : ℝ} {f : ℂ → F} (hR : 0 < R)
    (hd : DiffContOnCl ℂ f (ball c R)) (hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) :
    ‖deriv f c‖ ≤ C / R 
Arguments: {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {c : ℂ} {R C : ℝ} {f : ℂ → F} (hR : 0 < R) (hd : DiffContOnCl ℂ f (Metric.ball c R)) (hC : ∀ z ∈ Metric.sphere c R, ‖f z‖ ≤ C)
Goal: ‖deriv f c‖ ≤ C / R
-- Complex.liouville_theorem_aux
theorem liouville_theorem_aux {f : ℂ → F} (hf : Differentiable ℂ f) (hb : IsBounded (range f))
    (z w : ℂ) : f z = f w 
Arguments: {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : ℂ → F} (hf : Differentiable ℂ f) (hb : Bornology.IsBounded (Set.range f)) (z w : ℂ)
Goal: f z = f w
-- Differentiable.apply_eq_apply_of_bounded
theorem apply_eq_apply_of_bounded {f : E → F} (hf : Differentiable ℂ f) (hb : IsBounded (range f))
    (z w : E) : f z = f w 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : E → F} (hf : Differentiable ℂ f) (hb : Bornology.IsBounded (Set.range f)) (z w : E)
Goal: f z = f w
-- Differentiable.exists_const_forall_eq_of_bounded
theorem exists_const_forall_eq_of_bounded {f : E → F} (hf : Differentiable ℂ f)
    (hb : IsBounded (range f)) : ∃ c, ∀ z, f z = c 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : E → F} (hf : Differentiable ℂ f) (hb : Bornology.IsBounded (Set.range f))
Goal: ∃ c, ∀ (z : E), f z = c
-- Differentiable.exists_eq_const_of_bounded
theorem exists_eq_const_of_bounded {f : E → F} (hf : Differentiable ℂ f)
    (hb : IsBounded (range f)) : ∃ c, f = const E c 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : E → F} (hf : Differentiable ℂ f) (hb : Bornology.IsBounded (Set.range f))
Goal: ∃ c, f = Function.const E c
-- Differentiable.eq_const_of_tendsto_cocompact
theorem eq_const_of_tendsto_cocompact [Nontrivial E] {f : E → F} (hf : Differentiable ℂ f) {c : F}
    (hb : Tendsto f (cocompact E) (𝓝 c)) : f = Function.const E c 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [Nontrivial E] {f : E → F} (hf : Differentiable ℂ f) {c : F} (hb : Filter.Tendsto f (Filter.cocompact E) (𝓝 c))
Goal: f = Function.const E c
-- Differentiable.apply_eq_of_tendsto_cocompact
theorem apply_eq_of_tendsto_cocompact [Nontrivial E] {f : E → F} (hf : Differentiable ℂ f) {c : F}
    (x : E) (hb : Tendsto f (cocompact E) (𝓝 c)) : f x = c 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [Nontrivial E] {f : E → F} (hf : Differentiable ℂ f) {c : F} (x : E) (hb : Filter.Tendsto f (Filter.cocompact E) (𝓝 c))
Goal: f x = c
-- Complex.isHomeomorphicTrivialFiberBundle_re
theorem isHomeomorphicTrivialFiberBundle_re : IsHomeomorphicTrivialFiberBundle ℝ re 
Goal: IsHomeomorphicTrivialFiberBundle ℝ Complex.re
-- Complex.isHomeomorphicTrivialFiberBundle_im
theorem isHomeomorphicTrivialFiberBundle_im : IsHomeomorphicTrivialFiberBundle ℝ im 
Goal: IsHomeomorphicTrivialFiberBundle ℝ Complex.im
-- Complex.isOpenMap_re
theorem isOpenMap_re : IsOpenMap re 
Goal: IsOpenMap Complex.re
-- Complex.isOpenMap_im
theorem isOpenMap_im : IsOpenMap im 
Goal: IsOpenMap Complex.im
-- Complex.quotientMap_re
theorem quotientMap_re : QuotientMap re 
Goal: QuotientMap Complex.re
-- Complex.quotientMap_im
theorem quotientMap_im : QuotientMap im 
Goal: QuotientMap Complex.im
-- Complex.interior_preimage_re
theorem interior_preimage_re (s : Set ℝ) : interior (re ⁻¹' s) = re ⁻¹' interior s 
Arguments: (s : Set ℝ)
Goal: interior (Complex.re ⁻¹' s) = Complex.re ⁻¹' interior s
-- Complex.interior_preimage_im
theorem interior_preimage_im (s : Set ℝ) : interior (im ⁻¹' s) = im ⁻¹' interior s 
Arguments: (s : Set ℝ)
Goal: interior (Complex.im ⁻¹' s) = Complex.im ⁻¹' interior s
-- Complex.closure_preimage_re
theorem closure_preimage_re (s : Set ℝ) : closure (re ⁻¹' s) = re ⁻¹' closure s 
Arguments: (s : Set ℝ)
Goal: closure (Complex.re ⁻¹' s) = Complex.re ⁻¹' closure s
-- Complex.closure_preimage_im
theorem closure_preimage_im (s : Set ℝ) : closure (im ⁻¹' s) = im ⁻¹' closure s 
Arguments: (s : Set ℝ)
Goal: closure (Complex.im ⁻¹' s) = Complex.im ⁻¹' closure s
-- Complex.frontier_preimage_re
theorem frontier_preimage_re (s : Set ℝ) : frontier (re ⁻¹' s) = re ⁻¹' frontier s 
Arguments: (s : Set ℝ)
Goal: frontier (Complex.re ⁻¹' s) = Complex.re ⁻¹' frontier s
-- Complex.frontier_preimage_im
theorem frontier_preimage_im (s : Set ℝ) : frontier (im ⁻¹' s) = im ⁻¹' frontier s 
Arguments: (s : Set ℝ)
Goal: frontier (Complex.im ⁻¹' s) = Complex.im ⁻¹' frontier s
-- Complex.interior_setOf_re_le
@[simp]
theorem interior_setOf_re_le (a : ℝ) : interior { z : ℂ | z.re ≤ a } = { z | z.re < a } 
Arguments: (a : ℝ)
Goal: interior {z | z.re ≤ a} = {z | z.re < a}
-- Complex.interior_setOf_im_le
@[simp]
theorem interior_setOf_im_le (a : ℝ) : interior { z : ℂ | z.im ≤ a } = { z | z.im < a } 
Arguments: (a : ℝ)
Goal: interior {z | z.im ≤ a} = {z | z.im < a}
-- Complex.interior_setOf_le_re
@[simp]
theorem interior_setOf_le_re (a : ℝ) : interior { z : ℂ | a ≤ z.re } = { z | a < z.re } 
Arguments: (a : ℝ)
Goal: interior {z | a ≤ z.re} = {z | a < z.re}
-- Complex.interior_setOf_le_im
@[simp]
theorem interior_setOf_le_im (a : ℝ) : interior { z : ℂ | a ≤ z.im } = { z | a < z.im } 
Arguments: (a : ℝ)
Goal: interior {z | a ≤ z.im} = {z | a < z.im}
-- Complex.closure_setOf_re_lt
@[simp]
theorem closure_setOf_re_lt (a : ℝ) : closure { z : ℂ | z.re < a } = { z | z.re ≤ a } 
Arguments: (a : ℝ)
Goal: closure {z | z.re < a} = {z | z.re ≤ a}
-- Complex.closure_setOf_im_lt
@[simp]
theorem closure_setOf_im_lt (a : ℝ) : closure { z : ℂ | z.im < a } = { z | z.im ≤ a } 
Arguments: (a : ℝ)
Goal: closure {z | z.im < a} = {z | z.im ≤ a}
-- Complex.closure_setOf_lt_re
@[simp]
theorem closure_setOf_lt_re (a : ℝ) : closure { z : ℂ | a < z.re } = { z | a ≤ z.re } 
Arguments: (a : ℝ)
Goal: closure {z | a < z.re} = {z | a ≤ z.re}
-- Complex.closure_setOf_lt_im
@[simp]
theorem closure_setOf_lt_im (a : ℝ) : closure { z : ℂ | a < z.im } = { z | a ≤ z.im } 
Arguments: (a : ℝ)
Goal: closure {z | a < z.im} = {z | a ≤ z.im}
-- Complex.frontier_setOf_re_le
@[simp]
theorem frontier_setOf_re_le (a : ℝ) : frontier { z : ℂ | z.re ≤ a } = { z | z.re = a } 
Arguments: (a : ℝ)
Goal: frontier {z | z.re ≤ a} = {z | z.re = a}
-- Complex.frontier_setOf_im_le
@[simp]
theorem frontier_setOf_im_le (a : ℝ) : frontier { z : ℂ | z.im ≤ a } = { z | z.im = a } 
Arguments: (a : ℝ)
Goal: frontier {z | z.im ≤ a} = {z | z.im = a}
-- Complex.frontier_setOf_le_re
@[simp]
theorem frontier_setOf_le_re (a : ℝ) : frontier { z : ℂ | a ≤ z.re } = { z | z.re = a } 
Arguments: (a : ℝ)
Goal: frontier {z | a ≤ z.re} = {z | z.re = a}
-- Complex.frontier_setOf_le_im
@[simp]
theorem frontier_setOf_le_im (a : ℝ) : frontier { z : ℂ | a ≤ z.im } = { z | z.im = a } 
Arguments: (a : ℝ)
Goal: frontier {z | a ≤ z.im} = {z | z.im = a}
-- Complex.frontier_setOf_re_lt
@[simp]
theorem frontier_setOf_re_lt (a : ℝ) : frontier { z : ℂ | z.re < a } = { z | z.re = a } 
Arguments: (a : ℝ)
Goal: frontier {z | z.re < a} = {z | z.re = a}
-- Complex.frontier_setOf_im_lt
@[simp]
theorem frontier_setOf_im_lt (a : ℝ) : frontier { z : ℂ | z.im < a } = { z | z.im = a } 
Arguments: (a : ℝ)
Goal: frontier {z | z.im < a} = {z | z.im = a}
-- Complex.frontier_setOf_lt_re
@[simp]
theorem frontier_setOf_lt_re (a : ℝ) : frontier { z : ℂ | a < z.re } = { z | z.re = a } 
Arguments: (a : ℝ)
Goal: frontier {z | a < z.re} = {z | z.re = a}
-- Complex.frontier_setOf_lt_im
@[simp]
theorem frontier_setOf_lt_im (a : ℝ) : frontier { z : ℂ | a < z.im } = { z | z.im = a } 
Arguments: (a : ℝ)
Goal: frontier {z | a < z.im} = {z | z.im = a}
-- Complex.closure_reProdIm
theorem closure_reProdIm (s t : Set ℝ) : closure (s ×ℂ t) = closure s ×ℂ closure t 
Arguments: (s t : Set ℝ)
Goal: closure (s ×ℂ t) = closure s ×ℂ closure t
-- Complex.interior_reProdIm
theorem interior_reProdIm (s t : Set ℝ) : interior (s ×ℂ t) = interior s ×ℂ interior t 
Arguments: (s t : Set ℝ)
Goal: interior (s ×ℂ t) = interior s ×ℂ interior t
-- Complex.frontier_reProdIm
theorem frontier_reProdIm (s t : Set ℝ) :
    frontier (s ×ℂ t) = closure s ×ℂ frontier t ∪ frontier s ×ℂ closure t 
Arguments: (s t : Set ℝ)
Goal: frontier (s ×ℂ t) = closure s ×ℂ frontier t ∪ frontier s ×ℂ closure t
-- Complex.frontier_setOf_le_re_and_le_im
theorem frontier_setOf_le_re_and_le_im (a b : ℝ) :
    frontier { z | a ≤ re z ∧ b ≤ im z } = { z | a ≤ re z ∧ im z = b ∨ re z = a ∧ b ≤ im z } 
Arguments: (a b : ℝ)
Goal: frontier {z | a ≤ z.re ∧ b ≤ z.im} = {z | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ b ≤ z.im}
-- Complex.frontier_setOf_le_re_and_im_le
theorem frontier_setOf_le_re_and_im_le (a b : ℝ) :
    frontier { z | a ≤ re z ∧ im z ≤ b } = { z | a ≤ re z ∧ im z = b ∨ re z = a ∧ im z ≤ b } 
Arguments: (a b : ℝ)
Goal: frontier {z | a ≤ z.re ∧ z.im ≤ b} = {z | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ z.im ≤ b}
-- IsOpen.reProdIm
theorem IsOpen.reProdIm (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ×ℂ t) 
Arguments: {s t : Set ℝ} (hs : IsOpen s) (ht : IsOpen t)
Goal: IsOpen (s ×ℂ t)
-- IsClosed.reProdIm
theorem IsClosed.reProdIm (hs : IsClosed s) (ht : IsClosed t) : IsClosed (s ×ℂ t) 
Arguments: {s t : Set ℝ} (hs : IsClosed s) (ht : IsClosed t)
Goal: IsClosed (s ×ℂ t)
-- Bornology.IsBounded.reProdIm
theorem Bornology.IsBounded.reProdIm (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s ×ℂ t) 
Arguments: {s t : Set ℝ} (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t)
Goal: Bornology.IsBounded (s ×ℂ t)
-- Complex.isOpen_re_lt_EReal
lemma isOpen_re_lt_EReal (x : EReal) : IsOpen {z : ℂ | z.re < x} 
Arguments: (x : EReal)
Goal: IsOpen {z | ↑z.re < x}
-- Complex.isOpen_re_gt_EReal
lemma isOpen_re_gt_EReal (x : EReal) : IsOpen {z : ℂ | x < z.re} 
Arguments: (x : EReal)
Goal: IsOpen {z | x < ↑z.re}
-- Complex.isOpen_im_lt_EReal
lemma isOpen_im_lt_EReal (x : EReal) : IsOpen {z : ℂ | z.im < x} 
Arguments: (x : EReal)
Goal: IsOpen {z | ↑z.im < x}
-- Complex.isOpen_im_gt_EReal
lemma isOpen_im_gt_EReal (x : EReal) : IsOpen {z : ℂ | x < z.im} 
Arguments: (x : EReal)
Goal: IsOpen {z | x < ↑z.im}
-- approximatesLinearOn_empty
@[simp]
theorem approximatesLinearOn_empty (f : E → F) (f' : E →L[𝕜] F) (c : ℝ≥0) :
    ApproximatesLinearOn f f' ∅ c 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E → F) (f' : E →L[𝕜] F) (c : ℝ≥0)
Goal: ApproximatesLinearOn f f' ∅ c
-- ApproximatesLinearOn.mono_num
theorem mono_num (hc : c ≤ c') (hf : ApproximatesLinearOn f f' s c) :
    ApproximatesLinearOn f f' s c' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {s : Set E} {c c' : ℝ≥0} (hc : c ≤ c') (hf : ApproximatesLinearOn f f' s c)
Goal: ApproximatesLinearOn f f' s c'
-- ApproximatesLinearOn.mono_set
theorem mono_set (hst : s ⊆ t) (hf : ApproximatesLinearOn f f' t c) :
    ApproximatesLinearOn f f' s c 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {s t : Set E} {c : ℝ≥0} (hst : s ⊆ t) (hf : ApproximatesLinearOn f f' t c)
Goal: ApproximatesLinearOn f f' s c
-- ApproximatesLinearOn.approximatesLinearOn_iff_lipschitzOnWith
theorem approximatesLinearOn_iff_lipschitzOnWith {f : E → F} {f' : E →L[𝕜] F} {s : Set E}
    {c : ℝ≥0} : ApproximatesLinearOn f f' s c ↔ LipschitzOnWith c (f - ⇑f') s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {s : Set E} {c : ℝ≥0}
Goal: ApproximatesLinearOn f f' s c ↔ LipschitzOnWith c (f - ⇑f') s
-- ApproximatesLinearOn.lipschitz_sub
theorem lipschitz_sub (hf : ApproximatesLinearOn f f' s c) :
    LipschitzWith c fun x : s => f x - f' x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {s : Set E} {c : ℝ≥0} (hf : ApproximatesLinearOn f f' s c)
Goal: LipschitzWith c fun x => f ↑x - f' ↑x
-- ApproximatesLinearOn.lipschitz
protected theorem lipschitz (hf : ApproximatesLinearOn f f' s c) :
    LipschitzWith (‖f'‖₊ + c) (s.restrict f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {s : Set E} {c : ℝ≥0} (hf : ApproximatesLinearOn f f' s c)
Goal: LipschitzWith (‖f'‖₊ + c) (s.restrict f)
-- ApproximatesLinearOn.continuous
protected theorem continuous (hf : ApproximatesLinearOn f f' s c) : Continuous (s.restrict f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {s : Set E} {c : ℝ≥0} (hf : ApproximatesLinearOn f f' s c)
Goal: Continuous (s.restrict f)
-- ApproximatesLinearOn.continuousOn
protected theorem continuousOn (hf : ApproximatesLinearOn f f' s c) : ContinuousOn f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {s : Set E} {c : ℝ≥0} (hf : ApproximatesLinearOn f f' s c)
Goal: ContinuousOn f s
-- ApproximatesLinearOn.surjOn_closedBall_of_nonlinearRightInverse
theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f f' s c)
    (f'symm : f'.NonlinearRightInverse) {ε : ℝ} {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
    SurjOn f (closedBall b ε) (closedBall (f b) (((f'symm.nnnorm : ℝ)⁻¹ - c) * ε)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace E] {f : E → F} {s : Set E} {c : ℝ≥0} {f' : E →L[𝕜] F} (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse) {ε : ℝ} {b : E} (ε0 : 0 ≤ ε) (hε : Metric.closedBall b ε ⊆ s)
Goal: Set.SurjOn f (Metric.closedBall b ε) (Metric.closedBall (f b) (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε))
-- ApproximatesLinearOn.open_image
theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
    (hs : IsOpen s) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : IsOpen (f '' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace E] {f : E → F} {s : Set E} {c : ℝ≥0} {f' : E →L[𝕜] F} (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse) (hs : IsOpen s) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹)
Goal: IsOpen (f '' s)
-- ApproximatesLinearOn.image_mem_nhds
theorem image_mem_nhds (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
    {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : f '' s ∈ 𝓝 (f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace E] {f : E → F} {s : Set E} {c : ℝ≥0} {f' : E →L[𝕜] F} (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse) {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹)
Goal: f '' s ∈ 𝓝 (f x)
-- ApproximatesLinearOn.map_nhds_eq
theorem map_nhds_eq (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse) {x : E}
    (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : map f (𝓝 x) = 𝓝 (f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace E] {f : E → F} {s : Set E} {c : ℝ≥0} {f' : E →L[𝕜] F} (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse) {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹)
Goal: Filter.map f (𝓝 x) = 𝓝 (f x)
-- ApproximatesLinearOn.antilipschitz
protected theorem antilipschitz (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
    (hc : Subsingleton E ∨ c < N⁻¹) : AntilipschitzWith (N⁻¹ - c)⁻¹ (s.restrict f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E ≃L[𝕜] F} {s : Set E} {c : ℝ≥0} (hf : ApproximatesLinearOn f (↑f') s c) (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹)
Goal: AntilipschitzWith (‖↑f'.symm‖₊⁻¹ - c)⁻¹ (s.restrict f)
-- ApproximatesLinearOn.injective
protected theorem injective (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
    (hc : Subsingleton E ∨ c < N⁻¹) : Injective (s.restrict f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E ≃L[𝕜] F} {s : Set E} {c : ℝ≥0} (hf : ApproximatesLinearOn f (↑f') s c) (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹)
Goal: Function.Injective (s.restrict f)
-- ApproximatesLinearOn.injOn
protected theorem injOn (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
    (hc : Subsingleton E ∨ c < N⁻¹) : InjOn f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E ≃L[𝕜] F} {s : Set E} {c : ℝ≥0} (hf : ApproximatesLinearOn f (↑f') s c) (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹)
Goal: Set.InjOn f s