ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NormedSpace.expSeries_div_summable_of_mem_ball
theorem expSeries_div_summable_of_mem_ball [CompleteSpace 𝔸] (x : 𝔸)
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) : Summable fun n => x ^ n / n ! 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedDivisionRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: Summable fun n => x ^ n / ↑n.factorial
-- NormedSpace.expSeries_div_hasSum_exp_of_mem_ball
theorem expSeries_div_hasSum_exp_of_mem_ball [CompleteSpace 𝔸] (x : 𝔸)
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasSum (fun n => x ^ n / n !) (exp 𝕂 x) 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedDivisionRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasSum (fun n => x ^ n / ↑n.factorial) (NormedSpace.exp 𝕂 x)
-- NormedSpace.exp_neg_of_mem_ball
theorem exp_neg_of_mem_ball [CharZero 𝕂] [CompleteSpace 𝔸] {x : 𝔸}
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) : exp 𝕂 (-x) = (exp 𝕂 x)⁻¹ 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedDivisionRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CharZero 𝕂] [CompleteSpace 𝔸] {x : 𝔸} (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: NormedSpace.exp 𝕂 (-x) = (NormedSpace.exp 𝕂 x)⁻¹
-- NormedSpace.exp_add_of_mem_ball
theorem exp_add_of_mem_ball [CharZero 𝕂] {x y : 𝔸}
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius)
    (hy : y ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) : exp 𝕂 (x + y) = exp 𝕂 x * exp 𝕂 y 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] [CharZero 𝕂] {x y : 𝔸} (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius) (hy : y ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: NormedSpace.exp 𝕂 (x + y) = NormedSpace.exp 𝕂 x * NormedSpace.exp 𝕂 y
-- NormedSpace.expSeries_radius_eq_top
theorem expSeries_radius_eq_top : (expSeries 𝕂 𝔸).radius = ∞ 
Arguments: (𝕂 : Type u_1) (𝔸 : Type u_2) [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸]
Goal: (NormedSpace.expSeries 𝕂 𝔸).radius = ⊤
-- NormedSpace.expSeries_radius_pos
theorem expSeries_radius_pos : 0 < (expSeries 𝕂 𝔸).radius 
Arguments: (𝕂 : Type u_1) (𝔸 : Type u_2) [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸]
Goal: 0 < (NormedSpace.expSeries 𝕂 𝔸).radius
-- NormedSpace.norm_expSeries_summable
theorem norm_expSeries_summable (x : 𝔸) : Summable fun n => ‖expSeries 𝕂 𝔸 n fun _ => x‖ 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] (x : 𝔸)
Goal: Summable fun n => ‖(NormedSpace.expSeries 𝕂 𝔸 n) fun x_1 => x‖
-- NormedSpace.norm_expSeries_summable'
theorem norm_expSeries_summable' (x : 𝔸) : Summable fun n => ‖(n !⁻¹ : 𝕂) • x ^ n‖ 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] (x : 𝔸)
Goal: Summable fun n => ‖(↑n.factorial)⁻¹ • x ^ n‖
-- NormedSpace.expSeries_summable
theorem expSeries_summable (x : 𝔸) : Summable fun n => expSeries 𝕂 𝔸 n fun _ => x 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸)
Goal: Summable fun n => (NormedSpace.expSeries 𝕂 𝔸 n) fun x_1 => x
-- NormedSpace.expSeries_summable'
theorem expSeries_summable' (x : 𝔸) : Summable fun n => (n !⁻¹ : 𝕂) • x ^ n 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸)
Goal: Summable fun n => (↑n.factorial)⁻¹ • x ^ n
-- NormedSpace.expSeries_hasSum_exp
theorem expSeries_hasSum_exp (x : 𝔸) : HasSum (fun n => expSeries 𝕂 𝔸 n fun _ => x) (exp 𝕂 x) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸)
Goal: HasSum (fun n => (NormedSpace.expSeries 𝕂 𝔸 n) fun x_1 => x) (NormedSpace.exp 𝕂 x)
-- NormedSpace.exp_series_hasSum_exp'
theorem exp_series_hasSum_exp' (x : 𝔸) : HasSum (fun n => (n !⁻¹ : 𝕂) • x ^ n) (exp 𝕂 x) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸)
Goal: HasSum (fun n => (↑n.factorial)⁻¹ • x ^ n) (NormedSpace.exp 𝕂 x)
-- NormedSpace.exp_hasFPowerSeriesOnBall
theorem exp_hasFPowerSeriesOnBall : HasFPowerSeriesOnBall (exp 𝕂) (expSeries 𝕂 𝔸) 0 ∞ 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸]
Goal: HasFPowerSeriesOnBall (NormedSpace.exp 𝕂) (NormedSpace.expSeries 𝕂 𝔸) 0 ⊤
-- NormedSpace.exp_hasFPowerSeriesAt_zero
theorem exp_hasFPowerSeriesAt_zero : HasFPowerSeriesAt (exp 𝕂) (expSeries 𝕂 𝔸) 0 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸]
Goal: HasFPowerSeriesAt (NormedSpace.exp 𝕂) (NormedSpace.expSeries 𝕂 𝔸) 0
-- NormedSpace.exp_continuous
@[continuity]
theorem exp_continuous : Continuous (exp 𝕂 : 𝔸 → 𝔸) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸]
Goal: Continuous (NormedSpace.exp 𝕂)
-- Filter.Tendsto.exp
lemma _root_.Filter.Tendsto.exp {α : Type*} {l : Filter α} {f : α → 𝔸} {a : 𝔸}
    (hf : Tendsto f l (𝓝 a)) :
    Tendsto (fun x => exp 𝕂 (f x)) l (𝓝 (exp 𝕂 a)) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] {α : Type u_4} {l : Filter α} {f : α → 𝔸} {a : 𝔸} (hf : Filter.Tendsto f l (nhds a))
Goal: Filter.Tendsto (fun x => NormedSpace.exp 𝕂 (f x)) l (nhds (NormedSpace.exp 𝕂 a))
-- NormedSpace.exp_analytic
theorem exp_analytic (x : 𝔸) : AnalyticAt 𝕂 (exp 𝕂) x 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸)
Goal: AnalyticAt 𝕂 (NormedSpace.exp 𝕂) x
-- NormedSpace.exp_add_of_commute
theorem exp_add_of_commute {x y : 𝔸} (hxy : Commute x y) : exp 𝕂 (x + y) = exp 𝕂 x * exp 𝕂 y 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] {x y : 𝔸} (hxy : Commute x y)
Goal: NormedSpace.exp 𝕂 (x + y) = NormedSpace.exp 𝕂 x * NormedSpace.exp 𝕂 y
-- NormedSpace.isUnit_exp
theorem isUnit_exp (x : 𝔸) : IsUnit (exp 𝕂 x) 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸)
Goal: IsUnit (NormedSpace.exp 𝕂 x)
-- NormedSpace.invOf_exp
theorem invOf_exp (x : 𝔸) [Invertible (exp 𝕂 x)] : ⅟ (exp 𝕂 x) = exp 𝕂 (-x) 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) [Invertible (NormedSpace.exp 𝕂 x)]
Goal: ⅟(NormedSpace.exp 𝕂 x) = NormedSpace.exp 𝕂 (-x)
-- Ring.inverse_exp
theorem _root_.Ring.inverse_exp (x : 𝔸) : Ring.inverse (exp 𝕂 x) = exp 𝕂 (-x) 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸)
Goal: Ring.inverse (NormedSpace.exp 𝕂 x) = NormedSpace.exp 𝕂 (-x)
-- NormedSpace.exp_mem_unitary_of_mem_skewAdjoint
theorem exp_mem_unitary_of_mem_skewAdjoint [StarRing 𝔸] [ContinuousStar 𝔸] {x : 𝔸}
    (h : x ∈ skewAdjoint 𝔸) : exp 𝕂 x ∈ unitary 𝔸 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] {x : 𝔸} (h : x ∈ skewAdjoint 𝔸)
Goal: NormedSpace.exp 𝕂 x ∈ unitary 𝔸
-- NormedSpace.exp_sum_of_commute
theorem exp_sum_of_commute {ι} (s : Finset ι) (f : ι → 𝔸)
    (h : (s : Set ι).Pairwise fun i j => Commute (f i) (f j)) :
    exp 𝕂 (∑ i ∈ s, f i) =
      s.noncommProd (fun i => exp 𝕂 (f i)) fun i hi j hj _ => (h.of_refl hi hj).exp 𝕂 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] {ι : Type u_4} (s : Finset ι) (f : ι → 𝔸) (h : (↑s).Pairwise fun i j => Commute (f i) (f j))
Goal: NormedSpace.exp 𝕂 (∑ i ∈ s, f i) = s.noncommProd (fun i => NormedSpace.exp 𝕂 (f i)) ⋯
-- NormedSpace.exp_nsmul
theorem exp_nsmul (n : ℕ) (x : 𝔸) : exp 𝕂 (n • x) = exp 𝕂 x ^ n 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (n : ℕ) (x : 𝔸)
Goal: NormedSpace.exp 𝕂 (n • x) = NormedSpace.exp 𝕂 x ^ n
-- NormedSpace.map_exp
theorem map_exp {F} [FunLike F 𝔸 𝔹] [RingHomClass F 𝔸 𝔹] (f : F) (hf : Continuous f) (x : 𝔸) :
    f (exp 𝕂 x) = exp 𝕂 (f x) 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} {𝔹 : Type u_3} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [NormedRing 𝔹] [NormedAlgebra 𝕂 𝔹] [CompleteSpace 𝔸] {F : Type u_4} [FunLike F 𝔸 𝔹] [RingHomClass F 𝔸 𝔹] (f : F) (hf : Continuous ⇑f) (x : 𝔸)
Goal: f (NormedSpace.exp 𝕂 x) = NormedSpace.exp 𝕂 (f x)
-- NormedSpace.exp_smul
theorem exp_smul {G} [Monoid G] [MulSemiringAction G 𝔸] [ContinuousConstSMul G 𝔸] (g : G) (x : 𝔸) :
    exp 𝕂 (g • x) = g • exp 𝕂 x 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] {G : Type u_4} [Monoid G] [MulSemiringAction G 𝔸] [ContinuousConstSMul G 𝔸] (g : G) (x : 𝔸)
Goal: NormedSpace.exp 𝕂 (g • x) = g • NormedSpace.exp 𝕂 x
-- NormedSpace.exp_units_conj
theorem exp_units_conj (y : 𝔸ˣ) (x : 𝔸) : exp 𝕂 (y * x * ↑y⁻¹ : 𝔸) = y * exp 𝕂 x * ↑y⁻¹ 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (y : 𝔸ˣ) (x : 𝔸)
Goal: NormedSpace.exp 𝕂 (↑y * x * ↑y⁻¹) = ↑y * NormedSpace.exp 𝕂 x * ↑y⁻¹
-- NormedSpace.exp_units_conj'
theorem exp_units_conj' (y : 𝔸ˣ) (x : 𝔸) : exp 𝕂 (↑y⁻¹ * x * y) = ↑y⁻¹ * exp 𝕂 x * y 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (y : 𝔸ˣ) (x : 𝔸)
Goal: NormedSpace.exp 𝕂 (↑y⁻¹ * x * ↑y) = ↑y⁻¹ * NormedSpace.exp 𝕂 x * ↑y
-- Prod.fst_exp
@[simp]
theorem _root_.Prod.fst_exp [CompleteSpace 𝔹] (x : 𝔸 × 𝔹) : (exp 𝕂 x).fst = exp 𝕂 x.fst 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} {𝔹 : Type u_3} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [NormedRing 𝔹] [NormedAlgebra 𝕂 𝔹] [CompleteSpace 𝔸] [CompleteSpace 𝔹] (x : 𝔸 × 𝔹)
Goal: (NormedSpace.exp 𝕂 x).1 = NormedSpace.exp 𝕂 x.1
-- Prod.snd_exp
@[simp]
theorem _root_.Prod.snd_exp [CompleteSpace 𝔹] (x : 𝔸 × 𝔹) : (exp 𝕂 x).snd = exp 𝕂 x.snd 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} {𝔹 : Type u_3} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [NormedRing 𝔹] [NormedAlgebra 𝕂 𝔹] [CompleteSpace 𝔸] [CompleteSpace 𝔹] (x : 𝔸 × 𝔹)
Goal: (NormedSpace.exp 𝕂 x).2 = NormedSpace.exp 𝕂 x.2
-- Pi.exp_apply
@[simp]
theorem _root_.Pi.exp_apply {ι : Type*} {𝔸 : ι → Type*} [Finite ι] [∀ i, NormedRing (𝔸 i)]
    [∀ i, NormedAlgebra 𝕂 (𝔸 i)] [∀ i, CompleteSpace (𝔸 i)] (x : ∀ i, 𝔸 i) (i : ι) :
    exp 𝕂 x i = exp 𝕂 (x i) 
Arguments: (𝕂 : Type u_1) [RCLike 𝕂] {ι : Type u_4} {𝔸 : ι → Type u_5} [Finite ι] [(i : ι) → NormedRing (𝔸 i)] [(i : ι) → NormedAlgebra 𝕂 (𝔸 i)] [∀ (i : ι), CompleteSpace (𝔸 i)] (x : (i : ι) → 𝔸 i) (i : ι)
Goal: NormedSpace.exp 𝕂 x i = NormedSpace.exp 𝕂 (x i)
-- Pi.exp_def
theorem _root_.Pi.exp_def {ι : Type*} {𝔸 : ι → Type*} [Finite ι] [∀ i, NormedRing (𝔸 i)]
    [∀ i, NormedAlgebra 𝕂 (𝔸 i)] [∀ i, CompleteSpace (𝔸 i)] (x : ∀ i, 𝔸 i) :
    exp 𝕂 x = fun i => exp 𝕂 (x i) 
Arguments: (𝕂 : Type u_1) [RCLike 𝕂] {ι : Type u_4} {𝔸 : ι → Type u_5} [Finite ι] [(i : ι) → NormedRing (𝔸 i)] [(i : ι) → NormedAlgebra 𝕂 (𝔸 i)] [∀ (i : ι), CompleteSpace (𝔸 i)] (x : (i : ι) → 𝔸 i)
Goal: NormedSpace.exp 𝕂 x = fun i => NormedSpace.exp 𝕂 (x i)
-- Function.update_exp
theorem _root_.Function.update_exp {ι : Type*} {𝔸 : ι → Type*} [Finite ι] [DecidableEq ι]
    [∀ i, NormedRing (𝔸 i)] [∀ i, NormedAlgebra 𝕂 (𝔸 i)] [∀ i, CompleteSpace (𝔸 i)] (x : ∀ i, 𝔸 i)
    (j : ι) (xj : 𝔸 j) :
    Function.update (exp 𝕂 x) j (exp 𝕂 xj) = exp 𝕂 (Function.update x j xj) 
Arguments: (𝕂 : Type u_1) [RCLike 𝕂] {ι : Type u_4} {𝔸 : ι → Type u_5} [Finite ι] [DecidableEq ι] [(i : ι) → NormedRing (𝔸 i)] [(i : ι) → NormedAlgebra 𝕂 (𝔸 i)] [∀ (i : ι), CompleteSpace (𝔸 i)] (x : (i : ι) → 𝔸 i) (j : ι) (xj : 𝔸 j)
Goal: Function.update (NormedSpace.exp 𝕂 x) j (NormedSpace.exp 𝕂 xj) = NormedSpace.exp 𝕂 (Function.update x j xj)
-- NormedSpace.algebraMap_exp_comm
theorem algebraMap_exp_comm (x : 𝕂) : algebraMap 𝕂 𝔸 (exp 𝕂 x) = exp 𝕂 (algebraMap 𝕂 𝔸 x) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] (x : 𝕂)
Goal: (algebraMap 𝕂 𝔸) (NormedSpace.exp 𝕂 x) = NormedSpace.exp 𝕂 ((algebraMap 𝕂 𝔸) x)
-- NormedSpace.norm_expSeries_div_summable
theorem norm_expSeries_div_summable (x : 𝔸) : Summable fun n => ‖(x ^ n / n ! : 𝔸)‖ 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [RCLike 𝕂] [NormedDivisionRing 𝔸] [NormedAlgebra 𝕂 𝔸] (x : 𝔸)
Goal: Summable fun n => ‖x ^ n / ↑n.factorial‖
-- NormedSpace.expSeries_div_summable
theorem expSeries_div_summable (x : 𝔸) : Summable fun n => x ^ n / n ! 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [RCLike 𝕂] [NormedDivisionRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸)
Goal: Summable fun n => x ^ n / ↑n.factorial
-- NormedSpace.expSeries_div_hasSum_exp
theorem expSeries_div_hasSum_exp (x : 𝔸) : HasSum (fun n => x ^ n / n !) (exp 𝕂 x) 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [RCLike 𝕂] [NormedDivisionRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸)
Goal: HasSum (fun n => x ^ n / ↑n.factorial) (NormedSpace.exp 𝕂 x)
-- NormedSpace.exp_neg
theorem exp_neg (x : 𝔸) : exp 𝕂 (-x) = (exp 𝕂 x)⁻¹ 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedDivisionRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸)
Goal: NormedSpace.exp 𝕂 (-x) = (NormedSpace.exp 𝕂 x)⁻¹
-- NormedSpace.exp_zsmul
theorem exp_zsmul (z : ℤ) (x : 𝔸) : exp 𝕂 (z • x) = exp 𝕂 x ^ z 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedDivisionRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (z : ℤ) (x : 𝔸)
Goal: NormedSpace.exp 𝕂 (z • x) = NormedSpace.exp 𝕂 x ^ z
-- NormedSpace.exp_conj
theorem exp_conj (y : 𝔸) (x : 𝔸) (hy : y ≠ 0) : exp 𝕂 (y * x * y⁻¹) = y * exp 𝕂 x * y⁻¹ 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedDivisionRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (y x : 𝔸) (hy : y ≠ 0)
Goal: NormedSpace.exp 𝕂 (y * x * y⁻¹) = y * NormedSpace.exp 𝕂 x * y⁻¹
-- NormedSpace.exp_conj'
theorem exp_conj' (y : 𝔸) (x : 𝔸) (hy : y ≠ 0) : exp 𝕂 (y⁻¹ * x * y) = y⁻¹ * exp 𝕂 x * y 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedDivisionRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (y x : 𝔸) (hy : y ≠ 0)
Goal: NormedSpace.exp 𝕂 (y⁻¹ * x * y) = y⁻¹ * NormedSpace.exp 𝕂 x * y
-- NormedSpace.exp_add
theorem exp_add {x y : 𝔸} : exp 𝕂 (x + y) = exp 𝕂 x * exp 𝕂 y 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] {x y : 𝔸}
Goal: NormedSpace.exp 𝕂 (x + y) = NormedSpace.exp 𝕂 x * NormedSpace.exp 𝕂 y
-- NormedSpace.exp_sum
theorem exp_sum {ι} (s : Finset ι) (f : ι → 𝔸) : exp 𝕂 (∑ i ∈ s, f i) = ∏ i ∈ s, exp 𝕂 (f i) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [RCLike 𝕂] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] {ι : Type u_3} (s : Finset ι) (f : ι → 𝔸)
Goal: NormedSpace.exp 𝕂 (∑ i ∈ s, f i) = ∏ i ∈ s, NormedSpace.exp 𝕂 (f i)
-- NormedSpace.expSeries_eq_expSeries
theorem expSeries_eq_expSeries (n : ℕ) (x : 𝔸) :
    (expSeries 𝕂 𝔸 n fun _ => x) = expSeries 𝕂' 𝔸 n fun _ => x 
Arguments: (𝕂 : Type u_1) (𝕂' : Type u_2) (𝔸 : Type u_3) [Field 𝕂] [Field 𝕂'] [Ring 𝔸] [Algebra 𝕂 𝔸] [Algebra 𝕂' 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] (n : ℕ) (x : 𝔸)
Goal: ((NormedSpace.expSeries 𝕂 𝔸 n) fun x_1 => x) = (NormedSpace.expSeries 𝕂' 𝔸 n) fun x_1 => x
-- NormedSpace.exp_eq_exp
theorem exp_eq_exp : (exp 𝕂 : 𝔸 → 𝔸) = exp 𝕂' 
Arguments: (𝕂 : Type u_1) (𝕂' : Type u_2) (𝔸 : Type u_3) [Field 𝕂] [Field 𝕂'] [Ring 𝔸] [Algebra 𝕂 𝔸] [Algebra 𝕂' 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸]
Goal: NormedSpace.exp 𝕂 = NormedSpace.exp 𝕂'
-- NormedSpace.exp_ℝ_ℂ_eq_exp_ℂ_ℂ
theorem exp_ℝ_ℂ_eq_exp_ℂ_ℂ : (exp ℝ : ℂ → ℂ) = exp ℂ 
Goal: NormedSpace.exp ℝ = NormedSpace.exp ℂ
-- NormedSpace.of_real_exp_ℝ_ℝ
@[simp, norm_cast]
theorem of_real_exp_ℝ_ℝ (r : ℝ) : ↑(exp ℝ r) = exp ℂ (r : ℂ) 
Arguments: (r : ℝ)
Goal: ↑(NormedSpace.exp ℝ r) = NormedSpace.exp ℂ ↑r
-- isOpenMap_barycentric_coord
theorem isOpenMap_barycentric_coord [Nontrivial ι] (b : AffineBasis ι 𝕜 P) (i : ι) :
    IsOpenMap (b.coord i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} {E : Type u_3} {P : Type u_4} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MetricSpace P] [NormedAddTorsor E P] [Nontrivial ι] (b : AffineBasis ι 𝕜 P) (i : ι)
Goal: IsOpenMap ⇑(b.coord i)
-- continuous_barycentric_coord
@[continuity]
theorem continuous_barycentric_coord (i : ι) : Continuous (b.coord i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} {E : Type u_3} {P : Type u_4} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MetricSpace P] [NormedAddTorsor E P] [FiniteDimensional 𝕜 E] (b : AffineBasis ι 𝕜 P) (i : ι)
Goal: Continuous ⇑(b.coord i)
-- AffineBasis.interior_convexHull
theorem AffineBasis.interior_convexHull {ι E : Type*} [Finite ι] [NormedAddCommGroup E]
    [NormedSpace ℝ E] (b : AffineBasis ι ℝ E) :
    interior (convexHull ℝ (range b)) = {x | ∀ i, 0 < b.coord i x} 
Arguments: {ι : Type u_1} {E : Type u_2} [Finite ι] [NormedAddCommGroup E] [NormedSpace ℝ E] (b : AffineBasis ι ℝ E)
Goal: interior ((convexHull ℝ) (Set.range ⇑b)) = {x | ∀ (i : ι), 0 < (b.coord i) x}
-- IsOpen.exists_between_affineIndependent_span_eq_top
theorem IsOpen.exists_between_affineIndependent_span_eq_top {s u : Set P} (hu : IsOpen u)
    (hsu : s ⊆ u) (hne : s.Nonempty) (h : AffineIndependent ℝ ((↑) : s → P)) :
    ∃ t : Set P, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ ((↑) : t → P) ∧ affineSpan ℝ t = ⊤ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s u : Set P} (hu : IsOpen u) (hsu : s ⊆ u) (hne : s.Nonempty) (h : AffineIndependent ℝ Subtype.val)
Goal: ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤
-- IsOpen.exists_subset_affineIndependent_span_eq_top
theorem IsOpen.exists_subset_affineIndependent_span_eq_top {u : Set P} (hu : IsOpen u)
    (hne : u.Nonempty) : ∃ s ⊆ u, AffineIndependent ℝ ((↑) : s → P) ∧ affineSpan ℝ s = ⊤ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {u : Set P} (hu : IsOpen u) (hne : u.Nonempty)
Goal: ∃ s ⊆ u, AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤
-- IsOpen.affineSpan_eq_top
theorem IsOpen.affineSpan_eq_top {u : Set P} (hu : IsOpen u) (hne : u.Nonempty) :
    affineSpan ℝ u = ⊤ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {u : Set P} (hu : IsOpen u) (hne : u.Nonempty)
Goal: affineSpan ℝ u = ⊤
-- affineSpan_eq_top_of_nonempty_interior
theorem affineSpan_eq_top_of_nonempty_interior {s : Set V}
    (hs : (interior <| convexHull ℝ s).Nonempty) : affineSpan ℝ s = ⊤ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [NormedSpace ℝ V] {s : Set V} (hs : (interior ((convexHull ℝ) s)).Nonempty)
Goal: affineSpan ℝ s = ⊤
-- AffineBasis.centroid_mem_interior_convexHull
theorem AffineBasis.centroid_mem_interior_convexHull {ι} [Fintype ι] (b : AffineBasis ι ℝ V) :
    Finset.univ.centroid ℝ b ∈ interior (convexHull ℝ (range b)) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [NormedSpace ℝ V] {ι : Type u_3} [Fintype ι] (b : AffineBasis ι ℝ V)
Goal: Finset.centroid ℝ Finset.univ ⇑b ∈ interior ((convexHull ℝ) (Set.range ⇑b))
-- interior_convexHull_nonempty_iff_affineSpan_eq_top
theorem interior_convexHull_nonempty_iff_affineSpan_eq_top [FiniteDimensional ℝ V] {s : Set V} :
    (interior (convexHull ℝ s)).Nonempty ↔ affineSpan ℝ s = ⊤ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] {s : Set V}
Goal: (interior ((convexHull ℝ) s)).Nonempty ↔ affineSpan ℝ s = ⊤
-- Convex.interior_nonempty_iff_affineSpan_eq_top
theorem Convex.interior_nonempty_iff_affineSpan_eq_top [FiniteDimensional ℝ V] {s : Set V}
    (hs : Convex ℝ s) : (interior s).Nonempty ↔ affineSpan ℝ s = ⊤ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] {s : Set V} (hs : Convex ℝ s)
Goal: (interior s).Nonempty ↔ affineSpan ℝ s = ⊤
-- Complex.sin_arg
theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x 
Arguments: (x : ℂ)
Goal: Real.sin x.arg = x.im / Complex.abs x
-- Complex.cos_arg
theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x 
Arguments: {x : ℂ} (hx : x ≠ 0)
Goal: Real.cos x.arg = x.re / Complex.abs x
-- Complex.abs_mul_exp_arg_mul_I
@[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x 
Arguments: (x : ℂ)
Goal: ↑(Complex.abs x) * Complex.exp (↑x.arg * Complex.I) = x
-- Complex.abs_mul_cos_add_sin_mul_I
@[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x 
Arguments: (x : ℂ)
Goal: ↑(Complex.abs x) * (Complex.cos ↑x.arg + Complex.sin ↑x.arg * Complex.I) = x
-- Complex.abs_mul_cos_arg
@[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re 
Arguments: (x : ℂ)
Goal: Complex.abs x * Real.cos x.arg = x.re
-- Complex.abs_mul_sin_arg
@[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im 
Arguments: (x : ℂ)
Goal: Complex.abs x * Real.sin x.arg = x.im
-- Complex.abs_eq_one_iff
theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z 
Arguments: (z : ℂ)
Goal: Complex.abs z = 1 ↔ ∃ θ, Complex.exp (↑θ * Complex.I) = z
-- Complex.range_exp_mul_I
@[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 
Goal: (Set.range fun x => Complex.exp (↑x * Complex.I)) = Metric.sphere 0 1
-- Complex.arg_mul_cos_add_sin_mul_I
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
    arg (r * (cos θ + sin θ * I)) = θ 
Arguments: {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π)
Goal: (↑r * (Complex.cos ↑θ + Complex.sin ↑θ * Complex.I)).arg = θ
-- Complex.arg_cos_add_sin_mul_I
theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ 
Arguments: {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π)
Goal: (Complex.cos ↑θ + Complex.sin ↑θ * Complex.I).arg = θ
-- Complex.arg_exp_mul_I
lemma arg_exp_mul_I (θ : ℝ) :
    arg (exp (θ * I)) = toIocMod (mul_pos two_pos Real.pi_pos) (-π) θ 
Arguments: (θ : ℝ)
Goal: (Complex.exp (↑θ * Complex.I)).arg = toIocMod ⋯ (-π) θ
-- Complex.arg_zero
@[simp]
theorem arg_zero : arg 0 = 0 
Goal: Complex.arg 0 = 0
-- Complex.ext_abs_arg
theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y 
Arguments: {x y : ℂ} (h₁ : Complex.abs x = Complex.abs y) (h₂ : x.arg = y.arg)
Goal: x = y
-- Complex.ext_abs_arg_iff
theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y 
Arguments: {x y : ℂ}
Goal: x = y ↔ Complex.abs x = Complex.abs y ∧ x.arg = y.arg
-- Complex.arg_mem_Ioc
theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π 
Arguments: (z : ℂ)
Goal: z.arg ∈ Set.Ioc (-π) π
-- Complex.range_arg
@[simp]
theorem range_arg : Set.range arg = Set.Ioc (-π) π 
Goal: Set.range Complex.arg = Set.Ioc (-π) π
-- Complex.arg_le_pi
theorem arg_le_pi (x : ℂ) : arg x ≤ π 
Arguments: (x : ℂ)
Goal: x.arg ≤ π
-- Complex.neg_pi_lt_arg
theorem neg_pi_lt_arg (x : ℂ) : -π < arg x 
Arguments: (x : ℂ)
Goal: -π < x.arg
-- Complex.abs_arg_le_pi
theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π 
Arguments: (z : ℂ)
Goal: |z.arg| ≤ π
-- Complex.arg_nonneg_iff
@[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im 
Arguments: {z : ℂ}
Goal: 0 ≤ z.arg ↔ 0 ≤ z.im
-- Complex.arg_neg_iff
@[simp]
theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 
Arguments: {z : ℂ}
Goal: z.arg < 0 ↔ z.im < 0
-- Complex.arg_real_mul
theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x 
Arguments: (x : ℂ) {r : ℝ} (hr : 0 < r)
Goal: (↑r * x).arg = x.arg
-- Complex.arg_mul_real
theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x 
Arguments: {r : ℝ} (hr : 0 < r) (x : ℂ)
Goal: (x * ↑r).arg = x.arg
-- Complex.arg_eq_arg_iff
theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
    arg x = arg y ↔ (abs y / abs x : ℂ) * x = y 
Arguments: {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: x.arg = y.arg ↔ ↑(Complex.abs y) / ↑(Complex.abs x) * x = y
-- Complex.arg_one
@[simp]
theorem arg_one : arg 1 = 0 
Goal: Complex.arg 1 = 0
-- Complex.arg_neg_one
@[simp]
theorem arg_neg_one : arg (-1) = π 
Goal: (-1).arg = π
-- Complex.arg_I
@[simp]
theorem arg_I : arg I = π / 2 
Goal: Complex.I.arg = π / 2