ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- GaussianFourier.integral_cexp_neg_mul_sum_add
theorem integral_cexp_neg_mul_sum_add {ι : Type*} [Fintype ι] (hb : 0 < b.re) (c : ι → ℂ) :
    ∫ v : ι → ℝ, cexp (- b * ∑ i, (v i : ℂ) ^ 2 + ∑ i, c i * v i)
      = (π / b) ^ (Fintype.card ι / 2 : ℂ) * cexp ((∑ i, c i ^ 2) / (4 * b)) 
Arguments: {b : ℂ} {ι : Type u_2} [Fintype ι] (hb : 0 < b.re) (c : ι → ℂ)
Goal: ∫ (v : ι → ℝ), cexp (-b * ∑ i : ι, ↑(v i) ^ 2 + ∑ i : ι, c i * ↑(v i)) =
  (↑π / b) ^ (↑(Fintype.card ι) / 2) * cexp ((∑ i : ι, c i ^ 2) / (4 * b))
-- GaussianFourier.integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace
theorem integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace
    {ι : Type*} [Fintype ι] (hb : 0 < b.re) (c : ℂ) (w : EuclideanSpace ℝ ι) :
    ∫ v : EuclideanSpace ℝ ι, cexp (- b * ‖v‖^2 + c * ⟪w, v⟫) =
      (π / b) ^ (Fintype.card ι / 2 : ℂ) * cexp (c ^ 2 * ‖w‖^2 / (4 * b)) 
Arguments: {b : ℂ} {ι : Type u_2} [Fintype ι] (hb : 0 < b.re) (c : ℂ) (w : EuclideanSpace ℝ ι)
Goal: ∫ (v : EuclideanSpace ℝ ι), cexp (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫_ℝ) =
  (↑π / b) ^ (↑(Fintype.card ι) / 2) * cexp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))
-- GaussianFourier.integral_cexp_neg_mul_sq_norm_add
theorem integral_cexp_neg_mul_sq_norm_add
    (hb : 0 < b.re) (c : ℂ) (w : V) :
    ∫ v : V, cexp (- b * ‖v‖^2 + c * ⟪w, v⟫) =
      (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) * cexp (c ^ 2 * ‖w‖^2 / (4 * b)) 
Arguments: {b : ℂ} {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (hb : 0 < b.re) (c : ℂ) (w : V)
Goal: ∫ (v : V), cexp (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫_ℝ) =
  (↑π / b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2) * cexp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))
-- GaussianFourier.integral_cexp_neg_mul_sq_norm
theorem integral_cexp_neg_mul_sq_norm (hb : 0 < b.re) :
    ∫ v : V, cexp (- b * ‖v‖^2) = (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) 
Arguments: {b : ℂ} {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (hb : 0 < b.re)
Goal: ∫ (v : V), cexp (-b * ↑‖v‖ ^ 2) = (↑π / b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2)
-- GaussianFourier.integral_rexp_neg_mul_sq_norm
theorem integral_rexp_neg_mul_sq_norm {b : ℝ} (hb : 0 < b) :
    ∫ v : V, rexp (- b * ‖v‖^2) = (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℝ) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {b : ℝ} (hb : 0 < b)
Goal: ∫ (v : V), rexp (-b * ‖v‖ ^ 2) = (π / b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2)
-- fourierIntegral_gaussian_innerProductSpace'
theorem _root_.fourierIntegral_gaussian_innerProductSpace' (hb : 0 < b.re) (x w : V) :
    𝓕 (fun v ↦ cexp (- b * ‖v‖^2 + 2 * π * Complex.I * ⟪x, v⟫)) w =
      (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * ‖x - w‖ ^ 2 / b) 
Arguments: {b : ℂ} {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (hb : 0 < b.re) (x w : V)
Goal: 𝓕 (fun v => cexp (-b * ↑‖v‖ ^ 2 + 2 * ↑π * Complex.I * ↑⟪x, v⟫_ℝ)) w =
  (↑π / b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑‖x - w‖ ^ 2 / b)
-- fourierIntegral_gaussian_innerProductSpace
theorem _root_.fourierIntegral_gaussian_innerProductSpace (hb : 0 < b.re) (w : V) :
    𝓕 (fun v ↦ cexp (- b * ‖v‖^2)) w =
      (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * ‖w‖^2 / b) 
Arguments: {b : ℂ} {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (hb : 0 < b.re) (w : V)
Goal: 𝓕 (fun v => cexp (-b * ↑‖v‖ ^ 2)) w = (↑π / b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑‖w‖ ^ 2 / b)
-- DualNumber.exp_eps
@[simp]
theorem exp_eps : exp 𝕜 (eps : DualNumber R) = 1 + eps 
Arguments: (𝕜 : Type u_1) {R : Type u_2} [Field 𝕜] [CharZero 𝕜] [CommRing R] [Algebra 𝕜 R] [UniformSpace R] [TopologicalRing R] [T2Space R]
Goal: NormedSpace.exp 𝕜 DualNumber.eps = 1 + DualNumber.eps
-- DualNumber.exp_smul_eps
@[simp]
theorem exp_smul_eps (r : R) : exp 𝕜 (r • eps : DualNumber R) = 1 + r • eps 
Arguments: (𝕜 : Type u_1) {R : Type u_2} [Field 𝕜] [CharZero 𝕜] [CommRing R] [Algebra 𝕜 R] [UniformSpace R] [TopologicalRing R] [T2Space R] (r : R)
Goal: NormedSpace.exp 𝕜 (r • DualNumber.eps) = 1 + r • DualNumber.eps
-- Matrix.exp_diagonal
theorem exp_diagonal (v : m → 𝔸) : exp 𝕂 (diagonal v) = diagonal (exp 𝕂 v) 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] (v : m → 𝔸)
Goal: NormedSpace.exp 𝕂 (Matrix.diagonal v) = Matrix.diagonal (NormedSpace.exp 𝕂 v)
-- Matrix.exp_blockDiagonal
theorem exp_blockDiagonal (v : m → Matrix n n 𝔸) :
    exp 𝕂 (blockDiagonal v) = blockDiagonal (exp 𝕂 v) 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {n : Type u_3} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] (v : m → Matrix n n 𝔸)
Goal: NormedSpace.exp 𝕂 (Matrix.blockDiagonal v) = Matrix.blockDiagonal (NormedSpace.exp 𝕂 v)
-- Matrix.exp_blockDiagonal'
theorem exp_blockDiagonal' (v : ∀ i, Matrix (n' i) (n' i) 𝔸) :
    exp 𝕂 (blockDiagonal' v) = blockDiagonal' (exp 𝕂 v) 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {n' : m → Type u_5} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [(i : m) → Fintype (n' i)] [(i : m) → DecidableEq (n' i)] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] (v : (i : m) → Matrix (n' i) (n' i) 𝔸)
Goal: NormedSpace.exp 𝕂 (Matrix.blockDiagonal' v) = Matrix.blockDiagonal' (NormedSpace.exp 𝕂 v)
-- Matrix.exp_conjTranspose
theorem exp_conjTranspose [StarRing 𝔸] [ContinuousStar 𝔸] (A : Matrix m m 𝔸) :
    exp 𝕂 Aᴴ = (exp 𝕂 A)ᴴ 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] (A : Matrix m m 𝔸)
Goal: NormedSpace.exp 𝕂 Aᴴ = (NormedSpace.exp 𝕂 A)ᴴ
-- Matrix.IsHermitian.exp
theorem IsHermitian.exp [StarRing 𝔸] [ContinuousStar 𝔸] {A : Matrix m m 𝔸} (h : A.IsHermitian) :
    (exp 𝕂 A).IsHermitian 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] {A : Matrix m m 𝔸} (h : A.IsHermitian)
Goal: (NormedSpace.exp 𝕂 A).IsHermitian
-- Matrix.exp_transpose
theorem exp_transpose (A : Matrix m m 𝔸) : exp 𝕂 Aᵀ = (exp 𝕂 A)ᵀ 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [Field 𝕂] [CommRing 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] (A : Matrix m m 𝔸)
Goal: NormedSpace.exp 𝕂 Aᵀ = (NormedSpace.exp 𝕂 A)ᵀ
-- Matrix.IsSymm.exp
theorem IsSymm.exp {A : Matrix m m 𝔸} (h : A.IsSymm) : (exp 𝕂 A).IsSymm 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [Field 𝕂] [CommRing 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] {A : Matrix m m 𝔸} (h : A.IsSymm)
Goal: (NormedSpace.exp 𝕂 A).IsSymm
-- Matrix.exp_add_of_commute
nonrec theorem exp_add_of_commute (A B : Matrix m m 𝔸) (h : Commute A B) :
    exp 𝕂 (A + B) = exp 𝕂 A * exp 𝕂 B 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [RCLike 𝕂] [Fintype m] [DecidableEq m] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (A B : Matrix m m 𝔸) (h : Commute A B)
Goal: NormedSpace.exp 𝕂 (A + B) = NormedSpace.exp 𝕂 A * NormedSpace.exp 𝕂 B
-- Matrix.exp_sum_of_commute
nonrec theorem exp_sum_of_commute {ι} (s : Finset ι) (f : ι → Matrix m m 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) {m : Type u_2} {𝔸 : Type u_6} [RCLike 𝕂] [Fintype m] [DecidableEq m] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] {ι : Type u_7} (s : Finset ι) (f : ι → Matrix m m 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)) ⋯
-- Matrix.exp_nsmul
nonrec theorem exp_nsmul (n : ℕ) (A : Matrix m m 𝔸) : exp 𝕂 (n • A) = exp 𝕂 A ^ n 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [RCLike 𝕂] [Fintype m] [DecidableEq m] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (n : ℕ) (A : Matrix m m 𝔸)
Goal: NormedSpace.exp 𝕂 (n • A) = NormedSpace.exp 𝕂 A ^ n
-- Matrix.isUnit_exp
nonrec theorem isUnit_exp (A : Matrix m m 𝔸) : IsUnit (exp 𝕂 A) 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [RCLike 𝕂] [Fintype m] [DecidableEq m] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (A : Matrix m m 𝔸)
Goal: IsUnit (NormedSpace.exp 𝕂 A)
-- Matrix.exp_units_conj
nonrec theorem exp_units_conj (U : (Matrix m m 𝔸)ˣ) (A : Matrix m m 𝔸) :
    exp 𝕂 (U.val * A * (U⁻¹).val) = U.val * exp 𝕂 A * (U⁻¹).val 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [RCLike 𝕂] [Fintype m] [DecidableEq m] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (U : (Matrix m m 𝔸)ˣ) (A : Matrix m m 𝔸)
Goal: NormedSpace.exp 𝕂 (↑U * A * ↑U⁻¹) = ↑U * NormedSpace.exp 𝕂 A * ↑U⁻¹
-- Matrix.exp_units_conj'
theorem exp_units_conj' (U : (Matrix m m 𝔸)ˣ) (A : Matrix m m 𝔸) :
    exp 𝕂 ((U⁻¹).val * A * U.val) = (U⁻¹).val * exp 𝕂 A * U.val 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [RCLike 𝕂] [Fintype m] [DecidableEq m] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (U : (Matrix m m 𝔸)ˣ) (A : Matrix m m 𝔸)
Goal: NormedSpace.exp 𝕂 (↑U⁻¹ * A * ↑U) = ↑U⁻¹ * NormedSpace.exp 𝕂 A * ↑U
-- Matrix.exp_neg
theorem exp_neg (A : Matrix m m 𝔸) : exp 𝕂 (-A) = (exp 𝕂 A)⁻¹ 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [RCLike 𝕂] [Fintype m] [DecidableEq m] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (A : Matrix m m 𝔸)
Goal: NormedSpace.exp 𝕂 (-A) = (NormedSpace.exp 𝕂 A)⁻¹
-- Matrix.exp_zsmul
theorem exp_zsmul (z : ℤ) (A : Matrix m m 𝔸) : exp 𝕂 (z • A) = exp 𝕂 A ^ z 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [RCLike 𝕂] [Fintype m] [DecidableEq m] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (z : ℤ) (A : Matrix m m 𝔸)
Goal: NormedSpace.exp 𝕂 (z • A) = NormedSpace.exp 𝕂 A ^ z
-- Matrix.exp_conj
theorem exp_conj (U : Matrix m m 𝔸) (A : Matrix m m 𝔸) (hy : IsUnit U) :
    exp 𝕂 (U * A * U⁻¹) = U * exp 𝕂 A * U⁻¹ 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [RCLike 𝕂] [Fintype m] [DecidableEq m] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (U A : Matrix m m 𝔸) (hy : IsUnit U)
Goal: NormedSpace.exp 𝕂 (U * A * U⁻¹) = U * NormedSpace.exp 𝕂 A * U⁻¹
-- Matrix.exp_conj'
theorem exp_conj' (U : Matrix m m 𝔸) (A : Matrix m m 𝔸) (hy : IsUnit U) :
    exp 𝕂 (U⁻¹ * A * U) = U⁻¹ * exp 𝕂 A * U 
Arguments: (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [RCLike 𝕂] [Fintype m] [DecidableEq m] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (U A : Matrix m m 𝔸) (hy : IsUnit U)
Goal: NormedSpace.exp 𝕂 (U⁻¹ * A * U) = U⁻¹ * NormedSpace.exp 𝕂 A * U
-- NormedSpace.expSeries_apply_eq
theorem expSeries_apply_eq (x : 𝔸) (n : ℕ) :
    (expSeries 𝕂 𝔸 n fun _ => x) = (n !⁻¹ : 𝕂) • x ^ n 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] (x : 𝔸) (n : ℕ)
Goal: ((NormedSpace.expSeries 𝕂 𝔸 n) fun x_1 => x) = (↑n.factorial)⁻¹ • x ^ n
-- NormedSpace.expSeries_apply_eq'
theorem expSeries_apply_eq' (x : 𝔸) :
    (fun n => expSeries 𝕂 𝔸 n fun _ => x) = fun n => (n !⁻¹ : 𝕂) • x ^ n 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] (x : 𝔸)
Goal: (fun n => (NormedSpace.expSeries 𝕂 𝔸 n) fun x_1 => x) = fun n => (↑n.factorial)⁻¹ • x ^ n
-- NormedSpace.expSeries_sum_eq
theorem expSeries_sum_eq (x : 𝔸) : (expSeries 𝕂 𝔸).sum x = ∑' n : ℕ, (n !⁻¹ : 𝕂) • x ^ n 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] (x : 𝔸)
Goal: (NormedSpace.expSeries 𝕂 𝔸).sum x = ∑' (n : ℕ), (↑n.factorial)⁻¹ • x ^ n
-- NormedSpace.exp_eq_tsum
theorem exp_eq_tsum : exp 𝕂 = fun x : 𝔸 => ∑' n : ℕ, (n !⁻¹ : 𝕂) • x ^ n 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸]
Goal: NormedSpace.exp 𝕂 = fun x => ∑' (n : ℕ), (↑n.factorial)⁻¹ • x ^ n
-- NormedSpace.expSeries_apply_zero
theorem expSeries_apply_zero (n : ℕ) :
    (expSeries 𝕂 𝔸 n fun _ => (0 : 𝔸)) = Pi.single (f := fun _ => 𝔸) 0 1 n 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] (n : ℕ)
Goal: ((NormedSpace.expSeries 𝕂 𝔸 n) fun x => 0) = Pi.single 0 1 n
-- NormedSpace.exp_zero
@[simp]
theorem exp_zero : exp 𝕂 (0 : 𝔸) = 1 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸]
Goal: NormedSpace.exp 𝕂 0 = 1
-- NormedSpace.exp_op
@[simp]
theorem exp_op [T2Space 𝔸] (x : 𝔸) : exp 𝕂 (MulOpposite.op x) = MulOpposite.op (exp 𝕂 x) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [T2Space 𝔸] (x : 𝔸)
Goal: NormedSpace.exp 𝕂 (MulOpposite.op x) = MulOpposite.op (NormedSpace.exp 𝕂 x)
-- NormedSpace.exp_unop
@[simp]
theorem exp_unop [T2Space 𝔸] (x : 𝔸ᵐᵒᵖ) :
    exp 𝕂 (MulOpposite.unop x) = MulOpposite.unop (exp 𝕂 x) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [T2Space 𝔸] (x : 𝔸ᵐᵒᵖ)
Goal: NormedSpace.exp 𝕂 (MulOpposite.unop x) = MulOpposite.unop (NormedSpace.exp 𝕂 x)
-- NormedSpace.star_exp
theorem star_exp [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] (x : 𝔸) :
    star (exp 𝕂 x) = exp 𝕂 (star x) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] (x : 𝔸)
Goal: star (NormedSpace.exp 𝕂 x) = NormedSpace.exp 𝕂 (star x)
-- IsSelfAdjoint.exp
theorem _root_.IsSelfAdjoint.exp [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] {x : 𝔸}
    (h : IsSelfAdjoint x) : IsSelfAdjoint (exp 𝕂 x) 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] {x : 𝔸} (h : IsSelfAdjoint x)
Goal: IsSelfAdjoint (NormedSpace.exp 𝕂 x)
-- Commute.exp_right
theorem _root_.Commute.exp_right [T2Space 𝔸] {x y : 𝔸} (h : Commute x y) :
    Commute x (exp 𝕂 y) 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [T2Space 𝔸] {x y : 𝔸} (h : Commute x y)
Goal: Commute x (NormedSpace.exp 𝕂 y)
-- Commute.exp_left
theorem _root_.Commute.exp_left [T2Space 𝔸] {x y : 𝔸} (h : Commute x y) : Commute (exp 𝕂 x) y 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [T2Space 𝔸] {x y : 𝔸} (h : Commute x y)
Goal: Commute (NormedSpace.exp 𝕂 x) y
-- Commute.exp
theorem _root_.Commute.exp [T2Space 𝔸] {x y : 𝔸} (h : Commute x y) : Commute (exp 𝕂 x) (exp 𝕂 y) 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [Field 𝕂] [Ring 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [T2Space 𝔸] {x y : 𝔸} (h : Commute x y)
Goal: Commute (NormedSpace.exp 𝕂 x) (NormedSpace.exp 𝕂 y)
-- NormedSpace.expSeries_apply_eq_div
theorem expSeries_apply_eq_div (x : 𝔸) (n : ℕ) : (expSeries 𝕂 𝔸 n fun _ => x) = x ^ n / n ! 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [DivisionRing 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] (x : 𝔸) (n : ℕ)
Goal: ((NormedSpace.expSeries 𝕂 𝔸 n) fun x_1 => x) = x ^ n / ↑n.factorial
-- NormedSpace.expSeries_apply_eq_div'
theorem expSeries_apply_eq_div' (x : 𝔸) :
    (fun n => expSeries 𝕂 𝔸 n fun _ => x) = fun n => x ^ n / n ! 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [DivisionRing 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] (x : 𝔸)
Goal: (fun n => (NormedSpace.expSeries 𝕂 𝔸 n) fun x_1 => x) = fun n => x ^ n / ↑n.factorial
-- NormedSpace.expSeries_sum_eq_div
theorem expSeries_sum_eq_div (x : 𝔸) : (expSeries 𝕂 𝔸).sum x = ∑' n : ℕ, x ^ n / n ! 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [DivisionRing 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] (x : 𝔸)
Goal: (NormedSpace.expSeries 𝕂 𝔸).sum x = ∑' (n : ℕ), x ^ n / ↑n.factorial
-- NormedSpace.exp_eq_tsum_div
theorem exp_eq_tsum_div : exp 𝕂 = fun x : 𝔸 => ∑' n : ℕ, x ^ n / n ! 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [Field 𝕂] [DivisionRing 𝔸] [Algebra 𝕂 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸]
Goal: NormedSpace.exp 𝕂 = fun x => ∑' (n : ℕ), x ^ n / ↑n.factorial
-- NormedSpace.norm_expSeries_summable_of_mem_ball
theorem norm_expSeries_summable_of_mem_ball (x : 𝔸)
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    Summable fun n => ‖expSeries 𝕂 𝔸 n fun _ => x‖ 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] (x : 𝔸) (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: Summable fun n => ‖(NormedSpace.expSeries 𝕂 𝔸 n) fun x_1 => x‖
-- NormedSpace.norm_expSeries_summable_of_mem_ball'
theorem norm_expSeries_summable_of_mem_ball' (x : 𝔸)
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    Summable fun n => ‖(n !⁻¹ : 𝕂) • x ^ n‖ 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] (x : 𝔸) (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: Summable fun n => ‖(↑n.factorial)⁻¹ • x ^ n‖
-- NormedSpace.expSeries_summable_of_mem_ball
theorem expSeries_summable_of_mem_ball (x : 𝔸)
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    Summable fun n => expSeries 𝕂 𝔸 n fun _ => x 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: Summable fun n => (NormedSpace.expSeries 𝕂 𝔸 n) fun x_1 => x
-- NormedSpace.expSeries_summable_of_mem_ball'
theorem expSeries_summable_of_mem_ball' (x : 𝔸)
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    Summable fun n => (n !⁻¹ : 𝕂) • x ^ n 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: Summable fun n => (↑n.factorial)⁻¹ • x ^ n
-- NormedSpace.expSeries_hasSum_exp_of_mem_ball
theorem expSeries_hasSum_exp_of_mem_ball (x : 𝔸)
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasSum (fun n => expSeries 𝕂 𝔸 n fun _ => x) (exp 𝕂 x) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasSum (fun n => (NormedSpace.expSeries 𝕂 𝔸 n) fun x_1 => x) (NormedSpace.exp 𝕂 x)
-- NormedSpace.expSeries_hasSum_exp_of_mem_ball'
theorem expSeries_hasSum_exp_of_mem_ball' (x : 𝔸)
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasSum (fun n => (n !⁻¹ : 𝕂) • x ^ n) (exp 𝕂 x) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasSum (fun n => (↑n.factorial)⁻¹ • x ^ n) (NormedSpace.exp 𝕂 x)
-- NormedSpace.hasFPowerSeriesOnBall_exp_of_radius_pos
theorem hasFPowerSeriesOnBall_exp_of_radius_pos (h : 0 < (expSeries 𝕂 𝔸).radius) :
    HasFPowerSeriesOnBall (exp 𝕂) (expSeries 𝕂 𝔸) 0 (expSeries 𝕂 𝔸).radius 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (h : 0 < (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasFPowerSeriesOnBall (NormedSpace.exp 𝕂) (NormedSpace.expSeries 𝕂 𝔸) 0 (NormedSpace.expSeries 𝕂 𝔸).radius
-- NormedSpace.hasFPowerSeriesAt_exp_zero_of_radius_pos
theorem hasFPowerSeriesAt_exp_zero_of_radius_pos (h : 0 < (expSeries 𝕂 𝔸).radius) :
    HasFPowerSeriesAt (exp 𝕂) (expSeries 𝕂 𝔸) 0 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (h : 0 < (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: HasFPowerSeriesAt (NormedSpace.exp 𝕂) (NormedSpace.expSeries 𝕂 𝔸) 0
-- NormedSpace.continuousOn_exp
theorem continuousOn_exp : ContinuousOn (exp 𝕂 : 𝔸 → 𝔸) (EMetric.ball 0 (expSeries 𝕂 𝔸).radius) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸]
Goal: ContinuousOn (NormedSpace.exp 𝕂) (EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
-- NormedSpace.analyticAt_exp_of_mem_ball
theorem analyticAt_exp_of_mem_ball (x : 𝔸) (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    AnalyticAt 𝕂 (exp 𝕂) x 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (x : 𝔸) (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: AnalyticAt 𝕂 (NormedSpace.exp 𝕂) x
-- NormedSpace.exp_add_of_commute_of_mem_ball
theorem exp_add_of_commute_of_mem_ball [CharZero 𝕂] {x y : 𝔸} (hxy : Commute x y)
    (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] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] [CharZero 𝕂] {x y : 𝔸} (hxy : Commute x y) (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.isUnit_exp_of_mem_ball
theorem isUnit_exp_of_mem_ball [CharZero 𝕂] {x : 𝔸}
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) : IsUnit (exp 𝕂 x) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] [CharZero 𝕂] {x : 𝔸} (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: IsUnit (NormedSpace.exp 𝕂 x)
-- NormedSpace.invOf_exp_of_mem_ball
theorem invOf_exp_of_mem_ball [CharZero 𝕂] {x : 𝔸}
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) [Invertible (exp 𝕂 x)] :
    ⅟ (exp 𝕂 x) = exp 𝕂 (-x) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] [CharZero 𝕂] {x : 𝔸} (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius) [Invertible (NormedSpace.exp 𝕂 x)]
Goal: ⅟(NormedSpace.exp 𝕂 x) = NormedSpace.exp 𝕂 (-x)
-- NormedSpace.map_exp_of_mem_ball
theorem map_exp_of_mem_ball {F} [FunLike F 𝔸 𝔹] [RingHomClass F 𝔸 𝔹] (f : F) (hf : Continuous f)
    (x : 𝔸) (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    f (exp 𝕂 x) = exp 𝕂 (f x) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} {𝔹 : Type u_3} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedRing 𝔹] [NormedAlgebra 𝕂 𝔸] [NormedAlgebra 𝕂 𝔹] [CompleteSpace 𝔸] {F : Type u_4} [FunLike F 𝔸 𝔹] [RingHomClass F 𝔸 𝔹] (f : F) (hf : Continuous ⇑f) (x : 𝔸) (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: f (NormedSpace.exp 𝕂 x) = NormedSpace.exp 𝕂 (f x)
-- NormedSpace.algebraMap_exp_comm_of_mem_ball
theorem algebraMap_exp_comm_of_mem_ball [CompleteSpace 𝕂] (x : 𝕂)
    (hx : x ∈ EMetric.ball (0 : 𝕂) (expSeries 𝕂 𝕂).radius) :
    algebraMap 𝕂 𝔸 (exp 𝕂 x) = exp 𝕂 (algebraMap 𝕂 𝔸 x) 
Arguments: {𝕂 : Type u_1} {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝕂] (x : 𝕂) (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝕂).radius)
Goal: (algebraMap 𝕂 𝔸) (NormedSpace.exp 𝕂 x) = NormedSpace.exp 𝕂 ((algebraMap 𝕂 𝔸) x)
-- NormedSpace.norm_expSeries_div_summable_of_mem_ball
theorem norm_expSeries_div_summable_of_mem_ball (x : 𝔸)
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    Summable fun n => ‖x ^ n / (n ! : 𝔸)‖ 
Arguments: (𝕂 : Type u_1) {𝔸 : Type u_2} [NontriviallyNormedField 𝕂] [NormedDivisionRing 𝔸] [NormedAlgebra 𝕂 𝔸] (x : 𝔸) (hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius)
Goal: Summable fun n => ‖x ^ n / ↑n.factorial‖