ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- FormalMultilinearSeries.apply_order_ne_zero'
theorem apply_order_ne_zero' (hp : p.order ≠ 0) : p p.order ≠ 0 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} (hp : p.order ≠ 0)
Goal: p p.order ≠ 0
-- FormalMultilinearSeries.apply_eq_zero_of_lt_order
theorem apply_eq_zero_of_lt_order (hp : n < p.order) : p n = 0 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] {n : ℕ} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} (hp : n < p.order)
Goal: p n = 0
-- FormalMultilinearSeries.mkPiRing_coeff_eq
theorem mkPiRing_coeff_eq (p : FormalMultilinearSeries 𝕜 𝕜 E) (n : ℕ) :
    ContinuousMultilinearMap.mkPiRing 𝕜 (Fin n) (p.coeff n) = p n 
Arguments: {𝕜 : Type u} {E : Type v} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (p : FormalMultilinearSeries 𝕜 𝕜 E) (n : ℕ)
Goal: ContinuousMultilinearMap.mkPiRing 𝕜 (Fin n) (p.coeff n) = p n
-- FormalMultilinearSeries.apply_eq_prod_smul_coeff
@[simp]
theorem apply_eq_prod_smul_coeff : p n y = (∏ i, y i) • p.coeff n 
Arguments: {𝕜 : Type u} {E : Type v} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {n : ℕ} {y : Fin n → 𝕜}
Goal: (p n) y = (∏ i : Fin n, y i) • p.coeff n
-- FormalMultilinearSeries.coeff_eq_zero
theorem coeff_eq_zero : p.coeff n = 0 ↔ p n = 0 
Arguments: {𝕜 : Type u} {E : Type v} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {n : ℕ}
Goal: p.coeff n = 0 ↔ p n = 0
-- FormalMultilinearSeries.apply_eq_pow_smul_coeff
theorem apply_eq_pow_smul_coeff : (p n fun _ => z) = z ^ n • p.coeff n 
Arguments: {𝕜 : Type u} {E : Type v} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {n : ℕ} {z : 𝕜}
Goal: ((p n) fun x => z) = z ^ n • p.coeff n
-- FormalMultilinearSeries.norm_apply_eq_norm_coef
@[simp]
theorem norm_apply_eq_norm_coef : ‖p n‖ = ‖coeff p n‖ 
Arguments: {𝕜 : Type u} {E : Type v} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {n : ℕ}
Goal: ‖p n‖ = ‖p.coeff n‖
-- FormalMultilinearSeries.coeff_fslope
@[simp]
theorem coeff_fslope : p.fslope.coeff n = p.coeff (n + 1) 
Arguments: {𝕜 : Type u} {E : Type v} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {n : ℕ}
Goal: p.fslope.coeff n = p.coeff (n + 1)
-- FormalMultilinearSeries.coeff_iterate_fslope
@[simp]
theorem coeff_iterate_fslope (k n : ℕ) : (fslope^[k] p).coeff n = p.coeff (n + k) 
Arguments: {𝕜 : Type u} {E : Type v} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} (k n : ℕ)
Goal: (FormalMultilinearSeries.fslope^[k] p).coeff n = p.coeff (n + k)
-- constFormalMultilinearSeries_apply
@[simp]
theorem constFormalMultilinearSeries_apply [NontriviallyNormedField 𝕜] [NormedAddCommGroup E]
    [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {c : F} {n : ℕ} (hn : n ≠ 0) :
    constFormalMultilinearSeries 𝕜 E c n = 0 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {c : F} {n : ℕ} (hn : n ≠ 0)
Goal: constFormalMultilinearSeries 𝕜 E c n = 0
-- constFormalMultilinearSeries_zero
@[simp]
lemma constFormalMultilinearSeries_zero [NontriviallyNormedField 𝕜] [NormedAddCommGroup E ]
    [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] :
    constFormalMultilinearSeries 𝕜 E (0 : F) = 0 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
Goal: constFormalMultilinearSeries 𝕜 E 0 = 0
-- ContinuousLinearMap.fpowerSeries_apply_zero
theorem fpowerSeries_apply_zero (f : E →L[𝕜] F) (x : E) :
    f.fpowerSeries x 0 = ContinuousMultilinearMap.curry0 𝕜 _ (f x) 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E →L[𝕜] F) (x : E)
Goal: f.fpowerSeries x 0 = ContinuousMultilinearMap.curry0 𝕜 E (f x)
-- ContinuousLinearMap.fpowerSeries_apply_one
theorem fpowerSeries_apply_one (f : E →L[𝕜] F) (x : E) :
    f.fpowerSeries x 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm f 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E →L[𝕜] F) (x : E)
Goal: f.fpowerSeries x 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm f
-- ContinuousLinearMap.fpowerSeries_apply_add_two
theorem fpowerSeries_apply_add_two (f : E →L[𝕜] F) (x : E) (n : ℕ) : f.fpowerSeries x (n + 2) = 0 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E →L[𝕜] F) (x : E) (n : ℕ)
Goal: f.fpowerSeries x (n + 2) = 0
-- Complex.exists_root
theorem exists_root {f : ℂ[X]} (hf : 0 < degree f) : ∃ z : ℂ, IsRoot f z 
Arguments: {f : ℂ[X]} (hf : 0 < f.degree)
Goal: ∃ z, f.IsRoot z
-- Polynomial.Gal.splits_ℚ_ℂ
theorem splits_ℚ_ℂ {p : ℚ[X]} : Fact (p.Splits (algebraMap ℚ ℂ)) 
Arguments: {p : ℚ[X]}
Goal: Fact (Polynomial.Splits (algebraMap ℚ ℂ) p)
-- Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv
theorem card_complex_roots_eq_card_real_add_card_not_gal_inv (p : ℚ[X]) :
    (p.rootSet ℂ).toFinset.card =
      (p.rootSet ℝ).toFinset.card +
        (galActionHom p ℂ (restrict p ℂ
        (AlgEquiv.restrictScalars ℚ Complex.conjAe))).support.card 
Arguments: (p : ℚ[X])
Goal: (p.rootSet ℂ).toFinset.card =
  (p.rootSet ℝ).toFinset.card +
    ((Polynomial.Gal.galActionHom p ℂ)
          ((Polynomial.Gal.restrict p ℂ) (AlgEquiv.restrictScalars ℚ Complex.conjAe))).support.card
-- Polynomial.Gal.galActionHom_bijective_of_prime_degree
theorem galActionHom_bijective_of_prime_degree {p : ℚ[X]} (p_irr : Irreducible p)
    (p_deg : p.natDegree.Prime)
    (p_roots : Fintype.card (p.rootSet ℂ) = Fintype.card (p.rootSet ℝ) + 2) :
    Function.Bijective (galActionHom p ℂ) 
Arguments: {p : ℚ[X]} (p_irr : Irreducible p) (p_deg : Nat.Prime p.natDegree) (p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2)
Goal: Function.Bijective ⇑(Polynomial.Gal.galActionHom p ℂ)
-- Polynomial.Gal.galActionHom_bijective_of_prime_degree'
theorem galActionHom_bijective_of_prime_degree' {p : ℚ[X]} (p_irr : Irreducible p)
    (p_deg : p.natDegree.Prime)
    (p_roots1 : Fintype.card (p.rootSet ℝ) + 1 ≤ Fintype.card (p.rootSet ℂ))
    (p_roots2 : Fintype.card (p.rootSet ℂ) ≤ Fintype.card (p.rootSet ℝ) + 3) :
    Function.Bijective (galActionHom p ℂ) 
Arguments: {p : ℚ[X]} (p_irr : Irreducible p) (p_deg : Nat.Prime p.natDegree) (p_roots1 : Fintype.card ↑(p.rootSet ℝ) + 1 ≤ Fintype.card ↑(p.rootSet ℂ)) (p_roots2 : Fintype.card ↑(p.rootSet ℂ) ≤ Fintype.card ↑(p.rootSet ℝ) + 3)
Goal: Function.Bijective ⇑(Polynomial.Gal.galActionHom p ℂ)
-- Polynomial.mul_star_dvd_of_aeval_eq_zero_im_ne_zero
lemma Polynomial.mul_star_dvd_of_aeval_eq_zero_im_ne_zero (p : ℝ[X]) {z : ℂ} (h0 : aeval z p = 0)
    (hz : z.im ≠ 0) : (X - C ((starRingEnd ℂ) z)) * (X - C z) ∣ map (algebraMap ℝ ℂ) p 
Arguments: (p : ℝ[X]) {z : ℂ} (h0 : (Polynomial.aeval z) p = 0) (hz : z.im ≠ 0)
Goal: (Polynomial.X - Polynomial.C ((starRingEnd ℂ) z)) * (Polynomial.X - Polynomial.C z) ∣ Polynomial.map (algebraMap ℝ ℂ) p
-- Polynomial.quadratic_dvd_of_aeval_eq_zero_im_ne_zero
lemma Polynomial.quadratic_dvd_of_aeval_eq_zero_im_ne_zero (p : ℝ[X]) {z : ℂ} (h0 : aeval z p = 0)
    (hz : z.im ≠ 0) : X ^ 2 - C (2 * z.re) * X + C (‖z‖ ^ 2) ∣ p 
Arguments: (p : ℝ[X]) {z : ℂ} (h0 : (Polynomial.aeval z) p = 0) (hz : z.im ≠ 0)
Goal: Polynomial.X ^ 2 - Polynomial.C (2 * z.re) * Polynomial.X + Polynomial.C (‖z‖ ^ 2) ∣ p
-- Irreducible.degree_le_two
lemma Irreducible.degree_le_two {p : ℝ[X]} (hp : Irreducible p) : degree p ≤ 2 
Arguments: {p : ℝ[X]} (hp : Irreducible p)
Goal: p.degree ≤ 2
-- Irreducible.natDegree_le_two
lemma Irreducible.natDegree_le_two {p : ℝ[X]} (hp : Irreducible p) : natDegree p ≤ 2 
Arguments: {p : ℝ[X]} (hp : Irreducible p)
Goal: p.natDegree ≤ 2
-- Complex.norm_eq_abs
@[simp]
theorem norm_eq_abs (z : ℂ) : ‖z‖ = abs z 
Arguments: (z : ℂ)
Goal: ‖z‖ = Complex.abs z
-- Complex.norm_I
lemma norm_I : ‖I‖ = 1 
Goal: ‖Complex.I‖ = 1
-- Complex.norm_exp_ofReal_mul_I
theorem norm_exp_ofReal_mul_I (t : ℝ) : ‖exp (t * I)‖ = 1 
Arguments: (t : ℝ)
Goal: ‖Complex.exp (↑t * Complex.I)‖ = 1
-- Complex.dist_eq
theorem dist_eq (z w : ℂ) : dist z w = abs (z - w) 
Arguments: (z w : ℂ)
Goal: dist z w = Complex.abs (z - w)
-- Complex.dist_eq_re_im
theorem dist_eq_re_im (z w : ℂ) : dist z w = √((z.re - w.re) ^ 2 + (z.im - w.im) ^ 2) 
Arguments: (z w : ℂ)
Goal: dist z w = √((z.re - w.re) ^ 2 + (z.im - w.im) ^ 2)
-- Complex.dist_mk
@[simp]
theorem dist_mk (x₁ y₁ x₂ y₂ : ℝ) :
    dist (mk x₁ y₁) (mk x₂ y₂) = √((x₁ - x₂) ^ 2 + (y₁ - y₂) ^ 2) 
Arguments: (x₁ y₁ x₂ y₂ : ℝ)
Goal: dist { re := x₁, im := y₁ } { re := x₂, im := y₂ } = √((x₁ - x₂) ^ 2 + (y₁ - y₂) ^ 2)
-- Complex.dist_of_re_eq
theorem dist_of_re_eq {z w : ℂ} (h : z.re = w.re) : dist z w = dist z.im w.im 
Arguments: {z w : ℂ} (h : z.re = w.re)
Goal: dist z w = dist z.im w.im
-- Complex.nndist_of_re_eq
theorem nndist_of_re_eq {z w : ℂ} (h : z.re = w.re) : nndist z w = nndist z.im w.im 
Arguments: {z w : ℂ} (h : z.re = w.re)
Goal: nndist z w = nndist z.im w.im
-- Complex.edist_of_re_eq
theorem edist_of_re_eq {z w : ℂ} (h : z.re = w.re) : edist z w = edist z.im w.im 
Arguments: {z w : ℂ} (h : z.re = w.re)
Goal: edist z w = edist z.im w.im
-- Complex.dist_of_im_eq
theorem dist_of_im_eq {z w : ℂ} (h : z.im = w.im) : dist z w = dist z.re w.re 
Arguments: {z w : ℂ} (h : z.im = w.im)
Goal: dist z w = dist z.re w.re
-- Complex.nndist_of_im_eq
theorem nndist_of_im_eq {z w : ℂ} (h : z.im = w.im) : nndist z w = nndist z.re w.re 
Arguments: {z w : ℂ} (h : z.im = w.im)
Goal: nndist z w = nndist z.re w.re
-- Complex.edist_of_im_eq
theorem edist_of_im_eq {z w : ℂ} (h : z.im = w.im) : edist z w = edist z.re w.re 
Arguments: {z w : ℂ} (h : z.im = w.im)
Goal: edist z w = edist z.re w.re
-- Complex.dist_conj_self
theorem dist_conj_self (z : ℂ) : dist (conj z) z = 2 * |z.im| 
Arguments: (z : ℂ)
Goal: dist ((starRingEnd ℂ) z) z = 2 * |z.im|
-- Complex.nndist_conj_self
theorem nndist_conj_self (z : ℂ) : nndist (conj z) z = 2 * Real.nnabs z.im 
Arguments: (z : ℂ)
Goal: nndist ((starRingEnd ℂ) z) z = 2 * Real.nnabs z.im
-- Complex.dist_self_conj
theorem dist_self_conj (z : ℂ) : dist z (conj z) = 2 * |z.im| 
Arguments: (z : ℂ)
Goal: dist z ((starRingEnd ℂ) z) = 2 * |z.im|
-- Complex.nndist_self_conj
theorem nndist_self_conj (z : ℂ) : nndist z (conj z) = 2 * Real.nnabs z.im 
Arguments: (z : ℂ)
Goal: nndist z ((starRingEnd ℂ) z) = 2 * Real.nnabs z.im
-- Complex.comap_abs_nhds_zero
@[simp 1100]
theorem comap_abs_nhds_zero : comap abs (𝓝 0) = 𝓝 0 
Goal: Filter.comap (⇑Complex.abs) (nhds 0) = nhds 0
-- Complex.norm_real
theorem norm_real (r : ℝ) : ‖(r : ℂ)‖ = ‖r‖ 
Arguments: (r : ℝ)
Goal: ‖↑r‖ = ‖r‖
-- Complex.norm_rat
@[simp 1100]
theorem norm_rat (r : ℚ) : ‖(r : ℂ)‖ = |(r : ℝ)| 
Arguments: (r : ℚ)
Goal: ‖↑r‖ = |↑r|
-- Complex.norm_nat
@[simp 1100]
theorem norm_nat (n : ℕ) : ‖(n : ℂ)‖ = n 
Arguments: (n : ℕ)
Goal: ‖↑n‖ = ↑n
-- Complex.norm_int
@[simp 1100]
lemma norm_int {n : ℤ} : ‖(n : ℂ)‖ = |(n : ℝ)| 
Arguments: {n : ℤ}
Goal: ‖↑n‖ = |↑n|
-- Complex.norm_int_of_nonneg
theorem norm_int_of_nonneg {n : ℤ} (hn : 0 ≤ n) : ‖(n : ℂ)‖ = n 
Arguments: {n : ℤ} (hn : 0 ≤ n)
Goal: ‖↑n‖ = ↑n
-- Complex.normSq_eq_norm_sq
lemma normSq_eq_norm_sq (z : ℂ) : normSq z = ‖z‖ ^ 2 
Arguments: (z : ℂ)
Goal: Complex.normSq z = ‖z‖ ^ 2
-- Complex.continuous_abs
@[continuity]
theorem continuous_abs : Continuous abs 
Goal: Continuous ⇑Complex.abs
-- Complex.continuous_normSq
@[continuity]
theorem continuous_normSq : Continuous normSq 
Goal: Continuous ⇑Complex.normSq
-- Complex.nnnorm_real
@[simp, norm_cast]
theorem nnnorm_real (r : ℝ) : ‖(r : ℂ)‖₊ = ‖r‖₊ 
Arguments: (r : ℝ)
Goal: ‖↑r‖₊ = ‖r‖₊
-- Complex.nnnorm_nat
@[simp, norm_cast]
theorem nnnorm_nat (n : ℕ) : ‖(n : ℂ)‖₊ = n 
Arguments: (n : ℕ)
Goal: ‖↑n‖₊ = ↑n
-- Complex.nnnorm_int
@[simp, norm_cast]
theorem nnnorm_int (n : ℤ) : ‖(n : ℂ)‖₊ = ‖n‖₊ 
Arguments: (n : ℤ)
Goal: ‖↑n‖₊ = ‖n‖₊
-- Complex.nnnorm_eq_one_of_pow_eq_one
theorem nnnorm_eq_one_of_pow_eq_one {ζ : ℂ} {n : ℕ} (h : ζ ^ n = 1) (hn : n ≠ 0) : ‖ζ‖₊ = 1 
Arguments: {ζ : ℂ} {n : ℕ} (h : ζ ^ n = 1) (hn : n ≠ 0)
Goal: ‖ζ‖₊ = 1
-- Complex.norm_eq_one_of_pow_eq_one
theorem norm_eq_one_of_pow_eq_one {ζ : ℂ} {n : ℕ} (h : ζ ^ n = 1) (hn : n ≠ 0) : ‖ζ‖ = 1 
Arguments: {ζ : ℂ} {n : ℕ} (h : ζ ^ n = 1) (hn : n ≠ 0)
Goal: ‖ζ‖ = 1
-- Complex.le_of_eq_sum_of_eq_sum_norm
lemma le_of_eq_sum_of_eq_sum_norm {ι : Type*} {a b : ℝ} (f : ι → ℂ) (s : Finset ι) (ha₀ : 0 ≤ a)
    (ha : a = ∑ i ∈ s, f i) (hb : b = ∑ i ∈ s, (‖f i‖ : ℂ)) : a ≤ b 
Arguments: {ι : Type u_2} {a b : ℝ} (f : ι → ℂ) (s : Finset ι) (ha₀ : 0 ≤ a) (ha : ↑a = ∑ i ∈ s, f i) (hb : ↑b = ∑ i ∈ s, ↑‖f i‖)
Goal: a ≤ b
-- Complex.equivRealProd_apply_le
theorem equivRealProd_apply_le (z : ℂ) : ‖equivRealProd z‖ ≤ abs z 
Arguments: (z : ℂ)
Goal: ‖Complex.equivRealProd z‖ ≤ Complex.abs z
-- Complex.equivRealProd_apply_le'
theorem equivRealProd_apply_le' (z : ℂ) : ‖equivRealProd z‖ ≤ 1 * abs z 
Arguments: (z : ℂ)
Goal: ‖Complex.equivRealProd z‖ ≤ 1 * Complex.abs z
-- Complex.lipschitz_equivRealProd
theorem lipschitz_equivRealProd : LipschitzWith 1 equivRealProd 
Goal: LipschitzWith 1 ⇑Complex.equivRealProd
-- Complex.antilipschitz_equivRealProd
theorem antilipschitz_equivRealProd : AntilipschitzWith (NNReal.sqrt 2) equivRealProd 
Goal: AntilipschitzWith (NNReal.sqrt 2) ⇑Complex.equivRealProd
-- Complex.uniformEmbedding_equivRealProd
theorem uniformEmbedding_equivRealProd : UniformEmbedding equivRealProd 
Goal: UniformEmbedding ⇑Complex.equivRealProd
-- Complex.equivRealProdCLM_symm_apply_im
@[simps! (config := { simpRhs := true }) apply symm_apply_re symm_apply_im]
def equivRealProdCLM : ℂ ≃L[ℝ] ℝ × ℝ 
Goal: ∀ (a : ℝ × ℝ), (Complex.equivRealProdCLM.symm a).im = a.2
-- Complex.equivRealProdCLM_apply
@[simps! (config := { simpRhs := true }) apply symm_apply_re symm_apply_im]
def equivRealProdCLM : ℂ ≃L[ℝ] ℝ × ℝ 
Goal: ∀ (a : ℂ), Complex.equivRealProdCLM a = (a.re, a.im)
-- Complex.equivRealProdCLM_symm_apply_re
@[simps! (config := { simpRhs := true }) apply symm_apply_re symm_apply_im]
def equivRealProdCLM : ℂ ≃L[ℝ] ℝ × ℝ 
Goal: ∀ (a : ℝ × ℝ), (Complex.equivRealProdCLM.symm a).re = a.1
-- Complex.equivRealProdCLM_symm_apply
theorem equivRealProdCLM_symm_apply (p : ℝ × ℝ) :
    Complex.equivRealProdCLM.symm p = p.1 + p.2 * Complex.I 
Arguments: (p : ℝ × ℝ)
Goal: Complex.equivRealProdCLM.symm p = ↑p.1 + ↑p.2 * Complex.I
-- Complex.tendsto_abs_cocompact_atTop
theorem tendsto_abs_cocompact_atTop : Tendsto abs (cocompact ℂ) atTop 
Goal: Filter.Tendsto (⇑Complex.abs) (Filter.cocompact ℂ) Filter.atTop
-- Complex.tendsto_normSq_cocompact_atTop
theorem tendsto_normSq_cocompact_atTop : Tendsto normSq (cocompact ℂ) atTop 
Goal: Filter.Tendsto (⇑Complex.normSq) (Filter.cocompact ℂ) Filter.atTop
-- Complex.continuous_re
@[continuity, fun_prop]
theorem continuous_re : Continuous re 
Goal: Continuous Complex.re
-- Complex.reCLM_coe
@[simp]
theorem reCLM_coe : (reCLM : ℂ →ₗ[ℝ] ℝ) = reLm 
Goal: ↑Complex.reCLM = Complex.reLm
-- Complex.reCLM_apply
@[simp]
theorem reCLM_apply (z : ℂ) : (reCLM : ℂ → ℝ) z = z.re 
Arguments: (z : ℂ)
Goal: Complex.reCLM z = z.re
-- Complex.continuous_im
@[continuity, fun_prop]
theorem continuous_im : Continuous im 
Goal: Continuous Complex.im
-- Complex.imCLM_coe
@[simp]
theorem imCLM_coe : (imCLM : ℂ →ₗ[ℝ] ℝ) = imLm 
Goal: ↑Complex.imCLM = Complex.imLm
-- Complex.imCLM_apply
@[simp]
theorem imCLM_apply (z : ℂ) : (imCLM : ℂ → ℝ) z = z.im 
Arguments: (z : ℂ)
Goal: Complex.imCLM z = z.im
-- Complex.restrictScalars_one_smulRight'
theorem restrictScalars_one_smulRight' (x : E) :
    ContinuousLinearMap.restrictScalars ℝ ((1 : ℂ →L[ℂ] ℂ).smulRight x : ℂ →L[ℂ] E) =
      reCLM.smulRight x + I • imCLM.smulRight x 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℂ E] (x : E)
Goal: ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 x) =
  Complex.reCLM.smulRight x + Complex.I • Complex.imCLM.smulRight x
-- Complex.restrictScalars_one_smulRight
theorem restrictScalars_one_smulRight (x : ℂ) :
    ContinuousLinearMap.restrictScalars ℝ ((1 : ℂ →L[ℂ] ℂ).smulRight x : ℂ →L[ℂ] ℂ) =
    x • (1 : ℂ →L[ℝ] ℂ) 
Arguments: (x : ℂ)
Goal: ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 x) = x • 1
-- Complex.conjLIE_apply
@[simp]
theorem conjLIE_apply (z : ℂ) : conjLIE z = conj z 
Arguments: (z : ℂ)
Goal: Complex.conjLIE z = (starRingEnd ℂ) z
-- Complex.conjLIE_symm
@[simp]
theorem conjLIE_symm : conjLIE.symm = conjLIE 
Goal: Complex.conjLIE.symm = Complex.conjLIE
-- Complex.isometry_conj
theorem isometry_conj : Isometry (conj : ℂ → ℂ) 
Goal: Isometry ⇑(starRingEnd ℂ)
-- Complex.dist_conj_conj
@[simp]
theorem dist_conj_conj (z w : ℂ) : dist (conj z) (conj w) = dist z w 
Arguments: (z w : ℂ)
Goal: dist ((starRingEnd ℂ) z) ((starRingEnd ℂ) w) = dist z w
-- Complex.nndist_conj_conj
@[simp]
theorem nndist_conj_conj (z w : ℂ) : nndist (conj z) (conj w) = nndist z w 
Arguments: (z w : ℂ)
Goal: nndist ((starRingEnd ℂ) z) ((starRingEnd ℂ) w) = nndist z w
-- Complex.dist_conj_comm
theorem dist_conj_comm (z w : ℂ) : dist (conj z) w = dist z (conj w) 
Arguments: (z w : ℂ)
Goal: dist ((starRingEnd ℂ) z) w = dist z ((starRingEnd ℂ) w)
-- Complex.nndist_conj_comm
theorem nndist_conj_comm (z w : ℂ) : nndist (conj z) w = nndist z (conj w) 
Arguments: (z w : ℂ)
Goal: nndist ((starRingEnd ℂ) z) w = nndist z ((starRingEnd ℂ) w)
-- Complex.continuous_conj
@[continuity]
theorem continuous_conj : Continuous (conj : ℂ → ℂ) 
Goal: Continuous ⇑(starRingEnd ℂ)
-- Complex.ringHom_eq_id_or_conj_of_continuous
theorem ringHom_eq_id_or_conj_of_continuous {f : ℂ →+* ℂ} (hf : Continuous f) :
    f = RingHom.id ℂ ∨ f = conj 
Arguments: {f : ℂ →+* ℂ} (hf : Continuous ⇑f)
Goal: f = RingHom.id ℂ ∨ f = starRingEnd ℂ
-- Complex.conjCLE_coe
@[simp]
theorem conjCLE_coe : conjCLE.toLinearEquiv = conjAe.toLinearEquiv 
Goal: Complex.conjCLE.toLinearEquiv = Complex.conjAe.toLinearEquiv
-- Complex.conjCLE_apply
@[simp]
theorem conjCLE_apply (z : ℂ) : conjCLE z = conj z 
Arguments: (z : ℂ)
Goal: Complex.conjCLE z = (starRingEnd ℂ) z
-- Complex.isometry_ofReal
theorem isometry_ofReal : Isometry ((↑) : ℝ → ℂ) 
Goal: Isometry Complex.ofReal'
-- Complex.continuous_ofReal
@[continuity, fun_prop]
theorem continuous_ofReal : Continuous ((↑) : ℝ → ℂ) 
Goal: Continuous Complex.ofReal'
-- Filter.Tendsto.ofReal
lemma _root_.Filter.Tendsto.ofReal {α : Type*} {l : Filter α} {f : α → ℝ} {x : ℝ}
    (hf : Tendsto f l (𝓝 x)) : Tendsto (fun x ↦ (f x : ℂ)) l (𝓝 (x : ℂ)) 
Arguments: {α : Type u_2} {l : Filter α} {f : α → ℝ} {x : ℝ} (hf : Filter.Tendsto f l (nhds x))
Goal: Filter.Tendsto (fun x => ↑(f x)) l (nhds ↑x)
-- Complex.ringHom_eq_ofReal_of_continuous
theorem ringHom_eq_ofReal_of_continuous {f : ℝ →+* ℂ} (h : Continuous f) : f = Complex.ofReal 
Arguments: {f : ℝ →+* ℂ} (h : Continuous ⇑f)
Goal: f = Complex.ofReal
-- Complex.ofRealCLM_coe
@[simp]
theorem ofRealCLM_coe : (ofRealCLM : ℝ →ₗ[ℝ] ℂ) = ofRealAm.toLinearMap 
Goal: ↑Complex.ofRealCLM = Complex.ofRealAm.toLinearMap
-- Complex.ofRealCLM_apply
@[simp]
theorem ofRealCLM_apply (x : ℝ) : ofRealCLM x = x 
Arguments: (x : ℝ)
Goal: Complex.ofRealCLM x = ↑x
-- RCLike.re_eq_complex_re
theorem _root_.RCLike.re_eq_complex_re : ⇑(RCLike.re : ℂ →+ ℝ) = Complex.re 
Goal: ⇑RCLike.re = Complex.re
-- RCLike.im_eq_complex_im
theorem _root_.RCLike.im_eq_complex_im : ⇑(RCLike.im : ℂ →+ ℝ) = Complex.im 
Goal: ⇑RCLike.im = Complex.im
-- Complex.mul_conj'
lemma mul_conj' (z : ℂ) : z * conj z = ‖z‖ ^ 2 
Arguments: (z : ℂ)
Goal: z * (starRingEnd ℂ) z = ↑‖z‖ ^ 2
-- Complex.conj_mul'
lemma conj_mul' (z : ℂ) : conj z * z = ‖z‖ ^ 2 
Arguments: (z : ℂ)
Goal: (starRingEnd ℂ) z * z = ↑‖z‖ ^ 2
-- Complex.inv_eq_conj
lemma inv_eq_conj (hz : ‖z‖ = 1) : z⁻¹ = conj z 
Arguments: {z : ℂ} (hz : ‖z‖ = 1)
Goal: z⁻¹ = (starRingEnd ℂ) z
-- Complex.exists_norm_eq_mul_self
lemma exists_norm_eq_mul_self (z : ℂ) : ∃ c, ‖c‖ = 1 ∧ ‖z‖ = c * z 
Arguments: (z : ℂ)
Goal: ∃ c, ‖c‖ = 1 ∧ ↑‖z‖ = c * z
-- Complex.exists_norm_mul_eq_self
lemma exists_norm_mul_eq_self (z : ℂ) : ∃ c, ‖c‖ = 1 ∧ c * ‖z‖ = z 
Arguments: (z : ℂ)
Goal: ∃ c, ‖c‖ = 1 ∧ c * ↑‖z‖ = z
-- RCLike.complexRingEquiv_symm_apply
@[simps]
def _root_.RCLike.complexRingEquiv {𝕜 : Type*} [RCLike 𝕜]
    (h : RCLike.im (RCLike.I : 𝕜) = 1) : 𝕜 ≃+* ℂ 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1) (x : ℂ)
Goal: (RCLike.complexRingEquiv h).symm x = ↑x.re + ↑x.im * RCLike.I
-- RCLike.complexRingEquiv_apply
@[simps]
def _root_.RCLike.complexRingEquiv {𝕜 : Type*} [RCLike 𝕜]
    (h : RCLike.im (RCLike.I : 𝕜) = 1) : 𝕜 ≃+* ℂ 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1) (x : 𝕜)
Goal: (RCLike.complexRingEquiv h) x = ↑(RCLike.re x) + ↑(RCLike.im x) * Complex.I
-- RCLike.complexLinearIsometryEquiv_toFun
@[simps]
def _root_.RCLike.complexLinearIsometryEquiv {𝕜 : Type*} [RCLike 𝕜]
    (h : RCLike.im (RCLike.I : 𝕜) = 1) : 𝕜 ≃ₗᵢ[ℝ] ℂ 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1)
Goal: ∀ (a : 𝕜), (RCLike.complexLinearIsometryEquiv h) a = (RCLike.complexRingEquiv h).toFun a
-- RCLike.complexLinearIsometryEquiv_apply
@[simps]
def _root_.RCLike.complexLinearIsometryEquiv {𝕜 : Type*} [RCLike 𝕜]
    (h : RCLike.im (RCLike.I : 𝕜) = 1) : 𝕜 ≃ₗᵢ[ℝ] ℂ 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1)
Goal: ∀ (a : 𝕜), (RCLike.complexLinearIsometryEquiv h) a = (RCLike.complexRingEquiv h).toFun a
-- RCLike.complexLinearIsometryEquiv_symm_apply
@[simps]
def _root_.RCLike.complexLinearIsometryEquiv {𝕜 : Type*} [RCLike 𝕜]
    (h : RCLike.im (RCLike.I : 𝕜) = 1) : 𝕜 ≃ₗᵢ[ℝ] ℂ 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1)
Goal: ∀ (a : ℂ), (RCLike.complexLinearIsometryEquiv h).symm a = (RCLike.complexRingEquiv h).invFun a
-- RCLike.complexLinearIsometryEquiv_invFun
@[simps]
def _root_.RCLike.complexLinearIsometryEquiv {𝕜 : Type*} [RCLike 𝕜]
    (h : RCLike.im (RCLike.I : 𝕜) = 1) : 𝕜 ≃ₗᵢ[ℝ] ℂ 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] (h : RCLike.im RCLike.I = 1)
Goal: ∀ (a : ℂ), (RCLike.complexLinearIsometryEquiv h).invFun a = (RCLike.complexRingEquiv h).invFun a
-- Complex.eq_coe_norm_of_nonneg
theorem eq_coe_norm_of_nonneg {z : ℂ} (hz : 0 ≤ z) : z = ↑‖z‖ 
Arguments: {z : ℂ} (hz : 0 ≤ z)
Goal: z = ↑‖z‖
-- Complex.orderClosedTopology
lemma orderClosedTopology : OrderClosedTopology ℂ 
Goal: OrderClosedTopology ℂ
-- RCLike.re_to_complex
@[simp]
theorem re_to_complex {x : ℂ} : reC x = x.re 
Arguments: {x : ℂ}
Goal: RCLike.re x = x.re
-- RCLike.im_to_complex
@[simp]
theorem im_to_complex {x : ℂ} : imC x = x.im 
Arguments: {x : ℂ}
Goal: RCLike.im x = x.im
-- RCLike.I_to_complex
@[simp]
theorem I_to_complex : IC = Complex.I 
Goal: RCLike.I = Complex.I
-- RCLike.normSq_to_complex
@[simp]
theorem normSq_to_complex {x : ℂ} : norm_sqC x = Complex.normSq x 
Arguments: {x : ℂ}
Goal: RCLike.normSq x = Complex.normSq x
-- RCLike.hasSum_conj
@[simp]
theorem hasSum_conj {f : α → 𝕜} {x : 𝕜} : HasSum (fun x => conj (f x)) x ↔ HasSum f (conj x) 
Arguments: {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} {x : 𝕜}
Goal: HasSum (fun x => (starRingEnd 𝕜) (f x)) x ↔ HasSum f ((starRingEnd 𝕜) x)
-- RCLike.hasSum_conj'
theorem hasSum_conj' {f : α → 𝕜} {x : 𝕜} : HasSum (fun x => conj (f x)) (conj x) ↔ HasSum f x 
Arguments: {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} {x : 𝕜}
Goal: HasSum (fun x => (starRingEnd 𝕜) (f x)) ((starRingEnd 𝕜) x) ↔ HasSum f x