ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- StarModule.instOrderedSMul
lemma _root_.StarModule.instOrderedSMul {A : Type*} [NonUnitalRing A] [StarRing A] [PartialOrder A]
    [StarOrderedRing A] [Module K A] [StarModule K A] [IsScalarTower K A A] [SMulCommClass K A A] :
    OrderedSMul K A 
Arguments: {K : Type u_1} [RCLike K] {A : Type u_3} [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A] [Module K A] [StarModule K A] [IsScalarTower K A A] [SMulCommClass K A A]
Goal: OrderedSMul K A
-- RCLike.re_to_real
@[simp, rclike_simps]
theorem re_to_real {x : ℝ} : reR x = x 
Arguments: {x : ℝ}
Goal: RCLike.re x = x
-- RCLike.im_to_real
@[simp, rclike_simps]
theorem im_to_real {x : ℝ} : imR x = 0 
Arguments: {x : ℝ}
Goal: RCLike.im x = 0
-- RCLike.conj_to_real
@[rclike_simps]
theorem conj_to_real {x : ℝ} : conj x = x 
Arguments: {x : ℝ}
Goal: (starRingEnd ℝ) x = x
-- RCLike.I_to_real
@[simp, rclike_simps]
theorem I_to_real : IR = 0 
Goal: RCLike.I = 0
-- RCLike.normSq_to_real
@[simp, rclike_simps]
theorem normSq_to_real {x : ℝ} : normSq x = x * x 
Arguments: {x : ℝ}
Goal: RCLike.normSq x = x * x
-- RCLike.ofReal_real_eq_id
@[simp]
theorem ofReal_real_eq_id : @ofReal ℝ _ = id 
Goal: RCLike.ofReal = id
-- RCLike.reLm_coe
@[simp, rclike_simps]
theorem reLm_coe : (reLm : K → ℝ) = re 
Arguments: {K : Type u_1} [RCLike K]
Goal: ⇑RCLike.reLm = ⇑RCLike.re
-- RCLike.reCLM_coe
@[simp, rclike_simps, norm_cast]
theorem reCLM_coe : ((reCLM : K →L[ℝ] ℝ) : K →ₗ[ℝ] ℝ) = reLm 
Arguments: {K : Type u_1} [RCLike K]
Goal: ↑RCLike.reCLM = RCLike.reLm
-- RCLike.reCLM_apply
@[simp, rclike_simps]
theorem reCLM_apply : ((reCLM : K →L[ℝ] ℝ) : K → ℝ) = re 
Arguments: {K : Type u_1} [RCLike K]
Goal: ⇑RCLike.reCLM = ⇑RCLike.re
-- RCLike.continuous_re
@[continuity, fun_prop]
theorem continuous_re : Continuous (re : K → ℝ) 
Arguments: {K : Type u_1} [RCLike K]
Goal: Continuous ⇑RCLike.re
-- RCLike.imLm_coe
@[simp, rclike_simps]
theorem imLm_coe : (imLm : K → ℝ) = im 
Arguments: {K : Type u_1} [RCLike K]
Goal: ⇑RCLike.imLm = ⇑RCLike.im
-- RCLike.imCLM_coe
@[simp, rclike_simps, norm_cast]
theorem imCLM_coe : ((imCLM : K →L[ℝ] ℝ) : K →ₗ[ℝ] ℝ) = imLm 
Arguments: {K : Type u_1} [RCLike K]
Goal: ↑RCLike.imCLM = RCLike.imLm
-- RCLike.imCLM_apply
@[simp, rclike_simps]
theorem imCLM_apply : ((imCLM : K →L[ℝ] ℝ) : K → ℝ) = im 
Arguments: {K : Type u_1} [RCLike K]
Goal: ⇑RCLike.imCLM = ⇑RCLike.im
-- RCLike.continuous_im
@[continuity, fun_prop]
theorem continuous_im : Continuous (im : K → ℝ) 
Arguments: {K : Type u_1} [RCLike K]
Goal: Continuous ⇑RCLike.im
-- RCLike.conjAe_coe
@[simp, rclike_simps]
theorem conjAe_coe : (conjAe : K → K) = conj 
Arguments: {K : Type u_1} [RCLike K]
Goal: ⇑RCLike.conjAe = ⇑(starRingEnd K)
-- RCLike.conjLIE_apply
@[simp, rclike_simps]
theorem conjLIE_apply : (conjLIE : K → K) = conj 
Arguments: {K : Type u_1} [RCLike K]
Goal: ⇑RCLike.conjLIE = ⇑(starRingEnd K)
-- RCLike.conjCLE_coe
@[simp, rclike_simps]
theorem conjCLE_coe : (@conjCLE K _).toLinearEquiv = conjAe.toLinearEquiv 
Arguments: {K : Type u_1} [RCLike K]
Goal: RCLike.conjCLE.toLinearEquiv = RCLike.conjAe.toLinearEquiv
-- RCLike.conjCLE_apply
@[simp, rclike_simps]
theorem conjCLE_apply : (conjCLE : K → K) = conj 
Arguments: {K : Type u_1} [RCLike K]
Goal: ⇑RCLike.conjCLE = ⇑(starRingEnd K)
-- RCLike.continuous_conj
@[continuity]
theorem continuous_conj : Continuous (conj : K → K) 
Arguments: {K : Type u_1} [RCLike K]
Goal: Continuous ⇑(starRingEnd K)
-- RCLike.ofRealAm_coe
@[simp, rclike_simps]
theorem ofRealAm_coe : (ofRealAm : ℝ → K) = ofReal 
Arguments: {K : Type u_1} [RCLike K]
Goal: ⇑RCLike.ofRealAm = RCLike.ofReal
-- RCLike.ofRealLI_apply
@[simp, rclike_simps]
theorem ofRealLI_apply : (ofRealLI : ℝ → K) = ofReal 
Arguments: {K : Type u_1} [RCLike K]
Goal: ⇑RCLike.ofRealLI = RCLike.ofReal
-- RCLike.ofRealCLM_coe
@[simp, rclike_simps]
theorem ofRealCLM_coe : (@ofRealCLM K _ : ℝ →ₗ[ℝ] K) = ofRealAm.toLinearMap 
Arguments: {K : Type u_1} [RCLike K]
Goal: ↑RCLike.ofRealCLM = RCLike.ofRealAm.toLinearMap
-- RCLike.ofRealCLM_apply
@[simp, rclike_simps]
theorem ofRealCLM_apply : (ofRealCLM : ℝ → K) = ofReal 
Arguments: {K : Type u_1} [RCLike K]
Goal: ⇑RCLike.ofRealCLM = RCLike.ofReal
-- RCLike.continuous_ofReal
@[continuity, fun_prop]
theorem continuous_ofReal : Continuous (ofReal : ℝ → K) 
Arguments: {K : Type u_1} [RCLike K]
Goal: Continuous RCLike.ofReal
-- RCLike.continuous_normSq
@[continuity]
theorem continuous_normSq : Continuous (normSq : K → ℝ) 
Arguments: {K : Type u_1} [RCLike K]
Goal: Continuous ⇑RCLike.normSq
-- RCLike.im_eq_zero
lemma im_eq_zero (h : I = (0 : K)) (z : K) : im z = 0 
Arguments: {K : Type u_1} [RCLike K] (h : RCLike.I = 0) (z : K)
Goal: RCLike.im z = 0
-- RCLike.realRingEquiv_symm_apply
@[simps]
def realRingEquiv (h : I = (0 : K)) : K ≃+* ℝ 
Arguments: {K : Type u_1} [RCLike K] (h : RCLike.I = 0)
Goal: ∀ (a : ℝ), (RCLike.realRingEquiv h).symm a = ↑a
-- RCLike.realRingEquiv_apply
@[simps]
def realRingEquiv (h : I = (0 : K)) : K ≃+* ℝ 
Arguments: {K : Type u_1} [RCLike K] (h : RCLike.I = 0) (a : K)
Goal: (RCLike.realRingEquiv h) a = RCLike.re a
-- RCLike.realLinearIsometryEquiv_apply
@[simps]
noncomputable def realLinearIsometryEquiv (h : I = (0 : K)) : K ≃ₗᵢ[ℝ] ℝ 
Arguments: {K : Type u_1} [RCLike K] (h : RCLike.I = 0)
Goal: ∀ (a : K), (RCLike.realLinearIsometryEquiv h) a = (RCLike.realRingEquiv h).toFun a
-- RCLike.realLinearIsometryEquiv_toFun
@[simps]
noncomputable def realLinearIsometryEquiv (h : I = (0 : K)) : K ≃ₗᵢ[ℝ] ℝ 
Arguments: {K : Type u_1} [RCLike K] (h : RCLike.I = 0)
Goal: ∀ (a : K), (RCLike.realLinearIsometryEquiv h) a = (RCLike.realRingEquiv h).toFun a
-- RCLike.realLinearIsometryEquiv_invFun
@[simps]
noncomputable def realLinearIsometryEquiv (h : I = (0 : K)) : K ≃ₗᵢ[ℝ] ℝ 
Arguments: {K : Type u_1} [RCLike K] (h : RCLike.I = 0)
Goal: ∀ (a : ℝ), (RCLike.realLinearIsometryEquiv h).invFun a = (RCLike.realRingEquiv h).invFun a
-- RCLike.realLinearIsometryEquiv_symm_apply
@[simps]
noncomputable def realLinearIsometryEquiv (h : I = (0 : K)) : K ≃ₗᵢ[ℝ] ℝ 
Arguments: {K : Type u_1} [RCLike K] (h : RCLike.I = 0)
Goal: ∀ (a : ℝ), (RCLike.realLinearIsometryEquiv h).symm a = (RCLike.realRingEquiv h).invFun a
-- HasSum.hasSum_at_zero
theorem hasSum_at_zero (a : ℕ → E) : HasSum (fun n => (0 : 𝕜) ^ n • a n) (a 0) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (a : ℕ → E)
Goal: HasSum (fun n => 0 ^ n • a n) (a 0)
-- HasSum.exists_hasSum_smul_of_apply_eq_zero
theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m • a m) s)
    (ha : ∀ k < n, a k = 0) : ∃ t : E, z ^ n • t = s ∧ HasSum (fun m => z ^ m • a (m + n)) t 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : E} {n : ℕ} {z : 𝕜} {a : ℕ → E} (hs : HasSum (fun m => z ^ m • a m) s) (ha : ∀ k < n, a k = 0)
Goal: ∃ t, z ^ n • t = s ∧ HasSum (fun m => z ^ m • a (m + n)) t
-- HasFPowerSeriesAt.has_fpower_series_dslope_fslope
theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p z₀) :
    HasFPowerSeriesAt (dslope f z₀) p.fslope z₀ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜} (hp : HasFPowerSeriesAt f p z₀)
Goal: HasFPowerSeriesAt (dslope f z₀) p.fslope z₀
-- HasFPowerSeriesAt.has_fpower_series_iterate_dslope_fslope
theorem has_fpower_series_iterate_dslope_fslope (n : ℕ) (hp : HasFPowerSeriesAt f p z₀) :
    HasFPowerSeriesAt ((swap dslope z₀)^[n] f) (fslope^[n] p) z₀ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜} (n : ℕ) (hp : HasFPowerSeriesAt f p z₀)
Goal: HasFPowerSeriesAt ((Function.swap dslope z₀)^[n] f) (FormalMultilinearSeries.fslope^[n] p) z₀
-- HasFPowerSeriesAt.iterate_dslope_fslope_ne_zero
theorem iterate_dslope_fslope_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) :
    (swap dslope z₀)^[p.order] f z₀ ≠ 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜} (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0)
Goal: (Function.swap dslope z₀)^[p.order] f z₀ ≠ 0
-- HasFPowerSeriesAt.eq_pow_order_mul_iterate_dslope
theorem eq_pow_order_mul_iterate_dslope (hp : HasFPowerSeriesAt f p z₀) :
    ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ p.order • (swap dslope z₀)^[p.order] f z 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜} (hp : HasFPowerSeriesAt f p z₀)
Goal: ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ p.order • (Function.swap dslope z₀)^[p.order] f z
-- HasFPowerSeriesAt.locally_ne_zero
theorem locally_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) : ∀ᶠ z in 𝓝[≠] z₀, f z ≠ 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜} (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0)
Goal: ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0
-- HasFPowerSeriesAt.locally_zero_iff
theorem locally_zero_iff (hp : HasFPowerSeriesAt f p z₀) : (∀ᶠ z in 𝓝 z₀, f z = 0) ↔ p = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜} (hp : HasFPowerSeriesAt f p z₀)
Goal: (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ↔ p = 0
-- AnalyticAt.eventually_eq_zero_or_eventually_ne_zero
theorem eventually_eq_zero_or_eventually_ne_zero (hf : AnalyticAt 𝕜 f z₀) :
    (∀ᶠ z in 𝓝 z₀, f z = 0) ∨ ∀ᶠ z in 𝓝[≠] z₀, f z ≠ 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {z₀ : 𝕜} (hf : AnalyticAt 𝕜 f z₀)
Goal: (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0
-- AnalyticAt.eventually_eq_or_eventually_ne
theorem eventually_eq_or_eventually_ne (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀) :
    (∀ᶠ z in 𝓝 z₀, f z = g z) ∨ ∀ᶠ z in 𝓝[≠] z₀, f z ≠ g z 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : 𝕜 → E} {z₀ : 𝕜} (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀)
Goal: (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = g z) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ g z
-- AnalyticAt.frequently_zero_iff_eventually_zero
theorem frequently_zero_iff_eventually_zero {f : 𝕜 → E} {w : 𝕜} (hf : AnalyticAt 𝕜 f w) :
    (∃ᶠ z in 𝓝[≠] w, f z = 0) ↔ ∀ᶠ z in 𝓝 w, f z = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {w : 𝕜} (hf : AnalyticAt 𝕜 f w)
Goal: (∃ᶠ (z : 𝕜) in 𝓝[≠] w, f z = 0) ↔ ∀ᶠ (z : 𝕜) in 𝓝 w, f z = 0
-- AnalyticAt.frequently_eq_iff_eventually_eq
theorem frequently_eq_iff_eventually_eq (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀) :
    (∃ᶠ z in 𝓝[≠] z₀, f z = g z) ↔ ∀ᶠ z in 𝓝 z₀, f z = g z 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : 𝕜 → E} {z₀ : 𝕜} (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀)
Goal: (∃ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = g z) ↔ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = g z
-- AnalyticAt.unique_eventuallyEq_zpow_smul_nonzero
lemma unique_eventuallyEq_zpow_smul_nonzero {m n : ℤ}
    (hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z)
    (hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝[≠] z₀, f z = (z - z₀) ^ n • g z) :
    m = n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {z₀ : 𝕜} {m n : ℤ} (hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z) (hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ n • g z)
Goal: m = n
-- AnalyticAt.unique_eventuallyEq_pow_smul_nonzero
lemma unique_eventuallyEq_pow_smul_nonzero {m n : ℕ}
    (hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ m • g z)
    (hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ n • g z) :
    m = n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {z₀ : 𝕜} {m n : ℕ} (hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ m • g z) (hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z)
Goal: m = n
-- AnalyticAt.exists_eventuallyEq_pow_smul_nonzero_iff
theorem exists_eventuallyEq_pow_smul_nonzero_iff (hf : AnalyticAt 𝕜 f z₀) :
    (∃ (n : ℕ), ∃ (g : 𝕜 → E), AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧
    ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ n • g z) ↔ (¬∀ᶠ z in 𝓝 z₀, f z = 0) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {z₀ : 𝕜} (hf : AnalyticAt 𝕜 f z₀)
Goal: (∃ n g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z) ↔ ¬∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
-- AnalyticAt.order_eq_top_iff
lemma order_eq_top_iff (hf : AnalyticAt 𝕜 f z₀) : hf.order = ⊤ ↔ ∀ᶠ z in 𝓝 z₀, f z = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {z₀ : 𝕜} (hf : AnalyticAt 𝕜 f z₀)
Goal: hf.order = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
-- AnalyticAt.order_eq_nat_iff
lemma order_eq_nat_iff (hf : AnalyticAt 𝕜 f z₀) (n : ℕ) : hf.order = ↑n ↔
    ∃ (g : 𝕜 → E), AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ n • g z 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {z₀ : 𝕜} (hf : AnalyticAt 𝕜 f z₀) (n : ℕ)
Goal: hf.order = ↑n ↔ ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z
-- AnalyticOn.eqOn_zero_of_preconnected_of_frequently_eq_zero
theorem eqOn_zero_of_preconnected_of_frequently_eq_zero (hf : AnalyticOn 𝕜 f U)
    (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfw : ∃ᶠ z in 𝓝[≠] z₀, f z = 0) : EqOn f 0 U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {z₀ : 𝕜} {U : Set 𝕜} (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfw : ∃ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = 0)
Goal: Set.EqOn f 0 U
-- AnalyticOn.eqOn_zero_of_preconnected_of_mem_closure
theorem eqOn_zero_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U)
    (h₀ : z₀ ∈ U) (hfz₀ : z₀ ∈ closure ({z | f z = 0} \ {z₀})) : EqOn f 0 U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {z₀ : 𝕜} {U : Set 𝕜} (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfz₀ : z₀ ∈ closure ({z | f z = 0} \ {z₀}))
Goal: Set.EqOn f 0 U
-- AnalyticOn.eqOn_of_preconnected_of_frequently_eq
theorem eqOn_of_preconnected_of_frequently_eq (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U)
    (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : ∃ᶠ z in 𝓝[≠] z₀, f z = g z) : EqOn f g U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : 𝕜 → E} {z₀ : 𝕜} {U : Set 𝕜} (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U) (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : ∃ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = g z)
Goal: Set.EqOn f g U
-- AnalyticOn.eqOn_of_preconnected_of_mem_closure
theorem eqOn_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U)
    (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : z₀ ∈ closure ({z | f z = g z} \ {z₀})) :
    EqOn f g U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : 𝕜 → E} {z₀ : 𝕜} {U : Set 𝕜} (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U) (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : z₀ ∈ closure ({z | f z = g z} \ {z₀}))
Goal: Set.EqOn f g U
-- AnalyticOn.eq_of_frequently_eq
theorem eq_of_frequently_eq [ConnectedSpace 𝕜] (hf : AnalyticOn 𝕜 f univ) (hg : AnalyticOn 𝕜 g univ)
    (hfg : ∃ᶠ z in 𝓝[≠] z₀, f z = g z) : f = g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : 𝕜 → E} {z₀ : 𝕜} [ConnectedSpace 𝕜] (hf : AnalyticOn 𝕜 f Set.univ) (hg : AnalyticOn 𝕜 g Set.univ) (hfg : ∃ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = g z)
Goal: f = g
-- IsCoercive.bounded_below
theorem bounded_below (coercive : IsCoercive B) : ∃ C, 0 < C ∧ ∀ v, C * ‖v‖ ≤ ‖B♯ v‖ 
Arguments: {V : Type u} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [CompleteSpace V] {B : V →L[ℝ] V →L[ℝ] ℝ} (coercive : IsCoercive B)
Goal: ∃ C, 0 < C ∧ ∀ (v : V), C * ‖v‖ ≤ ‖(InnerProductSpace.continuousLinearMapOfBilin B) v‖
-- IsCoercive.antilipschitz
theorem antilipschitz (coercive : IsCoercive B) : ∃ C : ℝ≥0, 0 < C ∧ AntilipschitzWith C B♯ 
Arguments: {V : Type u} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [CompleteSpace V] {B : V →L[ℝ] V →L[ℝ] ℝ} (coercive : IsCoercive B)
Goal: ∃ C, 0 < C ∧ AntilipschitzWith C ⇑(InnerProductSpace.continuousLinearMapOfBilin B)
-- IsCoercive.ker_eq_bot
theorem ker_eq_bot (coercive : IsCoercive B) : ker B♯ = ⊥ 
Arguments: {V : Type u} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [CompleteSpace V] {B : V →L[ℝ] V →L[ℝ] ℝ} (coercive : IsCoercive B)
Goal: LinearMap.ker (InnerProductSpace.continuousLinearMapOfBilin B) = ⊥
-- IsCoercive.isClosed_range
theorem isClosed_range (coercive : IsCoercive B) : IsClosed (range B♯ : Set V) 
Arguments: {V : Type u} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [CompleteSpace V] {B : V →L[ℝ] V →L[ℝ] ℝ} (coercive : IsCoercive B)
Goal: IsClosed ↑(LinearMap.range (InnerProductSpace.continuousLinearMapOfBilin B))
-- IsCoercive.range_eq_top
theorem range_eq_top (coercive : IsCoercive B) : range B♯ = ⊤ 
Arguments: {V : Type u} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [CompleteSpace V] {B : V →L[ℝ] V →L[ℝ] ℝ} (coercive : IsCoercive B)
Goal: LinearMap.range (InnerProductSpace.continuousLinearMapOfBilin B) = ⊤
-- IsCoercive.continuousLinearEquivOfBilin_apply
@[simp]
theorem continuousLinearEquivOfBilin_apply (coercive : IsCoercive B) (v w : V) :
    ⟪coercive.continuousLinearEquivOfBilin v, w⟫_ℝ = B v w 
Arguments: {V : Type u} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [CompleteSpace V] {B : V →L[ℝ] V →L[ℝ] ℝ} (coercive : IsCoercive B) (v w : V)
Goal: ⟪coercive.continuousLinearEquivOfBilin v, w⟫_ℝ = (B v) w
-- IsCoercive.unique_continuousLinearEquivOfBilin
theorem unique_continuousLinearEquivOfBilin (coercive : IsCoercive B) {v f : V}
    (is_lax_milgram : ∀ w, ⟪f, w⟫_ℝ = B v w) : f = coercive.continuousLinearEquivOfBilin v 
Arguments: {V : Type u} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [CompleteSpace V] {B : V →L[ℝ] V →L[ℝ] ℝ} (coercive : IsCoercive B) {v f : V} (is_lax_milgram : ∀ (w : V), ⟪f, w⟫_ℝ = (B v) w)
Goal: f = coercive.continuousLinearEquivOfBilin v
-- ContinuousLinearMapWOT.ext_inner
@[ext]
lemma ext_inner {A B : E →WOT[𝕜] F} (h : ∀ x y, ⟪y, A x⟫_𝕜 = ⟪y, B x⟫_𝕜) : A = B 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] {A B : E →WOT[𝕜]F} (h : ∀ (x : E) (y : F), ⟪y, A x⟫_𝕜 = ⟪y, B x⟫_𝕜)
Goal: A = B
-- ContinuousLinearMapWOT.ext_inner_iff
lemma ext_inner_iff {A B : E →WOT[𝕜] F} : A = B ↔ ∀ x y, ⟪y, A x⟫_𝕜 = ⟪y, B x⟫_𝕜 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] {A B : E →WOT[𝕜]F}
Goal: A = B ↔ ∀ (x : E) (y : F), ⟪y, A x⟫_𝕜 = ⟪y, B x⟫_𝕜
-- ContinuousLinearMapWOT.tendsto_iff_forall_inner_apply_tendsto
lemma tendsto_iff_forall_inner_apply_tendsto [CompleteSpace F] {α : Type*} {l : Filter α}
    {f : α → E →WOT[𝕜] F} {A : E →WOT[𝕜] F} :
    Tendsto f l (𝓝 A) ↔ ∀ x y, Tendsto (fun a => ⟪y, (f a) x⟫_𝕜) l (𝓝 ⟪y, A x⟫_𝕜) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {α : Type u_4} {l : Filter α} {f : α → E →WOT[𝕜]F} {A : E →WOT[𝕜]F}
Goal: Filter.Tendsto f l (𝓝 A) ↔ ∀ (x : E) (y : F), Filter.Tendsto (fun a => ⟪y, (f a) x⟫_𝕜) l (𝓝 ⟪y, A x⟫_𝕜)
-- ContinuousLinearMapWOT.le_nhds_iff_forall_inner_apply_le_nhds
lemma le_nhds_iff_forall_inner_apply_le_nhds [CompleteSpace F] {l : Filter (E →WOT[𝕜] F)}
    {A : E →WOT[𝕜] F} : l ≤ 𝓝 A ↔ ∀ x y, l.map (fun T => ⟪y, T x⟫_𝕜) ≤ 𝓝 (⟪y, A x⟫_𝕜) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {l : Filter (E →WOT[𝕜]F)} {A : E →WOT[𝕜]F}
Goal: l ≤ 𝓝 A ↔ ∀ (x : E) (y : F), Filter.map (fun T => ⟪y, T x⟫_𝕜) l ≤ 𝓝 ⟪y, A x⟫_𝕜
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.existsNatOneDivLTMeasure_of_not_negative
private theorem existsNatOneDivLTMeasure_of_not_negative (hi : ¬s ≤[i] 0) :
    ∃ n : ℕ, ExistsOneDivLT s i n 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} (hi : ¬MeasureTheory.VectorMeasure.restrict s i ≤ MeasureTheory.VectorMeasure.restrict 0 i)
Goal: ∃ n, MeasureTheory.SignedMeasure.ExistsOneDivLT s i n
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.findExistsOneDivLT_spec
private theorem findExistsOneDivLT_spec (hi : ¬s ≤[i] 0) :
    ExistsOneDivLT s i (findExistsOneDivLT s i) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} (hi : ¬MeasureTheory.VectorMeasure.restrict s i ≤ MeasureTheory.VectorMeasure.restrict 0 i)
Goal: MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i)
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.findExistsOneDivLT_min
private theorem findExistsOneDivLT_min (hi : ¬s ≤[i] 0) {m : ℕ}
    (hm : m < findExistsOneDivLT s i) : ¬ExistsOneDivLT s i m 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} (hi : ¬MeasureTheory.VectorMeasure.restrict s i ≤ MeasureTheory.VectorMeasure.restrict 0 i) {m : ℕ} (hm : m < MeasureTheory.SignedMeasure.findExistsOneDivLT s i)
Goal: ¬MeasureTheory.SignedMeasure.ExistsOneDivLT s i m
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.someExistsOneDivLT_spec
private theorem someExistsOneDivLT_spec (hi : ¬s ≤[i] 0) :
    someExistsOneDivLT s i ⊆ i ∧
      MeasurableSet (someExistsOneDivLT s i) ∧
        (1 / (findExistsOneDivLT s i + 1) : ℝ) < s (someExistsOneDivLT s i) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} (hi : ¬MeasureTheory.VectorMeasure.restrict s i ≤ MeasureTheory.VectorMeasure.restrict 0 i)
Goal: MeasureTheory.SignedMeasure.someExistsOneDivLT s i ⊆ i ∧
  MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i) ∧
    1 / (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s i) + 1) <
      ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.someExistsOneDivLT_subset
private theorem someExistsOneDivLT_subset : someExistsOneDivLT s i ⊆ i 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α}
Goal: MeasureTheory.SignedMeasure.someExistsOneDivLT s i ⊆ i
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.someExistsOneDivLT_subset'
private theorem someExistsOneDivLT_subset' : someExistsOneDivLT s (i \ j) ⊆ i 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i j : Set α}
Goal: MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ j) ⊆ i
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.someExistsOneDivLT_measurableSet
private theorem someExistsOneDivLT_measurableSet : MeasurableSet (someExistsOneDivLT s i) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α}
Goal: MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)