ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_eq :
    ‖inclusionInDoubleDual 𝕜 E‖ = ‖ContinuousLinearMap.id 𝕜 (Dual 𝕜 E)‖ 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (E : Type u_2) [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: ‖NormedSpace.inclusionInDoubleDual 𝕜 E‖ = ‖ContinuousLinearMap.id 𝕜 (NormedSpace.Dual 𝕜 E)‖
-- NormedSpace.inclusionInDoubleDual_norm_le
theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (E : Type u_2) [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: ‖NormedSpace.inclusionInDoubleDual 𝕜 E‖ ≤ 1
-- NormedSpace.double_dual_bound
theorem double_dual_bound (x : E) : ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖ 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (E : Type u_2) [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E)
Goal: ‖(NormedSpace.inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖
-- NormedSpace.dualPairing_apply
@[simp]
theorem dualPairing_apply {v : Dual 𝕜 E} {x : E} : dualPairing 𝕜 E v x = v x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (E : Type u_2) [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {v : NormedSpace.Dual 𝕜 E} {x : E}
Goal: ((NormedSpace.dualPairing 𝕜 E) v) x = v x
-- NormedSpace.dualPairing_separatingLeft
theorem dualPairing_separatingLeft : (dualPairing 𝕜 E).SeparatingLeft 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (E : Type u_2) [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: (NormedSpace.dualPairing 𝕜 E).SeparatingLeft
-- NormedSpace.norm_le_dual_bound
theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
    ‖x‖ ≤ M 
Arguments: (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ (f : NormedSpace.Dual 𝕜 E), ‖f x‖ ≤ M * ‖f‖)
Goal: ‖x‖ ≤ M
-- NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : ∀ f : Dual 𝕜 E, f x = (0 : 𝕜)) : x = 0 
Arguments: (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} (h : ∀ (f : NormedSpace.Dual 𝕜 E), f x = 0)
Goal: x = 0
-- NormedSpace.eq_zero_iff_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 ↔ ∀ g : Dual 𝕜 E, g x = 0 
Arguments: (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E)
Goal: x = 0 ↔ ∀ (g : NormedSpace.Dual 𝕜 E), g x = 0
-- NormedSpace.eq_iff_forall_dual_eq
theorem eq_iff_forall_dual_eq {x y : E} : x = y ↔ ∀ g : Dual 𝕜 E, g x = g y 
Arguments: (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x y : E}
Goal: x = y ↔ ∀ (g : NormedSpace.Dual 𝕜 E), g x = g y
-- NormedSpace.mem_polar_iff
theorem mem_polar_iff {x' : Dual 𝕜 E} (s : Set E) : x' ∈ polar 𝕜 s ↔ ∀ z ∈ s, ‖x' z‖ ≤ 1 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {x' : NormedSpace.Dual 𝕜 E} (s : Set E)
Goal: x' ∈ NormedSpace.polar 𝕜 s ↔ ∀ z ∈ s, ‖x' z‖ ≤ 1
-- NormedSpace.polar_univ
@[simp]
theorem polar_univ : polar 𝕜 (univ : Set E) = {(0 : Dual 𝕜 E)} 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: NormedSpace.polar 𝕜 Set.univ = {0}
-- NormedSpace.isClosed_polar
theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (s : Set E)
Goal: IsClosed (NormedSpace.polar 𝕜 s)
-- NormedSpace.polar_closure
@[simp]
theorem polar_closure (s : Set E) : polar 𝕜 (closure s) = polar 𝕜 s 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (s : Set E)
Goal: NormedSpace.polar 𝕜 (closure s) = NormedSpace.polar 𝕜 s
-- NormedSpace.smul_mem_polar
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
    c⁻¹ • x' ∈ polar 𝕜 s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {x' : NormedSpace.Dual 𝕜 E} {c : 𝕜} (hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖)
Goal: c⁻¹ • x' ∈ NormedSpace.polar 𝕜 s
-- NormedSpace.polar_ball_subset_closedBall_div
theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
    polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r)
Goal: NormedSpace.polar 𝕜 (Metric.ball 0 r) ⊆ Metric.closedBall 0 (‖c‖ / r)
-- NormedSpace.closedBall_inv_subset_polar_closedBall
theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
    closedBall (0 : Dual 𝕜 E) r⁻¹ ⊆ polar 𝕜 (closedBall (0 : E) r) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
Goal: Metric.closedBall 0 r⁻¹ ⊆ NormedSpace.polar 𝕜 (Metric.closedBall 0 r)
-- NormedSpace.polar_closedBall
theorem polar_closedBall {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
    (hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ 
Arguments: {𝕜 : Type u_3} {E : Type u_4} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ} (hr : 0 < r)
Goal: NormedSpace.polar 𝕜 (Metric.closedBall 0 r) = Metric.closedBall 0 r⁻¹
-- NormedSpace.isBounded_polar_of_mem_nhds_zero
theorem isBounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
    IsBounded (polar 𝕜 s) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} (s_nhd : s ∈ 𝓝 0)
Goal: Bornology.IsBounded (NormedSpace.polar 𝕜 s)
-- Memℓp.all
theorem Memℓp.all (f : ∀ i, E i) : Memℓp f p 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] {p : ℝ≥0∞} [Finite α] (f : (i : α) → E i)
Goal: Memℓp f p
-- coe_equiv_lpPiLp
theorem coe_equiv_lpPiLp (f : lp E p) : Equiv.lpPiLp f = ⇑f 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] {p : ℝ≥0∞} [Finite α] (f : ↥(lp E p))
Goal: Equiv.lpPiLp f = ↑f
-- coe_equiv_lpPiLp_symm
theorem coe_equiv_lpPiLp_symm (f : PiLp p E) : (Equiv.lpPiLp.symm f : ∀ i, E i) = f 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] {p : ℝ≥0∞} [Finite α] (f : PiLp p E)
Goal: ↑(Equiv.lpPiLp.symm f) = f
-- coe_addEquiv_lpPiLp
theorem coe_addEquiv_lpPiLp  (f : lp E p) : AddEquiv.lpPiLp f = ⇑f 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] {p : ℝ≥0∞} [Finite α] (f : ↥(lp E p))
Goal: AddEquiv.lpPiLp f = ↑f
-- coe_addEquiv_lpPiLp_symm
theorem coe_addEquiv_lpPiLp_symm (f : PiLp p E) :
    (AddEquiv.lpPiLp.symm f : ∀ i, E i) = f 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] {p : ℝ≥0∞} [Finite α] (f : PiLp p E)
Goal: ↑(AddEquiv.lpPiLp.symm f) = f
-- equiv_lpPiLp_norm
theorem equiv_lpPiLp_norm [Fintype α] (f : lp E p) : ‖Equiv.lpPiLp f‖ = ‖f‖ 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] {p : ℝ≥0∞} [Fintype α] (f : ↥(lp E p))
Goal: ‖Equiv.lpPiLp f‖ = ‖f‖
-- coe_lpPiLpₗᵢ
theorem coe_lpPiLpₗᵢ [Fact (1 ≤ p)] (f : lp E p) : (lpPiLpₗᵢ E 𝕜 f : ∀ i, E i) = ⇑f 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] {p : ℝ≥0∞} [Fintype α] {𝕜 : Type u_3} [NontriviallyNormedField 𝕜] [(i : α) → NormedSpace 𝕜 (E i)] [Fact (1 ≤ p)] (f : ↥(lp E p))
Goal: (lpPiLpₗᵢ E 𝕜) f = ↑f
-- coe_lpPiLpₗᵢ_symm
theorem coe_lpPiLpₗᵢ_symm [Fact (1 ≤ p)] (f : PiLp p E) :
    ((lpPiLpₗᵢ E 𝕜).symm f : ∀ i, E i) = f 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] {p : ℝ≥0∞} [Fintype α] {𝕜 : Type u_3} [NontriviallyNormedField 𝕜] [(i : α) → NormedSpace 𝕜 (E i)] [Fact (1 ≤ p)] (f : PiLp p E)
Goal: ↑((lpPiLpₗᵢ E 𝕜).symm f) = f
-- coe_addEquiv_lpBCF
theorem coe_addEquiv_lpBCF (f : lp (fun _ : α ↦ E) ∞) : (AddEquiv.lpBCF f : α → E) = f 
Arguments: {α : Type u_1} {E : Type u_2} [TopologicalSpace α] [DiscreteTopology α] [NormedAddCommGroup E] (f : ↥(lp (fun x => E) ⊤))
Goal: ⇑(AddEquiv.lpBCF f) = ↑f
-- coe_addEquiv_lpBCF_symm
theorem coe_addEquiv_lpBCF_symm (f : α →ᵇ E) : (AddEquiv.lpBCF.symm f : α → E) = f 
Arguments: {α : Type u_1} {E : Type u_2} [TopologicalSpace α] [DiscreteTopology α] [NormedAddCommGroup E] (f : BoundedContinuousFunction α E)
Goal: ↑(AddEquiv.lpBCF.symm f) = ⇑f
-- coe_lpBCFₗᵢ
theorem coe_lpBCFₗᵢ (f : lp (fun _ : α ↦ E) ∞) : (lpBCFₗᵢ E 𝕜 f : α → E) = f 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_5} [TopologicalSpace α] [DiscreteTopology α] [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : ↥(lp (fun x => E) ⊤))
Goal: ⇑((lpBCFₗᵢ E 𝕜) f) = ↑f
-- coe_lpBCFₗᵢ_symm
theorem coe_lpBCFₗᵢ_symm (f : α →ᵇ E) : ((lpBCFₗᵢ E 𝕜).symm f : α → E) = f 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_5} [TopologicalSpace α] [DiscreteTopology α] [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : BoundedContinuousFunction α E)
Goal: ↑((lpBCFₗᵢ E 𝕜).symm f) = ⇑f
-- coe_ringEquiv_lpBCF
theorem coe_ringEquiv_lpBCF (f : lp (fun _ : α ↦ R) ∞) : (RingEquiv.lpBCF R f : α → R) = f 
Arguments: {α : Type u_1} {R : Type u_3} [TopologicalSpace α] [DiscreteTopology α] [NonUnitalNormedRing R] (f : ↥(lp (fun x => R) ⊤))
Goal: ⇑((RingEquiv.lpBCF R) f) = ↑f
-- coe_ringEquiv_lpBCF_symm
theorem coe_ringEquiv_lpBCF_symm (f : α →ᵇ R) : ((RingEquiv.lpBCF R).symm f : α → R) = f 
Arguments: {α : Type u_1} {R : Type u_3} [TopologicalSpace α] [DiscreteTopology α] [NonUnitalNormedRing R] (f : BoundedContinuousFunction α R)
Goal: ↑((RingEquiv.lpBCF R).symm f) = ⇑f
-- coe_algEquiv_lpBCF
theorem coe_algEquiv_lpBCF (f : lp (fun _ : α ↦ A) ∞) : (AlgEquiv.lpBCF α A 𝕜 f : α → A) = f 
Arguments: {α : Type u_1} {A : Type u_4} {𝕜 : Type u_5} [TopologicalSpace α] [DiscreteTopology α] [NormedRing A] [NormOneClass A] [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 A] (f : ↥(lp (fun x => A) ⊤))
Goal: ⇑((AlgEquiv.lpBCF α A 𝕜) f) = ↑f
-- coe_algEquiv_lpBCF_symm
theorem coe_algEquiv_lpBCF_symm (f : α →ᵇ A) : ((AlgEquiv.lpBCF α A 𝕜).symm f : α → A) = f 
Arguments: {α : Type u_1} {A : Type u_4} {𝕜 : Type u_5} [TopologicalSpace α] [DiscreteTopology α] [NormedRing A] [NormOneClass A] [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 A] (f : BoundedContinuousFunction α A)
Goal: ↑((AlgEquiv.lpBCF α A 𝕜).symm f) = ⇑f
-- NormedSpace.norm_smul_le
class NormedSpace (𝕜 : Type*) (E : Type*) [NormedField 𝕜] [SeminormedAddCommGroup E]
    extends Module 𝕜 E where
  norm_smul_le : ∀ (a : 𝕜) (b : E), ‖a • b‖ ≤ ‖a‖ * ‖b‖
Arguments: {𝕜 : Type u_6} {E : Type u_7} [NormedField 𝕜] [SeminormedAddCommGroup E] [self : NormedSpace 𝕜 E] (a : 𝕜) (b : E)
Goal: ‖a • b‖ ≤ ‖a‖ * ‖b‖
-- norm_zsmul
theorem norm_zsmul [NormedSpace 𝕜 E] (n : ℤ) (x : E) : ‖n • x‖ = ‖(n : 𝕜)‖ * ‖x‖ 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (n : ℤ) (x : E)
Goal: ‖n • x‖ = ‖↑n‖ * ‖x‖
-- eventually_nhds_norm_smul_sub_lt
theorem eventually_nhds_norm_smul_sub_lt (c : 𝕜) (x : E) {ε : ℝ} (h : 0 < ε) :
    ∀ᶠ y in 𝓝 x, ‖c • (y - x)‖ < ε 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (c : 𝕜) (x : E) {ε : ℝ} (h : 0 < ε)
Goal: ∀ᶠ (y : E) in 𝓝 x, ‖c • (y - x)‖ < ε
-- Filter.Tendsto.zero_smul_isBoundedUnder_le
theorem Filter.Tendsto.zero_smul_isBoundedUnder_le {f : α → 𝕜} {g : α → E} {l : Filter α}
    (hf : Tendsto f l (𝓝 0)) (hg : IsBoundedUnder (· ≤ ·) l (Norm.norm ∘ g)) :
    Tendsto (fun x => f x • g x) l (𝓝 0) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {α : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {f : α → 𝕜} {g : α → E} {l : Filter α} (hf : Filter.Tendsto f l (𝓝 0)) (hg : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ g))
Goal: Filter.Tendsto (fun x => f x • g x) l (𝓝 0)
-- Filter.IsBoundedUnder.smul_tendsto_zero
theorem Filter.IsBoundedUnder.smul_tendsto_zero {f : α → 𝕜} {g : α → E} {l : Filter α}
    (hf : IsBoundedUnder (· ≤ ·) l (norm ∘ f)) (hg : Tendsto g l (𝓝 0)) :
    Tendsto (fun x => f x • g x) l (𝓝 0) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {α : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {f : α → 𝕜} {g : α → E} {l : Filter α} (hf : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)) (hg : Filter.Tendsto g l (𝓝 0))
Goal: Filter.Tendsto (fun x => f x • g x) l (𝓝 0)
-- NormedSpace.exists_lt_norm
theorem NormedSpace.exists_lt_norm (c : ℝ) : ∃ x : E, c < ‖x‖ 
Arguments: (𝕜 : Type u_1) (E : Type u_3) [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [Nontrivial E] (c : ℝ)
Goal: ∃ x, c < ‖x‖
-- NormedSpace.unbounded_univ
protected theorem NormedSpace.unbounded_univ : ¬Bornology.IsBounded (univ : Set E) 
Arguments: (𝕜 : Type u_1) (E : Type u_3) [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [Nontrivial E]
Goal: ¬Bornology.IsBounded Set.univ
-- NormedSpace.cobounded_neBot
protected lemma NormedSpace.cobounded_neBot : NeBot (cobounded E) 
Arguments: (𝕜 : Type u_1) (E : Type u_3) [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [Nontrivial E]
Goal: (Bornology.cobounded E).NeBot
-- NormedSpace.noncompactSpace
protected theorem NormedSpace.noncompactSpace : NoncompactSpace E 
Arguments: (𝕜 : Type u_1) (E : Type u_3) [NormedField 𝕜] [Infinite 𝕜] [NormedAddCommGroup E] [Nontrivial E] [NormedSpace 𝕜 E]
Goal: NoncompactSpace E
-- NormedAlgebra.norm_smul_le
class NormedAlgebra (𝕜 : Type*) (𝕜' : Type*) [NormedField 𝕜] [SeminormedRing 𝕜'] extends
  Algebra 𝕜 𝕜' where
  norm_smul_le : ∀ (r : 𝕜) (x : 𝕜'), ‖r • x‖ ≤ ‖r‖ * ‖x‖
Arguments: {𝕜 : Type u_6} {𝕜' : Type u_7} [NormedField 𝕜] [SeminormedRing 𝕜'] [self : NormedAlgebra 𝕜 𝕜'] (r : 𝕜) (x : 𝕜')
Goal: ‖r • x‖ ≤ ‖r‖ * ‖x‖
-- norm_algebraMap
theorem norm_algebraMap (x : 𝕜) : ‖algebraMap 𝕜 𝕜' x‖ = ‖x‖ * ‖(1 : 𝕜')‖ 
Arguments: {𝕜 : Type u_1} (𝕜' : Type u_2) [NormedField 𝕜] [SeminormedRing 𝕜'] [NormedAlgebra 𝕜 𝕜'] (x : 𝕜)
Goal: ‖(algebraMap 𝕜 𝕜') x‖ = ‖x‖ * ‖1‖
-- nnnorm_algebraMap
theorem nnnorm_algebraMap (x : 𝕜) : ‖algebraMap 𝕜 𝕜' x‖₊ = ‖x‖₊ * ‖(1 : 𝕜')‖₊ 
Arguments: {𝕜 : Type u_1} (𝕜' : Type u_2) [NormedField 𝕜] [SeminormedRing 𝕜'] [NormedAlgebra 𝕜 𝕜'] (x : 𝕜)
Goal: ‖(algebraMap 𝕜 𝕜') x‖₊ = ‖x‖₊ * ‖1‖₊
-- norm_algebraMap'
@[simp]
theorem norm_algebraMap' [NormOneClass 𝕜'] (x : 𝕜) : ‖algebraMap 𝕜 𝕜' x‖ = ‖x‖ 
Arguments: {𝕜 : Type u_1} (𝕜' : Type u_2) [NormedField 𝕜] [SeminormedRing 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormOneClass 𝕜'] (x : 𝕜)
Goal: ‖(algebraMap 𝕜 𝕜') x‖ = ‖x‖
-- nnnorm_algebraMap'
@[simp]
theorem nnnorm_algebraMap' [NormOneClass 𝕜'] (x : 𝕜) : ‖algebraMap 𝕜 𝕜' x‖₊ = ‖x‖₊ 
Arguments: {𝕜 : Type u_1} (𝕜' : Type u_2) [NormedField 𝕜] [SeminormedRing 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormOneClass 𝕜'] (x : 𝕜)
Goal: ‖(algebraMap 𝕜 𝕜') x‖₊ = ‖x‖₊
-- norm_algebraMap_nnreal
@[simp]
theorem norm_algebraMap_nnreal (x : ℝ≥0) : ‖algebraMap ℝ≥0 𝕜' x‖ = x 
Arguments: (𝕜' : Type u_2) [SeminormedRing 𝕜'] [NormOneClass 𝕜'] [NormedAlgebra ℝ 𝕜'] (x : ℝ≥0)
Goal: ‖(algebraMap ℝ≥0 𝕜') x‖ = ↑x
-- nnnorm_algebraMap_nnreal
@[simp]
theorem nnnorm_algebraMap_nnreal (x : ℝ≥0) : ‖algebraMap ℝ≥0 𝕜' x‖₊ = x 
Arguments: (𝕜' : Type u_2) [SeminormedRing 𝕜'] [NormOneClass 𝕜'] [NormedAlgebra ℝ 𝕜'] (x : ℝ≥0)
Goal: ‖(algebraMap ℝ≥0 𝕜') x‖₊ = x
-- algebraMap_isometry
theorem algebraMap_isometry [NormOneClass 𝕜'] : Isometry (algebraMap 𝕜 𝕜') 
Arguments: (𝕜 : Type u_1) (𝕜' : Type u_2) [NormedField 𝕜] [SeminormedRing 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormOneClass 𝕜']
Goal: Isometry ⇑(algebraMap 𝕜 𝕜')
-- SeminormedAddCommGroup.Core.norm_smul
structure SeminormedAddCommGroup.Core (𝕜 : Type*) (E : Type*) [NormedField 𝕜] [AddCommGroup E]
    [Norm E] [Module 𝕜 E] : Prop where
  norm_nonneg (x : E) : 0 ≤ ‖x‖
  norm_smul (c : 𝕜) (x : E) : ‖c • x‖ = ‖c‖ * ‖x‖
  norm_triangle (x y : E) : ‖x + y‖ ≤ ‖x‖ + ‖y‖
Arguments: {𝕜 : Type u_6} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Norm E] [Module 𝕜 E] (self : SeminormedAddCommGroup.Core 𝕜 E) (c : 𝕜) (x : E)
Goal: ‖c • x‖ = ‖c‖ * ‖x‖
-- SeminormedAddCommGroup.Core.norm_nonneg
structure SeminormedAddCommGroup.Core (𝕜 : Type*) (E : Type*) [NormedField 𝕜] [AddCommGroup E]
    [Norm E] [Module 𝕜 E] : Prop where
  norm_nonneg (x : E) : 0 ≤ ‖x‖
  norm_smul (c : 𝕜) (x : E) : ‖c • x‖ = ‖c‖ * ‖x‖
  norm_triangle (x y : E) : ‖x + y‖ ≤ ‖x‖ + ‖y‖
Arguments: {𝕜 : Type u_6} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Norm E] [Module 𝕜 E] (self : SeminormedAddCommGroup.Core 𝕜 E) (x : E)
Goal: 0 ≤ ‖x‖
-- SeminormedAddCommGroup.Core.norm_triangle
structure SeminormedAddCommGroup.Core (𝕜 : Type*) (E : Type*) [NormedField 𝕜] [AddCommGroup E]
    [Norm E] [Module 𝕜 E] : Prop where
  norm_nonneg (x : E) : 0 ≤ ‖x‖
  norm_smul (c : 𝕜) (x : E) : ‖c • x‖ = ‖c‖ * ‖x‖
  norm_triangle (x y : E) : ‖x + y‖ ≤ ‖x‖ + ‖y‖
Arguments: {𝕜 : Type u_6} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Norm E] [Module 𝕜 E] (self : SeminormedAddCommGroup.Core 𝕜 E) (x y : E)
Goal: ‖x + y‖ ≤ ‖x‖ + ‖y‖
-- NormedSpace.Core.norm_eq_zero_iff
structure NormedSpace.Core (𝕜 : Type*) (E : Type*) [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
    [Norm E] extends SeminormedAddCommGroup.Core 𝕜 E : Prop where
  norm_eq_zero_iff (x : E) : ‖x‖ = 0 ↔ x = 0
Arguments: {𝕜 : Type u_6} {E : Type u_7} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [Norm E] (self : NormedSpace.Core 𝕜 E) (x : E)
Goal: ‖x‖ = 0 ↔ x = 0
-- ContinuousLinearMap.norm_map_tail_le
theorem ContinuousLinearMap.norm_map_tail_le
    (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G) (m : ∀ i, Ei i) :
    ‖f (m 0) (tail m)‖ ≤ ‖f‖ * ∏ i, ‖m i‖ 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei i.succ) G) (m : (i : Fin n.succ) → Ei i)
Goal: ‖(f (m 0)) (Fin.tail m)‖ ≤ ‖f‖ * ∏ i : Fin n.succ, ‖m i‖
-- ContinuousMultilinearMap.norm_map_init_le
theorem ContinuousMultilinearMap.norm_map_init_le
    (f : ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G))
    (m : ∀ i, Ei i) : ‖f (init m) (m (last n))‖ ≤ ‖f‖ * ∏ i, ‖m i‖ 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 (fun i => Ei i.castSucc) (Ei (Fin.last n) →L[𝕜] G)) (m : (i : Fin n.succ) → Ei i)
Goal: ‖(f (Fin.init m)) (m (Fin.last n))‖ ≤ ‖f‖ * ∏ i : Fin n.succ, ‖m i‖
-- ContinuousMultilinearMap.norm_map_cons_le
theorem ContinuousMultilinearMap.norm_map_cons_le (f : ContinuousMultilinearMap 𝕜 Ei G) (x : Ei 0)
    (m : ∀ i : Fin n, Ei i.succ) : ‖f (cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i, ‖m i‖ 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 Ei G) (x : Ei 0) (m : (i : Fin n) → Ei i.succ)
Goal: ‖f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖
-- ContinuousMultilinearMap.norm_map_snoc_le
theorem ContinuousMultilinearMap.norm_map_snoc_le (f : ContinuousMultilinearMap 𝕜 Ei G)
    (m : ∀ i : Fin n, Ei <| castSucc i) (x : Ei (last n)) :
    ‖f (snoc m x)‖ ≤ (‖f‖ * ∏ i, ‖m i‖) * ‖x‖ 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 Ei G) (m : (i : Fin n) → Ei i.castSucc) (x : Ei (Fin.last n))
Goal: ‖f (Fin.snoc m x)‖ ≤ (‖f‖ * ∏ i : Fin n, ‖m i‖) * ‖x‖
-- ContinuousLinearMap.uncurryLeft_apply
@[simp]
theorem ContinuousLinearMap.uncurryLeft_apply
    (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G) (m : ∀ i, Ei i) :
    f.uncurryLeft m = f (m 0) (tail m) 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei i.succ) G) (m : (i : Fin n.succ) → Ei i)
Goal: f.uncurryLeft m = (f (m 0)) (Fin.tail m)
-- ContinuousMultilinearMap.curryLeft_apply
@[simp]
theorem ContinuousMultilinearMap.curryLeft_apply (f : ContinuousMultilinearMap 𝕜 Ei G) (x : Ei 0)
    (m : ∀ i : Fin n, Ei i.succ) : f.curryLeft x m = f (cons x m) 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 Ei G) (x : Ei 0) (m : (i : Fin n) → Ei i.succ)
Goal: (f.curryLeft x) m = f (Fin.cons x m)
-- ContinuousLinearMap.curry_uncurryLeft
@[simp]
theorem ContinuousLinearMap.curry_uncurryLeft
    (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G) :
    f.uncurryLeft.curryLeft = f 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei i.succ) G)
Goal: f.uncurryLeft.curryLeft = f
-- ContinuousMultilinearMap.uncurry_curryLeft
@[simp]
theorem ContinuousMultilinearMap.uncurry_curryLeft (f : ContinuousMultilinearMap 𝕜 Ei G) :
    f.curryLeft.uncurryLeft = f 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 Ei G)
Goal: f.curryLeft.uncurryLeft = f
-- continuousMultilinearCurryLeftEquiv_apply
@[simp]
theorem continuousMultilinearCurryLeftEquiv_apply
    (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G) (v : ∀ i, Ei i) :
    continuousMultilinearCurryLeftEquiv 𝕜 Ei G f v = f (v 0) (tail v) 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei i.succ) G) (v : (i : Fin n.succ) → Ei i)
Goal: ((continuousMultilinearCurryLeftEquiv 𝕜 Ei G) f) v = (f (v 0)) (Fin.tail v)
-- continuousMultilinearCurryLeftEquiv_symm_apply
@[simp]
theorem continuousMultilinearCurryLeftEquiv_symm_apply (f : ContinuousMultilinearMap 𝕜 Ei G)
    (x : Ei 0) (v : ∀ i : Fin n, Ei i.succ) :
    (continuousMultilinearCurryLeftEquiv 𝕜 Ei G).symm f x v = f (cons x v) 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 Ei G) (x : Ei 0) (v : (i : Fin n) → Ei i.succ)
Goal: (((continuousMultilinearCurryLeftEquiv 𝕜 Ei G).symm f) x) v = f (Fin.cons x v)
-- ContinuousMultilinearMap.curryLeft_norm
@[simp]
theorem ContinuousMultilinearMap.curryLeft_norm (f : ContinuousMultilinearMap 𝕜 Ei G) :
    ‖f.curryLeft‖ = ‖f‖ 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 Ei G)
Goal: ‖f.curryLeft‖ = ‖f‖