ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContDiffBump.contDiffAt
protected theorem contDiffAt : ContDiffAt ℝ n f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) {x : E} {n : ℕ∞}
Goal: ContDiffAt ℝ n (↑f) x
-- ContDiffBump.contDiffWithinAt
protected theorem contDiffWithinAt {s : Set E} : ContDiffWithinAt ℝ n f s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) {x : E} {n : ℕ∞} {s : Set E}
Goal: ContDiffWithinAt ℝ n (↑f) s x
-- ContDiffBump.continuous
protected theorem continuous : Continuous f 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c)
Goal: Continuous ↑f
-- hasDerivAtFilter_iff_tendsto_slope
theorem hasDerivAtFilter_iff_tendsto_slope {x : 𝕜} {L : Filter 𝕜} :
    HasDerivAtFilter f f' x L ↔ Tendsto (slope f x) (L ⊓ 𝓟 {x}ᶜ) (𝓝 f') 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜}
Goal: HasDerivAtFilter f f' x L ↔ Filter.Tendsto (slope f x) (L ⊓ 𝓟 {x}ᶜ) (𝓝 f')
-- hasDerivWithinAt_iff_tendsto_slope
theorem hasDerivWithinAt_iff_tendsto_slope :
    HasDerivWithinAt f f' s x ↔ Tendsto (slope f x) (𝓝[s \ {x}] x) (𝓝 f') 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜}
Goal: HasDerivWithinAt f f' s x ↔ Filter.Tendsto (slope f x) (𝓝[s \ {x}] x) (𝓝 f')
-- hasDerivWithinAt_iff_tendsto_slope'
theorem hasDerivWithinAt_iff_tendsto_slope' (hs : x ∉ s) :
    HasDerivWithinAt f f' s x ↔ Tendsto (slope f x) (𝓝[s] x) (𝓝 f') 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (hs : x ∉ s)
Goal: HasDerivWithinAt f f' s x ↔ Filter.Tendsto (slope f x) (𝓝[s] x) (𝓝 f')
-- hasDerivAt_iff_tendsto_slope
theorem hasDerivAt_iff_tendsto_slope : HasDerivAt f f' x ↔ Tendsto (slope f x) (𝓝[≠] x) (𝓝 f') 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜}
Goal: HasDerivAt f f' x ↔ Filter.Tendsto (slope f x) (𝓝[≠] x) (𝓝 f')
-- hasDerivAt_iff_tendsto_slope_zero
theorem hasDerivAt_iff_tendsto_slope_zero :
    HasDerivAt f f' x ↔ Tendsto (fun t ↦ t⁻¹ • (f (x + t) - f x)) (𝓝[≠] 0) (𝓝 f') 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜}
Goal: HasDerivAt f f' x ↔ Filter.Tendsto (fun t => t⁻¹ • (f (x + t) - f x)) (𝓝[≠] 0) (𝓝 f')
-- HasDerivAt.tendsto_slope_zero_right
theorem HasDerivAt.tendsto_slope_zero_right [PartialOrder 𝕜] (h : HasDerivAt f f' x) :
    Tendsto (fun t ↦ t⁻¹ • (f (x + t) - f x)) (𝓝[>] 0) (𝓝 f') 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} [PartialOrder 𝕜] (h : HasDerivAt f f' x)
Goal: Filter.Tendsto (fun t => t⁻¹ • (f (x + t) - f x)) (𝓝[>] 0) (𝓝 f')
-- HasDerivAt.tendsto_slope_zero_left
theorem HasDerivAt.tendsto_slope_zero_left [PartialOrder 𝕜] (h : HasDerivAt f f' x) :
    Tendsto (fun t ↦ t⁻¹ • (f (x + t) - f x)) (𝓝[<] 0) (𝓝 f') 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} [PartialOrder 𝕜] (h : HasDerivAt f f' x)
Goal: Filter.Tendsto (fun t => t⁻¹ • (f (x + t) - f x)) (𝓝[<] 0) (𝓝 f')
-- range_derivWithin_subset_closure_span_image
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t))
Goal: Set.range (derivWithin f s) ⊆ closure ↑(Submodule.span 𝕜 (f '' t))
-- range_deriv_subset_closure_span_image
theorem range_deriv_subset_closure_span_image
    (f : 𝕜 → F) {t : Set 𝕜} (h : Dense t) :
    range (deriv f) ⊆ closure (Submodule.span 𝕜 (f '' t)) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : 𝕜 → F) {t : Set 𝕜} (h : Dense t)
Goal: Set.range (deriv f) ⊆ closure ↑(Submodule.span 𝕜 (f '' t))
-- isSeparable_range_derivWithin
theorem isSeparable_range_derivWithin [SeparableSpace 𝕜] (f : 𝕜 → F) (s : Set 𝕜) :
    IsSeparable (range (derivWithin f s)) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace.SeparableSpace 𝕜] (f : 𝕜 → F) (s : Set 𝕜)
Goal: TopologicalSpace.IsSeparable (Set.range (derivWithin f s))
-- isSeparable_range_deriv
theorem isSeparable_range_deriv [SeparableSpace 𝕜] (f : 𝕜 → F) :
    IsSeparable (range (deriv f)) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace.SeparableSpace 𝕜] (f : 𝕜 → F)
Goal: TopologicalSpace.IsSeparable (Set.range (deriv f))
-- HasDerivWithinAt.limsup_slope_le
theorem HasDerivWithinAt.limsup_slope_le (hf : HasDerivWithinAt f f' s x) (hr : f' < r) :
    ∀ᶠ z in 𝓝[s \ {x}] x, slope f x z < r 
Arguments: {f : ℝ → ℝ} {f' : ℝ} {s : Set ℝ} {x r : ℝ} (hf : HasDerivWithinAt f f' s x) (hr : f' < r)
Goal: ∀ᶠ (z : ℝ) in 𝓝[s \ {x}] x, slope f x z < r
-- HasDerivWithinAt.limsup_slope_le'
theorem HasDerivWithinAt.limsup_slope_le' (hf : HasDerivWithinAt f f' s x) (hs : x ∉ s)
    (hr : f' < r) : ∀ᶠ z in 𝓝[s] x, slope f x z < r 
Arguments: {f : ℝ → ℝ} {f' : ℝ} {s : Set ℝ} {x r : ℝ} (hf : HasDerivWithinAt f f' s x) (hs : x ∉ s) (hr : f' < r)
Goal: ∀ᶠ (z : ℝ) in 𝓝[s] x, slope f x z < r
-- HasDerivWithinAt.liminf_right_slope_le
theorem HasDerivWithinAt.liminf_right_slope_le (hf : HasDerivWithinAt f f' (Ici x) x)
    (hr : f' < r) : ∃ᶠ z in 𝓝[>] x, slope f x z < r 
Arguments: {f : ℝ → ℝ} {f' x r : ℝ} (hf : HasDerivWithinAt f f' (Set.Ici x) x) (hr : f' < r)
Goal: ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
-- HasDerivWithinAt.limsup_norm_slope_le
theorem HasDerivWithinAt.limsup_norm_slope_le (hf : HasDerivWithinAt f f' s x) (hr : ‖f'‖ < r) :
    ∀ᶠ z in 𝓝[s] x, ‖z - x‖⁻¹ * ‖f z - f x‖ < r 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {f' : E} {s : Set ℝ} {x r : ℝ} (hf : HasDerivWithinAt f f' s x) (hr : ‖f'‖ < r)
Goal: ∀ᶠ (z : ℝ) in 𝓝[s] x, ‖z - x‖⁻¹ * ‖f z - f x‖ < r
-- HasDerivWithinAt.limsup_slope_norm_le
theorem HasDerivWithinAt.limsup_slope_norm_le (hf : HasDerivWithinAt f f' s x) (hr : ‖f'‖ < r) :
    ∀ᶠ z in 𝓝[s] x, ‖z - x‖⁻¹ * (‖f z‖ - ‖f x‖) < r 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {f' : E} {s : Set ℝ} {x r : ℝ} (hf : HasDerivWithinAt f f' s x) (hr : ‖f'‖ < r)
Goal: ∀ᶠ (z : ℝ) in 𝓝[s] x, ‖z - x‖⁻¹ * (‖f z‖ - ‖f x‖) < r
-- HasDerivWithinAt.liminf_right_norm_slope_le
theorem HasDerivWithinAt.liminf_right_norm_slope_le (hf : HasDerivWithinAt f f' (Ici x) x)
    (hr : ‖f'‖ < r) : ∃ᶠ z in 𝓝[>] x, ‖z - x‖⁻¹ * ‖f z - f x‖ < r 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {f' : E} {x r : ℝ} (hf : HasDerivWithinAt f f' (Set.Ici x) x) (hr : ‖f'‖ < r)
Goal: ∃ᶠ (z : ℝ) in 𝓝[>] x, ‖z - x‖⁻¹ * ‖f z - f x‖ < r
-- HasDerivWithinAt.liminf_right_slope_norm_le
theorem HasDerivWithinAt.liminf_right_slope_norm_le (hf : HasDerivWithinAt f f' (Ici x) x)
    (hr : ‖f'‖ < r) : ∃ᶠ z in 𝓝[>] x, (z - x)⁻¹ * (‖f z‖ - ‖f x‖) < r 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {f' : E} {x r : ℝ} (hf : HasDerivWithinAt f f' (Set.Ici x) x) (hr : ‖f'‖ < r)
Goal: ∃ᶠ (z : ℝ) in 𝓝[>] x, (z - x)⁻¹ * (‖f z‖ - ‖f x‖) < r
-- ContinuousMultilinearMap.continuous_eval
theorem ContinuousMultilinearMap.continuous_eval {𝕜 ι : Type*} {E : ι → Type*} {F : Type*}
    [NormedField 𝕜] [Finite ι] [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
    [TopologicalSpace F] [AddCommGroup F] [TopologicalAddGroup F] [Module 𝕜 F] :
    Continuous fun p : ContinuousMultilinearMap 𝕜 E F × ∀ i, E i => p.1 p.2 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [Finite ι] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [TopologicalSpace F] [AddCommGroup F] [TopologicalAddGroup F] [Module 𝕜 F]
Goal: Continuous fun p => p.1 p.2
-- MultilinearMap.norm_map_coord_zero
lemma norm_map_coord_zero (hf : Continuous f) {m : ∀ i, E i} {i : ι} (hi : ‖m i‖ = 0) :
    ‖f m‖ = 0 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (hf : Continuous ⇑f) {m : (i : ι) → E i} {i : ι} (hi : ‖m i‖ = 0)
Goal: ‖f m‖ = 0
-- MultilinearMap.bound_of_shell_of_norm_map_coord_zero
theorem bound_of_shell_of_norm_map_coord_zero (hf₀ : ∀ {m i}, ‖m i‖ = 0 → ‖f m‖ = 0)
    {ε : ι → ℝ} {C : ℝ} (hε : ∀ i, 0 < ε i) {c : ι → 𝕜} (hc : ∀ i, 1 < ‖c i‖)
    (hf : ∀ m : ∀ i, E i, (∀ i, ε i / ‖c i‖ ≤ ‖m i‖) → (∀ i, ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖)
    (m : ∀ i, E i) : ‖f m‖ ≤ C * ∏ i, ‖m i‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (hf₀ : ∀ {m : (i : ι) → E i} {i : ι}, ‖m i‖ = 0 → ‖f m‖ = 0) {ε : ι → ℝ} {C : ℝ} (hε : ∀ (i : ι), 0 < ε i) {c : ι → 𝕜} (hc : ∀ (i : ι), 1 < ‖c i‖) (hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) (m : (i : ι) → E i)
Goal: ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
-- MultilinearMap.bound_of_shell_of_continuous
theorem bound_of_shell_of_continuous (hfc : Continuous f)
    {ε : ι → ℝ} {C : ℝ} (hε : ∀ i, 0 < ε i) {c : ι → 𝕜} (hc : ∀ i, 1 < ‖c i‖)
    (hf : ∀ m : ∀ i, E i, (∀ i, ε i / ‖c i‖ ≤ ‖m i‖) → (∀ i, ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖)
    (m : ∀ i, E i) : ‖f m‖ ≤ C * ∏ i, ‖m i‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (hfc : Continuous ⇑f) {ε : ι → ℝ} {C : ℝ} (hε : ∀ (i : ι), 0 < ε i) {c : ι → 𝕜} (hc : ∀ (i : ι), 1 < ‖c i‖) (hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) (m : (i : ι) → E i)
Goal: ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
-- MultilinearMap.exists_bound_of_continuous
theorem exists_bound_of_continuous (hf : Continuous f) :
    ∃ C : ℝ, 0 < C ∧ ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (hf : Continuous ⇑f)
Goal: ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
-- MultilinearMap.norm_image_sub_le_of_bound'
theorem norm_image_sub_le_of_bound' [DecidableEq ι] {C : ℝ} (hC : 0 ≤ C)
    (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) (m₁ m₂ : ∀ i, E i) :
    ‖f m₁ - f m₂‖ ≤ C * ∑ i, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) [DecidableEq ι] {C : ℝ} (hC : 0 ≤ C) (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) (m₁ m₂ : (i : ι) → E i)
Goal: ‖f m₁ - f m₂‖ ≤ C * ∑ i : ι, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖
-- MultilinearMap.norm_image_sub_le_of_bound
theorem norm_image_sub_le_of_bound {C : ℝ} (hC : 0 ≤ C) (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖)
    (m₁ m₂ : ∀ i, E i) :
    ‖f m₁ - f m₂‖ ≤ C * Fintype.card ι * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) {C : ℝ} (hC : 0 ≤ C) (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) (m₁ m₂ : (i : ι) → E i)
Goal: ‖f m₁ - f m₂‖ ≤ C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖
-- MultilinearMap.continuous_of_bound
theorem continuous_of_bound (C : ℝ) (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) : Continuous f 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (C : ℝ) (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖)
Goal: Continuous ⇑f
-- MultilinearMap.coe_mkContinuous
@[simp]
theorem coe_mkContinuous (C : ℝ) (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) : ⇑(f.mkContinuous C H) = f 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (C : ℝ) (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖)
Goal: ⇑(f.mkContinuous C H) = ⇑f
-- MultilinearMap.restr_norm_le
theorem restr_norm_le {k n : ℕ} (f : (MultilinearMap 𝕜 (fun _ : Fin n => G) G' : _))
    (s : Finset (Fin n)) (hk : s.card = k) (z : G) {C : ℝ} (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖)
    (v : Fin k → G) : ‖f.restr s hk z v‖ ≤ C * ‖z‖ ^ (n - k) * ∏ i, ‖v i‖ 
Arguments: {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] {k n : ℕ} (f : MultilinearMap 𝕜 (fun x => G) G') (s : Finset (Fin n)) (hk : s.card = k) (z : G) {C : ℝ} (H : ∀ (m : Fin n → G), ‖f m‖ ≤ C * ∏ i : Fin n, ‖m i‖) (v : Fin k → G)
Goal: ‖(f.restr s hk z) v‖ ≤ C * ‖z‖ ^ (n - k) * ∏ i : Fin k, ‖v i‖
-- ContinuousMultilinearMap.bound
theorem bound : ∃ C : ℝ, 0 < C ∧ ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G)
Goal: ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
-- ContinuousMultilinearMap.norm_def
theorem norm_def : ‖f‖ = sInf { c | 0 ≤ (c : ℝ) ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖ } 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G)
Goal: ‖f‖ = sInf {c | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i : ι, ‖m i‖}
-- ContinuousMultilinearMap.bounds_nonempty
theorem bounds_nonempty {f : ContinuousMultilinearMap 𝕜 E G} :
    ∃ c, c ∈ { c | 0 ≤ c ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖ } 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ContinuousMultilinearMap 𝕜 E G}
Goal: ∃ c, c ∈ {c | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i : ι, ‖m i‖}
-- ContinuousMultilinearMap.bounds_bddBelow
theorem bounds_bddBelow {f : ContinuousMultilinearMap 𝕜 E G} :
    BddBelow { c | 0 ≤ c ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖ } 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ContinuousMultilinearMap 𝕜 E G}
Goal: BddBelow {c | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i : ι, ‖m i‖}
-- ContinuousMultilinearMap.isLeast_opNorm
theorem isLeast_opNorm : IsLeast {c : ℝ | 0 ≤ c ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖} ‖f‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G)
Goal: IsLeast {c | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i : ι, ‖m i‖} ‖f‖
-- ContinuousMultilinearMap.opNorm_nonneg
theorem opNorm_nonneg : 0 ≤ ‖f‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G)
Goal: 0 ≤ ‖f‖
-- ContinuousMultilinearMap.le_opNorm
theorem le_opNorm : ‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i)
Goal: ‖f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖
-- ContinuousMultilinearMap.le_mul_prod_of_le_opNorm_of_le
theorem le_mul_prod_of_le_opNorm_of_le {C : ℝ} {b : ι → ℝ} (hC : ‖f‖ ≤ C) (hm : ∀ i, ‖m i‖ ≤ b i) :
    ‖f m‖ ≤ C * ∏ i, b i 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ContinuousMultilinearMap 𝕜 E G} {m : (i : ι) → E i} {C : ℝ} {b : ι → ℝ} (hC : ‖f‖ ≤ C) (hm : ∀ (i : ι), ‖m i‖ ≤ b i)
Goal: ‖f m‖ ≤ C * ∏ i : ι, b i
-- ContinuousMultilinearMap.le_opNorm_mul_prod_of_le
theorem le_opNorm_mul_prod_of_le {b : ι → ℝ} (hm : ∀ i, ‖m i‖ ≤ b i) : ‖f m‖ ≤ ‖f‖ * ∏ i, b i 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {m : (i : ι) → E i} {b : ι → ℝ} (hm : ∀ (i : ι), ‖m i‖ ≤ b i)
Goal: ‖f m‖ ≤ ‖f‖ * ∏ i : ι, b i
-- ContinuousMultilinearMap.le_opNorm_mul_pow_card_of_le
theorem le_opNorm_mul_pow_card_of_le {b : ℝ} (hm : ‖m‖ ≤ b) :
    ‖f m‖ ≤ ‖f‖ * b ^ Fintype.card ι 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {m : (i : ι) → E i} {b : ℝ} (hm : ‖m‖ ≤ b)
Goal: ‖f m‖ ≤ ‖f‖ * b ^ Fintype.card ι
-- ContinuousMultilinearMap.le_opNorm_mul_pow_of_le
theorem le_opNorm_mul_pow_of_le {n : ℕ} {Ei : Fin n → Type*} [∀ i, SeminormedAddCommGroup (Ei i)]
    [∀ i, NormedSpace 𝕜 (Ei i)] (f : ContinuousMultilinearMap 𝕜 Ei G) {m : ∀ i, Ei i} {b : ℝ}
    (hm : ‖m‖ ≤ b) : ‖f m‖ ≤ ‖f‖ * b ^ n 
Arguments: {𝕜 : Type u} {G : Type wG} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {n : ℕ} {Ei : Fin n → Type u_1} [(i : Fin n) → SeminormedAddCommGroup (Ei i)] [(i : Fin n) → NormedSpace 𝕜 (Ei i)] (f : ContinuousMultilinearMap 𝕜 Ei G) {m : (i : Fin n) → Ei i} {b : ℝ} (hm : ‖m‖ ≤ b)
Goal: ‖f m‖ ≤ ‖f‖ * b ^ n
-- ContinuousMultilinearMap.le_of_opNorm_le
theorem le_of_opNorm_le {C : ℝ} (h : ‖f‖ ≤ C) : ‖f m‖ ≤ C * ∏ i, ‖m i‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ContinuousMultilinearMap 𝕜 E G} (m : (i : ι) → E i) {C : ℝ} (h : ‖f‖ ≤ C)
Goal: ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
-- ContinuousMultilinearMap.ratio_le_opNorm
theorem ratio_le_opNorm : (‖f m‖ / ∏ i, ‖m i‖) ≤ ‖f‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i)
Goal: ‖f m‖ / ∏ i : ι, ‖m i‖ ≤ ‖f‖
-- ContinuousMultilinearMap.unit_le_opNorm
theorem unit_le_opNorm (h : ‖m‖ ≤ 1) : ‖f m‖ ≤ ‖f‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i) (h : ‖m‖ ≤ 1)
Goal: ‖f m‖ ≤ ‖f‖
-- ContinuousMultilinearMap.opNorm_le_bound
theorem opNorm_le_bound {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ m, ‖f m‖ ≤ M * ∏ i, ‖m i‖) : ‖f‖ ≤ M 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ M * ∏ i : ι, ‖m i‖)
Goal: ‖f‖ ≤ M
-- ContinuousMultilinearMap.opNorm_le_iff
theorem opNorm_le_iff {C : ℝ} (hC : 0 ≤ C) : ‖f‖ ≤ C ↔ ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {C : ℝ} (hC : 0 ≤ C)
Goal: ‖f‖ ≤ C ↔ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
-- ContinuousMultilinearMap.opNorm_add_le
theorem opNorm_add_le : ‖f + g‖ ≤ ‖f‖ + ‖g‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f g : ContinuousMultilinearMap 𝕜 E G)
Goal: ‖f + g‖ ≤ ‖f‖ + ‖g‖
-- ContinuousMultilinearMap.opNorm_zero
theorem opNorm_zero : ‖(0 : ContinuousMultilinearMap 𝕜 E G)‖ = 0 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G]
Goal: ‖0‖ = 0
-- ContinuousMultilinearMap.opNorm_smul_le
theorem opNorm_smul_le (c : 𝕜') : ‖c • f‖ ≤ ‖c‖ * ‖f‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {𝕜' : Type u_1} [NormedField 𝕜'] [NormedSpace 𝕜' G] [SMulCommClass 𝕜 𝕜' G] (c : 𝕜')
Goal: ‖c • f‖ ≤ ‖c‖ * ‖f‖
-- ContinuousMultilinearMap.opNorm_neg
theorem opNorm_neg : ‖-f‖ = ‖f‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G)
Goal: ‖-f‖ = ‖f‖
-- _private.Mathlib.Analysis.NormedSpace.Multilinear.Basic.0.ContinuousMultilinearMap.uniformity_eq_seminorm
private lemma uniformity_eq_seminorm :
    𝓤 (ContinuousMultilinearMap 𝕜 E G) = ⨅ r > 0, 𝓟 {f | ‖f.1 - f.2‖ < r} 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G]
Goal: 𝓤 (ContinuousMultilinearMap 𝕜 E G) = ⨅ r, ⨅ (_ : r > 0), 𝓟 {f | ‖f.1 - f.2‖ < r}
-- ContinuousMultilinearMap.le_opNNNorm
theorem le_opNNNorm : ‖f m‖₊ ≤ ‖f‖₊ * ∏ i, ‖m i‖₊ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i)
Goal: ‖f m‖₊ ≤ ‖f‖₊ * ∏ i : ι, ‖m i‖₊