ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- lp.mem_lp_const_smul
theorem mem_lp_const_smul (c : 𝕜) (f : lp E p) : c • (f : PreLp E) ∈ lp E p 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] {𝕜 : Type u_3} [NormedRing 𝕜] [(i : α) → Module 𝕜 (E i)] [∀ (i : α), BoundedSMul 𝕜 (E i)] (c : 𝕜) (f : ↥(lp E p))
Goal: c • ↑f ∈ lp E p
-- lp.coe_lpSubmodule
theorem coe_lpSubmodule : (lpSubmodule E p 𝕜).toAddSubgroup = lp E p 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] {𝕜 : Type u_3} [NormedRing 𝕜] [(i : α) → Module 𝕜 (E i)] [∀ (i : α), BoundedSMul 𝕜 (E i)]
Goal: (lpSubmodule E p 𝕜).toAddSubgroup = lp E p
-- lp.coeFn_smul
@[simp]
theorem coeFn_smul (c : 𝕜) (f : lp E p) : ⇑(c • f) = c • ⇑f 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] {𝕜 : Type u_3} [NormedRing 𝕜] [(i : α) → Module 𝕜 (E i)] [∀ (i : α), BoundedSMul 𝕜 (E i)] (c : 𝕜) (f : ↥(lp E p))
Goal: ↑(c • f) = c • ↑f
-- lp.norm_const_smul_le
theorem norm_const_smul_le (hp : p ≠ 0) (c : 𝕜) (f : lp E p) : ‖c • f‖ ≤ ‖c‖ * ‖f‖ 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] {𝕜 : Type u_3} [NormedRing 𝕜] [(i : α) → Module 𝕜 (E i)] [∀ (i : α), BoundedSMul 𝕜 (E i)] (hp : p ≠ 0) (c : 𝕜) (f : ↥(lp E p))
Goal: ‖c • f‖ ≤ ‖c‖ * ‖f‖
-- lp.norm_const_smul
theorem norm_const_smul (hp : p ≠ 0) {c : 𝕜} (f : lp E p) : ‖c • f‖ = ‖c‖ * ‖f‖ 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] {𝕜 : Type u_3} [NormedDivisionRing 𝕜] [(i : α) → Module 𝕜 (E i)] [∀ (i : α), BoundedSMul 𝕜 (E i)] (hp : p ≠ 0) {c : 𝕜} (f : ↥(lp E p))
Goal: ‖c • f‖ = ‖c‖ * ‖f‖
-- Memℓp.star_mem
theorem _root_.Memℓp.star_mem {f : ∀ i, E i} (hf : Memℓp f p) : Memℓp (star f) p 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] [(i : α) → StarAddMonoid (E i)] [∀ (i : α), NormedStarGroup (E i)] {f : (i : α) → E i} (hf : Memℓp f p)
Goal: Memℓp (star f) p
-- Memℓp.star_iff
@[simp]
theorem _root_.Memℓp.star_iff {f : ∀ i, E i} : Memℓp (star f) p ↔ Memℓp f p 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] [(i : α) → StarAddMonoid (E i)] [∀ (i : α), NormedStarGroup (E i)] {f : (i : α) → E i}
Goal: Memℓp (star f) p ↔ Memℓp f p
-- lp.coeFn_star
@[simp]
theorem coeFn_star (f : lp E p) : ⇑(star f) = star (⇑f) 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] [(i : α) → StarAddMonoid (E i)] [∀ (i : α), NormedStarGroup (E i)] (f : ↥(lp E p))
Goal: ↑(star f) = star ↑f
-- lp.star_apply
@[simp]
protected theorem star_apply (f : lp E p) (i : α) : star f i = star (f i) 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] [(i : α) → StarAddMonoid (E i)] [∀ (i : α), NormedStarGroup (E i)] (f : ↥(lp E p)) (i : α)
Goal: ↑(star f) i = star (↑f i)
-- Memℓp.infty_mul
theorem _root_.Memℓp.infty_mul {f g : ∀ i, B i} (hf : Memℓp f ∞) (hg : Memℓp g ∞) :
    Memℓp (f * g) ∞ 
Arguments: {I : Type u_3} {B : I → Type u_4} [(i : I) → NonUnitalNormedRing (B i)] {f g : (i : I) → B i} (hf : Memℓp f ⊤) (hg : Memℓp g ⊤)
Goal: Memℓp (f * g) ⊤
-- lp.infty_coeFn_mul
@[simp]
theorem infty_coeFn_mul (f g : lp B ∞) : ⇑(f * g) = ⇑f * ⇑g 
Arguments: {I : Type u_3} {B : I → Type u_4} [(i : I) → NonUnitalNormedRing (B i)] (f g : ↥(lp B ⊤))
Goal: ↑(f * g) = ↑f * ↑g
-- one_memℓp_infty
theorem _root_.one_memℓp_infty : Memℓp (1 : ∀ i, B i) ∞ 
Arguments: {I : Type u_3} {B : I → Type u_4} [(i : I) → NormedRing (B i)] [∀ (i : I), NormOneClass (B i)]
Goal: Memℓp 1 ⊤
-- Memℓp.infty_pow
theorem _root_.Memℓp.infty_pow {f : ∀ i, B i} (hf : Memℓp f ∞) (n : ℕ) : Memℓp (f ^ n) ∞ 
Arguments: {I : Type u_3} {B : I → Type u_4} [(i : I) → NormedRing (B i)] [∀ (i : I), NormOneClass (B i)] {f : (i : I) → B i} (hf : Memℓp f ⊤) (n : ℕ)
Goal: Memℓp (f ^ n) ⊤
-- natCast_memℓp_infty
theorem _root_.natCast_memℓp_infty (n : ℕ) : Memℓp (n : ∀ i, B i) ∞ 
Arguments: {I : Type u_3} {B : I → Type u_4} [(i : I) → NormedRing (B i)] [∀ (i : I), NormOneClass (B i)] (n : ℕ)
Goal: Memℓp ↑n ⊤
-- intCast_memℓp_infty
theorem _root_.intCast_memℓp_infty (z : ℤ) : Memℓp (z : ∀ i, B i) ∞ 
Arguments: {I : Type u_3} {B : I → Type u_4} [(i : I) → NormedRing (B i)] [∀ (i : I), NormOneClass (B i)] (z : ℤ)
Goal: Memℓp ↑z ⊤
-- lp.infty_coeFn_one
@[simp]
theorem infty_coeFn_one : ⇑(1 : lp B ∞) = 1 
Arguments: {I : Type u_3} {B : I → Type u_4} [(i : I) → NormedRing (B i)] [∀ (i : I), NormOneClass (B i)]
Goal: ↑1 = 1
-- lp.infty_coeFn_pow
@[simp]
theorem infty_coeFn_pow (f : lp B ∞) (n : ℕ) : ⇑(f ^ n) = (⇑f) ^ n 
Arguments: {I : Type u_3} {B : I → Type u_4} [(i : I) → NormedRing (B i)] [∀ (i : I), NormOneClass (B i)] (f : ↥(lp B ⊤)) (n : ℕ)
Goal: ↑(f ^ n) = ↑f ^ n
-- lp.infty_coeFn_natCast
@[simp]
theorem infty_coeFn_natCast (n : ℕ) : ⇑(n : lp B ∞) = n 
Arguments: {I : Type u_3} {B : I → Type u_4} [(i : I) → NormedRing (B i)] [∀ (i : I), NormOneClass (B i)] (n : ℕ)
Goal: ↑↑n = ↑n
-- lp.infty_coeFn_intCast
@[simp]
theorem infty_coeFn_intCast (z : ℤ) : ⇑(z : lp B ∞) = z 
Arguments: {I : Type u_3} {B : I → Type u_4} [(i : I) → NormedRing (B i)] [∀ (i : I), NormOneClass (B i)] (z : ℤ)
Goal: ↑↑z = ↑z
-- algebraMap_memℓp_infty
theorem _root_.algebraMap_memℓp_infty (k : 𝕜) : Memℓp (algebraMap 𝕜 (∀ i, B i) k) ∞ 
Arguments: {I : Type u_3} {𝕜 : Type u_4} {B : I → Type u_5} [NormedField 𝕜] [(i : I) → NormedRing (B i)] [(i : I) → NormedAlgebra 𝕜 (B i)] [∀ (i : I), NormOneClass (B i)] (k : 𝕜)
Goal: Memℓp ((algebraMap 𝕜 ((i : I) → B i)) k) ⊤
-- lp.single_apply
protected theorem single_apply (p) (i : α) (a : E i) (j : α) :
    lp.single p i a j = if h : j = i then Eq.ndrec a h.symm else 0 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] [DecidableEq α] (p : ℝ≥0∞) (i : α) (a : E i) (j : α)
Goal: ↑(lp.single p i a) j = if h : j = i then ⋯ ▸ a else 0
-- lp.single_apply_self
protected theorem single_apply_self (p) (i : α) (a : E i) : lp.single p i a i = a 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] [DecidableEq α] (p : ℝ≥0∞) (i : α) (a : E i)
Goal: ↑(lp.single p i a) i = a
-- lp.single_apply_ne
protected theorem single_apply_ne (p) (i : α) (a : E i) {j : α} (hij : j ≠ i) :
    lp.single p i a j = 0 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] [DecidableEq α] (p : ℝ≥0∞) (i : α) (a : E i) {j : α} (hij : j ≠ i)
Goal: ↑(lp.single p i a) j = 0
-- lp.single_neg
@[simp]
protected theorem single_neg (p) (i : α) (a : E i) : lp.single p i (-a) = -lp.single p i a 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] [DecidableEq α] (p : ℝ≥0∞) (i : α) (a : E i)
Goal: lp.single p i (-a) = -lp.single p i a
-- lp.single_smul
@[simp]
protected theorem single_smul (p) (i : α) (a : E i) (c : 𝕜) :
    lp.single p i (c • a) = c • lp.single p i a 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] {𝕜 : Type u_3} [NormedRing 𝕜] [(i : α) → Module 𝕜 (E i)] [∀ (i : α), BoundedSMul 𝕜 (E i)] [DecidableEq α] (p : ℝ≥0∞) (i : α) (a : E i) (c : 𝕜)
Goal: lp.single p i (c • a) = c • lp.single p i a
-- lp.norm_sum_single
protected theorem norm_sum_single (hp : 0 < p.toReal) (f : ∀ i, E i) (s : Finset α) :
    ‖∑ i ∈ s, lp.single p i (f i)‖ ^ p.toReal = ∑ i ∈ s, ‖f i‖ ^ p.toReal 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] [DecidableEq α] (hp : 0 < p.toReal) (f : (i : α) → E i) (s : Finset α)
Goal: ‖∑ i ∈ s, lp.single p i (f i)‖ ^ p.toReal = ∑ i ∈ s, ‖f i‖ ^ p.toReal
-- lp.norm_single
protected theorem norm_single (hp : 0 < p.toReal) (f : ∀ i, E i) (i : α) :
    ‖lp.single p i (f i)‖ = ‖f i‖ 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] [DecidableEq α] (hp : 0 < p.toReal) (f : (i : α) → E i) (i : α)
Goal: ‖lp.single p i (f i)‖ = ‖f i‖
-- lp.norm_sub_norm_compl_sub_single
protected theorem norm_sub_norm_compl_sub_single (hp : 0 < p.toReal) (f : lp E p) (s : Finset α) :
    ‖f‖ ^ p.toReal - ‖f - ∑ i ∈ s, lp.single p i (f i)‖ ^ p.toReal =
      ∑ i ∈ s, ‖f i‖ ^ p.toReal 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] [DecidableEq α] (hp : 0 < p.toReal) (f : ↥(lp E p)) (s : Finset α)
Goal: ‖f‖ ^ p.toReal - ‖f - ∑ i ∈ s, lp.single p i (↑f i)‖ ^ p.toReal = ∑ i ∈ s, ‖↑f i‖ ^ p.toReal
-- lp.norm_compl_sum_single
protected theorem norm_compl_sum_single (hp : 0 < p.toReal) (f : lp E p) (s : Finset α) :
    ‖f - ∑ i ∈ s, lp.single p i (f i)‖ ^ p.toReal = ‖f‖ ^ p.toReal - ∑ i ∈ s, ‖f i‖ ^ p.toReal 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] [DecidableEq α] (hp : 0 < p.toReal) (f : ↥(lp E p)) (s : Finset α)
Goal: ‖f - ∑ i ∈ s, lp.single p i (↑f i)‖ ^ p.toReal = ‖f‖ ^ p.toReal - ∑ i ∈ s, ‖↑f i‖ ^ p.toReal
-- lp.hasSum_single
protected theorem hasSum_single [Fact (1 ≤ p)] (hp : p ≠ ⊤) (f : lp E p) :
    HasSum (fun i : α => lp.single p i (f i : E i)) f 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] [DecidableEq α] [Fact (1 ≤ p)] (hp : p ≠ ⊤) (f : ↥(lp E p))
Goal: HasSum (fun i => lp.single p i (↑f i)) f
-- lp.uniformContinuous_coe
theorem uniformContinuous_coe [_i : Fact (1 ≤ p)] :
    UniformContinuous (α := lp E p) ((↑) : lp E p → ∀ i, E i) 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] [_i : Fact (1 ≤ p)]
Goal: UniformContinuous Subtype.val
-- lp.norm_apply_le_of_tendsto
theorem norm_apply_le_of_tendsto {C : ℝ} {F : ι → lp E ∞} (hCF : ∀ᶠ k in l, ‖F k‖ ≤ C)
    {f : ∀ a, E a} (hf : Tendsto (id fun i => F i : ι → ∀ a, E a) l (𝓝 f)) (a : α) : ‖f a‖ ≤ C 
Arguments: {α : Type u_1} {E : α → Type u_2} [(i : α) → NormedAddCommGroup (E i)] {ι : Type u_3} {l : Filter ι} [l.NeBot] {C : ℝ} {F : ι → ↥(lp E ⊤)} (hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C) {f : (a : α) → E a} (hf : Filter.Tendsto (id fun i => ↑(F i)) l (nhds f)) (a : α)
Goal: ‖f a‖ ≤ C
-- lp.sum_rpow_le_of_tendsto
theorem sum_rpow_le_of_tendsto (hp : p ≠ ∞) {C : ℝ} {F : ι → lp E p} (hCF : ∀ᶠ k in l, ‖F k‖ ≤ C)
    {f : ∀ a, E a} (hf : Tendsto (id fun i => F i : ι → ∀ a, E a) l (𝓝 f)) (s : Finset α) :
    ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C ^ p.toReal 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] {ι : Type u_3} {l : Filter ι} [l.NeBot] [_i : Fact (1 ≤ p)] (hp : p ≠ ⊤) {C : ℝ} {F : ι → ↥(lp E p)} (hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C) {f : (a : α) → E a} (hf : Filter.Tendsto (id fun i => ↑(F i)) l (nhds f)) (s : Finset α)
Goal: ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C ^ p.toReal
-- lp.norm_le_of_tendsto
theorem norm_le_of_tendsto {C : ℝ} {F : ι → lp E p} (hCF : ∀ᶠ k in l, ‖F k‖ ≤ C) {f : lp E p}
    (hf : Tendsto (id fun i => F i : ι → ∀ a, E a) l (𝓝 f)) : ‖f‖ ≤ C 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] {ι : Type u_3} {l : Filter ι} [l.NeBot] [_i : Fact (1 ≤ p)] {C : ℝ} {F : ι → ↥(lp E p)} (hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C) {f : ↥(lp E p)} (hf : Filter.Tendsto (id fun i => ↑(F i)) l (nhds ↑f))
Goal: ‖f‖ ≤ C
-- lp.memℓp_of_tendsto
theorem memℓp_of_tendsto {F : ι → lp E p} (hF : Bornology.IsBounded (Set.range F)) {f : ∀ a, E a}
    (hf : Tendsto (id fun i => F i : ι → ∀ a, E a) l (𝓝 f)) : Memℓp f p 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] {ι : Type u_3} {l : Filter ι} [l.NeBot] [_i : Fact (1 ≤ p)] {F : ι → ↥(lp E p)} (hF : Bornology.IsBounded (Set.range F)) {f : (a : α) → E a} (hf : Filter.Tendsto (id fun i => ↑(F i)) l (nhds f))
Goal: Memℓp f p
-- lp.tendsto_lp_of_tendsto_pi
theorem tendsto_lp_of_tendsto_pi {F : ℕ → lp E p} (hF : CauchySeq F) {f : lp E p}
    (hf : Tendsto (id fun i => F i : ℕ → ∀ a, E a) atTop (𝓝 f)) : Tendsto F atTop (𝓝 f) 
Arguments: {α : Type u_1} {E : α → Type u_2} {p : ℝ≥0∞} [(i : α) → NormedAddCommGroup (E i)] [_i : Fact (1 ≤ p)] {F : ℕ → ↥(lp E p)} (hF : CauchySeq F) {f : ↥(lp E p)} (hf : Filter.Tendsto (id fun i => ↑(F i)) Filter.atTop (nhds ↑f))
Goal: Filter.Tendsto F Filter.atTop (nhds f)
-- LipschitzWith.uniformly_bounded
lemma LipschitzWith.uniformly_bounded [PseudoMetricSpace α] (g : α → ι → ℝ) {K : ℝ≥0}
    (hg : ∀ i, LipschitzWith K (g · i)) (a₀ : α) (hga₀b : Memℓp (g a₀) ∞) (a : α) :
    Memℓp (g a) ∞ 
Arguments: {α : Type u_1} {ι : Type u_3} [PseudoMetricSpace α] (g : α → ι → ℝ) {K : ℝ≥0} (hg : ∀ (i : ι), LipschitzWith K fun x => g x i) (a₀ : α) (hga₀b : Memℓp (g a₀) ⊤) (a : α)
Goal: Memℓp (g a) ⊤
-- LipschitzOnWith.coordinate
theorem LipschitzOnWith.coordinate [PseudoMetricSpace α] (f : α → ℓ^∞(ι)) (s : Set α) (K : ℝ≥0) :
    LipschitzOnWith K f s ↔ ∀ i : ι, LipschitzOnWith K (fun a : α ↦ f a i) s 
Arguments: {α : Type u_1} {ι : Type u_3} [PseudoMetricSpace α] (f : α → ↥(lp (fun i => ℝ) ⊤)) (s : Set α) (K : ℝ≥0)
Goal: LipschitzOnWith K f s ↔ ∀ (i : ι), LipschitzOnWith K (fun a => ↑(f a) i) s
-- LipschitzWith.coordinate
theorem LipschitzWith.coordinate [PseudoMetricSpace α] {f : α → ℓ^∞(ι)} (K : ℝ≥0) :
    LipschitzWith K f ↔ ∀ i : ι, LipschitzWith K (fun a : α ↦ f a i) 
Arguments: {α : Type u_1} {ι : Type u_3} [PseudoMetricSpace α] {f : α → ↥(lp (fun i => ℝ) ⊤)} (K : ℝ≥0)
Goal: LipschitzWith K f ↔ ∀ (i : ι), LipschitzWith K fun a => ↑(f a) i
-- taylorWithin_succ
theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
    taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
      PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
      (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ)
Goal: taylorWithin f (n + 1) s x₀ =
  taylorWithin f n s x₀ +
    (PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))
      ((PolynomialModule.single ℝ (n + 1)) (taylorCoeffWithin f (n + 1) s x₀))
-- taylorWithinEval_succ
@[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
    taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
      (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ)
Goal: taylorWithinEval f (n + 1) s x₀ x =
  taylorWithinEval f n s x₀ x + (((↑n + 1) * ↑n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀
-- taylor_within_zero_eval
@[simp]
theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) :
    taylorWithinEval f 0 s x₀ x = f x₀ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ)
Goal: taylorWithinEval f 0 s x₀ x = f x₀
-- taylorWithinEval_self
@[simp]
theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
    taylorWithinEval f n s x₀ x₀ = f x₀ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ)
Goal: taylorWithinEval f n s x₀ x₀ = f x₀
-- taylor_within_apply
theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
    taylorWithinEval f n s x₀ x =
      ∑ k ∈ Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ)
Goal: taylorWithinEval f n s x₀ x = ∑ k ∈ Finset.range (n + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀
-- continuousOn_taylorWithinEval
theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
    (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
    ContinuousOn (fun t => taylorWithinEval f n s t x) s 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ (↑n) f s)
Goal: ContinuousOn (fun t => taylorWithinEval f n s t x) s
-- monomial_has_deriv_aux
theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
    HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t 
Arguments: (t x : ℝ) (n : ℕ)
Goal: HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(↑n + 1) * (x - t) ^ n) t
-- hasDerivWithinAt_taylor_coeff_within
theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
    (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
    (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
    HasDerivWithinAt
      (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z)
      ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -
        ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y)
Goal: HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z)
  ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -
    ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y)
  t y
-- hasDerivWithinAt_taylorWithinEval
theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
    (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
    (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s)
    (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) :
    HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
      (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ (↑n) f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y)
Goal: HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s'
  y
-- taylorWithinEval_hasDerivAt_Ioo
theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b)
    (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b))
    (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
    HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
      (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Set.Ioo a b) (hf : ContDiffOn ℝ (↑n) f (Set.Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Set.Icc a b)) (Set.Ioo a b))
Goal: HasDerivAt (fun y => taylorWithinEval f n (Set.Icc a b) y x)
  (((↑n !)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Set.Icc a b) t) t
-- hasDerivWithinAt_taylorWithinEval_at_Icc
theorem hasDerivWithinAt_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ}
    (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b))
    (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
    HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
      (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Set.Icc a b) (hf : ContDiffOn ℝ (↑n) f (Set.Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Set.Icc a b)) (Set.Icc a b))
Goal: HasDerivWithinAt (fun y => taylorWithinEval f n (Set.Icc a b) y x)
  (((↑n !)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Set.Icc a b) t) (Set.Icc a b) t
-- taylor_mean_remainder
theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x)
    (hf : ContDiffOn ℝ n f (Icc x₀ x))
    (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x))
    (gcont : ContinuousOn g (Icc x₀ x))
    (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1)
    (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) :
    ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x =
    ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' 
Arguments: {f g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ (↑n) f (Set.Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Set.Icc x₀ x)) (Set.Ioo x₀ x)) (gcont : ContinuousOn g (Set.Icc x₀ x)) (gdiff : ∀ x_1 ∈ Set.Ioo x₀ x, HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 ∈ Set.Ioo x₀ x, g' x_1 ≠ 0)
Goal: ∃ x' ∈ Set.Ioo x₀ x,
  f x - taylorWithinEval f n (Set.Icc x₀ x) x₀ x =
    ((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Set.Icc x₀ x) x'
-- taylor_mean_remainder_lagrange
theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x)
    (hf : ContDiffOn ℝ n f (Icc x₀ x))
    (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) :
    ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x =
      iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! 
Arguments: {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ (↑n) f (Set.Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Set.Icc x₀ x)) (Set.Ioo x₀ x))
Goal: ∃ x' ∈ Set.Ioo x₀ x,
  f x - taylorWithinEval f n (Set.Icc x₀ x) x₀ x =
    iteratedDerivWithin (n + 1) f (Set.Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!
-- taylor_mean_remainder_cauchy
theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x)
    (hf : ContDiffOn ℝ n f (Icc x₀ x))
    (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) :
    ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x =
      iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) 
Arguments: {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ (↑n) f (Set.Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Set.Icc x₀ x)) (Set.Ioo x₀ x))
Goal: ∃ x' ∈ Set.Ioo x₀ x,
  f x - taylorWithinEval f n (Set.Icc x₀ x) x₀ x =
    iteratedDerivWithin (n + 1) f (Set.Icc x₀ x) x' * (x - x') ^ n / ↑n ! * (x - x₀)
-- taylor_mean_remainder_bound
theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b)
    (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b)
    (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) :
    ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (↑n + 1) f (Set.Icc a b)) (hx : x ∈ Set.Icc a b) (hC : ∀ y ∈ Set.Icc a b, ‖iteratedDerivWithin (n + 1) f (Set.Icc a b) y‖ ≤ C)
Goal: ‖f x - taylorWithinEval f n (Set.Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / ↑n !
-- exists_taylor_mean_remainder_bound
theorem exists_taylor_mean_remainder_bound {f : ℝ → E} {a b : ℝ} {n : ℕ} (hab : a ≤ b)
    (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) :
    ∃ C, ∀ x ∈ Icc a b, ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (↑n + 1) f (Set.Icc a b))
Goal: ∃ C, ∀ x ∈ Set.Icc a b, ‖f x - taylorWithinEval f n (Set.Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1)