mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞
Hcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s
Hdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s
⊢ ContDiffOn 𝕜 (↑n) f s

Tactic:
intro x hx m hm

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞
Hcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s
Hdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s
x : E
hx : x ∈ s
m : ℕ
hm : ↑m ≤ n
⊢ ∃ u ∈ 𝓝[insert x s] x, ∃ p, HasFTaylorSeriesUpToOn (↑m) f p u
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case zero_eq
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞
Hcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s
Hdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s
x : E
hx : x ∈ s
m : ℕ
hm : ↑m ≤ n
⊢ ∀ x ∈ s, (ftaylorSeriesWithin 𝕜 f s x 0).curry0 = f x

Tactic:
intro y

Goal state after:
case zero_eq
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞
Hcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s
Hdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x ↦ iteratedFDerivWithin 𝕜 m f s x) s
x : E
hx : x ∈ s
m : ℕ
hm : ↑m ≤ n
y : E
⊢ y ∈ s → (ftaylorSeriesWithin 𝕜 f s y 0).curry0 = f y
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case refine_1.zero_eq
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
h : ∀ (m : ℕ), AnalyticOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s
x : E
hx : x ∈ s
⊢ ∀ x ∈ s, (ftaylorSeriesWithin 𝕜 f s x 0).curry0 = f x

Tactic:
intro y

Goal state after:
case refine_1.zero_eq
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
h : ∀ (m : ℕ), AnalyticOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s
x : E
hx : x ∈ s
y : E
⊢ y ∈ s → (ftaylorSeriesWithin 𝕜 f s y 0).curry0 = f y
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case mp
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ
u : Set E
H : u ∈ 𝓝 x
f' : E → E →L[𝕜] F
h_fderiv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
h_cont_diff : ContDiffWithinAt 𝕜 (↑n) f' u x
t : Set E
htu : t ⊆ u
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
hxt : x ∈ t
⊢ ∀ x ∈ t, HasFDerivAt f (f' x) x

Tactic:
intro y

Goal state after:
case mp
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ
u : Set E
H : u ∈ 𝓝 x
f' : E → E →L[𝕜] F
h_fderiv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
h_cont_diff : ContDiffWithinAt 𝕜 (↑n) f' u x
t : Set E
htu : t ⊆ u
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
hxt : x ∈ t
y : E
⊢ y ∈ t → HasFDerivAt f (f' y) y
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case h
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
H : ContDiffOn 𝕜 ω f univ
i : ℕ
⊢ AnalyticOnNhd 𝕜 (fun x ↦ ftaylorSeriesWithin 𝕜 f univ x i) univ

Tactic:
← analyticOn_univ

Goal state after:
case h
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
H : ContDiffOn 𝕜 ω f univ
i : ℕ
⊢ AnalyticOn 𝕜 (fun x ↦ ftaylorSeriesWithin 𝕜 f univ x i) univ
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case inl.refine_1
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
x : Fin n
j : Fin (c.partSize (c.index 0))
hij : c.emb (c.index 0) j = x.succ
j_ne : j ≠ 0
je_ne' : ↑j ≠ 0
A : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
B : update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0) = c.partSize (c.index 0) - 1
⊢ Fin.cast ⋯ j ≠ 0

pp: case inl.refine_2
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
x : Fin n
j : Fin (c.partSize (c.index 0))
hij : c.emb (c.index 0) j = x.succ
j_ne : j ≠ 0
je_ne' : ↑j ≠ 0
A : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
B : update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0) = c.partSize (c.index 0) - 1
⊢ (c.emb (c.index 0) (Fin.cast ⋯ (Fin.cast ⋯ ((Fin.cast ⋯ j).pred ?inl.refine_1)).succ)).pred ⋯ = x

Tactic:
· simpa using j_ne

Goal state after:
case inl.refine_2
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
x : Fin n
j : Fin (c.partSize (c.index 0))
hij : c.emb (c.index 0) j = x.succ
j_ne : j ≠ 0
je_ne' : ↑j ≠ 0
A : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
B : update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0) = c.partSize (c.index 0) - 1
⊢ (c.emb (c.index 0) (Fin.cast ⋯ (Fin.cast ⋯ ((Fin.cast ⋯ j).pred ⋯)).succ)).pred ⋯ = x
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: case inr
𝔸 : Type u_2
inst✝⁵ : Ring 𝔸
inst✝⁴ : TopologicalSpace 𝔸
inst✝³ : IsTopologicalRing 𝔸
inst✝² : T2Space 𝔸
inst✝¹ : StarRing 𝔸
inst✝ : ContinuousStar 𝔸
x : 𝔸
val✝ : Algebra ℚ 𝔸
⊢ star (∑' (n : ℕ), (↑n !)⁻¹ • x ^ n) = ∑' (n : ℕ), star ((↑n !)⁻¹ • x ^ n)

Tactic:
← tsum_star

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p✝ : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
f : F [×c✝.length]→L[𝕜] G
p : (i : Fin c✝.length) → E [×c✝.partSize i]→L[𝕜] F
v : Fin n → E
i : Fin n
c : 𝕜
x : E
⊢ f (c✝.applyOrderedFinpartition p (update v i (c • x))) = c • f (c✝.applyOrderedFinpartition p (update v i x))

Tactic:
by
        simp only [applyOrderedFinpartition_update_right,
          ContinuousMultilinearMap.map_update_smul]

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝔸 : Type u_2
inst✝² : NormedCommRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
ι : Type u_3
s : Finset ι
f : ι → 𝔸
⊢ exp (∑ i ∈ s, f i) = ∏ i ∈ s, exp (f i)

Tactic:
by
  rw [exp_sum_of_commute, Finset.noncommProd_eq_prod]
  exact fun i _hi j _hj _ => Commute.all _ _

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
s : Set 𝕜
f : 𝕜 → F
hn : 0 < n
c : F
⊢ iteratedDerivWithin n (fun z ↦ c - f z) s x = iteratedDerivWithin n (fun z ↦ -f z) s x

Tactic:
obtain ⟨n, rfl⟩ := n.exists_eq_succ_of_ne_zero hn.ne'
  rw [iteratedDerivWithin_succ', iteratedDerivWithin_succ']
  congr 1 with y
  rw [derivWithin.fun_neg]
  exact derivWithin_const_sub _

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Exponential

Goal state before:
pp: case e'_12
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedRing 𝔸
inst✝² : CharZero 𝕂
inst✝¹ : NormedAlgebra 𝕂 𝔸
inst✝ : CompleteSpace 𝔸
h : 0 < (expSeries 𝕂 𝔸).radius
e_4✝ : inst✝³.toAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup.toAddCommGroup
e_5✝ : (NormedAlgebra.toNormedSpace 𝔸).toModule ≍ (NormedAlgebra.toNormedSpace 𝔸).toModule
x : 𝔸
⊢ x = (expSeries 𝕂 𝔸 1) fun x_1 ↦ x

Tactic:
simp [expSeries_apply_eq, Nat.factorial]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
α : Type u_5
H : Type u_6
inst✝ : NormedAddCommGroup H
l : Filter α
p₁ p₂ : α → FormalMultilinearSeries 𝕜 F G
q₁ q₂ : α → FormalMultilinearSeries 𝕜 E F
f : α → H
n : ℕ
hp_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖p₁ x k‖
hpf : ∀ k ≤ n, (fun a ↦ p₁ a k - p₂ a k) =O[l] f
hq₁_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₁ x k‖
hq₂_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₂ x k‖
hqf : ∀ k ≤ n, (fun a ↦ q₁ a k - q₂ a k) =O[l] f
⊢ (fun a ↦ (p₁ a).taylorComp (q₁ a) n - (p₂ a).taylorComp (q₂ a) n) =O[l] f

Tactic:
simp only [FormalMultilinearSeries.taylorComp, ← Finset.sum_sub_distrib]
  refine .fun_sum fun c _ ↦ ?_
  refine .trans (.of_norm_le fun _ ↦
    c.norm_compAlongOrderedFinpartition_sub_compAlongOrderedFinpartition_le ..) ?_
  refine .add ?_ ?_
  · have H₁ : (p₁ · c.length) =O[l] (1 : α → ℝ) := (hp_bdd _ c.length_le).isBigO_one ℝ
    have H₂ : ∀ m, (q₁ · (c.partSize m)) =O[l] (1 : α → ℝ) := fun m ↦
      (hq₁_bdd _ <| c.partSize_le _).isBigO_one ℝ
    have H₃ : ∀ m, (q₂ · (c.partSize m)) =O[l] (1 : α → ℝ) := fun m ↦
      (hq₂_bdd _ <| c.partSize_le _).isBigO_one ℝ
    have H₄ : ∀ m, (fun a ↦ q₁ a (c.partSize m) - q₂ a (c.partSize m)) =O[l] f := fun m ↦
      hqf _ <| c.partSize_le _
    rw [← Asymptotics.isBigO_pi] at H₂ H₃ H₄
    have H₅ := ((H₂.prod_left H₃).norm_left.pow (c.length - 1)).mul H₄.norm_norm
    simpa [mul_assoc] using! H₁.norm_left.mul <| H₅.const_mul_left c.length
  · have H₁ : (fun a ↦ p₁ a c.length - p₂ a c.length) =O[l] f := hpf _ c.length_le
    have H₂ : ∀ i, (q₂ · (c.partSize i)) =O[l] (1 : α → ℝ) := fun i ↦
      (hq₂_bdd _ <| c.partSize_le i).isBigO_one ℝ
    simpa using H₁.norm_norm.mul <| .finsetProd fun i _ ↦ (H₂ i).norm_left

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
x' : E
hx' : x' ∈ ball x (r' - s)
B : ball x' s ⊆ ball x r'
⊢ ∀ y ∈ ball x' s, ∀ z ∈ ball x' s, ‖f z - f y - L (z - y)‖ < ε * r

Tactic:
intro y

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
x' : E
hx' : x' ∈ ball x (r' - s)
B : ball x' s ⊆ ball x r'
y : E
⊢ y ∈ ball x' s → ∀ z ∈ ball x' s, ‖f z - f y - L (z - y)‖ < ε * r
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
c : 𝕜
hc : 1 < ‖c‖
r ε : ℝ
hε : 0 < ε
hr : 0 < r
x : E
L₁ L₂ : E →L[𝕜] F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
⊢ ∀ (x : E), r / 2 / ‖c‖ ≤ ‖x‖ → ‖x‖ < r / 2 → ‖(L₁ - L₂) x‖ ≤ 4 * ‖c‖ * ε * ‖x‖

Tactic:
intro y

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
c : 𝕜
hc : 1 < ‖c‖
r ε : ℝ
hε : 0 < ε
hr : 0 < r
x : E
L₁ L₂ : E →L[𝕜] F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
y : E
⊢ r / 2 / ‖c‖ ≤ ‖y‖ → ‖y‖ < r / 2 → ‖(L₁ - L₂) y‖ ≤ 4 * ‖c‖ * ε * ‖y‖
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_7
M : Type u_8
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
inst✝³ : AddLeftStrictMono R
inst✝² : AddCommGroup M
inst✝¹ : Nontrivial M
inst✝ : Module R M
C : ConvexCone R M
h : Submodule.span R ↑C = ⊤
⊢ C.IsReproducing

Tactic:
rw [IsReproducing, Set.eq_univ_iff_forall]

Goal state after:
R : Type u_7
M : Type u_8
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
inst✝³ : AddLeftStrictMono R
inst✝² : AddCommGroup M
inst✝¹ : Nontrivial M
inst✝ : Module R M
C : ConvexCone R M
h : Submodule.span R ↑C = ⊤
⊢ ∀ (x : M), x ∈ ↑C - ↑C
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)
F' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)
hI : Set.Icc (-|x|) |x| ⊆ Set.Ioo (-1) 1
⊢ ∀ y ∈ Set.Ioo (-1) 1, HasDerivAt F (F' y) y

Tactic:
intro y

Goal state after:
x : ℝ
h : |x| < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)
F' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)
hI : Set.Icc (-|x|) |x| ⊆ Set.Ioo (-1) 1
y : ℝ
⊢ y ∈ Set.Ioo (-1) 1 → HasDerivAt F (F' y) y
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h₀ : 0 ≤ x
h : x < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)
F' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)
⊢ ∀ y ∈ Set.Icc 0 x, HasDerivAt F (F' y) y

Tactic:
intro y

Goal state after:
x : ℝ
h₀ : 0 ≤ x
h : x < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)
F' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)
y : ℝ
⊢ y ∈ Set.Icc 0 x → HasDerivAt F (F' y) y
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h₀ : 0 ≤ x
h : x < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)
F' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)
A : ∀ y ∈ Set.Icc 0 x, HasDerivAt F (F' y) y
⊢ ∀ x_1 ∈ interior (Set.Icc 0 x), 0 ≤ F' x_1

Tactic:
intro y

Goal state after:
x : ℝ
h₀ : 0 ≤ x
h : x < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)
F' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)
A : ∀ y ∈ Set.Icc 0 x, HasDerivAt F (F' y) y
y : ℝ
⊢ y ∈ interior (Set.Icc 0 x) → 0 ≤ F' y
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
⊢ HasSum (fun k ↦ 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)) (log (1 + x) - log (1 - x))

Tactic:
set term := fun n : ℕ => -1 * ((-x) ^ (n + 1) / ((n : ℝ) + 1)) + x ^ (n + 1) / (n + 1)

Goal state after:
x : ℝ
h : |x| < 1
term : ℕ → ℝ := fun n ↦ -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)
⊢ HasSum (fun k ↦ 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)) (log (1 + x) - log (1 - x))
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0
int_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ∫⁻ (x : α), ↑(f x) ∂μ < ∫⁻ (x : α), ↑(f x) ∂μ + ε / 2
| ∫⁻ (x : α), ↑(f x) ∂μ + ε / 2

Tactic:
rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0
int_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ∫⁻ (x : α), ↑(f x) ∂μ < ∫⁻ (x : α), ↑(f x) ∂μ + ε / 2
| (⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ ↑(f x)), (SimpleFunc.map ENNReal.ofNNReal φ).lintegral μ) + ε / 2
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
x : E
a : ℝ
hs₀ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : ℝ⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s
hs₂ : Absorbs ℝ s {x}
hx : x ∉ a • s
⊢ a ≤ gauge s x

Tactic:
obtain ⟨r, hr, h⟩ := hs₂.exists_pos

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
x : E
a : ℝ
hs₀ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : ℝ⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s
hs₂ : Absorbs ℝ s {x}
hx : x ∉ a • s
r : ℝ
hr : r > 0
h : ∀ (c : ℝ), r ≤ ‖c‖ → {x} ⊆ c • s
⊢ a ≤ gauge s x
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
x : E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hc : Convex ℝ s
hs₀ : 0 ∈ s
ha : Absorbent ℝ s
h : gauge s x ≤ 1
r : ℝ
hr₀ : 0 ≤ r
hr₁ : r < 1
⊢ gauge s (r • x) < 1

Tactic:
gauge_smul_of_nonneg hr₀

Goal state after:
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
x : E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hc : Convex ℝ s
hs₀ : 0 ∈ s
ha : Absorbent ℝ s
h : gauge s x ≤ 1
r : ℝ
hr₀ : 0 ≤ r
hr₁ : r < 1
⊢ r • gauge s x < 1
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hs : s ∈ 𝓝 0
⊢ ContinuousAt (gauge s) 0

Tactic:
by
  rw [ContinuousAt, gauge_zero]
  exact tendsto_gauge_nhds_zero hs

Goal state after:
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s : Set E
x : E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hc : Convex ℝ s
hs₀ : s ∈ 𝓝 0
ha : Absorbent ℝ s
ε : ℝ
hε₀ : 0 < ε
this : ε • s ∩ -(ε • s) ∈ 𝓝 0
⊢ ∀ᶠ (a : E) in 𝓝 0, gauge s (x + a) ∈ Icc (gauge s x - ε) (gauge s x + ε)

Tactic:
filter_upwards [this] with y hy

Goal state after:
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s : Set E
x : E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hc : Convex ℝ s
hs₀ : s ∈ 𝓝 0
ha : Absorbent ℝ s
ε : ℝ
hε₀ : 0 < ε
this : ε • s ∩ -(ε • s) ∈ 𝓝 0
y : E
hy : y ∈ ε • s ∩ -(ε • s)
⊢ gauge s (x + y) ∈ Icc (gauge s x - ε) (gauge s x + ε)
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
p : Seminorm ℝ E
⊢ gauge (p.ball 0 1) = ⇑p

Tactic:
ext x
  obtain hp | hp := { r : ℝ | 0 < r ∧ x ∈ r • p.ball 0 1 }.eq_empty_or_nonempty
  · rw [gauge, hp, Real.sInf_empty]
    by_contra h
    have hpx : 0 < p x := (apply_nonneg _ _).lt_of_ne h
    have hpx₂ : 0 < 2 * p x := mul_pos zero_lt_two hpx
    refine hp.subset ⟨hpx₂, (2 * p x)⁻¹ • x, ?_, smul_inv_smul₀ hpx₂.ne' _⟩
    rw [p.mem_ball_zero, map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos (inv_pos.2 hpx₂),
      inv_mul_lt_iff₀ hpx₂, mul_one]
    exact lt_mul_of_one_lt_left hpx one_lt_two
  refine IsGLB.csInf_eq ⟨fun r => ?_, fun r hr => le_of_forall_pos_le_add fun ε hε => ?_⟩ hp
  · rintro ⟨hr, y, hy, rfl⟩
    rw [p.mem_ball_zero] at hy
    rw [map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos hr]
    exact mul_le_of_le_one_right hr.le hy.le
  · have hpε : 0 < p x + ε := by positivity
    refine hr ⟨hpε, (p x + ε)⁻¹ • x, ?_, smul_inv_smul₀ hpε.ne' _⟩
    rw [p.mem_ball_zero, map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos (inv_pos.2 hpε),
      inv_mul_lt_iff₀ hpε, mul_one]
    exact lt_add_of_pos_right _ hε

Goal state after:
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr✝ : 0 ≤ r
x : E
hr : 0 < r
⊢ gauge (ball 0 r) x = ‖x‖ / r

Tactic:
rw [← smul_unitBall_of_pos hr, gauge_smul_left, Pi.smul_apply, gauge_unit_ball, smul_eq_mul,
    abs_of_nonneg hr.le, div_eq_inv_mul]
    simp_rw [mem_ball_zero_iff, norm_neg]
    exact fun _ => id

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case right
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
L : F
r ε x r' : ℝ
rr' : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ Icc x (x + r'), ∀ z ∈ Icc x (x + r'), ‖f z - f y - (z - y) • L‖ ≤ ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
x' : ℝ
hx' : x' ∈ Ioo x (x + r' - s)
A : Icc x' (x' + s) ⊆ Icc x (x + r')
⊢ ∀ y ∈ Icc x' (x' + s), ∀ z ∈ Icc x' (x' + s), ‖f z - f y - (z - y) • L‖ ≤ ε * r

Tactic:
intro y

Goal state after:
case right
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
L : F
r ε x r' : ℝ
rr' : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ Icc x (x + r'), ∀ z ∈ Icc x (x + r'), ‖f z - f y - (z - y) • L‖ ≤ ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
x' : ℝ
hx' : x' ∈ Ioo x (x + r' - s)
A : Icc x' (x' + s) ⊆ Icc x (x + r')
y : ℝ
⊢ y ∈ Icc x' (x' + s) → ∀ z ∈ Icc x' (x' + s), ‖f z - f y - (z - y) • L‖ ≤ ε * r
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this : HasDerivWithinAt f (derivWithin f (Ici x) x) (Ici x) x
⊢ ∃ R > 0, ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ε

Tactic:
simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
⊢ ∃ R > 0, ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ε
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * (1 / 2) ^ e
J2 : ‖L e p r - L e' p' r‖ ≤ 4 * (1 / 2) ^ e
J3 : ‖L e' p' r - L e' p' q'‖ ≤ 4 * (1 / 2) ^ e
⊢ ‖L e p q - L e p r + (L e p r - L e' p' r) + (L e' p' r - L e' p' q')‖ ≤
    ‖L e p q - L e p r‖ + ‖L e p r - L e' p' r‖ + ‖L e' p' r - L e' p' q'‖

Tactic:
by
        grw [norm_add_le, norm_add_le]

Goal state after: