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

Goal state before:
pp: case pos.length
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✝
n : ℕ
c : OrderedFinpartition (n + 1)
h : range (c.emb 0) = {0}
A : c.length - 1 + 1 = c.length
⊢ { length := c.length - 1 + 1, partSize := cons 1 fun i ↦ c.partSize (Fin.cast ⋯ i.succ), partSize_pos := ⋯,
        emb := fun i ↦
          cases (motive := fun x ↦ Fin (cons 1 (fun i ↦ c.partSize (Fin.cast ⋯ i.succ)) x) → Fin (n + 1)) (fun x ↦ 0)
            (fun m ↦ succ ∘ fun j ↦ (c.emb (Fin.cast ⋯ m.succ) j).pred ⋯) i,
        emb_strictMono := ⋯, parts_strictMono := ⋯, disjoint := ⋯, cover := ⋯ }.length =
    c.length

Tactic:
exact A

Goal state after:
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), (↑n !)⁻¹ • star (x ^ n)

Tactic:
← star_inv_natCast_smul,

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

Goal state before:
pp: case pos.e_6.hf
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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
j : Fin n
v : Fin n → E
z : E
m : Fin c.length
h : m = c.index j
⊢ Injective (c.emb (c.index j))

Tactic:
· exact (c.emb_strictMono (c.index j)).injective

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
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
f g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑n) f s x
hg : ContDiffWithinAt 𝕜 (↑n) g s x
⊢ iteratedDerivWithin n (f + g) s x = iteratedDerivWithin n f s x + iteratedDerivWithin n g s x

Tactic:
by
  simp_rw [iteratedDerivWithin, iteratedFDerivWithin_add_apply hf hg h hx, add_apply]

Goal state after:
Module: Mathlib.Topology.MetricSpace.CauSeqFilter

Goal state before:
pp: β : Type v
inst✝¹ : NormedRing β
hn : IsAbsoluteValue norm
f : CauSeq β norm
inst✝ : IsComplete β norm
s : Set β
os : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
lfs : f.lim ∈ s
ε : ℝ
hε : ε > 0
hεs : Metric.ball f.lim ε ⊆ s
N : ℕ
hN : ∀ j ≥ N, ‖↑(const norm f.lim - f) j‖ < ε
b : ℕ
hb : b ≥ N
⊢ dist f.lim (↑f b) < ε

Tactic:
dist_eq_norm

Goal state after:
β : Type v
inst✝¹ : NormedRing β
hn : IsAbsoluteValue norm
f : CauSeq β norm
inst✝ : IsComplete β norm
s : Set β
os : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
lfs : f.lim ∈ s
ε : ℝ
hε : ε > 0
hεs : Metric.ball f.lim ε ⊆ s
N : ℕ
hN : ∀ j ≥ N, ‖↑(const norm f.lim - f) j‖ < ε
b : ℕ
hb : b ≥ N
⊢ ‖f.lim - ↑f b‖ < ε
Module: Mathlib.Topology.MetricSpace.CauSeqFilter

Goal state before:
pp: case right
β : Type v
inst✝ : NormedField β
f : CauSeq β norm
s : Set (β × β)
hs : s ∈ uniformity β
ε : ℝ
hε : ε > 0
hεs : ∀ ⦃a b : β⦄, dist a b < ε → (a, b) ∈ s
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, ‖↑f j - ↑f k‖ < ε
a b : β
a' : ℕ
ha'1 : a' ≥ N
ha'2 : ↑f a' = a
b' : ℕ
hb'1 : b' ≥ N
hb'2 : ↑f b' = b
⊢ dist (↑f a') (↑f b') < ε

Tactic:
dist_eq_norm

Goal state after:
case right
β : Type v
inst✝ : NormedField β
f : CauSeq β norm
s : Set (β × β)
hs : s ∈ uniformity β
ε : ℝ
hε : ε > 0
hεs : ∀ ⦃a b : β⦄, dist a b < ε → (a, b) ∈ s
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, ‖↑f j - ↑f k‖ < ε
a b : β
a' : ℕ
ha'1 : a' ≥ N
ha'2 : ↑f a' = a
b' : ℕ
hb'1 : b' ≥ N
hb'2 : ↑f b' = b
⊢ ‖↑f a' - ↑f b'‖ < ε
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set 𝕜
h : UniqueDiffOn 𝕜 s
f : 𝕜 → F
c : 𝕜
hs : Set.MapsTo (fun x ↦ c * x) s s
n : ℕ
ih :
  ∀ {x : 𝕜},
    x ∈ s →
      ContDiffOn 𝕜 (↑n) f s → iteratedDerivWithin n (fun x ↦ f (c * x)) s x = c ^ n • iteratedDerivWithin n f s (c * x)
x : 𝕜
hx : x ∈ s
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hcx : c * x ∈ s
h₀ : Set.EqOn (iteratedDerivWithin n (fun x ↦ f (c * x)) s) (fun x ↦ c ^ n • iteratedDerivWithin n f s (c * x)) s
h₁ : DifferentiableWithinAt 𝕜 (iteratedDerivWithin n f s) s (c * x)
h₂ : DifferentiableWithinAt 𝕜 (fun x ↦ iteratedDerivWithin n f s (c * x)) s x
⊢ iteratedDerivWithin (n + 1) (fun x ↦ f (c * x)) s x = c ^ (n + 1) • iteratedDerivWithin (n + 1) f s (c * x)

Tactic:
rw [iteratedDerivWithin_succ, derivWithin_congr h₀ (ih hx hf.of_succ),
      derivWithin_fun_const_smul (c ^ n) h₂, iteratedDerivWithin_succ,
      ← Function.comp_def, derivWithin.scomp x h₁ (by fun_prop) hs,
      derivWithin_const_mul _ differentiableWithinAt_id, derivWithin_id' _ _ (h _ hx),
      smul_smul, mul_one, pow_succ]

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

Goal state before:
pp: ⊢ rexp = NormedSpace.exp

Tactic:
by
  ext x; exact mod_cast congr_fun Complex.exp_eq_exp_ℂ x

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: case hmem
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
x y : E
h : (𝓝[>] 0 ⊓ 𝓟 {x_1 | x + ↑x_1 • y ∈ s}).NeBot
⊢ ∀ᶠ (n : ℝ≥0) in 𝓝[>] 0 ⊓ 𝓟 {t | x + ↑t • y ∈ s}, x + id n • y ∈ s

Tactic:
· simp [eventually_inf_principal, NNReal.smul_def]

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

Goal state before:
pp: case zero
𝕜 : Type u_1
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
hu : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
⊢ iteratedFDerivWithin 𝕜 0 (-f) s x = -iteratedFDerivWithin 𝕜 0 f s x

Tactic:
ext h

Goal state after:
case zero
𝕜 : Type u_1
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
hu : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
h : Fin 0 → E
⊢ (iteratedFDerivWithin 𝕜 0 (-f) s x) h = (-iteratedFDerivWithin 𝕜 0 f s x) h
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: case succ
𝕜 : Type u_1
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
hu : UniqueDiffOn 𝕜 s
i : ℕ
hi : ∀ {x : E}, x ∈ s → iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x
x : E
hx : x ∈ s
⊢ iteratedFDerivWithin 𝕜 (i + 1) (-f) s x = -iteratedFDerivWithin 𝕜 (i + 1) f s x

Tactic:
ext h

Goal state after:
case succ
𝕜 : Type u_1
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
hu : UniqueDiffOn 𝕜 s
i : ℕ
hi : ∀ {x : E}, x ∈ s → iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x
x : E
hx : x ∈ s
h : Fin (i + 1) → E
⊢ (iteratedFDerivWithin 𝕜 (i + 1) (-f) s x) h = (-iteratedFDerivWithin 𝕜 (i + 1) f s x) h
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: case neg.hf
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la : β
inst✝ : RegularSpace β
hf : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 la)
hab : a < b
⊢ ∀ x ∈ Ico a b, ∃ y, Tendsto f (𝓝[Ioo a b] x) (𝓝 y)

Tactic:
intro x x_in
    rcases eq_left_or_mem_Ioo_of_mem_Ico x_in with (rfl | h)
    · use la
      simpa [hab]
    · exact ⟨f x, hf x h⟩

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ¬ContinuousWithinAt f (Ici a) a
hcontra : DifferentiableWithinAt ℝ f (Ioi a) a
this : ContinuousWithinAt f (Ioi a) a
⊢ False

Tactic:
rw [← ContinuousWithinAt.sdiff_iff this] at hcont_at_a

Goal state after:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ¬ContinuousWithinAt f (Ici a \ Ioi a) a
hcontra : DifferentiableWithinAt ℝ f (Ioi a) a
this : ContinuousWithinAt f (Ioi a) a
⊢ False
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
⊢ False

Tactic:
have hdiff' : DifferentiableOn ℝ f (Ioc a b) := fun z hz => by
        refine DifferentiableWithinAt.mono (t := Ioi a) ?_ Ioc_subset_Ioi_self
        have : derivWithin f (Ioi a) z ≠ 0 := ne_of_gt <| by
          simp_all only [and_imp, mem_Ioc, max_lt_iff]
        exact differentiableWithinAt_of_derivWithin_ne_zero this

Goal state after:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
hdiff' : DifferentiableOn ℝ f (Ioc a b)
⊢ False
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case add
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f₁ f₂ : α →ₛ ℝ≥0
a✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)
h₁ :
  ∀ {ε : ℝ≥0∞},
    ε ≠ 0 → ∃ g, (∀ (x : α), f₁ x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(f₁ x) ∂μ + ε
h₂ :
  ∀ {ε : ℝ≥0∞},
    ε ≠ 0 → ∃ g, (∀ (x : α), f₂ x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(f₂ x) ∂μ + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
g₁ : α → ℝ≥0
f₁_le_g₁ : ∀ (x : α), f₁ x ≤ g₁ x
g₁cont : LowerSemicontinuous g₁
g₁int : ∫⁻ (x : α), ↑(g₁ x) ∂μ ≤ ∫⁻ (x : α), ↑(f₁ x) ∂μ + ε / 2
g₂ : α → ℝ≥0
f₂_le_g₂ : ∀ (x : α), f₂ x ≤ g₂ x
g₂cont : LowerSemicontinuous g₂
g₂int : ∫⁻ (x : α), ↑(g₂ x) ∂μ ≤ ∫⁻ (x : α), ↑(f₂ x) ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(g₁ x) + ↑(g₂ x) ∂μ ≤ ∫⁻ (x : α), ↑(f₁ x) + ↑(f₂ x) ∂μ + ε

Tactic:
lintegral_add_left f₁.measurable.coe_nnreal_ennreal,

Goal state after:
case add
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f₁ f₂ : α →ₛ ℝ≥0
a✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)
h₁ :
  ∀ {ε : ℝ≥0∞},
    ε ≠ 0 → ∃ g, (∀ (x : α), f₁ x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(f₁ x) ∂μ + ε
h₂ :
  ∀ {ε : ℝ≥0∞},
    ε ≠ 0 → ∃ g, (∀ (x : α), f₂ x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(f₂ x) ∂μ + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
g₁ : α → ℝ≥0
f₁_le_g₁ : ∀ (x : α), f₁ x ≤ g₁ x
g₁cont : LowerSemicontinuous g₁
g₁int : ∫⁻ (x : α), ↑(g₁ x) ∂μ ≤ ∫⁻ (x : α), ↑(f₁ x) ∂μ + ε / 2
g₂ : α → ℝ≥0
f₂_le_g₂ : ∀ (x : α), f₂ x ≤ g₂ x
g₂cont : LowerSemicontinuous g₂
g₂int : ∫⁻ (x : α), ↑(g₂ x) ∂μ ≤ ∫⁻ (x : α), ↑(f₂ x) ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(g₁ x) + ↑(g₂ x) ∂μ ≤ ∫⁻ (a : α), ↑(f₁ a) ∂μ + ∫⁻ (a : α), ↑(f₂ a) ∂μ + ε
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case add
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f₁ f₂ : α →ₛ ℝ≥0
a✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)
h₁ :
  ∀ {ε : ℝ≥0∞},
    ε ≠ 0 → ∃ g, (∀ (x : α), f₁ x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(f₁ x) ∂μ + ε
h₂ :
  ∀ {ε : ℝ≥0∞},
    ε ≠ 0 → ∃ g, (∀ (x : α), f₂ x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(f₂ x) ∂μ + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ g,
    (∀ (x : α), (f₁ + f₂) x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑((f₁ + f₂) x) ∂μ + ε

Tactic:
rcases h₁ (ENNReal.half_pos ε0).ne' with ⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩
    rcases h₂ (ENNReal.half_pos ε0).ne' with ⟨g₂, f₂_le_g₂, g₂cont, g₂int⟩
    refine
      ⟨fun x => g₁ x + g₂ x, fun x => add_le_add (f₁_le_g₁ x) (f₂_le_g₂ x), g₁cont.add g₂cont, ?_⟩
    simp only [SimpleFunc.coe_add, ENNReal.coe_add, Pi.add_apply]
    rw [lintegral_add_left f₁.measurable.coe_nnreal_ennreal,
      lintegral_add_left g₁cont.measurable.coe_nnreal_ennreal]
    convert! add_le_add g₁int g₂int using 1
    conv_lhs => rw [← ENNReal.add_halves ε]
    abel

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : ∃ i < a, ∀ ⦃x : ℝ⦄, x ∈ Ioo i a → deriv f x ∈ Iic (-1)
f' : ℝ → ℝ := f ∘ Neg.neg
⊢ deriv f' =ᶠ[𝓝[>] (-a)] -deriv f ∘ Neg.neg

Tactic:
EventuallyEq,

Goal state after:
f : ℝ → ℝ
a : ℝ
hf : ∃ i < a, ∀ ⦃x : ℝ⦄, x ∈ Ioo i a → deriv f x ∈ Iic (-1)
f' : ℝ → ℝ := f ∘ Neg.neg
⊢ ∀ᶠ (x : ℝ) in 𝓝[>] (-a), deriv f' x = (-deriv f ∘ Neg.neg) 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∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((eapproxDiff f n) x) ∂μ + δ n
⊢ ∀ (i : ℕ), AEMeasurable (fun x ↦ ↑((eapproxDiff f i) x)) μ

Tactic:
intro n; exact (SimpleFunc.measurable _).coe_nnreal_ennreal.aemeasurable

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
ε : ℝ
hε : 0 < ε
x : E
hx : DifferentiableAt 𝕜 f x
⊢ ∃ R > 0, ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ε

Tactic:
let δ := (ε / 2) / 2
  obtain ⟨R, R_pos, hR⟩ :
      ∃ R > 0, ∀ y ∈ ball x R, ‖f y - f x - fderiv 𝕜 f x (y - x)‖ ≤ δ * ‖y - x‖ :=
    eventually_nhds_iff_ball.1 <| hx.hasFDerivAt.isLittleO.bound <| by positivity
  refine ⟨R, R_pos, fun r hr => ?_⟩
  have : r ∈ Ioc (r / 2) r := right_mem_Ioc.2 <| half_lt_self hr.1
  refine ⟨r, this, fun y hy z hz => ?_⟩
  calc
    ‖f z - f y - (fderiv 𝕜 f x) (z - y)‖ =
        ‖f z - f x - (fderiv 𝕜 f x) (z - x) - (f y - f x - (fderiv 𝕜 f x) (y - x))‖ := by
      simp only [map_sub]; abel_nf
    _ ≤ ‖f z - f x - (fderiv 𝕜 f x) (z - x)‖ + ‖f y - f x - (fderiv 𝕜 f x) (y - x)‖ :=
      norm_sub_le _ _
    _ ≤ δ * ‖z - x‖ + δ * ‖y - x‖ :=
      add_le_add (hR _ (ball_subset_ball hr.2.le hz)) (hR _ (ball_subset_ball hr.2.le hy))
    _ ≤ δ * r + δ * r := by rw [mem_ball_iff_norm] at hz hy; gcongr
    _ = (ε / 2) * r := by ring
    _ < ε * r := by gcongr; exacts [hr.1, half_lt_self hε]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
⊢ Tendsto (fun t ↦ |x| * |x| ^ t / (1 - |x|)) atTop (𝓝 (|x| * 0 / (1 - |x|)))

Tactic:
refine (tendsto_const_nhds.mul ?_).div_const _

Goal state after:
x : ℝ
h : |x| < 1
⊢ Tendsto (HPow.hPow |x|) atTop (𝓝 0)
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
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] 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)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e ↦ L e (n e) (n e)
ε : ℝ
εpos : ε > 0
e : ℕ
he : (1 / 2) ^ e < ε / (12 * ‖c‖)
e' : ℕ
he' : e' ≥ e
⊢ dist (L0 e) (L0 e') < ε

Tactic:
dist_eq_norm

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
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] 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)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e ↦ L e (n e) (n e)
ε : ℝ
εpos : ε > 0
e : ℕ
he : (1 / 2) ^ e < ε / (12 * ‖c‖)
e' : ℕ
he' : e' ≥ e
⊢ ‖L0 e - L0 e'‖ < ε
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case add
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f₁ f₂ : α →ₛ ℝ≥0
a✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)
h₁ :
  ∫⁻ (x : α), ↑(f₁ x) ∂μ ≠ ∞ →
    ∀ {ε : ℝ≥0∞},
      ε ≠ 0 → ∃ g, (∀ (x : α), g x ≤ f₁ x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(f₁ x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε
h₂ :
  ∫⁻ (x : α), ↑(f₂ x) ∂μ ≠ ∞ →
    ∀ {ε : ℝ≥0∞},
      ε ≠ 0 → ∃ g, (∀ (x : α), g x ≤ f₂ x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(f₂ x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε
int_f : ∫⁻ (x : α), ↑((f₁ + f₂) x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
A : ∫⁻ (x : α), ↑(f₁ x) ∂μ + ∫⁻ (x : α), ↑(f₂ x) ∂μ ≠ ∞
g₁ : α → ℝ≥0
f₁_le_g₁ : ∀ (x : α), g₁ x ≤ f₁ x
g₁cont : UpperSemicontinuous g₁
g₁int : ∫⁻ (x : α), ↑(f₁ x) ∂μ ≤ ∫⁻ (x : α), ↑(g₁ x) ∂μ + ε / 2
g₂ : α → ℝ≥0
f₂_le_g₂ : ∀ (x : α), g₂ x ≤ f₂ x
g₂cont : UpperSemicontinuous g₂
g₂int : ∫⁻ (x : α), ↑(f₂ x) ∂μ ≤ ∫⁻ (x : α), ↑(g₂ x) ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(f₁ x) + ↑(f₂ x) ∂μ ≤ ∫⁻ (x : α), ↑(g₁ x) + ↑(g₂ x) ∂μ + ε

Tactic:
lintegral_add_left f₁.measurable.coe_nnreal_ennreal,

Goal state after:
case add
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f₁ f₂ : α →ₛ ℝ≥0
a✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)
h₁ :
  ∫⁻ (x : α), ↑(f₁ x) ∂μ ≠ ∞ →
    ∀ {ε : ℝ≥0∞},
      ε ≠ 0 → ∃ g, (∀ (x : α), g x ≤ f₁ x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(f₁ x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε
h₂ :
  ∫⁻ (x : α), ↑(f₂ x) ∂μ ≠ ∞ →
    ∀ {ε : ℝ≥0∞},
      ε ≠ 0 → ∃ g, (∀ (x : α), g x ≤ f₂ x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(f₂ x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε
int_f : ∫⁻ (x : α), ↑((f₁ + f₂) x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
A : ∫⁻ (x : α), ↑(f₁ x) ∂μ + ∫⁻ (x : α), ↑(f₂ x) ∂μ ≠ ∞
g₁ : α → ℝ≥0
f₁_le_g₁ : ∀ (x : α), g₁ x ≤ f₁ x
g₁cont : UpperSemicontinuous g₁
g₁int : ∫⁻ (x : α), ↑(f₁ x) ∂μ ≤ ∫⁻ (x : α), ↑(g₁ x) ∂μ + ε / 2
g₂ : α → ℝ≥0
f₂_le_g₂ : ∀ (x : α), g₂ x ≤ f₂ x
g₂cont : UpperSemicontinuous g₂
g₂int : ∫⁻ (x : α), ↑(f₂ x) ∂μ ≤ ∫⁻ (x : α), ↑(g₂ x) ∂μ + ε / 2
⊢ ∫⁻ (a : α), ↑(f₁ a) ∂μ + ∫⁻ (a : α), ↑(f₂ a) ∂μ ≤ ∫⁻ (x : α), ↑(g₁ x) + ↑(g₂ x) ∂μ + ε
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
a : α
ha : 0 ≤ a
ha' : 0 < a
x : E
⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ a • s} = (a⁻¹ • gauge s) x

Tactic:
Pi.smul_apply,

Goal state after:
case inr
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
a : α
ha : 0 ≤ a
ha' : 0 < a
x : E
⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ a • s} = a⁻¹ • gauge s x
Module: Mathlib.Analysis.RCLike.Extend

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
F : Type u_2
inst✝³ : AddCommGroup F
inst✝² : Module ℝ F
inst✝¹ : Module 𝕜 F
inst✝ : IsScalarTower ℝ 𝕜 F
m✝ : ℝ
x✝¹ : Dual ℝ F
x✝ : F
⊢ (m✝ • x✝¹).extendRCLike x✝ = ((RingHom.id ℝ) m✝ • x✝¹.extendRCLike) x✝

Tactic:
simp [extendRCLike_apply, real_smul_eq_coe_mul]

Goal state after:
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
F : Type u_2
inst✝³ : AddCommGroup F
inst✝² : Module ℝ F
inst✝¹ : Module 𝕜 F
inst✝ : IsScalarTower ℝ 𝕜 F
m✝ : ℝ
x✝¹ : Dual ℝ F
x✝ : F
⊢ (algebraMap ℝ 𝕜) m✝ * (algebraMap ℝ 𝕜) (x✝¹ x✝) - I * ((algebraMap ℝ 𝕜) m✝ * (algebraMap ℝ 𝕜) (x✝¹ (I • x✝))) =
    ↑m✝ * (↑(x✝¹ x✝) - I * ↑(x✝¹ (I • x✝)))
Module: Mathlib.Analysis.RCLike.Extend

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
F : Type u_2
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : Module ℝ F
inst✝ : IsScalarTower ℝ 𝕜 F
m✝ : ℝ
x✝¹ : StrongDual ℝ F
x✝ : F
⊢ (m✝ • x✝¹).extendRCLike x✝ = ((RingHom.id ℝ) m✝ • x✝¹.extendRCLike) x✝

Tactic:
simp [extendRCLike_apply, real_smul_eq_coe_mul]

Goal state after:
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
F : Type u_2
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : Module ℝ F
inst✝ : IsScalarTower ℝ 𝕜 F
m✝ : ℝ
x✝¹ : StrongDual ℝ F
x✝ : F
⊢ (algebraMap ℝ 𝕜) m✝ * (algebraMap ℝ 𝕜) (x✝¹ x✝) - I * ((algebraMap ℝ 𝕜) m✝ * (algebraMap ℝ 𝕜) (x✝¹ (I • x✝))) =
    ↑m✝ * (↑(x✝¹ x✝) - I * ↑(x✝¹ (I • x✝)))
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
⊢ (|r|⁻¹ • gauge (ball 0 1)) x = ‖x‖ / r

pp: case inr.symmetric
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr✝ : 0 ≤ r
x : E
hr : 0 < r
⊢ ∀ x ∈ ball 0 1, -x ∈ ball 0 1

Tactic:
Pi.smul_apply,

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

case inr.symmetric
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr✝ : 0 ≤ r
x : E
hr : 0 < r
⊢ ∀ x ∈ ball 0 1, -x ∈ ball 0 1
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_3
inst✝⁴ : Zero X
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
inst✝ : T1Space R
⊢ IsClosed (range _root_.toContinuousMap)

Tactic:
rw [range_toContinuousMap]
    exact isClosed_singleton.preimage <| continuous_eval_const 0

Goal state after: