mathlib-initiative/mathlib-tactics huggingface dataset:
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
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
i : Fin c.length
⊢ (q.compAlongOrderedFinpartition p (c.extend (some i))).curryLeft =
    ((compAlongOrderedFinpartitionL 𝕜 E F G c) (q c.length)).toContinuousLinearMap (fun i ↦ p (c.partSize i)) i ∘SL
      (p (c.partSize i + 1)).curryLeft

Tactic:
by
  ext e v
  simp? [OrderedFinpartition.extend, extendMiddle, applyOrderedFinpartition_apply] says
    simp only [OrderedFinpartition.extend, extendMiddle, ContinuousMultilinearMap.curryLeft_apply,
      Nat.succ_eq_add_one, FormalMultilinearSeries.compAlongOrderedFinpartition_apply,
      applyOrderedFinpartition_apply, ContinuousLinearMap.comp_apply,
      ContinuousMultilinearMap.toContinuousLinearMap_apply, compAlongOrderedFinpartitionL_apply,
      compAlongOrderFinpartition_apply]
  congr
  ext j
  rcases eq_or_ne j i with rfl | hij
  · simp only [↓reduceDIte, update_self, ContinuousMultilinearMap.curryLeft_apply,
      Nat.succ_eq_add_one]
    apply FormalMultilinearSeries.congr _ (by simp)
    intro a ha h'a
    match a with
    | 0 => simp
    | a + 1 => simp [cons]
  · simp only [hij, ↓reduceDIte, ne_eq, not_false_eq_true, update_of_ne]
    apply FormalMultilinearSeries.congr _ (by simp [hij])
    simp

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

Goal state before:
pp: s : Set ℝ
a : ℝ
h : 1 ∈ posTangentConeAt s a
ι : Type
l : Filter ι
hl : l.NeBot
c : ι → ℝ≥0
d : ι → ℝ
hd₀ : Tendsto d l (𝓝 0)
hd : ∀ᶠ (n : ι) in l, a + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 1)
this : Tendsto (fun x ↦ a + d x) l (𝓝 a)
n : ι
hcdn : 0 < c n • d n
⊢ a + d n ∈ s → a + d n ∈ Ioi a ∩ s

Tactic:
hdn

Goal state after:
s : Set ℝ
a : ℝ
h : 1 ∈ posTangentConeAt s a
ι : Type
l : Filter ι
hl : l.NeBot
c : ι → ℝ≥0
d : ι → ℝ
hd₀ : Tendsto d l (𝓝 0)
hd : ∀ᶠ (n : ι) in l, a + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 1)
this : Tendsto (fun x ↦ a + d x) l (𝓝 a)
n : ι
hcdn : 0 < c n • d n
hdn : a + d n ∈ s
⊢ a + d n ∈ Ioi a ∩ s
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝔸 : Type u_5
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
n : ℕ
f : 𝕜 → 𝔸
c : 𝔸
hf : ContDiffAt 𝕜 (↑n) f x
⊢ iteratedDeriv n (fun x ↦ c * f x) x = c * iteratedDeriv n f x

Tactic:
by
  simpa only [iteratedDerivWithin_univ] using
    iteratedDerivWithin_const_mul (Set.mem_univ x) uniqueDiffOn_univ c hf

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 : WithTop ℕ∞
g : F → G
f : E → F
hg : HasFTaylorSeriesUpToOn n g q t
hf : HasFTaylorSeriesUpToOn n f p s
h : MapsTo f s t
m : ℕ
hm : ↑m ≤ n
c : OrderedFinpartition m
x✝ : c ∈ Finset.univ
B : (F [×c.length]→L[𝕜] G) →L[𝕜] (E [×c.partSize i]→L[𝕜] F) [×c.length]→L[𝕜] E [×m]→L[𝕜] G :=
  compAlongOrderedFinpartitionL 𝕜 E F G c
⊢ ContinuousOn (fun a ↦ (q (f a)).compAlongOrderedFinpartition (p a) c) s

Tactic:
change ContinuousOn
      ((fun p ↦ B p.1 p.2) ∘ (fun x ↦ (q (f x) c.length, fun i ↦ p x (c.partSize i)))) s

Goal state after:
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 : WithTop ℕ∞
g : F → G
f : E → F
hg : HasFTaylorSeriesUpToOn n g q t
hf : HasFTaylorSeriesUpToOn n f p s
h : MapsTo f s t
m : ℕ
hm : ↑m ≤ n
c : OrderedFinpartition m
x✝ : c ∈ Finset.univ
B : (F [×c.length]→L[𝕜] G) →L[𝕜] (E [×c.partSize i]→L[𝕜] F) [×c.length]→L[𝕜] E [×m]→L[𝕜] G :=
  compAlongOrderedFinpartitionL 𝕜 E F G c
⊢ ContinuousOn ((fun p ↦ (B p.1) p.2) ∘ fun x ↦ (q (f x) c.length, fun i ↦ p x (c.partSize i))) s
Module: Mathlib.Topology.Order.Rolle

Goal state before:
pp: case pos
X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : (Icc a b).Nonempty
c : X
cmem : c ∈ Icc a b
cle : ∀ x ∈ Icc a b, f c ≤ f x
C : X
Cmem : C ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
hc : f c = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

pp: case neg
X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : (Icc a b).Nonempty
c : X
cmem : c ∈ Icc a b
cle : ∀ x ∈ Icc a b, f c ≤ f x
C : X
Cmem : C ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
hc : ¬f c = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

Tactic:
· by_cases hC : f C = f a
    · have : ∀ x ∈ Icc a b, f x = f a := fun x hx => le_antisymm (hC ▸ Cge x hx) (hc ▸ cle x hx)
      -- `f` is a constant, so we can take any point in `Ioo a b`
      rcases nonempty_Ioo.2 hab with ⟨c', hc'⟩
      refine ⟨c', hc', Or.inl fun x hx ↦ ?_⟩
      simp only [mem_ofPred_eq, this x hx, this c' (Ioo_subset_Icc_self hc'), le_rfl]
    · refine ⟨C, ⟨lt_of_le_of_ne Cmem.1 <| mt ?_ hC, lt_of_le_of_ne Cmem.2 <| mt ?_ hC⟩, Or.inr Cge⟩
      exacts [fun h => by rw [h], fun h => by rw [h, hfI]]

Goal state after:
case neg
X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : (Icc a b).Nonempty
c : X
cmem : c ∈ Icc a b
cle : ∀ x ∈ Icc a b, f c ≤ f x
C : X
Cmem : C ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
hc : ¬f c = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n : ℕ∞ω
inst✝ : CompleteSpace E
f : OpenPartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha : a ∈ f.target
hf₀' : HasFDerivAt (↑f) (↑f₀') (↑f.symm a)
hf : ContDiffAt 𝕜 ω (↑f) (↑f.symm a)
⊢ ContDiffAt 𝕜 ω (↑f.symm) a

Tactic:
| =>

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n : ℕ∞ω
inst✝ : CompleteSpace E
f : OpenPartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha : a ∈ f.target
hf₀' : HasFDerivAt (↑f) (↑f₀') (↑f.symm a)
hf : ContDiffAt 𝕜 ω (↑f) (↑f.symm a)
⊢ ContDiffAt 𝕜 ω (↑f.symm) a
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n✝ : ℕ∞ω
inst✝ : CompleteSpace E
f : OpenPartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha : a ∈ f.target
hf₀' : HasFDerivAt (↑f) (↑f₀') (↑f.symm a)
n : ℕ∞
hf : ContDiffAt 𝕜 (↑n) (↑f) (↑f.symm a)
⊢ ContDiffAt 𝕜 (↑n) (↑f.symm) a

Tactic:
| =>

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n✝ : ℕ∞ω
inst✝ : CompleteSpace E
f : OpenPartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha : a ∈ f.target
hf₀' : HasFDerivAt (↑f) (↑f₀') (↑f.symm a)
n : ℕ∞
hf : ContDiffAt 𝕜 (↑n) (↑f) (↑f.symm a)
⊢ ContDiffAt 𝕜 (↑n) (↑f.symm) a
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
𝕜' : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜 𝕜'
inst✝³ : NormedSpace 𝕜' E
inst✝² : IsScalarTower 𝕜 𝕜' E
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
h : ContDiffWithinAt 𝕜' ω f s x
⊢ ContDiffWithinAt 𝕜 ω f s x

Tactic:
| =>

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
𝕜' : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜 𝕜'
inst✝³ : NormedSpace 𝕜' E
inst✝² : IsScalarTower 𝕜 𝕜' E
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
h : ContDiffWithinAt 𝕜' ω f s x
⊢ ContDiffWithinAt 𝕜 ω f s x
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n✝ : ℕ∞ω
𝕜' : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜 𝕜'
inst✝³ : NormedSpace 𝕜' E
inst✝² : IsScalarTower 𝕜 𝕜' E
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
n : ℕ∞
h : ContDiffWithinAt 𝕜' (↑n) f s x
⊢ ContDiffWithinAt 𝕜 (↑n) f s x

Tactic:
| =>

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n✝ : ℕ∞ω
𝕜' : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜 𝕜'
inst✝³ : NormedSpace 𝕜' E
inst✝² : IsScalarTower 𝕜 𝕜' E
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
n : ℕ∞
h : ContDiffWithinAt 𝕜' (↑n) f s x
⊢ ContDiffWithinAt 𝕜 (↑n) f s x
Module: Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus

Goal state before:
pp: X : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure X
l : Filter X
inst✝ : l.IsMeasurablyGenerated
f : X → E
b : E
h : Tendsto f (l ⊓ ae μ) (𝓝 b)
hfm : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
s : ι → Set X
li : Filter ι
hs : Tendsto s li l.smallSets
m : ι → ℝ
hsμ : (fun i ↦ μ.real (s i)) =ᶠ[li] m
⊢ (fun s ↦ ∫ (x : X) in s, f x ∂μ - μ.real s • b) =o[l.smallSets] fun s ↦ μ.real s

Tactic:
refine isLittleO_iff.2 fun ε ε₀ => ?_

Goal state after:
X : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure X
l : Filter X
inst✝ : l.IsMeasurablyGenerated
f : X → E
b : E
h : Tendsto f (l ⊓ ae μ) (𝓝 b)
hfm : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
s : ι → Set X
li : Filter ι
hs : Tendsto s li l.smallSets
m : ι → ℝ
hsμ : (fun i ↦ μ.real (s i)) =ᶠ[li] m
ε : ℝ
ε₀ : 0 < ε
⊢ ∀ᶠ (x : Set X) in l.smallSets, ‖∫ (x : X) in x, f x ∂μ - μ.real x • b‖ ≤ ε * ‖μ.real x‖
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case refine_1
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
x : α
⊢ f x ≤ (fun x ↦ ∑' (n : ℕ), ↑(g n x)) x

Tactic:
rw [← tsum_eapproxDiff f hf]

Goal state after:
case refine_1
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
x : α
⊢ ∑' (n : ℕ), ↑((eapproxDiff f n) x) ≤ (fun x ↦ ∑' (n : ℕ), ↑(g n x)) x
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
⊢ x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}

Tactic:
have :
    ∀ e : ℕ, ∃ n : ℕ, ∀ p q, n ≤ p → n ≤ q →
      ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e) := by
    intro e
    have := mem_iInter.1 hx e
    rcases mem_iUnion.1 this with ⟨n, hn⟩
    refine ⟨n, fun p q hp hq => ?_⟩
    simp only [mem_iInter] at hn
    rcases mem_iUnion.1 (hn p hp q hq) with ⟨L, hL⟩
    exact ⟨L, exists_prop.mp <| mem_iUnion.1 hL⟩

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
this :
  ∀ (e : ℕ),
    ∃ n, ∀ (p q : ℕ), n ≤ p → n ≤ q → ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)
⊢ x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
term : ℕ → ℝ := fun n ↦ -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)
h_term_eq_goal : (term ∘ fun x ↦ 2 * x) = fun k ↦ 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)
m : ℕ
hm : Even (m + 1)
⊢ -1 * ((-x) ^ (m + 1) / (↑m + 1)) + x ^ (m + 1) / (↑m + 1) = 0

Tactic:
rw [Even.neg_pow hm, neg_one_mul, neg_add_cancel]

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

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
hs : Convex ℝ s
absorbs : Absorbent ℝ s
x y : E
ε : ℝ
hε : 0 < ε
⊢ gauge s (x + y) < gauge s x + gauge s y + ε

Tactic:
obtain ⟨a, ha, ha', x, hx, rfl⟩ :=
    exists_lt_of_gauge_lt absorbs (lt_add_of_pos_right (gauge s x) (half_pos hε))

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
hs : Convex ℝ s
absorbs : Absorbent ℝ s
y : E
ε : ℝ
hε : 0 < ε
a : ℝ
ha : 0 < a
x : E
hx : x ∈ s
ha' : a < gauge s ((fun x ↦ a • x) x) + ε / 2
⊢ gauge s ((fun x ↦ a • x) x + y) < gauge s ((fun x ↦ a • x) x) + gauge s y + ε
Module: Mathlib.Analysis.Convex.Cone.Extension

Goal state before:
pp: case refine_2
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
f : E →ₗ.[ℝ] ℝ
N : E → ℝ
N_hom : ∀ (c : ℝ), 0 < c → ∀ (x : E), N (c • x) = c * N x
N_add : ∀ (x y : E), N (x + y) ≤ N x + N y
hf : ∀ (x : ↥f.domain), ↑f x ≤ N ↑x
N_0 : N 0 = 0
s : PointedCone ℝ (E × ℝ) := { carrier := {p | N p.1 ≤ p.2}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
f' : E × ℝ →ₗ.[ℝ] ℝ := (-f).coprod (LinearMap.id.toPMap ⊤)
hf'_nonneg : ∀ (x : ↥f'.domain), ↑x ∈ s → 0 ≤ ↑f' x
hf'_dense : ∀ (y : E × ℝ), ∃ x, ↑x + y ∈ s
g : E × ℝ →ₗ[ℝ] ℝ
g_nonneg : ∀ x ∈ s, 0 ≤ g x
g_eq : ∀ (x : ↥f.domain) (y : ℝ), g (↑x, y) = y - ↑f x
x : E
⊢ (-g ∘ₗ inl ℝ E ℝ) x ≤ N x

Tactic:
· calc -g (x, 0) = g (0, N x) - g (x, N x) := by simp [← map_sub, ← map_neg]
      _ = N x - g (x, N x) := by simpa using g_eq 0 (N x)
      _ ≤ N x := by simpa using g_nonneg ⟨x, N x⟩ (le_refl (N x))

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

Goal state before:
pp: case inl
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
p : Seminorm ℝ E
x : E
hp : {r | 0 < r ∧ x ∈ r • p.ball 0 1} = ∅
h : ¬0 = p x
hpx : 0 < p x
hpx₂ : 0 < 2 * p x
⊢ |(2 * p x)⁻¹| * p x < 1

Tactic:
abs_of_pos (inv_pos.2 hpx₂),

Goal state after:
case inl
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
p : Seminorm ℝ E
x : E
hp : {r | 0 < r ∧ x ∈ r • p.ball 0 1} = ∅
h : ¬0 = p x
hpx : 0 < p x
hpx₂ : 0 < 2 * p x
⊢ (2 * p x)⁻¹ * p x < 1
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inl
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hr : 0 ≤ 0
⊢ gauge (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {0}) x = 0

Tactic:
singleton_zero,

Goal state after:
case inl
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hr : 0 ≤ 0
⊢ gauge (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] 0) x = 0
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✝ : ∀ ⦃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‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 ↦ ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ ‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
    ‖f z - f x - (z - x) • derivWithin f (Ici x) x - (f y - f x - (y - x) • derivWithin f (Ici x) x)‖

Tactic:
congr 1; simp only [sub_smul]; abel

Goal state after:
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
⊢ x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}

Tactic:
have :
    ∀ e : ℕ, ∃ n : ℕ, ∀ p q, n ≤ p → n ≤ q →
      ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e) := by
    intro e
    have := mem_iInter.1 hx e
    rcases mem_iUnion.1 this with ⟨n, hn⟩
    refine ⟨n, fun p q hp hq => ?_⟩
    simp only [mem_iInter] at hn
    rcases mem_iUnion.1 (hn p hp q hq) with ⟨L, hL⟩
    exact ⟨L, exists_prop.mp <| mem_iUnion.1 hL⟩

Goal state after:
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
this :
  ∀ (e : ℕ),
    ∃ n, ∀ (p q : ℕ), n ≤ p → n ≤ q → ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)
⊢ x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case h_add
X : Type u_1
E : Type u_3
F : Type u_4
inst✝¹¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
𝕜' : Type u_7
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : RCLike 𝕜'
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : NormedSpace ℝ F
σ : 𝕜 →+* 𝕜'
inst✝³ : RingHomIsometric σ
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hσ : ∀ (r : ℝ) (x : 𝕜), σ (r • x) = r • σ x
L : E →SL[σ] F
φ : X → E
φ_int : Integrable φ μ
f : X → E
⊢ ∀ ⦃g : X → E⦄,
    Disjoint (Function.support f) (Function.support g) →
      Integrable f μ →
        Integrable g μ →
          ∫ (x : X), L (f x) ∂μ = L (∫ (x : X), f x ∂μ) →
            ∫ (x : X), L (g x) ∂μ = L (∫ (x : X), g x ∂μ) → ∫ (x : X), L ((f + g) x) ∂μ = L (∫ (x : X), (f + g) x ∂μ)

Tactic:
g

Goal state after:
case h_add
X : Type u_1
E : Type u_3
F : Type u_4
inst✝¹¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
𝕜' : Type u_7
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : RCLike 𝕜'
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : NormedSpace ℝ F
σ : 𝕜 →+* 𝕜'
inst✝³ : RingHomIsometric σ
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hσ : ∀ (r : ℝ) (x : 𝕜), σ (r • x) = r • σ x
L : E →SL[σ] F
φ : X → E
φ_int : Integrable φ μ
f g : X → E
⊢ Disjoint (Function.support f) (Function.support g) →
    Integrable f μ →
      Integrable g μ →
        ∫ (x : X), L (f x) ∂μ = L (∫ (x : X), f x ∂μ) →
          ∫ (x : X), L (g x) ∂μ = L (∫ (x : X), g x ∂μ) → ∫ (x : X), L ((f + g) x) ∂μ = L (∫ (x : X), (f + g) x ∂μ)
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case h_ae
X : Type u_1
E : Type u_3
F : Type u_4
inst✝¹¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
𝕜' : Type u_7
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : RCLike 𝕜'
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : NormedSpace ℝ F
σ : 𝕜 →+* 𝕜'
inst✝³ : RingHomIsometric σ
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hσ : ∀ (r : ℝ) (x : 𝕜), σ (r • x) = r • σ x
L : E →SL[σ] F
φ : X → E
φ_int : Integrable φ μ
f : X → E
⊢ ∀ ⦃g : X → E⦄,
    f =ᵐ[μ] g →
      Integrable f μ → ∫ (x : X), L (f x) ∂μ = L (∫ (x : X), f x ∂μ) → ∫ (x : X), L (g x) ∂μ = L (∫ (x : X), g x ∂μ)

Tactic:
g

Goal state after:
case h_ae
X : Type u_1
E : Type u_3
F : Type u_4
inst✝¹¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
𝕜' : Type u_7
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : RCLike 𝕜'
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : NormedSpace ℝ F
σ : 𝕜 →+* 𝕜'
inst✝³ : RingHomIsometric σ
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hσ : ∀ (r : ℝ) (x : 𝕜), σ (r • x) = r • σ x
L : E →SL[σ] F
φ : X → E
φ_int : Integrable φ μ
f g : X → E
⊢ f =ᵐ[μ] g →
    Integrable f μ → ∫ (x : X), L (f x) ∂μ = L (∫ (x : X), f x ∂μ) → ∫ (x : X), L (g x) ∂μ = L (∫ (x : X), g x ∂μ)
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
hf : AEMeasurable f μ
g : X → E
f' : X → ℝ≥0 := AEMeasurable.mk f hf
⊢ ∫ (x : X), f' x • g x ∂μ = ∫ (x : X), f x • g x ∂μ

Tactic:
by
      apply integral_congr_ae
      filter_upwards [hf.ae_eq_mk] with x hx
      rw [hx]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
⊢ ContinuousOn (fun t ↦ (g t - g a, ∫ (u : ℝ) in a..t, (G' u).toReal)) (Icc a b)

Tactic:
rw [← uIcc_of_le hab] at G'int hcont ⊢

Goal state after:
g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g [[a, b]]
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict [[a, b]])
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
⊢ ContinuousOn (fun t ↦ (g t - g a, ∫ (u : ℝ) in a..t, (G' u).toReal)) [[a, b]]