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

Goal state before:
pp: case partSize.inl
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
o : Option (Fin c.length)
i : Fin c.length
⊢ update (c.extendMiddle i).partSize i ((c.extendMiddle i).partSize i - 1) i = c.partSize i

pp: case partSize.inr
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
o : Option (Fin c.length)
i : Fin c.length
j : Fin (c.extendMiddle i).length
hij : i ≠ j
⊢ update (c.extendMiddle i).partSize i ((c.extendMiddle i).partSize i - 1) j = c.partSize j

Tactic:
· simp [extendMiddle]

Goal state after:
case partSize.inr
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
o : Option (Fin c.length)
i : Fin c.length
j : Fin (c.extendMiddle i).length
hij : i ≠ j
⊢ update (c.extendMiddle i).partSize i ((c.extendMiddle i).partSize i - 1) j = c.partSize j
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case emb.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✝
n : ℕ
c : OrderedFinpartition n
o : Option (Fin c.length)
a : Fin c.length
⊢ ∀ (i : Fin (c.partSize a)),
    c.emb a i =
      if h : a = a then ((c.extendMiddle a).emb a (Fin.cast ⋯ ⟨↑i, ⋯⟩.succ)).pred ⋯
      else ((c.extendMiddle a).emb a (Fin.cast ⋯ ⟨↑i, ⋯⟩)).pred ⋯

Tactic:
· simp [extendMiddle]

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

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁵ : Field 𝕂
inst✝⁴ : Ring 𝔸
inst✝³ : Algebra 𝕂 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : CharZero 𝕂
⊢ exp = (expSeries 𝕂 𝔸).sum

Tactic:
ext x

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁵ : Field 𝕂
inst✝⁴ : Ring 𝔸
inst✝³ : Algebra 𝕂 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
⊢ exp x = (expSeries 𝕂 𝔸).sum x
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁵ : Field 𝕂
inst✝⁴ : Ring 𝔸
inst✝³ : Algebra 𝕂 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : CharZero 𝕂
⊢ (expSeries 𝕂 𝔸).sum = fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n

Tactic:
ext x

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁵ : Field 𝕂
inst✝⁴ : Ring 𝔸
inst✝³ : Algebra 𝕂 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
⊢ (expSeries 𝕂 𝔸).sum x = ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝔸 : Type u_2
inst✝³ : DivisionRing 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : CharZero 𝔸
⊢ (expSeries ℚ 𝔸).sum = fun x ↦ ∑' (n : ℕ), x ^ n / ↑n !

Tactic:
ext x

Goal state after:
𝔸 : Type u_2
inst✝³ : DivisionRing 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : CharZero 𝔸
x : 𝔸
⊢ (expSeries ℚ 𝔸).sum x = ∑' (n : ℕ), x ^ n / ↑n !
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:
simp_rw [iteratedDerivWithin, iteratedFDerivWithin_add_apply hf hg h hx, add_apply]

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
s : Set 𝕜
f : 𝕜 → F
c : 𝕜
⊢ iteratedDerivWithin n (fun z ↦ f (c + z)) s = fun x ↦ iteratedDerivWithin n f (c +ᵥ s) (c + x)

Tactic:
ext x

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
s : Set 𝕜
f : 𝕜 → F
c x : 𝕜
⊢ iteratedDerivWithin n (fun z ↦ f (c + z)) s x = iteratedDerivWithin n f (c +ᵥ s) (c + x)
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
s : Set 𝕜
f : 𝕜 → F
c : 𝕜
⊢ iteratedDerivWithin n (fun z ↦ f (z + c)) s = fun x ↦ iteratedDerivWithin n f (c +ᵥ s) (x + c)

Tactic:
ext x

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
s : Set 𝕜
f : 𝕜 → F
c x : 𝕜
⊢ iteratedDerivWithin n (fun z ↦ f (z + c)) s x = iteratedDerivWithin n f (c +ᵥ s) (x + c)
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
⊢ 1 = (continuousMultilinearCurryFin1 𝕂 𝔸 𝔸) (expSeries 𝕂 𝔸 1)

Tactic:
ext x

Goal state after:
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
⊢ 1 x = ((continuousMultilinearCurryFin1 𝕂 𝔸 𝔸) (expSeries 𝕂 𝔸 1)) x
Module: Mathlib.Analysis.SpecialFunctions.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedCommRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
this : ∀ᶠ (h : 𝔸) in 𝓝 0, h ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
⊢ (fun h ↦ exp x * (exp (0 + h) - exp 0 - (ContinuousLinearMap.id 𝕂 𝔸) h)) =ᶠ[𝓝 0] fun h ↦
    exp (x + h) - exp x - exp x • (ContinuousLinearMap.id 𝕂 𝔸) h

Tactic:
filter_upwards [this] with _ hh

Goal state after:
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedCommRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
this : ∀ᶠ (h : 𝔸) in 𝓝 0, h ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
a✝ : 𝔸
hh : a✝ ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
⊢ exp x * (exp (0 + a✝) - exp 0 - (ContinuousLinearMap.id 𝕂 𝔸) a✝) =
    exp (x + a✝) - exp x - exp x • (ContinuousLinearMap.id 𝕂 𝔸) a✝
Module: Mathlib.Analysis.SpecialFunctions.Exponential

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

Tactic:
ext x

Goal state after:
x : ℝ
⊢ rexp x = NormedSpace.exp x
Module: Mathlib.Analysis.SpecialFunctions.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
⊢ HasFDerivAt (fun u ↦ exp (u • x)) ((ContinuousLinearMap.smulRight 1 x).smulRight (exp (t • x))) t

Tactic:
convert! hasFDerivAt_exp_smul_const_of_mem_ball 𝕂 _ _ htx using 1
  ext t'
  change Commute (t' • x) (exp (t • x))
  exact (((Commute.refl x).smul_left t').smul_right t).exp_right

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

Goal state before:
pp: case hy
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
f' : StrongDual ℝ E
a : E
h : IsLocalMin f a
hf : HasFDerivAt f f' a
y : E
⊢ y ∈ univ

Tactic:
apply mem_univ

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

Goal state before:
pp: case hy'
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
f' : StrongDual ℝ E
a : E
h : IsLocalMin f a
hf : HasFDerivAt f f' a
y : E
⊢ -y ∈ univ

Tactic:
apply mem_univ

Goal state after:
Module: Mathlib.Topology.Order.Rolle

Goal state before:
pp: 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
hC : ¬f C = f a
h : a = C
⊢ f C = f a

Tactic:
by rw [h]

Goal state after:
Module: Mathlib.Topology.Order.Rolle

Goal state before:
pp: 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
h : a = c
⊢ f c = f a

Tactic:
by rw [h]

Goal state after:
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
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ∫⁻ (x : α), ↑(f x) ∂μ = ∞
⊢ ∫⁻ (x : α), ↑((fun x ↦ c) x) ∂μ ≤ ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ + ε

Tactic:
by
          simp only [f, _root_.top_add, le_top, h]

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

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : Tendsto (deriv f) (𝓝[<] a) atBot
f' : ℝ → ℝ := f ∘ Neg.neg
hderiv : deriv f' =ᶠ[𝓝[>] (-a)] -deriv f ∘ Neg.neg
⊢ ¬DifferentiableWithinAt ℝ f (Iio a) a

Tactic:
have hmain : ¬ DifferentiableWithinAt ℝ f' (Ioi (-a)) (-a) := by
    refine not_differentiableWithinAt_of_deriv_tendsto_atTop_Ioi f' <| Tendsto.congr' hderiv.symm ?_
    refine Tendsto.comp (g := -deriv f) ?_ tendsto_neg_nhdsGT_neg
    exact Tendsto.comp (g := Neg.neg) tendsto_neg_atBot_atTop hf

Goal state after:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (deriv f) (𝓝[<] a) atBot
f' : ℝ → ℝ := f ∘ Neg.neg
hderiv : deriv f' =ᶠ[𝓝[>] (-a)] -deriv f ∘ Neg.neg
hmain : ¬DifferentiableWithinAt ℝ f' (Ioi (-a)) (-a)
⊢ ¬DifferentiableWithinAt ℝ f (Iio a) a
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Tactic:
have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'

Goal state after:
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Tactic:
have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'

Goal state after:
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε
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 = ⊤
x c : M
hc : c ∈ ↑C
⊢ 0 ∈ ↑C - ↑C

Tactic:
exact ⟨c, hc, c, hc, sub_self c⟩

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

Goal state before:
pp: n : ℕ∞ω
x : ℝ
hx✝ : x ≠ 0
A : ∀ (y : ℝ), 0 < y → ContDiffAt ℝ n log y
hx : x < 0
this : ContDiffAt ℝ n (log ∘ fun y ↦ -y) x
⊢ log = log ∘ fun y ↦ -y

Tactic:
ext x

Goal state after:
n : ℕ∞ω
x✝ : ℝ
hx✝ : x✝ ≠ 0
A : ∀ (y : ℝ), 0 < y → ContDiffAt ℝ n log y
hx : x✝ < 0
this : ContDiffAt ℝ n (log ∘ fun y ↦ -y) x✝
x : ℝ
⊢ log x = (log ∘ fun y ↦ -y) x
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: case inr.inl
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 y : M
hy : y ∈ ↑C
z : M
hz : z ∈ ↑C
c : M
hc : c ∈ ↑C
⊢ ∃ x ∈ C, ∃ y ∈ C, x - y = 0

Tactic:
exact ⟨c, hc, c, hc, sub_self c⟩

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable (fun x ↦ ↑(f x)) μ
fmeas : AEMeasurable f μ
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ∞
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ∞
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ∞
Ig : ∫⁻ (a : α), ENNReal.ofReal (g a).toReal ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ (∫⁻ (a : α), ↑(f a) ∂μ + ↑δ).toReal = (∫⁻ (a : α), ↑(f a) ∂μ).toReal + ↑δ

Tactic:
rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal]

Goal state after:
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
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
Sp : Set ℝ := ↑f '' {x | ↑x + y ∈ s}
Sn : Set ℝ := ↑f '' {x | -↑x - y ∈ s}
⊢ ∀ x ∈ Sn, ∀ y ∈ Sp, x ≤ y

Tactic:
rintro a ⟨xn, hxn, rfl⟩ b ⟨xp, hxp, rfl⟩

Goal state after:
case refine_2
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
Sp : Set ℝ := ↑f '' {x | ↑x + y ∈ s}
Sn : Set ℝ := ↑f '' {x | -↑x - y ∈ s}
xn : ↥f.domain
hxn : xn ∈ {x | -↑x - y ∈ s}
xp : ↥f.domain
hxp : xp ∈ {x | ↑x + y ∈ s}
⊢ ↑f xn ≤ ↑f xp
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
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
Sp : Set ℝ := ↑f '' {x | ↑x + y ∈ s}
Sn : Set ℝ := ↑f '' {x | -↑x - y ∈ s}
xn : ↥f.domain
hxn : xn ∈ {x | -↑x - y ∈ s}
xp : ↥f.domain
hxp : xp ∈ {x | ↑x + y ∈ s}
this : ↑xp + -↑xn ∈ s
⊢ ↑f xn ≤ ↑f xp

Tactic:
← sub_eq_add_neg,

Goal state after:
case refine_2
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
Sp : Set ℝ := ↑f '' {x | ↑x + y ∈ s}
Sn : Set ℝ := ↑f '' {x | -↑x - y ∈ s}
xn : ↥f.domain
hxn : xn ∈ {x | -↑x - y ∈ s}
xp : ↥f.domain
hxp : xp ∈ {x | ↑x + y ∈ s}
this : ↑xp - ↑xn ∈ s
⊢ ↑f xn ≤ ↑f xp
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
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
c : ℝ
le_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c
c_le : ∀ (x : ↥f.domain), ↑x + y ∈ s → c ≤ ↑f x
x : E
hx : x ∈ f.domain
r : ℝ
hy' : r • y ∈ (LinearPMap.mkSpanSingleton y (-c) ⋯).domain
hz : x + r • y ∈ (f.supSpanSingleton y (-c) hy).domain
hzs : x + r • y ∈ s
⊢ 0 ≤ ↑f ⟨x, hx⟩ + -((RingHom.id ℝ) r • c)

Tactic:
← sub_eq_add_neg,

Goal state after:
case refine_2
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
c : ℝ
le_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c
c_le : ∀ (x : ↥f.domain), ↑x + y ∈ s → c ≤ ↑f x
x : E
hx : x ∈ f.domain
r : ℝ
hy' : r • y ∈ (LinearPMap.mkSpanSingleton y (-c) ⋯).domain
hz : x + r • y ∈ (f.supSpanSingleton y (-c) hy).domain
hzs : x + r • y ∈ s
⊢ 0 ≤ ↑f ⟨x, hx⟩ - (RingHom.id ℝ) r • c
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
⊢ gauge 0 = 0

Tactic:
ext x

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
x : E
⊢ gauge 0 x = 0 x
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
a : ℝ
hs₁ : Convex ℝ s
hs₀ : 0 ∈ s
hs₂ : Absorbent ℝ s
ha : 0 ≤ a
⊢ {x | gauge s x ≤ a} = ⋂ r, ⋂ (_ : a < r), r • s

Tactic:
ext x

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
a : ℝ
hs₁ : Convex ℝ s
hs₀ : 0 ∈ s
hs₂ : Absorbent ℝ s
ha : 0 ≤ a
x : E
⊢ x ∈ {x | gauge s x ≤ a} ↔ x ∈ ⋂ r, ⋂ (_ : a < r), r • s
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
⊢ gauge (a • s) = a⁻¹ • gauge s

Tactic:
ext x

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
⊢ gauge (a • s) x = (a⁻¹ • gauge s) x
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

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
p : Seminorm ℝ E
x : E
⊢ gauge (p.ball 0 1) x = p x
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
δ : ℝ≥0 := NNReal.mk (ε / 2) ⋯
δpos : 0 < δ
fp : α → ℝ≥0 := fun x ↦ (f x).toNNReal
int_fp : Integrable (fun x ↦ ↑(fp x)) μ
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ∞
gp_integrable : Integrable (fun x ↦ (gp x).toReal) μ
gpint : ∫ (x : α), (gp x).toReal ∂μ < ∫ (x : α), ↑(fp x) ∂μ + ↑δ
fm : α → ℝ≥0 := fun x ↦ (-f x).toNNReal
int_fm : Integrable (fun x ↦ ↑(fm x)) μ
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable (fun x ↦ ↑(gm x)) μ
gmint : ∫ (x : α), ↑(fm x) ∂μ - ↑δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := fun x ↦ ↑(gp x) - ↑↑(gm x)
ae_g : ∀ᵐ (x : α) ∂μ, (g x).toReal = (↑(gp x)).toReal - (↑↑(gm x)).toReal
⊢ Integrable (fun x ↦ (g x).toReal) μ

Tactic:
integrable_congr ae_g

Goal state after:
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
δ : ℝ≥0 := NNReal.mk (ε / 2) ⋯
δpos : 0 < δ
fp : α → ℝ≥0 := fun x ↦ (f x).toNNReal
int_fp : Integrable (fun x ↦ ↑(fp x)) μ
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ∞
gp_integrable : Integrable (fun x ↦ (gp x).toReal) μ
gpint : ∫ (x : α), (gp x).toReal ∂μ < ∫ (x : α), ↑(fp x) ∂μ + ↑δ
fm : α → ℝ≥0 := fun x ↦ (-f x).toNNReal
int_fm : Integrable (fun x ↦ ↑(fm x)) μ
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable (fun x ↦ ↑(gm x)) μ
gmint : ∫ (x : α), ↑(fm x) ∂μ - ↑δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := fun x ↦ ↑(gp x) - ↑↑(gm x)
ae_g : ∀ᵐ (x : α) ∂μ, (g x).toReal = (↑(gp x)).toReal - (↑↑(gm x)).toReal
⊢ Integrable (fun x ↦ (↑(gp x)).toReal - (↑↑(gm x)).toReal) μ
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s✝ t✝ : Set E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s✝
hs₂ : IsClosed s✝
ht₁ : Convex ℝ t✝
ht₂ : IsCompact t✝
disj : Disjoint s✝ t✝
f : StrongDual ℝ E
s t : ℝ
hs : ∀ a ∈ t✝, f a < s
st : s < t
ht : ∀ b ∈ s✝, t < f b
⊢ ∀ b ∈ t✝, -s < (-f) b

Tactic:
simpa using hs

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

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
f : E → F
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
⊢ Measurable (fderiv 𝕜 f)

Tactic:
refine measurable_of_isClosed fun s hs => ?_

Goal state after:
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
f : E → F
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
s : Set (E →L[𝕜] F)
hs : IsClosed s
⊢ MeasurableSet (fderiv 𝕜 f ⁻¹' s)