mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.pretrivializationAtlas
this : ContinuousOn (fun x ↦ (a.coordChange he' he x.1) x.2) ((e'.baseSet ∩ e.baseSet) ×ˢ univ)
⊢ EqOn (↑e ∘ ↑e'.symm) (fun x ↦ (x.1, (a.coordChange he' he x.1) x.2)) ((e'.baseSet ∩ e.baseSet) ×ˢ univ)

Tactic:
rintro ⟨b, f⟩ ⟨hb, -⟩

Goal state after:
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.pretrivializationAtlas
this : ContinuousOn (fun x ↦ (a.coordChange he' he x.1) x.2) ((e'.baseSet ∩ e.baseSet) ×ˢ univ)
b : B
f : F
hb : (b, f).1 ∈ e'.baseSet ∩ e.baseSet
⊢ (↑e ∘ ↑e'.symm) (b, f) = (fun x ↦ (x.1, (a.coordChange he' he x.1) x.2)) (b, f)
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁹ : NormedAddCommGroup E
inst✝¹⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁶ : TopologicalSpace M
inst✝¹⁵ : ChartedSpace H M
E' : Type u_5
inst✝¹⁴ : NormedAddCommGroup E'
inst✝¹³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝¹² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹¹ : TopologicalSpace M'
inst✝¹⁰ : ChartedSpace H' M'
F : Type u_11
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝⁶ : TopologicalSpace N
inst✝⁵ : ChartedSpace G N
F' : Type u_14
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
G' : Type u_15
inst✝² : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹ : TopologicalSpace N'
inst✝ : ChartedSpace G' N'
x : M
f : M → M'
g : N → N'
y : N
hf : MDiffAt f x
hg : MDiffAt g y
⊢ MDiffAt (Prod.map f g) (x, y)

Tactic:
rw [← mdifferentiableWithinAt_univ] at *
  convert! hf.prodMap hg
  exact univ_prod_univ.symm

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.Tangent

Goal state before:
pp: case hx
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H : Type u_4
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_6
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
M' : Type u_7
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
F : Type u_8
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
i : ↑(atlas H M)
x : M
hx : x ∈ (↑i).source
v : E
⊢ (↑((↑i).extend I) ∘ ↑((↑i).extend I).symm) (↑((↑i).extend I) x) = ↑((↑i).extend I) x

Tactic:
simp_rw [Function.comp_apply, i.1.extend_left_inv hx]

Goal state after:
Module: Mathlib.Geometry.Manifold.Diffeomorph

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹⁵ : NormedAddCommGroup E'
inst✝¹⁴ : NormedSpace 𝕜 E'
F : Type u_4
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜 F
H : Type u_5
inst✝¹¹ : TopologicalSpace H
H' : Type u_6
inst✝¹⁰ : TopologicalSpace H'
G : Type u_7
inst✝⁹ : TopologicalSpace G
G' : Type u_8
inst✝⁸ : TopologicalSpace G'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
J' : ModelWithCorners 𝕜 F G'
M : Type u_9
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
M' : Type u_10
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
N : Type u_11
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace G N
N' : Type u_12
inst✝¹ : TopologicalSpace N'
inst✝ : ChartedSpace G' N'
n : ℕ∞ω
e✝ e : E ≃L[𝕜] F
x : M
⊢ ↑(extChartAt I x) x ∈
    (extChartAt I x).target ∩
      ↑(extChartAt I x).symm ⁻¹'
        (univ ∩ ⇑(Equiv.refl M) ⁻¹' (extChartAt (I.transContinuousLinearEquiv e) ((Equiv.refl M) x)).source)

Tactic:
· exact
      ⟨(extChartAt I x).map_source (mem_extChartAt_source x), trivial, by simp only [mfld_simps]⟩

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.Atlas

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
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x y : M
hy : y ∈ (extChartAt I x).source
⊢ mfderiv[range ↑I] ↑(extChartAt I x).symm (↑(extChartAt I x) y) ∘SL mfderiv% ↑(extChartAt I x) y =
    ContinuousLinearMap.id 𝕜 (TangentSpace I y)

Tactic:
have : y = (extChartAt I x).symm (extChartAt I x y) := ((extChartAt I x).left_inv hy).symm
  convert! mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt ((extChartAt I x).map_source hy)
  rw [(extChartAt I x).left_inv (by simpa using hy)]

Goal state after:
Module: Mathlib.Geometry.Manifold.Diffeomorph

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
H : Type u_5
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_9
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
n : ℕ∞ω
M' : Type u_13
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
⊢ ⇑(sumComm I M n M') ∘ Sum.inr = Sum.inl

Tactic:
by
  ext
  exact Sum.swap_inr

Goal state after:
Module: Mathlib.Topology.ShrinkingLemma

Goal state before:
pp: ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
x : X
hx : x ∈ s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
j : ι
hj : ¬j = i
hsi' : x ∈ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
⊢ x ∈ (v.toFun j)ᶜ

Tactic:
rw [Set.mem_iInter₂] at hsi'

Goal state after:
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
x : X
hx : x ∈ s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
j : ι
hj : ¬j = i
hsi' : ∀ (i_1 : ι), ¬i_1 = i → x ∈ (v.toFun i_1)ᶜ
⊢ x ∈ (v.toFun j)ᶜ
Module: Mathlib.Topology.MetricSpace.PartitionOfUnity

Goal state before:
pp: ι : Type u_1
X : Type u_2
inst✝ : EMetricSpace X
K U : ι → Set X
hK : ∀ (i : ι), IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (K i)
hU : ∀ (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
hKU : ∀ (i : ι), K i ⊆ U i
hfin : LocallyFinite K
x : X
this : ∀ (i : ι), x ∈ K i → ∀ᶠ (p : ℝ≥0∞ × X) in 𝓝 0 ×ˢ 𝓝 x, closedEBall p.2 p.1 ⊆ U i
⊢ ∀ᶠ (p : ℝ≥0∞ × X) in 𝓝 0 ×ˢ 𝓝 x, ∀ (i : ι), p.2 ∈ K i → closedEBall p.2 p.1 ⊆ U i

Tactic:
by
    apply mp_mem ((eventually_all_finite (hfin.point_finite x)).2 this)
      (mp_mem (@tendsto_snd ℝ≥0∞ _ (𝓝 0) _ _ (hfin.iInter_compl_mem_nhds hK x)) _)
    apply univ_mem'
    rintro ⟨r, y⟩ hxy hyU i hi
    simp only [mem_iInter, mem_compl_iff, not_imp_not, mem_preimage] at hxy
    exact hyU _ (hxy _ hi)

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp

Goal state before:
pp: α : Type u_1
G : Type u_2
p : ℝ≥0∞
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup G
f : α → G
hf : MemLp f p μ
hf_meas : StronglyMeasurable f
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
⊢ FinStronglyMeasurable f μ

Tactic:
borelize G

Goal state after:
α : Type u_1
G : Type u_2
p : ℝ≥0∞
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup G
f : α → G
hf : MemLp f p μ
hf_meas : StronglyMeasurable f
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
⊢ FinStronglyMeasurable f μ
Module: Mathlib.Order.Filter.Ring

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : MulZeroClass β
inst✝² : Preorder β
inst✝¹ : PosMulMono β
inst✝ : MulPosMono β
l : Filter α
f₁ f₂ g₁ g₂ : α → β
hf : f₁ ≤ᶠ[l] f₂
hg : g₁ ≤ᶠ[l] g₂
hg₀ : 0 ≤ᶠ[l] g₁
hf₀ : 0 ≤ᶠ[l] f₂
⊢ f₁ * g₁ ≤ᶠ[l] f₂ * g₂

Tactic:
by
  filter_upwards [hf, hg, hg₀, hf₀] with x using _root_.mul_le_mul

Goal state after:
Module: Mathlib.Topology.ShrinkingLemma

Goal state before:
pp: ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
hs : IsCompact s
uo : ∀ (i : ι), IsOpen (u i)
uf : ∀ x ∈ s, {i | x ∈ u i}.Finite
us : s ⊆ ⋃ i, u i
⊢ ∃ v,
    s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ (∀ (i : ι), closure (v i) ⊆ u i) ∧ ∀ (i : ι), IsCompact (closure (v i))

Tactic:
by
  have : Nonempty (PartialRefinement u s (fun w => IsCompact (closure w))) :=
    ⟨⟨u, ∅, uo, us, False.elim, False.elim, fun _ => rfl⟩⟩
  have : ∀ c : Set (PartialRefinement u s (fun w => IsCompact (closure w))),
      IsChain (· ≤ ·) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub :=
    fun c hc ne => ⟨.chainSup c hc ne uf us, fun v hv => PartialRefinement.le_chainSup _ _ _ _ hv⟩
  rcases zorn_le_nonempty this with ⟨v, hv⟩
  suffices ∀ i, i ∈ v.carrier from
    ⟨v, v.subset_iUnion, fun i => v.isOpen _, fun i => v.closure_subset (this i), ?_⟩
  · intro i
    exact v.pred_of_mem (this i)
  · intro i
    by_contra! hi
    rcases exists_gt_t2space v hs i hi with ⟨v', hlt, _⟩
    exact hv.not_lt hlt

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.ProperSpace

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : ProperSpace E
s : Set E
x : E
hx : AccPt x (𝓟 s)
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → E
hvx : ∀ (n : ℕ), v n ≠ x
hvu : ∀ (n : ℕ), v n ∈ closedBall x (u n)
hvs : ∀ (n : ℕ), v n ∈ s
d : ℕ → E := fun n ↦ v n - x
M : ∀ (n : ℕ), x + d n ∈ s \ {x}
r : 𝕜
hr : 1 < ‖r‖
c : ℕ → 𝕜
c_ne : ∀ (n : ℕ), c n ≠ 0
c_le : ∀ (n : ℕ), ‖c n • d n‖ < 1
le_c : ∀ (n : ℕ), 1 / ‖r‖ ≤ ‖c n • d n‖
hc : ∀ (n : ℕ), ‖c n‖⁻¹ ≤ 1⁻¹ * ‖r‖ * ‖d n‖
l : E
l_mem : l ∈ closedBall 0 1 \ ball 0 (1 / ‖r‖)
φ : ℕ → ℕ
φ_strict : StrictMono φ
hφ : Tendsto ((fun n ↦ c n • d n) ∘ φ) atTop (𝓝 l)
⊢ l ∈ {0}ᶜ

Tactic:
push _ ∈ _

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : ProperSpace E
s : Set E
x : E
hx : AccPt x (𝓟 s)
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → E
hvx : ∀ (n : ℕ), v n ≠ x
hvu : ∀ (n : ℕ), v n ∈ closedBall x (u n)
hvs : ∀ (n : ℕ), v n ∈ s
d : ℕ → E := fun n ↦ v n - x
M : ∀ (n : ℕ), x + d n ∈ s \ {x}
r : 𝕜
hr : 1 < ‖r‖
c : ℕ → 𝕜
c_ne : ∀ (n : ℕ), c n ≠ 0
c_le : ∀ (n : ℕ), ‖c n • d n‖ < 1
le_c : ∀ (n : ℕ), 1 / ‖r‖ ≤ ‖c n • d n‖
hc : ∀ (n : ℕ), ‖c n‖⁻¹ ≤ 1⁻¹ * ‖r‖ * ‖d n‖
l : E
l_mem : l ∈ closedBall 0 1 \ ball 0 (1 / ‖r‖)
φ : ℕ → ℕ
φ_strict : StrictMono φ
hφ : Tendsto ((fun n ↦ c n • d n) ∘ φ) atTop (𝓝 l)
⊢ ¬l = 0
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
h'f : HasCompactSupport f
hg : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] g
v : E
K : Set E := cthickening ‖v‖ (tsupport f)
K_compact : IsCompact K
t : ℝ
ht : t ∈ Ioc 0 1
t_pos : 0 < t
x : E
hx : x ∉ K
A : f x = 0
B : f (x + t • v) = 0
y : E
_hy : y ∈ K
⊢ 0 ≤ ↑C * ‖v‖ * ‖g y‖

Tactic:
by positivity

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Seq

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
hx : ∀ ε > 0, ∃ b ∈ s, dist x b < ε
c : 𝕜
hc : 1 < ‖c‖
n : ℕ
⊢ (1 / (2 * ‖c‖)) ^ n > 0

Tactic:
by positivity

Goal state after:
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ ∫ (x : E), (f (x + t • v) - f x) * g x ∂μ = ∫ (x : E), f x * (g (x + t • -v) - g x) ∂μ

Tactic:
simp_rw [_root_.sub_mul, _root_.mul_sub]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ ∫ (x : E), f (x + t • v) * g x - f x * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) - f x * g x ∂μ
Module: Mathlib.Analysis.Calculus.TangentCone.Seq

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
c : 𝕜
hc : 1 < ‖c‖
d : ℕ → E
hds : ∀ (n : ℕ), x + d n ∈ s
hd : ∀ (n : ℕ), ‖d n‖ < (1 / (2 * ‖c‖)) ^ n
N : ℕ
x✝ : True
n : ℕ
hn : N ≤ n
⊢ 0 < ‖c‖ ^ n

Tactic:
by positivity

Goal state after:
Module: Mathlib.Order.SuccPred.IntervalSucc

Goal state before:
pp: case inr.refine_2
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : LinearOrder β
f : α → β
hf : Monotone f
m n : α
hmn : m ≤ n
k : α
hmk : m ≤ k
⊢ ⋃ i ∈ Ico m k, Ioc (f i) (f (succ i)) = Ioc (f m) (f k) →
    ⋃ i ∈ Ico m (succ k), Ioc (f i) (f (succ i)) = Ioc (f m) (f (succ k))

Tactic:
ihk

Goal state after:
case inr.refine_2
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : LinearOrder β
f : α → β
hf : Monotone f
m n : α
hmn : m ≤ n
k : α
hmk : m ≤ k
ihk : ⋃ i ∈ Ico m k, Ioc (f i) (f (succ i)) = Ioc (f m) (f k)
⊢ ⋃ i ∈ Ico m (succ k), Ioc (f i) (f (succ i)) = Ioc (f m) (f (succ k))
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
B : ∀ i ∈ s, Integrable (fun x ↦ a i * ((fderiv ℝ g x) (v i) * f x)) μ
⊢ ∫ (x : E), ∑ x_1 ∈ s, a x_1 * ((fderiv ℝ g x) (v x_1) * f x) ∂μ =
    ∑ i ∈ s, a i * ∫ (x : E), (fderiv ℝ g x) (v i) * f x ∂μ

Tactic:
integral_finsetSum s B,

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
B : ∀ i ∈ s, Integrable (fun x ↦ a i * ((fderiv ℝ g x) (v i) * f x)) μ
⊢ ∑ i ∈ s, ∫ (a_1 : E), a i * ((fderiv ℝ g a_1) (v i) * f a_1) ∂μ =
    ∑ i ∈ s, a i * ∫ (x : E), (fderiv ℝ g x) (v i) * f x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
G : ℕ → ℝ → ℝ
hGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))
hG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))
hG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.ofReal (f b - f a)
hG'₀ : liminf (fun n ↦ ∫⁻ (x : ℝ) in Icc a b, ‖G n x‖ₑ) atTop ≠ ⊤
integrable_f_deriv : Integrable (deriv f) (volume.restrict (Icc a b))
this : f a ≤ f b
⊢ ∫ (x : ℝ) in a..b, deriv f x ∈ uIcc 0 (f b - f a)

Tactic:
uIcc_of_le (by linarith),

Goal state after:
f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
G : ℕ → ℝ → ℝ
hGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))
hG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))
hG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.ofReal (f b - f a)
hG'₀ : liminf (fun n ↦ ∫⁻ (x : ℝ) in Icc a b, ‖G n x‖ₑ) atTop ≠ ⊤
integrable_f_deriv : Integrable (deriv f) (volume.restrict (Icc a b))
this : f a ≤ f b
⊢ ∫ (x : ℝ) in a..b, deriv f x ∈ Icc 0 (f b - f a)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
G : ℕ → ℝ → ℝ
hGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))
hG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))
hG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.ofReal (f b - f a)
hG'₀ : liminf (fun n ↦ ∫⁻ (x : ℝ) in Icc a b, ‖G n x‖ₑ) atTop ≠ ⊤
integrable_f_deriv : Integrable (deriv f) (volume.restrict (Icc a b))
this : f a ≤ f b
⊢ ∫ (x : ℝ) in a..b, deriv f x ∈ uIcc 0 (f b - f a)

Tactic:
rw [uIcc_of_le (by linarith), mem_Icc]

Goal state after:
f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
G : ℕ → ℝ → ℝ
hGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))
hG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))
hG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.ofReal (f b - f a)
hG'₀ : liminf (fun n ↦ ∫⁻ (x : ℝ) in Icc a b, ‖G n x‖ₑ) atTop ≠ ⊤
integrable_f_deriv : Integrable (deriv f) (volume.restrict (Icc a b))
this : f a ≤ f b
⊢ 0 ≤ ∫ (x : ℝ) in a..b, deriv f x ∧ ∫ (x : ℝ) in a..b, deriv f x ≤ f b - f a
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.LebesgueDifferentiationThm

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E
hf : LocallyIntegrable f volume
hg : ∀ (x y : ℝ), IntervalIntegrable f volume x y
LDT : ∀ᵐ (x : ℝ), Tendsto (fun a ↦ ⨍ (y : ℝ) in a, f y) ((vitaliFamily volume 1).filterAt x) (𝓝 (f x))
h : ∀ {a b : ℝ}, ∫ (t : ℝ) in Ioc a b, f t = ∫ (t : ℝ) in Icc a b, f t
x : ℝ
hx : Tendsto (fun a ↦ ⨍ (y : ℝ) in a, f y) ((vitaliFamily volume 1).filterAt x) (𝓝 (f x))
c y : ℝ
hy : y ∈ Ioi x
⊢ slope (fun x ↦ ∫ (t : ℝ) in c..x, f t) x y = ((fun a ↦ ⨍ (y : ℝ) in a, f y) ∘ fun y ↦ Icc x y) y

Tactic:
replace hy : x ≤ y := hy.le

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E
hf : LocallyIntegrable f volume
hg : ∀ (x y : ℝ), IntervalIntegrable f volume x y
LDT : ∀ᵐ (x : ℝ), Tendsto (fun a ↦ ⨍ (y : ℝ) in a, f y) ((vitaliFamily volume 1).filterAt x) (𝓝 (f x))
h : ∀ {a b : ℝ}, ∫ (t : ℝ) in Ioc a b, f t = ∫ (t : ℝ) in Icc a b, f t
x : ℝ
hx : Tendsto (fun a ↦ ⨍ (y : ℝ) in a, f y) ((vitaliFamily volume 1).filterAt x) (𝓝 (f x))
c y : ℝ
hy : x ≤ y
⊢ slope (fun x ↦ ∫ (t : ℝ) in c..x, f t) x y = ((fun a ↦ ⨍ (y : ℝ) in a, f y) ∘ fun y ↦ Icc x y) y
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: F : Type u_2
inst✝ : SeminormedAddCommGroup F
a b : ℝ
f g : ℝ → F
hf : AbsolutelyContinuousOnInterval f a b
hg : AbsolutelyContinuousOnInterval g a b
t : ℕ × (ℕ → ℝ × ℝ)
i : ℕ
hi : i ∈ Finset.range t.1
⊢ 0 ≤ dist ((f + g) (t.2 i).1) ((f + g) (t.2 i).2)

Tactic:
by positivity

Goal state after:
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C : ℝ≥0
inst✝ : FiniteDimensional ℝ E
f : E → F
hf : LipschitzWith C f
s : Set E
hs : sphere 0 1 ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
L : E →L[ℝ] F
x : E
hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
ε : ℝ
εpos : 0 < ε
⊢ 0 < ε / (↑C + ‖L‖ + 1)

Tactic:
by positivity

Goal state after:
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: F : Type u_2
inst✝ : SeminormedAddCommGroup F
a b : ℝ
f : ℝ → F
hf : AbsolutelyContinuousOnInterval f a b
t : ℕ × (ℕ → ℝ × ℝ)
i : ℕ
hi : i ∈ Finset.range t.1
⊢ 0 ≤ dist ((-f) (t.2 i).1) ((-f) (t.2 i).2)

Tactic:
by positivity

Goal state after:
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C : ℝ≥0
inst✝ : FiniteDimensional ℝ E
f : E → F
hf : LipschitzWith C f
s : Set E
hs : sphere 0 1 ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
L : E →L[ℝ] F
x : E
hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
ε : ℝ
εpos : 0 < ε
⊢ ↑C + ‖L‖ + 1 ≠ 0

Tactic:
by positivity

Goal state after:
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: F : Type u_2
inst✝³ : SeminormedAddCommGroup F
a b : ℝ
f : ℝ → F
M : Type u_3
inst✝² : SeminormedRing M
inst✝¹ : Module M F
inst✝ : NormSMulClass M F
α : M
hf : AbsolutelyContinuousOnInterval f a b
t : ℕ × (ℕ → ℝ × ℝ)
i : ℕ
hi : i ∈ Finset.range t.1
⊢ 0 ≤ dist ((fun x ↦ α • f x) (t.2 i).1) ((fun x ↦ α • f x) (t.2 i).2)

Tactic:
by positivity

Goal state after:
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
a b : ℝ
f : ℝ → X
K : ℝ≥0
hfK : LipschitzOnWith K f (uIcc a b)
ε : ℝ
hε : ε > 0
⊢ ε / (↑K + 1) > 0

Tactic:
by positivity

Goal state after: