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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(n + 1)) g t
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hC : ∀ i ≤ n + 1, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n + 1 → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑n.succ
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
I : ∀ i ∈ Finset.range (n + 1), ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ ↑i ! * C * D ^ i
J : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
⊢ ‖iteratedFDerivWithin 𝕜 n
        (fun y ↦ ((ContinuousLinearMap.compL 𝕜 E Fu Gu) (fderivWithin 𝕜 g t (f y))) (fderivWithin 𝕜 f s y)) s x‖ ≤
    ∑ i ∈ Finset.range (n + 1),
      ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖

Tactic:
by
      exact (ContinuousLinearMap.compL 𝕜 E Fu Gu).norm_iteratedFDerivWithin_le_of_bilinear_of_le_one
        ((hg.fderivWithin ht (by simp)).comp (hf.of_le M.le) hst) (hf.fderivWithin hs (by simp))
        hs hx le_rfl (ContinuousLinearMap.norm_compL_le 𝕜 E Fu Gu)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
hf : ContDiffWithinAt 𝕜 ω f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ IsSymmSndFDerivWithinAt 𝕜 f s x

Tactic:
isSymmSndFDerivWithinAt_iff_iteratedFDerivWithin hs hx

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
hf : ContDiffWithinAt 𝕜 ω f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ ContinuousMultilinearMap.domDomCongr Fin.revPerm (iteratedFDerivWithin 𝕜 2 f s x) = iteratedFDerivWithin 𝕜 2 f s x
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : μ u ≤ μ s + ε / 2
R : α → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : μ (s \ ⋃ x ∈ t0, closedBall x (r0 x)) = 0
t0_disj : t0.PairwiseDisjoint fun x ↦ closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
s's : s' ⊆ s
N : ℕ
τ : ℝ
hτ : 1 < τ
H : IsEmpty (SatelliteConfig α N τ)
v : Set α
s'v : v ⊇ s'
v_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] v
μv : μ v ≤ μ s' + ε / 2 / ↑N
r1 : α → ℝ
hr1 : ∀ x ∈ s', r1 x ∈ f x ∩ Ioo 0 1 ∧ closedBall x (r1 x) ⊆ v
q : BallPackage (↑s') α := { c := fun x ↦ ↑x, r := fun x ↦ r1 ↑x, rpos := ⋯, r_bound := 1, r_le := ⋯ }
S : Fin N → Set ↑s'
S_disj : ∀ (i : Fin N), (S i).PairwiseDisjoint fun j ↦ closedBall (q.c j) (q.r j)
hS : range q.c ⊆ ⋃ i, ⋃ j ∈ S i, ball (q.c j) (q.r j)
S_count : ∀ (i : Fin N), (S i).Countable
r : α → ℝ := fun x ↦ if x ∈ s' then r1 x else r0 x
r_t0 : ∀ x ∈ t0, r x = r0 x
x : α
hx : x ∈ s
h'x : x ∈ s'
A : x ∈ range q.c
⊢ ∃ i, ∃ y ∈ S i, x ∈ ball (↑y) (r1 ↑y)

Tactic:
simpa only [mem_iUnion, mem_image, bex_def] using hS A

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V W V₁ W₁ : E → E
x : E
hV : V₁ =ᶠ[𝓝 x] V
hW : W₁ =ᶠ[𝓝 x] W
⊢ lieBracket 𝕜 V₁ W₁ =ᶠ[𝓝 x] lieBracket 𝕜 V W

Tactic:
filter_upwards [hV.eventuallyEq_nhds, hW.eventuallyEq_nhds] with y hVy hWy

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V W V₁ W₁ : E → E
x : E
hV : V₁ =ᶠ[𝓝 x] V
hW : W₁ =ᶠ[𝓝 x] W
y : E
hVy : V₁ =ᶠ[𝓝 y] V
hWy : W₁ =ᶠ[𝓝 y] W
⊢ lieBracket 𝕜 V₁ W₁ y = lieBracket 𝕜 V W y
Module: Mathlib.Analysis.Calculus.FDeriv.ContinuousAlternatingMap

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
F : Type u_4
G : Type u_5
H : Type u_6
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : NormedAddCommGroup H
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
fg : G [⋀^ι]→L[𝕜] H × (F →L[𝕜] G)
H₁ :
  HasStrictFDerivAt (fun fg ↦ fg.1.compContinuousLinearMap fg.2)
    (ContinuousMultilinearMap.compContinuousLinearMapL (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2).2 ∘SL
        ContinuousLinearMap.fst 𝕜 (ContinuousMultilinearMap 𝕜 (fun x ↦ G) H) (ι → F →L[𝕜] G) +
      (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2).1.fderivCompContinuousLinearMap
          (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2).2 ∘SL
        ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 (fun x ↦ G) H) (ι → F →L[𝕜] G))
    (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2)
⊢ HasStrictFDerivAt (toContinuousMultilinearMap ∘ fun fg ↦ fg.1.compContinuousLinearMap fg.2)
    (toContinuousMultilinearMapCLM 𝕜 ∘SL
      (compContinuousLinearMapCLM fg.2 ∘SL ContinuousLinearMap.fst 𝕜 (G [⋀^ι]→L[𝕜] H) (F →L[𝕜] G) +
        fg.1.fderivCompContinuousLinearMap fg.2 ∘SL ContinuousLinearMap.snd 𝕜 (G [⋀^ι]→L[𝕜] H) (F →L[𝕜] G)))
    fg

Tactic:
have H₂ := ((toContinuousMultilinearMapCLM 𝕜).hasStrictFDerivAt (x := fg.1))

Goal state after:
𝕜 : Type u_1
ι : Type u_2
F : Type u_4
G : Type u_5
H : Type u_6
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : NormedAddCommGroup H
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
fg : G [⋀^ι]→L[𝕜] H × (F →L[𝕜] G)
H₁ :
  HasStrictFDerivAt (fun fg ↦ fg.1.compContinuousLinearMap fg.2)
    (ContinuousMultilinearMap.compContinuousLinearMapL (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2).2 ∘SL
        ContinuousLinearMap.fst 𝕜 (ContinuousMultilinearMap 𝕜 (fun x ↦ G) H) (ι → F →L[𝕜] G) +
      (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2).1.fderivCompContinuousLinearMap
          (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2).2 ∘SL
        ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 (fun x ↦ G) H) (ι → F →L[𝕜] G))
    (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2)
H₂ : HasStrictFDerivAt (⇑(toContinuousMultilinearMapCLM 𝕜)) (toContinuousMultilinearMapCLM 𝕜) fg.1
⊢ HasStrictFDerivAt (toContinuousMultilinearMap ∘ fun fg ↦ fg.1.compContinuousLinearMap fg.2)
    (toContinuousMultilinearMapCLM 𝕜 ∘SL
      (compContinuousLinearMapCLM fg.2 ∘SL ContinuousLinearMap.fst 𝕜 (G [⋀^ι]→L[𝕜] H) (F →L[𝕜] G) +
        fg.1.fderivCompContinuousLinearMap fg.2 ∘SL ContinuousLinearMap.snd 𝕜 (G [⋀^ι]→L[𝕜] H) (F →L[𝕜] G)))
    fg
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
hx : 0 < ‖x‖
h : DifferentiableAt ℝ (fun x ↦ ‖x‖) 0
this✝ : DifferentiableAt ℝ (fun t ↦ ‖t • x‖) 0
this : DifferentiableAt ℝ (fun x ↦ |x|) 0
⊢ False

Tactic:
exact not_differentiableAt_abs_zero this

Goal state after:
Module: Mathlib.Analysis.Calculus.ImplicitFunction.Bivariate

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : IsRCLikeNormedField 𝕜
E₁ : Type u_2
inst✝⁸ : NormedAddCommGroup E₁
inst✝⁷ : NormedSpace 𝕜 E₁
inst✝⁶ : CompleteSpace E₁
E₂ : Type u_3
inst✝⁵ : NormedAddCommGroup E₂
inst✝⁴ : NormedSpace 𝕜 E₂
inst✝³ : CompleteSpace E₂
F : Type u_4
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
u : E₁ × E₂
f : E₁ → E₂ → F
f₁ : E₁ → E₂ → E₁ →L[𝕜] F
f₂ : E₁ → E₂ → E₂ →L[𝕜] F
df₁ : ∀ᶠ (v : E₁ × E₂) in 𝓝 u, HasFDerivAt (fun x ↦ f x v.2) (f₁ v.1 v.2) v.1
df₂ : ∀ᶠ (v : E₁ × E₂) in 𝓝 u, HasFDerivAt (fun x ↦ f v.1 x) (f₂ v.1 v.2) v.2
cf₁ : ContinuousAt (↿f₁) u
cf₂ : ContinuousAt (↿f₂) u
if₂u : (f₂ u.1 u.2).IsInvertible
⊢ HasStrictFDerivAt (implicitFunctionOfBivariate df₁ df₂ cf₁ cf₂ if₂u) (-(f₂ u.1 u.2).inverse ∘SL f₁ u.1 u.2) u.1

Tactic:
simpa using! HasStrictFDerivAt.hasStrictFDerivAt_implicitFunctionOfProdDomain
    (hasStrictFDerivAt_uncurry_coprod df₁ df₂ cf₁ cf₂) (by simpa using! if₂u)

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

Goal state before:
pp: case intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
φ : E → ℝ
x₀ : E
φ' : StrongDual ℝ E
ι : Type u_3
inst✝ : Finite ι
f : ι → E → ℝ
f' : ι → StrongDual ℝ E
hextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀
hf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀
hφ' : HasStrictFDerivAt φ φ' x₀
val✝ : Fintype ι
⊢ ¬LinearIndependent ℝ (Option.elim' φ' f')

Tactic:
rw [Fintype.linearIndependent_iff]

Goal state after:
case intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
φ : E → ℝ
x₀ : E
φ' : StrongDual ℝ E
ι : Type u_3
inst✝ : Finite ι
f : ι → E → ℝ
f' : ι → StrongDual ℝ E
hextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀
hf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀
hφ' : HasStrictFDerivAt φ φ' x₀
val✝ : Fintype ι
⊢ ¬∀ (g : Option ι → ℝ), ∑ i, g i • Option.elim' φ' f' i = 0 → ∀ (i : Option ι), g i = 0
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: case hf'_nonneg
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
⊢ ∀ x ∈ interior (Icc a b), 0 ≤ deriv f x

Tactic:
rw [interior_Icc]
      intro z hz
      simpa [(hff' z hz).deriv] using hf' z hz

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

Goal state before:
pp: case inr
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : b < a
⊢ IntervalIntegrable (fun x ↦ f' x • g (f x)) volume a b ↔ IntervalIntegrable g volume (f a) (f b)

Tactic:
rw [IntervalIntegrable.symm_iff, intervalIntegrable_iff_integrableOn_Icc_of_le hab.le,
      integrableOn_Icc_deriv_smul_iff_of_deriv_nonpos,
      intervalIntegrable_iff_integrableOn_Icc_of_le]
    · apply M right_mem_uIcc left_mem_uIcc hab.le
    · rwa [uIcc_of_ge hab.le] at hf
    · grind
    · grind
    · exact hab.le

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

Goal state before:
pp: case hf'_nonpos
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, f' x ≤ 0
hab : a ≤ b
⊢ ∀ x ∈ interior (Icc a b), deriv f x ≤ 0

Tactic:
rw [interior_Icc]
      intro z hz
      simpa [(hff' z hz).deriv] using hf' z hz

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
f f' : ℝ → E
a : ℝ
m : E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hcont✝ : ContinuousWithinAt f (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Ioi a) volume
hf : Tendsto f atTop (𝓝 m)
hcont : ContinuousOn f (Ici a)
⊢ (fun x ↦ f x - f a) =ᶠ[atTop] fun i ↦ ∫ (x : ℝ) in a..id i, f' x

Tactic:
filter_upwards [Ioi_mem_atTop a] with x hx

Goal state after:
E : Type u_1
f f' : ℝ → E
a : ℝ
m : E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hcont✝ : ContinuousWithinAt f (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Ioi a) volume
hf : Tendsto f atTop (𝓝 m)
hcont : ContinuousOn f (Ici a)
x : ℝ
hx : x ∈ Ioi a
⊢ f x - f a = ∫ (x : ℝ) in a..id x, f' x
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: case refine_1
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
x : ℝ
⊢ IntegrableOn g' (Ioc a (id x)) volume

Tactic:
exact intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)
      (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
⊢ (fun x ↦ g x - g a) =ᶠ[atTop] fun i ↦ ∫ (x : ℝ) in a..id i, ‖g' x‖

Tactic:
filter_upwards [Ioi_mem_atTop a] with x hx

Goal state after:
g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
x : ℝ
hx : x ∈ Ioi a
⊢ g x - g a = ∫ (x : ℝ) in a..id x, ‖g' x‖
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
a : ℝ
hf : ContinuousOn f (Ici a)
hft : Tendsto f atTop atTop
hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioi a)
hg1 : IntegrableOn g (f '' Ici a) volume
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) (Ioi a) volume
eq : ∀ (b : ℝ), a < b → ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u
⊢ ∫ (x : ℝ) in Ioi a, f' x • (g ∘ f) x = ∫ (u : ℝ) in Ioi (f a), g u

Tactic:
have t2 := intervalIntegral_tendsto_integral_Ioi _ hg2 tendsto_id

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
a : ℝ
hf : ContinuousOn f (Ici a)
hft : Tendsto f atTop atTop
hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioi a)
hg1 : IntegrableOn g (f '' Ici a) volume
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) (Ioi a) volume
eq : ∀ (b : ℝ), a < b → ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u
t2 : Tendsto (fun i ↦ ∫ (x : ℝ) in a..id i, f' x • (g ∘ f) x) atTop (𝓝 (∫ (x : ℝ) in Ioi a, f' x • (g ∘ f) x ∂volume))
⊢ ∫ (x : ℝ) in Ioi a, f' x • (g ∘ f) x = ∫ (u : ℝ) in Ioi (f a), g u
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : CommMonoid α
f : ι → β → α
g : β → α
inst✝ : UniformSpace α
⊢ HasProdUniformly f g ↔ TendstoUniformly (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop

Tactic:
by
  simpa [HasProdUniformly, HasProd, ← UniformFun.ofFun_prod, Finset.prod_fn] using!
    UniformFun.tendsto_iff_tendstoUniformly

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

Goal state before:
pp: H : Type u_1
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
⊢ ∀ (e : OpenPartialHomeomorph H H),
    (∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ G.members ∩ G'.members) → e ∈ G.members ∩ G'.members

Tactic:
intro e hx

Goal state after:
H : Type u_1
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : OpenPartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ G.members ∩ G'.members
⊢ e ∈ G.members ∩ G'.members
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: case neg
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace M'
cm : ChartedSpace H M
cm' : ChartedSpace H M'
h : IsEmpty H
⊢ ChartedSpace H (M ⊕ M')

Tactic:
have : IsEmpty M := isEmpty_of_chartedSpace H

Goal state after:
case neg
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace M'
cm : ChartedSpace H M
cm' : ChartedSpace H M'
h : IsEmpty H
this : IsEmpty M
⊢ ChartedSpace H (M ⊕ M')
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case e'_5
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
s : Set E
hs : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
u : E
hu : u ∈ s
v : E
hv : v ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • u = a • u

pp: case e'_6
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
s : Set E
hs : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
u : E
hu : u ∈ s
v : E
hv : v ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ b • v = b • v

Tactic:
· rw [← @algebraMap_smul (R := ℝ) (A := 𝕜), ← @algebraMap_smul (R := ℝ) (A := 𝕜)]

Goal state after:
case e'_6
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
s : Set E
hs : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
u : E
hu : u ∈ s
v : E
hv : v ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ b • v = b • v
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case e'_6
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
h : IsRCLikeNormedField 𝕜
s : Set E
hs : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
u : E
hu : u ∈ s
v : E
hv : v ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ b • v = b • v

Tactic:
· rw [← @algebraMap_smul (R := ℝ) (A := 𝕜), ← @algebraMap_smul (R := ℝ) (A := 𝕜)]

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

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✝
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
H : Type u_5
inst✝ : TopologicalSpace H
φ : PartialEquiv H E
hsource : φ.source = univ
htarget : Convex ℝ φ.target
hcont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑φ
hcont_inv : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, inst✝] ↑φ.symm
hint : (interior φ.target).Nonempty
this : range ↑φ = φ.target
⊢ Convex ℝ φ.target

Tactic:
exact htarget.convex_isRCLikeNormedField

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

Goal state before:
pp: case refine_2
H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : OpenPartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
this : P ((f ∘ ↑e.symm) ∘ ↑e) (↑e ⁻¹' ↑e.symm ⁻¹' s) x
⊢ ↑e ⁻¹' ↑e.symm ⁻¹' s =ᶠ[𝓝 x] s

Tactic:
· rw [eventuallyEq_set]
    refine eventually_of_mem (e.open_source.mem_nhds hxe) fun x' hx' ↦ ?_
    simp_rw [mem_preimage, e.left_inv hx']

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

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
H : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : LocallyCompactSpace E
I : ModelWithCorners 𝕜 E H
⊢ LocallyCompactSpace H

Tactic:
have : ∀ x : H, (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (I x) ∧ IsCompact s)
      fun s => I.symm '' (s ∩ range I) := fun x ↦ by
    rw [← I.symm_map_nhdsWithin_range]
    exact ((compact_basis_nhds (I x)).inf_principal _).map _
  refine .of_hasBasis this ?_
  rintro x s ⟨-, hsc⟩
  exact (hsc.inter_right I.isClosed_range).image I.continuous_symm

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.ExtChartAt

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
n : WithTop ℕ∞
I : ModelWithCorners 𝕜 E H
e e' : OpenPartialHomeomorph M H
inst✝ : ChartedSpace H M
he : e ∈ maximalAtlas I n M
he' : e' ∈ maximalAtlas I n M
⊢ ContDiffOn 𝕜 n (↑(extendCoordChange e e')) (extendCoordChange e e').source

Tactic:
rw [I.extendCoordChange_source, I.image_eq]
  exact (StructureGroupoid.compatible_of_mem_maximalAtlas he he').1

Goal state after: