mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case inl
α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i), {p.index k}
i : Fin N
jx : Ordinal.{u}
jx_lt : jx < p.lastStep
jxi : p.color jx = ↑i
hx : p.index jx ∈ s i
jy : Ordinal.{u}
jy_lt : jy < p.lastStep
jyi : p.color jy = ↑i
hy : p.index jy ∈ s i
x_ne_y : p.index jx ≠ p.index jy
jxy : jx ≤ jy
H : jx < jy
⊢ jx < jy

pp: case inr
α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i), {p.index k}
i : Fin N
jx : Ordinal.{u}
jx_lt : jx < p.lastStep
jxi : p.color jx = ↑i
hx : p.index jx ∈ s i
jy_lt : jx < p.lastStep
jyi : p.color jx = ↑i
hy : p.index jx ∈ s i
x_ne_y : p.index jx ≠ p.index jx
jxy : jx ≤ jx
⊢ jx < jx

Tactic:
· { exact H }

Goal state after:
case inr
α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i), {p.index k}
i : Fin N
jx : Ordinal.{u}
jx_lt : jx < p.lastStep
jxi : p.color jx = ↑i
hx : p.index jx ∈ s i
jy_lt : jx < p.lastStep
jyi : p.color jx = ↑i
hy : p.index jx ∈ s i
x_ne_y : p.index jx ≠ p.index jx
jxy : jx ≤ jx
⊢ jx < jx
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case a
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
s : Set G
h's : IsCompact (closure[inst✝⁸] s)
ν : Measure G := ⋯
⊢ μ'.haarScalarFactor μ • μ s ≤ μ' s

Tactic:
· calc
    ν s ≤ ν ((toMeasurable μ' s) ∩ (closure s)) :=
      measure_mono <| subset_inter (subset_toMeasurable μ' s) subset_closure
    _ = μ' ((toMeasurable μ' s) ∩ (closure s)) := by
      apply (measure_isMulInvariant_eq_smul_of_isCompact_closure_of_measurableSet _ _ _ _).symm
      · exact (measurableSet_toMeasurable μ' s).inter isClosed_closure.measurableSet
      · exact h's.closure_of_subset inter_subset_right
    _ ≤ μ' (toMeasurable μ' s) := measure_mono inter_subset_left
    _ = μ' s := measure_toMeasurable s

Goal state after:
Module: Mathlib.Analysis.Calculus.Conformal.NormedSpace

Goal state before:
pp: case pos
X : Type u_1
Y : Type u_2
inst✝³ : NormedAddCommGroup X
inst✝² : NormedAddCommGroup Y
inst✝¹ : NormedSpace ℝ X
inst✝ : NormedSpace ℝ Y
f : X → Y
x : X
H : IsConformalMap (fderiv ℝ f x)
h : DifferentiableAt ℝ f x
⊢ ConformalAt f x

pp: case neg
X : Type u_1
Y : Type u_2
inst✝³ : NormedAddCommGroup X
inst✝² : NormedAddCommGroup Y
inst✝¹ : NormedSpace ℝ X
inst✝ : NormedSpace ℝ Y
f : X → Y
x : X
H : IsConformalMap (fderiv ℝ f x)
h : ¬DifferentiableAt ℝ f x
⊢ ConformalAt f x

Tactic:
· exact ⟨fderiv ℝ f x, h.hasFDerivAt, H⟩

Goal state after:
case neg
X : Type u_1
Y : Type u_2
inst✝³ : NormedAddCommGroup X
inst✝² : NormedAddCommGroup Y
inst✝¹ : NormedSpace ℝ X
inst✝ : NormedSpace ℝ Y
f : X → Y
x : X
H : IsConformalMap (fderiv ℝ f x)
h : ¬DifferentiableAt ℝ f x
⊢ ConformalAt f x
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝⁴ : MetricSpace α
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : 0 < μ s
h✝ : Nonempty α
Npos : N ≠ 0
o : Set α
so : s ⊆ o
omeas : MeasurableSet o
μo : μ o = μ s
a : BallPackage (↑s) α := { c := fun x ↦ ↑x, r := fun x ↦ r ↑x, rpos := ⋯, r_bound := 1, r_le := ⋯ }
u : Fin N → Set ↑s
hu : ∀ (i : Fin N), (u i).PairwiseDisjoint fun j ↦ closedBall (a.c j) (a.r j)
hu' : range a.c ⊆ ⋃ i, ⋃ j ∈ u i, ball (a.c j) (a.r j)
u_count : ∀ (i : Fin N), (u i).Countable
v : Fin N → Set α := fun i ↦ ⋃ x ∈ u i, closedBall (↑x) (r ↑x)
A : s = ⋃ i, s ∩ v i
⊢ μ s ≤ ∑ i, μ (s ∩ v i)

Tactic:
by
        conv_lhs => rw [A]
        apply measure_iUnion_fintype_le

Goal state after:
Module: Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
i j : Fin N.succ
inej : i ≠ j
hi : ‖a.c i‖ ≤ 2
hj : 2 < ‖a.c j‖
ah : Pairwise fun i j ↦ a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨ a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
hcrj : ‖a.c j‖ ≤ a.r j + 1
I : a.r i ≤ 2
J : (1 - δ / 4) * τ ≤ 1
H : a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i
C : a.r j ≤ 4
⊢ (1 - δ / 4) * a.r j ≤ (1 - δ / 4) * (τ * a.r i)

Tactic:
gcongr; exact H.2

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
α : ↑I
f : E → F
a : E
⊢ ContDiffPointwiseHolderAt 0 α f a ↔ ContDiffAt ℝ 0 f a ∧ (fun x ↦ f x - f a) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ ↑α

Tactic:
by
  simp only [contDiffPointwiseHolderAt_iff, Nat.cast_zero, and_congr_right_iff]
  intro hfc
  simp only [iteratedFDeriv_zero_eq_comp, Function.comp_def, ← map_sub]
  rw [← isBigO_norm_left]
  simp_rw [LinearIsometryEquiv.norm_map, isBigO_norm_left]

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

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoMetricSpace X
inst✝ : MetricSpace Y
r : ℝ≥0
f : X → Y
x₁ x₂ : X
h : f x₁ = f x₂
⊢ f x₁ = f x₂

Tactic:
assumption

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise

Goal state before:
pp: E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F → G
hg : ContDiffPointwiseHolderAt k α g (f a)
hf : ContDiffPointwiseHolderAt k α f a
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
⊢ (fun x ↦
      (ftaylorSeries ℝ g (f x)).taylorComp (ftaylorSeries ℝ f x) k -
        (ftaylorSeries ℝ g (f a)).taylorComp (ftaylorSeries ℝ f a) k) =O[𝓝 a]
    fun x ↦ ‖x - a‖ ^ ↑α

Tactic:
apply FormalMultilinearSeries.taylorComp_sub_taylorComp_isBigO

Goal state after:
case hp_bdd
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F → G
hg : ContDiffPointwiseHolderAt k α g (f a)
hf : ContDiffPointwiseHolderAt k α f a
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
⊢ ∀ k_1 ≤ k, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 a) fun x ↦ ‖ftaylorSeries ℝ g (f x) k_1‖

case hpf
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F → G
hg : ContDiffPointwiseHolderAt k α g (f a)
hf : ContDiffPointwiseHolderAt k α f a
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
⊢ ∀ k_1 ≤ k, (fun a_2 ↦ ftaylorSeries ℝ g (f a_2) k_1 - ftaylorSeries ℝ g (f a) k_1) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ ↑α

case hq₁_bdd
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F → G
hg : ContDiffPointwiseHolderAt k α g (f a)
hf : ContDiffPointwiseHolderAt k α f a
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
⊢ ∀ k_1 ≤ k, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 a) fun x ↦ ‖ftaylorSeries ℝ f x k_1‖

case hq₂_bdd
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F → G
hg : ContDiffPointwiseHolderAt k α g (f a)
hf : ContDiffPointwiseHolderAt k α f a
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
⊢ ∀ k_1 ≤ k, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 a) fun x ↦ ‖ftaylorSeries ℝ f a k_1‖

case hqf
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F → G
hg : ContDiffPointwiseHolderAt k α g (f a)
hf : ContDiffPointwiseHolderAt k α f a
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
⊢ ∀ k_1 ≤ k, (fun a_2 ↦ ftaylorSeries ℝ f a_2 k_1 - ftaylorSeries ℝ f a k_1) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ ↑α
Module: Mathlib.Analysis.Calculus.FDeriv.Pi

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
inst✝³ : DecidableEq ι
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
x : (i : ι) → E i
i : ι
y : E i
⊢ HasFDerivAt (Function.update x i) (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id 𝕜 (E i)))) y

Tactic:
rw [hasFDerivAt_pi]
  intro j
  rcases eq_or_ne j i with rfl | hij
  · simpa using! hasFDerivAt_id _
  · simpa [hij] using! hasFDerivAt_const _ _

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case refine_4
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
r : α → ℝ
hr : ∀ x ∈ s', r x ∈ f x ∩ Ioo 0 1 ∧ Disjoint B (closedBall x (r x))
v : Finset α
vs' : ↑v ⊆ s'
hμv : μ (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤ ↑N / (↑N + 1) * μ s'
hv : (↑v).PairwiseDisjoint fun x ↦ closedBall x (r x)
⊢ μ (s \ ⋃ p ∈ t ∪ Finset.image (fun x ↦ (x, r x)) v, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * μ s'

Tactic:
convert! hμv using 2

Goal state after:
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
r : α → ℝ
hr : ∀ x ∈ s', r x ∈ f x ∩ Ioo 0 1 ∧ Disjoint B (closedBall x (r x))
v : Finset α
vs' : ↑v ⊆ s'
hμv : μ (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤ ↑N / (↑N + 1) * μ s'
hv : (↑v).PairwiseDisjoint fun x ↦ closedBall x (r x)
⊢ s \ ⋃ p ∈ t ∪ Finset.image (fun x ↦ (x, r x)) v, closedBall p.1 p.2 = s' \ ⋃ x ∈ v, closedBall x (r x)
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
⊢ ∀ᶠ (x : ℝ) in 𝓝 x₀, sign (f x) = sign (x - x₀)

Tactic:
by
  rw [← nhdsNE_sup_pure x₀, eventually_sup]
  refine ⟨?_, by simpa⟩
  have h_tendsto := hasDerivAt_iff_tendsto_slope.mp
    (differentiableAt_of_deriv_ne_zero <| ne_of_gt hf).hasDerivAt
  filter_upwards [(h_tendsto.eventually <| eventually_gt_nhds hf),
    self_mem_nhdsWithin] with x hx₀ hx₁
  rw [mem_compl_iff, mem_singleton_iff, ← Ne.eq_def] at hx₁
  obtain (hx' | hx') := hx₁.lt_or_gt
  · rw [sign_neg (neg_of_slope_pos hx' hx₀ hx), sign_neg (sub_neg.mpr hx')]
  · rw [sign_pos (pos_of_slope_pos hx' hx₀ hx), sign_pos (sub_pos.mpr hx')]

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

Goal state before:
pp: case a.e'_6
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)
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ D ^ i * D ^ (n - i + 1) = D ^ (n + 1)

Tactic:
← pow_add

Goal state after:
case a.e'_6
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)
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ D ^ (i + (n - i + 1)) = D ^ (n + 1)
Module: Mathlib.Analysis.Calculus.DifferentialForm.Basic

Goal state before:
pp: case hsymm.hn
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
r : WithTop ℕ∞
s : Set E
x : E
ω : F → F [⋀^Fin n]→L[𝕜] G
f : E → F
t : Set F
hω : DifferentiableWithinAt 𝕜 ω t (f x)
hf : ContDiffWithinAt 𝕜 r f s x
hr : minSmoothness 𝕜 2 ≤ r
hs : UniqueDiffOn 𝕜 s
hxc : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
hxs : x ∈ s
hst : MapsTo f s t
hdf : DifferentiableWithinAt 𝕜 f s x
hd2f : DifferentiableWithinAt 𝕜 (fderivWithin 𝕜 f s) s x
⊢ minSmoothness 𝕜 2 ≤ r

Tactic:
assumption

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

Goal state before:
pp: case hsymm.hs
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
r : WithTop ℕ∞
s : Set E
x : E
ω : F → F [⋀^Fin n]→L[𝕜] G
f : E → F
t : Set F
hω : DifferentiableWithinAt 𝕜 ω t (f x)
hf : ContDiffWithinAt 𝕜 r f s x
hr : minSmoothness 𝕜 2 ≤ r
hs : UniqueDiffOn 𝕜 s
hxc : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
hxs : x ∈ s
hst : MapsTo f s t
hdf : DifferentiableWithinAt 𝕜 f s x
hd2f : DifferentiableWithinAt 𝕜 (fderivWithin 𝕜 f s) s x
⊢ UniqueDiffOn 𝕜 s

Tactic:
assumption

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

Goal state before:
pp: case hsymm.hx
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
r : WithTop ℕ∞
s : Set E
x : E
ω : F → F [⋀^Fin n]→L[𝕜] G
f : E → F
t : Set F
hω : DifferentiableWithinAt 𝕜 ω t (f x)
hf : ContDiffWithinAt 𝕜 r f s x
hr : minSmoothness 𝕜 2 ≤ r
hs : UniqueDiffOn 𝕜 s
hxc : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
hxs : x ∈ s
hst : MapsTo f s t
hdf : DifferentiableWithinAt 𝕜 f s x
hd2f : DifferentiableWithinAt 𝕜 (fderivWithin 𝕜 f s) s x
⊢ x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)

Tactic:
assumption

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

Goal state before:
pp: case hsymm.h'x
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
r : WithTop ℕ∞
s : Set E
x : E
ω : F → F [⋀^Fin n]→L[𝕜] G
f : E → F
t : Set F
hω : DifferentiableWithinAt 𝕜 ω t (f x)
hf : ContDiffWithinAt 𝕜 r f s x
hr : minSmoothness 𝕜 2 ≤ r
hs : UniqueDiffOn 𝕜 s
hxc : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
hxs : x ∈ s
hst : MapsTo f s t
hdf : DifferentiableWithinAt 𝕜 f s x
hd2f : DifferentiableWithinAt 𝕜 (fderivWithin 𝕜 f s) s x
⊢ x ∈ s

Tactic:
assumption

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
W : E → E
⊢ lieBracket 𝕜 0 W = 0

Tactic:
by simp [← lieBracketWithin_univ]

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
W : E → E
⊢ lieBracket 𝕜 W 0 = 0

Tactic:
by simp [← lieBracketWithin_univ]

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

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
V W : E → E
s : Set E
x : E
f : E → F
hf : ContDiffWithinAt 𝕜 2 f s x
hsymm : IsSymmSndFDerivWithinAt 𝕜 f s x
hs : UniqueDiffOn 𝕜 s
hxs : x ∈ s
hW : DifferentiableWithinAt 𝕜 W s x
hV : DifferentiableWithinAt 𝕜 V s x
H₀ : DifferentiableWithinAt 𝕜 (fderivWithin 𝕜 f s) s x
H₁ : UniqueDiffWithinAt 𝕜 s x
⊢ (fderivWithin 𝕜 f s x) (lieBracketWithin 𝕜 V W s x) =
    (fderivWithin 𝕜 f s x ∘SL fderivWithin 𝕜 W s x + (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x).flip (W x)) (V x) -
      (fderivWithin 𝕜 f s x ∘SL fderivWithin 𝕜 V s x + (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x).flip (V x)) (W x)

Tactic:
assumption

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
V W : E → E
s : Set E
x : E
f : E → F
hf : ContDiffWithinAt 𝕜 2 f s x
hsymm : IsSymmSndFDerivWithinAt 𝕜 f s x
hs : UniqueDiffOn 𝕜 s
hxs : x ∈ s
hW : DifferentiableWithinAt 𝕜 W s x
hV : DifferentiableWithinAt 𝕜 V s x
H₀ : DifferentiableWithinAt 𝕜 (fderivWithin 𝕜 f s) s x
H₁ : UniqueDiffWithinAt 𝕜 s x
⊢ (fderivWithin 𝕜 f s x) (lieBracketWithin 𝕜 V W s x) =
    (fderivWithin 𝕜 f s x ∘SL fderivWithin 𝕜 W s x + (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x).flip (W x)) (V x) -
      (fderivWithin 𝕜 f s x ∘SL fderivWithin 𝕜 V s x + (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x).flip (V x)) (W x)
Module: Mathlib.Analysis.Calculus.VectorField

Goal state before:
pp: case hxs
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
V W : E → E
s : Set E
x : E
f : E → F
hf : ContDiffWithinAt 𝕜 2 f s x
hsymm : IsSymmSndFDerivWithinAt 𝕜 f s x
hs : UniqueDiffOn 𝕜 s
hxs : x ∈ s
hW : DifferentiableWithinAt 𝕜 W s x
hV : DifferentiableWithinAt 𝕜 V s x
H₀ : DifferentiableWithinAt 𝕜 (fderivWithin 𝕜 f s) s x
H₁ : UniqueDiffWithinAt 𝕜 s x
⊢ UniqueDiffWithinAt 𝕜 s x

Tactic:
assumption

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

Goal state before:
pp: case hc
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
V W : E → E
s : Set E
x : E
f : E → F
hf : ContDiffWithinAt 𝕜 2 f s x
hsymm : IsSymmSndFDerivWithinAt 𝕜 f s x
hs : UniqueDiffOn 𝕜 s
hxs : x ∈ s
hW : DifferentiableWithinAt 𝕜 W s x
hV : DifferentiableWithinAt 𝕜 V s x
H₀ : DifferentiableWithinAt 𝕜 (fderivWithin 𝕜 f s) s x
H₁ : UniqueDiffWithinAt 𝕜 s x
⊢ DifferentiableWithinAt 𝕜 (fderivWithin 𝕜 f s) s x

Tactic:
assumption

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

Goal state before:
pp: case hu
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
V W : E → E
s : Set E
x : E
f : E → F
hf : ContDiffWithinAt 𝕜 2 f s x
hsymm : IsSymmSndFDerivWithinAt 𝕜 f s x
hs : UniqueDiffOn 𝕜 s
hxs : x ∈ s
hW : DifferentiableWithinAt 𝕜 W s x
hV : DifferentiableWithinAt 𝕜 V s x
H₀ : DifferentiableWithinAt 𝕜 (fderivWithin 𝕜 f s) s x
H₁ : UniqueDiffWithinAt 𝕜 s x
⊢ DifferentiableWithinAt 𝕜 V s x

Tactic:
assumption

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

Goal state before:
pp: case hu
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
V W : E → E
s : Set E
x : E
f : E → F
hf : ContDiffWithinAt 𝕜 2 f s x
hsymm : IsSymmSndFDerivWithinAt 𝕜 f s x
hs : UniqueDiffOn 𝕜 s
hxs : x ∈ s
hW : DifferentiableWithinAt 𝕜 W s x
hV : DifferentiableWithinAt 𝕜 V s x
H₀ : DifferentiableWithinAt 𝕜 (fderivWithin 𝕜 f s) s x
H₁ : UniqueDiffWithinAt 𝕜 s x
⊢ DifferentiableWithinAt 𝕜 W s x

Tactic:
assumption

Goal state after: