mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.Conformal.NormedSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedAddCommGroup Y
inst✝³ : NormedAddCommGroup Z
inst✝² : NormedSpace ℝ X
inst✝¹ : NormedSpace ℝ Y
inst✝ : NormedSpace ℝ Z
f : X → Y
g : Y → Z
x : X
hg : ConformalAt g (f x)
hf : ConformalAt f x
⊢ ConformalAt (g ∘ f) x

Tactic:
rcases hf with ⟨f', hf₁, cf⟩

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedAddCommGroup Y
inst✝³ : NormedAddCommGroup Z
inst✝² : NormedSpace ℝ X
inst✝¹ : NormedSpace ℝ Y
inst✝ : NormedSpace ℝ Z
f : X → Y
g : Y → Z
x : X
hg : ConformalAt g (f x)
f' : X →L[ℝ] Y
hf₁ : HasFDerivAt f f' x
cf : IsConformalMap f'
⊢ ConformalAt (g ∘ f) x
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : Group G
inst✝¹⁰ : IsTopologicalGroup G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : BorelSpace G
H : Type u_2
inst✝⁷ : Group H
inst✝⁶ : TopologicalSpace H
inst✝⁵ : IsTopologicalGroup H
inst✝⁴ : CompactSpace H
inst✝³ : MeasurableSpace H
inst✝² : BorelSpace H
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
ν : Measure H
inst✝ : ν.IsHaarMeasure
f : G →* H
hcont : Continuous[inst✝¹², inst✝⁶] ⇑f
hsurj : Surjective ⇑f
huniv : μ univ = ν univ
this✝ : IsFiniteMeasure μ
this : (map (⇑f) μ).IsHaarMeasure
C : ℝ≥0 := (map (⇑f) μ).haarScalarFactor ν
hC : map (⇑f) μ = C • ν
⊢ C • ν univ = ν univ

Tactic:
← smul_apply,

Goal state after:
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : Group G
inst✝¹⁰ : IsTopologicalGroup G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : BorelSpace G
H : Type u_2
inst✝⁷ : Group H
inst✝⁶ : TopologicalSpace H
inst✝⁵ : IsTopologicalGroup H
inst✝⁴ : CompactSpace H
inst✝³ : MeasurableSpace H
inst✝² : BorelSpace H
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
ν : Measure H
inst✝ : ν.IsHaarMeasure
f : G →* H
hcont : Continuous[inst✝¹², inst✝⁶] ⇑f
hsurj : Surjective ⇑f
huniv : μ univ = ν univ
this✝ : IsFiniteMeasure μ
this : (map (⇑f) μ).IsHaarMeasure
C : ℝ≥0 := (map (⇑f) μ).haarScalarFactor ν
hC : map (⇑f) μ = C • ν
⊢ (C • ν) univ = ν univ
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
S : ∑ _i, μ s / ↑N ≤ ∑ i, μ (s ∩ v i)
i : Fin N
hi : μ s / (↑N + 1) < μ (s ∩ v i)
B : μ (o ∩ v i) = ∑' (x : ↑(u i)), μ (o ∩ closedBall (↑↑x) (r ↑↑x))
w : Finset ↑(u i)
hw : μ s / (↑N + 1) < ∑ x ∈ w, μ (o ∩ closedBall (↑↑x) (r ↑↑x))
l : α
hl : l ∈ ↑(Finset.image (fun x ↦ ↑↑x) w)
k' : ↑(u i)
left✝ : k' ∈ w
hk : ↑↑k' ∈ ↑(Finset.image (fun x ↦ ↑↑x) w)
hkl : ↑↑k' ≠ l
⊢ ∃ l' ∈ w, ↑↑l' = l

Tactic:
simpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hl

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : PseudoEMetricSpace X
inst✝¹ : PseudoEMetricSpace Y
inst✝ : PseudoEMetricSpace Z
s : Set X
Cg rg : ℝ≥0
g : Y → Z
t : Set Y
hg : HolderOnWith Cg rg g t
Cf rf : ℝ≥0
f : X → Y
hf : HolderOnWith Cf rf f s
hst : MapsTo f s t
x : X
hx : x ∈ s
⊢ ∀ y ∈ s, edist ((g ∘ f) x) ((g ∘ f) y) ≤ ↑(Cg * Cf ^ ↑rg) * edist x y ^ ↑(rg * rf)

Tactic:
y

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : PseudoEMetricSpace X
inst✝¹ : PseudoEMetricSpace Y
inst✝ : PseudoEMetricSpace Z
s : Set X
Cg rg : ℝ≥0
g : Y → Z
t : Set Y
hg : HolderOnWith Cg rg g t
Cf rf : ℝ≥0
f : X → Y
hf : HolderOnWith Cf rf f s
hst : MapsTo f s t
x : X
hx : x ∈ s
y : X
⊢ y ∈ s → edist ((g ∘ f) x) ((g ∘ f) y) ≤ ↑(Cg * Cf ^ ↑rg) * edist x y ^ ↑(rg * rf)
Module: Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace

Goal state before:
pp: case inr.inr
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
c' : Fin N.succ → E := fun i ↦ if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
norm_c'_le : ∀ (i : Fin N.succ), ‖c' i‖ ≤ 2
i j : Fin N.succ
inej : i ≠ j
hij : ‖a.c i‖ ≤ ‖a.c j‖
Hj : 2 < ‖a.c j‖
H'j : ‖a.c j‖ ≤ 2 ↔ False
Hi : 2 < ‖a.c i‖
H'i : ‖a.c i‖ ≤ 2 ↔ False
⊢ 1 - δ ≤ ‖c' i - c' j‖

Tactic:
simp_rw [c', H'i, if_false, H'j, if_false]

Goal state after:
case inr.inr
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
c' : Fin N.succ → E := fun i ↦ if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
norm_c'_le : ∀ (i : Fin N.succ), ‖c' i‖ ≤ 2
i j : Fin N.succ
inej : i ≠ j
hij : ‖a.c i‖ ≤ ‖a.c j‖
Hj : 2 < ‖a.c j‖
H'j : ‖a.c j‖ ≤ 2 ↔ False
Hi : 2 < ‖a.c i‖
H'i : ‖a.c i‖ ≤ 2 ↔ False
⊢ 1 - δ ≤ ‖(2 / ‖a.c i‖) • a.c i - (2 / ‖a.c j‖) • a.c j‖
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C₁ C₂ s t₁ t₂ : ℝ≥0
A : Set X
hf₁ : HolderOnWith C₁ r f A
hf₂ : HolderOnWith C₂ s f A
ht : t₁ + t₂ = 1
x : X
hx : x ∈ A
⊢ ∀ y ∈ A, edist (f x) (f y) ≤ ↑(C₁ ^ ↑t₁ * C₂ ^ ↑t₂) * edist x y ^ ↑(r * t₁ + s * t₂)

Tactic:
y

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C₁ C₂ s t₁ t₂ : ℝ≥0
A : Set X
hf₁ : HolderOnWith C₁ r f A
hf₂ : HolderOnWith C₂ s f A
ht : t₁ + t₂ = 1
x : X
hx : x ∈ A
y : X
⊢ y ∈ A → edist (f x) (f y) ≤ ↑(C₁ ^ ↑t₁ * C₂ ^ ↑t₂) * edist x y ^ ↑(r * t₁ + s * t₂)
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C D : ℝ≥0
A : Set X
hA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D
hf : HolderOnWith C r f A
x : X
hx : x ∈ A
⊢ ∀ y ∈ A, edist (f x) (f y) ≤ ↑(C * D ^ ↑r) * edist x y ^ ↑0

Tactic:
y

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C D : ℝ≥0
A : Set X
hA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D
hf : HolderOnWith C r f A
x : X
hx : x ∈ A
y : X
⊢ y ∈ A → edist (f x) (f y) ≤ ↑(C * D ^ ↑r) * edist x y ^ ↑0
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C D s : ℝ≥0
A : Set X
hA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D
hf : HolderOnWith C r f A
hsr : ↑s ≤ ↑r
ht : 0 < s
hr : 0 < ↑r
θ₁ : ℝ≥0 := NNReal.mk (↑s / ↑r) ⋯
θ₂ : ℝ≥0 := NNReal.mk (1 - ↑s / ↑r) ⋯
hθ : θ₁ + θ₂ = 1
hθt : r * θ₁ + 0 * θ₂ = s
⊢ C * D ^ (↑r - ↑s) = C * D ^ (↑r * ↑θ₂)

Tactic:
congr

Goal state after:
case e_a.e_a
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C D s : ℝ≥0
A : Set X
hA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D
hf : HolderOnWith C r f A
hsr : ↑s ≤ ↑r
ht : 0 < s
hr : 0 < ↑r
θ₁ : ℝ≥0 := NNReal.mk (↑s / ↑r) ⋯
θ₂ : ℝ≥0 := NNReal.mk (1 - ↑s / ↑r) ⋯
hθ : θ₁ + θ₂ = 1
hθt : r * θ₁ + 0 * θ₂ = s
⊢ ↑r - ↑s = ↑r * ↑θ₂
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C₁ C₂ : ℝ≥0
A : Set X
hf : HolderOnWith C₁ r f A
hC : C₁ ≤ C₂
x : X
hx : x ∈ A
⊢ ∀ y ∈ A, edist (f x) (f y) ≤ ↑C₂ * edist x y ^ ↑r

Tactic:
y

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C₁ C₂ : ℝ≥0
A : Set X
hf : HolderOnWith C₁ r f A
hC : C₁ ≤ C₂
x : X
hx : x ∈ A
y : X
⊢ y ∈ A → edist (f x) (f y) ≤ ↑C₂ * edist x y ^ ↑r
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
k : ℕ
α : ↑I
f : E → F
a : E
g : E → F
hf : ContDiffPointwiseHolderAt k α f a
hfg : f =ᶠ[𝓝 a] g
⊢ (fun x ↦ iteratedFDeriv ℝ k g x - iteratedFDeriv ℝ k g a) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ ↑α

Tactic:
by
    refine EventuallyEq.trans_isBigO (.sub ?_ ?_) hf.isBigO
    · exact hfg.symm.iteratedFDeriv ℝ _
    · rw [hfg.symm.iteratedFDeriv ℝ _ |>.self_of_nhds]

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

Goal state before:
pp: case refine_3.inr
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)
p : α × ℝ
hp : p ∈ t ∪ Finset.image (fun x ↦ (x, r x)) v
h'p : p ∈ Finset.image (fun x ↦ (x, r x)) v
⊢ p.2 ∈ f p.1

Tactic:
· rcases Finset.mem_image.1 h'p with ⟨p', p'v, rfl⟩
        exact (hr p' (vs' p'v)).1.1

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

Goal state before:
pp: 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
F : Finset (α × ℝ) → Finset (α × ℝ)
hF :
  ∀ (t : Finset (α × ℝ)),
    P t →
      t ⊆ F t ∧ P (F t) ∧ μ (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * μ (s \ ⋃ p ∈ t, closedBall p.1 p.2)
u : ℕ → Finset (α × ℝ) := fun n ↦ F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
n : ℕ
⊢ P (u n)

Tactic:
induction n with
    | zero =>
      simp only [P, u, Prod.forall, id, Function.iterate_zero]
      simp only [Finset.notMem_empty, IsEmpty.forall_iff, Finset.coe_empty, forall₂_true_iff,
        and_self_iff, pairwiseDisjoint_empty]
    | succ n IH =>
      rw [u_succ]
      exact (hF (u n) IH).2.1

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Alternating.Uncurry.Fin

Goal state before:
pp: case succ
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
g : E →L[𝕜] F
h : E →L[𝕜] E →L[𝕜] F
hsymm : ∀ (x y : E), (h x) y = (h y) x
n : ℕ
f : F [⋀^Fin (n + 1)]→L[𝕜] G
x : E
⊢ ∀ (y : E),
    ((ContinuousLinearMap.postcomp E (compContinuousLinearMapCLM g ∘SL f.curryLeft) ∘SL h) x) y =
      ((ContinuousLinearMap.postcomp E (compContinuousLinearMapCLM g ∘SL f.curryLeft) ∘SL h) y) x

Tactic:
y

Goal state after:
case succ
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
g : E →L[𝕜] F
h : E →L[𝕜] E →L[𝕜] F
hsymm : ∀ (x y : E), (h x) y = (h y) x
n : ℕ
f : F [⋀^Fin (n + 1)]→L[𝕜] G
x y : E
⊢ ((ContinuousLinearMap.postcomp E (compContinuousLinearMapCLM g ∘SL f.curryLeft) ∘SL h) x) y =
    ((ContinuousLinearMap.postcomp E (compContinuousLinearMapCLM g ∘SL f.curryLeft) ∘SL h) y) x
Module: Mathlib.Analysis.Calculus.FDeriv.ContinuousMultilinearMap

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : Type u_3
F : ι → Type u_4
G : ι → Type u_5
H : Type u_6
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : (i : ι) → NormedAddCommGroup (F i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (F i)
inst✝⁵ : (i : ι) → NormedAddCommGroup (G i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (G i)
inst✝³ : NormedAddCommGroup H
inst✝² : NormedSpace 𝕜 H
f : E → ContinuousMultilinearMap 𝕜 G H
f' : E →L[𝕜] ContinuousMultilinearMap 𝕜 G H
g : (i : ι) → E → F i →L[𝕜] G i
g' : (i : ι) → E →L[𝕜] F i →L[𝕜] G i
s : Set E
x : E
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
hf : HasFDerivWithinAt f f' s x
hg : ∀ (i : ι), HasFDerivWithinAt (g i) (g' i) s x
⊢ HasFDerivWithinAt (fun x ↦ (f x).compContinuousLinearMap fun x_1 ↦ g x_1 x)
    ((compContinuousLinearMapL fun x_1 ↦ g x_1 x) ∘SL f' +
      ((f x).fderivCompContinuousLinearMap fun x_1 ↦ g x_1 x) ∘SL ContinuousLinearMap.pi g')
    s x

Tactic:
convert!
    hasStrictFDerivAt_compContinuousLinearMap
          (f x, (g · x)) |>.hasFDerivAt |>.comp_hasFDerivWithinAt
      x (hf.prodMk (hasFDerivWithinAt_pi.2 hg))

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

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
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 ↦ ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
⊢ ‖f (x + h • v + h • w) - f (x + h • v) - h • (f' x) w - h ^ 2 • (f'' v) w - (h ^ 2 / 2) • (f'' w) w‖ ≤
    ε * ‖(‖v‖ + ‖w‖) * ‖w‖ * h ^ 2‖

Tactic:
have xt_mem : ∀ t ∈ Icc (0 : ℝ) 1, x + h • v + (t * h) • w ∈ interior s := by
    intro t ht
    have : x + h • v ∈ interior s := s_conv.add_smul_mem_interior xs hv ⟨hpos, h_lt_1.le⟩
    rw [← smul_smul]
    apply s_conv.interior.add_smul_mem this _ ht
    rw [add_assoc] at hw
    convert! s_conv.add_smul_mem_interior xs hw ⟨hpos, h_lt_1.le⟩ using 1
    module

Goal state after:
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
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 ↦ ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
xt_mem : ∀ t ∈ Icc 0 1, x + h • v + (t * h) • w ∈ interior s
⊢ ‖f (x + h • v + h • w) - f (x + h • v) - h • (f' x) w - h ^ 2 • (f'' v) w - (h ^ 2 / 2) • (f'' w) w‖ ≤
    ε * ‖(‖v‖ + ‖w‖) * ‖w‖ * h ^ 2‖
Module: Mathlib.Analysis.Calculus.VectorField

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
f : E → F
s : Set E
x : E
h'f : ContDiffWithinAt 𝕜 2 f s x
hf : (fderivWithin 𝕜 f s x).IsInvertible
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ ∃ N,
    ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y)) s x ∧
      ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y).symm) s x ∧
        (∀ᶠ (y : E) in 𝓝[s] x, ↑(N y) = fderivWithin 𝕜 f s y) ∧
          ∀ (v : E),
            (fderivWithin 𝕜 (fun y ↦ ↑(N y).symm) s x) v =
              -↑(N x).symm ∘SL (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v ∘SL ↑(N x).symm

Tactic:
rcases hf with ⟨M, hM⟩

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
f : E → F
s : Set E
x : E
h'f : ContDiffWithinAt 𝕜 2 f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
M : E ≃L[𝕜] F
hM : ↑M = fderivWithin 𝕜 f s x
⊢ ∃ N,
    ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y)) s x ∧
      ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y).symm) s x ∧
        (∀ᶠ (y : E) in 𝓝[s] x, ↑(N y) = fderivWithin 𝕜 f s y) ∧
          ∀ (v : E),
            (fderivWithin 𝕜 (fun y ↦ ↑(N y).symm) s x) v =
              -↑(N x).symm ∘SL (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v ∘SL ↑(N x).symm
Module: Mathlib.Analysis.Calculus.VectorField

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
s : Set E
inst✝ : CompleteSpace E
f : E → F
V W : F → F
x : E
t : Set F
hf : IsSymmSndFDerivWithinAt 𝕜 f s x
h'f : ContDiffWithinAt 𝕜 2 f s x
hV : DifferentiableWithinAt 𝕜 V t (f x)
hW : DifferentiableWithinAt 𝕜 W t (f x)
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
h : (fderivWithin 𝕜 f s x).IsInvertible
M : E → E ≃L[𝕜] F
M_symm_smooth : ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(M y).symm) s x
hM : ∀ᶠ (y : E) in 𝓝[s] x, ↑(M y) = fderivWithin 𝕜 f s y
M_diff :
  ∀ (v : E),
    (fderivWithin 𝕜 (fun y ↦ ↑(M y).symm) s x) v =
      -↑(M x).symm ∘SL (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v ∘SL ↑(M x).symm
⊢ pullbackWithin 𝕜 f (lieBracketWithin 𝕜 V W t) s x =
    lieBracketWithin 𝕜 (pullbackWithin 𝕜 f V s) (pullbackWithin 𝕜 f W s) s x

Tactic:
have hMx : M x = fderivWithin 𝕜 f s x := (mem_of_mem_nhdsWithin hx hM :)

Goal state after:
case pos
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
s : Set E
inst✝ : CompleteSpace E
f : E → F
V W : F → F
x : E
t : Set F
hf : IsSymmSndFDerivWithinAt 𝕜 f s x
h'f : ContDiffWithinAt 𝕜 2 f s x
hV : DifferentiableWithinAt 𝕜 V t (f x)
hW : DifferentiableWithinAt 𝕜 W t (f x)
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
h : (fderivWithin 𝕜 f s x).IsInvertible
M : E → E ≃L[𝕜] F
M_symm_smooth : ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(M y).symm) s x
hM : ∀ᶠ (y : E) in 𝓝[s] x, ↑(M y) = fderivWithin 𝕜 f s y
M_diff :
  ∀ (v : E),
    (fderivWithin 𝕜 (fun y ↦ ↑(M y).symm) s x) v =
      -↑(M x).symm ∘SL (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v ∘SL ↑(M x).symm
hMx : ↑(M x) = fderivWithin 𝕜 f s x
⊢ pullbackWithin 𝕜 f (lieBracketWithin 𝕜 V W t) s x =
    lieBracketWithin 𝕜 (pullbackWithin 𝕜 f V s) (pullbackWithin 𝕜 f W s) s x
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : t ≠ 0
h : HasStrictFDerivAt (fun x ↦ ‖x‖) f x
⊢ HasStrictFDerivAt (fun x ↦ ‖x‖) (↑(SignType.sign t) • f) (t • x)

Tactic:
by
  have h1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x) :=
    hasStrictFDerivAt_id (t • x) |>.const_smul t⁻¹
  have h2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x := h.const_smul |t|
  conv at h2 => enter [3]; rw [← one_smul ℝ x, ← inv_mul_cancel₀ ht, mul_smul]
  convert! h2.comp (t • x) h1 with y
  · rw [norm_smul, ← mul_assoc, norm_eq_abs, ← abs_mul, mul_inv_cancel₀ ht, abs_one, one_mul]
  ext y
  simp only [smul_apply, smul_eq_mul, comp_smulₛₗ, map_inv₀, RingHom.id_apply, comp_id]
  rw [eq_inv_mul_iff_mul_eq₀ ht, ← mul_assoc, self_mul_sign]

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

Goal state before:
pp: 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✝ : SFinite μ
x : α
⊢ Tendsto (fun r ↦ closedBall x r) (𝓝[>] 0) ((Besicovitch.vitaliFamily μ).filterAt x)

Tactic:
intro s hs
  simp only [mem_map]
  obtain ⟨ε, εpos, hε⟩ :
    ∃ (ε : ℝ), ε > 0 ∧
      ∀ a : Set α, a ∈ (Besicovitch.vitaliFamily μ).setsAt x → a ⊆ closedBall x ε → a ∈ s :=
    (VitaliFamily.mem_filterAt_iff _).1 hs
  filter_upwards [Ioc_mem_nhdsGT εpos] with _r hr
  apply hε
  · exact mem_image_of_mem _ hr.1
  · exact closedBall_subset_closedBall hr.2

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace F
f : F → 𝕜
f' x : F
⊢ HasFDerivWithinAt f ((toDual 𝕜 F) f') univ x ↔ HasFDerivAt f ((toDual 𝕜 F) f') x

Tactic:
exact hasFDerivWithinAt_univ

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

Goal state before:
pp: case hf
𝕂 : Type u_1
inst✝⁵ : RCLike 𝕂
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕂 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕂 F
inst✝ : CompleteSpace E
f : E → F
f' : E ≃L[𝕂] F
a : E
n : WithTop ℕ∞
hf : ContDiffAt 𝕂 n f a
hf' : HasFDerivAt f (↑f') a
hn : n ≠ 0
this : hf.localInverse hf' hn (f a) = a
⊢ ContDiffAt 𝕂 n (↑(toOpenPartialHomeomorph f hf hf' hn)) (↑(toOpenPartialHomeomorph f hf hf' hn).symm (f a))

Tactic:
· convert! hf

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

Goal state before:
pp: case hx
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : CompleteSpace E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : CompleteSpace F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : CompleteSpace G
φ : ImplicitFunctionData 𝕜 E F G
s : Set E
H : φ.implicitFunction (φ.leftFun φ.pt) = ↑φ.toOpenPartialHomeomorph.symm ∘ fun x ↦ (φ.leftFun φ.pt, x)
⊢ (φ.leftFun φ.pt, φ.rightFun φ.pt) ∈ φ.toOpenPartialHomeomorph.symm.source

Tactic:
· exact φ.toOpenPartialHomeomorph.mapsTo φ.pt_mem_toOpenPartialHomeomorph_source

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

Goal state before:
pp: a b : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hab : a < b
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : Tendsto f (𝓝[<] b) (𝓝 0)
hgb : Tendsto g (𝓝[<] b) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] b) l
hdnf : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
⊢ Tendsto (fun x ↦ f' (-x) * -1 / (g' (-x) * -1)) (𝓝[>] (-b)) ?m.203

Tactic:
simp only [neg_div_neg_eq, mul_one, mul_neg]
      exact hdiv.comp tendsto_neg_nhdsGT_neg

Goal state after: