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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G
isoD : Du ≃ₗᵢ[𝕜] D
isoE : Eu ≃ₗᵢ[𝕜] E
isoF : Fu ≃ₗᵢ[𝕜] F
isoG : Gu ≃ₗᵢ[𝕜] G
fu : Du → Eu := ⇑isoE.symm ∘ f ∘ ⇑isoD
gu : Du → Fu := ⇑isoF.symm ∘ g ∘ ⇑isoD
Bu₀ : Eu →L[𝕜] Fu →L[𝕜] G := ((B ∘SL ↑↑isoE).flip ∘SL ↑↑isoF).flip
Bu : Eu →L[𝕜] Fu →L[𝕜] Gu := ((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu)) ((compL 𝕜 Fu G Gu) ↑↑isoG.symm)) Bu₀
hBu : Bu = ((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu)) ((compL 𝕜 Fu G Gu) ↑↑isoG.symm)) Bu₀
Bu_eq : (fun y ↦ (Bu (fu y)) (gu y)) = ⇑isoG.symm ∘ (fun y ↦ (B (f y)) (g y)) ∘ ⇑isoD
Bu_le : ‖Bu‖ ≤ ‖B‖
su : Set Du := ⇑isoD ⁻¹' s
hsu : UniqueDiffOn 𝕜 su
xu : Du := isoD.symm x
hxu : xu ∈ su
xu_x : isoD xu = x
hfu : ContDiffOn 𝕜 (↑n) fu su
hgu : ContDiffOn 𝕜 (↑n) gu su
Nfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu su xu‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖
Ngu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu su xu‖ = ‖iteratedFDerivWithin 𝕜 i g s x‖
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (Bu (fu y)) (gu y)) su xu‖ =
    ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (B (f y)) (g y)) s x‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G
isoD : Du ≃ₗᵢ[𝕜] D
isoE : Eu ≃ₗᵢ[𝕜] E
isoF : Fu ≃ₗᵢ[𝕜] F
isoG : Gu ≃ₗᵢ[𝕜] G
fu : Du → Eu := ⇑isoE.symm ∘ f ∘ ⇑isoD
gu : Du → Fu := ⇑isoF.symm ∘ g ∘ ⇑isoD
Bu₀ : Eu →L[𝕜] Fu →L[𝕜] G := ((B ∘SL ↑↑isoE).flip ∘SL ↑↑isoF).flip
Bu : Eu →L[𝕜] Fu →L[𝕜] Gu := ((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu)) ((compL 𝕜 Fu G Gu) ↑↑isoG.symm)) Bu₀
hBu : Bu = ((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu)) ((compL 𝕜 Fu G Gu) ↑↑isoG.symm)) Bu₀
Bu_eq : (fun y ↦ (Bu (fu y)) (gu y)) = ⇑isoG.symm ∘ (fun y ↦ (B (f y)) (g y)) ∘ ⇑isoD
Bu_le : ‖Bu‖ ≤ ‖B‖
su : Set Du := ⇑isoD ⁻¹' s
hsu : UniqueDiffOn 𝕜 su
xu : Du := isoD.symm x
hxu : xu ∈ su
xu_x : isoD xu = x
hfu : ContDiffOn 𝕜 (↑n) fu su
hgu : ContDiffOn 𝕜 (↑n) gu su
Nfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu su xu‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖
Ngu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu su xu‖ = ‖iteratedFDerivWithin 𝕜 i g s x‖
NBu :
  ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (Bu (fu y)) (gu y)) su xu‖ =
    ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (B (f y)) (g y)) s x‖
this :
  ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (B (f y)) (g y)) s x‖ ≤
    ‖Bu‖ *
      ∑ x_1 ∈ Finset.range (n + 1),
        ↑(n.choose x_1) * ‖iteratedFDerivWithin 𝕜 x_1 f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - x_1) g s x‖
⊢ ‖Bu‖ *
      ∑ x_1 ∈ Finset.range (n + 1),
        ↑(n.choose x_1) * ‖iteratedFDerivWithin 𝕜 x_1 f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - x_1) g s x‖ ≤
    ‖B‖ *
      ∑ i ∈ Finset.range (n + 1),
        ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (B (f y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i ∈ Finset.range (n + 1),
        ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
hf : ContDiff 𝕜 N f
hg : ContDiff 𝕜 N g
x : D
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDeriv 𝕜 n (fun y ↦ (B (f y)) (g y)) x‖ ≤
    ‖B‖ * ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
hB : ‖B‖ ≤ 1
⊢ 0 ≤
    ∑ i ∈ Finset.range (n + 1),
      ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
hB : ‖B‖ ≤ 1
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (B (f y)) (g y)) s x‖ ≤
    ∑ i ∈ Finset.range (n + 1),
      ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
hf : ContDiff 𝕜 N f
hg : ContDiff 𝕜 N g
x : D
n : ℕ
hn : ↑n ≤ N
hB : ‖B‖ ≤ 1
⊢ ‖iteratedFDeriv 𝕜 n (fun y ↦ (B (f y)) (g y)) x‖ ≤
    ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
A : Type u_3
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
f g : E → A
N : ℕ∞ω
hf : ContDiff 𝕜 N f
hg : ContDiff 𝕜 N g
x : E
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDeriv 𝕜 n (fun y ↦ f y * g y) x‖ ≤
    ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖

Tactic:
by

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
⊢ ∀ (t : Finset (α × ℝ)),
    P t → ∃ u, t ⊆ u ∧ P u ∧ μ (s \ ⋃ p ∈ u, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * μ (s \ ⋃ p ∈ t, closedBall p.1 p.2)

Tactic:
by

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] ∅
n : ℕ
⊢ u n.succ = F (u n)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
f : 𝕜 → F
f' : F
inst✝ : TrivialStar 𝕜
L : Filter (𝕜 × 𝕜)
h : HasDerivAtFilter f f' L
⊢ HasDerivAtFilter (fun x ↦ star (f x)) (star f') L

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
f : 𝕜 → F
x : 𝕜
inst✝ : TrivialStar 𝕜
s : Set 𝕜
⊢ derivWithin (fun y ↦ star (f y)) s x = star (derivWithin f s x)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
x : 𝕜
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
f' : F
hf : HasDerivAt f f' x
⊢ HasDerivAt (star ∘ f ∘ ⇑(starRingEnd 𝕜)) (star f') ((starRingEnd 𝕜) x)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
x : 𝕜
f' : F
hf : HasDerivAt (star ∘ f ∘ ⇑(starRingEnd 𝕜)) f' x
⊢ HasDerivAt f (star f') ((starRingEnd 𝕜) x)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
x : 𝕜
f' : F
hf : HasDerivAt f (star f') ((starRingEnd 𝕜) x)
⊢ HasDerivAt (star ∘ f ∘ ⇑(starRingEnd 𝕜)) f' x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
x : 𝕜
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
hf : DifferentiableAt 𝕜 (star ∘ f ∘ ⇑(starRingEnd 𝕜)) x
⊢ DifferentiableAt 𝕜 f ((starRingEnd 𝕜) x)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
x : 𝕜
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
hf : DifferentiableAt 𝕜 f ((starRingEnd 𝕜) x)
⊢ DifferentiableAt 𝕜 (star ∘ f ∘ ⇑(starRingEnd 𝕜)) x

Tactic:
by

Goal state after:
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
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : ℕ∞ω
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
i : ι
u : Finset ι
hi : i ∉ u
IH :
  (∀ i ∈ u, ContDiffOn 𝕜 N (f i) s) →
    ∀ {n : ℕ},
      ↑n ≤ N →
        ‖iteratedFDerivWithin 𝕜 n (fun x ↦ ∏ j ∈ u, f j x) s x‖ ≤
          ∑ p ∈ u.sym n, ↑(↑p).countPerms * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
n : ℕ
hn : ↑n ≤ N
hf : ContDiffOn 𝕜 N (f i) s ∧ ∀ a ∈ u, ContDiffOn 𝕜 N (f a) s
⊢ ∀ (i_1 : ↥((insert i u).sym n)), i_1 ∈ Finset.univ ↔ (Finset.symInsertEquiv hi) i_1 ∈ Finset.univ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
⊢ deriv (star ∘ f ∘ ⇑(starRingEnd 𝕜)) = star ∘ deriv f ∘ ⇑(starRingEnd 𝕜)

Tactic:
by

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:
by

Goal state after:
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
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : ℕ∞ω
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
i : ι
u : Finset ι
hi : i ∉ u
IH :
  (∀ i ∈ u, ContDiffOn 𝕜 N (f i) s) →
    ∀ {n : ℕ},
      ↑n ≤ N →
        ‖iteratedFDerivWithin 𝕜 n (fun x ↦ ∏ j ∈ u, f j x) s x‖ ≤
          ∑ p ∈ u.sym n, ↑(↑p).countPerms * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
n : ℕ
hn : ↑n ≤ N
hf : ContDiffOn 𝕜 N (f i) s ∧ ∀ a ∈ u, ContDiffOn 𝕜 N (f a) s
⊢ ∀ i_1 ∈ Finset.univ,
    ↑(↑↑i_1).countPerms * ∏ j ∈ insert i u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑↑i_1) (f j) s x‖ =
      ((fun v ↦ ↑v.countPerms * ∏ j ∈ insert i u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j v) (f j) s x‖) ∘
          Sym.toMultiset ∘ Subtype.val ∘ ⇑(Finset.symInsertEquiv hi).symm)
        ((Finset.symInsertEquiv hi) i_1)

Tactic:
by

Goal state after:
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
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : ℕ∞ω
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
i : ι
u : Finset ι
hi : i ∉ u
IH :
  (∀ i ∈ u, ContDiffOn 𝕜 N (f i) s) →
    ∀ {n : ℕ},
      ↑n ≤ N →
        ‖iteratedFDerivWithin 𝕜 n (fun x ↦ ∏ j ∈ u, f j x) s x‖ ≤
          ∑ p ∈ u.sym n, ↑(↑p).countPerms * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
n : ℕ
hn : ↑n ≤ N
hf : ContDiffOn 𝕜 N (f i) s ∧ ∀ a ∈ u, ContDiffOn 𝕜 N (f a) s
m : Fin (n + 1)
a✝ : m ∈ Finset.univ
⊢ ↑(n - ↑m) ≤ ↑n

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : StarRing 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : StarAddMonoid F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : StarModule 𝕜 F
inst✝⁴ : ContinuousStar F
inst✝³ : StarAddMonoid E
inst✝² : StarModule 𝕜 E
inst✝¹ : ContinuousStar E
inst✝ : NormedStarGroup 𝕜
f : E → F
z : E
f' : E →L[𝕜] F
hf : HasFDerivAt f f' z
⊢ HasFDerivAt f f' (↑(starL 𝕜) (star z))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Alternating.Curry

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : E [⋀^Fin (n + 2)]→L[𝕜] F
x : E
x✝ : Fin n → E
⊢ Fin.cons x (Fin.cons x x✝) 0 = Fin.cons x (Fin.cons x x✝) 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Alternating.Curry

Goal state before:
pp: 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
g : F [⋀^Fin (n + 1)]→L[𝕜] G
f : E →L[𝕜] F
x : E
v : Fin n → E
i : Fin (n + 1)
⊢ ((fun i ↦ ↑((fun x ↦ f) i)) i) (Fin.cons x v i) = Fin.cons (f x) (fun i ↦ ((fun i ↦ ↑((fun x ↦ f) i)) i) (v i)) i

Tactic:
by

Goal state after:
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
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : ℕ∞ω
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
i : ι
u : Finset ι
hi : i ∉ u
n : ℕ
hn : ↑n ≤ N
hf : ContDiffOn 𝕜 N (f i) s ∧ ∀ a ∈ u, ContDiffOn 𝕜 N (f a) s
m : Fin (n + 1)
IH :
  ‖iteratedFDerivWithin 𝕜 (n - ↑m) (fun x ↦ ∏ j ∈ u, f j x) s x‖ ≤
    ∑ p ∈ u.sym (n - ↑m), ↑(↑p).countPerms * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
a✝ : m ∈ Finset.univ
p : Sym ι (n - ↑m)
hp : ∀ a ∈ p, a ∈ u
hip : i ∉ p
this :
  ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖ =
    ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑(Sym.fill i m p)) (f j) s x‖
⊢ ↑(n.choose ↑m) * ‖iteratedFDerivWithin 𝕜 (↑m) (f i) s x‖ *
      (↑(↑p).countPerms * ∏ x_1 ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count x_1 ↑p) (f x_1) s x‖) =
    ↑(n.choose ↑m * (↑p).countPerms) *
      (‖iteratedFDerivWithin 𝕜 (↑m) (f i) s x‖ *
        ∏ x_1 ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count x_1 ↑(Sym.fill i m p)) (f x_1) s x‖)

Tactic:
by

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)
Pu : ∀ (n : ℕ), P (u n)
⊢ ∀ (n : ℕ), μ (s \ ⋃ p ∈ ⋃ n, ↑(u n), closedBall p.1 p.2) ≤ μ (s \ ⋃ p ∈ u n, closedBall p.1 p.2)

Tactic:
by

Goal state after:
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
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
u : Finset ι
f : ι → E → A'
N : ℕ∞ω
hf : ∀ i ∈ u, ContDiffOn 𝕜 N (f i) s
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDerivWithin 𝕜 n (fun x ↦ ∏ j ∈ u, f j x) s x‖ ≤
    ∑ p ∈ u.sym n, ↑(↑p).countPerms * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖

Tactic:
by

Goal state after:
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
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
u : Finset ι
f : ι → E → A'
N : ℕ∞ω
hf : ∀ i ∈ u, ContDiff 𝕜 N (f i)
x : E
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDeriv 𝕜 n (fun x ↦ ∏ j ∈ u, f j x) x‖ ≤
    ∑ p ∈ u.sym n, ↑(↑p).countPerms * ∏ j ∈ u, ‖iteratedFDeriv 𝕜 (Multiset.count j ↑p) (f j) x‖

Tactic:
by

Goal state after:
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
⊢ 1 ≤ n + 1

Tactic:
by

Goal state after: