mathlib-initiative/mathlib-tactics huggingface dataset:
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 ≤
    ‖(if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i) - if ‖a.c j‖ ≤ 2 then a.c j else (2 / ‖a.c j‖) • a.c j‖

Tactic:
H'i,

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 ≤ ‖(if False then a.c i else (2 / ‖a.c i‖) • a.c i) - if ‖a.c j‖ ≤ 2 then a.c j else (2 / ‖a.c j‖) • a.c j‖
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 := ⋯
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋯
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := ⋯
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
q' : α
q'v : q' ∈ ↑v
hq : (q', r q') ∈ (fun x ↦ (x, r x)) '' ↑v
hpq : p ≠ (q', r q')
⊢ closedBall p.1 p.2 ⊆ B

Tactic:
hB,

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)
p : α × ℝ
hp : p ∈ ↑t
q' : α
q'v : q' ∈ ↑v
hq : (q', r q') ∈ (fun x ↦ (x, r x)) '' ↑v
hpq : p ≠ (q', r q')
⊢ closedBall p.1 p.2 ⊆ ⋃ p ∈ t, closedBall p.1 p.2
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]

Goal state after:
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
⊢ ∀ (i_1 : ι), HasFDerivAt (fun x_1 ↦ Function.update x i x_1 i_1) (Pi.single i (ContinuousLinearMap.id 𝕜 (E i)) i_1) y
Module: Mathlib.Analysis.Calculus.Deriv.Abs

Goal state before:
pp: h : DifferentiableAt ℝ abs 0
⊢ False

Tactic:
have h₁ : deriv abs (0 : ℝ) = 1 :=
    (uniqueDiffOn_Ici _ _ Set.self_mem_Ici).eq_deriv _ h.hasDerivAt.hasDerivWithinAt <|
      (hasDerivWithinAt_id _ _).congr_of_mem (fun _ h ↦ abs_of_nonneg h) Set.self_mem_Ici

Goal state after:
h : DifferentiableAt ℝ abs 0
h₁ : deriv abs 0 = 1
⊢ False
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: case empty
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
hf : ∀ i ∈ ∅, ContDiffOn 𝕜 N (f i) s
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDerivWithin 𝕜 n (fun x ↦ ∏ j ∈ ∅, f j x) s x‖ ≤
    ∑ p ∈ ∅.sym n, ↑(↑p).countPerms * ∏ j ∈ ∅, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖

Tactic:
cases n with
    | zero => simp [Sym.eq_nil_of_card_zero]
    | succ n => simp [iteratedFDerivWithin_succ_const]

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 exact_mod_cast n.sub_le m

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

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
h : ContinuousAt f a
hd₀ : DifferentiableOn ℝ f (Ioi a)
⊢ ContinuousOn f (Ioi a ∪ {a})

Tactic:
exact hd₀.continuousOn.union_continuousAt isOpen_Ioi (by simp [h])

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

Goal state before:
pp: f : ℝ → ℝ
x₀ : ℝ
h₀ : ∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (deriv f x) = sign (x - x₀)
⊢ ∀ᶠ (b : ℝ) in 𝓝[<] x₀, deriv f b < 0

Tactic:
filter_upwards [nhdsLT_le_nhdsNE _ h₀, self_mem_nhdsWithin] with x hx' (hx : x < x₀)
  rwa [← sub_neg, ← sign_eq_neg_one_iff, ← hx', sign_eq_neg_one_iff] at hx

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 : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : 1 / 2 ∈ Ioc 0 1
B : 1 / 2 ∈ Icc 0 1
h2v2w : x + 2 • v + 2 • w ∈ interior s
h2vww : x + (2 • v + w) + w ∈ interior s
h2v : x + 2 • v ∈ interior s
h2w : x + 2 • w ∈ interior s
hvw : x + (v + w) ∈ interior s
h2vw : x + (2 • v + w) ∈ interior s
hvww : x + (v + w) + w ∈ interior s
TA1 :
  (fun h ↦
      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -
        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]
    fun h ↦ h ^ 2
TA2 :
  (fun h ↦
      f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -
        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]
    fun h ↦ h ^ 2
⊢ (fun h ↦
      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -
        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]
    fun h ↦ h ^ 2

Tactic:
convert! TA1.sub TA2 using 1

Goal state after:
case e'_7
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 : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : 1 / 2 ∈ Ioc 0 1
B : 1 / 2 ∈ Icc 0 1
h2v2w : x + 2 • v + 2 • w ∈ interior s
h2vww : x + (2 • v + w) + w ∈ interior s
h2v : x + 2 • v ∈ interior s
h2w : x + 2 • w ∈ interior s
hvw : x + (v + w) ∈ interior s
h2vw : x + (2 • v + w) ∈ interior s
hvww : x + (v + w) + w ∈ interior s
TA1 :
  (fun h ↦
      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -
        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]
    fun h ↦ h ^ 2
TA2 :
  (fun h ↦
      f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -
        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]
    fun h ↦ h ^ 2
⊢ (fun h ↦
      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -
        h ^ 2 • (f'' v) w) =
    fun x_1 ↦
    f (x + x_1 • (2 • v + w) + x_1 • w) - f (x + x_1 • (2 • v + w)) - x_1 • (f' x) w - x_1 ^ 2 • (f'' (2 • v + w)) w -
        (x_1 ^ 2 / 2) • (f'' w) w -
      (f (x + x_1 • (v + w) + x_1 • w) - f (x + x_1 • (v + w)) - x_1 • (f' x) w - x_1 ^ 2 • (f'' (v + w)) w -
        (x_1 ^ 2 / 2) • (f'' w) w)
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 : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : (fun h ↦ h ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun h ↦ h ^ 2
this : (fun h ↦ 1 / h ^ 2) =O[𝓝[>] 0] fun h ↦ 1 / h ^ 2
C : (fun x ↦ (1 / x ^ 2) • x ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun x ↦ (1 / x ^ 2) • x ^ 2
h : ℝ
⊢ h ∈ Ioi 0 → (1 / h ^ 2) • h ^ 2 • ((f'' w) v - (f'' v) w) = (f'' w) v - (f'' v) w

Tactic:
(hpos : 0 < h)

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 : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : (fun h ↦ h ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun h ↦ h ^ 2
this : (fun h ↦ 1 / h ^ 2) =O[𝓝[>] 0] fun h ↦ 1 / h ^ 2
C : (fun x ↦ (1 / x ^ 2) • x ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun x ↦ (1 / x ^ 2) • x ^ 2
h : ℝ
hpos : 0 < h
⊢ (1 / h ^ 2) • h ^ 2 • ((f'' w) v - (f'' v) w) = (f'' w) v - (f'' v) w
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 : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : (fun h ↦ h ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun h ↦ h ^ 2
this : (fun h ↦ 1 / h ^ 2) =O[𝓝[>] 0] fun h ↦ 1 / h ^ 2
C : (fun x ↦ (1 / x ^ 2) • x ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun x ↦ (1 / x ^ 2) • x ^ 2
h : ℝ
⊢ h ∈ Ioi 0 → (1 / h ^ 2) • h ^ 2 = 1

Tactic:
(hpos : 0 < h)

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 : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : (fun h ↦ h ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun h ↦ h ^ 2
this : (fun h ↦ 1 / h ^ 2) =O[𝓝[>] 0] fun h ↦ 1 / h ^ 2
C : (fun x ↦ (1 / x ^ 2) • x ^ 2 • ((f'' w) v - (f'' v) w)) =o[𝓝[>] 0] fun x ↦ (1 / x ^ 2) • x ^ 2
h : ℝ
hpos : 0 < h
⊢ (1 / h ^ 2) • h ^ 2 = 1
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : WithTop ℕ∞
x : E
t : ℝ
ht : t ≠ 0
h : ContDiffAt ℝ n (fun x ↦ ‖x‖) x
h1 : ContDiffAt ℝ n (fun y ↦ t⁻¹ • y) (t • x)
h2 : ContDiffAt ℝ n (fun y ↦ |t| * ‖y‖) x
| ContDiffAt ℝ n (fun y ↦ |t| * ‖y‖) x

Tactic:
enter [4]; rw [← one_smul ℝ x, ← inv_mul_cancel₀ ht, mul_smul]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : WithTop ℕ∞
x : E
t : ℝ
ht : t ≠ 0
h : ContDiffAt ℝ n (fun x ↦ ‖x‖) x
h1 : ContDiffAt ℝ n (fun y ↦ t⁻¹ • y) (t • x)
h2 : ContDiffAt ℝ n (fun y ↦ |t| * ‖y‖) x
| t⁻¹ • t • x
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: case inr.inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : WithTop ℕ∞
x : E
t : ℝ
h : ContDiffAt ℝ n (fun x ↦ ‖x‖) (t • x)
hn : n ≠ 0
ht : t ≠ 0
⊢ ContDiffAt ℝ n (fun x ↦ ‖x‖) x

Tactic:
· exact contDiffAt_norm_smul_iff ht |>.2 h

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

Goal state before:
pp: 𝕜 : Type u_1
E₁ : Type u_2
E₂ : Type u_3
F : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E₁
inst✝⁵ : NormedSpace 𝕜 E₁
inst✝⁴ : NormedAddCommGroup E₂
inst✝³ : NormedSpace 𝕜 E₂
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : IsRCLikeNormedField 𝕜
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) v.1
df₂ : ∀ᶠ (v : E₁ × E₂) in 𝓝 u, HasFDerivAt (fun x ↦ f v.1 x) (↿f₂ v) v.2
cf₁ : ContinuousAt (↿f₁) u
cf₂ : ContinuousAt (↿f₂) u
⊢ (fun p ↦ ↿f p.1 - ↿f p.2 - ((↿f₁ u).coprod (↿f₂ u)) (p.1 - p.2)) =o[𝓝 (u, u)] fun p ↦ p.1 - p.2

Tactic:
unfold ContinuousAt at cf₁ cf₂

Goal state after:
𝕜 : Type u_1
E₁ : Type u_2
E₂ : Type u_3
F : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E₁
inst✝⁵ : NormedSpace 𝕜 E₁
inst✝⁴ : NormedAddCommGroup E₂
inst✝³ : NormedSpace 𝕜 E₂
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : IsRCLikeNormedField 𝕜
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) v.1
df₂ : ∀ᶠ (v : E₁ × E₂) in 𝓝 u, HasFDerivAt (fun x ↦ f v.1 x) (↿f₂ v) v.2
cf₁ : Tendsto (↿f₁) (𝓝 u) (𝓝 (↿f₁ u))
cf₂ : Tendsto (↿f₂) (𝓝 u) (𝓝 (↿f₂ u))
⊢ (fun p ↦ ↿f p.1 - ↿f p.2 - ((↿f₁ u).coprod (↿f₂ u)) (p.1 - p.2)) =o[𝓝 (u, u)] fun p ↦ p.1 - p.2
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
t : ℝ
h : DifferentiableAt ℝ (fun x ↦ ‖x‖) (t • x)
ht : t ≠ 0
⊢ DifferentiableAt ℝ (fun x ↦ ‖x‖) x

Tactic:
exact differentiableAt_norm_smul ht |>.2 h

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno

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
g : E → F
f : 𝕜 → E
s : Set 𝕜
t : Set E
x : 𝕜
hg : ContDiffWithinAt 𝕜 3 g t (f x)
hf : ContDiffWithinAt 𝕜 3 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
⊢ iteratedDerivWithin 3 (g ∘ f) s x =
    ((iteratedFDerivWithin 𝕜 3 g t (f x)) fun x_1 ↦ derivWithin f s x) +
          (iteratedFDerivWithin 𝕜 2 g t (f x)) ![iteratedDerivWithin 2 f s x, derivWithin f s x] +
        2 • (iteratedFDerivWithin 𝕜 2 g t (f x)) ![derivWithin f s x, iteratedDerivWithin 2 f s x] +
      (fderivWithin 𝕜 g t (f x)) (iteratedDerivWithin 3 f s x)

Tactic:
rw [iteratedDerivWithin_vcomp_eq_sum_orderedFinpartition hg hf ht hs hx hst le_rfl]
  simp only [← (OrderedFinpartition.extendEquiv 1).sum_comp,
    ← (OrderedFinpartition.extendEquiv 2).sum_comp, Fintype.sum_sigma,
    Fintype.sum_option, Nat.reduceAdd, OrderedFinpartition.extendEquiv_apply,
    OrderedFinpartition.extend_none, OrderedFinpartition.extend_some,
    OrderedFinpartition.extendMiddle_length, OrderedFinpartition.default_eq, Fintype.sum_unique,
    OrderedFinpartition.atomic_length, OrderedFinpartition.extendLeft_length, Fin.sum_univ_two]
  simp? [add_assoc, two_smul, iteratedFDerivWithin_one_apply (ht _ <| hst hx)] says
    simp only [OrderedFinpartition.extendLeft_partSize, OrderedFinpartition.extendLeft_length,
      OrderedFinpartition.atomic_length, Nat.reduceAdd, OrderedFinpartition.atomic_partSize,
      Fin.isValue, OrderedFinpartition.extendMiddle_partSize, Fin.cons_zero, Fin.update_cons_zero,
      Fin.cons_one, Fin.default_eq_zero, OrderedFinpartition.extendMiddle_length, Fin.cons_update,
      Fin.succ_zero_eq_one, update_self, update_idem,
      iteratedFDerivWithin_one_apply (ht _ <| hst hx), add_assoc, two_smul]
  have (j : _) : (Fin.cons 1 (Fin.cons 1 fun _ ↦ 1) : Fin 3 → ℕ) j = 1 := by
    fin_cases j <;> rfl
  congr <;> ext x <;> fin_cases x <;> simp [this]

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

Goal state before:
pp: s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx : x ∈ b
p q : ℝ
ps₁ : p ∈ s₁
qs₁ : q ∈ s₁
pq : p < q
hp : f p = f x
hq : f q = f x
⊢ ∃ p ∈ s₁, p ≠ x ∧ f p = f x

Tactic:
rcases eq_or_ne p x with h'p | h'p

Goal state after:
case inl
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx : x ∈ b
p q : ℝ
ps₁ : p ∈ s₁
qs₁ : q ∈ s₁
pq : p < q
hp : f p = f x
hq : f q = f x
h'p : p = x
⊢ ∃ p ∈ s₁, p ≠ x ∧ f p = f x

case inr
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx : x ∈ b
p q : ℝ
ps₁ : p ∈ s₁
qs₁ : q ∈ s₁
pq : p < q
hp : f p = f x
hq : f q = f x
h'p : p ≠ x
⊢ ∃ p ∈ s₁, p ≠ x ∧ f p = f x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case pos
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), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntegrableOn g [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] volume
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u

Tactic:
← intervalIntegrable_iff'

Goal state after:
case pos
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), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case pos
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), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
h_der :
  ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u

Tactic:
← intervalIntegrable_iff'

Goal state after:
case pos
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), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntervalIntegrable (fun x ↦ f' x • (g ∘ f) x) volume a b
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
h_der :
  ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in atBot, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in atBot, deriv g x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) atBot l
hdg : ∀ᶠ (x : ℝ) in atBot, DifferentiableAt ℝ g x
hdf' : ∀ᶠ (x : ℝ) in atBot, HasDerivAt f (deriv f x) x
⊢ Tendsto (fun x ↦ f x / g x) atBot l

Tactic:
have hdg' : ∀ᶠ x in atBot, HasDerivAt g (deriv g x) x :=
    hdg.mono fun _ => DifferentiableAt.hasDerivAt

Goal state after:
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in atBot, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in atBot, deriv g x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) atBot l
hdg : ∀ᶠ (x : ℝ) in atBot, DifferentiableAt ℝ g x
hdf' : ∀ᶠ (x : ℝ) in atBot, HasDerivAt f (deriv f x) x
hdg' : ∀ᶠ (x : ℝ) in atBot, HasDerivAt g (deriv g x) x
⊢ Tendsto (fun x ↦ f x / g x) atBot l
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝² : NormedAddCommGroup E
inst✝¹ : l.NeBot
inst✝ : l.IsCountablyGenerated
φ : ι → Set α
hφ : AECover μ l φ
f : α → E
I : ℝ≥0
hfm : AEStronglyMeasurable f μ
hbounded : ∀ᶠ (i : ι) in l, ∫⁻ (x : α) in φ i, ‖f x‖ₑ ∂μ ≤ ↑I
⊢ ∀ᶠ (i : ι) in l, ∫⁻ (x : α) in φ i, ‖f x‖ₑ ∂μ ≤ ENNReal.ofReal ↑I

Tactic:
simpa only [ENNReal.ofReal_coe_nnreal] using hbounded

Goal state after: