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

Goal state before:
pp: case pos.refine_1
α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : μ u ≤ μ s + ε / 2
R : α → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : μ (s \ ⋃ x ∈ t0, closedBall x (r0 x)) = 0
t0_disj : t0.PairwiseDisjoint fun x ↦ closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
s's : s' ⊆ s
N : ℕ
τ : ℝ
hτ : 1 < τ
H : IsEmpty (SatelliteConfig α N τ)
v : Set α
s'v : v ⊇ s'
v_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] v
μv : μ v ≤ μ s' + ε / 2 / ↑N
r1 : α → ℝ
hr1 : ∀ x ∈ s', r1 x ∈ f x ∩ Ioo 0 1 ∧ closedBall x (r1 x) ⊆ v
q : BallPackage (↑s') α := { c := fun x ↦ ↑x, r := fun x ↦ r1 ↑x, rpos := ⋯, r_bound := 1, r_le := ⋯ }
S : Fin N → Set ↑s'
S_disj : ∀ (i : Fin N), (S i).PairwiseDisjoint fun j ↦ closedBall (q.c j) (q.r j)
hS : range q.c ⊆ ⋃ i, ⋃ j ∈ S i, ball (q.c j) (q.r j)
S_count : ∀ (i : Fin N), (S i).Countable
r : α → ℝ := fun x ↦ if x ∈ s' then r1 x else r0 x
r_t0 : ∀ x ∈ t0, r x = r0 x
x : α
hx : x ∈ s
h'x : x ∈ s'
i : Fin N
y : ↑s'
ySi : y ∈ S i
xy : x ∈ ball (↑y) (r1 ↑y)
⊢ ↑y ∈ ⋃ i, Subtype.val '' S i

Tactic:
simp only [mem_iUnion, mem_image]

Goal state after:
case pos.refine_1
α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : μ u ≤ μ s + ε / 2
R : α → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : μ (s \ ⋃ x ∈ t0, closedBall x (r0 x)) = 0
t0_disj : t0.PairwiseDisjoint fun x ↦ closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
s's : s' ⊆ s
N : ℕ
τ : ℝ
hτ : 1 < τ
H : IsEmpty (SatelliteConfig α N τ)
v : Set α
s'v : v ⊇ s'
v_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] v
μv : μ v ≤ μ s' + ε / 2 / ↑N
r1 : α → ℝ
hr1 : ∀ x ∈ s', r1 x ∈ f x ∩ Ioo 0 1 ∧ closedBall x (r1 x) ⊆ v
q : BallPackage (↑s') α := { c := fun x ↦ ↑x, r := fun x ↦ r1 ↑x, rpos := ⋯, r_bound := 1, r_le := ⋯ }
S : Fin N → Set ↑s'
S_disj : ∀ (i : Fin N), (S i).PairwiseDisjoint fun j ↦ closedBall (q.c j) (q.r j)
hS : range q.c ⊆ ⋃ i, ⋃ j ∈ S i, ball (q.c j) (q.r j)
S_count : ∀ (i : Fin N), (S i).Countable
r : α → ℝ := fun x ↦ if x ∈ s' then r1 x else r0 x
r_t0 : ∀ x ∈ t0, r x = r0 x
x : α
hx : x ∈ s
h'x : x ∈ s'
i : Fin N
y : ↑s'
ySi : y ∈ S i
xy : x ∈ ball (↑y) (r1 ↑y)
⊢ ∃ i, ∃ x ∈ S i, ↑x = ↑y
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
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
M : E ≃L[𝕜] F
hM : ↑M = fderivWithin 𝕜 f s x
U : Set E := {y | ∃ N, ↑N = fderivWithin 𝕜 f s y}
hU : U ∈ 𝓝[s] x
N : E → E ≃L[𝕜] F := fun x ↦ if h : x ∈ U then Exists.choose h else M
eN : (fun y ↦ ↑(N y)) =ᶠ[𝓝[s] x] fun y ↦ fderivWithin 𝕜 f s y
e'N : ↑(N x) = fderivWithin 𝕜 f s x
hN : ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y)) s x
hN' : ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y).symm) s x
v : E
A' : ∀ (y : E), ((ContinuousLinearMap.compL 𝕜 F E F) ↑(N y)) ↑(N y).symm = ContinuousLinearMap.id 𝕜 F
⊢ (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:
have : fderivWithin 𝕜 (fun y ↦ ContinuousLinearMap.compL 𝕜 F E F (N y : E →L[𝕜] F)
      ((N y).symm : F →L[𝕜] E)) s x v = 0 := by
    simp [A', fderivWithin_const_apply]

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
U : Set E := {y | ∃ N, ↑N = fderivWithin 𝕜 f s y}
hU : U ∈ 𝓝[s] x
N : E → E ≃L[𝕜] F := fun x ↦ if h : x ∈ U then Exists.choose h else M
eN : (fun y ↦ ↑(N y)) =ᶠ[𝓝[s] x] fun y ↦ fderivWithin 𝕜 f s y
e'N : ↑(N x) = fderivWithin 𝕜 f s x
hN : ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y)) s x
hN' : ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y).symm) s x
v : E
A' : ∀ (y : E), ((ContinuousLinearMap.compL 𝕜 F E F) ↑(N y)) ↑(N y).symm = ContinuousLinearMap.id 𝕜 F
this : (fderivWithin 𝕜 (fun y ↦ ((ContinuousLinearMap.compL 𝕜 F E F) ↑(N y)) ↑(N y).symm) s x) v = 0
⊢ (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.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
⊢ x + (2 • v + w) ∈ interior s

Tactic:
convert! s_conv.interior.add_smul_sub_mem h2v h2v2w B using 1

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 : 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
⊢ x + (2 • v + w) = x + 2 • v + (1 / 2) • (x + 2 • v + 2 • w - (x + 2 • v))
Module: Mathlib.Analysis.Calculus.LineDeriv.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → F
x v : E
c : 𝕜
hc : c ≠ 0
H : LineDifferentiableAt 𝕜 f x v
⊢ lineDeriv 𝕜 f x (c • v) = c • lineDeriv 𝕜 f x v

pp: case neg
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → F
x v : E
c : 𝕜
hc : c ≠ 0
H : ¬LineDifferentiableAt 𝕜 f x v
⊢ lineDeriv 𝕜 f x (c • v) = c • lineDeriv 𝕜 f x v

Tactic:
· exact (H.hasLineDerivAt.smul c).lineDeriv

Goal state after:
case neg
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → F
x v : E
c : 𝕜
hc : c ≠ 0
H : ¬LineDifferentiableAt 𝕜 f x v
⊢ lineDeriv 𝕜 f x (c • v) = c • lineDeriv 𝕜 f x v
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‖) (t⁻¹ • t • x)
y : E
⊢ ‖y‖ = |t| * (‖t⁻¹‖ * ‖y‖)

Tactic:
← mul_assoc,

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‖) (t⁻¹ • t • x)
y : E
⊢ ‖y‖ = |t| * ‖t⁻¹‖ * ‖y‖
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: case e'_11
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
h1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
x✝ : E
⊢ ‖x✝‖ = |t| * (‖t⁻¹‖ * ‖x✝‖)

Tactic:
← mul_assoc,

Goal state after:
case e'_11
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
h1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
x✝ : E
⊢ ‖x✝‖ = |t| * ‖t⁻¹‖ * ‖x✝‖
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: case e'_12
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
h1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
e_8✝ : instAddCommGroup = normedAddCommGroup.toAddCommGroup
e_9✝ : Semiring.toModule ≍ RCLike.toInnerProductSpaceReal.toModule
y : E
⊢ t * (↑(SignType.sign t) * f y) = |t| * f y

Tactic:
← mul_assoc,

Goal state after:
case e'_12
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
h1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
e_8✝ : instAddCommGroup = normedAddCommGroup.toAddCommGroup
e_9✝ : Semiring.toModule ≍ RCLike.toInnerProductSpaceReal.toModule
y : E
⊢ t * ↑(SignType.sign t) * f y = |t| * f y
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case refine_2
α : Type u_1
inst✝⁵ : MetricSpace α
β : Type u
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SFinite μ
s : Set α
f : α → Set (Set α)
fsubset : ∀ x ∈ s, f x ⊆ (fun r ↦ closedBall x r) '' Ioi 0
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : α → Set ℝ := fun x ↦ {r | 0 < r ∧ closedBall x r ∈ f x}
A : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty
t : Set α
r : α → ℝ
left✝ : t.Countable
ts : t ⊆ s
tg : ∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1
μt : μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0
tdisj : t.PairwiseDisjoint fun x ↦ closedBall x (r x)
F : α → α × Set α := fun x ↦ (x, closedBall x (r x))
x : α
hx : x ∈ t
y : α
hy : y ∈ t
hxy : F x ≠ F y
⊢ Function.onFun Disjoint (fun p ↦ p.2) (F x) (F y)

Tactic:
exact tdisj hx hy (ne_of_apply_ne F hxy)

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

Goal state before:
pp: case e'_11
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : t ≠ 0
h : HasFDerivAt (fun x ↦ ‖x‖) f x
h1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
x✝ : E
⊢ ‖x✝‖ = |t| * (‖t⁻¹‖ * ‖x✝‖)

Tactic:
← mul_assoc,

Goal state after:
case e'_11
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : t ≠ 0
h : HasFDerivAt (fun x ↦ ‖x‖) f x
h1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
x✝ : E
⊢ ‖x✝‖ = |t| * ‖t⁻¹‖ * ‖x✝‖
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: case e'_12
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : t ≠ 0
h : HasFDerivAt (fun x ↦ ‖x‖) f x
h1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
e_8✝ : instAddCommGroup = normedAddCommGroup.toAddCommGroup
e_9✝ : Semiring.toModule ≍ RCLike.toInnerProductSpaceReal.toModule
y : E
⊢ t * (↑(SignType.sign t) * f y) = |t| * f y

Tactic:
← mul_assoc,

Goal state after:
case e'_12
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : t ≠ 0
h : HasFDerivAt (fun x ↦ ‖x‖) f x
h1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
e_8✝ : instAddCommGroup = normedAddCommGroup.toAddCommGroup
e_9✝ : Semiring.toModule ≍ RCLike.toInnerProductSpaceReal.toModule
y : E
⊢ t * ↑(SignType.sign t) * f y = |t| * f y
Module: Mathlib.Analysis.Normed.Module.Complemented

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
p q : Subspace 𝕜 E
h : IsCompl p q
hp : IsClosed ↑p
hq : IsClosed ↑q
⊢ IsTopCompl p q

Tactic:
by
  have := hp.completeSpace_coe; have := hq.completeSpace_coe
  rw [isTopCompl_iff_continuous_symm_prodEquivOfIsCompl h]
  exact (p.prodEquivOfIsCompl q h).continuous_symm (continuous_prodEquivOfIsCompl h)

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
t : ℝ
⊢ fderiv ℝ (fun x ↦ ‖x‖) (t • x) = ↑(SignType.sign t) • fderiv ℝ (fun x ↦ ‖x‖) x

Tactic:
cases subsingleton_or_nontrivial E

Goal state after:
case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
t : ℝ
h✝ : Subsingleton E
⊢ fderiv ℝ (fun x ↦ ‖x‖) (t • x) = ↑(SignType.sign t) • fderiv ℝ (fun x ↦ ‖x‖) x

case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
t : ℝ
h✝ : Nontrivial E
⊢ fderiv ℝ (fun x ↦ ‖x‖) (t • x) = ↑(SignType.sign t) • fderiv ℝ (fun x ↦ ‖x‖) x
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
x y : F
⊢ (starRingEnd 𝕜) ⟪∇ f y, x⟫ = (starRingEnd 𝕜) ((fderiv 𝕜 f y) x)

Tactic:
inner_gradient_left

Goal state after:
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
x y : F
⊢ (starRingEnd 𝕜) ((fderiv 𝕜 f y) x) = (starRingEnd 𝕜) ((fderiv 𝕜 f y) x)
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: 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

Tactic:
exact ⟨q, qs₁, (h'p.symm.le.trans_lt pq).ne', hq⟩

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

Goal state before:
pp: 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
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]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
cdsub : [[c, d]] ⊆ Ioo (min a b) (max a b)
hg_cont : ContinuousOn g (f '' [[c, d]])
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
⊢ HasDerivWithinAt (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x

Tactic:
have hJ : f '' [[c, d]] = J := (hf.mono (cdsub.trans Ioo_subset_Icc_self)).image_uIcc

Goal state after:
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
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]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
cdsub : [[c, d]] ⊆ Ioo (min a b) (max a b)
hg_cont : ContinuousOn g (f '' [[c, d]])
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hJ : f '' [[c, d]] = J
⊢ HasDerivWithinAt (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case inl.hf
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x
M : MonotoneOn f [[a, b]]
hab : a ≤ b
⊢ ContinuousOn f (Icc a b)

Tactic:
rwa [uIcc_of_le hab] at hf

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

Goal state before:
pp: case inl.hf
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
M : AntitoneOn f [[a, b]]
hab : a ≤ b
⊢ ContinuousOn f (Icc a b)

Tactic:
rwa [uIcc_of_le hab] at hf

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

Goal state before:
pp: case neg
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.IsCountablyGenerated
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
b : ι → ℝ
f : ℝ → E
hb : Tendsto b l atTop
a : ℝ
ha : IntegrableOn f (Ioi a) μ
this : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioi a, f x ∂μ - ∫ (x : ℝ) in a..b i, f x ∂μ = ∫ (x : ℝ) in Ioi (b i), f x ∂μ
⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in Ioi (b i), f x ∂μ) l (𝓝 0)

Tactic:
rw [← sub_self (∫ x in Ioi a, f x ∂μ)]

Goal state after:
case neg
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.IsCountablyGenerated
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
b : ι → ℝ
f : ℝ → E
hb : Tendsto b l atTop
a : ℝ
ha : IntegrableOn f (Ioi a) μ
this : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioi a, f x ∂μ - ∫ (x : ℝ) in a..b i, f x ∂μ = ∫ (x : ℝ) in Ioi (b i), f x ∂μ
⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in Ioi (b i), f x ∂μ) l (𝓝 (∫ (x : ℝ) in Ioi a, f x ∂μ - ∫ (x : ℝ) in Ioi a, f x ∂μ))
Module: Mathlib.Analysis.Calculus.LineDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
v : E
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
hg : Continuous (Function.uncurry g)
⊢ MeasurableSet {x | LineDifferentiableAt 𝕜 f x v}

Tactic:
exact measurable_prodMk_right (measurableSet_of_differentiableAt_with_param 𝕜 hg)

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

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

Tactic:
apply Tendsto.congr' _ (hf.const_sub _)

Goal state after:
E : Type u_1
f f' : ℝ → E
a : ℝ
m : E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hcont✝ : ContinuousWithinAt f (Iic a) a
hderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Iic a) volume
hf : Tendsto f atBot (𝓝 m)
hcont : ContinuousOn f (Iic a)
⊢ (fun x ↦ f a - f x) =ᶠ[atBot] fun i ↦ ∫ (x : ℝ) in id i..a, f' x
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: case e'_3
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
U : Set ℂ
z : ℂ
F : ι → ℂ → E
inst✝ : CompleteSpace E
u : ι → ℝ
hu : Summable u
hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hU : IsOpen U
hF_le : ∀ (i : ι), ∀ w ∈ U, ‖F i w‖ ≤ u i
hz : z ∈ U
hc : TendstoLocallyUniformlyOn (fun t x ↦ ∑ n ∈ t, F n x) (fun x ↦ ∑' (n : ι), F n x) atTop U
s : Finset ι
⊢ ∑ b ∈ s, deriv (F b) z = (deriv ∘ fun t x ↦ ∑ n ∈ t, F n x) s z

Tactic:
exact (deriv_fun_sum fun i _ => (hf i).differentiableAt (hU.mem_nhds hz)).symm

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

Goal state before:
pp: case inl
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : TopologicalSpace M'
cm : ChartedSpace H M
cm' : ChartedSpace H M'
inst✝ : Nonempty H
x : M
⊢ (ChartedSpace.chartAt x).lift_openEmbedding ⋯ ∈ (fun e ↦ e.lift_openEmbedding ⋯) '' ChartedSpace.atlas

Tactic:
use ChartedSpace.chartAt x, cm.chart_mem_atlas x

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

Goal state before:
pp: H : Type u
M : Type u_2
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
hs : Nonempty ↑e.source
inst✝¹ : HasGroupoid M G
inst✝ : ClosedUnderRestriction G
s : Opens M := { carrier := e.source, is_open' := ⋯ }
t : Opens H := { carrier := e.target, is_open' := ⋯ }
c' : OpenPartialHomeomorph (↥t) H
hc'✝ : c' ∈ atlas H ↥t
this : Nonempty ↥t
x : ↥t
hc' : c' = (chartAt H ↑x).subtypeRestr this
⊢ e.toHomeomorphSourceTarget.toOpenPartialHomeomorph ≫ₕ t.openPartialHomeomorphSubtypeCoe this ∈ maximalAtlas (↥s) G

Tactic:
let goal :=
    e.toHomeomorphSourceTarget.toOpenPartialHomeomorph ≫ₕ (t.openPartialHomeomorphSubtypeCoe this)

Goal state after:
H : Type u
M : Type u_2
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
G : StructureGroupoid H
e : OpenPartialHomeomorph M H
he : e ∈ atlas H M
hs : Nonempty ↑e.source
inst✝¹ : HasGroupoid M G
inst✝ : ClosedUnderRestriction G
s : Opens M := { carrier := e.source, is_open' := ⋯ }
t : Opens H := { carrier := e.target, is_open' := ⋯ }
c' : OpenPartialHomeomorph (↥t) H
hc'✝ : c' ∈ atlas H ↥t
this : Nonempty ↥t
x : ↥t
hc' : c' = (chartAt H ↑x).subtypeRestr this
goal : OpenPartialHomeomorph (↑e.source) H :=
  e.toHomeomorphSourceTarget.toOpenPartialHomeomorph ≫ₕ t.openPartialHomeomorphSubtypeCoe this
⊢ e.toHomeomorphSourceTarget.toOpenPartialHomeomorph ≫ₕ t.openPartialHomeomorphSubtypeCoe this ∈ maximalAtlas (↥s) G
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
x : H
s : Set H
⊢ map (↑I.symm ∘ ↑I) (𝓝[s] x) = 𝓝[s] x

Tactic:
I.symm_comp_self,

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
x : H
s : Set H
⊢ map id (𝓝[s] x) = 𝓝[s] x