mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
x : ℝ
hx : x ∈ Set.Icc (ak k) (ak (k + 1))
⊢ _root_.derivWithin (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b) x =
    _root_.derivWithin (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1)))) (Set.Icc (ak k) (ak (k + 1))) x

Tactic:
rw [← derivWithin_subset h3 (uniqueDiffOn_Icc h4 x hx) (h_ddf x (h3 hx))]

Goal state after:
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
x : ℝ
hx : x ∈ Set.Icc (ak k) (ak (k + 1))
⊢ _root_.derivWithin (_root_.derivWithin f (Set.Icc a b)) (Set.Icc (ak k) (ak (k + 1))) x =
    _root_.derivWithin (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1)))) (Set.Icc (ak k) (ak (k + 1))) x
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ≥0
hf : ∀ (x : X), f x ≤ 1
V : Set X
h : tsupport ⇑f ⊆ V
⊢ ↑(Λ f) ≤ (rieszMeasure Λ) V

Tactic:
by
  apply le_trans _ (measure_mono h)
  apply le_rieszMeasure_of_isCompact_tsupport_subset Λ hf f.hasCompactSupport
  exact subset_rfl

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : T2Space X
inst✝⁴ : LocallyCompactSpace X
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
μ ν : Measure X
inst✝¹ : μ.Regular
inst✝ : ν.Regular
x✝ : μ = ν
⊢ integralLinearMap μ = integralLinearMap ν

Tactic:
by congr

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case h.refine_2.inl
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : y m - ε < f x ∧ f x ≤ y m ∧ x ∈ tsupport ⇑f
hc : m < n
⊢ f x ≤ y n - ε ∨ y n < f x ∨ x ∉ tsupport ⇑f

Tactic:
left
      exact le_trans hx.2.1 (le_tsub_of_add_le_right (hy hc))

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case pos
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
n : Fin N
hn : n ∈ Finset.univ
x : X
hx : x ∈ tsupport ⇑(g n)
⊢ (g n • f) x ≤ ((y n + ε') • g n) x

Tactic:
rw [smul_eq_mul, mul_comm]

Goal state after:
case pos
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
n : Fin N
hn : n ∈ Finset.univ
x : X
hx : x ∈ tsupport ⇑(g n)
⊢ (f * g n) x ≤ ((y n + ε') • g n) x
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
⊢ innerProbChar 0 = 1

Tactic:
by simp [innerProbChar]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_3
inst✝⁷ : MeasurableSpace E
μ ν : Measure E
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : CompleteSpace E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : charFun μ = charFun ν
⊢ μ = ν

Tactic:
simp_rw [funext_iff, charFun_eq_integral_innerProbChar] at h

Goal state after:
E : Type u_3
inst✝⁷ : MeasurableSpace E
μ ν : Measure E
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : CompleteSpace E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ∀ (x : E), ∫ (v : E), (innerProbChar x) v ∂μ = ∫ (v : E), (innerProbChar x) v ∂ν
⊢ μ = ν
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : InnerProductSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
L : StrongDual ℝ E
⊢ charFun μ ((InnerProductSpace.toDual ℝ E).symm L) = charFunDual μ L

Tactic:
charFun_eq_charFunDual_toDualMap,

Goal state after:
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : InnerProductSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
L : StrongDual ℝ E
⊢ charFunDual μ ((InnerProductSpace.toDualMap ℝ E) ((InnerProductSpace.toDual ℝ E).symm L)) = charFunDual μ L
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
U : Set X
⊢ x ∈ U → IsOpen[inst✝¹] U → ∃ i, stoneCechUnit x ∈ i ∧ IsOpen i ∧ stoneCechUnit ⁻¹' i ⊆ U

Tactic:
hxU

Goal state after:
case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
U : Set X
hxU : x ∈ U
⊢ IsOpen[inst✝¹] U → ∃ i, stoneCechUnit x ∈ i ∧ IsOpen i ∧ stoneCechUnit ⁻¹' i ⊆ U
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
c : ℝ≥0∞
h : levyProkhorovEDist μ ν < c
B : Set Ω
B_mble : MeasurableSet B
⊢ ν B ≤ μ (thickening c.toReal B) + c

Tactic:
by
  obtain ⟨c', ⟨hc', lt_c⟩⟩ := sInf_lt_iff.mp h
  exact meas_le_of_le_of_forall_le_meas_thickening_add ν μ lt_c.le (hc' B B_mble).2

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
⊢ levyProkhorovEDist ν κ + (ε / 2 + (levyProkhorovEDist μ ν + ε / 2)) =
    levyProkhorovEDist μ ν + (levyProkhorovEDist ν κ + ε)

Tactic:
add_comm (ε / 2),

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
⊢ levyProkhorovEDist ν κ + (levyProkhorovEDist μ ν + ε / 2 + ε / 2) =
    levyProkhorovEDist μ ν + (levyProkhorovEDist ν κ + ε)
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν : Measure Ω
hLP : levyProkhorovEDist μ ν = 0
s : Set Ω
s_mble : MeasurableSet s
h_finite : ∃ δ > 0, ν (thickening δ s) ≠ ∞
⊢ μ s ≤ ν (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)

Tactic:
have key : Tendsto (fun ε ↦ ν (thickening ε.toReal s)) (𝓝[>] (0 : ℝ≥0∞)) (𝓝 (ν (closure s))) := by
    have aux : Tendsto ENNReal.toReal (𝓝[>] 0) (𝓝[>] 0) := by
      apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within (s := Ioi 0) ENNReal.toReal
      · exact tendsto_nhdsWithin_of_tendsto_nhds (continuousAt_toReal zero_ne_top).tendsto
      · filter_upwards [Ioo_mem_nhdsGT zero_lt_one] with x hx
        exact toReal_pos hx.1.ne' <| ne_top_of_lt hx.2
    exact (tendsto_measure_thickening h_finite).comp aux

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν : Measure Ω
hLP : levyProkhorovEDist μ ν = 0
s : Set Ω
s_mble : MeasurableSet s
h_finite : ∃ δ > 0, ν (thickening δ s) ≠ ∞
key :
  Tendsto (fun ε ↦ ν (thickening ε.toReal s)) (𝓝[>] 0)
    (𝓝 (ν (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)))
⊢ μ s ≤ ν (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case pos
X : Type u_1
inst✝⁷ : TopologicalSpace X
inst✝⁶ : T2Space X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
μ ν : Measure X
inst✝³ : LocallyCompactSpace X
inst✝² : ν.OuterRegular
inst✝¹ : IsFiniteMeasureOnCompacts ν
inst✝ : IsFiniteMeasureOnCompacts μ
hμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂μ ≤ ∫ (x : X), f x ∂ν
K : Set X
hK : IsCompact K
ε : NNReal
hε : 0 < ε
hν : ν K < ∞
hνK : ν K ≠ ∞
hμK : μ K ≠ ∞
V : Set X
pV1 : V ⊇ K
pV2 : IsOpen[inst✝⁷] V
pV3 : ν V ≤ ν K + ↑ε
VltTop : ν V < ∞
f : X →C_c ℝ
pf1 : EqOn (⇑f) 1 K
pf2 : tsupport ⇑f ⊆ V
pf3 : ∀ (x : X), f x ∈ Icc 0 1
x : X
hx : x ∈ tsupport ⇑f
⊢ f x ≤ V.indicator 1 x

Tactic:
simp [(pf2 hx), (pf3 x).2]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : T2Space X
inst✝⁴ : MeasurableSpace X
inst✝³ : BorelSpace X
μ ν : Measure X
inst✝² : LocallyCompactSpace X
inst✝¹ : μ.Regular
inst✝ : ν.Regular
x✝ : μ = ν
⊢ integralPositiveLinearMap μ = integralPositiveLinearMap ν

Tactic:
by congr

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ✝ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : CompactSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
⊢ (rieszMeasure Λ) univ < ∞

Tactic:
let o : C_c(X, ℝ) := ⟨1, HasCompactSupport.of_compactSpace 1⟩

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ✝ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : CompactSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
o : X →C_c ℝ := { toContinuousMap := 1, hasCompactSupport' := ⋯ }
⊢ (rieszMeasure Λ) univ < ∞
Module: Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar

Goal state before:
pp: G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
f : G → ℝ
φ : G ≃ₜ* G
e : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv
this✝ : (map (⇑e.symm) μ).IsHaarMeasure
this : (map (⇑e.symm) μ).Regular
⊢ mulEquivHaarChar φ • ∫ (a : G), f a ∂map (⇑φ ∘ ⇑e.symm) μ = mulEquivHaarChar φ • ∫ (a : G), f a ∂μ

Tactic:
show ⇑φ ∘ ⇑e.symm = id by ext; simp [e]

Goal state after:
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
f : G → ℝ
φ : G ≃ₜ* G
e : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv
this✝ : (map (⇑e.symm) μ).IsHaarMeasure
this : (map (⇑e.symm) μ).Regular
⊢ mulEquivHaarChar φ • ∫ (a : G), f a ∂map id μ = mulEquivHaarChar φ • ∫ (a : G), f a ∂μ
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case refine_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c d : E
γ₁ : Path a b
γ₂ : Path c d
s : Set (↑I × ↑I)
t : Set E
ω : E → E →L[ℝ] F
dω : E → E →L[ℝ] E →L[ℝ] F
φ : (↑γ₁).Homotopy ↑γ₂
hs : s.Countable
hφt : ∀ a_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, φ (a_1, b_1) ∈ t
hω : ∀ a_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, (a_1, b_1) ∉ s → HasFDerivWithinAt ω (dω (φ (a_1, b_1))) t (φ (a_1, b_1))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a_1 ∈ Ioo 0 1,
    ∀ b_1 ∈ Ioo 0 1,
      (a_1, b_1) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a_1, b_1)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a_1, b_1)), ((dω (φ (a_1, b_1))) u) v = ((dω (φ (a_1, b_1))) v) u
U : Set (ℝ × ℝ) := Ioo 0 1 ×ˢ Ioo 0 1
hU : U = Ioo 0 1 ×ˢ Ioo 0 1
hinterior : interior (Icc 0 1) = U
hunique : UniqueDiffOn ℝ (Icc 0 1)
hUopen : IsOpen U
hU_subset : U ⊆ Icc 0 1
hclosure : closure U = Icc 0 1
ψ : ℝ × ℝ → E := fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hcontdiff : ContDiffOn ℝ 2 ψ (Icc 0 1)
hψ : ψ = fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hψφ : ∀ (a_1 b_1 : ↑I), ψ (↑a_1, ↑b_1) = φ (a_1, b_1)
hψ_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ψ
hψUt : MapsTo ψ U t
dψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fderivWithin ℝ ψ (Icc 0 1)
s' : Set (ℝ × ℝ) := Prod.map Subtype.val Subtype.val '' s
hs' : s' = Prod.map Subtype.val Subtype.val '' s
hmem_s' : ∀ (x y : ↑I), (↑x, ↑y) ∈ s' ↔ (x, y) ∈ s
hs'c : s'.Countable
hdψ : ∀ a ∈ U, HasFDerivAt ψ (dψ a) a
d2ψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ →L[ℝ] E := fderivWithin ℝ dψ (Icc 0 1)
hd2ψ : ∀ a ∈ U, HasFDerivAt dψ (d2ψ a) a
hd2ψ_symm : ∀ a ∈ Icc 0 1, ∀ (x y : ℝ × ℝ), ((d2ψ a) x) y = ((d2ψ a) y) x
η : ℝ × ℝ → ℝ × ℝ →L[ℝ] F := fun a ↦ ω (ψ a) ∘SL dψ a
f : ℝ × ℝ → F := fun a ↦ (η a) (0, 1)
hf : ∀ a ∈ Icc 0 1, f a = (ω (ψ a)) (derivWithin (ψ ∘ fun x ↦ (a.1, x)) I a.2)
g : ℝ × ℝ → F := fun a ↦ -(η a) (1, 0)
hg : ∀ a ∈ Icc 0 1, g a = (ω (ψ a)) (-derivWithin (ψ ∘ fun x ↦ (x, a.2)) I a.1)
dη : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ →L[ℝ] F :=
  fun a ↦ (ContinuousLinearMap.compL ℝ (ℝ × ℝ) E F) (ω (ψ a)) ∘SL d2ψ a + (dω (ψ a)).bilinearComp (dψ a) (dψ a)
hdη : ∀ a ∈ U \ s', HasFDerivAt η (dη a) a
hdη_symm : ∀ a ∈ U \ s', ∀ (u v : ℝ × ℝ), ((dη a) u) v = ((dη a) v) u
f' : ℝ × ℝ → ℝ × ℝ →L[ℝ] F := fun a ↦ (ContinuousLinearMap.apply ℝ F) (0, 1) ∘SL dη a
hf' : ∀ a ∈ U \ s', HasFDerivAt f (f' a) a
g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] F := fun a ↦ -(ContinuousLinearMap.apply ℝ F) (1, 0) ∘SL dη a
hg' : ∀ a ∈ U \ s', HasFDerivAt g (g' a) a
hf'g' : (fun a ↦ (f' a) (1, 0) + (g' a) (0, 1)) =ᵐ[volume.restrict (Icc 0 1)] 0
this : ∫ (a : ℝ × ℝ) in Icc 0 1, (f' a) (1, 0) + (g' a) (0, 1) = 0
⊢ MapsTo ψ (Icc 0 1) (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)

Tactic:
rw [← hclosure]
        refine MapsTo.closure (fun a ha ↦ ?_) hψ_cont
        lift a to I × I using ⟨Ioo_subset_Icc_self ha.1, Ioo_subset_Icc_self ha.2⟩
        simpa [ψ] using hφt a.1 ha.1 a.2 ha.2

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ volume (ball x r) =
    ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (√π ^ finrank ℝ E / Gamma (↑(finrank ℝ E) / 2 + 1))

Tactic:
rw [← ((stdOrthonormalBasis ℝ E).measurePreserving_repr_symm).measure_preimage
      measurableSet_ball.nullMeasurableSet]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ volume (⇑(stdOrthonormalBasis ℝ E).repr.symm ⁻¹' ball x r) =
    ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (√π ^ finrank ℝ E / Gamma (↑(finrank ℝ E) / 2 + 1))
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
P : (i : ℕ) → Finpartition ⟨s i, ⋯⟩
n : ℕ
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
hs_disj : (↑(Finset.range n)).PairwiseDisjoint s'
Q : Finpartition ((Finset.range n).sup fun i ↦ ⟨s i, ⋯⟩) := Finpartition.combine P ⋯
⊢ ((Finset.range n).sup fun x ↦ ↑(s' x)) ⊆ ↑⟨⋃ i, s i, ⋯⟩

Tactic:
Finset.sup_set_eq_biUnion

Goal state after:
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
P : (i : ℕ) → Finpartition ⟨s i, ⋯⟩
n : ℕ
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
hs_disj : (↑(Finset.range n)).PairwiseDisjoint s'
Q : Finpartition ((Finset.range n).sup fun i ↦ ⟨s i, ⋯⟩) := Finpartition.combine P ⋯
⊢ ⋃ x ∈ Finset.range n, ↑(s' x) ⊆ ↑⟨⋃ i, s i, ⋯⟩
Module: Mathlib.MeasureTheory.Measure.Support

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : MeasurableSpace X
μ : Measure X
⊢ ∀ (x : X), (∃ᶠ (y : X) in 𝓝 x, y ∈ μ.support) → x ∈ μ.support

Tactic:
nhds_basis_opens _ |>.mem_measureSupport,

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : MeasurableSpace X
μ : Measure X
⊢ ∀ (x : X),
    (∃ᶠ (y : X) in 𝓝 x, ∀ (i : Set X), y ∈ i ∧ IsOpen[inst✝¹] i → 0 < μ i) →
      ∀ (i : Set X), x ∈ i ∧ IsOpen[inst✝¹] i → 0 < μ i
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s✝ : Set α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_sigma_subadd : m.IsSigmaSubadditive
this : MeasurableSpace α := (inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯).caratheodory
s : Set α
hs : s ∈ C
hs_meas : MeasurableSet s
⊢ (inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯) s ≤ m s

Tactic:
m.inducedOuterMeasure_eq hC m_sigma_subadd hs

Goal state after:
α : Type u_1
C : Set (Set α)
s✝ : Set α
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_sigma_subadd : m.IsSigmaSubadditive
this : MeasurableSpace α := (inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯).caratheodory
s : Set α
hs : s ∈ C
hs_meas : MeasurableSet s
⊢ m s ≤ m s
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
m : Measure X
hs : MeasurableSet s
h : ∀ (E : Set X), MeasurableSet E → E ⊆ s → ‖μ E‖ₑ ≤ m E
i : Finpartition ⟨s, ⋯⟩
⊢ ∑ x ∈ i.parts, m ↑x = m (⋃ x ∈ i.parts, ↑x)

Tactic:
refine (MeasureTheory.measure_biUnion_finset ?_ fun b _ => b.property).symm

Goal state after:
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
m : Measure X
hs : MeasurableSet s
h : ∀ (E : Set X), MeasurableSet E → E ⊆ s → ‖μ E‖ₑ ≤ m E
i : Finpartition ⟨s, ⋯⟩
⊢ (↑i.parts).PairwiseDisjoint Subtype.val
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
m : Measure X
hs : MeasurableSet s
h : ∀ (E : Set X), MeasurableSet E → E ⊆ s → ‖μ E‖ₑ ≤ m E
i : Finpartition ⟨s, ⋯⟩
⊢ ∑ x ∈ i.parts, m ↑x = m (i.parts.sup Subtype.val)

Tactic:
by
      rw [sup_set_eq_biUnion]
      refine (MeasureTheory.measure_biUnion_finset ?_ fun b _ => b.property).symm
      intro a ha b hb hab
      simpa [disjoint_iff, Subtype.ext_iff] using i.disjoint ha hb hab

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: case h₁
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝³ : TopologicalSpace V
inst✝² : ENormedAddCommMonoid V
inst✝¹ : T2Space V
μ ν : VectorMeasure X V
inst✝ : ContinuousAdd V
E : Set X
x✝ : MeasurableSet E
⊢ ‖μ E‖ₑ ≤ μ.variation E

Tactic:
exact enorm_measure_le_variation _ E

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: case h₂
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝³ : TopologicalSpace V
inst✝² : ENormedAddCommMonoid V
inst✝¹ : T2Space V
μ ν : VectorMeasure X V
inst✝ : ContinuousAdd V
E : Set X
x✝ : MeasurableSet E
⊢ ‖ν E‖ₑ ≤ ν.variation E

Tactic:
exact enorm_measure_le_variation _ E

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup V
μ ν : VectorMeasure X V
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
c : 𝕜
inst✝ : IsFiniteMeasure μ.variation
⊢ IsFiniteMeasure (c • μ).variation

Tactic:
simp only [variation_smul]
  infer_instance

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.AddContent

Goal state before:
pp: case succ
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
m : Set α → E
hm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s
inst✝ : IsFiniteMeasure μ
h'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t
h''m : ∀ (s : Set α), ¬MeasurableSet s → m s = 0
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Function.onFun Disjoint f)
n : ℕ
ih : m (⋃ i ∈ Finset.range n, f i) = ∑ i ∈ Finset.range n, m (f i)
⊢ m (⋃ i ∈ insert n (Finset.range n), f i) = ∑ x ∈ insert n (Finset.range n), m (f x)

Tactic:
rw [Finset.sum_insert (by simp)]

Goal state after:
case succ
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
m : Set α → E
hm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s
inst✝ : IsFiniteMeasure μ
h'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t
h''m : ∀ (s : Set α), ¬MeasurableSet s → m s = 0
f : ℕ → Set α
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Function.onFun Disjoint f)
n : ℕ
ih : m (⋃ i ∈ Finset.range n, f i) = ∑ i ∈ Finset.range n, m (f i)
⊢ m (⋃ i ∈ insert n (Finset.range n), f i) = m (f n) + ∑ x ∈ Finset.range n, m (f x)