mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case h
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
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
p : NNReal
hp : Λ f = ↑p
⊢ Λ f.nnrealPart.toReal = ↑p

Tactic:
show f.nnrealPart.toReal = f by ext z; simp [hf z],

Goal state after:
case h
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
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
p : NNReal
hp : Λ f = ↑p
⊢ Λ f = ↑p
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✝ : 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
⊢ Λ (g n) ≤ μ.real ↑(V n)

Tactic:
by
          apply (ENNReal.ofReal_le_iff_le_toReal _).mp
          · exact le_rieszMeasure_tsupport_subset Λ (fun x ↦ hg.2.2.1 n x) (hg.1 n)
          · rw [← lt_top_iff_ne_top]
            apply lt_of_le_of_lt (hV n).2.2
            rw [WithTop.add_lt_top]
            exact ⟨WithTop.lt_top_iff_ne_top.mpr (hE' n), ENNReal.ofReal_lt_top⟩

Goal state after:
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: ι : Type u_1
X : Type u_2
t : ι → TopologicalSpace X
ht : ∀ (i : ι), CompletelyRegularSpace X
this : TopologicalSpace X := ⨅ i, t i
x : X
K : Set X
hK : IsOpen[⨅ i, t i] K
hxK : x ∈ K
⊢ ∃ f, Continuous[⨅ i, t i, _] f ∧ f x = 0 ∧ EqOn f 1 Kᶜ

Tactic:
simp_rw [← hK.mem_nhds_iff, nhds_iInf, mem_iInf, exists_finite_iff_finset,
    Finset.coe_sort_coe] at hxK

Goal state after:
ι : Type u_1
X : Type u_2
t : ι → TopologicalSpace X
ht : ∀ (i : ι), CompletelyRegularSpace X
this : TopologicalSpace X := ⨅ i, t i
x : X
K : Set X
hK : IsOpen[⨅ i, t i] K
hxK : ∃ s V, (∀ (i : ↥s), V i ∈ 𝓝 x) ∧ K = ⋂ i, V i
⊢ ∃ f, Continuous[⨅ i, t i, _] f ∧ f x = 0 ∧ EqOn f 1 Kᶜ
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
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ μ (thickening (r.toReal + s.toReal) B) + (s + r) = μ (thickening (s.toReal + r.toReal) B) + (s + r)

Tactic:
by rw [add_comm r.toReal]

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

Goal state before:
pp: case h.refine_1.hs
ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) := ⋯
this✝ : IsPiSystem S
this : (i : ι) → PseudoMetricSpace (α i) := ⋯
u : Set ((i : ι) → α i)
u_open : IsOpen[Pi.topologicalSpace] u
x : (i : ι) → α i
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ι → ℝ
rpos : ∀ (i : ι), r i ∈ Ioo 0 ε
hr : ∀ (i : ι), ↑(μ i) (frontier (thickening (r i) {x i})) = 0
⊢ IsOpen[Pi.topologicalSpace] (univ.pi fun i ↦ ball (x i) (r i))

Tactic:
exact isOpen_set_pi finite_univ (by simp)

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✝ : 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, y n ≤ ↑N * b

Tactic:
have (n : Fin N) := calc y n
        _ ≤ a + ε' * N := by simp_all [y, show (n : ℝ) + 1 ≤ N by norm_cast; lia]
        _ = b := by simp [field, ε']

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✝ : 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)
this : ∀ (n : Fin N), y n ≤ b
⊢ ∑ n, y n ≤ ↑N * b
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
⊢ SeqContinuous toMeasure

Tactic:
intro μs ν hμs

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
⊢ Tendsto (toMeasure ∘ μs) atTop (𝓝 ν.toMeasure)
Module: Mathlib.MeasureTheory.Measure.Haar.DistribChar

Goal state before:
pp: G : Type u_1
A : Type u_2
inst✝⁶ : Group G
inst✝⁵ : AddCommGroup A
inst✝⁴ : DistribMulAction G A
inst✝³ : TopologicalSpace A
inst✝² : IsTopologicalAddGroup A
inst✝¹ : LocallyCompactSpace A
inst✝ : ContinuousConstSMul G A
g✝ : G
this✝ : MeasurableSpace A := borel A
this : BorelSpace A
g g' : G
⊢ ((DomMulAct.mk g' * DomMulAct.mk g) • addHaar).addHaarScalarFactor addHaar =
    (DomMulAct.mk g • addHaar).addHaarScalarFactor addHaar * (DomMulAct.mk g' • addHaar).addHaarScalarFactor addHaar

Tactic:
addHaarScalarFactor_eq_mul _ (DomMulAct.mk g' • addHaar (G := A))

Goal state after:
G : Type u_1
A : Type u_2
inst✝⁶ : Group G
inst✝⁵ : AddCommGroup A
inst✝⁴ : DistribMulAction G A
inst✝³ : TopologicalSpace A
inst✝² : IsTopologicalAddGroup A
inst✝¹ : LocallyCompactSpace A
inst✝ : ContinuousConstSMul G A
g✝ : G
this✝ : MeasurableSpace A := borel A
this : BorelSpace A
g g' : G
⊢ ((DomMulAct.mk g' * DomMulAct.mk g) • addHaar).addHaarScalarFactor (DomMulAct.mk g' • addHaar) *
      (DomMulAct.mk g' • addHaar).addHaarScalarFactor addHaar =
    (DomMulAct.mk g • addHaar).addHaarScalarFactor addHaar * (DomMulAct.mk g' • addHaar).addHaarScalarFactor addHaar
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
inst✝¹⁷ : Group A
inst✝¹⁶ : Group B
inst✝¹⁵ : Group C
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹¹ : IsTopologicalGroup A
inst✝¹⁰ : IsTopologicalGroup B
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : IsTopologicalGroup C
inst✝⁶ : LocallyCompactSpace B
inst✝⁵ : MeasurableSpace C
inst✝⁴ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝³ : T2Space B
inst✝² : MeasurableSpace B
inst✝¹ : BorelSpace B
U : Set B
hU : IsOpen U
inst✝ : DiscreteTopology A
h✝ : μC Set.univ * μA {1} < (H.inducedMeasure μA μC) U
ho : 0 < μA {1}
ht : μA {1} < ⊤
K : Set B
hKU : K ⊆ U
hK : IsCompact K
f : C(B, ℝ)
hf1 : Set.EqOn (⇑f) 1 K
hf2 : IsCompact (tsupport ⇑f)
hf3 : tsupport ⇑f ⊆ U
hf4 : ∀ (x : B), f x ∈ Set.Icc 0 1
h :
  μC Set.univ * μA {1} <
    ENNReal.ofReal (∫ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ∂μC)
hc : Set.InjOn (⇑ψ) U
b✝ : B
a : A
ha : a ∈ Function.support fun a ↦ f (b✝ * φ a)
b : A
hb : b ∈ Function.support fun a ↦ f (b✝ * φ a)
⊢ a = b

Tactic:
simpa [H.isClosedEmbedding.injective.eq_iff] using hc (hf3 (subset_tsupport _ ha))
      (hf3 (subset_tsupport _ hb)) (by simp [H.mulExact.apply_apply_eq_one])

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

Goal state before:
pp: Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
s : Set (ProbabilityMeasure Ω)
hs : IsOpen[coinduced toMeasure inferInstance] s
⊢ IsOpen[inferInstance] s

Tactic:
simpa using! hs.preimage continuous_ofMeasure_probabilityMeasure

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

Goal state before:
pp: case inl
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Subsingleton E
⊢ μ {0} = ENNReal.ofReal ((∫ (x : E), Real.exp (-‖x‖ ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))

Tactic:
← setIntegral_univ,

Goal state after:
case inl
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Subsingleton E
⊢ μ {0} = ENNReal.ofReal ((∫ (x : E) in Set.univ, Real.exp (-‖x‖ ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case e'_2
E : Type u_1
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
inst✝⁶ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : BorelSpace E
inst✝³ : T2Space E
inst✝² : ContinuousSMul ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
inst✝ : Nontrivial E
r : ℝ
F : Type u_1 := E
p : AddGroupNorm F := { toFun := g, map_zero' := h1, add_le' := h3, neg' := h2, eq_zero_of_map_eq_zero' := ⋯ }
this✝³ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝² : NormedSpace ℝ F := { toModule := inst✝⁷, norm_smul_le := ⋯ }
this✝¹ : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
⊢ {x | g x ≤ r} = ⇑φ ⁻¹' Metric.closedBall 0 r

Tactic:
Metric.closedBall,

Goal state after:
case e'_2
E : Type u_1
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
inst✝⁶ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : BorelSpace E
inst✝³ : T2Space E
inst✝² : ContinuousSMul ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
inst✝ : Nontrivial E
r : ℝ
F : Type u_1 := E
p : AddGroupNorm F := { toFun := g, map_zero' := h1, add_le' := h3, neg' := h2, eq_zero_of_map_eq_zero' := ⋯ }
this✝³ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝² : NormedSpace ℝ F := { toModule := inst✝⁷, norm_smul_le := ⋯ }
this✝¹ : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
⊢ {x | g x ≤ r} = ⇑φ ⁻¹' {y | dist y 0 ≤ r}
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case refine_4
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
u : E
hu : u ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ ∀ v ∈ tangentConeAt ℝ (range ⇑φ) x, ((dω x) u) v = ((dω x) v) u

Tactic:
v

Goal state after:
case refine_4
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
u : E
hu : u ∈ tangentConeAt ℝ (range ⇑φ) x
v : E
⊢ v ∈ tangentConeAt ℝ (range ⇑φ) x → ((dω x) u) v = ((dω x) v) u
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : IsTightMeasureSet S
y : E
this : ProperSpace E
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪y, x⟫_𝕜‖}) atTop (𝓝 0)

Tactic:
rw [isTightMeasureSet_iff_tendsto_measure_norm_gt] at h

Goal state after:
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪y, x⟫_𝕜‖}) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : Set X
hs : MeasurableSet s
ε : ℝ≥0
hε : 0 < ε
h : preVariationFun f s ≠ ∞
ε' : ℝ≥0 := min ε (preVariationFun f s).toNNReal
hε' : ↑ε' ≤ preVariationFun f s
this✝ : ε' ≤ ε
hw : preVariationFun f s ≠ 0
this : 0 < ε'
a : ℝ≥0∞ := preVariationFun f s - ↑ε'
ha : a < preVariationFun f s
P : Finpartition ⟨s, hs⟩
hP : a < ∑ p ∈ P.parts, f ↑p
⊢ a + ↑ε' ≤ ∑ p ∈ P.parts, f ↑p + ↑ε'

Tactic:
by
        exact (ENNReal.add_le_add_iff_right coe_ne_top).mpr (le_of_lt hP)

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

Goal state before:
pp: case h₁
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (f n ∆ t) < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n ↦ (μ (accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), μ.real (f n ∆ g n) < ε / (2 * (↑N + 1))
⊢ dist (μ (⋃ n, f n)).toReal (μ (⋃ n ∈ Finset.range (N + 1), f n)).toReal < ε / 2

Tactic:
simp only [Finset.mem_range, Nat.lt_add_one_iff]

Goal state after:
case h₁
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (f n ∆ t) < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n ↦ (μ (accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), μ.real (f n ∆ g n) < ε / (2 * (↑N + 1))
⊢ dist (μ (⋃ n, f n)).toReal (μ (⋃ n, ⋃ (_ : n ≤ N), f n)).toReal < ε / 2
Module: Mathlib.MeasureTheory.Measure.WithDensityFinite

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
t : Set α
ht : MeasurableSet t
⊢ ∞ * μ (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) = ∞ * μ.toFinite (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ)

Tactic:
by_cases hμt : μ (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) = 0

Goal state after:
case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
t : Set α
ht : MeasurableSet t
hμt : μ (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) = 0
⊢ ∞ * μ (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) = ∞ * μ.toFinite (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ)

case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
t : Set α
ht : MeasurableSet t
hμt : ¬μ (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) = 0
⊢ ∞ * μ (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ) = ∞ * μ.toFinite (t ∩ (μ.sigmaFiniteSetWRT μ)ᶜ)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsZeroOneMeasure μ
inst✝¹ : StandardBorelSpace α
inst✝ : NeZero μ
this : IsProbabilityMeasure μ
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y
⊢ ∃ x₀, μ = Measure.dirac x₀

Tactic:
let B := fun n => if h : μ (A n) = 1 then A n else (A n)ᶜ

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsZeroOneMeasure μ
inst✝¹ : StandardBorelSpace α
inst✝ : NeZero μ
this : IsProbabilityMeasure μ
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y
B : ℕ → Set α := fun n ↦ if h : μ (A n) = 1 then A n else (A n)ᶜ
⊢ ∃ x₀, μ = Measure.dirac x₀
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case ac
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : IsSetAlgebra 𝒜
S : μ.FiniteSpanningSetsIn 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
T : ℕ → Set X := accumulate S.set
T_mem : ∀ (n : ℕ), T n ∈ 𝒜
T_finite : ∀ (n : ℕ), μ (T n) < ∞
T_spanning : ⋃ n, T n = univ
mono : Monotone fun n ↦ T n ∩ s
this✝ : Filter.Tendsto (fun n ↦ ((⇑μ ∘ fun n ↦ T n ∩ s) n).toReal) Filter.atTop (nhds (μ (⋃ n, T n ∩ s)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist ((⇑μ ∘ fun n ↦ T n ∩ s) n).toReal (μ (⋃ n, T n ∩ s)).toReal < ε / 2
this : Fact (μ (T N) < ∞)
t : Set X
t_mem : t ∈ 𝒜
ht : (μ.restrict (T N)) (s ∆ t) < ENNReal.ofReal (ε / 2)
⊢ μ s - μ (s ∩ T N) < ENNReal.ofReal (ε / 2)

Tactic:
lt_ofReal_iff_toReal_lt (sub_ne_top hμs),

Goal state after:
case ac
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : IsSetAlgebra 𝒜
S : μ.FiniteSpanningSetsIn 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
T : ℕ → Set X := accumulate S.set
T_mem : ∀ (n : ℕ), T n ∈ 𝒜
T_finite : ∀ (n : ℕ), μ (T n) < ∞
T_spanning : ⋃ n, T n = univ
mono : Monotone fun n ↦ T n ∩ s
this✝ : Filter.Tendsto (fun n ↦ ((⇑μ ∘ fun n ↦ T n ∩ s) n).toReal) Filter.atTop (nhds (μ (⋃ n, T n ∩ s)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist ((⇑μ ∘ fun n ↦ T n ∩ s) n).toReal (μ (⋃ n, T n ∩ s)).toReal < ε / 2
this : Fact (μ (T N) < ∞)
t : Set X
t_mem : t ∈ 𝒜
ht : (μ.restrict (T N)) (s ∆ t) < ENNReal.ofReal (ε / 2)
⊢ (μ s - μ (s ∩ T N)).toReal < ε / 2
Module: Mathlib.MeasureTheory.SpecificCodomains.Pi

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p : ℝ≥0∞
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
f : X → E × F
h : MemLp (fun x ↦ (f x).1) p μ ∧ MemLp (fun x ↦ (f x).2) p μ
⊢ MemLp f p μ

Tactic:
have : f = (AddMonoidHom.inl E F) ∘ (fun x ↦ (f x).fst) +
        (AddMonoidHom.inr E F) ∘ (fun x ↦ (f x).snd) := by
      ext; all_goals simp

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p : ℝ≥0∞
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
f : X → E × F
h : MemLp (fun x ↦ (f x).1) p μ ∧ MemLp (fun x ↦ (f x).2) p μ
this : f = (⇑(AddMonoidHom.inl E F) ∘ fun x ↦ (f x).1) + ⇑(AddMonoidHom.inr E F) ∘ fun x ↦ (f x).2
⊢ MemLp f p μ
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Defs

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
V : Type u_2
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : ℕ → { t // MeasurableSet t }
hs : Pairwise (Function.onFun Disjoint (Subtype.val ∘ s))
⊢ (fun x ↦ ‖μ x‖ₑ) (⋃ i, ↑(s i)) ≤ ∑' (i : ℕ), (fun x ↦ ‖μ x‖ₑ) ↑(s i)

Tactic:
have hmeas : ∀ i, MeasurableSet (s i).val := fun i => (s i).prop

Goal state after:
X : Type u_1
mX : MeasurableSpace X
V : Type u_2
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : ℕ → { t // MeasurableSet t }
hs : Pairwise (Function.onFun Disjoint (Subtype.val ∘ s))
hmeas : ∀ (i : ℕ), MeasurableSet ↑(s i)
⊢ (fun x ↦ ‖μ x‖ₑ) (⋃ i, ↑(s i)) ≤ ∑' (i : ℕ), (fun x ↦ ‖μ x‖ₑ) ↑(s i)
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: case neg
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
E : Set X
hE : ¬MeasurableSet E
⊢ ‖μ E‖ₑ ≤ μ.variation E

pp: case pos
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
E : Set X
hE : MeasurableSet E
⊢ ‖μ E‖ₑ ≤ μ.variation E

Tactic:
· simp [hE]

Goal state after:
case pos
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
E : Set X
hE : MeasurableSet E
⊢ ‖μ E‖ₑ ≤ μ.variation E
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup V
μ : VectorMeasure X V
𝕜 : Type u_3
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
c : 𝕜
hc : c ≠ 0
⊢ ‖c‖₊ • (c⁻¹ • c • μ).variation ≤ ‖c‖₊ • ‖c⁻¹‖₊ • (c • μ).variation

Tactic:
by
    gcongr
    exact variation_smul_le

Goal state after: