mathlib-initiative/mathlib-tactics huggingface dataset:
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)
⊢ (f * g n) x ≤ ((y n + ε') • g n) x

Tactic:
apply mul_le_mul_of_nonneg_right ?_ (hg.2.2.1 n x).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✝ : 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 x ≤ y n + ε'
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case neg
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:
· simp [image_eq_zero_of_notMem_tsupport hx]

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

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
⊢ (∀ (x : X) (K : Set X), IsClosed[inst✝] K → x ∉ K → ∃ f, Continuous[inst✝, _] f ∧ f x = 0 ∧ EqOn f 1 K) = ?m.25

Tactic:
compl_surjective.forall,

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
⊢ (∀ (x : X) (x_1 : Set X), IsClosed[inst✝] x_1ᶜ → x ∉ x_1ᶜ → ∃ f, Continuous[inst✝, _] f ∧ f x = 0 ∧ EqOn f 1 x_1ᶜ) =
    ?m.25
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
⊢ ∀ {s : Set X} {a : X}, IsClosed[inst✝¹] s → a ∉ s → Disjoint (𝓝ˢ s) (𝓝 a)

Tactic:
intro s

Goal state after:
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
s : Set X
⊢ ∀ {a : X}, IsClosed[inst✝¹] s → a ∉ s → Disjoint (𝓝ˢ s) (𝓝 a)
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureProd

Goal state before:
pp: case refine_1
α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
μ : ProbabilityMeasure α × ProbabilityMeasure β
S : Set (Set (α × β)) :=
  {t | ∃ a b, MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0 ∧ t = a ×ˢ b}
a : Set α
b : Set β
ameas : MeasurableSet a
ha : μ.1 (frontier a) = 0
bmeas : MeasurableSet b
hb : μ.2 (frontier b) = 0
a' : Set α
b' : Set β
a'meas : MeasurableSet a'
ha' : μ.1 (frontier a') = 0
b'meas : MeasurableSet b'
hb' : μ.2 (frontier b') = 0
⊢ μ.1 (frontier (a ∩ a')) = 0

pp: case refine_2
α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
μ : ProbabilityMeasure α × ProbabilityMeasure β
S : Set (Set (α × β)) :=
  {t | ∃ a b, MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0 ∧ t = a ×ˢ b}
a : Set α
b : Set β
ameas : MeasurableSet a
ha : μ.1 (frontier a) = 0
bmeas : MeasurableSet b
hb : μ.2 (frontier b) = 0
a' : Set α
b' : Set β
a'meas : MeasurableSet a'
ha' : μ.1 (frontier a') = 0
b'meas : MeasurableSet b'
hb' : μ.2 (frontier b') = 0
⊢ μ.2 (frontier (b ∩ b')) = 0

Tactic:
· rw [null_iff_toMeasure_null] at ha ha' ⊢
      exact null_frontier_inter ha ha'

Goal state after:
case refine_2
α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
μ : ProbabilityMeasure α × ProbabilityMeasure β
S : Set (Set (α × β)) :=
  {t | ∃ a b, MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0 ∧ t = a ×ˢ b}
a : Set α
b : Set β
ameas : MeasurableSet a
ha : μ.1 (frontier a) = 0
bmeas : MeasurableSet b
hb : μ.2 (frontier b) = 0
a' : Set α
b' : Set β
a'meas : MeasurableSet a'
ha' : μ.1 (frontier a') = 0
b'meas : MeasurableSet b'
hb' : μ.2 (frontier b') = 0
⊢ μ.2 (frontier (b ∩ b')) = 0
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureProd

Goal state before:
pp: case h.refine_1
α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
μ : ProbabilityMeasure α × ProbabilityMeasure β
S : Set (Set (α × β)) :=
  {t | ∃ a b, MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0 ∧ t = a ×ˢ b}
this✝¹ : IsPiSystem S
this✝ : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
u : Set (α × β)
u_open : IsOpen[instTopologicalSpaceProd] u
x : α × β
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ℝ
hr : r ∈ Ioo 0 ε
r' : ℝ
hr' : r' ∈ Ioo 0 ε
μr : ↑μ.1 (frontier (ball x.1 r)) = 0
μr' : ↑μ.2 (frontier (ball x.2 r')) = 0
⊢ ball x.1 r ×ˢ ball x.2 r' ∈ 𝓝 x

Tactic:
exact (isOpen_ball.prod isOpen_ball).mem_nhds (by simp [hr.1, hr'.1])

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: case neg
A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
b : B
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
hb : U₀ ∈ nhds b
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
hS' : ¬μA.real S = 0
⊢ ∃ δ > 0, ENNReal.ofReal δ * ENNReal.ofReal (μA.real S) < ENNReal.ofReal ε

Tactic:
refine ⟨ε / 2 / μA.real S, by positivity, ?_⟩
          rwa [← ENNReal.ofReal_mul' measureReal_nonneg, ENNReal.ofReal_lt_ofReal_iff hε,
            div_mul_cancel₀ _ hS', half_lt_self_iff]

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
⊢ ∃ x ∈ {μ | μ.mass ≤ C}, ↑f ≤ 𝓝 x

Tactic:
have L (g : C_c(E, ℝ)) :
      ∃ x ∈ Icc (-C * ‖g.toBoundedContinuousFunction‖) (C * ‖g.toBoundedContinuousFunction‖),
      Tendsto (fun (μ : FiniteMeasure E) ↦ ∫ x, g x ∂μ) f (𝓝 x) := by
    simp only [Tendsto, ← Ultrafilter.coe_map]
    apply IsCompact.ultrafilter_le_nhds' isCompact_Icc
    simp only [neg_mul, Ultrafilter.mem_map]
    filter_upwards [hf] with μ hμ
    simp only [mem_preimage, mem_Icc]
    refine ⟨?_, ?_⟩
    · calc - (C * ‖g.toBoundedContinuousFunction‖)
      _ ≤ ∫ (x : E), - ‖g.toBoundedContinuousFunction‖ ∂μ := by
        simp only [integral_const, smul_eq_mul, mul_neg, neg_le_neg_iff]
        gcongr
        exact hμ
      _ ≤ ∫ x, g x ∂μ := by
        gcongr
        · simp
        · exact g.continuous.integrable_of_hasCompactSupport g.hasCompactSupport
        · intro x
          apply neg_le_of_abs_le
          exact g.toBoundedContinuousFunction.norm_coe_le_norm x
    · calc ∫ x, g x ∂μ
      _ ≤ ∫ (x : E), ‖g.toBoundedContinuousFunction‖ ∂μ := by
        gcongr
        · exact g.continuous.integrable_of_hasCompactSupport g.hasCompactSupport
        · simp
        · intro x
          apply le_of_abs_le
          exact g.toBoundedContinuousFunction.norm_coe_le_norm x
      _ ≤ C * ‖g.toBoundedContinuousFunction‖ := by
        simp only [integral_const, smul_eq_mul]
        gcongr
        exact hμ

Goal state after:
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
L :
  ∀ (g : E →C_c ℝ),
    ∃ x ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖),
      Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 x)
⊢ ∃ x ∈ {μ | μ.mass ≤ C}, ↑f ≤ 𝓝 x
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p
eq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℝ), (∑ i, |x i| ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ i, |0 i| ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℝ), (∑ i, |(-x) i| ^ p) ^ (1 / p) = (∑ i, |x i| ^ p) ^ (1 / p)
nm_add : ∀ (x y : ι → ℝ), (∑ i, |(x + y) i| ^ p) ^ (1 / p) ≤ (∑ i, |x i| ^ p) ^ (1 / p) + (∑ i, |y i| ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℝ), (∑ i, |(r • x) i| ^ p) ^ (1 / p) ≤ |r| * (∑ i, |x i| ^ p) ^ (1 / p)
x✝ : ι → ℝ
⊢ 0 ≤ ∑ i, |x✝ i| ^ p

Tactic:
exact Finset.sum_nonneg' (fun _ => rpow_nonneg (abs_nonneg _) _)

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

Goal state before:
pp: case h₂
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
C : ℝ≥0
K : Set E
hK : IsCompact K
f : ↑K → E := ⋯
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := ⋯
T : Set (FiniteMeasure ↑K) := ⋯
⊢ F '' T ⊆ {μ | μ.mass ≤ C ∧ μ Kᶜ = 0}

Tactic:
simp only [null_iff_toMeasure_null, image_subset_iff, preimage_ofPred_eq, toMeasure_map,
        ofPred_subset_ofPred, F, T]

Goal state after:
case h₂
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
C : ℝ≥0
K : Set E
hK : IsCompact K
f : ↑K → E := Subtype.val
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f
T : Set (FiniteMeasure ↑K) := {μ | μ.mass ≤ C}
⊢ ∀ (a : FiniteMeasure ↑K), a.mass ≤ C → (a.map f).mass ≤ C ∧ (Measure.map f ↑a) Kᶜ = 0
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
C : ℝ≥0
K : Set E
hK : IsCompact K
f : ↑K → E := Subtype.val
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f
T : Set (FiniteMeasure ↑K) := {μ | μ.mass ≤ C}
μ : FiniteMeasure ↑K
hμ : μ.mass ≤ C
⊢ ↑μ (f ⁻¹' Kᶜ) = 0

Tactic:
simp [f]

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

Goal state before:
pp: case e'_3.e'_5
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℂ), (∑ i, ‖x i‖ ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ i, ‖0 i‖ ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℂ), (∑ i, ‖(-x) i‖ ^ p) ^ (1 / p) = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
nm_add : ∀ (x y : ι → ℂ), (∑ i, ‖(x + y) i‖ ^ p) ^ (1 / p) ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p) + (∑ i, ‖y i‖ ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℂ), (∑ i, ‖(r • x) i‖ ^ p) ^ (1 / p) ≤ ‖r‖ * (∑ i, ‖x i‖ ^ p) ^ (1 / p)
⊢ (π * Real.Gamma (2 / p + 1)) ^ card ι = (∫ (x : ℂ), rexp (-‖x‖ ^ p)) ^ card ι

Tactic:
Complex.integral_exp_neg_rpow hp

Goal state after:
case e'_3.e'_5
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℂ), (∑ i, ‖x i‖ ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ i, ‖0 i‖ ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℂ), (∑ i, ‖(-x) i‖ ^ p) ^ (1 / p) = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
nm_add : ∀ (x y : ι → ℂ), (∑ i, ‖(x + y) i‖ ^ p) ^ (1 / p) ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p) + (∑ i, ‖y i‖ ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℂ), (∑ i, ‖(r • x) i‖ ^ p) ^ (1 / p) ≤ ‖r‖ * (∑ i, ‖x i‖ ^ p) ^ (1 / p)
⊢ (π * Real.Gamma (2 / p + 1)) ^ card ι = (π * Real.Gamma (2 / p + 1)) ^ card ι
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
this✝¹ : Tendsto (fun n ↦ ∑ i ∈ Finset.range (n + 1), ↑(ν i) univ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) univ))
n : ℕ
this✝ : ∑ i ∈ Finset.range (n + 1), ↑(ν i) univ = ↑(∑ i ∈ Finset.range (n + 1), ν i) univ
this :
  Tendsto (fun μ ↦ (∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i)).mass) (↑f)
    (𝓝 (∑ i ∈ Finset.range (n + 1), ν i).mass)
μ : FiniteMeasure E
hμ : μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n
⊢ μ ((partialSups K) n) ≤ C

Tactic:
exact le_trans (apply_mono _ (subset_univ _)) hμ.1

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case refine_4.a
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) := ⋯
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
u : E
hu : u ∈ tangentConeAt ℝ (range ⇑φ) x
v : E
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ x ∈ s

Tactic:
← hφs

Goal state after:
case refine_4.a
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
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ x ∈ range ⇑φ
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case refine_4.a
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) := ⋯
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
u : E
hu : u ∈ tangentConeAt ℝ (range ⇑φ) x
v : E
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ u ∈ tangentConeAt ℝ s x

Tactic:
← hφs

Goal state after:
case refine_4.a
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
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ u ∈ tangentConeAt ℝ (range ⇑φ) x
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case refine_4.a
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) := ⋯
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
u : E
hu : u ∈ tangentConeAt ℝ (range ⇑φ) x
v : E
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ v ∈ tangentConeAt ℝ s x

Tactic:
← hφs

Goal state after:
case refine_4.a
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
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ v ∈ tangentConeAt ℝ (range ⇑φ) x
Module: Mathlib.MeasureTheory.Measure.MeasuredSets

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
⊢ ∀ t ∈ C, ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t_1 ∈ C, μ (t_1 ∆ t) < ε

Tactic:
intro s

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s✝ : Set α
hs : MeasurableSet s✝
ε : ℝ≥0∞
hε : 0 < ε
s : Set α
⊢ s ∈ C → ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ C, μ (t ∆ s) < ε
Module: Mathlib.MeasureTheory.Measure.MeasuredSets

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
⊢ ∀ (t : Set α),
    MeasurableSet t →
      (∀ (ε : ℝ≥0∞), 0 < ε → ∃ t_1 ∈ C, μ (t_1 ∆ t) < ε) → ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t_1 ∈ C, μ (t_1 ∆ tᶜ) < ε

Tactic:
intro s

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s✝ : Set α
hs : MeasurableSet s✝
ε : ℝ≥0∞
hε : 0 < ε
s : Set α
⊢ MeasurableSet s → (∀ (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ C, μ (t ∆ s) < ε) → ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ C, μ (t ∆ sᶜ) < ε