mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: 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 ∈ Ioo 0 1, ∀ b ∈ Ioo 0 1, φ (a, b) ∈ t
hω : ∀ a ∈ Ioo 0 1, ∀ b ∈ Ioo 0 1, (a, b) ∉ s → HasFDerivWithinAt ω (dω (φ (a, b))) t (φ (a, b))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a ∈ Ioo 0 1,
    ∀ b ∈ Ioo 0 1,
      (a, b) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a, b)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a, b)), ((dω (φ (a, b))) u) v = ((dω (φ (a, b))) 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 b : ↑I), ψ (↑a, ↑b) = φ (a, b)
hψ_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ψ
a b : ℝ
ha : (a, b).1 ∈ Ioo 0 1
hb : (a, b).2 ∈ Ioo 0 1
⊢ ψ (a, b) ∈ t

Tactic:
lift a to I using Ioo_subset_Icc_self ha

Goal state after:
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 ∈ Ioo 0 1, ∀ b ∈ Ioo 0 1, φ (a, b) ∈ t
hω : ∀ a ∈ Ioo 0 1, ∀ b ∈ Ioo 0 1, (a, b) ∉ s → HasFDerivWithinAt ω (dω (φ (a, b))) t (φ (a, b))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a ∈ Ioo 0 1,
    ∀ b ∈ Ioo 0 1,
      (a, b) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a, b)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a, b)), ((dω (φ (a, b))) u) v = ((dω (φ (a, b))) 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 b : ↑I), ψ (↑a, ↑b) = φ (a, b)
hψ_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ψ
b : ℝ
a : { x // x ∈ I }
ha : (↑a, b).1 ∈ Ioo 0 1
hb : (↑a, b).2 ∈ Ioo 0 1
⊢ ψ (↑a, b) ∈ t
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
ε : ℝ≥0
εpos : 0 < ε
α : ℝ≥0
α_hyp : α < rieszContentAux Λ K + ε
f : X →C_c ℝ≥0
f_hyp : f ∈ {f | ∀ x ∈ K, 1 ≤ f x} ∧ Λ f = α
⊢ Λ f < rieszContentAux Λ K + ε

Tactic:
rw [f_hyp.right]

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
ε : ℝ≥0
εpos : 0 < ε
α : ℝ≥0
α_hyp : α < rieszContentAux Λ K + ε
f : X →C_c ℝ≥0
f_hyp : f ∈ {f | ∀ x ∈ K, 1 ≤ f x} ∧ Λ f = α
⊢ α < rieszContentAux Λ K + ε
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)
x : ℝ
hx : x ∈ Set.Icc (ak k) (ak (k + 1))
⊢ _root_.derivWithin f (Set.Icc a b) x = _root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))) x

Tactic:
← derivWithin_subset h3 (uniqueDiffOn_Icc h4 x hx) (h_df 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)
x : ℝ
hx : x ∈ Set.Icc (ak k) (ak (k + 1))
⊢ _root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))) x = _root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))) x
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
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
p : NNReal
hp : Λ f = ↑p
⊢ (rieszContent (toNNRealLinear Λ)).toFun { carrier := K, isCompact' := hK } ≤ p

Tactic:
apply csInf_le'

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
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
p : NNReal
hp : Λ f = ↑p
⊢ p ∈ ⇑(toNNRealLinear Λ) '' {f | ∀ x ∈ { carrier := K, isCompact' := hK }, 1 ≤ f x}
Module: Mathlib.MeasureTheory.Integral.TorusIntegral

Goal state before:
pp: n : ℕ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : (Fin n → ℂ) → E
c : Fin n → ℂ
R : Fin n → ℝ
hf : TorusIntegrable f c R
hg : TorusIntegrable g c R
⊢ (∯ (x : Fin n → ℂ) in T(c, R), f x + g x) = (∯ (x : Fin n → ℂ) in T(c, R), f x) + ∯ (x : Fin n → ℂ) in T(c, R), g x

Tactic:
simpa only [torusIntegral, smul_add, Pi.add_apply] using
    integral_add hf.function_integrable hg.function_integrable

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

Goal state before:
pp: 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)
n m : Fin N
h : n < m
⊢ a ≤ a

Tactic:
by rfl

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

Goal state before:
pp: 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)
n m : Fin N
h : n < m
⊢ ε ≤ ε

Tactic:
by rfl

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: case inl
α : Type u
s : Set (Set α)
i : Ordinal.{u_1}
x : Set α
hx : x ∈ generateMeasurableRec s i
h : i ≤ i
⊢ x ∈ generateMeasurableRec s i

Tactic:
exact hx

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

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
X : Type u_3
mX : MeasurableSpace X
μ : Measure X
inst✝ : BorelSpace E
f : X → E
hf : AEMeasurable f μ
r : ℝ
t : E
⊢ charFun (Measure.map (fun x ↦ r • f x) μ) t = charFun (Measure.map f μ) (r • t)

Tactic:
by
  rw [show (fun x ↦ r • (f x)) = (r • ·) ∘ f from rfl, ← AEMeasurable.map_map_of_aemeasurable,
    charFun_map_smul]
  all_goals fun_prop

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

Goal state before:
pp: case h.right.right
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
I' : Finset ι
V U : ↥I' → Set X
hUV : ∀ (i : ↥I'), U i ⊆ V i
fs : ↥I' → X → ↑I
hfs : ∀ (i : ↥I'), Continuous[t ↑i, _] (fs i)
hxfs : ∀ (i : ↥I'), fs i x✝ = 0
x : X
i : ↥I'
hfsU : fs i x = ⊤
hxi : x ∈ (V i)ᶜ
⊢ ∃ b, fs b x = ⊤

Tactic:
exists i

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

Goal state before:
pp: case mp
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
⊢ Tendsto diracProba L (𝓝 (diracProba x)) → Tendsto id L (𝓝 x)

pp: case mpr
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
⊢ Tendsto id L (𝓝 x) → Tendsto diracProba L (𝓝 (diracProba x))

Tactic:
· contrapose
    exact not_tendsto_diracProba_of_not_tendsto L

Goal state after:
case mpr
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
⊢ Tendsto id L (𝓝 x) → Tendsto diracProba L (𝓝 (diracProba x))
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureProd

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
ν : FiniteMeasure β
⊢ prod 0 ν = 0

Tactic:
rw [← mass_zero_iff, mass_prod, zero_mass, zero_mul]

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

Goal state before:
pp: case hf
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
intble₁ : IntegrableOn (fun t ↦ μ.real {a | t ≤ f a}) (Ioc 0 ‖f‖) volume
intble₂ : IntegrableOn (fun t ↦ ν.real (thickening ε {a | t ≤ f a})) (Ioc 0 ‖f‖) volume
⊢ Integrable (fun x ↦ ν.real (thickening ε {a | x ≤ f a})) (volume.restrict (Ioc 0 ‖f‖))

pp: case hg
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
intble₁ : IntegrableOn (fun t ↦ μ.real {a | t ≤ f a}) (Ioc 0 ‖f‖) volume
intble₂ : IntegrableOn (fun t ↦ ν.real (thickening ε {a | t ≤ f a})) (Ioc 0 ‖f‖) volume
⊢ Integrable (fun x ↦ ε) (volume.restrict (Ioc 0 ‖f‖))

Tactic:
· exact intble₂

Goal state after:
case hg
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
intble₁ : IntegrableOn (fun t ↦ μ.real {a | t ≤ f a}) (Ioc 0 ‖f‖) volume
intble₂ : IntegrableOn (fun t ↦ ν.real (thickening ε {a | t ≤ f a})) (Ioc 0 ‖f‖) volume
⊢ Integrable (fun x ↦ ε) (volume.restrict (Ioc 0 ‖f‖))
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: case inr
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
key : (Function.support fun a ↦ f (b * φ a)).Subsingleton
a : A
ha : (Function.support fun a ↦ f (b * φ a)) = {a}
⊢ 0 < (μA {1}).toReal

Tactic:
exact ENNReal.toReal_pos ho.ne' ht.ne

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 := 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.map f).mass ≤ C ∧ ↑μ (f ⁻¹' Kᶜ) = 0

Tactic:
refine ⟨(mass_map_le _ _).trans hμ, by simp [f]⟩

Goal state after:
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 → 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)
nm_zero : ‖toLp (ENNReal.ofReal p) 0‖ = 0
⊢ volume {x | ∑ i, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Tactic:
have eq_zero (x : ι → ℂ) : ‖toLp (.ofReal p) x‖ = 0 ↔ x = 0 :=
    norm_eq_zero.trans (toLp_eq_zero _)

Goal state after:
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)
nm_zero : ‖toLp (ENNReal.ofReal p) 0‖ = 0
eq_zero : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = 0 ↔ x = 0
⊢ volume {x | ∑ i, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case e'_2
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
this :
  ∫ᶜ (x : E) in Path.segment a b, ω x + ∫ᶜ (x : E) in φ.evalAt 1, ω x =
    ∫ᶜ (x : E) in Path.segment a c, ω x + ∫ᶜ (x : E) in φ.evalAt 0, ω x
⊢ ∫ᶜ (x : E) in Path.segment b c, ω x = ∫ᶜ (x : E) in Path.segment ((lineMap a b) 1) ((lineMap a c) 1), ω x

Tactic:
rw [← Path.cast_segment (lineMap_apply_one a b) (lineMap_apply_one a c), curveIntegral_cast]

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
eq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖toLp (ENNReal.ofReal p) 0‖ = 0
⊢ volume {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) ≤ r} =
    ENNReal.ofReal r ^ (2 * card ι) *
      ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Tactic:
have eq_zero (x : ι → ℂ) : ‖toLp (.ofReal p) x‖ = 0 ↔ x = 0 :=
    norm_eq_zero.trans (toLp_eq_zero _)

Goal state after:
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
eq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖toLp (ENNReal.ofReal p) 0‖ = 0
eq_zero : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = 0 ↔ x = 0
⊢ volume {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) ≤ r} =
    ENNReal.ofReal r ^ (2 * card ι) *
      ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: case h
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_2
ι : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype ι
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis ι 𝕜 E
h : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪b i, x⟫_𝕜‖}) atTop (𝓝 0)
hE : Nontrivial E
h_rank : 0 < ↑(Fintype.card ι)
this✝ : Nonempty ι
this : ProperSpace E
r : ℝ
μ : Measure E
hμS : μ ∈ S
x : E
hx : r < ‖x‖
hx' : r < √↑(Fintype.card ι) * ⨆ i, ‖⟪b i, x⟫_𝕜‖
⊢ ∃ i, r / √↑(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖

Tactic:
rw [← div_lt_iff₀' (by positivity)] at hx'

Goal state after:
case h
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_2
ι : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype ι
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis ι 𝕜 E
h : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪b i, x⟫_𝕜‖}) atTop (𝓝 0)
hE : Nontrivial E
h_rank : 0 < ↑(Fintype.card ι)
this✝ : Nonempty ι
this : ProperSpace E
r : ℝ
μ : Measure E
hμS : μ ∈ S
x : E
hx : r < ‖x‖
hx' : r / √↑(Fintype.card ι) < ⨆ i, ‖⟪b i, x⟫_𝕜‖
⊢ ∃ i, r / √↑(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
hf : IsSigmaSubadditiveSetFun f
hf' : f ∅ = 0
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
b : ℝ≥0∞
Q : Finpartition ⟨⋃ i, s i, ⋯⟩
hQ : b < ∑ p ∈ Q.parts, f ↑p
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
P : (i : ℕ) → Finpartition (s' i) := fun i ↦ Q.restrict ⋯
⊢ ∑' (i : ℕ), ∑ q ∈ Q.parts, f (↑q ⊓ ↑(s' i)) = ∑' (i : ℕ), ∑ p ∈ (P i).parts, f ↑p

Tactic:
by
          congr 1; funext i
          exact (Q.sum_restrict _ (fun p => f p) hf').symm

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : NormedAlgebra 𝕜 A
m𝕜 : MeasurableSpace 𝕜
μ : Measure 𝕜
⊢ resolventTransform μ = fun a ↦ ∫ (x : 𝕜), resolvent a x ∂μ

Tactic:
by rfl

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : NormedAlgebra 𝕜 A
m𝕜 : MeasurableSpace 𝕜
μ : Measure 𝕜
a : A
⊢ resolventTransform μ a = ∫ (x : 𝕜), resolvent a x ∂μ

Tactic:
by rfl

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
⊢ ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2) ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ

Tactic:
grw [ball_subset_ball, ball_infDist_subset_compl]

Goal state after:
case h
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
⊢ infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2 ≤ infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: case inr.inr.inl
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s = 1
hμt : μ t = 1
h✝ : μ (s ∩ t) = 0
⊢ μ (s ∩ t)ᶜ ≤ 0

Tactic:
calc
  _ = μ (sᶜ ∪ tᶜ) := by simp [compl_inter]
  _ ≤ μ sᶜ + μ tᶜ := measure_union_le _ _
  _ ≤ 0 := by
    rw [measure_compl hs (by simp), measure_univ hμs, hμs, tsub_self,
      measure_compl ht (by simp), measure_univ hμt, hμt, tsub_self]
    simp

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_sigma_subadd : m.IsSigmaSubadditive
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
⊢ (OuterMeasure.ofFunction ⇑m ⋯) s = m s

Tactic:
refine le_antisymm (OuterMeasure.ofFunction_le s) ?_

Goal state after:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_sigma_subadd : m.IsSigmaSubadditive
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
⊢ m s ≤ (OuterMeasure.ofFunction ⇑m ⋯) s
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup V
μ ν : VectorMeasure X V
inst✝ : Finite X
this : Fintype X := Fintype.ofFinite X
⊢ μ.variation Set.univ < ∞

Tactic:
simp only [variation_apply, preVariation_apply, MeasurableSet.univ, ennrealToMeasure_apply,
      ennrealPreVariation_apply, preVariationFun, ↓reduceDIte, ← sup_univ_eq_ciSup]

Goal state after:
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup V
μ ν : VectorMeasure X V
inst✝ : Finite X
this : Fintype X := Fintype.ofFinite X
⊢ (Finset.univ.sup fun P ↦ ∑ x ∈ P.parts, ‖μ ↑x‖ₑ) < ∞
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case insert
ι : Type u_1
X : Type u_2
E : Type u_3
F : Type u_4
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
μ : VectorMeasure X F
f : X → E
s : Set ι
t : ι → Set X
a✝¹ : ι
s✝ : Set ι
a✝ : a✝¹ ∉ s✝
h's : s✝.Finite
hf : (∀ i ∈ s✝, MeasurableSet (t i)) → (∀ i ∈ s✝, μ.IntegrableOn f (t i)) → μ.IntegrableOn f (⋃ i ∈ s✝, t i)
ht : ∀ i ∈ insert a✝¹ s✝, MeasurableSet (t i)
h't : ∀ i ∈ insert a✝¹ s✝, μ.IntegrableOn f (t i)
⊢ μ.IntegrableOn f (⋃ i ∈ insert a✝¹ s✝, t i)

Tactic:
| insert _ h's hf =>

Goal state after: