mathlib-initiative/mathlib-tactics huggingface dataset:
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,
    (∀ (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)

Tactic:
by

Goal state after:
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_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)
this :
  (((∫ (x : ℝ) in 0..1, g (x, 1)) - ∫ (x : ℝ) in 0..1, g (x, 0)) + ∫ (y : ℝ) in 0..1, f (1, y)) -
      ∫ (y : ℝ) in 0..1, f (0, y) =
    0
hfi :
  ∀ (s : ↑I),
    ∫ (t : ℝ) in 0..1, f (↑s, t) = ∫ᶜ (x : E) in { toContinuousMap := φ.curry s, source' := ⋯, target' := ⋯ }, ω x
hf₀ : ∫ (t : ℝ) in 0..1, f (0, t) = ∫ᶜ (x : E) in γ₁, ω x
⊢ ∫ (t : ℝ) in 0..1, f (1, t) = curveIntegral ω γ₂

Tactic:
by

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)
⊢ Λ (∑ n, g n • f) = ∑ n, Λ (g n • f)

Tactic:
by

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, Λ ((y n + ε') • g n) = ∑ n, (y n + ε') * Λ (g n)

Tactic:
by

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 + ε') * Λ (g n) = ∑ n, (|a| + y n + ε') * Λ (g n) - |a| * ∑ n, Λ (g n)

Tactic:
by

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)
⊢ ∫ (x : X), f x ∂μ + ε' * (2 * μ.real K + |a| + b + ε') ≤ ∫ (x : X), f x ∂μ + ε

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
⊢ ContDiff ℝ (↑↑n) (HSMul.hSMul (-(2 * Real.pi)⁻¹))

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
k : ℕ
hk : ↑k ≤ ↑n
⊢ ↑k ≤ ↑n

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
⊢ ContDiff ℝ (↑n) (charFun μ)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ∞
hint : ∀ (k : ℕ), MemLp id (↑k) μ
⊢ ContDiff ℝ (↑n) (HSMul.hSMul (-(2 * Real.pi)⁻¹))

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ∞
hint : ∀ (k : ℕ), MemLp id (↑k) μ
k : ℕ
hk : ↑k ≤ n
⊢ ↑k ≤ ↑k

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ∞
hint : ∀ (k : ℕ), MemLp id (↑k) μ
⊢ ContDiff ℝ (↑n) (charFun μ)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
⊢ AEStronglyMeasurable id μ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] (charFun μ)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
t : E
hint : MemLp id (↑n) μ
x : Fin n → E
h : innerₗ E = ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)
k : ℕ
hk : ↑k ≤ ↑n
⊢ ↑k ≤ ↑n

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
t : E
hint : MemLp id (↑n) μ
x : Fin n → E
h : innerₗ E = ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)
k : ℕ
hk : ↑k ≤ ↑n
⊢ Integrable (fun x ↦ ‖x‖ ^ k * ‖1 x‖) μ

Tactic:
by

Goal state after:
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_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)
this :
  (((∫ (x : ℝ) in 0..1, g (x, 1)) - ∫ (x : ℝ) in 0..1, g (x, 0)) + ∫ (y : ℝ) in 0..1, f (1, y)) -
      ∫ (y : ℝ) in 0..1, f (0, y) =
    0
hfi :
  ∀ (s : ↑I),
    ∫ (t : ℝ) in 0..1, f (↑s, t) = ∫ᶜ (x : E) in { toContinuousMap := φ.curry s, source' := ⋯, target' := ⋯ }, ω x
hf₀ : ∫ (t : ℝ) in 0..1, f (0, t) = ∫ᶜ (x : E) in γ₁, ω x
hf₁ : ∫ (t : ℝ) in 0..1, f (1, t) = curveIntegral ω γ₂
t : ↑I
⊢ ∫ᶜ (x : E) in φ.evalAt t, ω x = -∫ (s : ℝ) in 0..1, g (s, ↑t)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
t : E
hint : MemLp id (↑n) μ
x : Fin n → E
h : innerₗ E = ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)
hint' : ∀ (k : ℕ), ↑k ≤ ↑n → Integrable (fun x ↦ ‖x‖ ^ k * ‖1 x‖) μ
⊢ AEStronglyMeasurable 1 μ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
t : E
hint : MemLp id (↑n) μ
x : Fin n → E
h : innerₗ E = ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)
hint' : ∀ (k : ℕ), ↑k ≤ ↑n → Integrable (fun x ↦ ‖x‖ ^ k * ‖1 x‖) μ
⊢ Integrable (fun v ↦ ‖v‖ ^ n * ‖1 v‖) μ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
t : E
hint : MemLp id (↑n) μ
x : Fin n → E
⊢ (iteratedFDeriv ℝ n (charFun μ) t) x = I ^ n * ∫ (y : E), ↑(∏ i, ⟪y, x i⟫) * cexp (↑⟪y, t⟫ * I) ∂μ

Tactic:
by

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

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
n : ℕ
t : ℝ
hint : MemLp id (↑n) μ
⊢ iteratedDeriv n (charFun μ) t = I ^ n * ∫ (x : ℝ), ↑x ^ n * cexp (↑t * ↑x * I) ∂μ

Tactic:
by

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

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
⊢ iteratedDeriv n (charFun μ) 0 = I ^ n * ↑(∫ (x : ℝ), x ^ n ∂μ)

Tactic:
by

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

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
t : ℝ
k : ℕ
hkn : k ∈ Finset.range (n + 1)
⊢ ↑k ≤ ↑n

Tactic:
by

Goal state after:
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

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

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
t : ℝ
⊢ taylorWithinEval (charFun μ) n univ 0 t = ∑ k ∈ Finset.range (n + 1), (↑k !)⁻¹ * (↑t * I) ^ k * ↑(∫ (x : ℝ), x ^ k ∂μ)

Tactic:
by

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

Goal state before:
pp: F : Type u_2
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace ℝ F
⊢ probCharDual 0 = 1

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
hint : MemLp id 2 (Measure.map X P)
t : ℝ
⊢ taylorWithinEval (charFun (Measure.map X P)) 2 univ 0 t =
    1 + ↑(∫ (x : Ω), X x ∂P) * ↑t * I - ↑(∫ (x : Ω), (X ^ 2) x ∂P) * ↑t ^ 2 / 2

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
t : ℝ
⊢ AEStronglyMeasurable id (Measure.map X P)

Tactic:
by

Goal state after:
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_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)
this :
  (((∫ (x : ℝ) in 0..1, g (x, 1)) - ∫ (x : ℝ) in 0..1, g (x, 0)) + ∫ (y : ℝ) in 0..1, f (1, y)) -
      ∫ (y : ℝ) in 0..1, f (0, y) =
    0
⊢ ∫ᶜ (x : E) in γ₁, ω x + ∫ᶜ (x : E) in φ.evalAt 1, ω x = ∫ᶜ (x : E) in γ₂, ω x + ∫ᶜ (x : E) in φ.evalAt 0, ω x

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
t : ℝ
⊢ taylorWithinEval (charFun (Measure.map X P)) 2 univ 0 t = 1 - ↑t ^ 2 / 2

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
⊢ AEMeasurable X P

Tactic:
by

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

Goal state before:
pp: W : Type u_1
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : Module ℝ W
inst✝⁹ : TopologicalSpace W
e : AddChar ℝ Circle
V : Type u_2
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
inst✝⁶ : PseudoEMetricSpace V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : CompleteSpace V
inst✝² : SecondCountableTopology V
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
he' : e ≠ 1
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
hL : Continuous fun p ↦ (L p.1) p.2
P P' : Measure V
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
h : ∀ (w : W), ∫ (v : V), (char he hL w) v ∂P = ∫ (v : V), (char he hL w) v ∂P'
⊢ P = P'

Tactic:
by

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

Goal state before:
pp: μ : Measure ℝ
t : ℝ
⊢ charFun μ t = ∫ (x : ℝ), cexp (↑t * ↑x * I) ∂μ

Tactic:
by

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

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
μ : Measure E
⊢ charFun μ 0 = ↑(μ.real Set.univ)

Tactic:
by

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

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
t : E
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
⊢ charFun 0 t = 0

Tactic:
by

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

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
μ : Measure E
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
t : E
⊢ charFun μ (-t) = (starRingEnd ℂ) (charFun μ t)

Tactic:
by

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

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
μ : Measure E
t : E
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
⊢ charFun μ t = ∫ (v : E), (innerProbChar t) v ∂μ

Tactic:
by

Goal state after: