mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: case neg
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := (Homeomorph.mulLeft₀ c hc).symm.toMeasurableEquiv
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x ↦ c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal (∫ (x : ℝ) in s', |c⁻¹| * gaussianPDFReal μ v (e x)) =
    (gaussianReal (c * μ) (NNReal.mk (c ^ 2) ⋯ * v)) s'

Tactic:
gaussianReal_apply_eq_integral _ _ s'

Goal state after:
case neg
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := (Homeomorph.mulLeft₀ c hc).symm.toMeasurableEquiv
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x ↦ c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal (∫ (x : ℝ) in s', |c⁻¹| * gaussianPDFReal μ v (e x)) =
    ENNReal.ofReal (∫ (x : ℝ) in s', gaussianPDFReal (c * μ) (NNReal.mk (c ^ 2) ⋯ * v) x)

μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := (Homeomorph.mulLeft₀ c hc).symm.toMeasurableEquiv
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x ↦ c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ NNReal.mk (c ^ 2) ⋯ * v ≠ 0
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := (Homeomorph.mulLeft₀ c hc).symm.toMeasurableEquiv
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x ↦ c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ NNReal.mk (c ^ 2) ⋯ * v ≠ 0

pp: case neg
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := (Homeomorph.mulLeft₀ c hc).symm.toMeasurableEquiv
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x ↦ c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal (∫ (x : ℝ) in s', |c⁻¹| * gaussianPDFReal μ v (e x)) =
    ENNReal.ofReal (∫ (x : ℝ) in s', gaussianPDFReal (c * μ) (NNReal.mk (c ^ 2) ⋯ * v) x)

Tactic:
· simp only [ne_eq, mul_eq_zero, hv, or_false]
    rw [← NNReal.coe_inj]
    simp [hc]

Goal state after:
case neg
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := (Homeomorph.mulLeft₀ c hc).symm.toMeasurableEquiv
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x ↦ c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal (∫ (x : ℝ) in s', |c⁻¹| * gaussianPDFReal μ v (e x)) =
    ENNReal.ofReal (∫ (x : ℝ) in s', gaussianPDFReal (c * μ) (NNReal.mk (c ^ 2) ⋯ * v) x)
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := (Homeomorph.mulLeft₀ c hc).symm.toMeasurableEquiv
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x ↦ c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
x : ℝ
this : |c⁻¹| * |c| = 1
⊢ |c⁻¹| * (|c| * gaussianPDFReal (c * μ) (NNReal.mk (c ^ 2) ⋯ * v) x) =
    gaussianPDFReal (c * μ) (NNReal.mk (c ^ 2) ⋯ * v) x

Tactic:
← mul_assoc,

Goal state after:
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := (Homeomorph.mulLeft₀ c hc).symm.toMeasurableEquiv
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x ↦ c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
x : ℝ
this : |c⁻¹| * |c| = 1
⊢ |c⁻¹| * |c| * gaussianPDFReal (c * μ) (NNReal.mk (c ^ 2) ⋯ * v) x =
    gaussianPDFReal (c * μ) (NNReal.mk (c ^ 2) ⋯ * v) x
Module: Mathlib.Probability.Distributions.Gaussian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsGaussian μ
L : StrongDual ℝ E
⊢ Integrable (⇑L) μ

Tactic:
← memLp_one_iff_integrable

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsGaussian μ
L : StrongDual ℝ E
⊢ MemLp (⇑L) 1 μ
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: case pos
μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
x y : ℝ
h : MemLp id 2 μ
⊢ x * (y * cov[fun u ↦ u, fun u ↦ u; μ]) = x * y * Var[id; μ]

Tactic:
← mul_assoc,

Goal state after:
case pos
μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
x y : ℝ
h : MemLp id 2 μ
⊢ x * y * cov[fun u ↦ u, fun u ↦ u; μ] = x * y * Var[id; μ]
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : InnerProductSpace ℝ E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
μ : Measure E
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : SecondCountableTopology F
inst✝² : CompleteSpace F
inst✝¹ : CompleteSpace E
inst✝ : IsFiniteMeasure μ
h : MemLp id 2 μ
L : E →L[ℝ] F
u v : F
⊢ ∫ (z : F), ⟪u, z - ∫ (x : F), x ∂Measure.map (⇑L) μ⟫ * ⟪v, z - ∫ (x : F), x ∂Measure.map (⇑L) μ⟫ ∂Measure.map (⇑L) μ =
    ∫ (z : E),
      ⟪(ContinuousLinearMap.adjoint L) u, z - ∫ (x : E), x ∂μ⟫ *
        ⟪(ContinuousLinearMap.adjoint L) v, z - ∫ (x : E), x ∂μ⟫ ∂μ

Tactic:
L.integral_id_map (h.integrable (by simp))

Goal state after:
E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : InnerProductSpace ℝ E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
μ : Measure E
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : SecondCountableTopology F
inst✝² : CompleteSpace F
inst✝¹ : CompleteSpace E
inst✝ : IsFiniteMeasure μ
h : MemLp id 2 μ
L : E →L[ℝ] F
u v : F
⊢ ∫ (z : F), ⟪u, z - L (∫ (x : E), x ∂μ)⟫ * ⟪v, z - L (∫ (x : E), x ∂μ)⟫ ∂Measure.map (⇑L) μ =
    ∫ (z : E),
      ⟪(ContinuousLinearMap.adjoint L) u, z - ∫ (x : E), x ∂μ⟫ *
        ⟪(ContinuousLinearMap.adjoint L) v, z - ∫ (x : E), x ∂μ⟫ ∂μ
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : ¬MemLp id 2 μ
x y : E
⊢ ⟪0 x, y⟫ =
    (((uncenteredCovarianceBilinDual μ).bilinearComp (toDualMap ℝ E).toContinuousLinearMap
          (toDualMap ℝ E).toContinuousLinearMap)
        x)
      y

Tactic:
simp [hμ, uncenteredCovarianceBilinDual_of_not_memLp]

Goal state after:
Module: Mathlib.Probability.Moments.CovarianceBilinDual

Goal state before:
pp: case refine_2
E : Type u_1
inst✝² : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝¹ : NormedSpace ℝ E
inst✝ : OpensMeasurableSpace E
L₁ L₂ : StrongDual ℝ E
h : MemLp id 2 μ
⊢ Integrable (fun x ↦ ‖x‖ * (‖L₂‖ * ‖x‖)) μ

Tactic:
← mul_assoc,

Goal state after:
case refine_2
E : Type u_1
inst✝² : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝¹ : NormedSpace ℝ E
inst✝ : OpensMeasurableSpace E
L₁ L₂ : StrongDual ℝ E
h : MemLp id 2 μ
⊢ Integrable (fun x ↦ ‖x‖ * ‖L₂‖ * ‖x‖) μ
Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
hc_pos : 0 < c
hc_lt_top : c < ∞
n : ℕ
hn : μ {x | normThreshold a n < ‖x‖} ≤ μ {x | ‖x‖ ≤ a} * ((1 - μ {x | ‖x‖ ≤ a}) / μ {x | ‖x‖ ≤ a}) ^ 2 ^ n
h_mul_le : c * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2
⊢ μ {x | normThreshold a (n + 1) < ‖x‖} = c⁻¹ * (c * μ {x | normThreshold a (n + 1) < ‖x‖})

Tactic:
← mul_assoc,

Goal state after:
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
hc_pos : 0 < c
hc_lt_top : c < ∞
n : ℕ
hn : μ {x | normThreshold a n < ‖x‖} ≤ μ {x | ‖x‖ ≤ a} * ((1 - μ {x | ‖x‖ ≤ a}) / μ {x | ‖x‖ ≤ a}) ^ 2 ^ n
h_mul_le : c * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2
⊢ μ {x | normThreshold a (n + 1) < ‖x‖} = c⁻¹ * c * μ {x | normThreshold a (n + 1) < ‖x‖}
Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
hc_pos : 0 < c
hc_lt_top : c < ∞
n : ℕ
hn : μ {x | normThreshold a n < ‖x‖} ≤ μ {x | ‖x‖ ≤ a} * ((1 - μ {x | ‖x‖ ≤ a}) / μ {x | ‖x‖ ≤ a}) ^ 2 ^ n
h_mul_le : c * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2
⊢ c⁻¹ * (c ^ 2 * ((1 - c) / c) ^ (2 ^ n * 2)) = c * ((1 - c) / c) ^ 2 ^ (n + 1)

Tactic:
← mul_assoc,

Goal state after:
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
hc_pos : 0 < c
hc_lt_top : c < ∞
n : ℕ
hn : μ {x | normThreshold a n < ‖x‖} ≤ μ {x | ‖x‖ ≤ a} * ((1 - μ {x | ‖x‖ ≤ a}) / μ {x | ‖x‖ ≤ a}) ^ 2 ^ n
h_mul_le : c * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2
⊢ c⁻¹ * c ^ 2 * ((1 - c) / c) ^ (2 ^ n * 2) = c * ((1 - c) / c) ^ 2 ^ (n + 1)
Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
hc_pos : 0 < c
hc_lt_top : c < ∞
n : ℕ
hn : μ {x | normThreshold a n < ‖x‖} ≤ μ {x | ‖x‖ ≤ a} * ((1 - μ {x | ‖x‖ ≤ a}) / μ {x | ‖x‖ ≤ a}) ^ 2 ^ n
h_mul_le : c * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2
⊢ c⁻¹ * (c * c) * ((1 - c) / c) ^ (2 ^ n * 2) = c * ((1 - c) / c) ^ 2 ^ (n + 1)

Tactic:
← mul_assoc,

Goal state after:
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
hc_pos : 0 < c
hc_lt_top : c < ∞
n : ℕ
hn : μ {x | normThreshold a n < ‖x‖} ≤ μ {x | ‖x‖ ≤ a} * ((1 - μ {x | ‖x‖ ≤ a}) / μ {x | ‖x‖ ≤ a}) ^ 2 ^ n
h_mul_le : c * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2
⊢ c⁻¹ * c * c * ((1 - c) / c) ^ (2 ^ n * 2) = c * ((1 - c) / c) ^ 2 ^ (n + 1)
Module: Mathlib.Probability.Distributions.Gaussian.Multivariate

Goal state before:
pp: ι : Type u_1
inst✝⁵ : Fintype ι
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
b : OrthonormalBasis ι ℝ E
⊢ (fun x ↦ ∑ i, x i • b i) = ⇑((EuclideanSpace.basisFun ι ℝ).equiv b (Equiv.refl ι)) ∘ toLp 2

Tactic:
← b.equiv_apply_euclideanSpace

Goal state after:
ι : Type u_1
inst✝⁵ : Fintype ι
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
b : OrthonormalBasis ι ℝ E
⊢ (fun x ↦ ((EuclideanSpace.basisFun ι ℝ).equiv b (Equiv.refl ι)) (toLp 2 x)) =
    ⇑((EuclideanSpace.basisFun ι ℝ).equiv b (Equiv.refl ι)) ∘ toLp 2
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝⁶ : Preorder T
inst✝⁵ : MeasurableSpace E
F : Type u_4
G : Type u_5
inst✝⁴ : MeasurableSpace G
inst✝³ : FunLike F E G
inst✝² : AddGroup E
inst✝¹ : SubtractionMonoid G
inst✝ : AddMonoidHomClass F E G
f : F
hf : Measurable ⇑f
hX : HasIndepIncrements X P
⊢ HasIndepIncrements (fun t ω ↦ f (X t ω)) P

Tactic:
by
  intro n t ht
  simp_rw [← map_sub]
  exact (hX n t ht).comp (fun _ ↦ f) (fun _ ↦ hf)

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic

Goal state before:
pp: S : Type u_1
T : Type u_2
Ω : Type u_3
E : Type u_4
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : MeasurableSpace E
inst✝⁷ : BorelSpace E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : SecondCountableTopology E
F : Type u_6
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
inst✝ : SecondCountableTopology F
Y : S → Ω → F
hX : IsGaussianProcess X P
I : Finset S
J : S → Finset T
L : (s : S) → (↥(J s) → E) →L[ℝ] F
hL : ∀ (s : S) (ω : Ω), Y s ω = (L s) ((J s).restrict fun x ↦ X x ω)
c : ℝ
x : ↥(I.biUnion J) → E
x✝ : ↥I
⊢ ((L ↑x✝) fun t ↦ (c • x) ⟨↑t, ⋯⟩) = ((RingHom.id ℝ) c • fun s ↦ (L ↑s) fun t ↦ x ⟨↑t, ⋯⟩) x✝

Tactic:
simp [← Pi.smul_def]

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: ι : Type u_2
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
hL : ∀ (i : ι), (L i).toBilinForm.IsPosSemidef
x : StrongDual ℝ ((i : ι) → E i)
⊢ 0 ≤ ((diagonalStrongDualPi L).toBilinForm x) x

Tactic:
rw [toBilinForm_diagonalStrongDualPi_apply]
    exact sum_nonneg fun i _ ↦ (hL i).nonneg _

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
⊢ Measurable fun a r ↦ (preCDF ρ r a).toReal

Tactic:
rw [measurable_pi_iff]
  exact fun _ ↦ measurable_preCDF.ennreal_toReal

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
r : ℚ
s : Set α
hs : MeasurableSet s
x✝ : ρ.fst s < ∞
⊢ ∫⁻ (x : α) in s, preCDF ρ r x ∂ρ.fst ≤ ∫⁻ (x : α) in s, 1 ∂ρ.fst

Tactic:
rw [setLIntegral_preCDF_fst ρ r hs]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
r : ℚ
s : Set α
hs : MeasurableSet s
x✝ : ρ.fst s < ∞
⊢ (ρ.IicSnd ↑r) s ≤ ∫⁻ (x : α) in s, 1 ∂ρ.fst
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
⊢ Integrable (fun t ↦ ⨅ r, f (a, t) ↑r) (ν a)

Tactic:
← memLp_one_iff_integrable

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
⊢ MemLp (fun t ↦ ⨅ r, f (a, t) ↑r) 1 (ν a)
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
⊢ ∀ (s : Set β), MeasurableSet s → (ν a) s < ∞ → IntegrableOn (fun a_3 ↦ f (a, a_3) q) s (ν a)

pp: case refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
⊢ ∀ (s : Set β), MeasurableSet s → (ν a) s < ∞ → ∫ (x : β) in s, ⨅ r, f (a, x) ↑r ∂ν a = ∫ (x : β) in s, f (a, x) q ∂ν a

Tactic:
· exact fun _ _ _ ↦ (hf.integrable a _).integrableOn

Goal state after:
case refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
⊢ ∀ (s : Set β), MeasurableSet s → (ν a) s < ∞ → ∫ (x : β) in s, ⨅ r, f (a, x) ↑r ∂ν a = ∫ (x : β) in s, f (a, x) q ∂ν a
Module: Mathlib.Probability.Kernel.Composition.Lemmas

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
κ : Kernel β γ
inst✝ : IsSFiniteKernel κ
s : Set (α × γ)
hs : MeasurableSet s
⊢ ∫⁻ (x : α), ∫⁻ (a : β), (κ a) (Prod.mk x ⁻¹' s) ∂ν ∂μ = ∫⁻ (a : α × β), ((Kernel.id ∥ₖ κ) a) s ∂μ.prod ν

Tactic:
MeasureTheory.lintegral_prod

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
κ : Kernel β γ
inst✝ : IsSFiniteKernel κ
s : Set (α × γ)
hs : MeasurableSet s
⊢ ∫⁻ (x : α), ∫⁻ (a : β), (κ a) (Prod.mk x ⁻¹' s) ∂ν ∂μ = ∫⁻ (x : α), ∫⁻ (y : β), ((Kernel.id ∥ₖ κ) (x, y)) s ∂ν ∂μ

case hf
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
κ : Kernel β γ
inst✝ : IsSFiniteKernel κ
s : Set (α × γ)
hs : MeasurableSet s
⊢ AEMeasurable (fun a ↦ ((Kernel.id ∥ₖ κ) a) s) (μ.prod ν)
Module: Mathlib.Probability.Kernel.Disintegration.Basic

Goal state before:
pp: α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
ρ : Measure (α × Ω)
ρCond : Kernel α Ω
inst✝² : ρ.IsCondKernel ρCond
inst✝¹ : IsFiniteMeasure ρ
inst✝ : MeasurableSingletonClass α
x : α
hx : ρ.fst {x} ≠ 0
s : Set Ω
hs : MeasurableSet s
this✝ : IsSFiniteKernel ρCond
this : ∀ (a : α), (ρCond a) (Prod.mk a ⁻¹' {x} ×ˢ s) = {x}.indicator (fun x ↦ (ρCond x) s) a
⊢ (ρCond x) s = (ρ.fst {x})⁻¹ * (ρ.fst {x} * (ρCond x) s)

Tactic:
← mul_assoc,

Goal state after:
α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
ρ : Measure (α × Ω)
ρCond : Kernel α Ω
inst✝² : ρ.IsCondKernel ρCond
inst✝¹ : IsFiniteMeasure ρ
inst✝ : MeasurableSingletonClass α
x : α
hx : ρ.fst {x} ≠ 0
s : Set Ω
hs : MeasurableSet s
this✝ : IsSFiniteKernel ρCond
this : ∀ (a : α), (ρCond a) (Prod.mk a ⁻¹' {x} ×ˢ s) = {x}.indicator (fun x ↦ (ρCond x) s) a
⊢ (ρCond x) s = (ρ.fst {x})⁻¹ * ρ.fst {x} * (ρCond x) s
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
⊢ Integrable (fun x ↦ κ.densityProcess ν n a x s) (ν a)

Tactic:
← memLp_one_iff_integrable

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
⊢ MemLp (fun x ↦ κ.densityProcess ν n a x s) 1 (ν a)
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ Integrable (fun x ↦ κ.density ν a x s) (ν a)

Tactic:
← memLp_one_iff_integrable

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ MemLp (fun x ↦ κ.density ν a x s) 1 (ν a)
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: case pos.refine_2
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
Y : α → Ω
mβ : MeasurableSpace β
X : α → β
hY : AEMeasurable Y μ
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x ↦ (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
hX : AEMeasurable X μ
⊢ Measure.map X μ ⊗ₘ κ = Measure.map (AEMeasurable.mk X hX) μ ⊗ₘ κ

Tactic:
· rw [Measure.map_congr hX.ae_eq_mk]

Goal state after:
Module: Mathlib.Probability.Kernel.IonescuTulcea.Maps

Goal state before:
pp: case neg
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrder ι
inst✝² : DecidableLE ι
X : ι → Type u_2
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : (i : ι) → MeasurableSpace (X i)
m n : ι
i : ↥(Iic n)
h : ¬↑i ≤ m
⊢ Measurable fun c ↦ IicProdIoc m n c i

Tactic:
simpa [IicProdIoc, h] using measurable_snd.eval

Goal state after:
Module: Mathlib.Probability.Distributions.SetBernoulli

Goal state before:
pp: ι : Type u_1
u : Set ι
p : ↑I
inst✝ : Countable ι
i : ι
hi : i ∉ u
⊢ setBer(u, p) {s | i ∈ s} =
    (infinitePi fun i ↦ toNNReal p • dirac (i ∈ u) + toNNReal (σ p) • dirac False) (cylinder {i} {fun x ↦ True})

Tactic:
by
      rw [setBernoulli_apply']; congr!; ext; simp [funext_iff]

Goal state after:
Module: Mathlib.Probability.Distributions.SetBernoulli

Goal state before:
pp: ι : Type u_1
s : Set ι
p : ↑I
inst✝ : Countable ι
u : Finset ι
hsu : s ⊆ ↑u
⊢ setBer(↑u, p) {s} =
    ∏' (i : ι), ((if i ∈ u ↔ i ∈ s then ↑(toNNReal p) else 0) + if i ∈ s then 0 else ↑(toNNReal (σ p)))

Tactic:
simp [setBernoulli_apply, Set.image_singleton, Set.indicator]

Goal state after:
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝ : (n : ℕ) → MeasurableSpace (X n)
μ : (n : ℕ) → Measure ((i : ↥(Iic n)) → X ↑i)
n : ℕ
x : (i : ↥(Iic ((Iic n).sup id))) → X ↑i
i : ↥(Iic n)
⊢ ↑(Iic ((Iic n).sup id)) = ↑(Iic n)

Tactic:
sup_Iic n

Goal state after:
X : ℕ → Type u_1
inst✝ : (n : ℕ) → MeasurableSpace (X n)
μ : (n : ℕ) → Measure ((i : ↥(Iic n)) → X ↑i)
n : ℕ
x : (i : ↥(Iic ((Iic n).sup id))) → X ↑i
i : ↥(Iic n)
⊢ ↑(Iic n) = ↑(Iic n)
Module: Mathlib.Probability.Decision.Risk.Defs

Goal state before:
pp: Θ : Type u_1
𝓧 : Type u_2
𝓨 : Type u_3
mΘ : MeasurableSpace Θ
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
ℓ : Θ → 𝓨 → ℝ≥0∞
P : Kernel Θ 𝓧
π : Measure Θ
inst✝ : IsEmpty 𝓧
⊢ bayesRisk ℓ P π = 0

Tactic:
simp [bayesRisk]

Goal state after: