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

Goal state before:
pp: case e_μ
E : Type u_1
F : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
μ✝ : Measure E
inst✝³ : IsGaussian μ✝
inst✝² : SecondCountableTopology E
μ ν : Measure E
inst✝¹ : IsGaussian μ
inst✝ : IsGaussian ν
L : StrongDual ℝ E
this : ∫ (x : E), L x ∂μ ∗ ν = ∫ (x : ℝ), x ∂Measure.map (⇑L) μ ∗ Measure.map (⇑L) ν
⊢ ∫ (x : E), L x ∂μ + ∫ (x : E), L x ∂ν =
    ∫ (x : ℝ), x ∂gaussianReal (∫ (x : E), L x ∂μ + ∫ (x : E), L x ∂ν) (Var[⇑L; μ].toNNReal + Var[⇑L; ν].toNNReal)

pp: case e_v
E : Type u_1
F : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
μ✝ : Measure E
inst✝³ : IsGaussian μ✝
inst✝² : SecondCountableTopology E
μ ν : Measure E
inst✝¹ : IsGaussian μ
inst✝ : IsGaussian ν
L : StrongDual ℝ E
this : ∫ (x : E), L x ∂μ ∗ ν = ∫ (x : ℝ), x ∂Measure.map (⇑L) μ ∗ Measure.map (⇑L) ν
⊢ Var[⇑L; μ].toNNReal + Var[⇑L; ν].toNNReal =
    Var[id; gaussianReal (∫ (x : E), L x ∂μ + ∫ (x : E), L x ∂ν) (Var[⇑L; μ].toNNReal + Var[⇑L; ν].toNNReal)].toNNReal

Tactic:
<;>

Goal state after:
case e_μ
E : Type u_1
F : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
μ✝ : Measure E
inst✝³ : IsGaussian μ✝
inst✝² : SecondCountableTopology E
μ ν : Measure E
inst✝¹ : IsGaussian μ
inst✝ : IsGaussian ν
L : StrongDual ℝ E
this : ∫ (x : E), L x ∂μ ∗ ν = ∫ (x : ℝ), x ∂Measure.map (⇑L) μ ∗ Measure.map (⇑L) ν
⊢ ∫ (x : E), L x ∂μ + ∫ (x : E), L x ∂ν =
    ∫ (x : ℝ), x ∂gaussianReal (∫ (x : E), L x ∂μ + ∫ (x : E), L x ∂ν) (Var[⇑L; μ].toNNReal + Var[⇑L; ν].toNNReal)

case e_v
E : Type u_1
F : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
μ✝ : Measure E
inst✝³ : IsGaussian μ✝
inst✝² : SecondCountableTopology E
μ ν : Measure E
inst✝¹ : IsGaussian μ
inst✝ : IsGaussian ν
L : StrongDual ℝ E
this : ∫ (x : E), L x ∂μ ∗ ν = ∫ (x : ℝ), x ∂Measure.map (⇑L) μ ∗ Measure.map (⇑L) ν
⊢ Var[⇑L; μ].toNNReal + Var[⇑L; ν].toNNReal =
    Var[id; gaussianReal (∫ (x : E), L x ∂μ + ∫ (x : E), L x ∂ν) (Var[⇑L; μ].toNNReal + Var[⇑L; ν].toNNReal)].toNNReal
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
⊢ ∫ (x_1 : E), ⟪x, x_1⟫ * ⟪y, x_1⟫ ∂μ =
    ⟪(continuousLinearMapOfBilin
          ((uncenteredCovarianceBilinDual μ).bilinearComp (toDualMap ℝ E).toContinuousLinearMap
            (toDualMap ℝ E).toContinuousLinearMap))
        x,
      y⟫

Tactic:
← covarianceOperator_inner hμ

Goal state after:
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
⊢ ⟪(covarianceOperator μ) x, y⟫ =
    ⟪(continuousLinearMapOfBilin
          ((uncenteredCovarianceBilinDual μ).bilinearComp (toDualMap ℝ E).toContinuousLinearMap
            (toDualMap ℝ E).toContinuousLinearMap))
        x,
      y⟫
Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: a : ℝ
n : ℕ
⊢ normThreshold a (n + 1) ^ 2 ≤ a ^ 2 * (1 + √2) ^ 2 * 2 ^ (n + 2)

Tactic:
normThreshold_eq,

Goal state after:
a : ℝ
n : ℕ
⊢ (a * (1 + √2) * (√2 ^ (n + 1 + 1) - 1)) ^ 2 ≤ a ^ 2 * (1 + √2) ^ 2 * 2 ^ (n + 2)
Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: case ha
a : ℝ
n : ℕ
⊢ 1 ≤ √2

pp: case h₁
a : ℝ
n : ℕ
⊢ 0 ≤ n

Tactic:
<;>

Goal state after:
case ha
a : ℝ
n : ℕ
⊢ 1 ≤ √2

case h₁
a : ℝ
n : ℕ
⊢ 0 ≤ n
Module: Mathlib.Probability.Distributions.Gaussian.CharFun

Goal state before:
pp: case left
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology E
inst✝⁴ : CompleteSpace E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
m : E
f : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
hf : f.toBilinForm.IsPosSemidef
h' : IsGaussian μ
n : StrongDual ℝ E → ℤ
h :
  ∀ (L : StrongDual ℝ E),
    ↑(n L) =
      (↑(L (∫ (x : E), id x ∂μ)) * I - ↑(((covarianceBilinDual μ) L) L) / 2 - ↑(L m) * I + ↑((f L) L) / 2) /
        (2 * ↑π * I)
this✝¹ : Continuous n
this✝ : ∀ (x : StrongDual ℝ E), n = Function.const (StrongDual ℝ E) (n x)
this : ∀ (L : StrongDual ℝ E), n L = 0
hn : ∀ (L : StrongDual ℝ E), ((covarianceBilinDual μ) L) L = (f L) L ∧ L (∫ (x : E), x ∂μ) = L m
⊢ m = ∫ (x : E), x ∂μ

Tactic:
SeparatingDual.eq_iff_forall_dual_eq (R := ℝ)

Goal state after:
case left
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology E
inst✝⁴ : CompleteSpace E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
m : E
f : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
hf : f.toBilinForm.IsPosSemidef
h' : IsGaussian μ
n : StrongDual ℝ E → ℤ
h :
  ∀ (L : StrongDual ℝ E),
    ↑(n L) =
      (↑(L (∫ (x : E), id x ∂μ)) * I - ↑(((covarianceBilinDual μ) L) L) / 2 - ↑(L m) * I + ↑((f L) L) / 2) /
        (2 * ↑π * I)
this✝¹ : Continuous n
this✝ : ∀ (x : StrongDual ℝ E), n = Function.const (StrongDual ℝ E) (n x)
this : ∀ (L : StrongDual ℝ E), n L = 0
hn : ∀ (L : StrongDual ℝ E), ((covarianceBilinDual μ) L) L = (f L) L ∧ L (∫ (x : E), x ∂μ) = L m
⊢ ∀ (g : StrongDual ℝ E), g m = g (∫ (x : E), x ∂μ)
Module: Mathlib.Probability.Distributions.Gaussian.Multivariate

Goal state before:
pp: case hv
ι : Type u_2
inst✝ : DecidableEq ι
I J : Finset ι
μ : EuclideanSpace ℝ ↥I
S : Matrix ↥I ↥I ℝ
hS : S.PosSemidef
hJI : J ⊆ I
i j : ↥J
this :
  ∀ (i : ↥J),
    (fun u ↦ ⟪(EuclideanSpace.basisFun ↥J ℝ).toBasis i, u⟫) ∘ ⇑(EuclideanSpace.restrict₂ hJI) = fun u ↦ u.ofLp ⟨↑i, ⋯⟩
⊢ cov[fun u ↦ u.ofLp ⟨↑i, ⋯⟩, fun u ↦ u.ofLp ⟨↑j, ⋯⟩; multivariateGaussian μ S] =
    ((covarianceBilin
          (multivariateGaussian ((EuclideanSpace.restrict₂ hJI) μ) (S.submatrix (fun i ↦ ⟨↑i, ⋯⟩) fun i ↦ ⟨↑i, ⋯⟩)))
        ((EuclideanSpace.basisFun ↥J ℝ).toBasis i))
      ((EuclideanSpace.basisFun ↥J ℝ).toBasis j)

Tactic:
covariance_eval_multivariateGaussian hS,

Goal state after:
case hv
ι : Type u_2
inst✝ : DecidableEq ι
I J : Finset ι
μ : EuclideanSpace ℝ ↥I
S : Matrix ↥I ↥I ℝ
hS : S.PosSemidef
hJI : J ⊆ I
i j : ↥J
this :
  ∀ (i : ↥J),
    (fun u ↦ ⟪(EuclideanSpace.basisFun ↥J ℝ).toBasis i, u⟫) ∘ ⇑(EuclideanSpace.restrict₂ hJI) = fun u ↦ u.ofLp ⟨↑i, ⋯⟩
⊢ S ⟨↑i, ⋯⟩ ⟨↑j, ⋯⟩ =
    ((covarianceBilin
          (multivariateGaussian ((EuclideanSpace.restrict₂ hJI) μ) (S.submatrix (fun i ↦ ⟨↑i, ⋯⟩) fun i ↦ ⟨↑i, ⋯⟩)))
        ((EuclideanSpace.basisFun ↥J ℝ).toBasis i))
      ((EuclideanSpace.basisFun ↥J ℝ).toBasis j)
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I J : Finset ℝ≥0
hJI : J ⊆ I
⊢ projectiveFamily J = Measure.map (Finset.restrict₂ hJI) (projectiveFamily I)

Tactic:
nth_rw 2 [projectiveFamily]

Goal state after:
I J : Finset ℝ≥0
hJI : J ⊆ I
⊢ projectiveFamily J =
    Measure.map (Finset.restrict₂ hJI)
      (Measure.map (⇑(MeasurableEquiv.toLp 2 (↥I → ℝ)).symm) (multivariateGaussian 0 (covMatrix I)))
Module: Mathlib.Probability.Independence.CharacteristicFunction

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : InnerProductSpace ℝ E
μ ν : Measure E
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
⊢ charFun (Measure.map (fun p ↦ p.1 + p.2) (μ.prod ν)) = charFun μ * charFun ν

Tactic:
rw [IndepFun.charFun_map_fun_add_eq_mul, measurePreserving_fst.map_eq,
    measurePreserving_snd.map_eq]

Goal state after:
case mX
E : Type u_2
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : InnerProductSpace ℝ E
μ ν : Measure E
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
⊢ AEMeasurable Prod.fst (μ.prod ν)

case mY
E : Type u_2
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : InnerProductSpace ℝ E
μ ν : Measure E
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
⊢ AEMeasurable Prod.snd (μ.prod ν)

case hXY
E : Type u_2
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : BorelSpace E
inst✝³ : SecondCountableTopology E
inst✝² : InnerProductSpace ℝ E
μ ν : Measure E
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
⊢ Prod.fst ⟂ᵢ[μ.prod ν] Prod.snd
Module: Mathlib.Probability.Independence.CharacteristicFunction

Goal state before:
pp: case empty
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
s : Finset ι
E : Type u_3
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
X : ι → Ω → E
inst✝ : NormedSpace ℝ E
this : IsProbabilityMeasure P
mX : ∀ i ∈ ∅, AEMeasurable (X i) P
hX : iIndepFun (∅.restrict X) P
⊢ charFunDual (Measure.map (∑ i ∈ ∅, X i) P) = ∏ i ∈ ∅, charFunDual (Measure.map (X i) P)

Tactic:
ext; simp [show (0 : Ω → E) = fun _ ↦ 0 from rfl]

Goal state after:
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
h : ∀ (t : T), X t =ᵐ[P] Y t
I : Finset T
⊢ ∀ᵐ (ω : Ω) ∂P, ∀ (i : ↥I), X (↑i) ω = Y (↑i) ω

Tactic:
rw [MeasureTheory.ae_all_iff]

Goal state after:
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
h : ∀ (t : T), X t =ᵐ[P] Y t
I : Finset T
⊢ ∀ (i : ↥I), ∀ᵐ (a : Ω) ∂P, X (↑i) a = Y (↑i) a
Module: Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence

Goal state before:
pp: case fst
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
S : Type u_4
X : S → Ω → E
Y : T → Ω → E
inst✝ : NormedSpace ℝ E
hXY : IsGaussianProcess (Sum.elim X Y) P
mX : ∀ (s : S), AEMeasurable (X s) P
mY : ∀ (t : T), AEMeasurable (Y t) P
h : ∀ (s : S) (t : T) (L₁ L₂ : StrongDual ℝ E), cov[⇑L₁ ∘ X s, ⇑L₂ ∘ Y t; P] = 0
this : IsProbabilityMeasure P
I : Finset S
J : Finset T
c : ℝ
x : ↥(I.disjSum J) → E
x✝ : ↥I
⊢ (fun s ↦ (c • x) ⟨Sum.inl ↑s, ⋯⟩, fun t ↦ (c • x) ⟨Sum.inr ↑t, ⋯⟩).1 x✝ =
    ((RingHom.id ℝ) c • (fun s ↦ x ⟨Sum.inl ↑s, ⋯⟩, fun t ↦ x ⟨Sum.inr ↑t, ⋯⟩)).1 x✝

pp: case snd
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
S : Type u_4
X : S → Ω → E
Y : T → Ω → E
inst✝ : NormedSpace ℝ E
hXY : IsGaussianProcess (Sum.elim X Y) P
mX : ∀ (s : S), AEMeasurable (X s) P
mY : ∀ (t : T), AEMeasurable (Y t) P
h : ∀ (s : S) (t : T) (L₁ L₂ : StrongDual ℝ E), cov[⇑L₁ ∘ X s, ⇑L₂ ∘ Y t; P] = 0
this : IsProbabilityMeasure P
I : Finset S
J : Finset T
c : ℝ
x : ↥(I.disjSum J) → E
x✝ : ↥J
⊢ (fun s ↦ (c • x) ⟨Sum.inl ↑s, ⋯⟩, fun t ↦ (c • x) ⟨Sum.inr ↑t, ⋯⟩).2 x✝ =
    ((RingHom.id ℝ) c • (fun s ↦ x ⟨Sum.inl ↑s, ⋯⟩, fun t ↦ x ⟨Sum.inr ↑t, ⋯⟩)).2 x✝

Tactic:
<;>

Goal state after:
case fst
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
S : Type u_4
X : S → Ω → E
Y : T → Ω → E
inst✝ : NormedSpace ℝ E
hXY : IsGaussianProcess (Sum.elim X Y) P
mX : ∀ (s : S), AEMeasurable (X s) P
mY : ∀ (t : T), AEMeasurable (Y t) P
h : ∀ (s : S) (t : T) (L₁ L₂ : StrongDual ℝ E), cov[⇑L₁ ∘ X s, ⇑L₂ ∘ Y t; P] = 0
this : IsProbabilityMeasure P
I : Finset S
J : Finset T
c : ℝ
x : ↥(I.disjSum J) → E
x✝ : ↥I
⊢ (fun s ↦ (c • x) ⟨Sum.inl ↑s, ⋯⟩, fun t ↦ (c • x) ⟨Sum.inr ↑t, ⋯⟩).1 x✝ =
    ((RingHom.id ℝ) c • (fun s ↦ x ⟨Sum.inl ↑s, ⋯⟩, fun t ↦ x ⟨Sum.inr ↑t, ⋯⟩)).1 x✝

case snd
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
S : Type u_4
X : S → Ω → E
Y : T → Ω → E
inst✝ : NormedSpace ℝ E
hXY : IsGaussianProcess (Sum.elim X Y) P
mX : ∀ (s : S), AEMeasurable (X s) P
mY : ∀ (t : T), AEMeasurable (Y t) P
h : ∀ (s : S) (t : T) (L₁ L₂ : StrongDual ℝ E), cov[⇑L₁ ∘ X s, ⇑L₂ ∘ Y t; P] = 0
this : IsProbabilityMeasure P
I : Finset S
J : Finset T
c : ℝ
x : ↥(I.disjSum J) → E
x✝ : ↥J
⊢ (fun s ↦ (c • x) ⟨Sum.inl ↑s, ⋯⟩, fun t ↦ (c • x) ⟨Sum.inr ↑t, ⋯⟩).2 x✝ =
    ((RingHom.id ℝ) c • (fun s ↦ x ⟨Sum.inl ↑s, ⋯⟩, fun t ↦ x ⟨Sum.inr ↑t, ⋯⟩)).2 x✝
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: case mp
α : Type u_1
f : α → ℚ → ℝ
a : α
⊢ IsRatStieltjesPoint (fun p ↦ f p.2) ((), a) → IsRatStieltjesPoint f a

pp: case mpr
α : Type u_1
f : α → ℚ → ℝ
a : α
⊢ IsRatStieltjesPoint f a → IsRatStieltjesPoint (fun p ↦ f p.2) ((), a)

Tactic:
<;>

Goal state after:
case mp
α : Type u_1
f : α → ℚ → ℝ
a : α
⊢ IsRatStieltjesPoint (fun p ↦ f p.2) ((), a) → IsRatStieltjesPoint f a

case mpr
α : Type u_1
f : α → ℚ → ℝ
a : α
⊢ IsRatStieltjesPoint f a → IsRatStieltjesPoint (fun p ↦ f p.2) ((), a)
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case h
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
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✝ L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ ((i : ι) → E i)) :=
  LinearMap.mk₂ ℝ (fun x y ↦ ∑ i, ((L i) (x ∘SL single ℝ E i)) (y ∘SL single ℝ E i)) ⋯ ⋯ ⋯ ⋯
x y : StrongDual ℝ ((i : ι) → E i)
i : ι
a✝ : i ∈ univ
⊢ ‖((L i) (x ∘SL single ℝ E i)) (y ∘SL single ℝ E i)‖ ≤ ‖L i‖ * ‖x‖ * ‖y‖

Tactic:
grw [le_opNorm₂]

Goal state after:
case h
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
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✝ L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ ((i : ι) → E i)) :=
  LinearMap.mk₂ ℝ (fun x y ↦ ∑ i, ((L i) (x ∘SL single ℝ E i)) (y ∘SL single ℝ E i)) ⋯ ⋯ ⋯ ⋯
x y : StrongDual ℝ ((i : ι) → E i)
i : ι
a✝ : i ∈ univ
⊢ ‖L i‖ * ‖x ∘SL single ℝ E i‖ * ‖y ∘SL single ℝ E i‖ ≤ ‖L i‖ * ‖x‖ * ‖y‖
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case hbc
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
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✝ L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ ((i : ι) → E i)) := ⋯
x y : StrongDual ℝ ((i : ι) → E i)
i : ι
a✝ : i ∈ univ
⊢ ‖x ∘SL single ℝ E i‖ ≤ ‖x‖

pp: case h₂
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
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✝ L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ ((i : ι) → E i)) := ⋯
x y : StrongDual ℝ ((i : ι) → E i)
i : ι
a✝ : i ∈ univ
⊢ ‖y ∘SL single ℝ E i‖ ≤ ‖y‖

Tactic:
<;>

Goal state after:
case hbc
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
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✝ L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ ((i : ι) → E i)) := ⋯
x y : StrongDual ℝ ((i : ι) → E i)
i : ι
a✝ : i ∈ univ
⊢ ‖x ∘SL single ℝ E i‖ ≤ ‖x‖

case h₂
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
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✝ L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ ((i : ι) → E i)) := ⋯
x y : StrongDual ℝ ((i : ι) → E i)
i : ι
a✝ : i ∈ univ
⊢ ‖y ∘SL single ℝ E i‖ ≤ ‖y‖
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: case pos
q : ℚ
h✝ : q < 0
⊢ 0 ≤ 1

pp: case neg
q : ℚ
h✝ : ¬q < 0
⊢ 1 ≤ 1

Tactic:
<;>

Goal state after:
case pos
q : ℚ
h✝ : q < 0
⊢ 0 ≤ 1

case neg
q : ℚ
h✝ : ¬q < 0
⊢ 1 ≤ 1
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case h₁
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
L₁✝ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂✝ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
L₁ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ (E × F)) := ⋯
x y : StrongDual ℝ (E × F)
⊢ ‖(L₁ (x ∘SL inl ℝ E F)) (y ∘SL inl ℝ E F)‖ ≤ ‖L₁‖ * ‖x‖ * ‖y‖

Tactic:
grw [le_opNorm₂]

Goal state after:
case h₁
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
L₁✝ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂✝ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
L₁ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ (E × F)) :=
  LinearMap.mk₂ ℝ (fun x y ↦ (L₁ (x ∘SL inl ℝ E F)) (y ∘SL inl ℝ E F) + (L₂ (x ∘SL inr ℝ E F)) (y ∘SL inr ℝ E F)) ⋯ ⋯ ⋯
    ⋯
x y : StrongDual ℝ (E × F)
⊢ ‖L₁‖ * ‖x ∘SL inl ℝ E F‖ * ‖y ∘SL inl ℝ E F‖ ≤ ‖L₁‖ * ‖x‖ * ‖y‖
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case hbc
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
L₁✝ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂✝ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
L₁ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ (E × F)) := ⋯
x y : StrongDual ℝ (E × F)
⊢ ‖x ∘SL inl ℝ E F‖ ≤ ‖x‖

pp: case h₂
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
L₁✝ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂✝ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
L₁ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ (E × F)) := ⋯
x y : StrongDual ℝ (E × F)
⊢ ‖y ∘SL inl ℝ E F‖ ≤ ‖y‖

Tactic:
<;>

Goal state after:
case hbc
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
L₁✝ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂✝ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
L₁ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ (E × F)) := ⋯
x y : StrongDual ℝ (E × F)
⊢ ‖x ∘SL inl ℝ E F‖ ≤ ‖x‖

case h₂
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
L₁✝ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂✝ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
L₁ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ (E × F)) := ⋯
x y : StrongDual ℝ (E × F)
⊢ ‖y ∘SL inl ℝ E F‖ ≤ ‖y‖
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case e_f.e_6
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
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
y : ℝ
⊢ y ≤ x ↔ ∀ (a : ℚ), x < ↑a → y ≤ ↑a

Tactic:
exact le_iff_forall_lt_rat_imp_le

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

Goal state before:
pp: case h₂
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
L₁✝ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂✝ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
L₁ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ (E × F)) := ⋯
x y : StrongDual ℝ (E × F)
⊢ ‖(L₂ (x ∘SL inr ℝ E F)) (y ∘SL inr ℝ E F)‖ ≤ ‖L₂‖ * ‖x‖ * ‖y‖

Tactic:
grw [le_opNorm₂]

Goal state after:
case h₂
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
L₁✝ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂✝ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
L₁ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ (E × F)) :=
  LinearMap.mk₂ ℝ (fun x y ↦ (L₁ (x ∘SL inl ℝ E F)) (y ∘SL inl ℝ E F) + (L₂ (x ∘SL inr ℝ E F)) (y ∘SL inr ℝ E F)) ⋯ ⋯ ⋯
    ⋯
x y : StrongDual ℝ (E × F)
⊢ ‖L₂‖ * ‖x ∘SL inr ℝ E F‖ * ‖y ∘SL inr ℝ E F‖ ≤ ‖L₂‖ * ‖x‖ * ‖y‖