/-
 - Created in 2024 by Gaëtan Serré
-/

/-
- https://github.com/gaetanserre/SBS-Proofs
-/

import Mathlib.Data.EReal.Basic
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

import SBSProofs.Utils
import SBSProofs.Measure
import SBSProofs.RKHS.Basic

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y)

open scoped RealInnerProductSpace
open BigOperators Finset ENNReal NNReal MeasureTheory RKHS

set_option maxHeartbeats 400000

/-
  We defined measures μ and π (ν is considered as the standard Lebesgue measure) along with their densities (finite and non-zero on the entire space)
-/
variable {d : ℕ} (hd : d ≠ 0) {Ω : Set (Vector ℝ d)}

variable [MeasureSpace Ω]

variable (μ π : DensityMeasure Ω) [IsProbabilityMeasure μ.toMeasure] [IsProbabilityMeasure π.toMeasure]

variable (hdμ : ∀x, μ.d x ≠ 0) (hdπ : ∀x, π.d x ≠ 0)

/-
  We define a RKHS of Ω → ℝ functions.
-/
variable (H₀ : Set (Ω → ℝ)) [NormedAddCommGroup H₀] [InnerProductSpace ℝ H₀] [s : RKHS H₀]

/--
We consider that the left-hand side of the equivalence holds for all x. In the future, we want to take into account that it only holds for almost all x w.r.t. μ.
-/
def positive_definite_kernel := ∀ (f : Fin d → Ω → ℝ), (0 ≤ ∫ x in Set.univ, (∫ x' in Set.univ, (∑ i ∈ Set.univ, f i x * s.k x x' * f i x') ∂μ.toMeasure) ∂μ.toMeasure) ∧ (∫ x in Set.univ, (∫ x' in Set.univ, (∑ i ∈ Set.univ, f i x * s.k x x' * f i x') ∂μ.toMeasure) ∂μ.toMeasure = 0 ↔ ∀i, ∀ᵐ x ∂ μ.toMeasure, f i x = 0)

variable (h_kernel_positive : positive_definite_kernel μ H₀)

/-===============================KERNEL STEIN DISCREPANCY===============================-/
/-
Here, we prove that KSD(μ | π) is a valid discrepancy measure and that π is the only fixed point of Φₜ(μ).
-/

/-
  From here, as the derivative of multivariate function are hard to define and to manipulate (defining the gradient, the divergence...), we define the gradient of *f* as follows:
  f  : Ω → ℝ
  df : Fin d → Ω → ℝ
       i ↦ x ↦ ∂xⁱ f(x)

  For vector-valued function, we defined them as follows:
  f  : Fin d → Ω → ℝ
       i ↦ x ↦ f(x)ⁱ
  df : Fin d → Ω → ℝ
       i ↦ x ↦ ∂xⁱ f(x)ⁱ

  Also, we assume some simple lemmas using the above formalism. Sometimes, these lemmas are not rigorously defined but, in our framework, it is more than enough.
-/

/- dk : x ↦ i ↦ y ↦ ∂xⁱ k(x, y) -/
variable (dk : Ω → Fin d → Ω → ℝ)

/- d_ln_π : i ↦ x ↦ ∂xⁱ ln (μ(x) / π(x)) -/
variable (d_ln_π : Fin d → Ω → ℝ)

/-
  Definition of the steepest direction ϕ and its derivative.
-/
variable (ϕ : product_RKHS H₀ hd) (dϕ : Fin d → Ω → ℝ)

/- As Ω is supposed to be compact, we will use this assumption only when the function is trivially integrable (e.g. continuous on compact). -/
axiom is_integrable_H₀ : ∀ (f : Ω → ℝ), Integrable f μ.toMeasure
axiom is_integrable_H₀_volume : ∀ (f : Ω → ℝ), Integrable f
axiom is_measurable_H₀ : ∀ (f : Ω → ℝ), Measurable f
axiom is_measurable_H₀_enn : ∀ (f : Ω → ℝ≥0∞), Measurable f

/-
d_ln_π_μ : i ↦ x ↦ ∂xⁱ ln (π(x) / μ(x))
-/
variable (d_ln_π_μ : Fin d → Ω → ℝ)

/-
Simple derivative rule: if the derivative is 0 ∀x, then the function is constant.
-/
variable (hd_ln_π_μ : ∀ (ν : Measure Ω), (∀i, ∀ᵐ x ∂ν, d_ln_π_μ i x = 0) → (∃ c, ∀ᵐ x ∂ν, log (μ.d x / π.d x) = c))

/-
dπ' : i ↦ x ↦ ∂xⁱ π(x)
-/
variable (dπ' : Fin d → Ω → ℝ)

/-
Log-trick: ∂xⁱ ln (π(x)) * π(x) = ∂xⁱ π(x).
-/
variable (hπ' : ∀x, ∀i, ENNReal.toReal (π.d x) * d_ln_π i x = dπ' i x)


variable [Norm Ω]
/--
  Stein class of measure. f is in the Stein class of μ if, ∀i ∈ Set.univ, lim_(‖x‖ → ∞) μ(x) * ϕ(x)ⁱ = 0.
-/
def SteinClass (f : Fin d → Ω → ℝ) := ∀ x, tends_to_infty (λ (x : Ω) ↦ ‖x‖) → ∀i, ENNReal.toReal (μ.d x) * f i x = 0


/-
  Kernel Stein Discrepancy
-/
variable (KSD : Measure Ω → Measure Ω → ℝ)

/--
KSD(μ | π) = ⟪∇ln π/μ, Pμ ∇ln π/μ⟫_L²(μ). We assume here that KSD is also equal to ∫ x, ∑ i ∈ Set.univ, (d_ln_π i x * ϕ i x + dϕ i x) ∂μ.
-/
def is_ksd := KSD μ.toMeasure π.toMeasure = (∫ x in Set.univ, (∫ x' in Set.univ, (∑ i ∈ Set.univ, d_ln_π_μ i x * s.k x x' * d_ln_π_μ i x') ∂μ.toMeasure) ∂μ.toMeasure) ∧ (KSD μ.toMeasure π.toMeasure = ∫ x, ∑ i ∈ Set.univ, (d_ln_π i x * (ϕ i).1 x + dϕ i x) ∂μ.toMeasure)

/-
  KSD(μ | π) is originally defined as ‖ϕ^⋆‖²_H, it is therefore non-negative.
-/
def is_ksd_norm := KSD μ.toMeasure π.toMeasure = ‖ϕ‖^2

/-
  ϕ is in the Stein class of π
-/
variable (hstein : SteinClass π (λ i ↦ (ϕ i).1))

include hπ' h_kernel_positive hstein hd_ln_π_μ hdπ hdμ in
/--
  We show that, if ϕ is in the Stein class of π, KSD is a valid discrepancy measure i.e. μ = π ↔ KSD(μ | π) = 0.
-/
lemma KSD_is_valid_discrepancy (hksd : is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD) : μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0 :=
Goals accomplished!by
Goals accomplished!
Goals accomplished! 🐙
  
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0constructor
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mpr
KSD μ.toMeasure π.toMeasure = 0 → μ.toMeasure = π.toMeasure
  -- μ = π ↦ KSD(μ | π) = 0.
  ·
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mpr
KSD μ.toMeasure π.toMeasure = 0 → μ.toMeasure = π.toMeasureintro h
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
mp
KSD μ.toMeasure π.toMeasure = 0

    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
mp
KSD μ.toMeasure π.toMeasure = 0hksd.right
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) ∂μ.toMeasure = 0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) ∂μ.toMeasure = 0

    -- ∑ i, f i + g i = ∑ i, f i + ∑ i, g i.
    have split_sum : ∀x, ∑ i ∈ Set.univ, (d_ln_π i x * (ϕ i).1 x + dϕ i x) = (∑ i ∈ Set.univ, d_ln_π i x * (ϕ i).1 x) + (∑ i ∈ Set.univ, dϕ i x) := λ x ↦ sum_add_distrib
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) ∂μ.toMeasure = 0
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0simp_rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) ∂μ.toMeasure = 0split_sum
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x ∂μ.toMeasure = 0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x ∂μ.toMeasure = 0

    -- Split the integral of sum into sum of integral.
    have h1 : Integrable (λ x ↦ (∑ i ∈ Set.univ, d_ln_π i x * (ϕ i).1 x)) μ.toMeasure := is_integrable_H₀ μ _
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x ∂μ.toMeasure = 0
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0have h2 : Integrable (λ x ↦ (∑ i ∈ Set.univ, dϕ i x)) μ.toMeasure := is_integrable_H₀ μ _
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x ∂μ.toMeasure = 0
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x ∂μ.toMeasure = 0integral_add (h1) h2
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
mp
∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂μ.toMeasure =
  0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
mp
∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂μ.toMeasure =
  0

    -- Make the `Set.univ` appears for using the density later.
    have int_univ : ∫ a, ∑ i ∈ Set.univ, d_ln_π i a * (ϕ i).1 a ∂μ.toMeasure = ∫ a in Set.univ, ∑ i ∈ Set.univ, d_ln_π i a * (ϕ i).1 a ∂μ.toMeasure := 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
mp
∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂μ.toMeasure =
  0by
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
mp
∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂μ.toMeasure =
  0simp
Goals accomplished!
Goals accomplished! 🐙
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
mp
∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂μ.toMeasure =
  0int_univ
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
mp
∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂μ.toMeasure =
  0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
mp
∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂μ.toMeasure =
  0

    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
mp
∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂μ.toMeasure =
  0remove_univ_integral μ.toMeasure (λ x ↦ ∑ i ∈ Set.univ, d_ln_π i x * (ϕ i).1 x)
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x ∂μ.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂μ.toMeasure =
  0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x ∂μ.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂μ.toMeasure =
  0

    -- Replace μ by π in the integration.
    rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x ∂μ.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂μ.toMeasure =
  0h
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x ∂π.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x ∂π.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0

    -- Replace by its density.
    have hi := is_integrable_H₀ π (λ x ↦ ∑ i ∈ Set.univ, d_ln_π i x * (ϕ i).1 x)
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x ∂π.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0

    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mp
∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x ∂π.toMeasure +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0π.density_integration hi (is_measurable_H₀_enn _) (is_measurable_H₀_enn _) (is_integrable_H₀_volume _)
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mp
(∫ (x : ↑Ω), (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mp
(∫ (x : ↑Ω), (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0

    -- Get ENNReal.toReal (π.d x) in the sum (a * ∑ b = ∑ b * a).
    have mul_dist : ∀x, ENNReal.toReal (π.d x) * (∑ i ∈ Set.univ, (λ i ↦ d_ln_π i x * (ϕ i).1 x) i) = ∑ i ∈ Set.univ, (λ i ↦ d_ln_π i x * (ϕ i).1 x) i * ENNReal.toReal (π.d x) := 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mp
(∫ (x : ↑Ω), (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0by
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal{
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal

      
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toRealhave mul_dist_sum : ∀ (a : ℝ), ∀ (f : Fin d → ℝ), (∑ i ∈ Set.univ, f i) * a = ∑ i ∈ Set.univ, f i * a := λ a f ↦ sum_mul (Set.toFinset Set.univ) (λ i ↦ f i) a
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist_sum: ∀ (a : ℝ) (f : Fin d → ℝ), (∑ i ∈ Set.univ.toFinset, f i) * a = ∑ i ∈ Set.univ.toFinset, f i * a
∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
      
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toRealintro x
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist_sum: ∀ (a : ℝ) (f : Fin d → ℝ), (∑ i ∈ Set.univ.toFinset, f i) * a = ∑ i ∈ Set.univ.toFinset, f i * a
x: ↑Ω
(π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
  ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
      
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toRealrw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist_sum: ∀ (a : ℝ) (f : Fin d → ℝ), (∑ i ∈ Set.univ.toFinset, f i) * a = ∑ i ∈ Set.univ.toFinset, f i * a
x: ↑Ω
(π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
  ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toRealmul_comm
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist_sum: ∀ (a : ℝ) (f : Fin d → ℝ), (∑ i ∈ Set.univ.toFinset, f i) * a = ∑ i ∈ Set.univ.toFinset, f i * a
x: ↑Ω
(∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i) * (π.d x).toReal =
  ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist_sum: ∀ (a : ℝ) (f : Fin d → ℝ), (∑ i ∈ Set.univ.toFinset, f i) * a = ∑ i ∈ Set.univ.toFinset, f i * a
x: ↑Ω
(∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i) * (π.d x).toReal =
  ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
      
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toRealexact mul_dist_sum (ENNReal.toReal (π.d x)) (λ i ↦ d_ln_π i x * (ϕ i).1 x)
Goals accomplished!
Goals accomplished! 🐙
    }
Goals accomplished!
Goals accomplished! 🐙
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0simp_rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mp
(∫ (x : ↑Ω), (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0mul_dist
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, d_ln_π x_1 x * ↑(ϕ x_1) x * (π.d x).toReal) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, d_ln_π x_1 x * ↑(ϕ x_1) x * (π.d x).toReal) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0

    -- Make the product ENNReal.toReal (π.d x) * d_ln_π i x appears to use the ln derivative rule.
    have mul_comm : ∀x, ∀i, d_ln_π i x * (ϕ i).1 x * ENNReal.toReal (π.d x) = ENNReal.toReal (π.d x) * d_ln_π i x * (ϕ i).1 x := λ x i ↦ (mul_rotate (ENNReal.toReal (π.d x)) (d_ln_π i x) ((ϕ i).1 x)).symm
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, d_ln_π x_1 x * ↑(ϕ x_1) x * (π.d x).toReal) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0simp_rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, d_ln_π x_1 x * ↑(ϕ x_1) x * (π.d x).toReal) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0mul_comm,
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, (π.d x).toReal * d_ln_π x_1 x * ↑(ϕ x_1) x) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, d_ln_π x_1 x * ↑(ϕ x_1) x * (π.d x).toReal) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0hπ'
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0

    -- Make the `Set.univ` appears to use the density.
    have int_univ : ∫ a, ∑ i ∈ Set.univ, dϕ i a ∂π.toMeasure = ∫ a in Set.univ, ∑ i ∈ Set.univ, dϕ i a ∂π.toMeasure := 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0by
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0simp
Goals accomplished!
Goals accomplished! 🐙
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0int_univ
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0

    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  0remove_univ_integral π.toMeasure (λ x ↦ ∑ i ∈ Set.univ, dϕ i x)
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i x ∂π.toMeasure =
  0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i x ∂π.toMeasure =
  0
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0have hi := is_integrable_H₀ π (λ x ↦ ∑ i ∈ Set.univ, dϕ i x)
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi✝: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i x ∂π.toMeasure =
  0
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi✝: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i x ∂π.toMeasure =
  0π.density_integration hi (is_measurable_H₀_enn _) (is_measurable_H₀_enn _) (is_integrable_H₀_volume _)
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi✝: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (x : ↑Ω), (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, dϕ i x =
  0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi✝: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (x : ↑Ω), (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, dϕ i x =
  0

    -- Use the integration by parts on the right-hand side integral.

    rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi✝: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    ∫ (x : ↑Ω), (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, dϕ i x =
  0mv_integration_by_parts volume (λ x ↦ ENNReal.toReal (π.d x)) (λ i ↦ (ϕ i).1) dπ' dϕ (hstein)
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi✝: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    -∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, dπ' i x * ↑(ϕ i) x =
  0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: μ.toMeasure = π.toMeasure
split_sum: ∀ (x : ↑Ω),
  ∑ i ∈ Set.univ.toFinset, (d_ln_π i x * ↑(ϕ i) x + dϕ i x) =
    ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x + ∑ i ∈ Set.univ.toFinset, dϕ i x
h1: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) μ.toMeasure
h2: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) μ.toMeasure
int_univ✝: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, d_ln_π i a * ↑(ϕ i) a ∂μ.toMeasure
hi✝: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, d_ln_π i x * ↑(ϕ i) x) π.toMeasure
mul_dist: ∀ (x : ↑Ω),
  (π.d x).toReal * ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i =
    ∑ i ∈ Set.univ.toFinset, (fun i => d_ln_π i x * ↑(ϕ i) x) i * (π.d x).toReal
mul_comm: ∀ (x : ↑Ω) (i : Fin d), d_ln_π i x * ↑(ϕ i) x * (π.d x).toReal = (π.d x).toReal * d_ln_π i x * ↑(ϕ i) x
int_univ: ∫ (a : ↑Ω), ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure =
  ∫ (a : ↑Ω) in Set.univ, ∑ i ∈ Set.univ.toFinset, dϕ i a ∂π.toMeasure
hi: Integrable (fun x => ∑ i ∈ Set.univ.toFinset, dϕ i x) π.toMeasure
mp
(∫ (x : ↑Ω), ∑ x_1 ∈ Set.univ.toFinset, dπ' x_1 x * ↑(ϕ x_1) x) +
    -∫ (x : ↑Ω), ∑ i ∈ Set.univ.toFinset, dπ' i x * ↑(ϕ i) x =
  0
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
mp
μ.toMeasure = π.toMeasure → KSD μ.toMeasure π.toMeasure = 0simp
Goals accomplished!
Goals accomplished! 🐙

  -- KSD(μ | π) = 0 ↦ μ = π.
  intro h
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: KSD μ.toMeasure π.toMeasure = 0
mpr
μ.toMeasure = π.toMeasure
  
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: KSD μ.toMeasure π.toMeasure = 0
mpr
μ.toMeasure = π.toMeasurehksd.left
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
mpr
μ.toMeasure = π.toMeasure]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
mpr
μ.toMeasure = π.toMeasure at h
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
mpr
μ.toMeasure = π.toMeasure

  -- We use the fact that the kernel is positive-definite that implies that d_ln_π_μ = 0.
  have d_ln_π_μ_eq_0 := (h_kernel_positive d_ln_π_μ).right.mp h
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
mpr
μ.toMeasure = π.toMeasure

  -- Simple derivative rule: ∂x f x = 0 → f x = c
  specialize hd_ln_π_μ μ.toMeasure d_ln_π_μ_eq_0
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
hd_ln_π_μ: ∃ c, ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
mpr
μ.toMeasure = π.toMeasure

  
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0rcases hd_ln_π_μ with ⟨c, h⟩
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
mpr.intro
μ.toMeasure = π.toMeasure

  -- We show that, since, for almost all x, log (μ.d x / π.d x) = c, then μ.d x = exp(c) * π.d x.
  have dμ_propor : ∀ᵐ x ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x := 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
mpr.intro
μ.toMeasure = π.toMeasureby
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x{
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
    -- We show that the two sets are equal to show that the complement of right-hand side one is of null measure.
    have eq_sets : {x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x} := 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d xby
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
{x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}{
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
{x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}
      
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
{x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}ext x
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
h
x ∈ {x | log (μ.d x / π.d x) = c} ↔ x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}
      
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
{x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}constructor
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
h.mp
x ∈ {x | log (μ.d x / π.d x) = c} → x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
h.mpr
x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x} → x ∈ {x | log (μ.d x / π.d x) = c}
      
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
{x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}·
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
h.mp
x ∈ {x | log (μ.d x / π.d x) = c} → x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x} 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
h.mp
x ∈ {x | log (μ.d x / π.d x) = c} → x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
h.mpr
x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x} → x ∈ {x | log (μ.d x / π.d x) = c}intro (hx : log (μ.d x / π.d x) = c)
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
h.mp
x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}
        
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
h.mp
x ∈ {x | log (μ.d x / π.d x) = c} → x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}have frac_neq_zero : μ.d x / π.d x ≠ 0 := 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
h.mp
x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}by
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
μ.d x / π.d x ≠ 0{
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
μ.d x / π.d x ≠ 0
          
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
μ.d x / π.d x ≠ 0have frac_pos : 0 < μ.d x / π.d x := ENNReal.div_pos_iff.mpr ⟨hdμ x, π.d_neq_top x⟩
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_pos: 0 < μ.d x / π.d x
μ.d x / π.d x ≠ 0
          
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
μ.d x / π.d x ≠ 0exact zero_lt_iff.mp frac_pos
Goals accomplished!
Goals accomplished! 🐙
        }
Goals accomplished!
Goals accomplished! 🐙

        
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
h.mp
x ∈ {x | log (μ.d x / π.d x) = c} → x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}have frac_finite : μ.d x / π.d x ≠ ∞ := 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
h.mp
x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}by
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
μ.d x / π.d x ≠ ⊤{
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
μ.d x / π.d x ≠ ⊤
          
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
μ.d x / π.d x ≠ ⊤by_contra h
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝¹: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h✝: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
h: μ.d x / π.d x = ⊤
False
          
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
μ.d x / π.d x ≠ ⊤rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝¹: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h✝: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
h: μ.d x / π.d x = ⊤
Falsediv_eq_top
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝¹: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h✝: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
h: μ.d x ≠ 0 ∧ π.d x = 0 ∨ μ.d x = ⊤ ∧ π.d x ≠ ⊤
False]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝¹: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h✝: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
h: μ.d x ≠ 0 ∧ π.d x = 0 ∨ μ.d x = ⊤ ∧ π.d x ≠ ⊤
False at h
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝¹: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h✝: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
h: μ.d x ≠ 0 ∧ π.d x = 0 ∨ μ.d x = ⊤ ∧ π.d x ≠ ⊤
False
          
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
μ.d x / π.d x ≠ ⊤cases h with
            
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝¹: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h✝: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
h: μ.d x ≠ 0 ∧ π.d x = 0 ∨ μ.d x = ⊤ ∧ π.d x ≠ ⊤
False| inl hp =>
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝¹: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h✝: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
h: μ.d x ≠ 0 ∧ π.d x = 0 ∨ μ.d x = ⊤ ∧ π.d x ≠ ⊤
False{
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
hp: μ.d x ≠ 0 ∧ π.d x = 0
inl
False
              
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
hp: μ.d x ≠ 0 ∧ π.d x = 0
inl
Falsercases hp with ⟨_, hpr⟩
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
left✝: μ.d x ≠ 0
hpr: π.d x = 0
inl.intro
False
              
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
hp: μ.d x ≠ 0 ∧ π.d x = 0
inl
Falseexact hdπ x hpr
Goals accomplished!
Goals accomplished! 🐙
            }
Goals accomplished!
Goals accomplished! 🐙
            
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝¹: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h✝: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
h: μ.d x ≠ 0 ∧ π.d x = 0 ∨ μ.d x = ⊤ ∧ π.d x ≠ ⊤
False| inr hq =>
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝¹: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h✝: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
h: μ.d x ≠ 0 ∧ π.d x = 0 ∨ μ.d x = ⊤ ∧ π.d x ≠ ⊤
False{
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
hq: μ.d x = ⊤ ∧ π.d x ≠ ⊤
inr
False
              
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
hq: μ.d x = ⊤ ∧ π.d x ≠ ⊤
inr
Falsercases hq with ⟨hql, _⟩
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
hql: μ.d x = ⊤
right✝: π.d x ≠ ⊤
inr.intro
False
              
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
hq: μ.d x = ⊤ ∧ π.d x ≠ ⊤
inr
Falseexact μ.d_neq_top x hql
Goals accomplished!
Goals accomplished! 🐙
            }
Goals accomplished!
Goals accomplished! 🐙
        }
Goals accomplished!
Goals accomplished! 🐙

        
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
h.mp
x ∈ {x | log (μ.d x / π.d x) = c} → x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}have cancel_log_exp : μ.d x / π.d x = ENNReal.ofReal (Real.exp c) := cancel_log_exp (μ.d x / π.d x) frac_neq_zero frac_finite c hx
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: log (μ.d x / π.d x) = c
frac_neq_zero: μ.d x / π.d x ≠ 0
frac_finite: μ.d x / π.d x ≠ ⊤
cancel_log_exp: μ.d x / π.d x = ENNReal.ofReal (Real.exp c)
h.mp
x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}
        
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
h.mp
x ∈ {x | log (μ.d x / π.d x) = c} → x ∈ {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}simp [←cancel_log_exp, ENNReal.div_eq_inv_mul, mul_right_comm (π.d x)⁻¹ (μ.d x) (π.d x), ENNReal.inv_mul_cancel (hdπ x) (π.d_neq_top x)]
Goals accomplished!
Goals accomplished! 🐙

      
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
{x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}intro (hx : μ.d x = ENNReal.ofReal (Real.exp c) * π.d x)
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
h.mpr
x ∈ {x | log (μ.d x / π.d x) = c}

      
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
{x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}have log_exp := enn_comp_log_exp c
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
h.mpr
x ∈ {x | log (μ.d x / π.d x) = c}

      
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
{x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}have frac : μ.d x / π.d x = ENNReal.ofReal (Real.exp c) := 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
h.mpr
x ∈ {x | log (μ.d x / π.d x) = c}by
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
μ.d x / π.d x = ENNReal.ofReal (Real.exp c){
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
μ.d x / π.d x = ENNReal.ofReal (Real.exp c)
        
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
μ.d x / π.d x = ENNReal.ofReal (Real.exp c)rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
μ.d x / π.d x = ENNReal.ofReal (Real.exp c)hx,
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
ENNReal.ofReal (Real.exp c) * π.d x / π.d x = ENNReal.ofReal (Real.exp c) 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
μ.d x / π.d x = ENNReal.ofReal (Real.exp c)mul_div_assoc (ENNReal.ofReal (Real.exp c)) (π.d x) (π.d x)
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
ENNReal.ofReal (Real.exp c) * (π.d x / π.d x) = ENNReal.ofReal (Real.exp c)]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
ENNReal.ofReal (Real.exp c) * (π.d x / π.d x) = ENNReal.ofReal (Real.exp c)
        
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
μ.d x / π.d x = ENNReal.ofReal (Real.exp c)rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
ENNReal.ofReal (Real.exp c) * (π.d x / π.d x) = ENNReal.ofReal (Real.exp c)show (π.d x) / (π.d x) = (π.d x) * (π.d x)⁻¹ by 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
ENNReal.ofReal (Real.exp c) * (π.d x / π.d x) = ENNReal.ofReal (Real.exp c)rfl
Goals accomplished!
Goals accomplished! 🐙]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
ENNReal.ofReal (Real.exp c) * (π.d x * (π.d x)⁻¹) = ENNReal.ofReal (Real.exp c)
        
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
μ.d x / π.d x = ENNReal.ofReal (Real.exp c)rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
ENNReal.ofReal (Real.exp c) * (π.d x * (π.d x)⁻¹) = ENNReal.ofReal (Real.exp c)ENNReal.mul_inv_cancel (hdπ x) (π.d_neq_top x)
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
ENNReal.ofReal (Real.exp c) * 1 = ENNReal.ofReal (Real.exp c)]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
ENNReal.ofReal (Real.exp c) * 1 = ENNReal.ofReal (Real.exp c)
        
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
μ.d x / π.d x = ENNReal.ofReal (Real.exp c)simp
Goals accomplished!
Goals accomplished! 🐙
      }
Goals accomplished!
Goals accomplished! 🐙

      
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
{x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}rwa [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (ENNReal.ofReal (Real.exp c)) = c
frac: μ.d x / π.d x = ENNReal.ofReal (Real.exp c)
h.mpr
x ∈ {x | log (μ.d x / π.d x) = c}←frac
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (μ.d x / π.d x) = c
frac: μ.d x / π.d x = ENNReal.ofReal (Real.exp c)
h.mpr
x ∈ {x | log (μ.d x / π.d x) = c}]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
x: ↑Ω
hx: μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
log_exp: log (μ.d x / π.d x) = c
frac: μ.d x / π.d x = ENNReal.ofReal (Real.exp c)
h.mpr
x ∈ {x | log (μ.d x / π.d x) = c} at log_exp
Goals accomplished!
Goals accomplished! 🐙
    }
Goals accomplished!
Goals accomplished! 🐙
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d xhave null_measure : μ {x | log (μ.d x / π.d x) = c}ᶜ = 0 := h
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
eq_sets: {x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}
null_measure: μ.toMeasure {x | log (μ.d x / π.d x) = c}ᶜ = 0
∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d xrwa [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
eq_sets: {x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}
null_measure: μ.toMeasure {x | log (μ.d x / π.d x) = c}ᶜ = 0
∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d xcongrArg compl eq_sets
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
eq_sets: {x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}
null_measure: μ.toMeasure {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}ᶜ = 0
∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
eq_sets: {x | log (μ.d x / π.d x) = c} = {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}
null_measure: μ.toMeasure {x | μ.d x = ENNReal.ofReal (Real.exp c) * π.d x}ᶜ = 0
∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x at null_measure
Goals accomplished!
Goals accomplished! 🐙
  }
Goals accomplished!
Goals accomplished! 🐙

  -- We show by contradiction that ENNReal.ofReal (Real.exp c) = 1. If it is ≠ 1, this implies a contradiction as μ.d x = ENNReal.ofReal (Real.exp c) * π.d x and ∫⁻ x, μ.d x ∂ν = 1.
  have exp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 := 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
mpr.intro
μ.toMeasure = π.toMeasureby
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1{
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1by_contra hc
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ¬ENNReal.ofReal (Real.exp c) = 1
False;
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ¬ENNReal.ofReal (Real.exp c) = 1
False 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1push_neg at hc
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
False
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1have univ_eq_one_μ : ∫⁻ _, 1 ∂μ.toMeasure = 1 := 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
Falseby
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
Falsesimp
Goals accomplished!
Goals accomplished! 🐙
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1have univ_eq_one_π : ∫⁻ _, 1 ∂π.toMeasure = 1 := 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_μ: ∫⁻ (x : ↑Ω), 1 ∂μ.toMeasure = 1
Falseby
Goals accomplished!
Goals accomplished! 🐙 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_μ: ∫⁻ (x : ↑Ω), 1 ∂μ.toMeasure = 1
Falsesimp
Goals accomplished!
Goals accomplished! 🐙

    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_μ: ∫⁻ (x : ↑Ω), 1 ∂μ.toMeasure = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
Falseμ.density_lintegration (λ _ ↦ 1) (is_measurable_H₀_enn (λ _ ↦ 1))
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_μ: ∫⁻ (x : ↑Ω), μ.d x * 1 = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
False]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_μ: ∫⁻ (x : ↑Ω), μ.d x * 1 = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
False at univ_eq_one_μ
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_μ: ∫⁻ (x : ↑Ω), μ.d x * 1 = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
False

    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1have ae_μ_to_ae_volume := (DensityMeasure.ae_density_measure_iff_ae_volume (ae_of_all _ hdμ)).mp dμ_propor
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_μ: ∫⁻ (x : ↑Ω), μ.d x * 1 = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
False

    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1simp_rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_μ: ∫⁻ (x : ↑Ω), μ.d x * 1 = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
Falsemul_one
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (x : ↑Ω), μ.d x = 1
False] at univ_eq_one_μ
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (x : ↑Ω), μ.d x = 1
False
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (x : ↑Ω), μ.d x = 1
Falselintegral_congr_ae ae_μ_to_ae_volume
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
False]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
False at univ_eq_one_μ
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
False

    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), 1 ∂π.toMeasure = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
Falseπ.density_lintegration (λ _ ↦ 1) (is_measurable_H₀_enn (λ _ ↦ 1))
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x * 1 = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
False]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x * 1 = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
False at univ_eq_one_π
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x * 1 = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
False
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1simp_rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x * 1 = 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
Falsemul_one
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x = 1
False] at univ_eq_one_π
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x = 1
False

    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x = 1
Falselintegral_const_mul (ENNReal.ofReal (Real.exp c)) (π.d_measurable),
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ENNReal.ofReal (Real.exp c) * ∫⁻ (a : ↑Ω), π.d a = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x = 1
False 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x = 1
Falseuniv_eq_one_π,
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ENNReal.ofReal (Real.exp c) * 1 = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x = 1
False 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ∫⁻ (a : ↑Ω), ENNReal.ofReal (Real.exp c) * π.d a = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x = 1
Falsemul_one
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ENNReal.ofReal (Real.exp c) = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x = 1
False]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ENNReal.ofReal (Real.exp c) = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x = 1
False at univ_eq_one_μ
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
hc: ENNReal.ofReal (Real.exp c) ≠ 1
ae_μ_to_ae_volume: μ.d =ᶠ[ae volume] fun x => ENNReal.ofReal (Real.exp c) * π.d x
univ_eq_one_μ: ENNReal.ofReal (Real.exp c) = 1
univ_eq_one_π: ∫⁻ (x : ↑Ω), π.d x = 1
False
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
ENNReal.ofReal (Real.exp c) = 1exact hc univ_eq_one_μ
Goals accomplished!
Goals accomplished! 🐙
  }
Goals accomplished!
Goals accomplished! 🐙

  -- We rewrite μ = π as ∀s, ∫⁻ x in s, μ.d ∂ν = ∀s, ∫⁻ x in s, π.d ∂ν and use μ.d = 1 * π.d.
  simp_rw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
mpr.intro
μ.toMeasure = π.toMeasureexp_c_eq_one,
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = 1 * π.d x
mpr.intro
μ.toMeasure = π.toMeasure 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = ENNReal.ofReal (Real.exp c) * π.d x
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
mpr.intro
μ.toMeasure = π.toMeasureone_mul
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = π.d x
mpr.intro
μ.toMeasure = π.toMeasure] at dμ_propor
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = π.d x
mpr.intro
μ.toMeasure = π.toMeasure
  
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0refine DensityMeasure.densities_ae_eq_iff_eq_measure.mp ?_
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ, dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
h✝: ∫ (x : ↑Ω) in Set.univ,
    ∫ (x' : ↑Ω) in Set.univ,
      ∑ i ∈ Set.univ.toFinset, d_ln_π_μ i x * k H₀ x x' * d_ln_π_μ i x' ∂μ.toMeasure ∂μ.toMeasure =
  0
d_ln_π_μ_eq_0: ∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, d_ln_π_μ i x = 0
c: ℝ
h: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, log (μ.d x / π.d x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: ∀ᵐ (x : ↑Ω) ∂μ.toMeasure, μ.d x = π.d x
mpr.intro
μ.d =ᶠ[ae volume] π.d
  
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0exact (μ.ae_density_measure_iff_ae_volume (ae_of_all _ hdμ)).mp dμ_propor
Goals accomplished!
Goals accomplished! 🐙

include hd_ln_π_μ dπ' hπ' hstein hdμ hdπ h_kernel_positive
/--
  π is the only fixed point of Φₜ(μ). We proved that by showing that, if μ = π, ϕ^* = 0 and ϕ^* ≠ 0 otherwise.
-/
lemma π_unique_fixed_point (hksd : is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD) (ksd_norm : is_ksd_norm hd μ π H₀ ϕ KSD) : μ.toMeasure = π.toMeasure ↔ ∀ i, ϕ i = 0 :=
Goals accomplished!by
Goals accomplished!
Goals accomplished! 🐙
  
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
μ.toMeasure = π.toMeasure ↔ ∀ (i : Fin d), ϕ i = 0have KSD_discrepancy := KSD_is_valid_discrepancy hd μ π hdμ hdπ H₀ h_kernel_positive d_ln_π ϕ dϕ d_ln_π_μ hd_ln_π_μ dπ' hπ' KSD hstein hksd
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
μ.toMeasure = π.toMeasure ↔ ∀ (i : Fin d), ϕ i = 0
  
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
μ.toMeasure = π.toMeasure ↔ ∀ (i : Fin d), ϕ i = 0constructor
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mp
μ.toMeasure = π.toMeasure → ∀ (i : Fin d), ϕ i = 0
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mpr
(∀ (i : Fin d), ϕ i = 0) → μ.toMeasure = π.toMeasure
  
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
μ.toMeasure = π.toMeasure ↔ ∀ (i : Fin d), ϕ i = 0{
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mp
μ.toMeasure = π.toMeasure → ∀ (i : Fin d), ϕ i = 0
    -- μ = π → ϕ^* = 0
    intro μ_eq_π
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
μ_eq_π: μ.toMeasure = π.toMeasure
mp
∀ (i : Fin d), ϕ i = 0

    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mp
μ.toMeasure = π.toMeasure → ∀ (i : Fin d), ϕ i = 0
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mpr
(∀ (i : Fin d), ϕ i = 0) → μ.toMeasure = π.toMeasurerw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
μ_eq_π: μ.toMeasure = π.toMeasure
mp
∀ (i : Fin d), ϕ i = 0ksd_norm,
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ ‖ϕ‖ ^ 2 = 0
μ_eq_π: μ.toMeasure = π.toMeasure
mp
∀ (i : Fin d), ϕ i = 0 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
μ_eq_π: μ.toMeasure = π.toMeasure
mp
∀ (i : Fin d), ϕ i = 0sq_eq_zero_iff
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ ‖ϕ‖ = 0
μ_eq_π: μ.toMeasure = π.toMeasure
mp
∀ (i : Fin d), ϕ i = 0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ ‖ϕ‖ = 0
μ_eq_π: μ.toMeasure = π.toMeasure
mp
∀ (i : Fin d), ϕ i = 0 at KSD_discrepancy
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ ‖ϕ‖ = 0
μ_eq_π: μ.toMeasure = π.toMeasure
mp
∀ (i : Fin d), ϕ i = 0
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mp
μ.toMeasure = π.toMeasure → ∀ (i : Fin d), ϕ i = 0
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mpr
(∀ (i : Fin d), ϕ i = 0) → μ.toMeasure = π.toMeasurerw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ ‖ϕ‖ = 0
μ_eq_π: μ.toMeasure = π.toMeasure
mp
∀ (i : Fin d), ϕ i = 0←norm_eq_zero_iff
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ ‖ϕ‖ = 0
μ_eq_π: μ.toMeasure = π.toMeasure
mp
‖ϕ‖ = 0]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ ‖ϕ‖ = 0
μ_eq_π: μ.toMeasure = π.toMeasure
mp
‖ϕ‖ = 0
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mp
μ.toMeasure = π.toMeasure → ∀ (i : Fin d), ϕ i = 0
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mpr
(∀ (i : Fin d), ϕ i = 0) → μ.toMeasure = π.toMeasureexact KSD_discrepancy.mp μ_eq_π
Goals accomplished!
Goals accomplished! 🐙
  }
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mpr
(∀ (i : Fin d), ϕ i = 0) → μ.toMeasure = π.toMeasure
  
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
μ.toMeasure = π.toMeasure ↔ ∀ (i : Fin d), ϕ i = 0{
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mpr
(∀ (i : Fin d), ϕ i = 0) → μ.toMeasure = π.toMeasure
    -- ϕ^* ≠ 0 → μ = π
    intro phi_norm
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: ∀ (i : Fin d), ϕ i = 0
mpr
μ.toMeasure = π.toMeasure
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mpr
(∀ (i : Fin d), ϕ i = 0) → μ.toMeasure = π.toMeasurerw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: ∀ (i : Fin d), ϕ i = 0
mpr
μ.toMeasure = π.toMeasure←norm_eq_zero_iff
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: ‖ϕ‖ = 0
mpr
μ.toMeasure = π.toMeasure]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: ‖ϕ‖ = 0
mpr
μ.toMeasure = π.toMeasure at phi_norm
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: ‖ϕ‖ = 0
mpr
μ.toMeasure = π.toMeasure
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mpr
(∀ (i : Fin d), ϕ i = 0) → μ.toMeasure = π.toMeasureby_contra h
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: ‖ϕ‖ = 0
h: ¬μ.toMeasure = π.toMeasure
mpr
False;
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: ‖ϕ‖ = 0
h: ¬μ.toMeasure = π.toMeasure
mpr
False
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mpr
(∀ (i : Fin d), ϕ i = 0) → μ.toMeasure = π.toMeasurerw [
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: ‖ϕ‖ = 0
h: ¬μ.toMeasure = π.toMeasure
mpr
False←sq_eq_zero_iff,
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: ‖ϕ‖ ^ 2 = 0
h: ¬μ.toMeasure = π.toMeasure
mpr
False 
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: ‖ϕ‖ = 0
h: ¬μ.toMeasure = π.toMeasure
mpr
False←ksd_norm
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: KSD μ.toMeasure π.toMeasure = 0
h: ¬μ.toMeasure = π.toMeasure
mpr
False]
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: KSD μ.toMeasure π.toMeasure = 0
h: ¬μ.toMeasure = π.toMeasure
mpr
False at phi_norm
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
phi_norm: KSD μ.toMeasure π.toMeasure = 0
h: ¬μ.toMeasure = π.toMeasure
mpr
False
    
Goals accomplished!
d: ℕ
hd: d ≠ 0
Ω: Set (Vector ℝ d)
inst✝⁵: MeasureSpace ↑Ω
μ, π: DensityMeasure ↑Ω
inst✝⁴: IsProbabilityMeasure μ.toMeasure
inst✝³: IsProbabilityMeasure π.toMeasure
hdμ: ∀ (x : ↑Ω), μ.d x ≠ 0
hdπ: ∀ (x : ↑Ω), π.d x ≠ 0
H₀: Set (↑Ω → ℝ)
inst✝²: NormedAddCommGroup ↑H₀
inst✝¹: InnerProductSpace ℝ ↑H₀
s: RKHS H₀
h_kernel_positive: positive_definite_kernel μ H₀
d_ln_π: Fin d → ↑Ω → ℝ
ϕ: product_RKHS H₀ hd
dϕ, d_ln_π_μ: Fin d → ↑Ω → ℝ
hd_ln_π_μ: ∀ (ν : Measure ↑Ω), (∀ (i : Fin d), ∀ᵐ (x : ↑Ω) ∂ν, d_ln_π_μ i x = 0) → ∃ c, ∀ᵐ (x : ↑Ω) ∂ν, log (μ.d x / π.d x) = c
dπ': Fin d → ↑Ω → ℝ
hπ': ∀ (x : ↑Ω) (i : Fin d), (π.d x).toReal * d_ln_π i x = dπ' i x
inst✝: Norm ↑Ω
KSD: Measure ↑Ω → Measure ↑Ω → ℝ
hstein: SteinClass π fun i => ↑(ϕ i)
hksd: is_ksd hd μ π H₀ d_ln_π ϕ dϕ d_ln_π_μ KSD
ksd_norm: is_ksd_norm hd μ π H₀ ϕ KSD
KSD_discrepancy: μ.toMeasure = π.toMeasure ↔ KSD μ.toMeasure π.toMeasure = 0
mpr
(∀ (i : Fin d), ϕ i = 0) → μ.toMeasure = π.toMeasureexact h (KSD_discrepancy.mpr phi_norm)
Goals accomplished!
Goals accomplished! 🐙
  }
Goals accomplished!
Goals accomplished! 🐙