| import OAI.Analysis.FiniteFactor.Coupling.Density |
| import OAI.Analysis.FiniteFactor.Coupling.Parameters |
| import Mathlib.MeasureTheory.Function.L2Space |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory |
| noncomputable section |
|
|
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsProbabilityMeasure μ] |
| variable {ι : Type*} [Fintype ι] |
|
|
| def mixtureDensity (q : ℝ) (π : ι → ℝ) (d : ι → X → ℝ) (x : X) : ℝ := |
| q + ∑ i, π i * d i x |
|
|
| theorem mixtureDensity_memLp (q : ℝ) (π : ι → ℝ) (d : ι → X → ℝ) |
| (hd : ∀ i, MemLp (d i) 2 μ) : MemLp (mixtureDensity q π d) 2 μ := |
| (memLp_const q).add (memLp_finsetSum Finset.univ (fun i _ => (hd i).const_mul (π i))) |
|
|
| theorem mixtureDensity_integral (q : ℝ) (π : ι → ℝ) (d : ι → X → ℝ) |
| (hd : ∀ i, Integrable (d i) μ) (hm : ∀ i, ∫ x, d i x ∂μ = 1) |
| (hπ : q + ∑ i, π i = 1) : ∫ x, mixtureDensity q π d x ∂μ = 1 := by |
| have hi := integral_add (integrable_const q) |
| (integrable_finsetSum Finset.univ (fun i _ => (hd i).const_mul (π i))) |
| unfold mixtureDensity |
| rw [hi, integral_finsetSum Finset.univ (fun i _ => (hd i).const_mul (π i))] |
| simp_rw [integral_const_mul, hm, mul_one] |
| simpa using hπ |
|
|
| theorem mixtureDensity_energy_le (q : ℝ) (π : ι → ℝ) (d : ι → X → ℝ) |
| (hd : ∀ i, MemLp (d i) 2 μ) (hm : ∀ i, ∫ x, d i x ∂μ = 1) |
| (hπ : q + ∑ i, π i = 1) (E : ι → ℝ) |
| (hE : ∀ i, (∫ x, (d i x - 1)^2 ∂μ) ≤ E i) : |
| (∫ x, mixtureDensity q π d x ^ 2 ∂μ) ≤ |
| 1 + (Fintype.card ι : ℝ) * ∑ i, (π i)^2 * E i := by |
| classical |
| have hd1 (i) : Integrable (d i) μ := (hd i).integrable (by norm_num) |
| have hh := mixtureDensity_memLp q π d hd |
| have hm' := mixtureDensity_integral q π d hd1 hm hπ |
| have heq (x : X) : mixtureDensity q π d x - 1 = ∑ i, π i * (d i x - 1) := by |
| simp only [mixtureDensity, mul_sub, mul_one, Finset.sum_sub_distrib] |
| linarith |
| have hbound (x : X) : (mixtureDensity q π d x - 1)^2 ≤ |
| (Fintype.card ι : ℝ) * ∑ i, (π i)^2 * (d i x - 1)^2 := by |
| rw [heq] |
| simpa [mul_pow] using |
| (sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := fun i => π i * (d i x - 1))) |
| have hsub (i) : MemLp (fun x => d i x - 1) 2 μ := (hd i).sub (memLp_const 1) |
| have his (i) : Integrable (fun x => (π i)^2 * (d i x - 1)^2) μ := |
| (hsub i).integrable_sq.const_mul _ |
| have hi := integral_mono ((hh.sub (memLp_const 1)).integrable_sq) |
| ((integrable_finsetSum Finset.univ (fun i _ => his i)).const_mul (Fintype.card ι : ℝ)) hbound |
| rw [integral_const_mul, integral_finsetSum Finset.univ (fun i _ => his i)] at hi |
| simp_rw [integral_const_mul] at hi |
| have hsum : (∑ i, (π i)^2 * ∫ x, (d i x - 1)^2 ∂μ) ≤ ∑ i, (π i)^2 * E i := by |
| exact Finset.sum_le_sum fun i _ => mul_le_mul_of_nonneg_left (hE i) (sq_nonneg _) |
| have he := density_energy_identity (hh.integrable (by norm_num)) hh.integrable_sq hm' |
| have hc := mul_le_mul_of_nonneg_left hsum (Nat.cast_nonneg (Fintype.card ι) : (0 : ℝ) ≤ _) |
| dsimp only [Pi.sub_apply] at hi |
| linarith |
|
|
| def unitIntervalMeasure : Measure ℝ := volume.restrict (Set.Ico 0 1) |
|
|
| instance : IsProbabilityMeasure unitIntervalMeasure := by |
| constructor |
| simp [unitIntervalMeasure, Real.volume_Ico] |
|
|
| theorem unitIntervalMeasure_interval {a b : ℝ} (ha : 0 ≤ a) (hab : a ≤ b) (hb : b ≤ 1) : |
| unitIntervalMeasure.real (Set.Ico a b) = b-a := by |
| rw [unitIntervalMeasure, measureReal_restrict_apply measurableSet_Ico, |
| Set.inter_eq_left.mpr (Set.Ico_subset_Ico ha hb), Real.volume_real_Ico_of_le hab] |
|
|
| namespace CouplingParameters |
| variable (p : CouplingParameters) |
|
|
| def endpointLength (right : Bool) (i : Fin 5) : ℝ := |
| if right then p.rightLength i else p.leftLength i |
|
|
| def endpointInterval (right : Bool) (i : Fin 5) : Set ℝ := |
| if right then Set.Ico (1-p.rightLength i) 1 else Set.Ico 0 (p.leftLength i) |
|
|
| def endpointComponentDensity (right : Bool) (i : Fin 5) : ℝ → ℝ := |
| intervalDensity (p.endpointInterval right i) (p.endpointLength right i) |
|
|
| def endpointDensity (right : Bool) : ℝ → ℝ := |
| mixtureDensity p.diagonalMass p.replacementMass (p.endpointComponentDensity right) |
|
|
| theorem endpointLength_pos (right : Bool) (i : Fin 5) : 0 < p.endpointLength right i := by |
| cases right <;> simp [endpointLength, leftLength_pos, rightLength_pos] |
|
|
| theorem endpointLength_le (right : Bool) (i : Fin 5) : p.endpointLength right i ≤ 1 := by |
| have := p.lengths_sum_le i |
| have := p.leftLength_pos i |
| have := p.rightLength_pos i |
| cases right <;> simp [endpointLength] <;> linarith |
|
|
| theorem endpointInterval_measurable (right : Bool) (i : Fin 5) : |
| MeasurableSet (p.endpointInterval right i) := by |
| cases right <;> exact measurableSet_Ico |
|
|
| theorem endpointInterval_mass (right : Bool) (i : Fin 5) : |
| unitIntervalMeasure.real (p.endpointInterval right i) = p.endpointLength right i := by |
| have hpos := p.endpointLength_pos right i |
| have hle := p.endpointLength_le right i |
| cases right |
| · simpa [endpointInterval, endpointLength] using |
| unitIntervalMeasure_interval (a := 0) (by norm_num) hpos.le hle |
| · have h₀ : 0 ≤ 1-p.rightLength i := by simpa [endpointLength] using sub_nonneg.mpr hle |
| have h₁ : 1-p.rightLength i ≤ 1 := by have := p.rightLength_pos i; linarith |
| simpa [endpointInterval, endpointLength] using |
| unitIntervalMeasure_interval h₀ h₁ le_rfl |
|
|
| theorem endpointDensity_memLp (right : Bool) : MemLp (p.endpointDensity right) 2 unitIntervalMeasure := by |
| apply mixtureDensity_memLp |
| intro i |
| exact (memLp_const _).indicator (p.endpointInterval_measurable right i) |
|
|
| theorem endpointDensity_integral (right : Bool) : |
| ∫ t, p.endpointDensity right t ∂unitIntervalMeasure = 1 := by |
| apply mixtureDensity_integral |
| · intro i |
| exact intervalDensity_integrable (p.endpointInterval_measurable right i) _ |
| · intro i |
| exact intervalDensity_integral (p.endpointInterval_measurable right i) |
| (p.endpointLength_pos right i) (p.endpointInterval_mass right i) |
| · exact p.mass_sum |
|
|
| theorem endpointDensity_lower (right : Bool) (t : ℝ) : |
| p.diagonalMass ≤ p.endpointDensity right t := by |
| apply le_add_of_nonneg_right |
| exact Finset.sum_nonneg fun i _ => mul_nonneg (p.replacementMass_nonneg i) |
| (intervalDensity_nonneg (p.endpointLength_pos right i).le t) |
|
|
| end CouplingParameters |
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
