openAI-math-dataset / lean /OAI /Probability /FreeIsing /Estimates /DiscrepancyDensityModification.lean
| import OAI.Probability.FreeIsing.Energy.DiscrepancyRowEnergy | |
| import OAI.Probability.FreeIsing.Estimates.ProgressivePrimitiveFubini | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped NNReal | |
| namespace Problem367.DiscrepancyEnergy | |
| variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Countable ι] | |
| theorem density_congr_ae {μ : Measure Ω} | |
| {c c' b b' : ι → Ω → ℝ} {X Y m m' k k' : Ω → ℝ} | |
| (hc : ∀ i, c i =ᵐ[μ] b i) (hc' : ∀ i, c' i =ᵐ[μ] b' i) | |
| (hX : X =ᵐ[μ] Y) (hm : m =ᵐ[μ] k) (hm' : m' =ᵐ[μ] k') : | |
| density c c' X m m' =ᵐ[μ] density b b' Y k k' := by | |
| filter_upwards [ae_all_iff.mpr hc, ae_all_iff.mpr hc', hX, hm, hm'] | |
| with ω hω hω' hXω hmω hmω' | |
| simp only [density, hω, hω', hXω, hmω, hmω'] | |
| theorem isStronglyProgressive_density | |
| {F : Filtration ℝ≥0 mΩ} | |
| {c c' : ι → ℝ≥0 → Ω → ℝ} {X m m' : ℝ≥0 → Ω → ℝ} | |
| (hc : ∀ i, IsStronglyProgressive F (c i)) | |
| (hc' : ∀ i, IsStronglyProgressive F (c' i)) | |
| (hX : IsStronglyProgressive F X) | |
| (hm : IsStronglyProgressive F m) (hm' : IsStronglyProgressive F m') : | |
| IsStronglyProgressive F (fun t => | |
| density (fun i => c i t) (fun i => c' i t) (X t) (m t) (m' t)) := by | |
| intro T | |
| let : MeasurableSpace Ω := F T | |
| have hnoise : StronglyMeasurable (fun p : Set.Iic T × Ω => | |
| ∑' i : ι, (c i p.1 p.2 - c' i p.1 p.2) ^ 2) := StronglyMeasurable.tsum | |
| (fun i => ((hc i T).sub (hc' i T)).pow 2) | |
| have hleft : StronglyMeasurable (fun p : Set.Iic T × Ω => | |
| ∑' i : ι, c i p.1 p.2 ^ 2) := StronglyMeasurable.tsum (fun i => (hc i T).pow 2) | |
| have hright : StronglyMeasurable (fun p : Set.Iic T × Ω => | |
| ∑' i : ι, c' i p.1 p.2 ^ 2) := StronglyMeasurable.tsum (fun i => (hc' i T).pow 2) | |
| exact hnoise.add (((stronglyMeasurable_const.mul (hX T)).mul | |
| (((hm T).mul hleft).sub ((hm' T).mul hright)))) | |
| theorem ae_all_primitive_density_eq_of_modification {μ : Measure Ω} [SFinite μ] | |
| {F G : Filtration ℝ≥0 mΩ} | |
| {c c' b b' : ι → ℝ≥0 → Ω → ℝ} | |
| {X Y m m' k k' : ℝ≥0 → Ω → ℝ} | |
| (hc : ∀ i, IsStronglyProgressive F (c i)) | |
| (hc' : ∀ i, IsStronglyProgressive F (c' i)) | |
| (hX : IsStronglyProgressive F X) | |
| (hm : IsStronglyProgressive F m) (hm' : IsStronglyProgressive F m') | |
| (hb : ∀ i, IsStronglyProgressive G (b i)) | |
| (hb' : ∀ i, IsStronglyProgressive G (b' i)) | |
| (hY : IsStronglyProgressive G Y) | |
| (hk : IsStronglyProgressive G k) (hk' : IsStronglyProgressive G k') | |
| (hcb : ∀ i t, c i t =ᵐ[μ] b i t) | |
| (hcb' : ∀ i t, c' i t =ᵐ[μ] b' i t) | |
| (hXY : ∀ t, X t =ᵐ[μ] Y t) | |
| (hmk : ∀ t, m t =ᵐ[μ] k t) (hmk' : ∀ t, m' t =ᵐ[μ] k' t) : | |
| ∀ᵐ ω ∂μ, ∀ t, | |
| nnrealTimePrimitive (fun s => density (fun i => c i s) | |
| (fun i => c' i s) (X s) (m s) (m' s)) t ω = | |
| nnrealTimePrimitive (fun s => density (fun i => b i s) | |
| (fun i => b' i s) (Y s) (k s) (k' s)) t ω := by | |
| apply ae_all_nnrealTimePrimitive_eq_of_modification | |
| (isStronglyProgressive_density hc hc' hX hm hm') | |
| (isStronglyProgressive_density hb hb' hY hk hk') | |
| intro t | |
| exact density_congr_ae (fun i => hcb i t) (fun i => hcb' i t) | |
| (hXY t) (hmk t) (hmk' t) | |
| end Problem367.DiscrepancyEnergy | |
| end | |
| end OAI | |