openAI-math-dataset / lean /OAI /Probability /LogConcave /OraclePrograms /CenteringMeanCircuitFull.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.Sampling.VelocityJointLip | |
| import OAI.Probability.LogConcave.Complexity.KernelAnalyticNormalizedBudget | |
| import OAI.Probability.LogConcave.Sampling.StationaryMeanTarget | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling | |
| open Set MeasureTheory Quadrature ProbabilityTheory | |
| open scoped Classical BigOperators NNReal | |
| theorem centeringMeanCircuit_full_rms (n : ℕ) : | |
| ∃A B Ap Ah Lp Lh : ℝ≥0,∃C J D : ℝ,0 ≤ C ∧ 1 ≤ J ∧ 1 ≤ D ∧ ∃k : ℕ, | |
| ∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0},∀hF : Primitive F lam, | |
| ∀(x : Point d) {r : ℝ},∀hr : 0<r,r ≤ 1 → 0<lam → | |
| ∀hl : (lam:ℝ)*r^2 ≤ 1/2,(lam:ℝ)*r ≤ 1 → 1 ≤ d → | |
| ∀{T h : ℝ},∀hT0 : 0<T,∀hT1 : T<1,∀hh : 0<h,h ≤ Real.log 2 → h ≤ logMeshLength T → | |
| Ap*probabilityMeanLipschitz lam r ≤ 1/2 → Ah*probabilityMeanLipschitz lam r ≤ 1/2 → | |
| Lp*probabilityMeanLipschitz lam r ≤ 1/2 → Lh*probabilityMeanLipschitz lam r ≤ 1/2 → | |
| A*probabilityMeanLipschitz lam r ≤ 1/2 → B*probabilityMeanLipschitz lam r ≤ 1/2 → | |
| ∀N : ℕ,∀R : ℝ,0<R → R^2 ≤ 1-T^2 → ∀m : ℕ,1 ≤ m → ∀ψ : ℝ,0<ψ → ψ*m ≤ 1 → | |
| ∀s : ℝ,0<s → ∀nc Nc : ℕ,0<nc → | |
| centeringRowBudget nc*velocityJointLip n m A lam r T h ψ s ≤ 1/2 → | |
| let e := probabilityEndpoint hT0 hT1 hh (n+1) | |
| let E := velocityNormalizedBudget d k n m N lam Ap Ah Lp r R T h ψ C J D | |
| let K := velocityJointLip n m A lam r T h ψ s | |
| let Z := centerStateEnvelope nc Nc ((lam:ℝ)*r/s) R (centeringRowBudget nc*K:ℝ≥0) E | |
| ∃X : StationaryPath (gibbs (centeringPotential F x r T)) | |
| (skewLieField (centeringPotential F x r T) | |
| (jointSkew (centeringKernel hF x hr hl hT0.le hT1) s)), | |
| (∀i,Integrable (fun y => ‖stationaryPicard nc Nc | |
| (centeringVelocityJoint F x r T h ψ s n m N e) y i-X.path y (probabilityNodes nc i)‖^2) | |
| (gibbs (centeringPotential F x r T))) ∧ | |
| (∀i,(∫y,‖stationaryPicard nc Nc | |
| (centeringVelocityJoint F x r T h ψ s n m N e) y i-X.path y (probabilityNodes nc i)‖^2 | |
| ∂gibbs (centeringPotential F x r T)) ≤ (circuitD F x)^2*Z) ∧ | |
| Measurable (stationaryMeanTarget hF x hr hl hT0.le hT1 X) ∧ | |
| (gibbs (centeringPotential F x r T)).map (stationaryMeanTarget hF x hr hl hT0.le hT1 X)= | |
| (stdGaussian (Point d)).map (fun g => primitiveExpectedField F x r+s • g) ∧ | |
| Integrable (fun y => ‖centeringMeanCircuit F x r T h ψ s n m N nc Nc e y- | |
| stationaryMeanTarget hF x hr hl hT0.le hT1 X y‖^2) (gibbs (centeringPotential F x r T)) ∧ | |
| (∫y,‖centeringMeanCircuit F x r T h ψ s n m N nc Nc e y- | |
| stationaryMeanTarget hF x hr hl hT0.le hT1 X y‖^2 ∂gibbs (centeringPotential F x r T)) ≤ | |
| (s*circuitD F x)^2*centerMeanEnvelope nc ((lam:ℝ)*r/s) R Z := by | |
| obtain ⟨A,B,hLip⟩ := centeringVelocityJoint_lipschitz n | |
| obtain ⟨Ap,Ah,Lp,Lh,C,J,D,hC,hJ,hD,k,hRms⟩ := centeringVelocityJoint_full_rms n | |
| refine ⟨A,B,Ap,Ah,Lp,Lh,C,J,D,hC,hJ,hD,k,?_⟩ | |
| intro d F lam hF x r hr hr1 hlam hl hL hd T h hT0 hT1 hh hsmall hlarge hp hharm hlp hlh ha hb N R hR hRT m hm ψ hψ hψm s hs nc Nc hnc hq e E K Z | |
| have hhat := hLip hF x hr hl hT0 hT1 hh hψ hs m hψm ha hb N e | |
| have hv := hRms hF x hr hlam hl hd hT0 hT1 hh hsmall hlarge hp hharm hlp hlh N R hR hRT m hm ψ hψ hψm s | |
| have hE : 0 ≤ E := velocityNormalizedBudget_nonneg d k n m N lam Ap Ah Lp r R T h ψ C J D hR.le | |
| have hZ : 0 ≤ Z := centerStateEnvelope_nonneg nc Nc _ _ _ E (NNReal.coe_nonneg _) hE | |
| have hvel := velocityRmsBudget_normalized hF x Ap Ah Lp (C:=C) (J:=J) (D:=D) hr hr1 hR hT0 hT1 hh hψ hs hC (show 0≤J by linarith only [hJ]) hl hL k n m N | |
| obtain ⟨X⟩ := exists_centering_stationaryPath hF x hr hlam hl hT0.le hT1 hs | |
| have hstate (i : Fin (nc+1)) := centeringPicard_state_rms hF x hr hlam hl hR hRT hT0.le hT1 hs X | |
| n m N nc Nc hnc e hhat hq (show 0 ≤ (((lam:ℝ)*r/s)*circuitD F x)^2*E by positivity) | |
| hv.1 (hv.2.trans hvel) i | |
| have hstate' (i : Fin (nc+1)) : (∫y,‖stationaryPicard nc Nc | |
| (centeringVelocityJoint F x r T h ψ s n m N e) y i-X.path y (probabilityNodes nc i)‖^2 | |
| ∂gibbs (centeringPotential F x r T)) ≤ (circuitD F x)^2*Z := | |
| (hstate i).2.trans (centeringPicardBudget_normalized F x nc Nc lam K r R s E hE) | |
| have hmean := centeringMeanCircuit_rms hF x hr hlam hl hR hRT hT0.le hT1 hs X | |
| n m N nc Nc hnc e hhat (show 0 ≤ (circuitD F x)^2*Z by positivity) | |
| (fun i => (hstate i).1) hstate' | |
| refine ⟨X,(fun i => (hstate i).1),hstate',?_, | |
| stationaryMeanTarget_law hF x hr hlam hl hT0.le hT1 hs X,hmean.1,?_⟩ | |
| · have he : stationaryMeanTarget hF x hr hl hT0.le hT1 X= | |
| fun y => primitiveExpectedField F x r+s • (productPointEquiv d d (X.path y 1)).2 := by | |
| funext y | |
| exact stationaryMeanTarget_identity hF x hr hlam hl hT0.le hT1 hs X y | |
| rw [he] | |
| exact measurable_const.add (((productPointEquiv d d).continuous.measurable.comp (X.measurable_slice 1)).snd.const_smul s) | |
| · exact hmean.2.trans (centeringMeanBudget_normalized F x nc lam r R hs hT0.le hT1.le hZ) | |
| end LogConcaveSampling | |
| end | |
| end | |
| end OAI | |