| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.CascadeRecursion |
| import OAI.Probability.SKBarriers.Scalar.ThermalMoments |
|
|
| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| section CascadeProbability |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
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| omit [NormedAddCommGroup E] [NormedSpace ℝ E] in |
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| theorem cascadeMoment_nonneg (n : ℕ) (m : Fin n → ℝ) |
| (f g : CascadeSpace E n → ℝ) (hg : ∀ z, 0 ≤ g z) (x : E) : |
| 0 ≤ cascadeMoment n m f g x := by |
| induction n with |
| | zero => exact hg x |
| | succ n ih => |
| apply ih |
| intro z |
| exact integral_nonneg (fun y => hg (z,y)) |
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|
| theorem cascadeMoment_const (n : ℕ) (m : Fin n → ℝ) |
| (f : CascadeSpace E n → ℝ) (hf : BoundedDerivs f) (a : ℝ) : |
| cascadeMoment n m f (fun _ => a) = fun _ => a := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| have h : gaussianAverage (m (Fin.last n)) f (fun _ => a) = fun _ => a := by |
| funext z |
| let := gaussianStepLaw_probability hf (m (Fin.last n)) z |
| simp only [gaussianAverage,integral_const,probReal_univ,one_smul] |
| rw [cascadeMoment,h] |
| exact ih _ _ (hf.gaussianStep _) |
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|
| omit [NormedAddCommGroup E] [NormedSpace ℝ E] in |
| theorem cascadeMoment_const_mul (n : ℕ) (m : Fin n → ℝ) |
| (f g : CascadeSpace E n → ℝ) (a : ℝ) : |
| cascadeMoment n m f (fun z => a*g z) = fun x => a*cascadeMoment n m f g x := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| have h : gaussianAverage (m (Fin.last n)) f (fun z => a*g z) = |
| fun z => a*gaussianAverage (m (Fin.last n)) f g z := by |
| funext z; exact integral_const_mul a _ |
| rw [cascadeMoment,h,ih] |
| rfl |
|
|
| theorem cascadeMoment_sum {T : Type} [Fintype T] (n : ℕ) (m : Fin n → ℝ) |
| (f : CascadeSpace E n → ℝ) (hf : BoundedDerivs f) |
| (g : T → CascadeSpace E n → ℝ) (hg : ∀ t, Continuous (g t)) |
| (C : T → ℝ) (hC : ∀ t, 0 ≤ C t) (hb : ∀ t z, ‖g t z‖ ≤ C t) : |
| cascadeMoment n m f (fun z => ∑ t, g t z) = |
| fun x => ∑ t, cascadeMoment n m f (g t) x := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| have h : gaussianAverage (m (Fin.last n)) f (fun z => ∑ t, g t z) = |
| fun z => ∑ t, gaussianAverage (m (Fin.last n)) f (g t) z := by |
| funext z |
| let := gaussianStepLaw_probability hf (m (Fin.last n)) z |
| apply integral_finsetSum |
| intro t _ |
| exact Integrable.of_bound ((hg t).comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| (C t) (ae_of_all _ (fun y => hb t (z,y))) |
| rw [cascadeMoment,h] |
| exact ih _ _ (hf.gaussianStep _) _ |
| (fun t => gaussianAverage_continuous hf _ (hg t) (hC t) (hb t)) |
| (fun t z => gaussianAverage_norm_le hf _ (hb t) z) |
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| variable {S : Type} [Fintype S] [Nonempty S] |
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| def cascadeSpinWeight (n : ℕ) (m : Fin n → ℝ) |
| (c : S → ℝ) (L : S → CascadeSpace E n →L[ℝ] ℝ) (x : E) (s : S) : ℝ := |
| cascadeMoment n m (affineLogPartition c L) (fun z => affineGibbs c L z s) x |
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| theorem affineGibbs_continuous (c : S → ℝ) (L : S → E →L[ℝ] ℝ) (s : S) : |
| Continuous (fun x => affineGibbs c L x s) := by |
| exact (Real.continuous_exp.comp (continuous_const.add (L s).continuous)).div |
| (continuous_finsetSum _ (fun t _ => Real.continuous_exp.comp (continuous_const.add (L t).continuous))) |
| (fun x => (affinePartition_pos c L x).ne') |
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| theorem affineGibbs_le_one (c : S → ℝ) (L : S → E →L[ℝ] ℝ) (x : E) (s : S) : |
| affineGibbs c L x s ≤ 1 := by |
| rw [← affineGibbs_sum c L x] |
| exact Finset.single_le_sum (fun t _ => (affineGibbs_pos c L x t).le) (Finset.mem_univ s) |
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| theorem affineGibbs_norm_le_one (c : S → ℝ) (L : S → E →L[ℝ] ℝ) (x : E) (s : S) : |
| ‖affineGibbs c L x s‖ ≤ 1 := by |
| rw [Real.norm_eq_abs,abs_of_pos (affineGibbs_pos c L x s)] |
| exact affineGibbs_le_one c L x s |
|
|
| theorem cascadeSpinWeight_nonneg (n : ℕ) (m : Fin n → ℝ) |
| (c : S → ℝ) (L : S → CascadeSpace E n →L[ℝ] ℝ) (x : E) (s : S) : |
| 0 ≤ cascadeSpinWeight n m c L x s := |
| cascadeMoment_nonneg n m _ _ (fun z => (affineGibbs_pos c L z s).le) x |
|
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| theorem cascadeSpinWeight_sum (n : ℕ) (m : Fin n → ℝ) |
| (c : S → ℝ) (L : S → CascadeSpace E n →L[ℝ] ℝ) (x : E) : |
| ∑ s, cascadeSpinWeight n m c L x s = 1 := by |
| have H := congrFun (cascadeMoment_sum n m (affineLogPartition c L) |
| (affineLogPartition_boundedDerivs c L) (fun s z => affineGibbs c L z s) |
| (affineGibbs_continuous c L) (fun _ => 1) (by intro; norm_num) |
| (fun s z => affineGibbs_norm_le_one c L z s)) x |
| simp_rw [affineGibbs_sum] at H |
| rw [cascadeMoment_const n m _ (affineLogPartition_boundedDerivs c L)] at H |
| exact H.symm |
|
|
| theorem cascadeMoment_affineMoment (n : ℕ) (m : Fin n → ℝ) |
| (c : S → ℝ) (L : S → CascadeSpace E n →L[ℝ] ℝ) (g : S → ℝ) (x : E) : |
| cascadeMoment n m (affineLogPartition c L) (affineMoment c L g) x = |
| ∑ s, cascadeSpinWeight n m c L x s*g s := by |
| have H := congrFun (cascadeMoment_sum n m (affineLogPartition c L) |
| (affineLogPartition_boundedDerivs c L) (fun s z => g s*affineGibbs c L z s) |
| (fun s => continuous_const.mul (affineGibbs_continuous c L s)) (fun s => ‖g s‖) |
| (fun s => norm_nonneg _) (fun s z => by |
| rw [norm_mul]; exact (mul_le_mul_of_nonneg_left (affineGibbs_norm_le_one c L z s) (norm_nonneg _)).trans_eq (mul_one _))) x |
| simp_rw [cascadeMoment_const_mul] at H |
| change cascadeMoment n m (affineLogPartition c L) (fun z => ∑ s, affineGibbs c L z s*g s) x = _ |
| simpa only [cascadeSpinWeight,mul_comm] using H |
|
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| theorem cascadeSpinWeight_regular (n : ℕ) (m : Fin n → ℝ) |
| (c : S → ℝ) (L : S → CascadeSpace E n →L[ℝ] ℝ) (s : S) : |
| Continuous (fun x => cascadeSpinWeight n m c L x s) ∧ |
| ∀ x, ‖cascadeSpinWeight n m c L x s‖ ≤ 1 := |
| cascadeMoment_bounded_continuous n m _ (affineLogPartition_boundedDerivs c L) _ |
| (affineGibbs_continuous c L s) (by norm_num) (fun x => affineGibbs_norm_le_one c L x s) |
| end CascadeProbability |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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