| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.CascadeBlockLaw |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| section Generic |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
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| theorem BoundedDerivs.hasExpGrowth {f : E → ℝ} (hf : BoundedDerivs f) : HasExpGrowth f := by |
| obtain ⟨hd,C,D,hC,_hD,h₁,_h₂⟩ := hf |
| exact growth_of_fderiv_bound f (hd.differentiable (by norm_num)) hC h₁ |
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| theorem gaussianStep_add_prefix {f : E × ℝ → ℝ} (hf : BoundedDerivs f) |
| (c : E → ℝ) (m : ℝ) : |
| gaussianStep m (fun z => f z+c z.1) = fun x => gaussianStep m f x+c x := by |
| funext x |
| by_cases hm : m=0 |
| · subst m |
| simp only [gaussianStep,ite_true] |
| rw [integral_add (hf.hasExpGrowth.integrable_gaussian_section hf.1.continuous x) |
| (integrable_const _)] |
| simp |
| · simp only [gaussianStep,ite_eq_right hm,positiveGaussianLogStep,mul_add,Real.exp_add] |
| rw [integral_mul_const,Real.log_mul (integral_exp_pos (hf.exp_integrable m x)).ne' |
| (Real.exp_ne_zero _),Real.log_exp,add_div,mul_div_cancel_left₀ _ hm] |
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| omit [NormedAddCommGroup E] [NormedSpace ℝ E] in |
|
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| theorem gaussianStep_prefix (c : E → ℝ) (m : ℝ) : |
| gaussianStep m (fun z : E × ℝ => c z.1) = c := by |
| funext x |
| by_cases hm : m=0 |
| · simp [gaussianStep,hm] |
| · simp [gaussianStep,hm,positiveGaussianLogStep] |
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| theorem gaussianStep_sum_independent {I : Type} [Fintype I] [DecidableEq I] |
| (f : I → E × ℝ → ℝ) (hf : ∀ i, BoundedDerivs (f i)) (j : I) |
| (hi : ∀ i, i ≠ j → TranslationInvariant (f i) (0,1)) (m : ℝ) : |
| gaussianStep m (fun z => ∑ i, f i z) = fun x => ∑ i, gaussianStep m (f i) x := by |
| have he (i : I) (h : i ≠ j) (x : E) (y : ℝ) : f i (x,y) = f i (x,0) := by |
| simpa using hi i h (x,0) y |
| have hex : (fun z => ∑ i, f i z) = fun z => f j z+∑ i ∈ Finset.univ.erase j, f i (z.1,0) := by |
| funext z |
| rw [← Finset.add_sum_erase _ _ (Finset.mem_univ j)] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i h |
| exact he i (Finset.mem_erase.mp h).1 z.1 z.2 |
| rw [hex,gaussianStep_add_prefix (hf j) (fun x => ∑ i ∈ Finset.univ.erase j, f i (x,0)) m] |
| funext x |
| rw [← Finset.add_sum_erase _ _ (Finset.mem_univ j)] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i h |
| have H : f i = fun z => f i (z.1,0) := funext (fun z => he i (Finset.mem_erase.mp h).1 z.1 z.2) |
| rw [H,gaussianStep_prefix (fun y => f i (y,0)) m] |
| end Generic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
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| theorem hierarchyPressure_zero (n : ℕ) (f : ParameterSpace n → ℝ) |
| (hf : BoundedDerivs f) (x : ℝ) : |
| hierarchyPressure n (fun _ => 0) f x = ∫ z, f z ∂fiberGaussian n x := by |
| induction n with |
| | zero => simp [hierarchyPressure,fiberGaussian] |
| | succ n ih => |
| rw [hierarchyPressure,ih _ (hf.gaussianStep 0),fiberGaussian, |
| integral_prod _ (hf.hasExpGrowth.integrable_fiberGaussian (n+1) hf.1.continuous x)] |
| simp only [gaussianStep,ite_true] |
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| theorem hierarchyPressure_sum_independent {I : Type} [Fintype I] [DecidableEq I] |
| (n : ℕ) (m : Fin n → ℝ) (owner : Fin n → I) |
| (f : I → ParameterSpace n → ℝ) (hf : ∀ i, BoundedDerivs (f i)) |
| (hi : ∀ t i, i ≠ owner t → TranslationInvariant (f i) (coordinateAxis n t)) : |
| hierarchyPressure n m (fun z => ∑ i, f i z) = fun x => ∑ i, hierarchyPressure n m (f i) x := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| rw [hierarchyPressure,gaussianStep_sum_independent f hf (owner (Fin.last n)) |
| (by simpa only [coordinateAxis_last] using hi (Fin.last n))] |
| rw [ih (fun t => m t.castSucc) (fun t => owner t.castSucc) |
| (fun i => gaussianStep (m (Fin.last n)) (f i)) |
| (fun i => (hf i).gaussianStep _) (fun t i h => ?_)] |
| · rfl |
| · apply TranslationInvariant.gaussianStep |
| simpa only [coordinateAxis_castSucc] using hi t.castSucc i h |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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