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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | import Mathlib
import OAI.Probability.SKBarriers.Hierarchy.CascadeBlockLaw
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
section Generic
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
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
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]
omit [NormedAddCommGroup E] [NormedSpace ℝ E] in
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]
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
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]
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
end
end
end
end OAI
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