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7d9f027 | 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 | import Mathlib
import OAI.Probability.SKGap.Stability.FieldIntegralReduction
namespace OAI
section
noncomputable section
namespace SKGap
open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity
open scoped BigOperators Matrix.Norms.Frobenius ENNReal
instance conditionalGoeLaw_probability {ι : Type*} [Fintype ι] [DecidableEq ι] [Nonempty ι]
(r v : ℝ) (m z : ι → ℝ) : IsProbabilityMeasure (conditionalGoeLaw r v m z) := by
unfold conditionalGoeLaw
infer_instance
lemma conditionalFieldWeight_le_one {n : ℕ} [NeZero n] (j t σ : ℝ)
(E : Set (((Fin n×Fin n) → ℝ) × (Fin n → ℝ))) (y : Fin n → ℝ) :
conditionalFieldWeight j t σ E y ≤ 1 := prob_le_one
lemma conditionalFieldWeight_of_real_bound {n : ℕ} [NeZero n] {j t σ c : ℝ}
(E : Set (((Fin n×Fin n) → ℝ) × (Fin n → ℝ))) (y : Fin n → ℝ)
(hb : (conditionalGoeLaw (j/(n:ℝ)) (t+σ^2) (magnetization y) (empiricalObservation j t σ y)).real
{W | (W,y)∈E} ≤ c) : conditionalFieldWeight j t σ E y ≤ ENNReal.ofReal c := by
rw [← ENNReal.ofReal_toReal (show conditionalFieldWeight j t σ E y ≠ ⊤ from measure_ne_top _ _)]
exact ENNReal.ofReal_le_ofReal hb
lemma measurable_scalarIntegrand {n : ℕ} [NeZero n] (j t σ : ℝ) :
Measurable (scalarIntegrand (ι := Fin n) j t σ) := by
unfold scalarIntegrand productNormal scalarEmpirical scalarPsi scalarA scalarD scalarS scalarM scalarQMoment
simp only [integral_empiricalLaw]
unfold gaussianPDFReal
fun_prop
lemma scalar_density_prefactor_le_one {n : ℕ} [NeZero n] {j t σ : ℝ}
(hj : 0 ≤ j) (ht : 0 ≤ t) (hσ : 0 < σ) (y : Fin n → ℝ) :
sqrt (scalarS j (scalarEmpirical y t σ)/
(scalarS j (scalarEmpirical y t σ)+j*scalarQMoment (empiricalLaw y))) ≤ 1 := by
have hq : 0 ≤ j*scalarQMoment (empiricalLaw y) := mul_nonneg hj (scalarQMoment_bounds _).1
have hs : 0 < scalarS j (scalarEmpirical y t σ) := add_pos_of_pos_of_nonneg
(add_pos_of_nonneg_of_pos ht (sq_pos_of_pos hσ)) hq
exact sqrt_le_one.mpr ((div_le_one (add_pos_of_pos_of_nonneg hs hq)).mpr (le_add_of_nonneg_right hq))
lemma scalarIntegrand_nonneg {n : ℕ} [NeZero n] (j t σ : ℝ) (y : Fin n → ℝ) :
0 ≤ scalarIntegrand j t σ y := mul_nonneg (exp_pos _).le
(Finset.prod_nonneg (fun i _=>gaussianPDFReal_nonneg _ _ (y i)))
lemma scalar_weighted_density_bound {n : ℕ} [NeZero n] {j t σ : ℝ}
(hj : 0 ≤ j) (ht : 0 ≤ t) (hσ : 0 < σ) (y : Fin n → ℝ) {w b : ℝ≥0∞}
(hw : w ≤ b) :
ENNReal.ofReal (sqrt (scalarS j (scalarEmpirical y t σ)/
(scalarS j (scalarEmpirical y t σ)+j*scalarQMoment (empiricalLaw y)))*scalarIntegrand j t σ y)*w ≤
ENNReal.ofReal (scalarIntegrand j t σ y)*b := by
apply mul_le_mul' _ hw
apply ENNReal.ofReal_le_ofReal
exact mul_le_of_le_one_left (scalarIntegrand_nonneg j t σ y) (scalar_density_prefactor_le_one hj ht hσ y)
end SKGap
end
end
end OAI
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