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9b6e014 | 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 | import Mathlib
import OAI.Probability.Ballisticity.Entropy.FiniteLogStability
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter Function
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
lemma upperPairNeighborhood_mono {d : ℕ} (e : Direction d) (a : ℝ)
(x : Lattice d × Lattice d) {N R : ℕ} (hNR : N ≤ R) :
upperPairNeighborhood e a x N ⊆ upperPairNeighborhood e a x R := by
intro y hy
refine ⟨?_,hy.2⟩
have hball (z : Lattice d) : LatticeBall z N ⊆ LatticeBall z R := by
intro w hw i
exact (hw i).trans (by exact_mod_cast hNR)
exact hy.1.elim (fun h => Or.inl (hball _ h)) (fun h => Or.inr (hball _ h))
lemma finite_outward_upper_row_approximation {d : ℕ} {κ : ℝ≥0} (hκ : 0 < κ)
(e f : Direction d) (A : ℝ) (N : ℕ) {ε B : ℝ} (hε : 0 < ε) (hB : 0 ≤ B) :
∃ R : ℕ, ∀ (a : ℝ) (x : Lattice d × Lattice d), x ∈ PairAtHeight (realPosition (step e)) a →
∃ F : Environment d → ℝ,
@Measurable _ _ (rowSigma (upperPairNeighborhood e a x R)) _ F ∧
(∀ ω, 0 ≤ F ω ∧ F ω ≤ B) ∧
(∀ ω, (∀ y g, κ ≤ (ω y).1 g) →
|min (finiteLogExcess A (fun n => outwardKernelMass e f (n+1) x) N ω) B-F ω| ≤ ε) := by
have hloc (n : ℕ) := outwardLog_fixed_height_locality hκ e f (Nat.succ_pos n) hε
choose r hr using hloc
obtain ⟨R,hR⟩ := ((Finset.range (N+1)).image r).exists_le
refine ⟨R,?_⟩
intro a x hx
let q : ℕ → Environment d → ℝ := fun n ω => outwardKernelMass e f (n+1) x (upperPairPatch e a x (r n) ω)
let F : Environment d → ℝ := fun ω => min (finiteLogExcess A q N ω) B
refine ⟨F,?_,?_,?_⟩
· have hm : @Measurable _ _ (rowSigma (upperPairNeighborhood e a x R)) _ (finiteLogExcess A q N) := by
unfold finiteLogExcess
apply Measurable.ennreal_toReal
apply Measurable.iSup
intro n
have hnr : r n ≤ R := hR (r n) (Finset.mem_image.mpr ⟨n,Finset.mem_range.mpr n.isLt,rfl⟩)
have hp := (upperPairPatch_measurable e a x hx (r n)).mono
(rowSigma_mono (upperPairNeighborhood_mono e a x hnr)) le_rfl
have hq := (measurable_outwardKernelMass e f (n+1) x).comp hp
exact (hq.log.neg.sub_const _).ennreal_ofReal
exact hm.min measurable_const
· intro ω
exact ⟨le_min ENNReal.toReal_nonneg hB,min_le_right _ _⟩
· intro ω hω
have hd := finiteLogExcess_stability A (fun n => outwardKernelMass e f (n+1) x) q N ω hε.le
(fun n hn => (hr n a x hx ω hω).le)
have h₁ := (abs_le.mp hd).1
have h₂ := (abs_le.mp hd).2
apply abs_le.mpr
constructor
· have hh : min (finiteLogExcess A q N ω) B ≤ min (finiteLogExcess A (fun n => outwardKernelMass e f (n+1) x) N ω) B+ε := by
have ht := min_le_min (show finiteLogExcess A q N ω ≤ finiteLogExcess A (fun n => outwardKernelMass e f (n+1) x) N ω+ε by linarith) (show B ≤ B+ε by linarith)
simpa only [min_add_add_right] using ht
exact by dsimp [F]; linarith
· have hh : min (finiteLogExcess A (fun n => outwardKernelMass e f (n+1) x) N ω) B ≤ min (finiteLogExcess A q N ω) B+ε := by
have ht := min_le_min (show finiteLogExcess A (fun n => outwardKernelMass e f (n+1) x) N ω ≤ finiteLogExcess A q N ω+ε by linarith) (show B ≤ B+ε by linarith)
simpa only [min_add_add_right] using ht
exact by dsimp [F]; linarith
end DirectionalTransience
end
end
end OAI
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