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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 | import Mathlib
import OAI.Probability.Ballisticity.Entropy.BufferCostBounds
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter Function
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
lemma outwardKernelMass_lower {d : ℕ} (e f : Direction d) (H : ℕ)
(ω : Environment d) (x : Lattice d × Lattice d) {κ : ℝ≥0}
(hκ : ∀ y, κ ≤ (ω y).1 e) :
(κ:ℝ)^(2*H) ≤ outwardKernelMass e f H x ω := by
let S := {y : Lattice d × Lattice d | signedCoordinate f (x.2-x.1) ≤ signedCoordinate f (y.2-y.1)}
have hlow : ((κ:ℝ≥0∞)^2)^H ≤ rawPairEndpointLaw (realPosition (step e)) H ω x S := by
induction H with
| zero => simp [rawPairEndpointLaw_zero,S]
| succ H ih =>
have hh := rawPairEndpointLaw_upward e H ω x hκ S
rw [Measure.smul_apply,smul_eq_mul,Measure.map_apply (measurable_of_countable _) (Set.to_countable _).measurableSet] at hh
have hpre : pairUp e ⁻¹' S=S := by
ext y
simp only [S,Set.mem_preimage,Set.mem_ofPred,pairUp_preserves_gap]
rw [hpre] at hh
apply le_trans _ hh
rw [pow_succ']
exact mul_le_mul_right ih _
have hfinite : rawPairEndpointLaw (realPosition (step e)) H ω x S ≠ ⊤ :=
ne_top_of_le_ne_top ENNReal.one_ne_top ((measure_mono (Set.subset_univ _)).trans (rawPairEndpointLaw_total_le_one _ _ _ _))
have hh := ENNReal.toReal_mono hfinite hlow
simpa only [ENNReal.toReal_pow,ENNReal.coe_toReal,←pow_mul,outwardKernelMass,S] using hh
lemma rawPairEndpointLaw_upper_congr {d : ℕ} (e : Direction d) (H : ℕ)
(a : ℝ) (x : Lattice d × Lattice d) (hx : x ∈ PairAtHeight (realPosition (step e)) a)
(ω η : Environment d) (he : Set.EqOn ω η {y | a ≤ dot (realPosition y) (realPosition (step e))}) :
rawPairEndpointLaw (realPosition (step e)) H ω x = rawPairEndpointLaw (realPosition (step e)) H η x := by
have h₁ : Set.EqOn ω η (Strip (realPosition (step e)) x.1 H) := by
intro y hy
apply he
change a ≤ dot (realPosition y) (realPosition (step e))
simpa only [hx.1] using hy.1
have h₂ : Set.EqOn ω η (Strip (realPosition (step e)) x.2 H) := by
intro y hy
apply he
change a ≤ dot (realPosition y) (realPosition (step e))
simpa only [hx.2] using hy.1
unfold rawPairEndpointLaw
rw [hitKernel_congr _ _ ω η h₁,hitKernel_congr _ _ ω η h₂]
end DirectionalTransience
end
end
end OAI
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