| import Mathlib |
| import OAI.Probability.Ballisticity.Entropy.BufferCostBounds |
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| namespace OAI |
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| section |
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| section |
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| open MeasureTheory ProbabilityTheory Filter Function |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
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| 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 |
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| 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 |
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| end |
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| end |
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| end OAI |
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