| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.BufferRootRetention |
| import OAI.Probability.Ballisticity.Walk.BufferHistoryBracket |
| import OAI.Probability.Ballisticity.Estimates.RawPairMassComposition |
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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 buffer_all_height_retention {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| {R : ℝ} (hR : 0 ≤ R) (a ε α g : ℝ) {κ : ℝ≥0} |
| (hκpos : 0 < κ) (hκ1 : κ ≤ 1) (hg : 0 < g) (hg1 : g ≤ 1) |
| (hε : 0 ≤ ε) (hε1 : ε ≤ 1) (hα : 0 ≤ α) |
| (H : ℝ → ℕ) (hH : ∀ r, 0 < H r) |
| (x : Lattice d × Lattice d) (hx : x ∈ PairAtHeight (realPosition (step e)) a) |
| (ω : Environment d) (hκ : ∀ y u, κ ≤ (ω y).1 u) (t : ℕ) : |
| ∃ n : ℕ, let l := (bufferHistory e f hef R (signedCoordinate f (x.2-x.1)) ε α g κ H hH |
| (bufferRootNode e f hef R a x hx) ω n); |
| bufferWordHeight l ≤ t ∧ |
| ((κ:ℝ≥0∞)^(⌈R⌉₊+2)*(bufferRetentionCost R g κ)^n)*((κ:ℝ≥0∞)^2*ENNReal.ofReal g) ≤ |
| rawPairEndpointLaw (realPosition (step e)) (t+1) ω x |
| {y | signedCoordinate f (x.2-x.1) ≤ signedCoordinate f (y.2-y.1)} := by |
| let z := signedCoordinate f (x.2-x.1) |
| let root := bufferRootNode e f hef R a x hx |
| obtain ⟨n,hnt,hnt'⟩ := bufferHistory_bracket e f hef R z ε α g κ H hH root ω t |
| let l := bufferHistory e f hef R z ε α g κ H hH root ω n |
| change bufferWordHeight l ≤ t at hnt |
| let N := bufferNodeAt e f hef R z ε α g κ H hH root l |
| have ha : ω ∈ N.active := by |
| rw [bufferNodeAt_active_iff_history,bufferHistory_length] |
| exact ⟨Set.mem_univ _,rfl⟩ |
| have hv := bufferNodeAt_valid e f hef hR z ε α g hκpos hκ1 hg hg1 hε H hH root le_rfl |
| (bufferRootNode_valid e f hef R a x hx κ) l |
| obtain ⟨hp,hgap⟩ := hv ω hκ ha |
| have := hp |
| have hnR : 0 ≤ N.radius := hR.trans (bufferNodeAt_radius e f hef R z ε α g κ H hH root le_rfl l) |
| have hret := bufferNodeAt_root_retained e f hef hR a ε α g hκpos hκ1 hg hg1 hε hε1 hα |
| H hH x hx l ω hκ ha |
| have hj : t-bufferWordHeight l ≤ bufferFirstFailure (realPosition (step e)) f N.level |
| (z+(1-ε)*N.radius) g N.law (H N.radius) ω := by |
| change t < _ at hnt' |
| simp only [bufferHistory,bufferWordHeight] at hnt' |
| change t < (bufferNodeDecision e f z ε α g H N ω).1+1+bufferWordHeight l at hnt' |
| change t < bufferFirstFailure (realPosition (step e)) f N.level (z+(1-ε)*N.radius) g N.law (H N.radius) ω+1+bufferWordHeight l at hnt' |
| omega |
| have hm := bufferPartialStage_mass e f N.level z N.radius ε α g N.law (hH N.radius) ω |
| (fun y => hκ y e) hκ1 hg1 hnR hε hε1 hα hgap (t-bufferWordHeight l) hj |
| have hb := rawPair_gap_mass_composition e f (1+bufferWordHeight l) (t-bufferWordHeight l) ω x (N.law ω).val |
| ((κ:ℝ≥0∞)^(⌈R⌉₊+2)*(bufferRetentionCost R g κ)^l.length) ((κ:ℝ≥0∞)^2*ENNReal.ofReal g) z hret hm |
| have hh : 1+bufferWordHeight l+(t-bufferWordHeight l)=t+1 := by omega |
| have hlen : l.length=n := bufferHistory_length e f hef R z ε α g κ H hH root ω n |
| refine ⟨n,hnt,?_⟩ |
| simpa only [hh,hlen] using hb |
| end DirectionalTransience |
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| end |
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| end |
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| end OAI |
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