| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.BufferTree |
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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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| noncomputable def bufferWordHeight : List (ℕ × Bool) → ℕ |
| | [] => 0 |
| | p::l => p.1+1+bufferWordHeight l |
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| lemma bufferNodeAt_level {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (R₀ z₀ ε α g : ℝ) (κ : ℝ≥0) (H : ℝ → ℕ) (hH : ∀ r, 0 < H r) |
| (root : AdaptedBufferNode e) (l : List (ℕ × Bool)) : |
| (bufferNodeAt e f hef R₀ z₀ ε α g κ H hH root l).level = root.level+bufferWordHeight l := by |
| induction l with |
| | nil => simp [bufferNodeAt,bufferWordHeight] |
| | cons p l ih => |
| change (bufferNodeAt e f hef R₀ z₀ ε α g κ H hH root l).level+p.1+1 = _ |
| rw [ih,bufferWordHeight,Nat.cast_add,Nat.cast_add,Nat.cast_one] |
| ring |
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| lemma bufferNodeAt_radius_elapsed {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| {R₀ ε α M C : ℝ} (hR₀ : 0 < R₀) (hε : 0 ≤ ε) (hα : 0 ≤ α) (_ : 0 < M) |
| (hC0 : R₀ ≤ C) (hC : (1+α)*M ≤ C) |
| (z₀ g : ℝ) (κ : ℝ≥0) (H : ℝ → ℕ) (hH : ∀ r, 0 < H r) |
| (hplan : ∀ r : ℝ, R₀ ≤ r → r ≤ M*(H r+1)) |
| (root : AdaptedBufferNode e) (hr : root.radius=R₀) (ω : Environment d) |
| (l : List (ℕ × Bool)) |
| (ha : ω ∈ (bufferNodeAt e f hef R₀ z₀ ε α g κ H hH root l).active) : |
| (bufferNodeAt e f hef R₀ z₀ ε α g κ H hH root l).radius ≤ C*(bufferWordHeight l+1) := by |
| have hCpos : 0 < C := hR₀.trans_le hC0 |
| induction l with |
| | nil => simpa [bufferNodeAt,bufferWordHeight,hr] using hC0 |
| | cons p l ih => |
| let N := bufferNodeAt e f hef R₀ z₀ ε α g κ H hH root l |
| have hpa : ω ∈ N.active := ha.1 |
| have hb : BufferStageEvent (realPosition (step e)) f (H N.radius) z₀ N.radius ε α g (N.law ω).val ω ↔ p.2=true := ha.2.2 |
| have hk : bufferFirstFailure (realPosition (step e)) f N.level (z₀+(1-ε)*N.radius) g N.law (H N.radius) ω=p.1 := ha.2.1 |
| have hNr : R₀ ≤ N.radius := bufferNodeAt_radius e f hef R₀ z₀ ε α g κ H hH root hr.ge l |
| have hn : 0 ≤ N.radius := hR₀.le.trans hNr |
| have ih' := ih hpa |
| change bufferNextRadius R₀ N.radius ε α p.2 ≤ C*((bufferWordHeight (p::l):ℕ)+1) |
| unfold bufferNextRadius |
| apply max_le |
| · apply hC0.trans |
| have hh : (1:ℝ) ≤ (bufferWordHeight (p::l):ℝ)+1 := by |
| have := Nat.cast_nonneg (α := ℝ) (bufferWordHeight (p::l)); linarith |
| simpa only [mul_one] using mul_le_mul_of_nonneg_left hh hCpos.le |
| · cases hp : p.2 |
| · simp only [Bool.false_eq_true,↓reduceIte] |
| have hdec : (1-ε)*N.radius ≤ N.radius := by nlinarith |
| apply hdec.trans (ih'.trans _) |
| apply mul_le_mul_of_nonneg_left _ hCpos.le |
| simp only [bufferWordHeight,Nat.cast_add,Nat.cast_one] |
| have ht := Nat.cast_nonneg (α := ℝ) (bufferWordHeight l) |
| have hk := Nat.cast_nonneg (α := ℝ) p.1 |
| linarith |
| · simp only [↓reduceIte] |
| have hE := hb.mpr hp |
| have htop := (bufferStageEvent_iff_last_tests (realPosition (step e)) f N.level z₀ |
| N.radius ε α g N.law (hH N.radius) ω).mp hE |>.1 |
| have hj := bufferFirstFailure_eq_top_of_pass (realPosition (step e)) f N.level |
| (z₀+(1-ε)*N.radius) g N.law (hH N.radius) ω htop |
| have hkH : p.1=H N.radius := hk.symm.trans hj |
| have hh := mul_le_mul_of_nonneg_left (hplan N.radius hNr) (show 0 ≤ 1+α by linarith) |
| rw [←mul_assoc] at hh |
| apply hh.trans |
| rw [←hkH] |
| have hmul := mul_le_mul_of_nonneg_right hC (show 0 ≤ (p.1:ℝ)+1 by positivity) |
| apply hmul.trans |
| apply mul_le_mul_of_nonneg_left _ hCpos.le |
| simp only [bufferWordHeight,Nat.cast_add,Nat.cast_one] |
| have ht := Nat.cast_nonneg (α := ℝ) (bufferWordHeight l) |
| have hk := Nat.cast_nonneg (α := ℝ) p.1 |
| linarith |
| end DirectionalTransience |
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| end |
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| end |
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| end OAI |
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