| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.CountableProdBind |
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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 bufferNodeAt_root_retained (e f : Direction d) (hef : e.1 ≠ f.1) |
| (hR : 0 ≤ R) (a ε α g : ℝ) |
| (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) |
| (l : List (ℕ × Bool)) (ω : Environment d) (hκ : ∀ y u, κ ≤ (ω y).1 u) |
| (ha : ω ∈ (bufferNodeAt e f hef R (signedCoordinate f (x.2-x.1)) ε α g κ H hH |
| (bufferRootNode e f hef R a x hx) l).active) : |
| ((κ:ℝ≥0∞)^(⌈R⌉₊+2)*(bufferRetentionCost R g κ)^l.length) • |
| ((bufferNodeAt e f hef R (signedCoordinate f (x.2-x.1)) ε α g κ H hH |
| (bufferRootNode e f hef R a x hx) l).law ω).val ≤ |
| rawPairEndpointLaw (realPosition (step e)) (1+bufferWordHeight l) ω x := by |
| let root := bufferRootNode e f hef R a x hx |
| have hh := bufferNodeAt_retained e f hef hR (signedCoordinate f (x.2-x.1)) ε α g hκpos hκ1 hg hg1 hε hε1 hα |
| H hH root le_rfl (bufferRootNode_valid e f hef R a x hx κ) l ω hκ ha |
| have hs : (κ:ℝ≥0∞)^(⌈R⌉₊+2) • ((bufferRetentionCost R g κ)^l.length • |
| ((bufferNodeAt e f hef R (signedCoordinate f (x.2-x.1)) ε α g κ H hH root l).law ω).val) ≤ |
| (κ:ℝ≥0∞)^(⌈R⌉₊+2) • rawPairMixture (realPosition (step e)) (bufferWordHeight l) (root.law ω).val ω := by |
| intro U |
| simp only [Measure.smul_apply,smul_eq_mul] |
| have hhU := hh U |
| simp only [Measure.smul_apply,smul_eq_mul] at hhU |
| gcongr |
| rw [←rawPairMixture_smul] at hs |
| have hm := rawPairMixture_mono (realPosition (step e)) (bufferWordHeight l) ω |
| (bufferRootNode_retained e f hef R a x hx ω hκ) |
| have hf := hs.trans (hm.trans (rawPairEndpointLaw_comp_le e 1 (bufferWordHeight l) ω x)) |
| simpa only [smul_smul] using hf |
| end DirectionalTransience |
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| end |
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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 bufferStageRetainedLaw_baseline (e f : Direction d) |
| (a z₀ r ε α g : ℝ) (π : Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) a)) |
| (hH : 0 < H) (ω : Environment d) (κ : ℝ≥0) |
| (hr : 0 ≤ r) (hε : ε ≤ 1) (hα : 0 ≤ α) : |
| ∀ᵐ y ∂bufferStageRetainedLaw e f a z₀ r ε α g π H ω κ, |
| z₀ ≤ signedCoordinate f (y.2-y.1) := by |
| by_cases hE : BufferStageEvent (realPosition (step e)) f H z₀ r ε α g (π ω).val ω |
| · rw [bufferStageRetainedLaw_success e f a z₀ r ε α g π hH ω κ hE] |
| filter_upwards [retainedPairLaw_support (realPosition (step e)) f H (z₀+(1+α)*r) (π ω).val ω] with y hy |
| have : 0 ≤ (1+α)*r := by positivity |
| linarith |
| · filter_upwards [bufferStageRetainedLaw_failure_support e f a z₀ r ε α g π hH ω κ hE] with y hy |
| have : 0 ≤ (1-ε)*r := mul_nonneg (sub_nonneg.mpr hε) hr |
| linarith |
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| lemma bufferPartialStage_mass (e f : Direction d) |
| (a z₀ r ε α g : ℝ) (π : Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) a)) |
| (hH : 0 < H) (ω : Environment d) |
| (hκ : ∀ y, κ ≤ (ω y).1 e) (hκ1 : κ ≤ 1) (hg1 : g ≤ 1) |
| (hr : 0 ≤ r) (hε : 0 ≤ ε) (hε1 : ε ≤ 1) (hα : 0 ≤ α) |
| [IsProbabilityMeasure (π ω).val] |
| (hgap : ∀ᵐ x ∂(π ω).val, z₀+r ≤ signedCoordinate f (x.2-x.1)) |
| (j : ℕ) (hj : j ≤ bufferFirstFailure (realPosition (step e)) f a (z₀+(1-ε)*r) g π H ω) : |
| (κ : ℝ≥0∞)^2*ENNReal.ofReal g ≤ pairKernelMass (realPosition (step e)) f j z₀ (π ω).val ω := by |
| have hk1 : (κ:ℝ≥0∞) ≤ 1 := by exact_mod_cast hκ1 |
| have hc : (κ:ℝ≥0∞)^2*ENNReal.ofReal g ≤ ENNReal.ofReal g := by |
| calc |
| _ ≤ 1*ENNReal.ofReal g := by |
| gcongr |
| exact pow_le_one₀ (show (0:ℝ≥0∞) ≤ κ from zero_le) hk1 |
| _ = ENNReal.ofReal g := one_mul _ |
| by_cases hj0 : j=0 |
| · subst j |
| rw [pairKernelMass_zero_of_supported] |
| · exact hc.trans (by simpa using ENNReal.ofReal_le_ofReal hg1) |
| · filter_upwards [hgap] with x hx |
| linarith |
| rcases lt_or_eq_of_le hj with hjlt|hjeq |
| · have hp := before_bufferFirstFailure_pass (realPosition (step e)) f a (z₀+(1-ε)*r) g π H ω (Nat.pos_of_ne_zero hj0) hjlt |
| apply hc.trans (hp.trans _) |
| unfold pairKernelMass |
| apply lintegral_mono |
| intro x |
| apply measure_mono |
| intro y hy |
| change z₀+(1-ε)*r ≤ signedCoordinate f (y.2-y.1) at hy |
| change z₀ ≤ signedCoordinate f (y.2-y.1) |
| have : 0 ≤ (1-ε)*r := mul_nonneg (sub_nonneg.mpr hε1) hr |
| linarith |
| · subst j |
| let μ := bufferStageRetainedLaw e f a z₀ r ε α g π H ω κ |
| have hmass := bufferStageRetainedLaw_mass_lower e f a z₀ r ε α g π hH ω κ hκ1 hg1 hr hε hgap |
| have hs := bufferStageRetainedLaw_baseline e f a z₀ r ε α g π hH ω κ hr hε1 hα |
| have hm : μ Set.univ ≤ μ := by |
| apply measure_mono_ae |
| filter_upwards [hs] with y hy |
| exact fun _ => hy |
| have hh := hmass.trans (hm.trans ((bufferStageRetainedLaw_le e f a z₀ r ε α g π hH ω hκ) _)) |
| simpa only [rawPairMixture_apply,pairKernelMass] using hh |
| end DirectionalTransience |
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| end |
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| end |
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| end OAI |
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