| import OAI.Dynamics.StandardMap.EndpointObservations |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| open MeasureTheory Set Filter Metric |
| open scoped ENNReal Topology CompactlySupported Classical |
| namespace StandardMapEntropy |
| noncomputable def bridgeLeft (b : Bool) (j l : ArrayTestIndex) : EndpointObservation := |
| if b then testEndpoint j else shortfallEndpoint l |
| noncomputable def bridgeRight (b : Bool) (j l : ArrayTestIndex) : EndpointObservation := |
| if b then shortfallEndpoint l else testEndpoint j |
| def DyadicBridgeExclusions (d : DistanceArray) : Prop := |
| ∀ (u v : DyadicTime) (q : ℕ) (j l : ArrayTestIndex) (b : Bool), |
| (u:ℝ)<(v:ℝ) → (1:ℝ)/(2:ℝ)^q≤((v:ℝ)-(u:ℝ))/1000 → |
| (bridgeLeft b j l).inWindow u q → (bridgeRight b j l).inWindow v q → |
| 0<(bridgeLeft b j l).value d → 0<(bridgeRight b j l).value d → |
| d.val u v≤(999/1000:ℝ)*((v:ℝ)-(u:ℝ)) |
| namespace CriticalScaleSequence |
| variable (S : CriticalScaleSequence) (K : S.LimitLaws) |
| lemma endpoint_bridge_ae (select : Bool) (u v : DyadicTime) (huv : (u:ℝ)<(v:ℝ)) |
| (W H : EndpointObservation) (q : ℕ) (hq : (1:ℝ)/(2:ℝ)^q≤((v:ℝ)-(u:ℝ))/1000) |
| (hW : W.inWindow u q) (hH : H.inWindow v q) (j : ArrayTestIndex) |
| (htest : (∀d,W.value d=clippedTest j d) ∨ (∀d,H.value d=clippedTest j d)) : |
| ∀ᵐ d ∂K.selected S select,0< W.value d.val → 0< H.value d.val → |
| d.val.val u v≤(999/1000:ℝ)*((v:ℝ)-(u:ℝ)) := by |
| have hh (a c : ℚ) : ∀ᵐ d ∂K.selected S select, |
| 0<(a:ℝ) → 0<(c:ℝ) → ¬ d.val∈endpointBridgeEvent u v W H a c := by |
| by_cases ha : 0<(a:ℝ) |
| · by_cases hc : 0<(c:ℝ) |
| · have hn : ∀ᵐ d ∂K.selected S select,¬d.val∈endpointBridgeEvent u v W H a c := |
| ae_iff.mpr (by simpa only [not_not] using S.endpoint_bridge_null K select u v huv W H q hq hW hH j htest a c ha hc) |
| filter_upwards [hn] with d hd |
| exact fun _ _ => hd |
| · exact Eventually.of_forall fun d _ hc' => (hc hc').elim |
| · exact Eventually.of_forall fun d ha' _ => (ha ha').elim |
| filter_upwards [ae_all_iff.mpr (fun a => ae_all_iff.mpr (hh a))] with d hd |
| intro hw hh |
| obtain ⟨a,ha0,ha⟩ := exists_rat_btwn hw |
| obtain ⟨c,hc0,hc⟩ := exists_rat_btwn hh |
| by_contra hn |
| exact hd a c ha0 hc0 ⟨lt_of_not_ge hn,ha,hc⟩ |
| lemma dyadicBridge_aeselected (select : Bool) : ∀ᵐ d ∂K.selected S select,DyadicBridgeExclusions d.val := by |
| have hh (u v : DyadicTime) (q : ℕ) (j l : ArrayTestIndex) (b : Bool) : |
| ∀ᵐ d ∂K.selected S select, |
| (u:ℝ)<(v:ℝ) → (1:ℝ)/(2:ℝ)^q≤((v:ℝ)-(u:ℝ))/1000 → |
| (bridgeLeft b j l).inWindow u q → (bridgeRight b j l).inWindow v q → |
| 0<(bridgeLeft b j l).value d.val → 0<(bridgeRight b j l).value d.val → |
| d.val.val u v≤(999/1000:ℝ)*((v:ℝ)-(u:ℝ)) := by |
| by_cases h : (u:ℝ)<(v:ℝ) ∧ (1:ℝ)/(2:ℝ)^q≤((v:ℝ)-(u:ℝ))/1000 ∧ |
| (bridgeLeft b j l).inWindow u q ∧ (bridgeRight b j l).inWindow v q |
| · have he : (∀d,(bridgeLeft b j l).value d=clippedTest j d) ∨ |
| (∀d,(bridgeRight b j l).value d=clippedTest j d) := by |
| cases b <;> simp [bridgeLeft,bridgeRight] |
| filter_upwards [S.endpoint_bridge_ae K select u v h.1 _ _ q h.2.1 h.2.2.1 h.2.2.2 j he] with d hd |
| exact fun _ _ _ _ => hd |
| · exact Eventually.of_forall fun d h1 h2 h3 h4 => (h ⟨h1,h2,h3,h4⟩).elim |
| exact ae_all_iff.mpr fun u => ae_all_iff.mpr fun v => ae_all_iff.mpr fun q => |
| ae_all_iff.mpr fun j => ae_all_iff.mpr fun l => ae_all_iff.mpr (hh u v q j l) |
| end CriticalScaleSequence |
| end StandardMapEntropy |
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| end OAI |
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