File size: 3,655 Bytes
92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Dynamics.StandardMap.EndpointObservations
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
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
end OAI
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