openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Geometry /CountableRootPrefixAtoms.lean
| import OAI.NumberTheory.DukePrimeDegree.Entropy.RootPrefixFutureFibers | |
| namespace OAI | |
| /-! # Positive leaf mass of countable whole-future atoms | |
| The argument uses support of the actual conditional leaf kernel. No finite | |
| entropy of the countable observation is required. | |
| -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped Topology | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| namespace RootPrefixCoordinates | |
| variable {n : ℕ} {P₀ : Finset (Fin n × Fin n)} {D : Finset (Fin n)} | |
| {V A : Type*} [TopologicalSpace V] | |
| theorem wholeFutureAtom_mem_nhds (C : RootPrefixCoordinates n P₀ V) | |
| (c : EliminationChain P₀ D) (hP₀ : InitialRootPattern P₀) | |
| (hf : c.finalEntries = allPositiveRoots n) (g : G n) | |
| (T : X n → X n) (Q : X n → A) | |
| (z : (UpperRootTransverse n × V) × c.Time) | |
| (hgood : ∀ᶠ v in 𝓝 (completeRootChain n).zeroTime, ∀ k : ℕ, | |
| Q (T^[k] (rightAction (C.timeParam c hP₀ hf g z) ((completeRootChain n).word v))) = | |
| Q (T^[k] (C.timeParam c hP₀ hf g z))) : | |
| {t | ∀ k : ℕ, Q (T^[k] (C.timeParam c hP₀ hf g (z.1,t))) = | |
| Q (T^[k] (C.timeParam c hP₀ hf g z))} ∈ 𝓝 z.2 := by | |
| have hc : Tendsto (c.fullRootTime hP₀ hf ∘ c.remainingRebase hP₀ hf z.2) | |
| (𝓝 z.2) (𝓝 (completeRootChain n).zeroTime) := by | |
| have H := ((c.continuous_fullRootTime hP₀ hf).comp | |
| (c.remainingRebase hP₀ hf z.2).continuous).tendsto z.2 | |
| simpa only [Function.comp_apply,c.remainingRebase_self,c.fullRootTime_zero] using H | |
| filter_upwards [hc.eventually hgood] with t ht | |
| intro k | |
| rw [Function.comp_apply,c.word_fullRootTime] at ht | |
| rw [C.timeParam_rebased c hP₀ hf g z.1 z.2 t] | |
| exact ht k | |
| end RootPrefixCoordinates | |
| namespace RootPrefixBox | |
| variable {n : ℕ} {P₀ : Finset (Fin n × Fin n)} {D : Finset (Fin n)} | |
| {V A : Type*} [TopologicalSpace V] [SecondCountableTopology V] | |
| [MeasurableSpace V] [BorelSpace V] | |
| {C : RootPrefixCoordinates n P₀ V} {c : EliminationChain P₀ D} | |
| {hP₀ : InitialRootPattern P₀} {hf : c.finalEntries = allPositiveRoots n} {g : G n} | |
| omit [SecondCountableTopology V] in | |
| theorem ae_wholeFutureAtom_pos (B : RootPrefixBox C c hP₀ hf g) | |
| (μ : Measure (X n)) [IsFiniteMeasure μ] (T : X n → X n) (Q : X n → A) | |
| (hgood : ∀ᵐ x ∂μ, ∀ᶠ v in 𝓝 (completeRootChain n).zeroTime, ∀ k : ℕ, | |
| Q (T^[k] (rightAction x ((completeRootChain n).word v))) = Q (T^[k] x)) : | |
| ∀ᵐ z ∂B.coordinateMeasure μ, 0 < B.conditionalKernel μ z.1 | |
| {t | ∀ k : ℕ, Q (T^[k] (C.timeParam c hP₀ hf g (z.1,t))) = | |
| Q (T^[k] (C.timeParam c hP₀ hf g z))} := by | |
| filter_upwards [B.ae_coordinate_of_ambient μ hgood,B.ae_mem_conditionalKernel_support μ] | |
| with z hz hs | |
| apply ((B.conditionalKernel μ z.1).mem_support_iff_forall z.2).mp hs | |
| exact C.wholeFutureAtom_mem_nhds c hP₀ hf g T Q z hz | |
| end RootPrefixBox | |
| end DukePrimeDegree | |
| end | |
| end OAI | |