openAI-math-dataset / lean /OAI /NumberTheory /Jacobsthal /Estimates /ConsecutivePowerProjection.lean
| import OAI.NumberTheory.Jacobsthal.Paths.ConsecutivePairStop | |
| namespace OAI | |
| namespace Erdos970 | |
| section | |
| namespace Erdos970Dependency.MarkedVisits | |
| open Filter Set MeasureTheory ProbabilityTheory | |
| open scoped ProbabilityTheory ENNReal Classical | |
| open NumberTheoryLean.FinitePathMeasures | |
| noncomputable def historyGoodInput (a : ℕ) (v H : ℝ) : Set (RawHistory a) := | |
| rawLast a ⁻¹' regenerativeCostWindow v H | |
| lemma historyGoodInput_measurable (a : ℕ) (v H : ℝ) : MeasurableSet (historyGoodInput a v H) := | |
| (regenerativeCostWindow_measurable v H).preimage (rawLast_measurable a) | |
| lemma pairHitTest_first_eq (a : ℕ) (v H : ℝ) : | |
| pairHitTest a 0 (a+1) (by omega) v H = | |
| (rawPrefix (show a ≤ a+1 by omega) ⁻¹' historyGoodInput a v H) ∩ firstPrefixMarkEvent a true := by | |
| ext h | |
| change (_ ∧ _ ∧ _) ↔ ((_ ∧ _) ∧ _) | |
| constructor | |
| · rintro ⟨hR,hM,hW⟩ | |
| exact ⟨⟨hR,hW⟩,hM⟩ | |
| · rintro ⟨⟨hR,hW⟩,hM⟩ | |
| exact ⟨hR,hM,hW⟩ | |
| theorem pairStoppedHistory_zero_input_filter (a : ℕ) (v H : ℝ) : | |
| pairStoppedHistory a 0 v H= | |
| ((rawExtension a (a+1)).restrict (firstPrefixMarkEvent_measurable a true)) ∘ₖ | |
| stateFilter (historyGoodInput_measurable a v H) := by | |
| rw [pairStoppedHistory_zero] | |
| have he := restriction_comp_of_retained_prefix (Kernel.id : Kernel (RawHistory a) (RawHistory a)) | |
| (rawExtension a (a+1)) (rawPrefix_measurable (show a ≤ a+1 by omega)) | |
| (rawExtension_retains_prefix (show a ≤ a+1 by omega)) | |
| (historyGoodInput_measurable a v H) (firstPrefixMarkEvent_measurable a true) | |
| rw [Kernel.comp_id] at he | |
| ext past S hS | |
| have heS := congrArg (fun K : Kernel (RawHistory a) (RawHistory (a+1)) => K past S) he | |
| rw [Kernel.restrict_apply' _ _ _ hS] at heS | |
| rw [Kernel.restrict_apply' _ _ _ hS,pairHitTest_first_eq] | |
| exact heS | |
| theorem canonical_capture_projection (a : ℕ) (v H : ℝ) (past : RawHistory a) : | |
| (pairStoppedHistory a 0 v H past).map (rawLast (a+1))=consecutiveCapture v H (rawLast a past) := by | |
| rw [pairStoppedHistory_zero_input_filter,stateFilter_input,consecutiveCapture,stateFilter_input] | |
| by_cases hp : rawLast a past ∈ regenerativeCostWindow v H | |
| · have hh : past ∈ historyGoodInput a v H := hp | |
| rw [ite_eq_left hh,ite_eq_left hp,Kernel.restrict_apply,Kernel.restrict_apply] | |
| change ((rawExtension a (a+1) past).restrict ((rawLast (a+1)) ⁻¹' markedArrival)).map (rawLast (a+1))=_ | |
| rw [← Measure.restrict_map (rawLast_measurable (a+1)) markedArrival_measurable,rawExtension_last_step] | |
| · have hh : past ∉ historyGoodInput a v H := hp | |
| rw [ite_eq_right hh,ite_eq_right hp,Measure.map_zero] | |
| noncomputable def firstNoHitHistory (a : ℕ) (v H : ℝ) : Kernel (RawHistory a) (RawHistory (a+1)) := | |
| (rawExtension a (a+1)).restrict (pairHitTest_measurable a 0 (a+1) (by omega) v H).compl | |
| instance firstNoHitHistory_isFiniteKernel (a : ℕ) (v H : ℝ) : IsFiniteKernel (firstNoHitHistory a v H) := by | |
| unfold firstNoHitHistory | |
| infer_instance | |
| lemma firstHistory_partition (a : ℕ) (v H : ℝ) : | |
| pairStoppedHistory a 0 v H+firstNoHitHistory a v H=rawExtension a (a+1) := by | |
| ext past S hS | |
| rw [_root_.add_apply,Measure.add_apply,pairStoppedHistory_zero,firstNoHitHistory, | |
| Kernel.restrict_apply,Kernel.restrict_apply,← Measure.add_apply, | |
| Measure.restrict_add_restrict_compl (pairHitTest_measurable a 0 (a+1) (by omega) v H)] | |
| theorem canonical_first_continue_projection (a : ℕ) (v H : ℝ) (past : RawHistory a) : | |
| (firstNoHitHistory a v H past).map (rawLast (a+1))=consecutiveFirstContinue v H (rawLast a past) := by | |
| have hPart := congrArg (fun K : Kernel (RawHistory a) (RawHistory (a+1)) => | |
| (K past).map (rawLast (a+1))) (firstHistory_partition a v H) | |
| rw [_root_.add_apply,Measure.map_add _ _ (rawLast_measurable (a+1)), | |
| canonical_capture_projection,rawExtension_last_step] at hPart | |
| have hOther := congrArg (fun K : Kernel CostState CostState => K (rawLast a past)) (consecutive_first_partition v H) | |
| rw [_root_.add_apply] at hOther | |
| ext S _hS | |
| have he := congrArg (fun μ : Measure CostState => μ S) (hPart.trans hOther.symm) | |
| simp only [Measure.add_apply] at he | |
| exact (ENNReal.add_right_inj (measure_ne_top (consecutiveCapture v H (rawLast a past)) S)).mp he | |
| end Erdos970Dependency.MarkedVisits | |
| end | |
| section | |
| namespace Erdos970Dependency.MarkedVisits | |
| open Filter Set MeasureTheory ProbabilityTheory | |
| open scoped ProbabilityTheory ENNReal | |
| open NumberTheoryLean.FinitePathMeasures | |
| noncomputable def noHitPairHistory (a : ℕ) (v H : ℝ) : Kernel (RawHistory a) (RawHistory (a+2)) := | |
| (rawExtension a (a+2)).restrict (pairHitTest_measurable a 0 (a+2) (by omega) v H).compl | |
| instance noHitPairHistory_isFiniteKernel (a : ℕ) (v H : ℝ) : IsFiniteKernel (noHitPairHistory a v H) := by | |
| unfold noHitPairHistory | |
| infer_instance | |
| theorem noHitPairHistory_factor (a : ℕ) (v H : ℝ) : | |
| noHitPairHistory a v H=rawExtension (a+1) (a+2) ∘ₖ firstNoHitHistory a v H := by | |
| unfold noHitPairHistory firstNoHitHistory | |
| have he := rawExtension_restrict_comp (show a ≤ a+1 by omega) (show a+1 ≤ a+2 by omega) | |
| (pairHitTest_measurable a 0 (a+1) (by omega) v H).compl MeasurableSet.univ | |
| rw [Kernel.restrict_univ] at he | |
| ext past S hS | |
| have heS := congrArg (fun K : Kernel (RawHistory a) (RawHistory (a+2)) => K past S) he | |
| rw [Kernel.restrict_apply' _ _ _ hS] at heS | |
| rw [Kernel.restrict_apply' _ _ _ hS] | |
| have hSet : (rawPrefix (show a+1 ≤ a+2 by omega) ⁻¹' (pairHitTest a 0 (a+1) (by omega) v H)ᶜ) ∩ univ= | |
| (pairHitTest a 0 (a+2) (by omega) v H)ᶜ := by | |
| rw [inter_univ,preimage_compl,pairHitTest_prefix] | |
| rw [hSet] at heS | |
| exact heS | |
| theorem canonical_pair_continue_projection (a : ℕ) (v H : ℝ) (past : RawHistory a) : | |
| (noHitPairHistory a v H past).map (rawLast (a+2))=consecutiveContinue v H (rawLast a past) := by | |
| apply Measure.ext_of_lintegral | |
| intro F hF | |
| rw [lintegral_map hF (rawLast_measurable (a+2)),noHitPairHistory_factor, | |
| Kernel.lintegral_comp _ _ _ (g := fun h => F (rawLast (a+2) h)) (hF.comp (rawLast_measurable (a+2)))] | |
| have he (h : RawHistory (a+1)) : | |
| (∫⁻ y, F (rawLast (a+2) y) ∂rawExtension (a+1) (a+2) h)=∫⁻ z, F z ∂costKernel (rawLast (a+1) h) := by | |
| rw [← lintegral_map (g := rawLast (a+2)) hF (rawLast_measurable (a+2)),rawExtension_last_step] | |
| simp_rw [he] | |
| have hG : Measurable (fun z : CostState => ∫⁻ y, F y ∂costKernel z) := hF.lintegral_kernel | |
| rw [← lintegral_map (g := rawLast (a+1)) hG (rawLast_measurable (a+1)),canonical_first_continue_projection, | |
| consecutiveContinue,Kernel.lintegral_comp _ _ _ hF] | |
| lemma shifted_pair_continue_projection (a n b : ℕ) (hb : b=(a+2*n)+2) (v H : ℝ) | |
| (past : RawHistory (a+2*n)) : | |
| (((rawExtension (a+2*n) b).restrict | |
| (pairHitTest_measurable a n b (by omega) v H).compl) past).map (rawLast b)= | |
| consecutiveContinue v H (rawLast (a+2*n) past) := by | |
| subst b | |
| exact canonical_pair_continue_projection (a+2*n) v H past | |
| end Erdos970Dependency.MarkedVisits | |
| end | |
| section | |
| namespace Erdos970Dependency.MarkedVisits | |
| open Filter Set MeasureTheory ProbabilityTheory | |
| open scoped ProbabilityTheory ENNReal | |
| open NumberTheoryLean.FinitePathMeasures | |
| theorem pairSurvivingHistory_projection (a n : ℕ) (v H : ℝ) (past : RawHistory a) : | |
| (pairSurvivingHistory a n v H past).map (rawLast (a+2*n))= | |
| ((consecutiveContinue v H)^n) (rawLast a past) := by | |
| have (m : ℕ) : IsSFiniteKernel ((consecutiveContinue v H)^m) := kernel_power_sfinite _ m | |
| induction n with | |
| | zero => | |
| rw [pairSurvivingHistory_zero] | |
| change (Measure.dirac past).map (rawLast a)=Measure.dirac (rawLast a past) | |
| exact Measure.map_dirac' (rawLast_measurable a) past | |
| | succ n ih => | |
| apply Measure.ext_of_lintegral | |
| intro F hF | |
| rw [lintegral_map hF (rawLast_measurable _),pairSurvivingHistory_succ, | |
| Kernel.lintegral_comp _ _ _ (g := fun h => F (rawLast (a+2*(n+1)) h)) (hF.comp (rawLast_measurable _))] | |
| have he (h : RawHistory (a+2*n)) : | |
| (∫⁻ y, F (rawLast (a+2*(n+1)) y) ∂((rawExtension (a+2*n) (a+2*(n+1))).restrict | |
| (pairHitTest_measurable a n (a+2*(n+1)) (by omega) v H).compl) h)= | |
| ∫⁻ z, F z ∂consecutiveContinue v H (rawLast (a+2*n) h) := by | |
| rw [← lintegral_map (g := rawLast (a+2*(n+1))) hF (rawLast_measurable _), | |
| shifted_pair_continue_projection a n (a+2*(n+1)) (by omega) v H h] | |
| simp_rw [he] | |
| have hG : Measurable (fun z : CostState => ∫⁻ y, F y ∂consecutiveContinue v H z) := hF.lintegral_kernel | |
| rw [← lintegral_map (g := rawLast (a+2*n)) hG (rawLast_measurable _),ih,pow_succ'] | |
| change (∫⁻ z, ∫⁻ y, F y ∂consecutiveContinue v H z ∂((consecutiveContinue v H)^n) (rawLast a past))= | |
| ∫⁻ y, F y ∂(consecutiveContinue v H ∘ₖ ((consecutiveContinue v H)^n)) (rawLast a past) | |
| exact (Kernel.lintegral_comp _ _ _ hF).symm | |
| theorem pairStoppedHistory_projection (a n : ℕ) (v H : ℝ) (past : RawHistory a) : | |
| (pairStoppedHistory a n v H past).map (rawLast (a+2*n+1))= | |
| (consecutiveCapture v H ∘ₖ ((consecutiveContinue v H)^n)) (rawLast a past) := by | |
| have : IsSFiniteKernel ((consecutiveContinue v H)^n) := kernel_power_sfinite _ n | |
| apply Measure.ext_of_lintegral | |
| intro F hF | |
| rw [lintegral_map hF (rawLast_measurable _),pairStoppedHistory_factor, | |
| Kernel.lintegral_comp _ _ _ (g := fun h => F (rawLast (a+2*n+1) h)) (hF.comp (rawLast_measurable _))] | |
| have he (h : RawHistory (a+2*n)) : | |
| (∫⁻ y, F (rawLast (a+2*n+1) y) ∂((rawExtension (a+2*n) (a+2*n+1)).restrict | |
| (pairHitTest_measurable a n (a+2*n+1) le_rfl v H)) h)= | |
| ∫⁻ z, F z ∂consecutiveCapture v H (rawLast (a+2*n) h) := by | |
| have hP := canonical_capture_projection (a+2*n) v H h | |
| rw [pairStoppedHistory_zero] at hP | |
| rw [← lintegral_map (g := rawLast (a+2*n+1)) hF (rawLast_measurable _)] | |
| exact congrArg (fun μ : Measure CostState => ∫⁻ z, F z ∂μ) hP | |
| simp_rw [he] | |
| have hG : Measurable (fun z : CostState => ∫⁻ y, F y ∂consecutiveCapture v H z) := hF.lintegral_kernel | |
| rw [← lintegral_map (g := rawLast (a+2*n)) hG (rawLast_measurable _), | |
| pairSurvivingHistory_projection,Kernel.lintegral_comp _ _ _ hF] | |
| end Erdos970Dependency.MarkedVisits | |
| end | |
| end Erdos970 | |
| end OAI | |