openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /ConditionalCrossingAvoidance.lean
| import OAI.Analysis.IrrationalRotation.Analysis.CrossingAvoidance | |
| import OAI.Analysis.IrrationalRotation.Couplings.ConditionalKernelSupport | |
| import Mathlib.Probability.Independence.Conditional | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory | |
| open scoped BigOperators | |
| namespace Invariant233.CrossingAvoidance | |
| section KernelBounds | |
| variable {ι Ω Y : Type*} [Countable ι] [MeasurableSpace Ω] [MeasurableSpace Y] | |
| theorem kernel_iIndepSet_ae (κ : Kernel Y Ω) (ν : Measure Y) | |
| (events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i)) | |
| (hind : Kernel.iIndepSet events κ ν) : | |
| ∀ᵐ y ∂ν, iIndepSet events (κ y) := by | |
| have hall : ∀ᵐ y ∂ν, ∀ s : Finset ι, | |
| κ y (⋂ i ∈ s, events i) = ∏ i ∈ s, κ y (events i) := | |
| ae_all_iff.mpr (fun s => hind.meas_biInter s) | |
| filter_upwards [hall] with y hy | |
| exact (iIndepSet_iff_meas_biInter hmeas).mpr hy | |
| theorem kernel_avoid_le_exp (κ : Kernel Y Ω) (ν : Measure Y) | |
| (events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i)) | |
| (hind : Kernel.iIndepSet events κ ν) (selected : Y → Finset ι) (p : Y → ℝ) | |
| (hp : ∀ᵐ y ∂ν, ∀ i ∈ selected y, p y ≤ (κ y).real (events i)) : | |
| ∀ᵐ y ∂ν, (κ y).real (⋂ i ∈ selected y, (events i)ᶜ) ≤ | |
| Real.exp (-((selected y).card : ℝ) * p y) := by | |
| filter_upwards [kernel_iIndepSet_ae κ ν events hmeas hind, hp] with y hy hp | |
| let : IsProbabilityMeasure (κ y) := hy.isProbabilityMeasure | |
| exact measureReal_avoid_le_exp (κ y) events hy hmeas (selected y) (p y) hp | |
| theorem kernel_bad_le_exp_intensity (κ : Kernel Y Ω) (ν : Measure Y) | |
| (events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i)) | |
| (hind : Kernel.iIndepSet events κ ν) (selected : Y → Finset ι) | |
| (m ρ L : ℝ) (hρ : 0 ≤ ρ) (hL : L ≤ m * ρ) | |
| (hcard : ∀ᵐ y ∂ν, m / 40 ≤ ((selected y).card : ℝ)) | |
| (hp : ∀ᵐ y ∂ν, ∀ i ∈ selected y, ρ / 10 ≤ (κ y).real (events i)) | |
| (bad : Y → Set Ω) | |
| (hbad : ∀ᵐ y ∂ν, ∀ᵐ ω ∂κ y, | |
| ω ∈ bad y → ∀ i ∈ selected y, ω ∉ events i) : | |
| ∀ᵐ y ∂ν, (κ y).real (bad y) ≤ Real.exp (-L / 400) := by | |
| filter_upwards [kernel_iIndepSet_ae κ ν events hmeas hind, hcard, hp, hbad] | |
| with y hy hcard hp hbad | |
| let : IsProbabilityMeasure (κ y) := hy.isProbabilityMeasure | |
| have hsubset : bad y ≤ᵐ[κ y] ⋂ i ∈ selected y, (events i)ᶜ := by | |
| filter_upwards [hbad] with ω hω | |
| intro hb | |
| exact Set.mem_iInter.mpr (fun i => Set.mem_iInter.mpr (fun hi => hω hb i hi)) | |
| have hmono : (κ y).real (bad y) ≤ (κ y).real (⋂ i ∈ selected y, (events i)ᶜ) := | |
| ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hsubset) | |
| refine hmono.trans ?_ | |
| exact measureReal_le_exp_intensity (κ y) events hy hmeas (selected y) m ρ L | |
| hρ hcard hL hp _ (Set.Subset.refl _) | |
| theorem kernel_bad_le_exp_intensity_on (κ : Kernel Y Ω) (ν : Measure Y) | |
| (events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i)) | |
| (hind : Kernel.iIndepSet events κ ν) (selected : Y → Finset ι) | |
| (m ρ L : ℝ) (hρ : 0 ≤ ρ) (hL : L ≤ m * ρ) (active : Y → Prop) | |
| (hcard : ∀ᵐ y ∂ν, active y → m / 40 ≤ ((selected y).card : ℝ)) | |
| (hp : ∀ᵐ y ∂ν, active y → ∀ i ∈ selected y, ρ / 10 ≤ (κ y).real (events i)) | |
| (bad : Y → Set Ω) | |
| (hzero : ∀ᵐ y ∂ν, ¬ active y → κ y (bad y) = 0) | |
| (hbad : ∀ᵐ y ∂ν, ∀ᵐ ω ∂κ y, | |
| ω ∈ bad y → ∀ i ∈ selected y, ω ∉ events i) : | |
| ∀ᵐ y ∂ν, (κ y).real (bad y) ≤ Real.exp (-L / 400) := by | |
| filter_upwards [kernel_iIndepSet_ae κ ν events hmeas hind, hcard, hp, hzero, hbad] | |
| with y hy hcard hp hzero hbad | |
| let : IsProbabilityMeasure (κ y) := hy.isProbabilityMeasure | |
| by_cases ha : active y | |
| · have hsubset : bad y ≤ᵐ[κ y] ⋂ i ∈ selected y, (events i)ᶜ := by | |
| filter_upwards [hbad] with ω hω | |
| intro hb | |
| exact Set.mem_iInter.mpr (fun i => Set.mem_iInter.mpr (fun hi => hω hb i hi)) | |
| have hmono : (κ y).real (bad y) ≤ (κ y).real (⋂ i ∈ selected y, (events i)ᶜ) := | |
| ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hsubset) | |
| exact hmono.trans (measureReal_le_exp_intensity (κ y) events hy hmeas (selected y) m ρ L | |
| hρ (hcard ha) hL (hp ha) _ (Set.Subset.refl _)) | |
| · change (κ y (bad y)).toReal ≤ _ | |
| rw [hzero ha, ENNReal.toReal_zero] | |
| exact (Real.exp_pos _).le | |
| end KernelBounds | |
| section ConditionalExpectation | |
| variable {ι Ω : Type*} [Countable ι] | |
| theorem iCondIndepSet_ae_condExpKernel (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} | |
| [StandardBorelSpace Ω] (hm' : m' ≤ mΩ) (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i)) | |
| (hind : iCondIndepSet m' hm' events μ) : | |
| ∀ᵐ x ∂μ, iIndepSet events (condExpKernel μ m' x) := by | |
| exact ae_of_ae_trim hm' | |
| (@kernel_iIndepSet_ae ι Ω Ω ‹Countable ι› mΩ m' | |
| (condExpKernel μ m') (μ.trim hm') events hmeas hind) | |
| theorem conditional_bad_le_exp_intensity (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} | |
| [StandardBorelSpace Ω] (hm' : m' ≤ mΩ) (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i)) | |
| (hind : iCondIndepSet m' hm' events μ) (selected : Ω → Finset ι) | |
| (m ρ L : ℝ) (hρ : 0 ≤ ρ) (hL : L ≤ m * ρ) (active : Ω → Prop) | |
| (hcard : ∀ᵐ x ∂μ, active x → m / 40 ≤ ((selected x).card : ℝ)) | |
| (hp : ∀ᵐ x ∂μ, active x → ∀ i ∈ selected x, | |
| ρ / 10 ≤ (condExpKernel μ m' x).real (events i)) | |
| (bad : Set Ω) (hbadmeas : MeasurableSet bad) | |
| (hzero : ∀ᵐ x ∂μ, ¬ active x → condExpKernel μ m' x bad = 0) | |
| (hbad : ∀ᵐ x ∂μ, ∀ᵐ ω ∂condExpKernel μ m' x, | |
| ω ∈ bad → ∀ i ∈ selected x, ω ∉ events i) : | |
| ∀ᵐ x ∂μ, (μ[bad.indicator (fun _ => (1 : ℝ)) | m']) x ≤ | |
| Real.exp (-L / 400) := by | |
| filter_upwards [iCondIndepSet_ae_condExpKernel m' hm' μ events hmeas hind, | |
| hcard, hp, hzero, hbad, condExpKernel_ae_eq_condExp hm' hbadmeas] | |
| with x hx hcard hp hzero hbad heq | |
| rw [← heq] | |
| let : IsProbabilityMeasure (condExpKernel μ m' x) := hx.isProbabilityMeasure | |
| by_cases ha : active x | |
| · have hsubset : bad ≤ᵐ[condExpKernel μ m' x] ⋂ i ∈ selected x, (events i)ᶜ := by | |
| filter_upwards [hbad] with ω hω | |
| intro hb | |
| exact Set.mem_iInter.mpr (fun i => Set.mem_iInter.mpr (fun hi => hω hb i hi)) | |
| have hmono : (condExpKernel μ m' x).real bad ≤ | |
| (condExpKernel μ m' x).real (⋂ i ∈ selected x, (events i)ᶜ) := | |
| ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hsubset) | |
| exact hmono.trans (measureReal_le_exp_intensity (condExpKernel μ m' x) events hx hmeas | |
| (selected x) m ρ L hρ (hcard ha) hL (hp ha) _ (Set.Subset.refl _)) | |
| · change (condExpKernel μ m' x bad).toReal ≤ _ | |
| rw [hzero ha, ENNReal.toReal_zero] | |
| exact (Real.exp_pos _).le | |
| theorem conditional_bad_le_exp_of_endpoint {Y : Type*} | |
| (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] | |
| [MeasurableSpace Y] [MeasurableEq Y] | |
| (hm' : m' ≤ mΩ) (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (events : ι → Set Ω) (hmeas : ∀ i, MeasurableSet (events i)) | |
| (hind : iCondIndepSet m' hm' events μ) | |
| (endpoint : Ω → Y) (hendpoint : Measurable[m'] endpoint) | |
| (selected : Y → Finset ι) (active : Y → Prop) | |
| (m ρ L : ℝ) (hρ : 0 ≤ ρ) (hL : L ≤ m * ρ) | |
| (hcard : ∀ y, active y → m / 40 ≤ ((selected y).card : ℝ)) | |
| (hp : ∀ᵐ x ∂μ, active (endpoint x) → ∀ i ∈ selected (endpoint x), | |
| ρ / 10 ≤ (μ[(events i).indicator (fun _ => (1 : ℝ)) | m']) x) | |
| (bad : Set Ω) (hbadmeas : MeasurableSet bad) | |
| (hbad : ∀ᵐ ω ∂μ, ω ∈ bad → active (endpoint ω) ∧ | |
| ∀ i ∈ selected (endpoint ω), ω ∉ events i) : | |
| ∀ᵐ x ∂μ, (μ[bad.indicator (fun _ => (1 : ℝ)) | m']) x ≤ | |
| Real.exp (-L / 400) := by | |
| have heq : ∀ᵐ x ∂μ, ∀ i, (condExpKernel μ m' x).real (events i) = | |
| (μ[(events i).indicator (fun _ => (1 : ℝ)) | m']) x := | |
| ae_all_iff.mpr (fun i => condExpKernel_ae_eq_condExp hm' (hmeas i)) | |
| have hpKernel : ∀ᵐ x ∂μ, active (endpoint x) → ∀ i ∈ selected (endpoint x), | |
| ρ / 10 ≤ (condExpKernel μ m' x).real (events i) := by | |
| filter_upwards [hp, heq] with x hx heq | |
| intro ha i hi | |
| rw [heq i] | |
| exact hx ha i hi | |
| have hfixed := condExpKernel_ae_eq_of_measurable m' hm' μ endpoint hendpoint | |
| have hlift := ae_ae_condExpKernel_of_ae m' hm' μ hbad | |
| have hsupport : ∀ᵐ x ∂μ, ∀ᵐ ω ∂condExpKernel μ m' x, | |
| ω ∈ bad → active (endpoint x) ∧ ∀ i ∈ selected (endpoint x), ω ∉ events i := by | |
| filter_upwards [hfixed, hlift] with x hfixed hlift | |
| filter_upwards [hfixed, hlift] with ω hfixed hlift | |
| intro hb | |
| simpa only [hfixed] using hlift hb | |
| have hzero : ∀ᵐ x ∂μ, ¬ active (endpoint x) → condExpKernel μ m' x bad = 0 := by | |
| filter_upwards [hsupport] with x hx | |
| intro ha | |
| have hn : ∀ᵐ ω ∂condExpKernel μ m' x, ω ∉ bad := by | |
| filter_upwards [hx] with ω hω | |
| exact fun hb => ha (hω hb).1 | |
| simpa only [not_not, Set.ofPred_mem_eq] using (ae_iff.mp hn) | |
| have havoid : ∀ᵐ x ∂μ, ∀ᵐ ω ∂condExpKernel μ m' x, | |
| ω ∈ bad → ∀ i ∈ selected (endpoint x), ω ∉ events i := by | |
| filter_upwards [hsupport] with x hx | |
| filter_upwards [hx] with ω hω | |
| exact fun hb => (hω hb).2 | |
| exact conditional_bad_le_exp_intensity m' hm' μ events hmeas hind | |
| (fun x => selected (endpoint x)) m ρ L hρ hL (fun x => active (endpoint x)) | |
| (Filter.Eventually.of_forall (fun x => hcard (endpoint x))) hpKernel | |
| bad hbadmeas hzero havoid | |
| end ConditionalExpectation | |
| end Invariant233.CrossingAvoidance | |
| end OAI | |