| import OAI.Geometry.PinnedDistances.MobiusObstruction |
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| namespace OAI |
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| universe u1 u2 |
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| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped Topology |
| namespace WeakPinned.Bounded |
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| abbrev Quad (X : Type u1) := X × X × X × X |
| abbrev Expanded (X : Type u2) := Quad X × ((X × X) × (X × X)) |
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| lemma no_energy_law (μ ν : Measure ℂ) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] [NullSingletonClass μ] [NullSingletonClass ν] |
| (Q : Measure (Quad ℂ)) [IsProbabilityMeasure Q] |
| (hind : Q.map (fun q => (q.1,q.2.2))=μ.prod (Q.map (fun q => q.2.2))) |
| (hsupp : ∀ᵐ q ∂ Q,q.2.2.1∈μ.support) |
| (he : ∀ᵐ q ∂ Q,crossEnergy μ q.1 q.2.2.1=targetEnergy ν q.2.2.2) : False := by |
| let τ : Measure (ℂ × ℂ) := Q.map (fun q => q.2.2) |
| have : IsProbabilityMeasure τ := inferInstance |
| have hp := measurableSet_energy_eq μ ν |
| have he' : ∀ᵐ v ∂ μ.prod τ,crossEnergy μ v.1 v.2.1=targetEnergy ν v.2.2 := by |
| rw [← hind] |
| exact (ae_map_iff (by fun_prop) hp).mpr he |
| have hf : ∀ᵐ cd ∂ τ,∀ᵐ a ∂ μ,crossEnergy μ a cd.1=targetEnergy ν cd.2 := |
| (Measure.ae_ae_comm hp).mp ((Measure.ae_prod_iff_ae_ae hp).mp he') |
| have hs : ∀ᵐ cd ∂ τ,cd.1∈μ.support := |
| (ae_map_iff (by fun_prop) (μ.isClosed_support.measurableSet.preimage measurable_fst)).mpr hsupp |
| obtain ⟨cd,hc,hcd⟩ := (hs.and hf).exists |
| exact not_ae_energy_eq μ ν cd.1 cd.2 hc hcd |
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| lemma mobius_of_product (a b c d z v : ℂ) (hz : z≠a) |
| (hp : (z-a)*(v-b)=(c-a)*(d-b)) : v=mobius a b c d z := by |
| unfold mobius |
| have hh : v-b=(c-a)*(d-b)/(z-a) := (eq_div_iff (sub_ne_zero.mpr hz)).mpr (by simpa [mul_comm] using hp) |
| linear_combination hh |
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| lemma expanded_kernel_eq (μ ν : Measure ℂ) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] [NullSingletonClass μ] |
| (Q : Measure (Quad ℂ)) (R : Measure (Expanded ℂ)) |
| [IsProbabilityMeasure Q] [IsProbabilityMeasure R] |
| (he : R.map Prod.fst=Q) |
| (hm : R.map (fun r => (r.1,(r.2.1.1,r.2.2.1)))=Q.prod (μ.prod μ)) |
| (hbd : ∀ᵐ q ∂ Q,q.2.1≠q.2.2.2) |
| (hz : ∀ᵐ r ∂ R,(r.2.1.1-r.1.1)*(r.2.1.2-r.1.2.1)= |
| (r.1.2.2.1-r.1.1)*(r.1.2.2.2-r.1.2.1)) |
| (hw : ∀ᵐ r ∂ R,(r.2.2.1-r.1.1)*(r.2.2.2-r.1.2.1)= |
| (r.1.2.2.1-r.1.1)*(r.1.2.2.2-r.1.2.1)) : |
| ∀ᵐ r ∂ R,crossKernel r.1.1 r.1.2.2.1 (r.2.1.1,r.2.2.1)= |
| targetKernel r.1.2.2.2 (r.2.1.2,r.2.2.2) := by |
| have hpred : MeasurableSet {p : Quad ℂ × (ℂ × ℂ) | |
| p.2.1≠p.1.1 ∧ p.2.2≠p.1.1 ∧ p.2.2≠p.1.2.2.1} := by |
| measurability |
| have hpair : ∀ᵐ p ∂ Q.prod (μ.prod μ), |
| p.2.1≠p.1.1 ∧ p.2.2≠p.1.1 ∧ p.2.2≠p.1.2.2.1 := by |
| rw [Measure.ae_prod_iff_ae_ae hpred] |
| exact Eventually.of_forall fun e => by |
| filter_upwards [ae_pair_ne μ e.1,ae_pair_ne μ e.2.2.1] with zw ha hc |
| exact ⟨ha.1,ha.2,hc.2⟩ |
| rw [← hm] at hpair |
| have hp := (ae_map_iff (by fun_prop) hpred).mp hpair |
| have hd : ∀ᵐ r ∂ R,r.1.2.1≠r.1.2.2.2 := by |
| rw [← he] at hbd |
| exact (ae_map_iff measurable_fst.aemeasurable (by measurability)).mp hbd |
| filter_upwards [hp,hd,hz,hw] with r hp hd hz hw |
| have hz' := mobius_of_product r.1.1 r.1.2.1 r.1.2.2.1 r.1.2.2.2 _ _ hp.1 hz |
| have hw' := mobius_of_product r.1.1 r.1.2.1 r.1.2.2.1 r.1.2.2.2 _ _ hp.2.1 hw |
| have hh := (kernel_mobius r.1.1 r.1.2.1 r.1.2.2.1 r.1.2.2.2 |
| (r.2.1.1,r.2.2.1) hd hp.1 hp.2.1 hp.2.2).symm |
| simpa only [← hz',← hw'] using hh |
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| lemma expanded_energy_eq (μ ν : Measure ℂ) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] (Q : Measure (Quad ℂ)) (R : Measure (Expanded ℂ)) |
| [IsProbabilityMeasure Q] [IsProbabilityMeasure R] |
| (hm₁ : R.map (fun r => (r.1,(r.2.1.1,r.2.2.1)))=Q.prod (μ.prod μ)) |
| (hm₂ : R.map (fun r => (r.1,(r.2.1.2,r.2.2.2)))=Q.prod (ν.prod ν)) |
| (hk : ∀ᵐ r ∂ R,crossKernel r.1.1 r.1.2.2.1 (r.2.1.1,r.2.2.1)= |
| targetKernel r.1.2.2.2 (r.2.1.2,r.2.2.2)) : |
| ∀ᵐ q ∂ Q,crossEnergy μ q.1 q.2.2.1=targetEnergy ν q.2.2.2 := by |
| apply conditional_integral_eq R Q (μ.prod μ) (ν.prod ν) Prod.fst |
| (fun r => (r.2.1.1,r.2.2.1)) (fun r => (r.2.1.2,r.2.2.2)) |
| (by fun_prop) (by fun_prop) (by fun_prop) hm₁ hm₂ |
| (fun p => crossKernel p.1.1 p.1.2.2.1 p.2) |
| (fun p => targetKernel p.1.2.2.2 p.2) ?_ ?_ ?_ ?_ hk |
| · unfold crossKernel; fun_prop |
| · unfold targetKernel; fun_prop |
| · intro p |
| rw [Real.norm_eq_abs,abs_of_nonneg (crossKernel_nonneg _ _ _)] |
| exact crossKernel_le_one _ _ _ |
| · intro p |
| rw [Real.norm_eq_abs,abs_of_nonneg (targetKernel_nonneg _ _)] |
| exact targetKernel_le_one _ _ |
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| end WeakPinned.Bounded |
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| end |
| end |
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| end OAI |
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