| import OAI.Dynamics.EntropyCounterexample.MapData |
| import OAI.Dynamics.EntropyCounterexample.MapContinuity |
| import OAI.Dynamics.EntropyCounterexample.MapQuotientContinuity |
| import OAI.Geometry.SingularHomology.SkewProductHomology |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Problem340.Construction |
|
|
| def rawClockFunction (D : SphereData) (x : RawManifold) : ℝ := |
| clockIncrement D x.1 x.2.1 |
|
|
| def rawPerturbation (D : SphereData) (x : RawManifold) : ℂ := |
| readout D x.1 x.2.2 |
|
|
| def rawRegisterFunction (D : SphereData) (x : RawManifold) (i : Fin 3) : ℂ := |
| (periodicGain D x.1 : ℂ) * D.saturation (x.2.2 i) + input D i x.1 x.2.1 |
|
|
| theorem rawMap_eq_rawUpdate_coefficients (D : SphereData) : |
| rawMap D = rawUpdate (rawClockFunction D) (rawPerturbation D) |
| (rawRegisterFunction D) := rfl |
|
|
| theorem continuous_rawMap_of_rawCoefficients (D : SphereData) |
| (hh : Continuous (rawClockFunction D)) (hC : Continuous (rawPerturbation D)) |
| (hR : Continuous (rawRegisterFunction D)) : Continuous (rawMap D) := |
| continuous_rawUpdate hh hC hR |
|
|
| def rawMapHomotopy (D : SphereData) |
| (hh : Continuous (rawClockFunction D)) (hC : Continuous (rawPerturbation D)) |
| (hR : Continuous (rawRegisterFunction D)) : |
| ContinuousMap.Homotopy |
| ⟨(@rawNormalForm 3), continuous_rawNormalForm⟩ |
| ⟨rawMap D, continuous_rawMap_of_rawCoefficients D hh hC hR⟩ := |
| rawUpdateHomotopy hh hC hR |
|
|
| theorem realHomologyMap_rawMap_eq_normalForm (n : ℕ) (D : SphereData) |
| (hh : Continuous (rawClockFunction D)) (hC : Continuous (rawPerturbation D)) |
| (hR : Continuous (rawRegisterFunction D)) : |
| realHomologyMap n (rawMap D) (continuous_rawMap_of_rawCoefficients D hh hC hR) = |
| realHomologyMap n rawNormalForm continuous_rawNormalForm := |
| realHomologyMap_rawUpdate_eq_normalForm n 3 hh hC hR |
|
|
| theorem rawMap_eigenvector_of_sphere (n : ℕ) (D : SphereData) |
| (hh : Continuous (rawClockFunction D)) (hC : Continuous (rawPerturbation D)) |
| (hR : Continuous (rawRegisterFunction D)) |
| (c : ℝ) (v : RealSingularHomology n Sphere) (hv : v ≠ 0) |
| (heig : realHomologyMap n rawSphereSquare continuous_rawSphereSquare v = c • v) : |
| ∃ w : RealSingularHomology n RawManifold, w ≠ 0 ∧ |
| realHomologyMap n (rawMap D) (continuous_rawMap_of_rawCoefficients D hh hC hR) w = c • w := |
| rawUpdate_eigenvector_of_sphere n 3 hh hC hR c v hv heig |
|
|
| theorem continuous_rawCoefficients (D : SphereData) |
| (hh : Continuous (fun p : AddCircle (100 : ℝ) × Sphere => clockIncrement D p.1 p.2)) |
| (hβ : ∀ i, Continuous (periodicReadout i)) |
| (ha : Continuous (periodicGain D)) (hS : Continuous D.saturation) |
| (hU : ∀ i, Continuous (fun p : AddCircle (100 : ℝ) × Sphere => input D i p.1 p.2)) : |
| Continuous (rawClockFunction D) ∧ Continuous (rawPerturbation D) ∧ |
| Continuous (rawRegisterFunction D) := by |
| let p : RawManifold → AddCircle (100 : ℝ) × Sphere := fun x => (x.1, x.2.1) |
| have hp : Continuous p := continuous_fst.prodMk (continuous_fst.comp continuous_snd) |
| let clock : RawManifold → AddCircle (100 : ℝ) := Prod.fst |
| have ht : Continuous clock := continuous_fst |
| let regs (i : Fin 3) : RawManifold → Sphere := fun x => x.2.2 i |
| have hr (i : Fin 3) : Continuous (regs i) := |
| (continuous_apply i).comp (continuous_snd.comp continuous_snd) |
| constructor |
| · have H := hh.comp hp |
| exact H |
| constructor |
| · unfold rawPerturbation readout |
| apply continuous_const.mul |
| apply continuous_finsetSum |
| intro i _ |
| have H := (Complex.continuous_ofReal.comp ((hβ i).comp ht)).mul (hS.comp (hr i)) |
| exact H |
| · unfold rawRegisterFunction |
| apply continuous_pi |
| intro i |
| have H := ((Complex.continuous_ofReal.comp (ha.comp ht)).mul (hS.comp (hr i))).add |
| ((hU i).comp hp) |
| exact H |
|
|
| theorem continuous_concrete_rawCoefficients (L : ℝ) (hL : 2 < L) : |
| Continuous (rawClockFunction (concreteData L hL)) ∧ |
| Continuous (rawPerturbation (concreteData L hL)) ∧ |
| Continuous (rawRegisterFunction (concreteData L hL)) := by |
| let D := concreteData L hL |
| have hg : Continuous D.gate := SphereCutoffs.continuous_onePointClockGate |
| have hS : Continuous D.saturation := sphereSaturation_continuous (3 * L) _ |
| have hbase (i : Fin 3) : Continuous (fun x : ℝ × Sphere => baseInput D i x.1 x.2) := |
| continuous_baseInput_of_standard_cutoffs D rfl rfl (by change 1 < L; linarith) i |
| have ha : Continuous (periodicGain D) := by |
| let : Fact ((0 : ℝ) < 100) := ⟨by norm_num⟩ |
| apply AddCircle.liftIco_zero_continuous |
| · change gainProfile L 0 = gainProfile L 100 |
| rw [gainProfile_zero_outside (by norm_num), gainProfile_zero_outside (by norm_num)] |
| · exact (gainProfile_contDiff L).continuous.continuousOn |
| exact continuous_rawCoefficients D (continuous_clockIncrement D hg) |
| continuous_periodicReadout ha hS (fun i => continuous_input_of_baseInput D i (hbase i)) |
|
|
| theorem concreteMap_continuous_for_homotopy (L : ℝ) (hL : 2 < L) : |
| Continuous (concreteMap L hL) := by |
| obtain ⟨hh, hC, hR⟩ := continuous_concrete_rawCoefficients L hL |
| exact continuous_rawMap_of_rawCoefficients (concreteData L hL) hh hC hR |
|
|
| def concreteMapHomotopy (L : ℝ) (hL : 2 < L) : |
| ContinuousMap.Homotopy |
| ⟨(@rawNormalForm 3), continuous_rawNormalForm⟩ |
| ⟨concreteMap L hL, concreteMap_continuous_for_homotopy L hL⟩ := by |
| obtain ⟨hh, hC, hR⟩ := continuous_concrete_rawCoefficients L hL |
| exact rawMapHomotopy (concreteData L hL) hh hC hR |
|
|
| theorem realHomologyMap_concreteMap_eq_normalForm (n : ℕ) (L : ℝ) (hL : 2 < L) : |
| realHomologyMap n (concreteMap L hL) (concreteMap_continuous_for_homotopy L hL) = |
| realHomologyMap n rawNormalForm continuous_rawNormalForm := |
| (realHomologyMap_eq_of_homotopy n continuous_rawNormalForm |
| (concreteMap_continuous_for_homotopy L hL) (concreteMapHomotopy L hL)).symm |
|
|
| theorem concreteMap_eigenvector_of_sphere (n : ℕ) (L : ℝ) (hL : 2 < L) |
| (c : ℝ) (v : RealSingularHomology n Sphere) (hv : v ≠ 0) |
| (heig : realHomologyMap n rawSphereSquare continuous_rawSphereSquare v = c • v) : |
| ∃ w : RealSingularHomology n RawManifold, w ≠ 0 ∧ |
| realHomologyMap n (concreteMap L hL) (concreteMap_continuous_for_homotopy L hL) w = c • w := by |
| obtain ⟨hh, hC, hR⟩ := continuous_concrete_rawCoefficients L hL |
| exact rawMap_eigenvector_of_sphere n (concreteData L hL) hh hC hR c v hv heig |
|
|
| end Problem340.Construction |
|
|
| end |
|
|
| end OAI |
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|
