| import OAI.Dynamics.EntropyCounterexample.MapData |
| import OAI.Dynamics.EntropyCounterexample.MapContinuity |
| import OAI.Dynamics.EntropyCounterexample.MapQuotientContinuity |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Problem340.Construction |
|
|
| theorem continuous_concrete_baseInput (L : ℝ) (hL : 2 < L) (i : Fin 3) : |
| Continuous (fun x : ℝ × Sphere => baseInput (concreteData L hL) i x.1 x.2) := by |
| apply continuous_baseInput_of_standard_cutoffs _ rfl rfl |
| change 1 < L |
| linarith |
|
|
| theorem continuous_concrete_input (L : ℝ) (hL : 2 < L) (i : Fin 3) : |
| Continuous (fun x : AddCircle (100 : ℝ) × Sphere => |
| input (concreteData L hL) i x.1 x.2) := |
| continuous_input_of_baseInput _ i (continuous_concrete_baseInput L hL i) |
|
|
| theorem concrete_gain_seam (L : ℝ) (hL : 2 < L) : |
| (concreteData L hL).gain 0 = (concreteData L hL).gain 100 := by |
| change gainProfile L 0 = gainProfile L 100 |
| rw [gainProfile_zero_outside (by norm_num), gainProfile_zero_outside (by norm_num)] |
|
|
| theorem continuous_concrete_periodicGain (L : ℝ) (hL : 2 < L) : |
| Continuous (periodicGain (concreteData L hL)) := by |
| let : Fact ((0 : ℝ) < 100) := ⟨by norm_num⟩ |
| apply AddCircle.liftIco_zero_continuous |
| · exact concrete_gain_seam L hL |
| · exact (gainProfile_contDiff L).continuous.continuousOn |
|
|
| theorem continuous_concreteMap (L : ℝ) (hL : 2 < L) : |
| Continuous (concreteMap L hL) := by |
| apply continuous_rawMap_of_coefficients (concreteData L hL) |
| · exact SphereCutoffs.continuous_onePointClockGate |
| · exact continuous_concrete_input L hL |
| · exact sphereSaturation_continuous (3 * L) _ |
| · exact continuous_concrete_periodicGain L hL |
|
|
| end Problem340.Construction |
|
|
| end |
|
|
| end OAI |
|
|
