| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Currents.BorelAction |
| import OAI.Geometry.CAT0Fillings.Currents.CycleDerivative |
| import OAI.Geometry.CAT0Fillings.Prism.TimeCoordinate |
| import OAI.Geometry.CAT0Fillings.Calculus.LipschitzFTC |
|
|
| namespace OAI |
|
|
| section |
| section |
| open Filter Set |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
| open Set Filter MeasureTheory |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology NNReal |
| open Set MeasureTheory Filter |
| open scoped Topology ENNReal NNReal |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
|
|
| namespace CAT0Fillings |
| open Set MeasureTheory Filter |
| open scoped Topology NNReal ENNReal |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] |
| local notation "BL" => boundedLipSubmodule (X := X) |
| lemma IsMetricCurrent.cycle_time_FTC {k : ℕ} {T : Functional X (k+1)} |
| (hcur : IsMetricCurrent T) (hT : IntegerRectifiable T) (hz : IsCycle T) |
| (μ : Measure X) [IsFiniteMeasure μ] (hμ : Controls T μ) |
| (v : Fin (k+2) → ℝ × X → ℝ) (K : ℝ≥0) (hv : ∀ i, LipschitzWith K (v i)) |
| (a b : ℝ) : |
| (∫ t in a..b, ∑ i, (-1:ℝ)^i.val * BorelCoefficients.borelAction μ hcur |
| (fun x => deriv (fun s => v i (s,x)) t) |
| (fun j x => v (i.succAbove j) (t,x))) = |
| T (fun x => v 0 (b,x)) (fun j x => v j.succ (b,x)) - |
| T (fun x => v 0 (a,x)) (fun j x => v j.succ (a,x)) := by |
| let V (t : ℝ) (i : Fin (k+2)) : BL := |
| ⟨fun x => v i (t,x),boundedLip_of_lipschitz_compact (lipschitz_slice (hv i) t)⟩ |
| have hx (t i) : LipschitzWith K (V t i : X → ℝ) := lipschitz_slice (hv i) t |
| have ht (i x) : LipschitzWith K (fun t => (V t i : X → ℝ) x) := lipschitz_time (hv i) x |
| have hd : ∀ᵐ t ∂volume, HasDerivAt (fun s => hcur.fullMultilinear (V s)) |
| (∑ i, (-1:ℝ)^i.val * BorelCoefficients.borelAction μ hcur |
| (fun x => deriv (fun s => v i (s,x)) t) (fun j x => v (i.succAbove j) (t,x))) t := by |
| have ha : ∀ᵐ t ∂volume, ∀ i, ∀ᵐ x ∂μ, HasDerivAt (fun s => v i (s,x)) |
| (deriv (fun s => v i (s,x)) t) t := |
| ae_all_iff.mpr (fun i => ae_time_hasDeriv (hv i) μ) |
| filter_upwards [ha] with t ha |
| exact hcur.hasDeriv_cycle_path hT hz μ hμ V K hx ht t _ |
| (fun i => integrable_timeDeriv (hv i) μ t) ha |
| have heq : (∫ t in a..b, ∑ i, (-1:ℝ)^i.val * BorelCoefficients.borelAction μ hcur |
| (fun x => deriv (fun s => v i (s,x)) t) |
| (fun j x => v (i.succAbove j) (t,x))) = |
| ∫ t in a..b, deriv (fun s => hcur.fullMultilinear (V s)) t := by |
| apply intervalIntegral.integral_congr_ae |
| filter_upwards [hd] with t ht _ |
| exact ht.deriv.symm |
| rw [heq,LipschitzWith.integral_deriv_eq_sub (hcur.cycle_time_lipschitz hT hz V K hx ht)] |
| rfl |
|
|
| end CAT0Fillings |
|
|
| namespace CAT0Fillings |
| attribute [local instance] Classical.propDecidable |
|
|
| attribute [local instance] Classical.propDecidable |
| end CAT0Fillings |
|
|
| end |
| end |
|
|
| end OAI |
|
|
