| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Currents.ProductRule |
| import OAI.Geometry.CAT0Fillings.Currents.AffineCalculus |
|
|
| 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 ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
| 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 |
|
|
| namespace CAT0Fillings |
| open Set MeasureTheory Filter Matrix |
| open scoped Topology NNReal ENNReal |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] |
| local notation "BL" => boundedLipSubmodule (X := X) |
| lemma Foundations.IntegerRectifiable.cycle_update {k : ℕ} {T : Functional X (k+1)} |
| (hT : IntegerRectifiable T) (hz : IsCycle T) {b f : X → ℝ} |
| (hb : BoundedLip b) (hf : BoundedLip f) (π : Fin (k+1) → X → ℝ) |
| (hπ : ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)) (j : Fin (k+1)) : |
| T b (Function.update π j f) = |
| (-1:ℝ)^(j.val+1) * T f (Fin.cons b (j.removeNth π)) := by |
| have hp := Foundations.IntegerRectifiable.apply_permute hT |
| (Foundations.admissible_vecCons hb hf.1 (fun i => hπ (j.succAbove i))) j.cycleRange |
| change T b (Fin.cons f (j.removeNth π) ∘ j.cycleRange) = _ at hp |
| rw [Fin.cons_comp_cycleRange,Fin.insertNth_removeNth,Fin.sign_cycleRange] at hp |
| rw [hp,Foundations.IntegerRectifiable.cycle_product_rule hT hz hb hf _ (fun i => hπ (j.succAbove i))] |
| change ((((-1:ℤˣ)^j.val : ℤˣ) : ℤ) : ℝ) * |
| (-T f (Fin.cons b (j.removeNth π))) = _ |
| simp [pow_succ] |
|
|
| omit [MeasurableSpace X] [BorelSpace X] [CompactSpace X] in |
| lemma fin_removeNth_succ {α : Type*} {k : ℕ} (v : Fin (k+2) → α) (j : Fin (k+1)) : |
| j.succ.removeNth v = Fin.cons (v 0) (j.removeNth (fun i => v i.succ)) := by |
| funext i |
| refine Fin.cases ?_ (fun l => ?_) i |
| · simp [Fin.removeNth_apply] |
| · simp [Fin.removeNth_apply] |
|
|
| lemma IsMetricCurrent.fullMultilinear_cycle_update {k : ℕ} {T : Functional X (k+1)} |
| (hcur : IsMetricCurrent T) (hT : IntegerRectifiable T) (hz : IsCycle T) |
| (v : Fin (k+2) → BL) (i : Fin (k+2)) (g : BL) : |
| hcur.fullMultilinear (Function.update v i g) = |
| (-1:ℝ)^i.val * T g (fun j => (i.removeNth v j : X → ℝ)) := by |
| refine Fin.cases ?_ (fun j => ?_) i |
| · simp only [IsMetricCurrent.fullMultilinear,MultilinearMap.mk'_apply, |
| Function.update_self,bl_update_tail_zero,Fin.removeNth_zero] |
| simp only [Fin.val_zero,pow_zero,one_mul] |
| rfl |
| · change T ((Function.update v j.succ g) 0) |
| (fun l => ((Function.update v j.succ g) l.succ : X → ℝ)) = _ |
| rw [Function.update_of_ne (Fin.succ_ne_zero j).symm,bl_update_tail_succ] |
| rw [Foundations.IntegerRectifiable.cycle_update hT hz (v 0).property g.property _ |
| (fun l => (v l.succ).property.1),fin_removeNth_succ] |
| congr 2 |
| funext l |
| refine Fin.cases ?_ (fun m => ?_) l <;> rfl |
| |
| lemma IsMetricCurrent.hasDeriv_cycleAffine {k : ℕ} {T : Functional X (k+1)} |
| (hcur : IsMetricCurrent T) (hT : IntegerRectifiable T) (hz : IsCycle T) |
| (f g : Fin (k+2) → BL) (t : ℝ) : |
| HasDerivAt (fun s => T (f 0 + s • g 0) |
| (fun i => (f i.succ + s • g i.succ : BL))) |
| (∑ i, (-1:ℝ)^i.val * T (g i) |
| (fun j => ((i.removeNth (fun l => f l + t • g l)) j : X → ℝ))) t := by |
| have h := hcur.hasDeriv_fullAffine f g t |
| convert h using 1 |
| · rfl |
| · symm |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact hcur.fullMultilinear_cycle_update hT hz _ i _ |
| |
| end CAT0Fillings |
| |
| namespace CAT0Fillings |
| attribute [local instance] Classical.propDecidable |
| |
| attribute [local instance] Classical.propDecidable |
| end CAT0Fillings |
| |
| end |
| end |
| |
| end OAI |
| |
