| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Differentiation.FullRank |
| import OAI.Geometry.CAT0Fillings.Slicing.RectifiableCoarea |
| import OAI.Geometry.CAT0Fillings.Currents.PushBoundary |
| import OAI.Geometry.CAT0Fillings.Charts.Postcompose |
|
|
| namespace OAI |
|
|
| section |
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open Filter Set |
| open scoped Topology NNReal |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
|
|
| namespace CAT0Fillings |
| open Set MeasureTheory Filter BorelRestriction CurrentOperations |
| open scoped Topology |
|
|
| namespace IntegerChart |
| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] [CompactSpace X] [CompactSpace Y] |
| [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] |
| [Nonempty X] [Nonempty Y] {k : ℕ} (C : IntegerChart X k) |
| theorem push_integerRectifiable (hC : IsMetricCurrent C.action) |
| {f : X → Y} {K : ℝ≥0} (hf : LipschitzWith K f) : |
| IntegerRectifiable (pushCurrent f C.action) := by |
| classical |
| obtain ⟨L,U,hL,hU⟩ := C.bilipschitz |
| obtain ⟨t,J,hts,ht,hd,hbilip,hzero⟩ := |
| MetricDifferentiation.exists_lipschitz_chart_decomposition C.borel C.bounded.measure_lt_top.ne (hf.comp hL) |
| let E i := C.paramExtended '' t i |
| have hE i : MeasurableSet (E i) := C.measurableSet_paramExtended_image (ht i) (hts i) |
| have hEd : Pairwise (fun i j => Disjoint (E i) (E j)) := |
| fun i j hij => C.paramExtended_disjoint_images (hts i) (hts j) (hd hij) |
| let R i := C.restrictDomain (ht i) (hts i) |
| have hR i : IsMetricCurrent (R i).action := C.restrictDomain_isMetricCurrent _ _ hC |
| have hRact i : (R i).action = restrictCurrent hC (E i) := C.restrictDomain_action _ _ hC |
| let D i := (R i).postcompose hf (hbilip i).2 |
| have hDact i : (D i).action = pushCurrent f (R i).action := (R i).postcompose_action hf (hbilip i).2 |
| have hD i : IsMetricCurrent (D i).action := by rw [hDact]; exact pushCurrent_isMetricCurrent (hR i) hf |
| have hRs : Summable (fun i => mass (R i).action) := by |
| simp_rw [hRact] |
| exact summable_restriction_mass hC E hE hEd |
| have hDs : Summable (fun i => mass (D i).action) := by |
| apply (hRs.mul_left ((K : ℝ)^k)).of_nonneg_of_le (fun i => mass_nonneg _) |
| intro i |
| rw [hDact] |
| exact mass_pushCurrent_le (hR i) hf |
| have heq : pushCurrent f C.action = fun b π => ∑' i, (D i).action b π := by |
| funext b π |
| by_cases hab : Admissible b π |
| · simp_rw [hDact,pushCurrent_apply f _ hab,hRact] |
| rw [←restrictCurrent_iUnion hC E hE hEd, |
| ←C.restrictMultiplicity_action hC (MeasurableSet.iUnion hE)] |
| rw [action,action,ite_eq_left (admissible_comp hab hf),ite_eq_left (admissible_comp hab hf)] |
| apply integral_congr_ae |
| filter_upwards [ae_restrict_mem C.borel,hzero π hab.2] with z hz hzeroz |
| change (C.multiplicity z : ℝ)*C.scalar (b ∘ f) z*C.jacobian (fun i => π i ∘ f) z = |
| ↑((C.paramExtended ⁻¹' ⋃ i, E i).indicator C.multiplicity z)* |
| C.scalar (b ∘ f) z*C.jacobian (fun i => π i ∘ f) z |
| by_cases hzi : z ∈ ⋃ i, t i |
| · have him : z ∈ C.paramExtended ⁻¹' ⋃ i, E i := by |
| obtain ⟨i,hi⟩ := mem_iUnion.mp hzi |
| exact mem_iUnion.mpr ⟨i,⟨z,hi,rfl⟩⟩ |
| rw [indicator_of_mem him] |
| · have hj : C.jacobian (fun i => π i ∘ f) z = 0 := hzeroz hzi |
| rw [hj,mul_zero,mul_zero] |
| · rw [(pushCurrent_isMetricCurrent hC hf).offDomain b π hab] |
| symm |
| calc (∑' i, (D i).action b π) = ∑' (_i : ℕ), (0 : ℝ) := |
| tsum_congr fun i => (hD i).offDomain b π hab |
| _ = 0 := tsum_zero |
| rw [heq] |
| exact integerRectifiable_chartSum D hD hDs |
| |
| end IntegerChart |
| end CAT0Fillings |
| |
| namespace CAT0Fillings |
| open Set MeasureTheory Filter CurrentOperations |
| |
| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] [CompactSpace X] [CompactSpace Y] |
| [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace X] [BorelSpace Y] |
| [Nonempty X] [Nonempty Y] {k : ℕ} |
| |
| theorem integerRectifiable_push {T : Functional X k} (hT : IntegerRectifiable T) |
| {f : X → Y} {K : ℝ≥0} (hf : LipschitzWith K f) : |
| IntegerRectifiable (pushCurrent f T) := by |
| classical |
| obtain ⟨C,hd,hC,hCS,heq⟩ := hT |
| have hP i : IsMetricCurrent (pushCurrent f (C i).action) := pushCurrent_isMetricCurrent (hC i) hf |
| have hPI i : IntegerRectifiable (pushCurrent f (C i).action) := (C i).push_integerRectifiable (hC i) hf |
| have hPS : Summable (fun i => mass (pushCurrent f (C i).action)) := |
| (hCS.mul_left ((K : ℝ)^k)).of_nonneg_of_le (fun i => mass_nonneg _) (fun i => mass_pushCurrent_le (hC i) hf) |
| have hEq : pushCurrent f T = (fun b π => ∑' i, pushCurrent f (C i).action b π) := by |
| funext b π |
| by_cases hab : Admissible b π |
| · simp_rw [pushCurrent_apply f _ hab] |
| exact heq _ _ |
| · simp only [pushCurrent,ite_eq_right hab,tsum_zero] |
| rw [hEq] |
| exact integerRectifiable_tsum hP hPI hPS |
|
|
| theorem integral_push {T : Functional X k} (hT : IsIntegral k T) |
| {f : X → Y} {K : ℝ≥0} (hf : LipschitzWith K f) : |
| IsIntegral k (pushCurrent f T) := by |
| cases k with |
| | zero => exact ⟨pushCurrent_isMetricCurrent hT.1 hf,integerRectifiable_push hT.2 hf⟩ |
| | succ k => |
| refine ⟨pushCurrent_isMetricCurrent hT.1 hf,integerRectifiable_push hT.2.1 hf,?_,?_⟩ |
| · rw [pushCurrent_boundarySucc T hf] |
| exact pushCurrent_isMetricCurrent hT.2.2.1 hf |
| · rw [pushCurrent_boundarySucc T hf] |
| exact integerRectifiable_push hT.2.2.2 hf |
|
|
| end CAT0Fillings |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| universe u |
|
|
| end CAT0Fillings |
|
|
| 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 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
