| import Mathlib |
| import OAI.Geometry.IntegralFillings.Model |
|
|
| namespace OAI |
|
|
| section |
| 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 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 scoped Topology ENNReal NNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open scoped ENNReal NNReal Topology |
| open Set MeasureTheory Filter |
| open scoped Topology NNReal |
|
|
| namespace SharpIntegralFillings |
| open Set MeasureTheory Filter |
| open scoped Topology |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] {k : ℕ} |
| omit [BorelSpace X] in |
| lemma controls_neg_iff {T : Functional X k} {μ : Measure X} : |
| Controls (-T) μ ↔ Controls T μ := by |
| simp only [Controls,Pi.neg_apply,abs_neg] |
|
|
| omit [BorelSpace X] in |
| lemma mass_neg (T : Functional X k) : mass (-T) = mass T := by |
| simp only [mass,controls_neg_iff] |
|
|
| omit [BorelSpace X] in |
| lemma IsMetricCurrent.neg {T : Functional X k} (hT : IsMetricCurrent T) : |
| IsMetricCurrent (-T) := by |
| refine ⟨?_,?_,?_,?_,?_,?_⟩ |
| · intro b π hab |
| simp only [Pi.neg_apply,hT.offDomain b π hab,neg_zero] |
| · intro b c π a d hb hc hπ |
| simp only [Pi.neg_apply,hT.linearFirst b c π a d hb hc hπ] |
| ring |
| · intro b π i f a c hb hf |
| simp only [Pi.neg_apply,hT.linearCoord b π i f a c hb hf] |
| ring |
| · intro b π πs hb hπ hlim |
| exact (hT.sequentialContinuity b π πs hb hπ hlim).neg |
| · intro b π hab hl |
| simp only [Pi.neg_apply,hT.locality b π hab hl,neg_zero] |
| · obtain ⟨μ,hμ,hc⟩ := hT.finiteMass |
| exact ⟨μ,hμ,controls_neg_iff.mpr hc⟩ |
|
|
| namespace IntegerChart |
| variable (C : IntegerChart X k) |
| noncomputable def negate : IntegerChart X k := |
| { C with multiplicity := fun z => -C.multiplicity z |
| integrable := C.integrable.neg.congr (Eventually.of_forall fun _z => (Int.cast_neg _).symm) } |
|
|
| omit [MeasurableSpace X] [BorelSpace X] in |
| lemma negate_action : C.negate.action = -C.action := by |
| classical |
| funext b π |
| by_cases hab : Admissible b π |
| · simp only [action,ite_eq_left hab,Pi.neg_apply] |
| change (∫ z in C.domain,(-C.multiplicity z:ℤ)*C.scalar b z*C.jacobian π z) = |
| -(∫ z in C.domain,(C.multiplicity z:ℝ)*C.scalar b z*C.jacobian π z) |
| simp only [Int.cast_neg,neg_mul,integral_neg] |
| · simp only [action,ite_eq_right hab,Pi.neg_apply,neg_zero] |
| end IntegerChart |
|
|
| omit [BorelSpace X] in |
| lemma IntegerRectifiable.neg {T : Functional X k} (hI : IntegerRectifiable T) : |
| IntegerRectifiable (-T) := by |
| obtain ⟨C,hd,hC,hS,heq⟩ := hI |
| refine ⟨fun i => (C i).negate,hd,?_,?_,?_⟩ |
| · intro i |
| rw [IntegerChart.negate_action] |
| exact (hC i).neg |
| · simp_rw [IntegerChart.negate_action,mass_neg] |
| exact hS |
| · intro b π |
| simp only [IntegerChart.negate_action,Pi.neg_apply,heq,tsum_neg] |
|
|
| end SharpIntegralFillings |
| end |
|
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| end OAI |
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|
