| import Mathlib | |
| import OAI.Geometry.IntegralFillings.Currents.Pushforward | |
| namespace OAI | |
| section | |
| open Set Filter MeasureTheory | |
| open scoped Topology ENNReal NNReal | |
| namespace SharpIntegralFillings | |
| open Set MeasureTheory | |
| open CurrentOperations | |
| namespace IntegerChart | |
| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] {k : ℕ} | |
| (C : IntegerChart X k) {f : X → Y} {K J : ℝ≥0} | |
| noncomputable def postcompose (hf : LipschitzWith K f) | |
| (hJ : AntilipschitzWith J (f ∘ C.param)) : IntegerChart Y k where | |
| domain := C.domain | |
| borel := C.borel | |
| bounded := C.bounded | |
| param := f ∘ C.param | |
| bilipschitz := by | |
| obtain ⟨L,U,hL,_⟩ := C.bilipschitz | |
| exact ⟨K*L,J,hf.comp hL,hJ⟩ | |
| multiplicity := C.multiplicity | |
| integrable := C.integrable | |
| lemma postcompose_scalar (hf : LipschitzWith K f) | |
| (hJ : AntilipschitzWith J (f ∘ C.param)) (b : Y → ℝ) : | |
| (C.postcompose hf hJ).scalar b = C.scalar (b ∘ f) := by | |
| classical | |
| funext x | |
| change (if h : x ∈ C.domain then b (f (C.param ⟨x,h⟩)) else 0) = _ | |
| rfl | |
| lemma postcompose_jacobian (hf : LipschitzWith K f) | |
| (hJ : AntilipschitzWith J (f ∘ C.param)) (π : Fin k → Y → ℝ) : | |
| (C.postcompose hf hJ).jacobian π = C.jacobian (fun i => π i ∘ f) := by | |
| rfl | |
| lemma postcompose_action (hf : LipschitzWith K f) | |
| (hJ : AntilipschitzWith J (f ∘ C.param)) : | |
| (C.postcompose hf hJ).action = pushCurrent f C.action := by | |
| classical | |
| funext b π | |
| by_cases h : Admissible b π | |
| · rw [pushCurrent_apply f C.action h,action,ite_eq_left h, | |
| action,ite_eq_left (admissible_comp h hf)] | |
| change (∫ x in C.domain, (C.multiplicity x : ℝ)* | |
| (C.postcompose hf hJ).scalar b x*(C.postcompose hf hJ).jacobian π x) = _ | |
| rw [C.postcompose_scalar hf hJ,C.postcompose_jacobian hf hJ] | |
| · simp only [action,pushCurrent,ite_eq_right h] | |
| lemma postcompose_image (hf : LipschitzWith K f) | |
| (hJ : AntilipschitzWith J (f ∘ C.param)) : | |
| (C.postcompose hf hJ).image = f '' C.image := by | |
| exact range_comp f C.param | |
| end IntegerChart | |
| end SharpIntegralFillings | |
| end | |
| end OAI | |