File size: 2,083 Bytes
59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | 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
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