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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 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import Mathlib
import OAI.Geometry.IntegralFillings.Currents.Pushforward
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace SharpIntegralFillings
attribute [local instance] Classical.propDecidable
universe u
namespace CurrentOperations
variable {X : Type u} {Y : Type*} [MetricSpace X] [MetricSpace Y]
[MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y]
omit [BorelSpace X] in
lemma pushCurrent_id {k : ℕ} {T : Functional X k} (hT : IsMetricCurrent T) :
pushCurrent id T = T := by
funext b π
by_cases h : Admissible b π
· rw [pushCurrent_apply id T h]
rfl
· simp only [pushCurrent, ite_eq_right h, hT.offDomain b π h]
omit [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] in
lemma pushCurrent_comp {Z : Type*} [MetricSpace Z] {k : ℕ} {T : Functional X k}
{f : X → Y} {g : Y → Z} {L : ℝ≥0} (hg : LipschitzWith L g) :
pushCurrent (g ∘ f) T = pushCurrent g (pushCurrent f T) := by
funext b π
by_cases h : Admissible b π
· rw [pushCurrent_apply (g ∘ f) T h, pushCurrent_apply g (pushCurrent f T) h,
pushCurrent_apply f T (admissible_comp h hg)]
rfl
· simp only [pushCurrent, ite_eq_right h]
omit [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] in
lemma pushCurrent_boundarySucc {k : ℕ} (T : Functional X (k+1))
{f : X → Y} {K : ℝ≥0} (hf : LipschitzWith K f) :
boundarySucc (pushCurrent f T) = pushCurrent f (boundarySucc T) := by
funext b π
by_cases h : Admissible b π
· have hcons : Admissible (fun _ : Y => (1 : ℝ)) (Matrix.vecCons b π) := by
refine ⟨BoundedLip.const 1, ?_⟩
intro i
refine Fin.cases h.1.1 (fun j => h.2 j) i
rw [boundarySucc, ite_eq_left h, pushCurrent_apply f T hcons,
pushCurrent_apply f (boundarySucc T) h, boundarySucc,
ite_eq_left (admissible_comp h hf)]
congr 1
funext i x
exact Fin.cases rfl (fun _ => rfl) i
· simp only [boundarySucc, pushCurrent, ite_eq_right h]
omit [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] in
lemma pushCurrent_zero {k : ℕ} (f : X → Y) :
pushCurrent f (fun (_ : X → ℝ) (_ : Fin k → X → ℝ) => 0) = 0 := by
funext b π
simp only [pushCurrent, ite_self, Pi.zero_apply]
omit [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] in
lemma pushCurrent_cycle {k : ℕ} {T : Functional X (k+1)}
(hT : boundarySucc T = 0) {f : X → Y} {K : ℝ≥0} (hf : LipschitzWith K f) :
boundarySucc (pushCurrent f T) = 0 := by
rw [pushCurrent_boundarySucc T hf, hT]
exact pushCurrent_zero f
lemma mass_pushCurrent_le [CompactSpace X] {k : ℕ} {T : Functional X k}
(hT : IsMetricCurrent T) {f : X → Y} {K : ℝ≥0} (hf : LipschitzWith K f) :
mass (pushCurrent f T) ≤ (K : ℝ)^k * mass T := by
obtain ⟨μ, hμ, hc, hm⟩ := hT.exists_controls_mass_eq
let := hμ
have hfin : IsFiniteMeasure ((↑(K^k) : ℝ≥0∞) • μ.map f) := by
constructor
simp only [Measure.smul_apply, smul_eq_mul]
exact ENNReal.mul_lt_top ENNReal.coe_lt_top (measure_lt_top (μ.map f) univ)
calc
mass (pushCurrent f T) ≤ (((↑(K^k) : ℝ≥0∞) • μ.map f)).real univ :=
mass_le_measure hfin (pushCurrent_controls hT hc hf)
_ = (K : ℝ)^k * mass T := by
rw [hm]
simp only [Measure.real, Measure.smul_apply, smul_eq_mul, ENNReal.toReal_mul,
ENNReal.coe_toReal, NNReal.coe_pow, Measure.map_apply hf.continuous.measurable
MeasurableSet.univ, preimage_univ]
end CurrentOperations
end SharpIntegralFillings
end
end OAI
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