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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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import Mathlib
import OAI.Geometry.IntegralFillings.Currents.MassBounds
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 [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] in
lemma boundedLip_comp {b : Y → ℝ} (hb : BoundedLip b) {f : X → Y}
{K : ℝ≥0} (hf : LipschitzWith K f) : BoundedLip (b ∘ f) := by
obtain ⟨⟨L, hL⟩, M, hM⟩ := hb
exact ⟨⟨L*K, hL.comp hf⟩, M, fun x => hM (f x)⟩
omit [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] in
lemma admissible_comp {k : ℕ} {b : Y → ℝ} {π : Fin k → Y → ℝ}
(h : Admissible b π) {f : X → Y} {K : ℝ≥0} (hf : LipschitzWith K f) :
Admissible (b ∘ f) (fun i => π i ∘ f) := by
refine ⟨boundedLip_comp h.1 hf, fun i => ?_⟩
obtain ⟨L, hL⟩ := h.2 i
exact ⟨L*K, hL.comp hf⟩
noncomputable def pushCurrent {k : ℕ} (f : X → Y) (T : Functional X k) : Functional Y k :=
fun b π => if Admissible b π then T (b ∘ f) (fun i => π i ∘ f) else 0
omit [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] in
lemma pushCurrent_apply {k : ℕ} (f : X → Y) (T : Functional X k)
{b : Y → ℝ} {π : Fin k → Y → ℝ} (h : Admissible b π) :
pushCurrent f T b π = T (b ∘ f) (fun i => π i ∘ f) := ite_eq_left h
lemma pushCurrent_controls {k : ℕ} {T : Functional X k} (hT : IsMetricCurrent T)
{μ : Measure X} (hμ : Controls T μ) {f : X → Y} {K : ℝ≥0}
(hf : LipschitzWith K f) :
Controls (pushCurrent f T) ((↑(K^k) : ℝ≥0∞) • μ.map f) := by
intro b π hb hπ
rw [pushCurrent_apply f T ⟨hb, fun i => ⟨1, hπ i⟩⟩,
integral_smul_measure, integral_map hf.continuous.aemeasurable
hb.continuous.abs.aestronglyMeasurable]
have h := hT.mass_bound hμ (boundedLip_comp hb hf) (fun _ => K)
(fun i => by simpa only [one_mul] using (hπ i).comp hf)
simpa only [Finset.prod_const, Finset.card_univ, Fintype.card_fin,
ENNReal.coe_toReal, NNReal.coe_pow, smul_eq_mul, Function.comp_apply] using h
lemma pushCurrent_isMetricCurrent {k : ℕ} {T : Functional X k}
(hT : IsMetricCurrent T) {f : X → Y} {K : ℝ≥0} (hf : LipschitzWith K f) :
IsMetricCurrent (pushCurrent f T) where
offDomain := by intro b π h; exact ite_eq_right h
linearFirst := by
intro b c π a d hb hc hπ
rw [pushCurrent_apply f T ⟨(hb.const_mul a).add (hc.const_mul d), hπ⟩,
pushCurrent_apply f T ⟨hb, hπ⟩, pushCurrent_apply f T ⟨hc, hπ⟩]
exact hT.linearFirst (b ∘ f) (c ∘ f) (fun i => π i ∘ f) a d
(boundedLip_comp hb hf) (boundedLip_comp hc hf) (admissible_comp ⟨hb, hπ⟩ hf).2
linearCoord := by
intro b π i g a c h hg
have hπg : ∀ j, ∃ L : ℝ≥0, LipschitzWith L ((Function.update π i g) j) := by
intro j
by_cases hj : j = i
· simpa only [hj, Function.update_self] using hg
· simpa only [Function.update_of_ne hj] using h.2 j
have hl : ∃ L : ℝ≥0, LipschitzWith L (fun y => a * π i y + c * g y) := by
obtain ⟨L, hL⟩ := h.2 i
obtain ⟨M, hM⟩ := hg
exact ⟨_, (lipschitz_mul_real hL a).add (lipschitz_mul_real hM c)⟩
have hπac : ∀ j, ∃ L : ℝ≥0,
LipschitzWith L ((Function.update π i (fun y => a * π i y + c * g y)) j) := by
intro j
by_cases hj : j = i
· simpa only [hj, Function.update_self] using hl
· simpa only [Function.update_of_ne hj] using h.2 j
rw [pushCurrent_apply f T ⟨h.1, hπac⟩, pushCurrent_apply f T h,
pushCurrent_apply f T ⟨h.1, hπg⟩]
have h₁ : (fun j => (Function.update π i (fun y => a * π i y + c * g y)) j ∘ f) =
Function.update (fun j => π j ∘ f) i (fun x => a * π i (f x) + c * g (f x)) := by
funext j x
by_cases hji : j = i <;> simp [hji, Function.update_of_ne]
have h₂ : (fun j => (Function.update π i g) j ∘ f) =
Function.update (fun j => π j ∘ f) i (g ∘ f) := by
funext j x
by_cases hji : j = i <;> simp [hji, Function.update_of_ne]
rw [h₁, h₂]
obtain ⟨L, hL⟩ := hg
exact hT.linearCoord _ _ i _ a c (admissible_comp h hf) ⟨L*K, hL.comp hf⟩
sequentialContinuity := by
intro b π πs hb hLip hpt
choose L hL using hLip
have hπ (i) : LipschitzWith (L i) (π i) := by
apply lipschitzWith_iff_dist_le_mul.mpr
intro x y
exact le_of_tendsto ((hpt i x).dist (hpt i y))
(Eventually.of_forall fun j => (hL i j).dist_le_mul x y)
simp only [pushCurrent_apply f T ⟨hb, fun i => ⟨L i, hL i _⟩⟩,
pushCurrent_apply f T ⟨hb, fun i => ⟨L i, hπ i⟩⟩]
exact hT.sequentialContinuity _ _ _ (boundedLip_comp hb hf)
(fun i => ⟨L i*K, fun j => (hL i j).comp hf⟩) (fun i x => hpt i (f x))
locality := by
intro b π h hloc
rw [pushCurrent_apply f T h]
obtain ⟨i, U, c, hU, hbU, hc⟩ := hloc
apply hT.locality _ _ (admissible_comp h hf)
refine ⟨i, f ⁻¹' U, c, hU.preimage hf.continuous, ?_, ?_⟩
· intro x hx; exact hbU hx
· intro x hx; exact hc (f x) hx
finiteMass := by
obtain ⟨μ, hμ, hctrl⟩ := hT.finiteMass
let := hμ
refine ⟨(↑(K^k) : ℝ≥0∞) • μ.map f, ⟨?_⟩, pushCurrent_controls hT hctrl hf⟩
simp only [Measure.smul_apply, smul_eq_mul]
exact ENNReal.mul_lt_top ENNReal.coe_lt_top (measure_lt_top (μ.map f) univ)
end CurrentOperations
end SharpIntegralFillings
end
end OAI
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