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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 | import Mathlib
import OAI.Geometry.IntegralFillings.Model
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace SharpIntegralFillings
attribute [local instance] Classical.propDecidable
universe u
section Foundations
variable {X : Type u} [MetricSpace X] [MeasurableSpace X]
omit [MeasurableSpace X] in
lemma lipschitz_mul_real {f : X → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) (a : ℝ) :
LipschitzWith (⟨|a|, abs_nonneg a⟩ * K) (fun x => a * f x) := by
apply lipschitzWith_iff_dist_le_mul.mpr
intro x y
change dist (a * f x) (a * f y) ≤ (|a| * (K : ℝ)) * dist x y
rw [Real.dist_eq, ← mul_sub, abs_mul]
calc |a| * |f x - f y| ≤ |a| * ((K : ℝ) * dist x y) :=
mul_le_mul_of_nonneg_left (hf.dist_le_mul x y) (abs_nonneg a)
_ = _ := by ring
def lipSubmodule : Submodule ℝ (X → ℝ) where
carrier := {f | ∃ K : ℝ≥0, LipschitzWith K f}
zero_mem' := ⟨0, LipschitzWith.const 0⟩
add_mem' := by
rintro f g ⟨K, hK⟩ ⟨L, hL⟩
exact ⟨K + L, hK.add hL⟩
smul_mem' := by
intro a f hf
obtain ⟨K, hK⟩ := hf
exact ⟨_, lipschitz_mul_real hK a⟩
omit [MeasurableSpace X] in
lemma lip_coe_update {k : ℕ} (π : Fin k → lipSubmodule (X := X))
(i : Fin k) (f : lipSubmodule (X := X)) :
(fun j => ((Function.update π i f) j : X → ℝ)) =
Function.update (fun j => (π j : X → ℝ)) i (f : X → ℝ) := by
funext j
by_cases h : j = i <;> simp [h]
noncomputable def IsMetricCurrent.multilinear {k : ℕ} {T : Functional X k}
(hT : IsMetricCurrent T) (b : X → ℝ) (hb : BoundedLip b) :
MultilinearMap ℝ (fun _ : Fin k => lipSubmodule (X := X)) ℝ :=
MultilinearMap.mk' (fun π => T b (fun i => π i)) (by
intro π i f g
have h := hT.linearCoord b
(fun j => ((Function.update π i f) j : X → ℝ)) i (g : X → ℝ) 1 1
⟨hb, fun j => ((Function.update π i f) j).property⟩ g.property
simpa only [lip_coe_update, Function.update_self, one_mul,
Function.update_idem, Submodule.coe_add, Pi.add_def] using h) (by
intro π i a f
have h := hT.linearCoord b
(fun j => ((Function.update π i f) j : X → ℝ)) i (f : X → ℝ) a 0
⟨hb, fun j => ((Function.update π i f) j).property⟩ f.property
simpa only [lip_coe_update, Function.update_self, zero_mul, add_zero,
Function.update_idem, Submodule.coe_smul, Pi.smul_def, smul_eq_mul] using h)
lemma IsMetricCurrent.scale_coordinates {k : ℕ} {T : Functional X k}
(hT : IsMetricCurrent T) {b : X → ℝ} (hb : BoundedLip b)
{π : Fin k → X → ℝ} (hπ : ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i))
(a : Fin k → ℝ) :
T b (fun i x => a i * π i x) = (∏ i, a i) * T b π := by
let p : Fin k → lipSubmodule (X := X) := fun i => ⟨π i, hπ i⟩
exact (hT.multilinear b hb).map_smul_univ a p
lemma IsMetricCurrent.eq_zero_of_const_coord {k : ℕ} {T : Functional X k}
(hT : IsMetricCurrent T) {b : X → ℝ} {π : Fin k → X → ℝ}
(h : Admissible b π) (i : Fin k) (c : ℝ) (hc : ∀ x, π i x = c) :
T b π = 0 := by
apply hT.locality b π h
exact ⟨i, univ, c, isOpen_univ, subset_univ _, fun x _ => hc x⟩
lemma IsMetricCurrent.eq_zero_of_zero_lipschitz_coord {k : ℕ} {T : Functional X k}
(hT : IsMetricCurrent T) {b : X → ℝ} {π : Fin k → X → ℝ}
(h : Admissible b π) (i : Fin k) (hi : LipschitzWith 0 (π i)) :
T b π = 0 := by
cases isEmpty_or_nonempty X with
| inl hempty =>
let := hempty
exact hT.eq_zero_of_const_coord h i 0 (fun x => isEmptyElim x)
| inr hnonempty =>
obtain ⟨x₀⟩ := hnonempty
apply hT.eq_zero_of_const_coord h i (π i x₀)
intro x
apply dist_le_zero.mp
simpa using hi.dist_le_mul x x₀
end Foundations
end SharpIntegralFillings
end
end OAI
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