| import Mathlib |
| import OAI.Geometry.IntegralFillings.Model |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
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| 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 |
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| end OAI |
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