| 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 | |