| import Mathlib |
| import OAI.Geometry.IntegralFillings.Model |
|
|
| namespace OAI |
|
|
| section |
| open Filter Set |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
| open Set Filter MeasureTheory |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology NNReal |
| open Set MeasureTheory Filter |
| open scoped Topology ENNReal NNReal |
|
|
| namespace SharpIntegralFillings.Prism |
| abbrev Euc (k : ℕ) := EuclideanSpace ℝ (Fin k) |
|
|
| noncomputable def splitLinear (k : ℕ) : Euc (k+1) ≃ₗ[ℝ] ℝ × Euc k where |
| toFun z := (z 0, WithLp.toLp 2 (fun i => z i.succ)) |
| invFun p := WithLp.toLp 2 (Matrix.vecCons p.1 (WithLp.ofLp p.2)) |
| left_inv z := by ext i; exact Fin.cases rfl (fun _ => rfl) i |
| right_inv p := by aesop |
| map_add' z w := by ext <;> rfl |
| map_smul' a z := by ext <;> rfl |
|
|
| noncomputable def split (k : ℕ) : Euc (k+1) ≃L[ℝ] ℝ × Euc k := |
| (splitLinear k).toContinuousLinearEquiv |
|
|
| @[simp] lemma split_apply (k : ℕ) (z : Euc (k+1)) : |
| split k z = (z 0, WithLp.toLp 2 (fun i => z i.succ)) := rfl |
|
|
| @[simp] lemma split_symm_apply (k : ℕ) (p : ℝ × Euc k) : |
| (split k).symm p = WithLp.toLp 2 (Matrix.vecCons p.1 (WithLp.ofLp p.2)) := rfl |
|
|
| lemma split_measurePreserving (k : ℕ) : MeasurePreserving (split k) := by |
| have h := (MeasurePreserving.prod (MeasurePreserving.id (volume : Measure ℝ)) |
| (PiLp.volume_preserving_toLp (Fin k))).comp |
| ((volume_preserving_piFinSuccAbove (fun _ : Fin (k+1) => ℝ) 0).comp |
| (PiLp.volume_preserving_ofLp (Fin (k+1)))) |
| convert h using 1 |
| · funext z |
| apply Prod.ext |
| · rfl |
| · ext j |
| simp [Function.comp_def,MeasurableEquiv.piFinSuccAbove_apply,Fin.tail] |
|
|
| lemma split_symm_measurePreserving (k : ℕ) : MeasurePreserving (split k).symm := by |
| exact MeasurePreserving.symm (split k).toHomeomorph.toMeasurableEquiv |
| (split_measurePreserving k) |
|
|
| lemma integral_split (k : ℕ) (f : Euc (k+1) → ℝ) : |
| ∫ z, f z = ∫ p : ℝ × Euc k, f ((split k).symm p) := by |
| exact ((split_symm_measurePreserving k).integral_comp (split k).symm.toHomeomorph.measurableEmbedding f).symm |
|
|
| lemma split_single_zero (k : ℕ) : split k (EuclideanSpace.single 0 1) = (1,0) := by |
| ext <;> simp |
|
|
| lemma split_single_succ (k : ℕ) (j : Fin k) : |
| split k (EuclideanSpace.single j.succ 1) = (0,EuclideanSpace.single j 1) := by |
| ext <;> simp |
|
|
| end SharpIntegralFillings.Prism |
|
|
| end |
|
|
| end OAI |
|
|
