| import Mathlib |
| import OAI.Probability.LatticeSurface.Convexity.BoundedBase |
| import OAI.Probability.LatticeSurface.Geometry.CapReflection |
| import OAI.Probability.LatticeSurface.Geometry.CapUnpin |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Set |
|
|
| open scoped ENNReal |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V : Type*} |
|
|
| lemma volume_coordinate_slices [Fintype V] (pins : Set V) (x : V) (hx : x ∉ pins) |
| (K : Set (Free pins → ℝ)) (hK : MeasurableSet K) : |
| volume K = ∫⁻ t : ℝ, volume {h : Free (insert x pins) → ℝ | |
| (splitCoordinate pins x hx).symm (h,t) ∈ K} := by |
| have he := (preserving_splitCoordinate pins x hx).symm (splitCoordinate pins x hx) |
| rw [← he.measure_preimage hK.nullMeasurableSet] |
| change ((volume : Measure (Free (insert x pins) → ℝ)).prod (volume : Measure ℝ)) |
| ((splitCoordinate pins x hx).symm ⁻¹' K) = _ |
| exact Measure.prod_apply_symm (hK.preimage (splitCoordinate pins x hx).symm.measurable) |
|
|
| lemma coordinate_restore_update (pins : Set V) (x : V) (hx : x ∉ pins) (b : V → ℝ) |
| (h : Free (insert x pins) → ℝ) (t : ℝ) (y : V) : |
| restore pins b ((splitCoordinate pins x hx).symm (h,t)) y = |
| restore (insert x pins) (Function.update b x t) h y := by |
| classical |
| rw [coordinate_restore] |
| by_cases hy : y = x |
| · subst y |
| simp [restore] |
| · by_cases hp : y ∈ insert x pins <;> simp [restore, hy, hp] |
|
|
| lemma volume_body_coordinate [Fintype V] (G : SimpleGraph V) (pins : Set V) (b : V → ℝ) |
| (x : V) (hx : x ∉ pins) : |
| volume (conditionalBody G pins b) = ∫⁻ t : ℝ, |
| volume (conditionalBody G (insert x pins) (Function.update b x t)) := by |
| rw [volume_coordinate_slices pins x hx _ (closed_conditionalBody G pins b).measurableSet] |
| apply lintegral_congr |
| intro t |
| congr 1 |
| ext h |
| simp only [conditionalBody, mem_ofPred_eq, coordinate_restore_update] |
|
|
| end ExtremalEdges |
|
|
| end |
|
|
| end OAI |
|
|
