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dd14c26 | 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 | 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
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