File size: 2,967 Bytes
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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import Mathlib
import OAI.Probability.LatticeSurface.Convexity.BoundedBase
import OAI.Probability.LatticeSurface.Geometry.CapContract
import OAI.Probability.LatticeSurface.Geometry.CapReflection
import OAI.Probability.LatticeSurface.Geometry.CapSplit
import OAI.Probability.LatticeSurface.Geometry.CapUnpin
import OAI.Probability.LatticeSurface.Geometry.CapWeld
namespace OAI
noncomputable section
open MeasureTheory Set
open scoped ENNReal
namespace ExtremalEdges
attribute [local instance] Classical.propDecidable
variable {V : Type*}
lemma restore_zero_add (pins : Set V) (h k : Free pins → ℝ) (x : V) :
restore pins (fun _ => 0) (h+k) x =
restore pins (fun _ => 0) h x + restore pins (fun _ => 0) k x := by
by_cases hx : x ∈ pins <;> simp [restore, hx]
lemma restore_zero_smul (pins : Set V) (c : ℝ) (h : Free pins → ℝ) (x : V) :
restore pins (fun _ => 0) (c • h) x = c * restore pins (fun _ => 0) h x := by
by_cases hx : x ∈ pins <;> simp [restore, hx]
lemma convex_zeroBody (G : SimpleGraph V) (pins : Set V) :
Convex ℝ (conditionalBody G pins (fun _ => 0)) := by
intro h hh k hk s t hs ht hst x y hxy
simp only [restore_zero_add, restore_zero_smul]
calc
|s * restore pins (fun _ => 0) h x + t * restore pins (fun _ => 0) k x -
(s * restore pins (fun _ => 0) h y + t * restore pins (fun _ => 0) k y)| =
|s * (restore pins (fun _ => 0) h x-restore pins (fun _ => 0) h y) +
t * (restore pins (fun _ => 0) k x-restore pins (fun _ => 0) k y)| := by congr 1; ring
_ ≤ |s * (restore pins (fun _ => 0) h x-restore pins (fun _ => 0) h y)| +
|t * (restore pins (fun _ => 0) k x-restore pins (fun _ => 0) k y)| := abs_add_le _ _
_ ≤ s*1+t*1 := by rw [abs_mul, abs_mul, abs_of_nonneg hs, abs_of_nonneg ht]; gcongr; exacts [hh x y hxy,hk x y hxy]
_ = 1 := by linarith
lemma neg_zeroBody (G : SimpleGraph V) (pins : Set V)
{h : Free pins → ℝ} (hh : h ∈ conditionalBody G pins (fun _ => 0)) :
-h ∈ conditionalBody G pins (fun _ => 0) := by
have he := body_neg_preimage G pins (fun _ => 0)
simp only [neg_zero] at he
change h ∈ Neg.neg ⁻¹' conditionalBody G pins (fun _ => 0)
rw [← he]
exact hh
lemma compact_zeroBody [Fintype V] (G : SimpleGraph V) (hc : G.Connected)
(pins : Set V) (hp : pins.Nonempty) :
IsCompact (conditionalBody G pins (fun _ => 0)) := by
choose p hpin using hp
let path (x : V) : G.Walk x p := Classical.choice (hc x p)
let R (x : Free pins) : ℝ := (path x).length
apply isCompact_Icc.of_isClosed_subset (closed_conditionalBody G pins _) (s := Icc (-R) R)
intro h hh
constructor <;> intro x
· have hb := bound_along_walk G (restore pins (fun _ => 0) h) hh (path x)
simpa [restore, x.2, hpin, R] using (abs_le.mp hb).1
· have hb := bound_along_walk G (restore pins (fun _ => 0) h) hh (path x)
simpa [restore, x.2, hpin, R] using (abs_le.mp hb).2
end ExtremalEdges
end
end OAI
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