| 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 |
| |
