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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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 | import Mathlib
import OAI.Probability.LatticeSurface.Convexity.BoundedBase
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*}
def squash (w v x : V) : V := if x = w then v else x
def contract (G : SimpleGraph V) (w v : V) : SimpleGraph V := G.map (squash w v)
lemma graph_constraint_map (G : SimpleGraph V) (f : V → V) (h : V → ℝ) :
(∀ x y, (G.map f).Adj x y → |h x-h y| ≤ 1) ↔
(∀ x y, G.Adj x y → |h (f x)-h (f y)| ≤ 1) := by
constructor
· intro hh x y hxy
by_cases he : f x = f y
· simp [he]
· exact hh _ _ (SimpleGraph.map_adj_apply' hxy he)
· rintro hh x y ⟨hne,a,b,hab,rfl,rfl⟩
exact hh a b hab
lemma contracted_body (G : SimpleGraph V) (pins : Set V) (w v : V) :
conditionalBody (contract G w v) pins (fun _ => 0) = weldBody G pins w v := by
ext h
rw [conditionalBody, mem_ofPred_eq, contract, graph_constraint_map]
have hf : ∀ x, restore pins (fun _ => 0) h (squash w v x) = weldHeight pins w v h x := by
intro x
by_cases hx : x = w <;> simp [squash, weldHeight, hx]
simp only [hf, weldBody, mem_ofPred_eq]
lemma degree_map_singleton [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj]
(f : V → V) (a : V) (hf : ∀ x, f x = a → x = a) :
(G.map f).degree a ≤ G.degree a := by
classical
rw [← SimpleGraph.card_neighborFinset_eq_degree, ← SimpleGraph.card_neighborFinset_eq_degree]
refine (Finset.card_le_card (t := (G.neighborFinset a).image f) ?_).trans (Finset.card_image_le)
intro y hy
have hh : (G.map f).Adj a y := by simpa using hy
rcases hh with ⟨hne,x,z,hxz,hxa,hzy⟩
have hxa' := hf x hxa
subst x
exact Finset.mem_image.mpr ⟨z, by simpa using hxz, hzy⟩
def collapse (W : Finset V) (r : V → V) (x : V) : V := if x ∈ W then r x else x
lemma collapse_empty (r : V → V) : collapse ∅ r = id := by
funext x; simp [collapse]
lemma collapse_insert {W : Finset V} {w : V} {r : V → V}
(hw : w ∉ W) (hr : ∀ x ∈ insert w W, r x ∉ insert w W) :
collapse (insert w W) r = squash w (r w) ∘ collapse W r := by
classical
funext x
have hrr : ∀ y ∈ insert w W, r y ≠ w := fun y hy hh => hr y hy (by simp [hh])
by_cases hxw : x = w
· subst x; simp [collapse, squash, hw]
· by_cases hx : x ∈ W
· have hn := hrr x (Finset.mem_insert_of_mem hx)
simp [collapse, squash, hx, hxw, hn]
· simp [collapse, squash, hx, hxw]
lemma collapse_unique [DecidableEq V] {W : Finset V} {w : V} {r : V → V}
(_hw : w ∉ W) (hr : ∀ x ∈ W, r x ≠ w) :
∀ x, collapse W r x = w → x = w := by
intro x hh
by_cases hx : x ∈ W
· exact False.elim (hr x hx (by simpa [collapse, hx] using hh))
· simpa [collapse, hx] using hh
lemma volume_graph_body_congr [Fintype V] {G H : SimpleGraph V} (h : G = H)
(pins : Set V) (b : V → ℝ) :
volume (conditionalBody G pins b) = volume (conditionalBody H pins b) := by rw [h]
theorem volume_collapsed_body_le [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj]
(pins : Set V) (W : Finset V) (r : V → V) (D : ℕ)
(hwp : ∀ w ∈ W, w ∉ pins) (hr : ∀ w ∈ W, r w ∉ W)
(hd : ∀ w ∈ W, G.degree w ≤ D) :
volume (conditionalBody (G.map (collapse W r)) (pins ∪ (W : Set V)) (fun _ => 0)) ≤
(2 ^ D) ^ W.card * volume (conditionalBody G pins (fun _ => 0)) := by
classical
induction W using Finset.induction_on with
| empty =>
have hg : G.map (collapse ∅ r) = G := by simp [collapse_empty]
rw [hg, volume_body_pin_congr G (show pins ∪ (∅ : Finset V) = pins by simp) _]
simp
| @insert w W hw ih =>
have hwp' : ∀ x ∈ W, x ∉ pins := fun x hx => hwp x (Finset.mem_insert_of_mem hx)
have hr' : ∀ x ∈ W, r x ∉ W := fun x hx ht => hr x (Finset.mem_insert_of_mem hx)
(Finset.mem_insert_of_mem ht)
have hd' : ∀ x ∈ W, G.degree x ≤ D := fun x hx => hd x (Finset.mem_insert_of_mem hx)
have ihh := ih hwp' hr' hd'
let H := G.map (collapse W r)
let P : Set V := pins ∪ (insert w W : Finset V)
have hwP : w ∈ P := Or.inr (by simp)
have hvw : r w ≠ w := fun he => hr w (by simp) (by simp [he])
have heG : G.map (collapse (insert w W) r) = contract H w (r w) := by
rw [collapse_insert hw hr]
exact (SimpleGraph.map_map _ _ _).symm
have hPr : P \ {w} = pins ∪ (W : Set V) := by
ext x
have hnp := hwp w (by simp)
simp only [P, mem_sdiff, mem_union, Finset.mem_coe, Finset.mem_insert, mem_singleton_iff]
constructor
· rintro ⟨hp | hx,hn⟩
· exact Or.inl hp
· exact Or.inr (hx.resolve_left hn)
· intro hh
refine ⟨?_, ?_⟩
· exact hh.elim Or.inl (fun hx => Or.inr (Or.inr hx))
· intro he
subst x
exact hh.elim hnp hw
have hdeg : H.degree w ≤ D :=
(degree_map_singleton G (collapse W r) w (collapse_unique hw
(fun x hx hh => hr x (Finset.mem_insert_of_mem hx) (by simp [hh])))).trans
(hd w (by simp))
have hv := volume_weld_le H P w (r w) hwP hvw
rw [← contracted_body] at hv
rw [volume_body_pin_congr H hPr _] at hv
calc
volume (conditionalBody (G.map (collapse (insert w W) r))
(pins ∪ (insert w W : Finset V)) (fun _ => 0)) =
volume (conditionalBody (contract H w (r w)) P (fun _ => 0)) :=
volume_graph_body_congr heG _ _
_ ≤ 2 ^ H.degree w * volume (conditionalBody H (pins ∪ (W : Set V)) (fun _ => 0)) := hv
_ ≤ 2 ^ D * ((2 ^ D) ^ W.card * volume (conditionalBody G pins (fun _ => 0))) := by
apply mul_le_mul' ?_ ihh
exact pow_le_pow_right' (by norm_num) hdeg
_ = _ := by rw [Finset.card_insert_of_notMem hw, pow_succ]; ring
end ExtremalEdges
end
end OAI
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