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9b6e014 | 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 | import OAI.Probability.BenjaminiSchramm.NormalizedLayerTransport
namespace OAI
noncomputable section
open Set
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def adjustedLayerIndex (a : V → ℝ) (R u : ℝ) (x y : V) : ℤ :=
⌊G.orbitAdjustedHeight a x y/R+u⌋-⌊u⌋
@[simp] lemma adjustedLayerIndex_self (a : V → ℝ) (R u : ℝ) (x : V) :
G.adjustedLayerIndex a R u x x=0 := by simp [adjustedLayerIndex]
lemma adjustedLayerIndex_aut (a : V → ℝ)
(ha : ∀ (φ : G.Aut) x, a (φ.vertex x)=a x) (R u : ℝ)
(φ : G.Aut) (x y : V) :
G.adjustedLayerIndex a R u (φ.vertex x) (φ.vertex y)=G.adjustedLayerIndex a R u x y := by
simp only [adjustedLayerIndex,G.orbitAdjustedHeight_aut a ha]
lemma adjustedLayerIndex_change_base (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(a : V → ℝ) (R u : ℝ) (x y z : V) :
G.adjustedLayerIndex a R u x z = G.adjustedLayerIndex a R u x y+
G.adjustedLayerIndex a R (G.orbitAdjustedHeight a x y/R+u) y z := by
unfold adjustedLayerIndex
rw [G.orbitAdjustedHeight_change_base hc hl a x y z]
have he : (G.orbitAdjustedHeight a x y+G.orbitAdjustedHeight a y z)/R+u =
G.orbitAdjustedHeight a y z/R+(G.orbitAdjustedHeight a x y/R+u) := by ring
rw [he]
ring
lemma adjustedLayerIndex_periodic (a : V → ℝ) (R u : ℝ) (n : ℤ) (x y : V) :
G.adjustedLayerIndex a R (u+n) x y=G.adjustedLayerIndex a R u x y := by
unfold adjustedLayerIndex
rw [←add_assoc,Int.floor_add_intCast,Int.floor_add_intCast]
ring
lemma adjustedLayerIndex_height_bounds (a : V → ℝ) (R u : ℝ) (hR : 0 < R) (x y : V) :
R*((G.adjustedLayerIndex a R u x y : ℝ)-1) < G.orbitAdjustedHeight a x y ∧
G.orbitAdjustedHeight a x y < R*((G.adjustedLayerIndex a R u x y : ℝ)+1) := by
have h₁ := Int.floor_le (G.orbitAdjustedHeight a x y/R+u)
have h₂ := Int.lt_floor_add_one (G.orbitAdjustedHeight a x y/R+u)
have h₃ := Int.floor_le u
have h₄ := Int.lt_floor_add_one u
have he : (G.adjustedLayerIndex a R u x y : ℝ) =
(⌊G.orbitAdjustedHeight a x y/R+u⌋ : ℝ)-(⌊u⌋ : ℝ) := by
simp only [adjustedLayerIndex,Int.cast_sub]
rw [he]
constructor
· have hh : (⌊G.orbitAdjustedHeight a x y/R+u⌋ : ℝ)-(⌊u⌋ : ℝ)-1 <
G.orbitAdjustedHeight a x y/R := by linarith
have hm := (lt_div_iff₀ hR).mp hh
nlinarith
· have hh : G.orbitAdjustedHeight a x y/R <
(⌊G.orbitAdjustedHeight a x y/R+u⌋ : ℝ)-(⌊u⌋ : ℝ)+1 := by linarith
have hm := (div_lt_iff₀ hR).mp hh
nlinarith
lemma adjustedLayerIndex_edge_jump (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(a : V → ℝ) (R u : ℝ) (hR : 0 < R)
(hjump : ∀ e v w, v ∈ G.ends e → w ∈ G.ends e →
|G.orbitAdjustedHeight a v w| ≤ R)
(x v w : V) (e : E) (hv : v ∈ G.ends e) (hw : w ∈ G.ends e) :
G.adjustedLayerIndex a R u x v-1 ≤ G.adjustedLayerIndex a R u x w ∧
G.adjustedLayerIndex a R u x w ≤ G.adjustedLayerIndex a R u x v+1 := by
have hh := abs_le.mp (hjump e v w hv hw)
have hcoc := G.orbitAdjustedHeight_change_base hc hl a x v w
have hdiv : G.orbitAdjustedHeight a x v/R-1 ≤ G.orbitAdjustedHeight a x w/R ∧
G.orbitAdjustedHeight a x w/R ≤ G.orbitAdjustedHeight a x v/R+1 := by
constructor
· apply (le_div_iff₀ hR).mpr
have he : (G.orbitAdjustedHeight a x v/R-1)*R = G.orbitAdjustedHeight a x v-R := by
field_simp
rw [he]
linarith
· apply (div_le_iff₀ hR).mpr
have he : (G.orbitAdjustedHeight a x v/R+1)*R = G.orbitAdjustedHeight a x v+R := by
field_simp
rw [he]
linarith
have hlo := Int.floor_mono (add_le_add hdiv.1 (le_refl u))
have hhi := Int.floor_mono (add_le_add hdiv.2 (le_refl u))
have helo : G.orbitAdjustedHeight a x v/R-1+u = G.orbitAdjustedHeight a x v/R+u-1 := by ring
have hehi : G.orbitAdjustedHeight a x v/R+1+u = G.orbitAdjustedHeight a x v/R+u+1 := by ring
rw [helo,Int.floor_sub_one] at hlo
rw [hehi,Int.floor_add_one] at hhi
unfold adjustedLayerIndex
omega
end CriticalPercolation.BondGraph
end
end OAI
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