| import OAI.Probability.BenjaminiSchramm.NormalizedLayerTransport |
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| namespace OAI |
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| noncomputable section |
| open Set |
| open scoped Classical |
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| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
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| def adjustedLayerIndex (a : V → ℝ) (R u : ℝ) (x y : V) : ℤ := |
| ⌊G.orbitAdjustedHeight a x y/R+u⌋-⌊u⌋ |
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| @[simp] lemma adjustedLayerIndex_self (a : V → ℝ) (R u : ℝ) (x : V) : |
| G.adjustedLayerIndex a R u x x=0 := by simp [adjustedLayerIndex] |
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| 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] |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| end CriticalPercolation.BondGraph |
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| end |
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| end OAI |
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