File size: 4,216 Bytes
f61e547 | 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 | import OAI.Probability.FirstPassage.EndpointExtensions
namespace OAI
namespace PlanarFPP
open Set
lemma endpoint_coarse_columns (A : ℝ) (m J : ℕ) (x : ℤ)
(hx : |(x : ℝ)| ≤ A*(2 : ℝ)^m) :
|(entranceColumn x m J : ℝ)| ≤ (A+2)*(2 : ℝ)^m ∧
|(exitColumn x m J : ℝ)| ≤ (A+2)*(2 : ℝ)^m := by
have hl := entranceColumn_span x m J
have hr := exitColumn_span x m J
have hl' : (gridMesh m J : ℝ) ≤ (x : ℝ)-(entranceColumn x m J : ℝ) ∧
(x : ℝ)-(entranceColumn x m J : ℝ) < 2*(gridMesh m J : ℝ) := by exact_mod_cast hl
have hr' : (gridMesh m J : ℝ) ≤ (exitColumn x m J : ℝ)-(x : ℝ) ∧
(exitColumn x m J : ℝ)-(x : ℝ) < 2*(gridMesh m J : ℝ) := by exact_mod_cast hr
have hd := gridMesh_le_scale m J
have hd0 : (0 : ℝ) < gridMesh m J := by exact_mod_cast gridMesh_pos m J
have hx' := abs_le.mp hx
constructor <;> apply abs_le.mpr <;> constructor <;> linarith [hl'.1,hl'.2,hr'.1,hr'.2]
lemma endpointBox_mono (b A A' R R' : ℝ) (k m : ℕ) (hA : A ≤ A') (hR : R ≤ R') :
endpointBox b A R k m ⊆ endpointBox b A' R' k m := by
intro p hp
exact ⟨hp.1.trans (mul_le_mul_of_nonneg_right hA (by positivity)),
hp.2.trans (mul_le_mul_of_nonneg_right hR (by positivity))⟩
theorem gridConnections_extend_negative_path (w : Edge → ℝ) (b m₀ K A R a ε : ℝ)
(hK : 0 ≤ K) (hm : 0 < m₀) (hR : 0 ≤ R) (hε : 0 < ε)
(k m J : ℕ) (hJ : 1 ≤ J) (hJlast : J ≤ gridLastLevel m) (hlast : gridLastLevel m ≤ k)
(hg : w ∈ gridConnectionsEvent b m₀ K A R k m J)
(herror : 2*endpointExtensionError K k J+2*|a| ≤ ε*(2 : ℝ)^k/2)
(hscale : (2*R*|a| /m₀)*(2 : ℝ)^k ≤ (gridMesh m J : ℝ))
(p : Vertex) (ds : List Step) (hin : PathIn p ds (endpointBox b A R k m))
(hcost : defect w b m₀ a p ds < -ε*(2 : ℝ)^k) :
∃ X Y : ℤ, gridMesh m J ∣ X ∧ gridMesh m J ∣ Y ∧
|(X : ℝ)| ≤ (A+2)*(2 : ℝ)^m ∧ |(Y : ℝ)| ≤ (A+2)*(2 : ℝ)^m ∧
(gridMesh m J : ℝ) ≤ ((Y-X : ℤ) : ℝ) ∧
∃ q es, q.1=X ∧ (q+displacement es).1=Y ∧
PathIn q es (endpointBox b (A+4) (R+2*(K+200)) k m) ∧
defect w b m₀ a q es < -(ε/2)*(2 : ℝ)^k := by
let z := p+displacement ds
have hp : p ∈ endpointBox b A R k m := by simpa using hin 0 (by omega)
have hz : z ∈ endpointBox b A R k m := by simpa [z] using hin ds.length le_rfl
obtain ⟨l,_unused,hl,_hu,hld,_hud,hil,_hiu⟩ :=
gridConnections_endpoint_defects w b m₀ K A R hK k m J hJ hJlast hlast hg p hp
obtain ⟨_unused',u,_hl,hu,_hld,hud,_hil,hiu⟩ :=
gridConnections_endpoint_defects w b m₀ K A R hK k m J hJ hJlast hlast hg z hz
let q := endpointVertex b (transverse b p) (entranceColumn p.1 m J)
have hrev := negative_defect_reverse_span w hg.1 b m₀ a R ((2 : ℝ)^k) ε hm hR
(by positivity) hε.le p z ds rfl hp.2 hz.2 hcost
have hspan := extended_columns_span p.1 z.1 m J (2*R*|a| /m₀) ((2 : ℝ)^k)
(by convert hrev using 1; ring) hscale
refine ⟨entranceColumn p.1 m J,exitColumn z.1 m J,entranceColumn_dvd _ _ _,exitColumn_dvd _ _ _,
(endpoint_coarse_columns A m J p.1 hp.1).1,(endpoint_coarse_columns A m J z.1 hz.1).2,
hspan,q,l++ds++u,rfl,?_,?_,?_⟩
· rw [displacement_append,displacement_append,← add_assoc,← add_assoc]
change ((endpointVertex b (transverse b p) (entranceColumn p.1 m J)+displacement l+
displacement ds)+displacement u).1=exitColumn z.1 m J
rw [hl]
change (z+displacement u).1=exitColumn z.1 m J
rw [hu]
rfl
· have him : PathIn p ds (endpointBox b (A+4) (R+2*(K+200)) k m) := by
intro i hi
exact endpointBox_mono b A (A+4) R (R+2*(K+200)) k m (by linarith) (by linarith) (hin i hi)
apply pathIn_append _ _ _ _ (pathIn_append _ _ _ _ hil (by simpa [hl] using him))
simpa only [displacement_append,← add_assoc,hl] using hiu
· rw [defect_append,defect_append]
have he : q+displacement l=p := hl
have he' : q+displacement (l++ds)=z := by rw [displacement_append,← add_assoc,he]
rw [he,he']
have hb := hld a
have hc := hud a
change defect w b m₀ a q l ≤ _ at hb
linarith
end PlanarFPP
end OAI
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