File size: 3,794 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 83 84 85 86 87 88 89 90 91 92 93 94 95 | import OAI.Probability.FirstPassage.EndpointMesh
import OAI.Probability.FirstPassage.PathConcatenation
namespace OAI
namespace PlanarFPP
noncomputable def endpointVertex (b r : ℝ) (X : ℤ) : Vertex := (X,⌊b*X+r⌋)
lemma endpointVertex_error (b r : ℝ) (X : ℤ) :
-1 < transverse b (endpointVertex b r X)-r ∧
transverse b (endpointVertex b r X)-r ≤ 0 := by
have hl := Int.floor_le (b*X+r)
have hu := Int.lt_floor_add_one (b*X+r)
dsimp [transverse,endpointVertex]
constructor <;> linarith
lemma endpointVertex_close (b r : ℝ) (X : ℤ) :
|transverse b (endpointVertex b r X)-r| < 1 := by
have h := endpointVertex_error b r X
exact abs_lt.mpr ⟨h.1,by linarith [h.2]⟩
lemma endpointVertex_self (b : ℝ) (p : Vertex) :
endpointVertex b (transverse b p) p.1=p := by
apply Prod.ext
· rfl
· simp [endpointVertex,transverse]
lemma endpointVertex_displacement (b r : ℝ) (X Y : ℤ) :
|transverse b (endpointVertex b r Y-endpointVertex b r X)| < 1 := by
have hx := endpointVertex_error b r X
have hy := endpointVertex_error b r Y
have he : transverse b (endpointVertex b r Y-endpointVertex b r X) =
transverse b (endpointVertex b r Y)-transverse b (endpointVertex b r X) := by
simp [transverse]; ring
rw [he,abs_lt]
constructor <;> linarith [hx.1,hx.2,hy.1,hy.2]
lemma endpoint_path_defect (w : Edge → ℝ) (b r m₀ a E : ℝ) (X Y : ℤ)
(ds : List Step)
(hd : endpointVertex b r X+displacement ds=endpointVertex b r Y)
(hc : pathCost w (endpointVertex b r X) ds ≤ m₀*((Y-X : ℤ) : ℝ)+E) :
defect w b m₀ a (endpointVertex b r X) ds ≤ E+|a| := by
have hd' : displacement ds=endpointVertex b r Y-endpointVertex b r X := by
rw [← hd]; abel
have hr := (endpointVertex_displacement b r X Y).le
have hm : |a*transverse b (endpointVertex b r Y-endpointVertex b r X)| ≤ |a| := by
rw [abs_mul]
nlinarith [abs_nonneg a]
have hlo := (abs_le.mp hm).1
unfold defect support
rw [hd']
have hx : (endpointVertex b r Y-endpointVertex b r X).1=Y-X := rfl
rw [hx]
linarith
lemma negative_defect_reverse_span (w : Edge → ℝ) (hw : ∀ e, 0 ≤ w e)
(b m₀ a R s ε : ℝ) (hm : 0 < m₀) (_hR : 0 ≤ R) (hs : 0 ≤ s) (hε : 0 ≤ ε)
(p z : Vertex) (ds : List Step) (hd : p+displacement ds=z)
(hp : |transverse b p| ≤ R*s) (hz : |transverse b z| ≤ R*s)
(hc : defect w b m₀ a p ds < -ε*s) :
-2*R*|a| *s/m₀ ≤ ((z.1-p.1 : ℤ) : ℝ) := by
have hd' : displacement ds=z-p := by rw [← hd]; abel
have hr : |transverse b (z-p)| ≤ 2*R*s := by
have he : transverse b (z-p)=transverse b z-transverse b p := by simp [transverse]; ring
rw [he]
exact (abs_sub _ _).trans (by linarith)
have ha : a*transverse b (z-p) ≤ 2*R*|a| *s := by
have hh := mul_le_mul_of_nonneg_left hr (abs_nonneg a)
have hb := le_abs_self (a*transverse b (z-p))
rw [abs_mul] at hb
nlinarith
have hcost := pathCost_nonneg hw p ds
have heps := mul_nonneg hε hs
unfold defect support at hc
rw [hd'] at hc
change pathCost w p ds-(m₀*((z.1-p.1 : ℤ) : ℝ)+a*transverse b (z-p)) < -ε*s at hc
apply (div_le_iff₀ hm).mpr
nlinarith
lemma extended_columns_span (p z : ℤ) (m J : ℕ) (C s : ℝ)
(hrev : -C*s ≤ ((z-p : ℤ) : ℝ))
(hscale : C*s ≤ (gridMesh m J : ℝ)) :
(gridMesh m J : ℝ) ≤ ((exitColumn z m J-entranceColumn p m J : ℤ) : ℝ) := by
have hp := entranceColumn_span p m J
have hz := exitColumn_span z m J
have hp' : (gridMesh m J : ℝ) ≤ ((p-entranceColumn p m J : ℤ) : ℝ) := by exact_mod_cast hp.1
have hz' : (gridMesh m J : ℝ) ≤ ((exitColumn z m J-z : ℤ) : ℝ) := by exact_mod_cast hz.1
push_cast at hp' hz' hrev ⊢
linarith
end PlanarFPP
end OAI
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