File size: 3,711 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.ShapeDeficit
import OAI.Probability.FirstPassage.HighDensityOriented
namespace OAI
namespace PlanarFPP
open Filter
open scoped Topology
def crossingChain (initialPath : List Step) (trial connector : ℕ → List Step) : ℕ → List Step
| 0 => initialPath ++ trial 0
| n+1 => crossingChain initialPath trial connector n ++ (connector n ++ trial (n+1))
theorem crossingChain_endpoint (initialPath : List Step) (trial connector : ℕ → List Step)
(a z : ℕ → Vertex) (hp : displacement initialPath = a 0)
(ht : ∀ n, a n+displacement (trial n)=z n)
(hc : ∀ n, z n+displacement (connector n)=a (n+1)) (n : ℕ) :
displacement (crossingChain initialPath trial connector n)=z n := by
induction n with
| zero => simpa only [crossingChain,displacement_append,hp] using ht 0
| succ n ih =>
simp only [crossingChain,displacement_append,ih,← add_assoc,hc,ht]
theorem crossingChain_defect (w : Edge → ℝ) (b m a e : ℝ)
(initialPath : List Step) (trial connector : ℕ → List Step) (p z : ℕ → Vertex)
(hp : displacement initialPath=p 0)
(ht : ∀ n, p n+displacement (trial n)=z n)
(hc : ∀ n, z n+displacement (connector n)=p (n+1))
(htrial : ∀ n, defect w b m a (p n) (trial n) ≤ -e)
(hconnector : ∀ n, defect w b m a (z n) (connector n) ≤ e/2) (n : ℕ) :
defect w b m a 0 (crossingChain initialPath trial connector n) ≤
defect w b m a 0 initialPath-((n : ℝ)+1)*e+(n : ℝ)*e/2 := by
induction n with
| zero =>
simp only [crossingChain,defect_append,zero_add,hp]
have hh := htrial 0
norm_num
linarith
| succ n ih =>
rw [crossingChain,defect_append,crossingChain_endpoint initialPath trial connector p z hp ht hc,
zero_add,defect_append,hc]
have h1 := htrial (n+1)
have h2 := hconnector n
push_cast
linarith
theorem oriented_path_coordinates (v : ℕ → Vertex) (h0 : v 0=0)
(hs : ∀ n, v (n+1)=v n+(1,0) ∨ v (n+1)=v n+(0,1)) (n : ℕ) :
(v n).1+(v n).2=(n : ℤ) ∧ 0 ≤ (v n).1 ∧ 0 ≤ (v n).2 ∧ (v n).2 ≤ n := by
induction n with
| zero => simp [h0]
| succ n ih =>
rcases hs n with he | he <;> rw [he] <;> simp only [Prod.fst_add,Prod.snd_add] <;>
push_cast <;> omega
theorem UniformShape.no_crossing_chain (w : Edge → ℝ) (hw : ∀ e, 0 ≤ w e)
(μ : Plane → ℝ) (hu : UniformShape w μ) (b m a e B : ℝ) (he : 0 < e) (hB : 0 < B)
(hf : ∀ v, m*v 0+a*(v 1-b*v 0) ≤ μ v)
(initialPath : List Step) (trial connector : ℕ → List Step) (p z : ℕ → Vertex)
(hp : displacement initialPath=p 0)
(ht : ∀ n, p n+displacement (trial n)=z n)
(hc : ∀ n, z n+displacement (connector n)=p (n+1))
(htrial : ∀ n, defect w b m a (p n) (trial n) ≤ -e)
(hconnector : ∀ n, defect w b m a (z n) (connector n) ≤ e/2)
(hescape : Tendsto (fun n => ‖latticeEmbed (z n)‖) atTop atTop)
(hbound : ∀ n, ‖latticeEmbed (z n)‖ ≤ B*((n : ℝ)+1)) : False := by
apply hu.no_linear_deficit w hw μ (fun v => m*v 0+a*(v 1-b*v 0)) hf z
(crossingChain initialPath trial connector) (defect w b m a 0 initialPath) (e/2) B
(by linarith) hB hescape hbound
(crossingChain_endpoint initialPath trial connector p z hp ht hc)
intro n
have hh := crossingChain_defect w b m a e initialPath trial connector p z hp ht hc htrial hconnector n
rw [defect,crossingChain_endpoint initialPath trial connector p z hp ht hc] at hh
simp only [support,transverse] at hh
change pathCost w 0 (crossingChain initialPath trial connector n) ≤
m*(z n).1+a*((z n).2-b*(z n).1)+defect w b m a 0 initialPath-e/2*n
linarith
end PlanarFPP
end OAI
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