File size: 2,937 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 | import OAI.Probability.FirstPassage.CrossingChain
import OAI.Probability.FirstPassage.RareFastGeometry
namespace OAI
namespace PlanarFPP
open Filter
open scoped Topology
theorem crossing_endpoints_escape_linear (b σ S : ℝ) (n d : ℤ)
(hb : |b| ≤ 1) (hσ : |σ|=1) (hS : 0 < S) (hn : 0 < n) (hd : 0 ≤ d)
(v z : ℕ → Vertex) (hv0 : v 0=0)
(hstep : ∀ t, v (t+1)=v t+(1,0) ∨ v (t+1)=v t+(0,1))
(hx : ∀ t, (z t).1=blockX n d (v t)+n)
(hr : ∀ t, |transverse b (z t)-blockY σ S (v t)| ≤ S/4) :
Tendsto (fun t => ‖latticeEmbed (z t)‖) atTop atTop ∧
∀ t, ‖latticeEmbed (z t)‖ ≤ (2*((n : ℝ)+d)+3*S)*((t : ℝ)+1) := by
have hn' : (1 : ℝ) ≤ n := by exact_mod_cast hn
have hd' : (0 : ℝ) ≤ d := by exact_mod_cast hd
have hcoords (t : ℕ) := oriented_path_coordinates v hv0 hstep t
have hxt (t : ℕ) : ((z t).1 : ℝ) = (t : ℝ)*((n : ℝ)+d)+n := by
rw [hx]
simp only [blockX,blockLayer,Int.cast_add,Int.cast_mul]
have hh := congrArg (fun a : ℤ => (a : ℝ)) (hcoords t).1
push_cast at hh
rw [hh]
have hxpos (t : ℕ) : (t : ℝ)+1 ≤ ((z t).1 : ℝ) := by
rw [hxt]
nlinarith [Nat.cast_nonneg (α := ℝ) t]
have hnlow (t : ℕ) : (t : ℝ) ≤ ‖latticeEmbed (z t)‖ := by
have hh := plane_coordinate_le_norm (latticeEmbed (z t)) 0
change |((z t).1 : ℝ)| ≤ _ at hh
have hp : (0 : ℝ) ≤ (z t).1 := by linarith [Nat.cast_nonneg (α := ℝ) t,hxpos t]
rw [abs_of_nonneg hp] at hh
linarith [hxpos t]
refine ⟨tendsto_atTop_mono hnlow tendsto_natCast_atTop_atTop,?_⟩
intro t
have hlevel : (0 : ℝ) ≤ blockLevel (v t) ∧ (blockLevel (v t) : ℝ) ≤ 2*t := by
have hh := hcoords t
constructor <;> dsimp [blockLevel] <;> exact_mod_cast (by omega : _)
have hy : |blockY σ S (v t)| ≤ 2*(t : ℝ)*S := by
unfold blockY
rw [abs_mul,abs_mul,hσ,one_mul,abs_of_nonneg hlevel.1,abs_of_pos hS]
exact mul_le_mul_of_nonneg_right hlevel.2 hS.le
have hrbound : |transverse b (z t)| ≤ 2*(t : ℝ)*S+S/4 := by
have hh := abs_add_le (transverse b (z t)-blockY σ S (v t)) (blockY σ S (v t))
rw [sub_add_cancel] at hh
linarith [hr t]
have hpos : (0 : ℝ) ≤ (z t).1 := by linarith [hxpos t,Nat.cast_nonneg (α := ℝ) t]
have hyy : |((z t).2 : ℝ)| ≤ |transverse b (z t)|+((z t).1 : ℝ) := by
have hh := abs_add_le (transverse b (z t)) (b*((z t).1 : ℝ))
have he : transverse b (z t)+b*((z t).1 : ℝ)=((z t).2 : ℝ) := by unfold transverse; ring
rw [he,abs_mul,abs_of_nonneg hpos] at hh
have hm := mul_le_mul_of_nonneg_right hb hpos
linarith
have hnorm := plane_norm_le_coordinates (latticeEmbed (z t))
change ‖latticeEmbed (z t)‖ ≤ |((z t).1 : ℝ)|+|((z t).2 : ℝ)| at hnorm
rw [abs_of_nonneg hpos] at hnorm
have ht : (0 : ℝ) ≤ t := Nat.cast_nonneg _
rw [hxt] at hyy hnorm
nlinarith
end PlanarFPP
end OAI
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