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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 96 97 98 99 100 101 102 103 104 105 | import OAI.Probability.FirstPassage.BKDyadicPaths
import Mathlib.Combinatorics.Colex
namespace OAI
namespace PlanarFPP
open Finset
noncomputable def dyadicPositions (t : ℕ) (x : Finset (Edge × Fin t)) (e : Edge) : Finset ℕ :=
((univ : Finset (Fin t)).filter (fun h => (e,h) ∈ x)).image (fun h => t-(h.val+1))
noncomputable def dyadicRank (t : ℕ) (x : Finset (Edge × Fin t)) (e : Edge) : ℕ :=
∑ a ∈ dyadicPositions t x e, 2^a
lemma dyadicPositions_subset (t : ℕ) (x : Finset (Edge × Fin t)) (e : Edge) :
dyadicPositions t x e ⊆ range t := by
intro a ha
obtain ⟨h,_,rfl⟩ := mem_image.mp ha
rw [mem_range]
have := h.isLt
omega
lemma mem_dyadicPositions (t : ℕ) (x : Finset (Edge × Fin t)) (e : Edge) (h : Fin t) :
t-(h.val+1) ∈ dyadicPositions t x e ↔ (e,h) ∈ x := by
constructor
· intro hm
obtain ⟨a,ha,he⟩ := mem_image.mp hm
have heq : a=h := by apply Fin.ext; have := a.isLt; have := h.isLt; omega
subst a
exact (mem_filter.mp ha).2
· intro hm
exact mem_image.mpr ⟨h,mem_filter.mpr ⟨mem_univ _,hm⟩,rfl⟩
lemma sum_two_powers_range (t : ℕ) : (∑ a ∈ range t, 2^a)+1=2^t := by
induction t with
| zero => simp
| succ t ih => rw [sum_range_succ,pow_succ]; omega
lemma dyadicRank_lt (t : ℕ) (x : Finset (Edge × Fin t)) (e : Edge) :
dyadicRank t x e < 2^t := by
have h := sum_le_sum_of_subset_of_nonneg (f := fun a : ℕ => 2^a) (dyadicPositions_subset t x e)
(fun _ _ _ => Nat.zero_le _)
have hg := sum_two_powers_range t
unfold dyadicRank
omega
lemma dyadicRank_eq_iff (t : ℕ) (x y : Finset (Edge × Fin t)) (e : Edge) :
dyadicRank t x e = dyadicRank t y e ↔ ∀ h : Fin t, (e,h) ∈ x ↔ (e,h) ∈ y := by
constructor
· intro he
have hp : dyadicPositions t x e = dyadicPositions t y e := by
have hh := congrArg (fun n : ℕ => n.bitIndices.toFinset) he
simpa [dyadicRank] using hh
intro h
rw [← mem_dyadicPositions t x e h,← mem_dyadicPositions t y e h,hp]
· intro he
have hp : dyadicPositions t x e = dyadicPositions t y e := by
unfold dyadicPositions
congr 1
ext h
simp [he h]
simp [dyadicRank,hp]
lemma dyadicRanks_injective (S : Finset Edge) (t : ℕ) {x y : Finset (Edge × Fin t)}
(hx : x ⊆ S ×ˢ univ) (hy : y ⊆ S ×ˢ univ)
(h : ∀ e ∈ S, dyadicRank t x e = dyadicRank t y e) : x=y := by
ext ⟨e,a⟩
by_cases he : e ∈ S
· exact (dyadicRank_eq_iff t x y e).mp (h e he) a
· have hn : (e,a) ∉ x := fun hm => he (mem_product.mp (hx hm)).1
have hn' : (e,a) ∉ y := fun hm => he (mem_product.mp (hy hm)).1
simp [hn,hn']
lemma dyadicRank_sum (t : ℕ) (x : Finset (Edge × Fin t)) (e : Edge) :
dyadicRank t x e = ∑ h : Fin t, if (e,h) ∈ x then 2^(t-(h.val+1)) else 0 := by
unfold dyadicRank dyadicPositions
rw [sum_image]
· simp only [sum_filter]
· intro a _ b _ hab
change t-(a.val+1)=t-(b.val+1) at hab
apply Fin.ext
have := a.isLt
have := b.isLt
omega
lemma dyadicSurvival_eq_rank (t : ℕ) (x : Finset (Edge × Fin t)) (e : Edge) :
dyadicSurvival t x e = (dyadicRank t x e : ℝ)/(2:ℝ)^t := by
rw [dyadicRank_sum]
simp only [Nat.cast_sum,Nat.cast_ite,Nat.cast_pow,Nat.cast_ofNat,Nat.cast_zero]
rw [Finset.sum_div]
unfold dyadicSurvival
apply sum_congr rfl
intro h _
by_cases hm : (e,h) ∈ x
· simp only [hm,↓reduceIte]
have hh : t=(t-(h.val+1))+(h.val+1) := by omega
have hp : (2:ℝ)^t = 2^(t-(h.val+1))*2^(h.val+1) := by rw [← pow_add,← hh]
rw [hp]
field_simp
· simp [hm]
end PlanarFPP
end OAI
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