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9b6e014 | 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 | import Mathlib
import OAI.Probability.BinarySweep.Trajectories.IsolatedCancellation
import OAI.Probability.BinarySweep.Conditional.PlacementAlternation
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps
attribute [local instance] Classical.propDecidable
open Sparse
variable {b h : ℕ} {bits : Fin b → ℕ} {I : Type*} [Fintype I] [DecidableEq I]
def centeredEndpointKernel (H : PathFamily bits h) (z : ℝ)
(e : I → GridSlot bits × GridSlot bits) : ℝ :=
(gridSize bits:ℝ)⁻¹ ^ Fintype.card I *
alternatingSubsetSum Finset.univ (gridSize bits) (fun A => endpointProbability H z A e)
lemma centeredEndpointKernel_isolated (H : PathFamily bits h) (z : ℝ)
(e : I → GridSlot bits × GridSlot bits) (i : I)
(hh : ∀j k, (fun a : {a : Fin b // a≠j} => H.position j.castSucc k a) ≠ endpointLine (e i) j)
(ha : ∀j a, a≠i → endpointLine (e a) j≠endpointLine (e i) j) :
centeredEndpointKernel H z e=0 := by
unfold centeredEndpointKernel
rw [alternatingSubsetSum_of_isolated Finset.univ
(by exact_mod_cast (gridSize_pos bits).ne') _ i (Finset.mem_univ _) ?_,mul_zero]
intro A hA
exact endpointProbability_insert_isolated H z A e i hh
(fun j a haA => ha j a (Finset.mem_erase.mp (hA haA)).1)
def pathEndpoints (H : PathFamily bits h) (k : Fin h) : GridSlot bits × GridSlot bits :=
(H.position 0 k,H.position (Fin.last b) k)
lemma endpointLine_path (H : PathFamily bits h) (k : Fin h) (j : Fin b) :
endpointLine (pathEndpoints H k) j=(fun a : {a : Fin b // a≠j} => H.position j.castSucc k a) := by
unfold endpointLine pathEndpoints
rw [independentPosition_of_path (fun t => H.position t k) (fun j i hi => H.changes_only_stage j k i hi)]
def shareLine (e f : GridSlot bits × GridSlot bits) : Prop :=
∃j, endpointLine e j=endpointLine f j
lemma shareLine_symmetric :
∀ ⦃first second : GridSlot bits × GridSlot bits⦄,
shareLine first second → shareLine second first := by
rintro e f ⟨j,hj⟩
exact ⟨j,hj.symm⟩
lemma shareLine_probability (f : GridSlot bits × GridSlot bits) :
finiteProbability uniformWeight (fun e => shareLine (bits:=bits) e f) ≤
∑j : Fin b, (1:ℝ)/Fintype.card (GridOutside bits j) := by
have he := finiteProbability_union uniformWeight uniformWeight_nonneg
(fun (j : Fin b) (e : GridSlot bits × GridSlot bits) => endpointLine e j=endpointLine f j)
have hj (j : Fin b) : finiteProbability uniformWeight
(fun e : GridSlot bits × GridSlot bits => endpointLine e j=endpointLine f j)=
(1:ℝ)/Fintype.card (GridOutside bits j) :=
independent_line_uniform j (endpointLine f j)
simpa only [hj,shareLine] using he
lemma centeredEndpointKernel_support (H : PathFamily bits h) (z : ℝ)
(e : I → GridSlot bits × GridSlot bits) (he : centeredEndpointKernel H z e≠0) :
incidentSet shareLine (pathEndpoints H) e=Finset.univ := by
apply Finset.eq_univ_of_forall
intro i
by_contra hi
have hn : ¬((∃a, i≠a ∧ shareLine (e i) (e a)) ∨ ∃k, shareLine (e i) (pathEndpoints H k)) := by
simpa only [incidentSet,Finset.mem_filter,Finset.mem_univ,true_and] using hi
apply he
apply centeredEndpointKernel_isolated H z e i
· intro j k hk
apply hn
exact Or.inr ⟨k,j,hk.symm.trans (endpointLine_path H k j).symm⟩
· intro j a hai hj
exact hn (Or.inl ⟨a,Ne.symm hai,j,hj.symm⟩)
theorem centeredEndpointKernel_exceptional_probability (H : PathFamily bits h) (z : ℝ)
(hsmall : ((Fintype.card I+h:ℕ):ℝ) *
(∑j : Fin b, (1:ℝ)/Fintype.card (GridOutside bits j)) ≤ 1) :
productProbability (uniformWeight (Ω:=GridSlot bits × GridSlot bits))
(fun e : I → GridSlot bits × GridSlot bits => centeredEndpointKernel H z e≠0) ≤
(2:ℝ)^Fintype.card I *
(((Fintype.card I+h:ℕ):ℝ) * (∑j : Fin b, (1:ℝ)/Fintype.card (GridOutside bits j))) ^
((Fintype.card I+1)/2) := by
have hh := sharing_probability_le (I:=I) uniformWeight uniformWeight_nonneg uniformWeight_sum
shareLine shareLine_symmetric (pathEndpoints H) _ (by positivity)
shareLine_probability (Fintype.card I) (by simpa only [Fintype.card_fin] using hsmall)
simp only [Fintype.card_fin] at hh
apply (productProbability_mono uniformWeight_nonneg ?_).trans hh
intro e he
rw [centeredEndpointKernel_support H z e he,Finset.card_univ]
end BinaryCoordinateSweeps
end
end OAI
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