File size: 3,288 Bytes
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 | import Mathlib
import OAI.Probability.BinarySweep.Trajectories.PathSign
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps
attribute [local instance] Classical.propDecidable
variable {b h : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h)
lemma exists_untouched_line (j : Fin b) (hh : h<Fintype.card (GridOutside bits j)) :
∃y : GridOutside bits j, lineHoles H j y=0 := by
by_contra hn
push Not at hn
have he := Finset.sum_le_sum (s:=Finset.univ)
(fun y _ => Nat.one_le_iff_ne_zero.mpr (hn y))
simp only [Finset.sum_const,Finset.card_univ,smul_eq_mul,mul_one,sum_lineHoles] at he
omega
lemma signed_path_weight_zero_sparse (z : ℝ) (j : Fin b) (hj : 0<bits j)
(hh : h<Fintype.card (GridOutside bits j)) :
(∑g : GridChoices bits, if pathEvent H g then
gridWeight bits z g*realSign (gridSweep bits g) else 0)=0 := by
obtain ⟨y,hy⟩ := exists_untouched_line H j hh
exact signed_path_weight_zero_empty H z j hj y hy
lemma two_outside_embedding {X Y : Type*} [Fintype X] [Fintype Y]
(e : X ↪ Y) (h : Fintype.card X+1<Fintype.card Y) :
∃u v : Y, u≠v ∧ u∉Set.range e ∧ v∉Set.range e := by
let S := {y : Y // y∉Set.range e}
have hc : Fintype.card S=Fintype.card Y-Fintype.card X := by
dsimp only [S]
rw [Fintype.card_subtype_compl,Fintype.card_range]
have hS : 1<Fintype.card S := by omega
have : Nontrivial S := Fintype.one_lt_card_iff_nontrivial.mp hS
obtain ⟨u,v,huv⟩ := exists_pair_ne S
exact ⟨u.val,v.val,(fun he => huv (Subtype.ext he)),u.property,v.property⟩
lemma exists_line_two_free (j : Fin b)
(hh : h+Fintype.card (GridOutside bits j)<gridSize bits) :
∃y : GridOutside bits j, ∃u v : Slot (bits j), u≠v ∧
u∉Set.range (lineOutput H j y) ∧ v∉Set.range (lineOutput H j y) := by
have hfull : Fintype.card (GridOutside bits j)*2^bits j=gridSize bits := by
have he := Fintype.card_congr (Equiv.piSplitAt j (fun i => Slot (bits i)))
change Fintype.card (GridSlot bits)=Fintype.card (Slot (bits j) × GridOutside bits j) at he
have hs : Fintype.card (Slot (bits j))=2^bits j := by simp [Slot]
have hg : Fintype.card (GridSlot bits)=gridSize bits := by simp [GridSlot,gridSize,Slot]
simpa only [Fintype.card_prod,hs,hg,mul_comm] using he.symm
have he : ∃y : GridOutside bits j, lineHoles H j y+1<2^bits j := by
by_contra hn
push Not at hn
have hf := Finset.sum_le_sum (s:=Finset.univ) (fun y _ => hn y)
simp only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,smul_eq_mul, mul_one,sum_lineHoles,hfull] at hf
omega
obtain ⟨y,hy⟩ := he
refine ⟨y,?_⟩
apply two_outside_embedding (lineOutput H j y)
have hc : Fintype.card (LinePaths H j y)=lineHoles H j y := by
unfold LinePaths lineHoles
congr 1
rw [hc]
simpa [Slot] using hy
lemma signed_path_weight_zero_few (j : Fin b)
(hh : h+Fintype.card (GridOutside bits j)<gridSize bits) :
(∑g : GridChoices bits, if pathEvent H g then
gridWeight bits 0 g*realSign (gridSweep bits g) else 0)=0 := by
obtain ⟨y,u,v,huv,hu,hv⟩ := exists_line_two_free H j hh
exact signed_path_weight_zero_uniform H j y huv hu hv
end BinaryCoordinateSweeps
end
end OAI
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