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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 | import Mathlib
import OAI.Probability.BinarySweep.SparseBounds.ConditionalCentered
import OAI.Probability.BinarySweep.Trajectories.PathSign
import OAI.Probability.BinarySweep.FiniteLaws.RemainingSign
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 conditional_full_sign (z : ℝ) :
(∑g : ConditionalChoices H, conditionalChoiceWeight H z g*realSign (gridSweep bits g.val)) =
(∑g : GridChoices bits, if pathEvent H g then
gridWeight bits z g*realSign (gridSweep bits g) else 0)/conditionalNormalizer H z := by
unfold conditionalChoiceWeight
simp_rw [div_mul_eq_mul_div]
rw [←Finset.sum_div]
congr 1
have he := sum_dite_zero (pathEvent H)
(fun g _ => gridWeight bits z g*realSign (gridSweep bits g))
refine Eq.trans ?_ he.symm
apply Finset.sum_congr
· ext g; simp only [Finset.mem_univ]
intro g _
rfl
theorem conditional_sign_zero (z : ℝ)
(hs : (∑g : GridChoices bits, if pathEvent H g then
gridWeight bits z g*realSign (gridSweep bits g) else 0)=0) :
(∑a, conditionalGroupLaw H z a*realSign a)=0 := by
change (∑a, conditionalGroupLaw H z a • realSign a)=0
rw [conditionalGroupLaw_expectation]
by_cases hn : Nonempty (ConditionalChoices H)
· obtain ⟨g₀⟩ := hn
simp only [smul_eq_mul]
have he (g : ConditionalChoices H) :
conditionalChoiceWeight H z g *
(realSign (remainingPerm H g₀.val g₀.property)/realSign (gridSweep bits g₀.val)*
realSign (gridSweep bits g.val)) =
(realSign (remainingPerm H g₀.val g₀.property)/realSign (gridSweep bits g₀.val))*
(conditionalChoiceWeight H z g*realSign (gridSweep bits g.val)) := by ring
have hf (g : ConditionalChoices H) :
conditionalChoiceWeight H z g*realSign (remainingPerm H g.val g.property) =
(realSign (remainingPerm H g₀.val g₀.property)/realSign (gridSweep bits g₀.val))*
(conditionalChoiceWeight H z g*realSign (gridSweep bits g.val)) := by
rw [remaining_realSign_factor H g g₀]
exact he g
simp_rw [hf]
rw [←Finset.mul_sum,conditional_full_sign H z,hs,zero_div,mul_zero]
· have : IsEmpty (ConditionalChoices H) := not_nonempty_iff.mp hn
exact Finset.sum_eq_zero (fun g _ => isEmptyElim g)
lemma conditional_sign_zero_empty (z : ℝ) (j : Fin b) (hj : 0<bits j)
(y : GridOutside bits j) (hy : lineHoles H j y=0) :
(∑a, conditionalGroupLaw H z a*realSign a)=0 :=
conditional_sign_zero H z (signed_path_weight_zero_empty H z j hj y hy)
lemma conditional_sign_zero_uniform (j : Fin b) (y : GridOutside bits j)
{u v : Slot (bits j)} (huv : u≠v)
(hu : u∉Set.range (lineOutput H j y)) (hv : v∉Set.range (lineOutput H j y)) :
(∑a, conditionalGroupLaw H 0 a*realSign a)=0 :=
conditional_sign_zero H 0 (signed_path_weight_zero_uniform H j y huv hu hv)
end BinaryCoordinateSweeps
end
end OAI
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