| 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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|
