| import Mathlib |
| import OAI.RepresentationTheory.Saxl.Main |
| import OAI.RepresentationTheory.UniversalSquare.Support.RowChecker |
|
|
| namespace OAI |
|
|
| /-! Fast Row Check. -/ |
|
|
| section |
|
|
| namespace UniversalTensorSquare |
|
|
| def quickBand (M r : ℕ) (rs : List ℕ) : Bool := |
| (List.range 4).any fun k => |
| let d := k+1 |
| let q := testWidth M d |
| decide (d*q+7+d ≤ 2*M-1) && decide (2*M-1 ≤ (rs.take d).sum) && |
| decide (r ≤ ((List.range q).map fun j => (rs.countP (fun a => decide (j < a))-d)/2).sum) |
|
|
| lemma quickBand_sound {M r : ℕ} {rs : List ℕ} (h : quickBand M r rs = true) : |
| RowsBand M r rs := by |
| obtain ⟨k,hk,hh⟩ := List.any_eq_true.mp h |
| simp only [Bool.and_eq_true,decide_eq_true_eq] at hh |
| exact ⟨k,hk,hh.1.1,hh.1.2,hh.2⟩ |
|
|
| def quickResidual (M r : ℕ) (qs : List (List ℕ)) (t : RowTree) (rs : List ℕ) : Bool := |
| let ts := transposeRows rs |
| t.find rs || t.find ts || quickBand M r rs || quickBand M r ts || |
| qs.any (fun q => decide (RowsDom rs q) || decide (RowsDom ts q)) |
|
|
| lemma quickResidual_sound {M r : ℕ} {qs : List (List ℕ)} {t : RowTree} {rs : List ℕ} |
| (h : quickResidual M r qs t rs = true) : TreeResidual M r qs t rs := by |
| simp only [quickResidual,Bool.or_eq_true] at h |
| rcases h with (((h | h) | h) | h) | h |
| · exact Or.inr (Or.inl h) |
| · exact Or.inr (Or.inr h) |
| · exact Or.inl (Or.inl (quickBand_sound h)) |
| · exact Or.inl (Or.inr (Or.inl (quickBand_sound h))) |
| · obtain ⟨q,hq,h⟩ := List.any_eq_true.mp h |
| simp only [Bool.or_eq_true,decide_eq_true_eq] at h |
| exact Or.inl (Or.inr (Or.inr (Or.inl ⟨q,hq,h⟩))) |
|
|
| lemma quickCheckedRows (M r : ℕ) (qs : List (List ℕ)) (t : RowTree) |
| (fuel n upper : ℕ) (A B : List (ℕ × ℕ)) |
| (h : checkRowPrefixes (quickResidual M r qs t) fuel n upper A B [] = true) : |
| ∀ rs ∈ constrainedRows fuel n upper A B, TreeResidual M r qs t rs := by |
| intro rs hrs |
| apply quickResidual_sound |
| simpa only [List.reverse_nil,List.nil_append] using |
| checkRowPrefixes_sound (quickResidual M r qs t) fuel n upper A B [] h rs hrs |
|
|
| end UniversalTensorSquare |
| end |
|
|
| end OAI |
|
|
