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e3e0a0d | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 | import Mathlib
import OAI.RepresentationTheory.Saxl.Main
import OAI.RepresentationTheory.UniversalSquare.Support.FastRowCheck
namespace OAI
/-! Union Row Checker. -/
section
namespace UniversalTensorSquare
def liveRegions (fuel n upper : ℕ) (regions : List PrefixRegion) : List PrefixRegion :=
regions.filter fun R => !quickRejected fuel n upper R.1 R.2
def nextRegions (a : ℕ) (regions : List PrefixRegion) : List PrefixRegion :=
regions.filterMap fun R => if quickAllowed a R.1 R.2 then
some (clipBudget R.1 a,rowBudget R.2 a) else none
def checkRegionPrefixes (P : List ℕ → Bool) : ℕ → ℕ → ℕ →
List PrefixRegion → List ℕ → Bool
| fuel,n,upper,regions,pref =>
let alive := liveRegions fuel n upper regions
if alive.isEmpty then true else
match fuel with
| 0 => if n = 0 then P pref.reverse else true
| k+1 =>
let lo := max 1 (n/(k+1))
(List.range' lo (min n upper+1-lo)).all fun a =>
if n ≤ (k+1)*a then
checkRegionPrefixes P k (n-a) a (nextRegions a alive) (a::pref)
else true
lemma checkRegionPrefixes_sound (P : List ℕ → Bool) (fuel n upper : ℕ)
(regions : List PrefixRegion) (pref : List ℕ)
(hc : checkRegionPrefixes P fuel n upper regions pref = true) :
∀ A B, (A,B) ∈ regions → ∀ rs ∈ constrainedRows fuel n upper A B,
P (pref.reverse ++ rs) = true := by
induction fuel generalizing n upper regions pref with
| zero =>
intro A B hR rs hrs
rw [constrainedRows] at hrs
split_ifs at hrs with hrej hn
· simp at hrs
· simp only [List.mem_singleton] at hrs
subst rs
have hmem : (A,B) ∈ liveRegions 0 n upper regions := by
apply List.mem_filter.mpr
exact ⟨hR,by simpa only [Bool.not_eq_true',quickRejected_iff,
Bool.eq_false_iff,ne_eq] using hrej⟩
have hne : ¬(liveRegions 0 n upper regions).isEmpty := by
simp only [List.isEmpty_iff]
intro he
simp only [he,List.not_mem_nil] at hmem
simpa only [checkRegionPrefixes,ite_eq_right hne,ite_eq_left hn,List.append_nil] using hc
· simp at hrs
| succ k ih =>
intro A B hR rs hrs
rw [constrainedRows] at hrs
split_ifs at hrs with hrej
· simp at hrs
· have hmem : (A,B) ∈ liveRegions (k+1) n upper regions := by
apply List.mem_filter.mpr
exact ⟨hR,by simpa only [Bool.not_eq_true',quickRejected_iff,
Bool.eq_false_iff,ne_eq] using hrej⟩
have hne : ¬(liveRegions (k+1) n upper regions).isEmpty := by
simp only [List.isEmpty_iff]
intro he
simp only [he,List.not_mem_nil] at hmem
obtain ⟨a,ha,hr⟩ := List.mem_flatMap.mp hrs
split_ifs at hr with hallowed
· obtain ⟨ys,hy,rfl⟩ := List.mem_map.mp hr
have hlo : max 1 (n/(k+1)) ≤ a := by
have hh := Nat.div_le_of_le_mul hallowed.2
have ha' := (List.mem_range'_1.mp ha).1
exact max_le ha' hh
have hai : a ∈ List.range' (max 1 (n/(k+1)))
(min n upper+1-max 1 (n/(k+1))) := by
have ha' := List.mem_range'_1.mp ha
apply List.mem_range'_1.mpr
refine ⟨hlo, ?_⟩
have he : max 1 (n/(k+1)) + (min n upper+1-max 1 (n/(k+1))) = min n upper+1 := by
apply Nat.add_sub_of_le
omega
rw [he]
omega
rw [checkRegionPrefixes] at hc
simp only [ite_eq_right hne] at hc
have hh := List.all_eq_true.mp hc a hai
rw [ite_eq_left hallowed.2] at hh
have hnext : (clipBudget A a,rowBudget B a) ∈
nextRegions a (liveRegions (k+1) n upper regions) := by
apply List.mem_filterMap.mpr
refine ⟨(A,B),hmem,?_⟩
simp only [quickAllowed_iff,ite_eq_left hallowed.1]
have hi := ih (n-a) a (nextRegions a (liveRegions (k+1) n upper regions))
(a::pref) hh _ _ hnext ys hy
simpa only [List.reverse_cons,List.append_assoc,List.singleton_append] using hi
· simp at hr
lemma checkedRegion_diagram {n M r H W : ℕ} {qs : List (List ℕ)} {t : RowTree}
{regions : List PrefixRegion}
(check : ∀ h ≤ H, checkRegionPrefixes (quickResidual M r qs t) h n W regions [] = true)
(μ : YoungDiagram) (hn : μ.card = n) {A B : List (ℕ × ℕ)}
(hR : (A,B) ∈ regions) (hA : RowBounds μ.transpose A) (hB : RowBounds μ B)
(hH : firstBound n A ≤ H) (hW : firstBound n B ≤ W) :
TreeResidual M r qs t μ.rowLens := by
have hu : ∀ a ∈ μ.rowLens, a ≤ W := by
intro a ha
obtain ⟨i,hi,he⟩ := List.getElem_of_mem ha
rw [← he, YoungDiagram.get_rowLens]
exact (μ.rowLen_anti _ _ (Nat.zero_le _)).trans ((firstBound_sound μ hn hB).trans hW)
have hl : μ.rowLens.length ≤ H := by
rw [YoungDiagram.length_rowLens]
have hh := (firstBound_sound μ.transpose ((Saxl.transpose_card μ).trans hn) hA).trans hH
simpa only [YoungDiagram.rowLen_transpose] using hh
have hmem : μ.rowLens ∈ constrainedRows μ.rowLens.length n W A B := by
apply mem_constrainedRows _ _ _ _ _ _ rfl
· exact (Saxl.card_eq_sum_rowLens μ).symm.trans hn
· exact μ.pos_of_mem_rowLens
· exact hu
· exact μ.rowLens_sorted
· intro p hp
rw [clippedSum_rowLens]
exact (hA p hp).2
· intro p hp
rw [← rowPrefix_eq_take]
exact (hB p hp).2
apply quickResidual_sound
have hh := checkRegionPrefixes_sound (quickResidual M r qs t) μ.rowLens.length n W
regions [] (check _ hl) A B hR μ.rowLens hmem
simpa only [List.reverse_nil,List.nil_append] using hh
end UniversalTensorSquare
end
end OAI
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