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8df23f4 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 | import OAI.Computability.BinPacking.PCP.PreprocessingLazyWords
namespace OAI
namespace BinPackingGames.Foundations.PCP.PreprocessingOverlayWords
open PortTables GraphTables PreprocessingOverlayTables
open Complexity
variable {n d e : Nat}
abbrev row := @PreprocessingLazyWords.row
abbrev trueRelation := PreprocessingLazyWords.trueRelation
def oldReverseMap (d e j : Nat) : Nat := (d + e) * (j / d) + j % d
def expanderReverseMap (d e j : Nat) : Nat := (d + e) * (j / e) + d + j % e
def oldRow (G : PortTables.Table n d) (e : Nat) (v : Fin n) (p : Fin d) :
DartRow n (n * (d + e)) :=
⟨v, rowIndex n (d + e) ((rotation G (v, p)).1,
overlayPorts d e (.inl (rotation G (v, p)).2)),
G.relations[rowIndex n d (v, p)]⟩
def expanderRow (d : Nat) (H : ExpanderTables.Table n e) (v : Fin n) (p : Fin e) :
DartRow n (n * (d + e)) :=
⟨v, rowIndex n (d + e) ((ExpanderTables.lookup H (v, p)).1,
overlayPorts d e (.inr (ExpanderTables.lookup H (v, p)).2)), trueRelation⟩
private theorem relation_ext {r s : RelationTable}
(h : ∀ a b, relationAt r a b = relationAt s a b) : r = s := by
apply Vector.ext
intro i hi
simpa only [relationAt, Prod.eta, Equiv.apply_symm_apply, Fin.getElem_fin] using
h (relationIndex.symm ⟨i, hi⟩).1 (relationIndex.symm ⟨i, hi⟩).2
theorem row_overlay_old (G : PortTables.Table n d) (H : ExpanderTables.Table n e)
(v : Fin n) (p : Fin d) :
row (overlay G H) v (overlayPorts d e (.inl p)) = oldRow G e v p := by
have hr : (overlay G H).reverseIndex[rowIndex n (d + e) (v, overlayPorts d e (.inl p))] =
rowIndex n (d + e) ((rotation G (v, p)).1,
overlayPorts d e (.inl (rotation G (v, p)).2)) := by
rw [← rowIndex_rotation, overlay_rotation_old]
have hp : (overlay G H).relations[rowIndex n (d + e) (v, overlayPorts d e (.inl p))] =
G.relations[rowIndex n d (v, p)] := by
apply relation_ext
exact overlay_accepts_old G H v p
exact congrArg₂ (DartRow.mk v) hr hp
theorem row_overlay_expander (G : PortTables.Table n d) (H : ExpanderTables.Table n e)
(v : Fin n) (p : Fin e) :
row (overlay G H) v (overlayPorts d e (.inr p)) = expanderRow d H v p := by
have hr : (overlay G H).reverseIndex[rowIndex n (d + e) (v, overlayPorts d e (.inr p))] =
rowIndex n (d + e) ((ExpanderTables.lookup H (v, p)).1,
overlayPorts d e (.inr (ExpanderTables.lookup H (v, p)).2)) := by
rw [← rowIndex_rotation, overlay_rotation_expander]
have hp : (overlay G H).relations[rowIndex n (d + e) (v, overlayPorts d e (.inr p))] =
trueRelation := by
apply relation_ext
intro a b
simpa only [PortTables.accepts, relationAt, trueRelation,
PreprocessingLazyWords.trueRelation, Fin.getElem_fin, Vector.getElem_replicate] using
overlay_accepts_expander G H v p a b
exact congrArg₂ (DartRow.mk v) hr hp
theorem oldRow_reverse_val (G : PortTables.Table n d) (v : Fin n) (p : Fin d) :
(oldRow G e v p).reverseIndex.val = oldReverseMap d e (row G v p).reverseIndex.val := by
simp only [oldRow, rowIndex_val, overlayPorts_inl_val, oldReverseMap, row,
PreprocessingLazyWords.row]
change (G.reverseIndex[rowIndex n d (v, p)]).val % d +
(d + e) * ((G.reverseIndex[rowIndex n d (v, p)]).val / d) = _
omega
theorem expanderRow_reverse_val (H : ExpanderTables.Table n e) (v : Fin n) (p : Fin e) :
(expanderRow d H v p).reverseIndex.val =
expanderReverseMap d e (ExpanderTables.reverseIndex H (ExpanderTables.rowIndex n e (v, p))).val := by
simp only [expanderRow, rowIndex_val, overlayPorts_inr_val, expanderReverseMap]
change d + (ExpanderTables.reverseIndex H (ExpanderTables.rowIndex n e (v, p))).val % e +
(d + e) * ((ExpanderTables.reverseIndex H (ExpanderTables.rowIndex n e (v, p))).val / e) = _
omega
theorem oldRow_words (G : PortTables.Table n d) (v : Fin n) (p : Fin d) :
rowWords (oldRow G e v p) =
[v.val, oldReverseMap d e (row G v p).reverseIndex.val] ++
relationWords (row G v p).relation := by
rw [rowWords, oldRow_reverse_val]
rfl
theorem expanderRow_words (H : ExpanderTables.Table n e) (v : Fin n) (p : Fin e) :
rowWords (expanderRow d H v p) =
[v.val, expanderReverseMap d e
(ExpanderTables.reverseIndex H (ExpanderTables.rowIndex n e (v, p))).val] ++
List.replicate 4096 1 := by
rw [rowWords, expanderRow_reverse_val]
exact congrArg (fun rs => [v.val, expanderReverseMap d e
(ExpanderTables.reverseIndex H (ExpanderTables.rowIndex n e (v, p))).val] ++ rs)
PreprocessingLazyWords.relationWords_true
theorem ofFn_overlayPorts {α : Type*} (f : Fin (d + e) → α) :
List.ofFn f = List.ofFn (fun p => f (overlayPorts d e (.inl p))) ++
List.ofFn (fun p => f (overlayPorts d e (.inr p))) := by
rw [List.ofFn_add]
rfl
def vertexRows (G : PortTables.Table n d) (H : ExpanderTables.Table n e) (v : Fin n) :
List (DartRow n (n * (d + e))) :=
List.ofFn (oldRow G e v) ++ List.ofFn (expanderRow d H v)
theorem vertexRows_length (G : PortTables.Table n d) (H : ExpanderTables.Table n e) (v : Fin n) :
(vertexRows G H v).length = d + e := by
simp only [vertexRows, List.length_append, List.length_ofFn]
theorem flatRows_overlay (G : PortTables.Table n d) (H : ExpanderTables.Table n e) :
(flatRows (overlay G H)).toList = (List.ofFn (vertexRows G H)).flatten := by
rw [PreprocessingLazyWords.flatRows_list]
apply congrArg List.flatten
congr 1
funext v
rw [ofFn_overlayPorts]
simp only [row_overlay_old, row_overlay_expander]
rfl
theorem vertexRows_words (G : PortTables.Table n d) (H : ExpanderTables.Table n e) (v : Fin n) :
(vertexRows G H v).flatMap rowWords =
(List.ofFn fun p : Fin d =>
[v.val, oldReverseMap d e (row G v p).reverseIndex.val] ++
relationWords (row G v p).relation).flatten ++
(List.ofFn fun p : Fin e =>
[v.val, expanderReverseMap d e
(ExpanderTables.reverseIndex H (ExpanderTables.rowIndex n e (v, p))).val] ++
List.replicate 4096 1).flatten := by
have ho : rowWords ∘ oldRow G e v = fun p : Fin d =>
[v.val, oldReverseMap d e (row G v p).reverseIndex.val] ++
relationWords (row G v p).relation := by
funext p
exact oldRow_words G v p
have he : rowWords ∘ expanderRow d H v = fun p : Fin e =>
[v.val, expanderReverseMap d e
(ExpanderTables.reverseIndex H (ExpanderTables.rowIndex n e (v, p))).val] ++
List.replicate 4096 1 := by
funext p
exact expanderRow_words H v p
simp only [vertexRows, List.flatMap_def, List.map_append, List.map_ofFn, List.flatten_append]
rw [ho, he]
theorem tableWords_overlay (G : PortTables.Table n d) (H : ExpanderTables.Table n e) :
PortTables.tableWords (overlay G H) =
[n, n * (d + e)] ++ (List.ofFn (vertexRows G H)).flatten.flatMap rowWords := by
rw [PortTables.tableWords_eq, flatRows_overlay]
theorem tableBits_overlay (G : PortTables.Table n d) (H : ExpanderTables.Table n e) :
PortTables.tableBits (overlay G H) = encodeWords
([n, n * (d + e)] ++ (List.ofFn (vertexRows G H)).flatten.flatMap rowWords) := by
change encodeWords (PortTables.tableWords (overlay G H)) = _
rw [tableWords_overlay]
theorem rotationWords_vertices (H : ExpanderTables.Table n e) :
ExpanderTableWords.rotationWords H =
(List.ofFn fun v : Fin n => List.ofFn fun p : Fin e =>
(ExpanderTables.reverseIndex H (ExpanderTables.rowIndex n e (v, p))).val).flatten := by
have hflat : ExpanderTableWords.rotationWords H =
List.ofFn (fun i : Fin (n * e) => (ExpanderTables.reverseIndex H i).val) := by
apply List.ext_getElem
· simp [ExpanderTableWords.rotationWords_length]
· intro i hi hj
rw [ExpanderTableWords.rotationWords_getElem H i (by simpa using hi)]
simp only [List.getElem_ofFn]
rw [hflat, PreprocessingLazyWords.ofFn_rowIndex]
rfl
end BinPackingGames.Foundations.PCP.PreprocessingOverlayWords
end OAI
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