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import OAI.Computability.MaxCut.PCP.PreprocessingPaddingOffsets
namespace OAI
namespace MaxCutGames.Foundations.PCP.PreprocessingInternalRows
open DegreeReplacement PreprocessingCloudIndex PreprocessingRegularTables
variable (t : GraphTables.Table) (padding : Fin t.vertices → Nat) {q : Nat}
variable (tables : ∀ v, ExpanderTables.Table (cloudSize t v + padding v) q)
/-- The one actual cloud-table lookup used by this internal row. -/
def localStep (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
Fin (cloudSize t v + padding v) × Fin q :=
ExpanderTables.lookup (tables v) (paddedCloudRank t padding v x, p)
/-- Decode the returned cloud index into its actual global padded dart. -/
def destination (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
PaddedCloud t padding v :=
paddedCloudSelect t padding v (localStep t padding tables v x p).1
def rowIndex (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
Fin (vertexCount t padding * (q + 1)) :=
PortTables.rowIndex (vertexCount t padding) (q + 1)
(vertexOrder t padding x.val, portOrder q (.inl p))
/-- This is a row read from the actual serialized output table. -/
def row (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
GraphTables.DartRow (vertexCount t padding) (vertexCount t padding * (q + 1)) :=
(PortTables.flatRows (ofCloudTables t padding tables))[rowIndex t padding v x p]
/-- The fixed, complete 64-label equality predicate. -/
def equalityRelation : GraphTables.RelationTable :=
GraphTables.relationOf (fun a b => decide (a = b))
@[simp] theorem equalityRelation_apply (a b : GraphTables.Label) :
GraphTables.relationAt equalityRelation a b = decide (a = b) :=
GraphTables.relationAt_relationOf _ a b
/-- The reindexed local cloud graph is exactly the stored local lookup. -/
theorem cloud_rotation (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
(cloudGraphs t padding tables v).rot (x, p) =
(destination t padding tables v x p, (localStep t padding tables v x p).2) := by
have h := GraphTransport.reindex_rot (ExpanderTables.graph (tables v))
(paddedCloudEquiv t padding v).symm (Equiv.refl _)
(paddedCloudRank t padding v x) p
simpa only [cloudGraphs, ExpanderTables.graph_rot, Equiv.refl_apply,
paddedCloudRank, paddedCloudSelect, Equiv.symm_apply_apply, destination, localStep] using h
/-- The actual global replacement rotation uses that same lookup and select. -/
theorem source_rotation (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
(sourcePortGraph t padding tables).rot (x.val, .inl p) =
((destination t padding tables v x p).val,
.inl (localStep t padding tables v x p).2) := by
rcases x with ⟨x, hx⟩
subst v
exact congrArg
(fun z : PaddedCloud t padding
((paddedGraph (GraphTables.semantics t) (fun v => Fin (padding v))).tail x) × Fin q =>
(z.1.val, (Sum.inl z.2 : Fin q ⊕ Unit)))
(cloud_rotation t padding tables _ ⟨x, rfl⟩ p)
/-- Full output rotation, retaining the actual return port. -/
theorem rotation_internal (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
PortTables.rotation (ofCloudTables t padding tables)
(vertexOrder t padding x.val, portOrder q (.inl p)) =
(vertexOrder t padding (destination t padding tables v x p).val,
portOrder q (.inl (localStep t padding tables v x p).2)) := by
rw [rotation_ofCloudTables, source_rotation t padding tables v x p]
/-- The stored global reverse index is the row-major encoding of the destination. -/
theorem reverseIndex_internal (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
((ofCloudTables t padding tables).reverseIndex[rowIndex t padding v x p]).val =
(q + 1) * (vertexOrder t padding (destination t padding tables v x p).val).val +
(localStep t padding tables v x p).2.val := by
unfold rowIndex
rw [← PortTables.rowIndex_rotation, rotation_internal t padding tables v x p,
PortTables.rowIndex_val, portOrder_internal]
dsimp only
omega
private theorem relation_ext_inline_PreprocessingInternalRows {r s : GraphTables.RelationTable}
(h : ∀ a b, GraphTables.relationAt r a b = GraphTables.relationAt s a b) : r = s := by
apply Vector.ext
intro i hi
simpa only [GraphTables.relationAt, Prod.eta, Equiv.apply_symm_apply,
Fin.getElem_fin] using
h (GraphTables.relationIndex.symm ⟨i, hi⟩).1
(GraphTables.relationIndex.symm ⟨i, hi⟩).2
/-- Equality is the entire stored relation vector, not only one tested entry. -/
theorem relations_internal (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
(ofCloudTables t padding tables).relations[rowIndex t padding v x p] = equalityRelation := by
apply relation_ext_inline_PreprocessingInternalRows
intro a b
rw [equalityRelation_apply]
exact accepts_internal t padding tables x.val p a b
@[simp] theorem row_tail (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
(row t padding tables v x p).tail.val = (vertexOrder t padding x.val).val := by
simp only [row, rowIndex, PortTables.flatRows, Vector.getElem_ofFn,
Fin.getElem_fin, Fin.eta, Equiv.symm_apply_apply]
theorem row_reverse (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
(row t padding tables v x p).reverseIndex.val =
(q + 1) * (vertexOrder t padding (destination t padding tables v x p).val).val +
(localStep t padding tables v x p).2.val := by
simpa only [row, PortTables.flatRows, Vector.getElem_ofFn, Fin.getElem_fin] using
reverseIndex_internal t padding tables v x p
@[simp] theorem row_relation (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
(row t padding tables v x p).relation = equalityRelation := by
simpa only [row, PortTables.flatRows, Vector.getElem_ofFn, Fin.getElem_fin] using
relations_internal t padding tables v x p
@[simp] theorem localStep_original (v : Fin t.vertices) (x : Cloud (GraphTables.semantics t) v)
(p : Fin q) :
localStep t padding tables v (paddedOld t padding v x) p =
ExpanderTables.lookup (tables v) (Fin.castAdd (padding v) (cloudRank t v x), p) := rfl
@[simp] theorem localStep_dummy (v : Fin t.vertices) (j : Fin (padding v)) (p : Fin q) :
localStep t padding tables v (paddedNew t padding v j) p =
ExpanderTables.lookup (tables v) (Fin.natAdd (cloudSize t v) j, p) := rfl
@[simp] theorem row_tail_original (v : Fin t.vertices) (x : Cloud (GraphTables.semantics t) v)
(p : Fin q) :
(row t padding tables v (paddedOld t padding v x) p).tail.val = x.val.val := by
rw [row_tail]
rfl
@[simp] theorem row_tail_dummy (v : Fin t.vertices) (j : Fin (padding v)) (p : Fin q) :
(row t padding tables v (paddedNew t padding v j) p).tail.val =
t.darts + PreprocessingPaddingOffsets.offset padding v.val + j.val := by
rw [row_tail]
exact PreprocessingPaddingOffsets.vertexOrder_dummy_val t padding v j
/-- The global target number computed from an arbitrary returned cloud index. -/
def selectedIndex (v : Fin t.vertices) (i : Fin (cloudSize t v + padding v)) : Nat :=
if hi : i.val < cloudSize t v then
(cloudSelect t v ⟨i.val, hi⟩).val.val
else t.darts + PreprocessingPaddingOffsets.offset padding v.val + (i.val - cloudSize t v)
/-- Rank/select destination decoding agrees with the prefix-offset formula. -/
theorem selectedIndex_eq (v : Fin t.vertices) (i : Fin (cloudSize t v + padding v)) :
(vertexOrder t padding (paddedCloudSelect t padding v i).val).val =
selectedIndex t padding v i := by
unfold selectedIndex
split
· rename_i hi
have he : i = Fin.castAdd (padding v) (⟨i.val, hi⟩ : Fin (cloudSize t v)) := Fin.ext rfl
refine (congrArg (fun j =>
(vertexOrder t padding (paddedCloudSelect t padding v j).val).val) he).trans ?_
rw [paddedCloudSelect_castAdd]
rfl
· rename_i hi
have hj : i.val - cloudSize t v < padding v := by have h := i.isLt; omega
have he : i = Fin.natAdd (cloudSize t v) (⟨i.val - cloudSize t v, hj⟩ : Fin (padding v)) := by
apply Fin.ext
dsimp only [Fin.val_natAdd]
omega
refine (congrArg (fun j =>
(vertexOrder t padding (paddedCloudSelect t padding v j).val).val) he).trans ?_
rw [paddedCloudSelect_natAdd]
change (vertexOrder t padding
(Sum.inr ⟨v, (⟨i.val - cloudSize t v, hj⟩ : Fin (padding v))⟩)).val = _
exact PreprocessingPaddingOffsets.vertexOrder_dummy_val t padding v _
/-- Old target indices select the original dart from its actual cloud list. -/
theorem selectedIndex_old (v : Fin t.vertices) (i : Fin (cloudSize t v + padding v))
(hi : i.val < cloudSize t v) :
selectedIndex t padding v i = (cloudSelect t v ⟨i.val, hi⟩).val.val := by
simp only [selectedIndex, dite_eq_left hi]
/-- Dummy target indices use the checked prefix offset and local index. -/
theorem selectedIndex_dummy (v : Fin t.vertices) (i : Fin (cloudSize t v + padding v))
(hi : cloudSize t v ≤ i.val) :
selectedIndex t padding v i =
t.darts + PreprocessingPaddingOffsets.offset padding v.val + (i.val - cloudSize t v) := by
simp only [selectedIndex, dite_eq_right (Nat.not_lt_of_ge hi)]
/-- Closed arithmetic reverse field for the row machine. -/
theorem row_reverse_selectedIndex (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
(row t padding tables v x p).reverseIndex.val =
(q + 1) * selectedIndex t padding v (localStep t padding tables v x p).1 +
(localStep t padding tables v x p).2.val := by
rw [row_reverse]
unfold destination
rw [selectedIndex_eq]
/-- All 4098 words of the actual internal row, in the existing codec. -/
theorem rowWords_eq (v : Fin t.vertices) (x : PaddedCloud t padding v) (p : Fin q) :
GraphTables.rowWords (row t padding tables v x p) =
[(vertexOrder t padding x.val).val,
(q + 1) * selectedIndex t padding v (localStep t padding tables v x p).1 +
(localStep t padding tables v x p).2.val] ++
GraphTables.relationWords equalityRelation := by
unfold GraphTables.rowWords
rw [row_tail, row_reverse_selectedIndex, row_relation]
end MaxCutGames.Foundations.PCP.PreprocessingInternalRows
end OAI
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