| import Mathlib |
| import OAI.Computability.MaxCut.PCP.PoweringOpinionTables |
|
|
| namespace OAI |
|
|
| /-! Exact tail and reverse-index fields of the actual powering-table rows. |
| The reverse address is affine in the live starting vertex; the coefficient |
| and offset depend only on the fixed degree, word, and orientation. -/ |
|
|
| namespace MaxCutGames.Foundations.PCP.PoweringRowHeaderSemantics |
|
|
| open PoweringWalks PoweringEnumeration |
| open MaxCutGames.Foundations.Complexity |
|
|
| variable {vertices d : Nat} |
|
|
| def blockSize (d n : Nat) : Nat := 2 * d ^ (n + 1) |
|
|
| def reverseOffset (d n : Nat) (ports : Fin (n + 1) → Fin d) (direction : Bool) : Nat := |
| 2 * (wordEquiv d (n + 1) ports).val + (if direction then 0 else 1) |
|
|
| def reverseValue (d n : Nat) (vertex : Fin vertices) |
| (ports : Fin (n + 1) → Fin d) (direction : Bool) : Nat := |
| blockSize d n * vertex.val + reverseOffset d n ports direction |
|
|
| def tailVertex (input : PortTables.Table vertices d) (n : Nat) (vertex : Fin vertices) |
| (ports : Fin (n + 1) → Fin d) (direction : Bool) : Fin vertices := |
| if direction then wordEnd (PortTables.portGraph input) (n + 1) vertex ports else vertex |
|
|
| def headerWords (input : PortTables.Table vertices d) (n : Nat) (vertex : Fin vertices) |
| (ports : Fin (n + 1) → Fin d) (direction : Bool) : List Nat := |
| [(tailVertex input n vertex ports direction).val, reverseValue d n vertex ports direction] |
|
|
| / |
| theorem table_tail (input : PortTables.Table vertices d) (n : Nat) (vertex : Fin vertices) |
| (ports : Fin (n + 1) → Fin d) (direction : Bool) : |
| (PoweringTables.table input n).rows[encodeDart vertices d n (direction, vertex, ports)].tail = |
| tailVertex input n vertex ports direction := by |
| change (GenericGraphTables.semantics (PoweringTables.table input n)).tail |
| (encodeDart vertices d n (direction, vertex, ports)) = _ |
| rw [PoweringTables.semantics_table] |
| change (PoweringTables.mathematicalGraph input n).tail |
| ((dartEquiv vertices d n).symm (dartEquiv vertices d n (direction, vertex, ports))) = _ |
| rw [Equiv.symm_apply_apply] |
| rfl |
|
|
| / |
| theorem table_reverseValue (input : PortTables.Table vertices d) (n : Nat) |
| (vertex : Fin vertices) (ports : Fin (n + 1) → Fin d) (direction : Bool) : |
| (PoweringTables.table input n).rows[encodeDart vertices d n |
| (direction, vertex, ports)].reverseIndex.val = reverseValue d n vertex ports direction := by |
| change (GenericGraphTables.reverseAt (PoweringTables.table input n).rows |
| (encodeDart vertices d n (direction, vertex, ports))).val = _ |
| rw [PoweringTables.table_reverse, encodeDart, dartEquiv_val] |
| cases direction <;> |
| simp [reverseValue, blockSize, reverseOffset, orientationEquiv, Nat.add_assoc] <;> decide |
|
|
| theorem headerWords_eq (input : PortTables.Table vertices d) (n : Nat) (vertex : Fin vertices) |
| (ports : Fin (n + 1) → Fin d) (direction : Bool) : |
| headerWords input n vertex ports direction = |
| [(PoweringTables.table input n).rows[encodeDart vertices d n |
| (direction, vertex, ports)].tail.val, |
| (PoweringTables.table input n).rows[encodeDart vertices d n |
| (direction, vertex, ports)].reverseIndex.val] := by |
| rw [table_tail, table_reverseValue] |
| rfl |
|
|
| / |
| serialized row, preserving the rest of the output tape. -/ |
| theorem header_relation_bits (input : PortTables.Table vertices d) (n : Nat) |
| (vertex : Fin vertices) (ports : Fin (n + 1) → Fin d) (direction : Bool) |
| (suffix : List Bool) : |
| encodeWords (headerWords input n vertex ports direction) ++ |
| (encodeWords (GenericGraphTables.relationWords |
| (PoweringTables.table input n).rows[encodeDart vertices d n |
| (direction, vertex, ports)].relation) ++ suffix) = |
| encodeWords (GenericGraphTables.rowWords |
| (PoweringTables.table input n).rows[encodeDart vertices d n |
| (direction, vertex, ports)]) ++ suffix := by |
| rw [headerWords_eq, ← List.append_assoc, ← encodeWords_append] |
| rfl |
|
|
| end MaxCutGames.Foundations.PCP.PoweringRowHeaderSemantics |
|
|
| end OAI |
|
|
