File size: 3,694 Bytes
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 | import OAI.Computability.BinPacking.PCP.PoweringTableSemantics
namespace OAI
namespace BinPackingGames.Foundations.PCP.PoweringRowHeaderSemantics
open PoweringWalks PoweringEnumeration
open BinPackingGames.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
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 BinPackingGames.Foundations.PCP.PoweringRowHeaderSemantics
end OAI
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