File size: 2,864 Bytes
ea76aac | 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 | import Mathlib
import OAI.Computability.MinUncut.PCP.PreprocessingTableSpectral
namespace OAI
section
namespace MinUncutGames.Foundations.PCP.PortTables
variable {n d : Nat}
def HasPortTails (rows : GraphTables.Rows n (n * d)) : Prop :=
∀ i : Fin (n * d), rows[i].tail = ((rowIndex n d).symm i).1
instance (rows : GraphTables.Rows n (n * d)) : Decidable (HasPortTails rows) := by
unfold HasPortTails
infer_instance
@[simp] theorem flatRows_hasPortTails (table : Table n d) : HasPortTails (flatRows table) := by
intro i
simp [flatRows]
def ofRows (rows : GraphTables.Rows n (n * d)) (valid : GraphTables.Valid rows) : Table n d where
reverseIndex := Vector.ofFn (fun i => rows[i].reverseIndex)
relations := Vector.ofFn (fun i => rows[i].relation)
involutive := by
intro i
simp only [Fin.getElem_fin, Vector.getElem_ofFn]
exact valid.1 i
transpose := by
intro i a b
simp only [Fin.getElem_fin, Vector.getElem_ofFn]
exact valid.2 i a b
private theorem table_ext {table table' : Table n d}
(hr : table.reverseIndex = table'.reverseIndex)
(hp : table.relations = table'.relations) : table = table' := by
cases table
cases table'
cases hr
cases hp
rfl
@[simp] theorem ofRows_flatRows (table : Table n d) :
ofRows (flatRows table) (flatRows_valid table) = table := by
apply table_ext
· apply Vector.ext
intro i hi
simp [ofRows, flatRows]
· apply Vector.ext
intro i hi
simp [ofRows, flatRows]
def extractInput (ports : Nat) : GraphTables.Table → Option (Input ports)
| ⟨vertices, darts, rows, valid⟩ =>
if count_ok : darts = vertices * ports then
let rows' : GraphTables.Rows vertices (vertices * ports) := count_ok ▸ rows
let valid' : GraphTables.Valid rows' := by
cases count_ok
exact valid
if HasPortTails rows' then
some ⟨vertices, ofRows rows' valid'⟩
else none
else none
@[simp] theorem extractInput_graphTable (table : Table n d) :
extractInput d (graphTable table) = some (⟨n, table⟩ : Input d) := by
simp [extractInput, graphTable]
def decodeInputBits (ports : Nat) (bits : List Bool) : Option (Input ports) :=
GraphTables.decodeTableBits bits >>= extractInput ports
@[simp] theorem decodeInputBits_inputBits {ports : Nat} (input : Input ports) :
decodeInputBits ports (inputBits input) = some input := by
rcases input with ⟨vertices, table⟩
simp [decodeInputBits, inputBits, tableBits]
def encoding (ports : Nat) : Computability.Encoding (Input ports) Bool where
encode := inputBits
decode := decodeInputBits ports
decode_encode := decodeInputBits_inputBits
theorem inputBits_injective (ports : Nat) : Function.Injective (@inputBits ports) :=
(encoding ports).encode_injective
end MinUncutGames.Foundations.PCP.PortTables
end
end OAI
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