| import OAI.Combinatorics.ContingencyTables.Machines.ListWordProgram |
| import OAI.Combinatorics.ContingencyTables.Transport.DyadicAcceptance |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| namespace ContingencyTables.DensePrograms |
| open MatchingFPRAS MatchingFPRAS.TreeTyped ProfilePrograms |
|
|
| def dyadicWordAccept (z : List Bool×ℕ) : Bool := |
| decide (StackAddition.value z.1<2^(z.1.length-z.2)) |
|
|
| / |
| no binary integer supplied as a gap can cause an unbounded expansion. -/ |
| def dyadicWordAcceptRealizer : Realizer dyadicWordAccept := by |
| let N : Realizer (fun z : List Bool×ℕ => z.1.length-z.2) := |
| composition (pair (composition first listLength) second) subtract |
| let B := composition (composition (pair first N) takeFast) bitDenominatorRealizer |
| exact (composition (pair (composition first bitValue) B) less).congr (by |
| intro z |
| simp only [Function.comp_apply,bitDenominator_eq,List.length_take,dyadicWordAccept] |
| rw [min_eq_left (Nat.sub_le _ _)]) |
|
|
| theorem polynomial_dyadicWordAccept : PolynomialTime dyadicWordAcceptRealizer := by |
| have hn : PolynomialTime (composition |
| (pair (composition (first (A := List Bool) (B := ℕ)) listLength) second) subtract) := |
| polynomial_composition (polynomial_pair |
| (polynomial_composition polynomial_first polynomial_listLength) polynomial_second) |
| polynomial_subtract |
| have hb := polynomial_composition (polynomial_composition |
| (polynomial_pair polynomial_first hn) polynomial_takeFast) polynomial_bitDenominator |
| unfold dyadicWordAcceptRealizer |
| exact polynomial_congr _ (polynomial_composition (polynomial_pair |
| (polynomial_composition polynomial_first polynomial_bitValue) hb) polynomial_less) |
|
|
| theorem dyadicWordAccept_ofFn (gap k : ℕ) (bits : Fin k → Bool) : |
| dyadicWordAccept (List.ofFn bits,gap)= |
| decide (FairBits.acceptDyadic gap k (FairBits.binaryWordEquiv k bits)) := by |
| simp only [dyadicWordAccept,List.length_ofFn,bitValue_eq_word,FairBits.acceptDyadic] |
|
|
| end ContingencyTables.DensePrograms |
|
|
| end OAI |
|
|
