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97c3405 | 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 | import OAI.Combinatorics.ContingencyTables.Machines.ListWordProgram
import OAI.Combinatorics.ContingencyTables.Transport.DyadicAcceptance
namespace OAI
/-! # Polynomial binary evaluation of the finite Metropolis coin -/
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))
/-- The power of two is built from a prefix of the supplied word. Thus
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
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