File size: 4,141 Bytes
d5b7741 | 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 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactIntegerChoice
import OAI.Combinatorics.ContingencyTables.Machines.ListWordProgram
import OAI.Combinatorics.MatchingCount.Complexity.TreePolynomialSums
namespace OAI
/-! # Binary execution of an exact integer-weighted rejection trial -/
namespace ContingencyTables.ExactIntegerPrograms
open MatchingFPRAS MatchingFPRAS.TreeTyped ProfilePrograms
/-- Fractions share the total integer weight before canonical reduction. -/
def weightsCode (xs : List ℕ) : List TreeFraction.Fraction :=
xs.map (fun w => TreeFraction.normalize (w,xs.sum))
def weightsCodeRealizer : Realizer weightsCode :=
composition (f := fun xs : List ℕ => (xs.sum,xs))
(g := fun z : ℕ×List ℕ => z.2.map (fun w => TreeFraction.normalize (w,z.1)))
(pair listSum identity)
(mapWith (composition (pair second first) TreeFraction.normalizeRealizer))
lemma polynomial_weightsCode : PolynomialTime weightsCodeRealizer :=
polynomial_composition (polynomial_pair polynomial_listSum polynomial_identity)
(polynomial_mapWith (polynomial_composition (polynomial_pair polynomial_second polynomial_first)
TreeFraction.polynomial_normalize))
abbrev Input := List ℕ×List Bool
def choiceInput (z : Input) : List TreeFraction.Fraction×TreeFraction.Fraction :=
(weightsCode z.1,(StackAddition.value z.2,z.1.sum))
def choiceInputRealizer : Realizer choiceInput :=
pair (composition first weightsCodeRealizer)
(pair (composition second bitValue) (composition first listSum))
lemma polynomial_choiceInput : PolynomialTime choiceInputRealizer :=
polynomial_pair (polynomial_composition polynomial_first polynomial_weightsCode)
(polynomial_pair (polynomial_composition polynomial_second polynomial_bitValue)
(polynomial_composition polynomial_first polynomial_listSum))
/-- One trial returns an index or requests another independent word.
All integers are processed in binary; no weight is expanded into copies. -/
def weightedIndex (z : Input) : Option ℕ :=
if StackAddition.value z.2<z.1.sum then some (ExecutableChoice.choose (choiceInput z)) else none
def weightedIndexRealizer : Realizer weightedIndex :=
(choose (composition (pair (composition second bitValue) (composition first listSum)) less)
(composition (composition choiceInputRealizer ExecutableChoice.chooseFast) TreeTyped.some)
(constant none)).congr (by
intro z
simp only [Function.comp_apply,decide_eq_true_eq,weightedIndex])
lemma polynomial_weightedIndex : PolynomialTime weightedIndexRealizer := by
unfold weightedIndexRealizer
apply polynomial_congr
exact polynomial_choose
(polynomial_composition (polynomial_pair
(polynomial_composition polynomial_second polynomial_bitValue)
(polynomial_composition polynomial_first polynomial_listSum)) polynomial_less)
(polynomial_composition
(polynomial_composition polynomial_choiceInput ExecutableChoice.polynomial_chooseFast)
polynomial_some) (polynomial_constant none)
noncomputable section
open ExactIntegerChoice
open scoped BigOperators Classical
lemma weightsCode_ofFn {r : ℕ} (w : Fin r → ℕ) :
weightsCode (List.ofFn w)=List.ofFn (fun i => TreeFraction.repr (normalized w i)) := by
simp only [weightsCode,List.sum_ofFn,List.map_ofFn,TreeFraction.normalize_repr]
rfl
/-- The compiled binary trial is exactly the integer-grid experiment. -/
lemma weightedIndex_ofFn {r k : ℕ} (w : Fin r → ℕ) (h : 0<totalWeight w)
(hQ : totalWeight w≤2^k) (bits : Fin k → Bool) :
weightedIndex (List.ofFn w,List.ofFn bits)=
(weightedTrial w h hQ (FairBits.binaryWordEquiv k bits)).map Fin.val := by
unfold weightedIndex choiceInput weightedTrial intervalTrial
rw [weightsCode_ofFn,bitValue_eq_word,List.sum_ofFn]
change (if (FairBits.binaryWordEquiv k bits).val<totalWeight w then _ else _)=_
split_ifs with hb
· exact congrArg Option.some (ExecutableChoice.choose_eq (normalized w) (normalized_nonnegative w)
(normalized_total w h) h ⟨(FairBits.binaryWordEquiv k bits).val,hb⟩)
· rfl
end
end ContingencyTables.ExactIntegerPrograms
end OAI
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