File size: 2,829 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 | import OAI.Combinatorics.ContingencyTables.Machines.CompletionEnumerationProgram
namespace OAI
/-! # Enumerating only the short proposal word -/
namespace ContingencyTables.DensePrograms
open MatchingFPRAS MatchingFPRAS.TreeTyped
open scoped BigOperators Classical
def digitBool (a : ℕ) : Bool := decide (a=1)
def digitBoolRealizer : Realizer digitBool := composition (pair identity (constant 1)) natEqual
lemma polynomial_digitBool : PolynomialTime digitBoolRealizer :=
polynomial_composition (polynomial_pair polynomial_identity (polynomial_constant 1))
polynomial_natEqual
def booleanWords (xs : List Bool) : List (List Bool) :=
(gridWords (xs,2)).map (List.map digitBool)
def booleanWordsRealizer : Realizer booleanWords :=
composition (composition (pair identity (constant 2)) gridWordsRealizer)
(map (map digitBoolRealizer))
lemma work_booleanWords : WorkBound booleanWordsRealizer
(fun xs => gridStateCount (xs,2)+weight xs) := by
have hi : WorkBound (pair (identity (A := List Bool)) (constant 2))
(fun xs => gridStateCount (xs,2)+weight xs) :=
work_of_polynomial (polynomial_pair polynomial_identity (polynomial_constant 2))
Polynomial.X (fun xs => by simp)
have hg : WorkBound (composition (pair (identity (A := List Bool)) (constant 2))
gridWordsRealizer) (fun xs => gridStateCount (xs,2)+weight xs) := by
apply work_composition hi work_gridWords (Polynomial.X+10)
intro xs
simp only [Polynomial.eval_add,Polynomial.eval_X,Polynomial.eval_ofNat,weight_prod,id_eq]
have hc : weight (2:ℕ)≤9 := by simpa using weight_nat_numeric 2
omega
unfold booleanWordsRealizer
exact work_then_polynomial hg (polynomial_map (polynomial_map polynomial_digitBool))
def booleanEnumeration (s : ℕ) : Fin (2^s) ≃ (Fin s → Bool) :=
finFunctionFinEquiv.symm.trans (Equiv.piCongrRight (fun _ => finTwoEquiv))
lemma booleanWords_of_length (xs : List Bool) (s : ℕ) (hs : xs.length=s) :
booleanWords xs=List.ofFn (fun k => List.ofFn (booleanEnumeration s k)) := by
unfold booleanWords
rw [gridWords_finFunctions xs 1 s hs,List.map_ofFn]
apply congrArg List.ofFn
funext k
dsimp only [Function.comp_apply]
rw [List.map_ofFn]
apply congrArg List.ofFn
funext i
change digitBool ((finFunctionFinEquiv.symm k) i).val=
finTwoEquiv ((finFunctionFinEquiv.symm k) i)
have hb (a : Fin 2) : digitBool a.val=finTwoEquiv a := by
fin_cases a <;> rfl
exact hb _
lemma sum_booleanWords (xs : List Bool) (s : ℕ) (hs : xs.length=s)
(w : List Bool → ℕ) :
((booleanWords xs).map w).sum=∑ bits : Fin s → Bool,w (List.ofFn bits) := by
rw [booleanWords_of_length xs s hs,List.map_ofFn,List.sum_ofFn]
exact (booleanEnumeration s).sum_comp (fun bits => w (List.ofFn bits))
end ContingencyTables.DensePrograms
end OAI
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