File size: 3,077 Bytes
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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | import OAI.Combinatorics.ContingencyTables.Dense.DenseAxisSemantics
import OAI.Combinatorics.ContingencyTables.Dense.DenseAxisBits
namespace OAI
/-! # Binary-word decoding for a dense coordinate proposal -/
namespace ContingencyTables.DensePrograms
open MatchingFPRAS MatchingFPRAS.TreeTyped ProfilePrograms GridBoundary
abbrev AxisWordData := (ℕ×List ℕ)×List Bool
def axisWordStep (z : AxisWordData) : Option (List ℕ) :=
axisListStep ((z.1,StackAddition.value z.2/2),decide (StackAddition.value z.2%2=1))
def axisWordStepRealizer : Realizer axisWordStep := by
let V : Realizer (fun z : AxisWordData => StackAddition.value z.2) := composition second bitValue
let I := composition (pair V (constant 2)) quotient
let S := composition (pair (composition (pair V (constant 2)) remainder) (constant 1)) natEqual
exact composition (pair (pair first I) S) axisListStepRealizer
theorem polynomial_axisWordStep : PolynomialTime axisWordStepRealizer := by
have hv : PolynomialTime (composition
(second (A := ℕ×List ℕ) (B := List Bool)) bitValue) :=
polynomial_composition polynomial_second polynomial_bitValue
have hi := polynomial_composition (polynomial_pair hv (polynomial_constant 2)) polynomial_quotient
have hs := polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_pair hv (polynomial_constant 2)) polynomial_remainder)
(polynomial_constant 1)) polynomial_natEqual
unfold axisWordStepRealizer
exact polynomial_composition (polynomial_pair (polynomial_pair polynomial_first hi) hs)
polynomial_axisListStep
lemma directionCode_first {q : ℕ} (v : Fin (q*2)) :
((directionCodeEquiv q).symm v).1.val=v.val/2 := rfl
lemma directionCode_second {q : ℕ} (v : Fin (q*2)) :
((directionCodeEquiv q).symm v).2=decide (v.val%2=1) := by
have hf (b : Fin 2) : finTwoEquiv b=decide (b.val=1) := by
fin_cases b <;> rfl
exact hf _
/-- A binary index gives exactly the signed coordinate code used by
the finite dense-chain transition; unused indices still reject. -/
theorem axisWordStep_eq {q K k : ℕ} (a : Grid (I := Fin q) K) (bits : Fin k → Bool) :
axisWordStep ((K,gridNatCode a),List.ofFn bits)=
(axisCodeStep a (FairBits.binaryWordEquiv k bits)).map gridNatCode := by
rw [axisWordStep,bitValue_eq_word]
let v := FairBits.binaryWordEquiv k bits
change axisListStep (((K,gridNatCode a),v.val/2),decide (v.val%2=1))=
(axisCodeStep a v).map gridNatCode
by_cases hv : v.val<q*2
· let u : Fin (q*2) := ⟨v.val,hv⟩
let dir := (directionCodeEquiv q).symm u
have hfirst : dir.1.val=v.val/2 := directionCode_first u
have hsecond : dir.2=decide (v.val%2=1) := directionCode_second u
rw [←hfirst,←hsecond,axisListStep_eq]
simp only [axisCodeStep,dite_eq_left hv]
rfl
· have hi : ¬v.val/2<q := by omega
have hlen : (gridNatCode a).length=q := by simp only [gridNatCode,List.length_ofFn]
simp only [axisListStep,hlen,ite_eq_right hi,axisCodeStep,dite_eq_right hv,Option.map_none]
end ContingencyTables.DensePrograms
end OAI
|