| 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 _ |
|
|
| / |
| 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 |
|
|
