| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Graphs.Related2 |
| import OAI.Combinatorics.MatchingCount.Complexity.TreeChoiceCosts |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.ExecutableHierarchy |
| open TreeTyped TreeFraction Polynomial |
|
|
| def level (x : ℕ×Fraction) : ℕ := min x.1 ((x.2.1/x.2.2).size-1) |
| def levelRealizer : Realizer level := composition (pair first |
| (composition (composition (composition second quotient) bitSize) predecessor)) minNat |
| lemma polynomial_level : PolynomialTime levelRealizer := by |
| exact polynomial_composition (polynomial_pair polynomial_first |
| (polynomial_composition (polynomial_composition (polynomial_composition polynomial_second |
| polynomial_quotient) polynomial_bitSize) polynomial_predecessor)) polynomial_minNat |
| lemma findGreatest_lt_size (H s : ℕ) : Nat.findGreatest (fun k=>k<s) H=min H (s-1) := by |
| apply Nat.findGreatest_eq_iff.mpr |
| refine ⟨min_le_left _ _,?_,?_⟩ |
| · intro h |
| have hle:=min_le_right H (s-1) |
| omega |
| · intro k hk hkH hks |
| have hmin:=le_min hkH (show k ≤ s-1 by omega) |
| omega |
| lemma level_val (H : ℕ) (x : Fraction) : level (H,x)=(NumberedHierarchy.level H (val x)).val := by |
| have hp : (fun k=>(2:ℚ)^k≤val x)=(fun k=>k<(x.1/x.2).size) := by |
| funext k |
| apply propext |
| rw [Nat.lt_size] |
| rw [←Rat.natFloor_natCast_div_natCast] |
| have h:=Nat.le_floor_iff' (R:=ℚ) (n:=2^k) (a:=val x) (by positivity) |
| simpa only [val,Nat.cast_pow,Nat.cast_ofNat] using h.symm |
| simp only [NumberedHierarchy.level,level,hp,findGreatest_lt_size] |
| lemma level_repr (H : ℕ) (q : ℚ) (hq : 0≤q) : |
| level (H,TreeFraction.repr q)=(NumberedHierarchy.level H q).val := by |
| rw [level_val,TreeFraction.val_repr hq] |
|
|
| def threshold (x : List Bool×ℕ) : Fraction := (bitDenominator (x.1.take x.2),1) |
| def thresholdRealizer : Realizer threshold := pair (composition takeFast bitDenominatorRealizer) (constant 1) |
| lemma polynomial_threshold : PolynomialTime thresholdRealizer := by |
| unfold thresholdRealizer |
| exact polynomial_pair (polynomial_composition polynomial_takeFast polynomial_bitDenominator) (polynomial_constant 1) |
| lemma threshold_repr (xs : List Bool) (k : ℕ) (hk : k≤xs.length) : |
| threshold (xs,k)=TreeFraction.repr ((2:ℚ)^k) := by |
| simp only [threshold,bitDenominator_eq,List.length_take,min_eq_left hk] |
| have h : val ((2^k,1):Fraction)=(2:ℚ)^k := by simp [val] |
| rw [←h,←normalize_repr] |
| simp [TreeFraction.normalize] |
| end MatchingFPRAS.ExecutableHierarchy |
|
|
| end OAI |
|
|
