File size: 4,199 Bytes
e5a97c6 | 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 89 90 91 92 93 94 95 96 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.TreeLabelTable
import OAI.Combinatorics.MatchingCount.Algorithm.OnlineScheduleNat
namespace OAI
namespace MatchingFPRAS.TreeFraction
open TreeTyped Polynomial
def natCeil (x : Fraction) : ℕ := (x.1+x.2-1)/x.2
def natCeilRealizer : Realizer natCeil :=
(composition (pair (composition (composition (pair first second) TreeTyped.add) predecessor) second) quotient).congr (fun _=>rfl)
lemma polynomial_natCeil : PolynomialTime natCeilRealizer := by
unfold natCeilRealizer; apply polynomial_congr
have := polynomial_predecessor
have := polynomial_quotient
tree_cost
lemma natCeil_eq (x : Fraction) : natCeil x=⌈val x⌉₊ := by
by_cases hd : x.2=0
· simp [natCeil,val,hd]
have hd' : (0:ℚ)<x.2 := by exact_mod_cast (Nat.pos_of_ne_zero hd)
have h (m : ℕ) : ⌈val x⌉₊ ≤ m ↔ natCeil x ≤ m := by
rw [Nat.ceil_le]
unfold val natCeil
rw [div_le_iff₀ hd',←Nat.cast_mul,Nat.cast_le,Nat.div_le_iff_le_mul (Nat.pos_of_ne_zero hd)]
omega
exact Nat.le_antisymm ((h _).mp le_rfl) ((h _).mpr le_rfl)
lemma natCeil_repr (q : ℚ) (hq : 0≤q) : natCeil (repr q)=⌈q⌉₊ := by
rw [natCeil_eq,val_repr hq]
def inverse (x : Fraction) : Fraction := normalize (x.2,x.1)
def inverseRealizer : Realizer inverse := composition (pair second first) normalizeRealizer
lemma polynomial_inverse : PolynomialTime inverseRealizer := by
unfold inverseRealizer; exact polynomial_composition (polynomial_pair polynomial_second polynomial_first) polynomial_normalize
lemma inverse_repr (q : ℚ) (hq : 0≤q) : inverse (repr q)=repr q⁻¹ := by
unfold inverse
rw [normalize_repr]
congr 1
change (q.den:ℚ)/q.num.natAbs=q⁻¹
calc
_ = (val (repr q))⁻¹ := by simp [val,repr]
_ = q⁻¹ := by rw [val_repr hq]
end MatchingFPRAS.TreeFraction
namespace MatchingFPRAS.OnlineSchedule
open TreeTyped TreeFraction Polynomial
abbrev RootInput := ℕ×List Bool
def rootCandidate (x : RootInput×ℕ) : Bool := decide (x.1.1<(x.2+1)^2)
def rootCandidateRealizer : Realizer rootCandidate :=
(composition (pair first.fst (composition (composition second successor) (fixedPower 2))) TreeTyped.less).congr (fun _=>rfl)
lemma polynomial_rootCandidate : PolynomialTime rootCandidateRealizer := by
unfold rootCandidateRealizer Realizer.fst; apply polynomial_congr
have := polynomial_fixedPower 2
have := polynomial_successor
tree_cost
def findRoot (x : RootInput) : ℕ :=
listMin ((List.range (x.2.length+1)).filter (fun i=>rootCandidate (x,i)),x.2.length)
def findRootRealizer : Realizer findRoot := by
let B : Realizer (fun x : RootInput=>x.2.length) := composition second listLength
let R := composition (composition B successor) range
let F := composition (pair identity R) (filterWith rootCandidateRealizer)
exact (composition (pair F B) listMinRealizer).congr (fun _=>rfl)
lemma polynomial_findRoot : PolynomialTime findRootRealizer := by
unfold findRootRealizer
dsimp only
apply polynomial_congr
apply polynomial_composition _ polynomial_listMin
apply polynomial_pair _ (polynomial_composition polynomial_second polynomial_listLength)
apply polynomial_composition _ (polynomial_filterWithList polynomial_rootCandidate)
apply polynomial_pair polynomial_identity
apply polynomial_boundedRange
(polynomial_composition (polynomial_composition polynomial_second polynomial_listLength) polynomial_successor)
(X+1)
intro x
have h:=list_length_le_weight x.2
change x.2.length+1≤(X+1).eval (weight x)
simp only [Polynomial.eval_add,Polynomial.eval_X,Polynomial.eval_one]
rw [weight_prod]
omega
lemma findRoot_eq (n : ℕ) (fuel : List Bool) (hf : n.sqrt≤fuel.length) :
findRoot (n,fuel)=n.sqrt := by
apply Nat.le_antisymm
· apply listMin_le
simp only [List.mem_filter,List.mem_range,rootCandidate,decide_eq_true_eq]
exact ⟨by omega,by simpa [pow_two,Nat.succ_eq_add_one] using Nat.lt_succ_sqrt n⟩
· apply le_listMin hf
intro i hi
simp only [List.mem_filter,List.mem_range,rootCandidate,decide_eq_true_eq,pow_two] at hi
have h:=Nat.sqrt_lt.mpr hi.2
omega
end MatchingFPRAS.OnlineSchedule
end OAI
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