File size: 3,943 Bytes
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import OAI.Combinatorics.MatchingCount.Algorithm.TreeGeneratedPrepared
import OAI.Combinatorics.MatchingCount.Machines.TreeStart
import OAI.Combinatorics.MatchingCount.Sampling.TreePreparedTop
namespace OAI
namespace MatchingFPRAS.ExecutableSource
open TreeTyped TreeFraction TreeTyped.Numbered ExecutableBottleneck ExecutableHierarchy ExecutableRefresh Enlargement RealGeometry NumericLadder
open ExecutableLadder UnitRefresh Replica.Ladder Transport
attribute [local instance] NumberedHierarchy.labelOrder
attribute [local instance] Transport.sym2Order Transport.bagEdgeOrder Transport.bagEdgeFintype
attribute [local instance] UnitRefresh.Computable.allowedDec Transport.prodOrder Transport.sumOrder
abbrev GeneratedTopInput := List (List Bool×Bits)×GeneratedInput
def generatedTop (x : GeneratedTopInput) : Finset ℕ :=
preparedTop (x.1,(generatedPrepared x.2,exactStart x.2.1.1))
def generatedTopRealizer : Realizer generatedTop := (composition
(pair first (pair (composition second generatedPreparedRealizer)
(composition second.fst.fst exactStartRealizer))) preparedTopRealizer).congr (fun _=>rfl)
lemma polynomial_generatedTop : PolynomialTime generatedTopRealizer := by
unfold generatedTopRealizer
apply polynomial_congr
dsimp only [Realizer.fst, Realizer.snd]
have := polynomial_generatedPrepared
have := polynomial_exactStart
have := polynomial_preparedTop
tree_cost
lemma bagWeight_nonneg {n : ℕ} [Nontrivial (Fin n)]
(w : Fin n→Fin n→ℚ) (hw : ∀ a b,0<w a b) (K D : ℕ)
(e : BagEdge (RationalSource.bags w K)) : 0≤RationalSource.bagWeight w K D e :=
add_nonneg (RationalSource.realPortion_nonneg w hw K e)
(div_nonneg (by unfold RationalSource.bagHeight; positivity) (Nat.cast_nonneg D))
lemma generatedTop_source {n : ℕ} [Nontrivial (Fin n)] (hn : 2≤n) (he : Even n)
(w : Fin n→Fin n→ℚ) (hw : ∀ a b,0<w a b) (K D : ℕ)
(hu : ∀ a b,a≠b→BottleneckAlgorithm.edge w a b≤(4:ℚ)^K)
(ft fq : List Bool) (hq : 0<fq.length) {k r : ℕ}
(bits : Fin r→ScanBits (SourceVertex (Fin n) K) k) :
generatedTop (List.ofFn (fun i=>(List.ofFn (bits i).1,localBitsRep (bits i).2)),
(((((matrixRep w,List.replicate (2*K+2) false),List.range n),List.replicate (2*K) false),D),(ft,fq)))=
Numbered.set (ladderRun (RationalSource.bags_adjacent w K hu) (RationalSource.tree_acyclic w K)
(fun l : RationalSource.Label w K=>l.1.val) (RationalSource.parent w K)
(RationalSource.bags_ranked w K hu) (RationalSource.bags_nonempty w K)
(RationalSource.parent_rank w K) hq (RationalSource.bagWeight w K D)
(RationalSource.startMatching w K he) k r (fun _=>RationalSource.startMatching w K he) bits
(toLex ((0 : Fin (ft.length+1)),(⟨0,hq⟩ : Fin fq.length)))).val := by
unfold generatedTop
dsimp only
rw [generatedPrepared_source hn w (fun a b=>(hw a b).le) K D ft fq,
exactStart_source hn he w (fun a b=>(hw a b).le) K]
exact preparedTop_source (RationalSource.bags_adjacent w K hu) (RationalSource.tree_acyclic w K)
(fun l : RationalSource.Label w K=>l.1.val) (RationalSource.parent w K)
(RationalSource.bags_ranked w K hu) (RationalSource.bags_nonempty w K)
(RationalSource.parent_rank w K) ft fq hq (RationalSource.bagWeight w K D)
(bagWeight_nonneg w hw K D) (RationalSource.startMatching w K he) _
(setupBags_source hn w (fun a b=>(hw a b).le) K) bits
open TreePost Turing
theorem generatedTop_post : ∃ p : Polynomial ℕ,∀ x : GeneratedTopInput,∀ t : ℕ,
p.eval (packed x).length≤t → ∀ coins : Fin t→Bool,
(run (machine generatedTopRealizer.program) (packed x) coins).state=none ∧
(run (machine generatedTopRealizer.program) (packed x) coins).tape.right₀=
ListBlank.mk (packed (generatedTop x)) :=
polynomial_compiles generatedTopRealizer polynomial_generatedTop
end MatchingFPRAS.ExecutableSource
end OAI
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