| import Mathlib |
| 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 |
|
|
