File size: 3,884 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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.TreeMatrix
import OAI.Combinatorics.MatchingCount.Machines.TreeEnumeration
import OAI.Combinatorics.MatchingCount.Machines.TreeClosureReindex
namespace OAI
namespace MatchingFPRAS.ExecutableSource
open TreeTyped TreeFraction ExecutableBottleneck Enlargement RealGeometry
attribute [local instance] Transport.prodOrder Transport.sumOrder
lemma vertices_eq_ofFn_length (n : ℕ) (fuel : List Bool) (m : ℕ) (h : 4*fuel.length=m) :
vertices (List.range n,fuel)=List.ofFn (fun i=>vertexRepr
(UnitRefresh.orderedEnum (Vertex (Fin n) (LogicalEdge (Fin n)) m) i)) := by
subst m
exact vertices_eq_ofFn n fuel
abbrev BuildInput := (Matrix×List Bool)×List Nat
def prepare (x : BuildInput) : PathTableInput := (((x.1.1,close x.1.1),x.1.2),x.2)
def prepareRealizer : Realizer prepare := (pair (pair (pair first.fst
(composition first.fst closeRealizer)) first.snd) second).congr (fun _=>rfl)
lemma polynomial_prepare : PolynomialTime prepareRealizer := by
unfold prepareRealizer
apply polynomial_congr
have := polynomial_close
dsimp only [Realizer.fst,Realizer.snd]
tree_cost
def buildMatrix (x : BuildInput) : Matrix := materialize (prepare x)
def buildMatrixRealizer : Realizer buildMatrix := composition prepareRealizer materializeRealizer
lemma polynomial_buildMatrix : PolynomialTime buildMatrixRealizer :=
polynomial_composition polynomial_prepare polynomial_materialize
def buildClosure (x : BuildInput) : Matrix := close (buildMatrix x)
def buildClosureRealizer : Realizer buildClosure := composition buildMatrixRealizer closeRealizer
lemma polynomial_buildClosure : PolynomialTime buildClosureRealizer :=
polynomial_composition polynomial_buildMatrix polynomial_close
lemma source_activity_nonneg {n : ℕ} [Nontrivial (Fin n)] (hn : 2≤n) (w : Fin n→Fin n→ℚ)
(hw : ∀ i j,0≤w i j) (K : ℕ) (a b : SourceVertex (Fin n) K) :
0≤RationalSource.activity w K a b := by
unfold RationalSource.activity
apply Finset.sum_nonneg
intro q _
split_ifs
· exact pathActivity_nonneg hn w hw K q.1 q.2
· exact le_rfl
lemma buildMatrix_repr {n : ℕ} [Nontrivial (Fin n)] (hn : 2≤n)
(w : Fin n→Fin n→ℚ) (hw : ∀ i j,0≤w i j) (K : ℕ) :
buildMatrix ((matrixRep w,List.replicate (2*K+2) false),List.range n)=
matrixRep (fun i j=>RationalSource.activity w K
(UnitRefresh.orderedEnum (SourceVertex (Fin n) K) i)
(UnitRefresh.orderedEnum (SourceVertex (Fin n) K) j)) := by
unfold buildMatrix prepare materialize
rw [close_matrixRep w hw]
rw [show BottleneckAlgorithm.close w n=RationalSource.logicalB w by simp [RationalSource.logicalB]]
change activityMatrix (pathTable (((matrixRep w,matrixRep (RationalSource.logicalB w)),
List.replicate (2*K+2) false),List.range n),
vertices (List.range n,List.replicate (2*K+2) false))=_
have hv : vertices (List.range n,List.replicate (2*K+2) false)=
List.ofFn (fun i=>vertexRepr (UnitRefresh.orderedEnum (SourceVertex (Fin n) K) i)) := by
exact vertices_eq_ofFn_length n _ _ (by simp)
rw [hv]
exact activityMatrix_source_repr hn w hw K _ (List.length_replicate ..) _
lemma buildClosure_repr {n : ℕ} [Nontrivial (Fin n)] (hn : 2≤n)
(w : Fin n→Fin n→ℚ) (hw : ∀ i j,0≤w i j) (K : ℕ) :
buildClosure ((matrixRep w,List.replicate (2*K+2) false),List.range n)=
matrixRep (fun i j=>RationalSource.realB w K
(UnitRefresh.orderedEnum (SourceVertex (Fin n) K) i)
(UnitRefresh.orderedEnum (SourceVertex (Fin n) K) j)) := by
unfold buildClosure
rw [buildMatrix_repr hn w hw K,close_matrixRep _ (fun _ _=>source_activity_nonneg hn w hw K _ _)]
rw [BottleneckAlgorithm.close_reindex (UnitRefresh.orderedEnum (SourceVertex (Fin n) K))
UnitRefresh.orderedEnum_lt]
rfl
end MatchingFPRAS.ExecutableSource
end OAI
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