File size: 9,033 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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.TreeBindings
namespace OAI
namespace MatchingFPRAS.ExecutableSource
open TreeTyped TreeFraction TreeTyped.Numbered ExecutableBottleneck ExecutableHierarchy ExecutableRefresh Enlargement RealGeometry NumericLadder
open ExecutableLadder UnitRefresh Replica.Ladder Transport NumericMarking ExecutableStatistics ExecutableScaling
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 GeneratedUpdateData := GeneratedStageData×Vector
abbrev GeneratedUpdateInput := GeneratedUpdateData×List SampleBits
def generatedUpdateArg : Realizer (fun x : GeneratedUpdateData=>(x.1.1.1,x.2)) :=
(TreeTyped.pair (first (A:=GeneratedStageData) (B:=Vector)).fst.fst second).congr (fun _=>rfl)
lemma polynomial_generatedUpdateArg : PolynomialTime generatedUpdateArg := by
unfold generatedUpdateArg
apply polynomial_congr
dsimp only [Realizer.fst, Realizer.snd]
tree_cost
def generatedUpdateSetup (x : GeneratedUpdateData) : UpdateConfig×((Reweights×(ℕ×List ℕ))×SampleContext) :=
(generatedUpdateConfig (x.1.1.1,x.2),generatedStageSetup x.1)
def generatedUpdateSetupRealizer : Realizer generatedUpdateSetup := (TreeTyped.pair
(composition generatedUpdateArg generatedUpdateConfigRealizer)
(composition (first (A:=GeneratedStageData) (B:=Vector)) generatedStageSetupRealizer)).congr (fun _=>rfl)
lemma polynomial_generatedUpdateSetup : PolynomialTime generatedUpdateSetupRealizer :=
polynomial_congr _ (TreeTyped.polynomial_pair
(polynomial_composition polynomial_generatedUpdateArg polynomial_generatedUpdateConfig)
(polynomial_composition polynomial_first polynomial_generatedStageSetup))
def updateWith {A : Type} [Codec A]
{f : A→UpdateConfig×((Reweights×(ℕ×List ℕ))×SampleContext)} (F : Realizer f) :
Realizer (fun x : A×List SampleBits=>stageUpdate ((f x.1).1,((f x.1).2,x.2))) :=
(composition (TreeTyped.pair (composition (first (A:=A) (B:=List SampleBits)) F) second)
((composition (TreeTyped.pair first.fst (TreeTyped.pair first.snd second)) stageUpdateRealizer))).congr (fun _=>rfl)
lemma polynomial_updateWith {A : Type} [Codec A]
{f : A→UpdateConfig×((Reweights×(ℕ×List ℕ))×SampleContext)}
{F : Realizer f} (hF : PolynomialTime F) : PolynomialTime (updateWith F) := by
unfold updateWith
apply polynomial_congr
dsimp only [Realizer.fst, Realizer.snd]
have := polynomial_stageUpdate
tree_cost
def generatedUpdate (x : GeneratedUpdateInput) : Fraction×Vector :=
stageUpdate ((generatedUpdateSetup x.1).1,((generatedUpdateSetup x.1).2,x.2))
def generatedUpdateRealizer : Realizer generatedUpdate := updateWith generatedUpdateSetupRealizer
lemma polynomial_generatedUpdate : PolynomialTime generatedUpdateRealizer :=
polynomial_updateWith polynomial_generatedUpdateSetup
variable {n : ℕ} [Nontrivial (Fin n)]
noncomputable def generatedSourceSample (he : Even n) (w : Fin n→Fin n→ℚ)
(hw : ∀ a b,0<w a b) (K D : ℕ) (hD : 0<D)
(hu : ∀ a b,a≠b→BottleneckAlgorithm.edge w a b≤(4:ℚ)^K)
(ft fq : List Bool) (hq : 0<fq.length) {k r km : ℕ}
(bits : Fin r→ScanBits (SourceVertex (Fin n) K) k)
(mb : BagEdge (RationalSource.bags w K)→Fin km→Bool) : RationalSource.Marked w K :=
TableMarking.reveal (BagEdge.Incident (bags:=RationalSource.bags w K))
(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))))
(fun e=>(RationalSource.markingTable w hw K D hD e).draw km (mb e))
lemma numericMarks_selected {E : Type} [Fintype E] [LinearOrder E]
(s : Finset E) (m : E→Fin 3) (de : DecidableEq E) :
numericMarks s (@selectedMarks E de s (fun e=>m e.val))=numericMarks s m := by
unfold numericMarks
apply List.map_congr_left
intro e he
have hm : e∈s := (Finset.mem_sort _).mp he
simp only [selectedMarks,hm,↓reduceDIte]
lemma generatedMarked_sourceSample (hn : 2≤n) (he : Even n) (w : Fin n→Fin n→ℚ)
(hw : ∀ a b,0<w a b) (K D : ℕ) (hD : 0<D)
(hu : ∀ a b,a≠b→BottleneckAlgorithm.edge w a b≤(4:ℚ)^K)
(ft fq : List Bool) (hq : 0<fq.length) {k r km : ℕ}
(bits : Fin r→ScanBits (SourceVertex (Fin n) K) k)
(mb : BagEdge (RationalSource.bags w K)→Fin km→Bool) :
let x := generatedSourceSample he w hw K D hD hu ft fq hq bits mb
generatedMarked
((((((matrixRep w,List.replicate (2*K+2) false),List.range n),List.replicate (2*K) false),D),(ft,fq)),
(List.ofFn (fun z=>(List.ofFn (bits z).1,localBitsRep (bits z).2)),
(List.ofFn (orderedEnum (BagEdge (RationalSource.bags w K)))).map (fun e=>(index e,List.ofFn (mb e)))))=
numericMarks x.1.val (selectedMarks x.1.val x.2) := by
dsimp only
rw [generatedMarked_source hn he w hw K D hD hu ft fq hq bits mb]
dsimp only [generatedSourceSample,TableMarking.reveal]
exact (numericMarks_selected (E:=BagEdge (RationalSource.bags w K)) _ _ _).symm
lemma generatedSamples_source (hn : 2≤n) (he : Even n) (w : Fin n→Fin n→ℚ)
(hw : ∀ a b,0<w a b) (K D : ℕ) (hD : 0<D)
(hu : ∀ a b,a≠b→BottleneckAlgorithm.edge w a b≤(4:ℚ)^K)
(ft fq : List Bool) (hq : 0<fq.length) {S k r km : ℕ}
(bits : Fin S→Fin r→ScanBits (SourceVertex (Fin n) K) k)
(mb : Fin S→BagEdge (RationalSource.bags w K)→Fin km→Bool) :
sampleMany
(generatedContext (((((matrixRep w,List.replicate (2*K+2) false),List.range n),List.replicate (2*K) false),D),(ft,fq)),
List.ofFn (fun i=>(List.ofFn (fun z=>(List.ofFn (bits i z).1,localBitsRep (bits i z).2)),
(List.ofFn (orderedEnum (BagEdge (RationalSource.bags w K)))).map (fun e=>(index e,List.ofFn (mb i e))))))=
sourceSamples w K (fun i=>generatedSourceSample he w hw K D hD hu ft fq hq (bits i) (mb i)) := by
unfold sampleMany sourceSamples
rw [List.map_ofFn]
apply congrArg List.ofFn
funext i
have hm := generatedMarked_sourceSample hn he w hw K D hD hu ft fq hq (bits i) (mb i)
simpa only [generatedMarked,Function.comp_def] using hm
lemma generatedUpdate_source {G : GraphInput} [Nontrivial (Fin G.n)] (hn : 2≤G.n)
(he : Even G.n) {j : ℕ} (s : FiniteTrial.ScaleState G j) (K : ℕ)
(hD : 0<FiniteTrial.sourceD (G:=G) K)
(hu : ∀ a b,a≠b→BottleneckAlgorithm.edge s.activity a b≤(4:ℚ)^K)
(ft fq : List Bool) (hq : 0<fq.length) {S k r km : ℕ}
(bits : Fin S→Fin r→ScanBits (SourceVertex (Fin G.n) K) k)
(mb : Fin S→BagEdge (RationalSource.bags s.activity K)→Fin km→Bool) :
let x := fun i=>generatedSourceSample he s.activity s.activity_pos K
(FiniteTrial.sourceD (G:=G) K) hD hu ft fq hq (bits i) (mb i)
let keys := List.ofFn (orderedEnum (BagEdge (RationalSource.bags s.activity K)))
generatedUpdate
((((((((matrixRep s.activity,List.replicate (2*K+2) false),List.range G.n),List.replicate (2*K) false),
FiniteTrial.sourceD (G:=G) K),(ft,fq)),matrixRep s.nextActivity),vectorRep s.scale),
List.ofFn (fun i=>(List.ofFn (fun z=>(List.ofFn (bits i z).1,localBitsRep (bits i z).2)),
keys.map (fun e=>(index e,List.ofFn (mb i e))))))=
(TreeFraction.repr (s.ratioEstimate x),vectorRep (s.update hn x).scale) := by
dsimp only
let x := fun i=>generatedSourceSample he s.activity s.activity_pos K
(FiniteTrial.sourceD (G:=G) K) hD hu ft fq hq (bits i) (mb i)
unfold generatedUpdate generatedUpdateSetup
rw [generatedUpdateConfig_source hn s K]
unfold stageUpdate generatedStageSetup
dsimp only
rw [generatedSamples_source hn he s.activity s.activity_pos K (FiniteTrial.sourceD (G:=G) K) hD hu ft fq hq bits mb]
unfold generatedObservables
dsimp only
rw [sourceReweights_source hn s.activity s.nextActivity (fun a b=>(s.activity_pos a b).le)
(fun a b=>(s.nextActivity_pos a b).le) K,
sourceAllIndices_source hn s.activity (fun a b=>(s.activity_pos a b).le) K]
exact scaleState_update_rep hn s x
open TreePost Turing
theorem generatedUpdate_post : ∃ p : Polynomial ℕ,∀ x : GeneratedUpdateInput,∀ t : ℕ,
p.eval (packed x).length≤t → ∀ coins : Fin t→Bool,
(run (machine generatedUpdateRealizer.program) (packed x) coins).state=none ∧
(run (machine generatedUpdateRealizer.program) (packed x) coins).tape.right₀=
ListBlank.mk (packed (generatedUpdate x)) :=
polynomial_compiles generatedUpdateRealizer polynomial_generatedUpdate
end MatchingFPRAS.ExecutableSource
end OAI
|