| 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 ( 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 | |