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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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Tapes.GuardBits
import OAI.Combinatorics.MatchingCount.Algorithm.OnlineGuard
import OAI.Combinatorics.MatchingCount.Algorithm.OnlineTrialLaw
namespace OAI
namespace MatchingFPRAS.OnlineGuard
open PhysicalStream TreeTyped TreeFraction ExecutableSource ExecutableScaling OnlineSchedule OnlineStage
open Transport UnitRefresh Enlargement Variation TreeTyped.Numbered
open scoped Classical
noncomputable local instance {n : ℕ} [Nontrivial (Fin n)] (w : Fin n→Fin n→ℝ) (K : ℕ) :
Fintype (SourceMarked w K) := Fintype.ofFinite _
attribute [local instance] NumberedHierarchy.labelOrder
attribute [local instance] Transport.sym2Order Transport.bagEdgeOrder Transport.bagEdgeFintype
attribute [local instance] UnitRefresh.Computable.allowedDec Transport.prodOrder Transport.sumOrder
variable {G : GraphInput} [Nontrivial (Fin G.n)]
lemma guard_many_mean (p : Trial.Parameters G) (s : FiniteTrial.ScaleState G p.K) (h : Bool→ℚ) :
(test (fixedData p s,p.S)).mean h=
qaverage (fun b=>h (decide (FiniteTrial.guardPass p s (s.drawGuard p b)))) := by
classical
have hD : 0<FiniteTrial.sourceD (G:=G) p.K := by
rw [FiniteTrial.sourceD_eq]; exact Enlargement.sourceD_one p.K
have hq : 0<copies (n:=G.n) p.K (FiniteTrial.sourceD (G:=G) p.K) := by
simpa only [copies,vertexCount,FiniteTrial.sourceD_eq,Trial.Parameters.q,Trial.Parameters.N,Trial.Parameters.D] using p.q_pos
let raw := realRawDraw s.activity p.K (FiniteTrial.sourceD (G:=G) p.K) (denominator p)
s.activity_pos hD (s.cap le_rfl) hq p.even (1/denominator p)
have hr : imageLaw raw=(s.runLaw p le_rfl).mass := by
rw [realRawDraw_law]
dsimp only [FiniteTrial.ScaleState.runLaw]
apply finiteLaw_parameters
· exact FiniteTrial.sourceD_eq p.K
· simp only [denominator,Nat.cast_mul,Nat.cast_ofNat]
have hp : imageLaw (fun bs : Fin 16→Raw s.activity p.K (FiniteTrial.sourceD (G:=G) p.K)
(denominator p) (1/denominator p)=>fun i=>raw (bs i))=imageLaw (s.drawGuard p) := by
rw [imageLaw_componentwise,hr,FiniteTrial.ScaleState.drawGuard_law]
rfl
unfold test
rw [Experiment.mean_map,sourceRequest_source]
unfold fixedRequest
simp only [Trial.Parameters.N]
rw [OnlineSample.mean_many]
apply Eq.trans _ (qaverage_eq_of_imageLaw_eq _ _ hp (fun x=>h (decide (FiniteTrial.guardPass p s x))))
apply qaverage_congr
intro bs
apply congrArg h
have hcert (b : Raw s.activity p.K (FiniteTrial.sourceD (G:=G) p.K) (denominator p) (1/denominator p)) :
sparseCertificate (fixedData p s,OnlineSample.sampleRep
(fun i=>index (orderedEnum (BagEdge (RationalSource.bags s.activity p.K)) i)) b)=
TreeFraction.repr (s.certificate p.K (raw b)) := by
have hc:=sparseCertificate_source p.hn p.even s p.K (FiniteTrial.sourceD (G:=G) p.K)
hD (s.cap le_rfl)
(List.replicate (tierCount (n:=G.n) p.K (FiniteTrial.sourceD (G:=G) p.K)) false)
(List.replicate (copies (n:=G.n) p.K (FiniteTrial.sourceD (G:=G) p.K)) false)
(by simpa using hq) b.1 (fun e=>b.2 ((orderedEnum _).symm e))
dsimp only at hc
simp only [List.map_ofFn] at hc
change sparseCertificate (fixedData p s,_) = _ at hc
rw [generated_eq_raw] at hc
rw [FiniteTrial.ScaleState.certificate,realRawDraw_inverse]
simpa only [OnlineSample.sampleRep,OnlineSample.scanRep,OnlineSample.localRep,
List.map_ofFn,Function.comp_def,Equiv.symm_apply_apply,NumericLadder.localBitsRep] using hc
apply Bool.eq_iff_iff.mpr
rw [decide_eq_true_eq]
simp only [accept,List.any_eq_true,List.mem_ofFn,FiniteTrial.guardPass]
constructor
· rintro ⟨x,⟨i,rfl⟩,hi⟩
rw [hcert,positive_repr (s.certificate_range p.K _).1,decide_eq_true_eq] at hi
exact ⟨i,hi⟩
· rintro ⟨i,hi⟩
refine ⟨_,⟨i,rfl⟩,?_⟩
rw [hcert,positive_repr (s.certificate_range p.K _).1,decide_eq_true_eq]
exact hi
lemma checked_mean (p : Trial.Parameters G) (x : (FiniteTrial.layers p p.K le_rfl).Ω) (h : Fraction→ℚ) :
(checked (OnlineTrial.represent p (FiniteTrial.layers p p.K le_rfl) x)).mean h=
qaverage (fun b=>h (TreeFraction.repr (FiniteTrial.guardedOutput p ⟨x,((FiniteTrial.layers p p.K le_rfl).state x).drawGuard p b⟩))) := by
unfold checked OnlineTrial.represent
rw [Experiment.mean_map,guard_many_mean]
apply qaverage_congr
intro b
simp only [finish,decide_eq_true_eq,FiniteTrial.guardedOutput,OnlineTrial.product,FiniteTrial.output]
split_ifs <;> rfl
lemma trial_mean (p : Trial.Parameters G) (h : Fraction→ℚ) :
(trial (List.replicate p.K false,OnlineTrial.represent p (FiniteTrial.firstLayer p) PUnit.unit)).mean h=
qaverage (fun b=>h (TreeFraction.repr (FiniteTrial.guardedBitOutput p b))) := by
unfold trial
rw [Experiment.mean_bind,OnlineTrial.sweep_mean p p.K le_rfl]
simp_rw [checked_mean]
change _=qaverage ((fun c : (Fin (p.K*FiniteTrial.stageBits p)→Bool)×(Fin (FiniteTrial.guardBits p)→Bool)=>
h (TreeFraction.repr (FiniteTrial.guardedOutput p ⟨FiniteTrial.drawLayer p p.K le_rfl c.1,
((FiniteTrial.layers p p.K le_rfl).state (FiniteTrial.drawLayer p p.K le_rfl c.1)).drawGuard p c.2⟩))) ∘
pairBitsEquiv (p.K*FiniteTrial.stageBits p) (FiniteTrial.guardBits p))
rw [qaverage_equiv,qaverage_prod]
end MatchingFPRAS.OnlineGuard
end OAI
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