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97c3405 | 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 | import OAI.Combinatorics.ContingencyTables.Counting.CompanionCanonicalPhaseLaw
import OAI.Combinatorics.ContingencyTables.Counting.CompanionRestartSupport
namespace OAI
/-! # Representation preservation for every executed counting phase
The restart support statement is obtained from its actual finite-word law.
The remaining trajectory and observation words may be unused padding for
that law; their presence still supplies the required precision markers.
-/
namespace ContingencyTables.CompanionCanonicalPhaseSupport
open MatchingFPRAS CommonBasesFPRAS FiniteObservableTransport
open CompanionFactorialApplication CompanionComputedInitial CompanionComputedProposal
open CompanionEvaluatedWeightProgram CompanionStoredRepresentation CompanionHistoryReadback
open CompanionCanonicalPhaseLaw CompanionOracleDensity GridBoundary
open scoped Classical
noncomputable section
variable {m n : ℕ}
lemma restart_prefix_le (A : Instance (Fin m) (Fin n)) (C K H N R L W j : ℕ)
(epsilon : TreeFraction.Fraction) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W)
(v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K)
(hj : j < H)
(hwidth : RootedPhaseHistory.totalWidth A (rawSettings A C K H N R L epsilon hK hW v) ≤ W) :
CompanionComputedInitial.wordLength A
(CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
CompanionRestartProgram.sumBits A C K j
(CompanionInitializedRestartProgram.parameters A H
(CompanionTargetMixing.restartTime A H L) N R L epsilon) ≤ W := by
let s := rawSettings A C K H N R L epsilon hK hW v
calc
_ = RootedRestartWords.initialStageBits A C K H L s.k R s.h j s.xi :=
CompanionOuterWidths.initialWidth_eq A C K H N R L j epsilon
_ ≤ RootedPhaseHistory.width A s j := Nat.le_add_right _ _
_ ≤ RootedPhaseHistory.totalWidth A s := RootedPhaseHistory.width_le A s j hj
_ ≤ W := hwidth
theorem output_rep (A : Instance (Fin m) (Fin n)) (C K H N R L W : ℕ)
(epsilon : TreeFraction.Fraction) (hC : 0 < C) (hK : 1 ≤ K)
(hW : 100*A.d*K ≤ A.W) (hB : (K:ℝ)=binWidth A C)
(hxi : 0 < TreeFraction.val (CountingPhaseParametersProgram.noise
(CompanionInitializedRestartProgram.parameters A H
(CompanionTargetMixing.restartTime A H L) N R L epsilon)))
(z : View A→ℕ) (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K)
(initial : CompanionPhaseFinishProgram.Stored) (history : List CompanionPhaseFinishProgram.Stored)
(hj : history.length < H) (hphase : history.length+1 ≤ rationalHeight A C)
(hwidth : RootedPhaseHistory.totalWidth A (rawSettings A C K H N R L epsilon hK hW v) ≤ W)
(hdegree : ∀ i,i ≤ history.length+1 → PowerOfTwoApproximation.rootDegree
(roundingTolerance (innerTolerance
((rawSettings A C K H N R L epsilon hK hW v).xi/2) i)) ≤ W)
(hdepth : ∀ i,i ≤ history.length+1 → PowerOfTwoApproximation.depth
(roundingTolerance (innerTolerance
((rawSettings A C K H N R L epsilon hK hW v).xi/2) i)) ≤ W)
(hinit : StoredRep (CanonicalInitialEncoding.copyOrder (capacities A)).length initial)
(hh : ∀ q∈history,StoredRep (CanonicalInitialEncoding.copyOrder (capacities A)).length q)
(bits : Fin W→Bool) :
StoredRep (CanonicalInitialEncoding.copyOrder (capacities A)).length
(CompanionOuterPhaseProgram.output
(((CompanionRestartInitializer.data A C K H
(CompanionTargetMixing.restartTime A H L) N R L epsilon z v,initial),history),List.ofFn bits)) := by
let t := CompanionTargetMixing.restartTime A H L
let j := history.length
let p := CompanionInitializedRestartProgram.parameters A H t N R L epsilon
let a := wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))
let b := CompanionRestartProgram.sumBits A C K j p
let padding := W-(a+b)
let w := matrixAt (CanonicalInitialEncoding.copyOrder (capacities A)).length initial history
let E := estimateAt initial history
have hprefix : a+b ≤ W := restart_prefix_le A C K H N R L W j epsilon hK hW v hj hwidth
have hlen : a+(b+padding)=W := by
rw [←Nat.add_assoc]
exact Nat.add_sub_of_le hprefix
have hl := CompanionInitializedRestartProgram.output_padded_sameLaw A C K H t N R L j padding
hC (by omega) hK hW hB epsilon hxi z v w (fun i => E (i.val+1))
(fun i => by
unfold CompanionRestartStageProgram.oracleData
rw [RootedOracleConfigProgram.rootDegree_data]
exact (hdegree (i.val+1) (by omega)).trans_eq hlen.symm)
(fun i => by
unfold CompanionRestartStageProgram.oracleData
rw [RootedOracleConfigProgram.depth_data]
exact (hdepth (i.val+1) (by omega)).trans_eq hlen.symm)
have hl' := SameLaw.word_length hlen
(fun bs => CompanionInitializedRestartProgram.output
(CompanionInitializedRestartProgram.data A C K H t N R L j epsilon z v w
(fun i => E (i.val+1)) bs)) _ hl
obtain ⟨sourceBits,hs⟩ := hl'.exists_preimage bits
have hd := outer_data_readback A C K H t N R L epsilon z v initial history hinit hh (List.ofFn bits)
apply CompanionInitializedRestartProgram.outer_output_rep_of_supported A _ hinit hh
refine ⟨CompanionInitializedRestartProgram.model A C K H t N R L j epsilon hK hW v w sourceBits,?_⟩
rw [←hd]
change CompanionInitializedRestartProgram.output
(CompanionOuterPhaseProgram.restartInput
(CompanionOuterPhaseProgram.data A C K H t N R L j epsilon z v w E (List.ofFn bits)))=_
rw [CompanionOuterPhaseProgram.restartInput_data]
exact hs.symm
end
end ContingencyTables.CompanionCanonicalPhaseSupport
end OAI
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