openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Repetition /CleanPhysicalProjectedMeeting.lean
| import OAI.Computability.PerfectCompleteness.Decoding.CleanPhysicalDecoderLaw | |
| import OAI.Computability.PerfectCompleteness.Decoding.ProjectedRawDecoderLaw | |
| import OAI.Computability.PerfectCompleteness.Foundations.WholeArrayRawAssembly | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CleanPhysicalProjectedMeeting | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| abbrev F2 := ZMod 2 | |
| attribute [local instance] CleanPhysicalDecoderLaw.nativeLeftFintype RightDecoder.scalarFintype | |
| section TapeCast | |
| variable {branch rows repeats : Nat → Nat} {n t : Nat} | |
| private theorem evaluate_uncastTape | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (upper lower : Nodes branch n) (cut : OwnInputReference.Cut upper lower) | |
| {k : Nat} (hk : Nodes.height lower = k) | |
| (tape : WholeArraySampler.Tape rows repeats | |
| ((Nodes.path upper).append (hk ▸ cut.path)) slots) : | |
| WholeArraySampler.evaluate rows repeats ((Nodes.path upper).append cut.path) slots | |
| (CleanPhysicalDecoderLaw.uncastTape slots upper lower cut hk tape) = | |
| WholeArraySampler.evaluate rows repeats ((Nodes.path upper).append (hk ▸ cut.path)) slots tape := by | |
| cases hk | |
| rfl | |
| end TapeCast | |
| variable {branch rows repeats : Nat → Nat} {n h t v m : Nat} [NeZero m] | |
| {upper : Nodes branch n} {d : HierarchicalFrozenTables.LowerNodes upper (h + 1)} | |
| (S : CleanDecoderRate.Setup branch rows repeats n h t v m upper d) | |
| abbrev lower := CleanPhysicalDecoderLaw.lower S | |
| def rawSample (W : Submodule F2 (Block rows upper)) (x : CleanDecoderRate.Sample S) : | |
| OwnInputReference.RawSample (CleanNativeReplay.rightSlots S x) rows upper (lower S) W repeats S.cut := | |
| WholeArrayInteriorOwnInputLaw.exposedRead rows repeats (CleanNativeReplay.rightSlots S x) | |
| upper (lower S) S.cut W | |
| (CleanPhysicalDecoderLaw.uncastTape (CleanNativeReplay.rightSlots S x) upper (lower S) S.cut | |
| (CleanDecoderContext.lowerHeight upper d) (CleanPhysicalDecoderLaw.rightWholeTape S x)) | |
| abbrev visible (W : Submodule F2 (Block rows upper)) (x : CleanDecoderRate.Sample S) := | |
| (rawSample S W x).2 | |
| theorem raw_arrays (W : Submodule F2 (Block rows upper)) (x : CleanDecoderRate.Sample S) : | |
| OwnInputRawAssembly.arrays (CleanNativeReplay.rightSlots S x) rows upper (lower S) W | |
| repeats S.cut (rawSample S W x) = CleanPhysicalDecoderLaw.rightArrays S x := by | |
| have hraw := WholeArrayRawAssembly.arrays_exposedRead rows repeats | |
| (CleanNativeReplay.rightSlots S x) upper (lower S) S.cut W | |
| (CleanPhysicalDecoderLaw.uncastTape (CleanNativeReplay.rightSlots S x) upper (lower S) S.cut | |
| (CleanDecoderContext.lowerHeight upper d) (CleanPhysicalDecoderLaw.rightWholeTape S x)) | |
| exact hraw.trans (evaluate_uncastTape (CleanNativeReplay.rightSlots S x) upper (lower S) S.cut | |
| (CleanDecoderContext.lowerHeight upper d) (CleanPhysicalDecoderLaw.rightWholeTape S x)) | |
| theorem projected_raw_arrays (W : Submodule F2 (Block rows upper)) | |
| (x : CleanDecoderRate.Sample S) : | |
| ProjectedRawAssembly.arrays (CleanNativeReplay.leftSlots S x) (CleanNativeReplay.rightSlots S x) | |
| (CleanPhysicalDecoderLaw.physicalProjection S x) rows upper (lower S) W repeats S.cut | |
| (rawSample S W x) = CleanPhysicalDecoderLaw.leftArrays S x := by | |
| unfold ProjectedRawAssembly.arrays | |
| rw [raw_arrays] | |
| exact (CleanPhysicalCanonicalQuery.arrays_pullback rows repeats (CleanDecoderRate.path S) | |
| S.outside S.placeholder S.clauses S.designated x S.exterior).symm | |
| theorem rightInput_eq_readInput (W : Submodule F2 (Block rows upper)) | |
| (a : Block rows (lower S)) (x : CleanDecoderRate.Sample S) : | |
| CleanPhysicalDecoderLaw.rightInput S W a x = | |
| OwnInputReference.readInput (CleanNativeReplay.rightSlots S x) rows upper (lower S) W a | |
| (OwnInputRawAssembly.scalarEval (CleanNativeReplay.rightSlots S x) upper (lower S) | |
| repeats S.cut) (visible S W x) := | |
| CleanPhysicalDecoderLaw.rightInput_eq_ownInput S W a x | |
| variable {Sample : OriginalDecoderMark.SlotFamily branch n t → Type*} | |
| [∀ slots, Fintype (Sample slots)] | |
| (F : OriginalDecoderMark.Family branch rows n t Sample) (κ : ℝ) | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t)) | |
| (r : Nat) (ρ : ℝ) (A : ManyGoodRows.RowMap (Block rows upper) r) | |
| (a : Block rows (lower S)) (threshold : ℝ) | |
| theorem leftLaw_eq_visible (x : CleanDecoderRate.Sample S) : | |
| CleanPhysicalDecoderLaw.leftLaw S F κ σ r ρ A x = | |
| HierarchicalProjectedMeeting.leftLaw | |
| (CleanNativeReplay.leftSlots S x) (CleanNativeReplay.rightSlots S x) | |
| (CleanPhysicalDecoderLaw.physicalProjection S x) upper (h + 1) | |
| (HierarchicalProjectedRawPrediction.visibleBackground | |
| (CleanNativeReplay.leftSlots S x) (CleanNativeReplay.rightSlots S x) | |
| (CleanPhysicalDecoderLaw.physicalProjection S x) rows upper (lower S) A repeats S.cut | |
| (visible S (LinearMap.ker A) x)) | |
| (F.original (CleanNativeReplay.leftSlots S x)) | |
| (F.arrays (CleanNativeReplay.leftSlots S x)) | |
| (F.lowerEvent (CleanNativeReplay.leftSlots S x)) κ σ ρ A | |
| (HierarchicalProjectedRawPrediction.known (CleanNativeReplay.rightSlots S x) | |
| rows upper (lower S) A repeats S.cut (visible S (LinearMap.ker A) x)) := by | |
| change CleanPhysicalDecoderLaw.adviceLaw F (CleanNativeReplay.leftSlots S x) upper (h + 1) κ | |
| (CleanPhysicalDecoderLaw.leftArrays S x) σ r ρ A = _ | |
| rw [← projected_raw_arrays S (LinearMap.ker A) x] | |
| exact ProjectedRawDecoderLaw.adviceLaw_eq_leftLaw | |
| (CleanNativeReplay.leftSlots S x) (CleanNativeReplay.rightSlots S x) | |
| (CleanPhysicalDecoderLaw.physicalProjection S x) rows upper (lower S) (h + 1) | |
| (F.original (CleanNativeReplay.leftSlots S x)) | |
| (F.arrays (CleanNativeReplay.leftSlots S x)) | |
| (F.lowerEvent (CleanNativeReplay.leftSlots S x)) κ σ ρ A repeats S.cut | |
| (rawSample S (LinearMap.ker A) x) | |
| def visiblePairKernel (x : CleanDecoderRate.Sample S) := | |
| (HierarchicalProjectedMeeting.leftLaw | |
| (CleanNativeReplay.leftSlots S x) (CleanNativeReplay.rightSlots S x) | |
| (CleanPhysicalDecoderLaw.physicalProjection S x) upper (h + 1) | |
| (HierarchicalProjectedRawPrediction.visibleBackground | |
| (CleanNativeReplay.leftSlots S x) (CleanNativeReplay.rightSlots S x) | |
| (CleanPhysicalDecoderLaw.physicalProjection S x) rows upper (lower S) A repeats S.cut | |
| (visible S (LinearMap.ker A) x)) | |
| (F.original (CleanNativeReplay.leftSlots S x)) | |
| (F.arrays (CleanNativeReplay.leftSlots S x)) | |
| (F.lowerEvent (CleanNativeReplay.leftSlots S x)) κ σ ρ A | |
| (HierarchicalProjectedRawPrediction.known (CleanNativeReplay.rightSlots S x) | |
| rows upper (lower S) A repeats S.cut (visible S (LinearMap.ker A) x))).product | |
| (RightDecoder.law (CleanNativeReplay.rightSlots S x) rows upper (lower S) | |
| (LinearMap.ker A) a repeats S.cut σ | |
| (OwnInputReference.readInput (CleanNativeReplay.rightSlots S x) rows upper (lower S) | |
| (LinearMap.ker A) a | |
| (OwnInputRawAssembly.scalarEval (CleanNativeReplay.rightSlots S x) upper (lower S) | |
| repeats S.cut) (visible S (LinearMap.ker A) x)) threshold) | |
| theorem pairKernel_eq_visible (x : CleanDecoderRate.Sample S) : | |
| CleanPhysicalDecoderLaw.pairKernel S F κ σ r ρ A (LinearMap.ker A) a threshold x = | |
| visiblePairKernel S F κ σ r ρ A a threshold x := by | |
| unfold CleanPhysicalDecoderLaw.pairKernel visiblePairKernel | |
| rw [leftLaw_eq_visible] | |
| unfold CleanPhysicalDecoderLaw.rightLaw | |
| rw [rightInput_eq_readInput] | |
| theorem meeting_probability_eq (cube : Nat) (hcube : 0 < cube) | |
| (x : CleanDecoderRate.Sample S) | |
| (hx : (CleanDecoderRate.rawLaw S cube hcube).weight x ≠ 0) : | |
| (visiblePairKernel S F κ σ r ρ A a threshold x).probability | |
| (fun answer => decide ((ProjectedNodeEmbedding.intoOriginal | |
| (CleanPhysicalDecoderLaw.physicalProjection S x) upper).dualMap answer.1 = answer.2)) = | |
| (CleanDecoderPairLaw.pairKernel S σ (CleanPhysicalDecoderLaw.useful S F κ) | |
| r ρ A (LinearMap.ker A) a threshold x).probability | |
| (fun answer => decide (answer.2 = DecoderSourcePullback.upperTarget | |
| (CleanDecoderOddLists.fullProjection (CleanDecoderPairLaw.context S x) | |
| (CleanDecoderPairLaw.occurrences S x)) upper answer.1)) := by | |
| rw [← pairKernel_eq_visible] | |
| have hevent : | |
| (fun answer : CleanPhysicalDecoderLaw.Answers | |
| (CleanNativeReplay.leftSlots S x) (CleanNativeReplay.rightSlots S x) upper => | |
| decide ((ProjectedNodeEmbedding.intoOriginal | |
| (CleanPhysicalDecoderLaw.physicalProjection S x) upper).dualMap answer.1 = answer.2)) = | |
| (fun answer => decide (answer.2 = DecoderSourcePullback.upperTarget | |
| (CleanPhysicalDecoderLaw.physicalProjection S x) upper answer.1)) := by | |
| funext answer | |
| rw [ProjectedNodeEmbedding.dualMap_eq_upperTarget] | |
| exact decide_eq_decide.mpr eq_comm | |
| rw [hevent] | |
| exact CleanPhysicalDecoderLaw.meeting_probability_eq S F κ σ r ρ A (LinearMap.ker A) | |
| a threshold cube hcube x hx | |
| end | |
| end PerfectCompleteness.CleanPhysicalProjectedMeeting | |
| end | |
| end OAI | |