openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /LowerCutBucketLocality.lean
| import OAI.Computability.PerfectCompleteness.Decoding.LowerCutRowsLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.WholeCutReplayLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.WholeReplaySubtreeLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.LowerCutBucketLocality | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open WholeCutGrouping WholeArraySubtreeSplit | |
| abbrev F2 := ZMod 2 | |
| variable {branch : Nat → Nat} {n k m t : Nat} | |
| def rootBuckets (rows repeats : Nat → Nat) : | |
| {k m : Nat} → (pDown : Path branch k (m + 1)) → m + 1 < k → | |
| (slots : Slots branch k → Fin t → MixedSupport.Slot) → | |
| Exterior rows repeats pDown slots → | |
| LowerCutCallSplit.Assembled rows repeats pDown slots → | |
| BucketSampler.Direction (rows k) → H slots | |
| | _, _, .refl _, hproper, _, _, _, _ => False.elim ((Nat.lt_irrefl _) hproper) | |
| | _, _, .step i pDown, _, slots, exterior, assembled, direction => | |
| LowerCutScalar.reconstruct F2 repeats (.step i pDown) (LeafDomain slots) | |
| (fun terminal => assembled.1 (.inl (direction, terminal))) (exterior.1 direction) | |
| def buckets (rows repeats : Nat → Nat) : | |
| {n k m : Nat} → (pUp : Path branch n k) → (pDown : Path branch k (m + 1)) → | |
| m + 1 < k → (slots : Slots branch n → Fin t → MixedSupport.Slot) → | |
| Exterior rows repeats (pUp.append pDown) slots → | |
| LowerCutCallSplit.Assembled rows repeats (pUp.append pDown) slots → | |
| BucketSampler.Direction (rows k) → H (subtreeSlots pUp slots) | |
| | _, _, _, .refl _, pDown, hproper, slots, exterior, assembled => | |
| rootBuckets rows repeats pDown hproper slots exterior assembled | |
| | _, _, _, .step i pUp, pDown, hproper, slots, exterior, assembled => | |
| buckets rows repeats pUp pDown hproper (childSlots slots i) exterior.2.1 | |
| (fun call => assembled.1 (.inr call), assembled.2) | |
| theorem buckets_congr_otherCalls (rows repeats : Nat → Nat) (pUp : Path branch n k) : | |
| ∀ {m : Nat} (pDown : Path branch k (m + 1)) (hproper : m + 1 < k) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (exterior : Exterior rows repeats (pUp.append pDown) slots) | |
| (x y : LowerCutCallSplit.Assembled rows repeats (pUp.append pDown) slots), | |
| (∀ call, (∀ row, call ≠ WholeCutCalls.directRow rows repeats (pUp.append pDown) row) → | |
| y.1 call = x.1 call) → | |
| buckets rows repeats pUp pDown hproper slots exterior y = | |
| buckets rows repeats pUp pDown hproper slots exterior x := by | |
| induction pUp with | |
| | refl k => | |
| intro m pDown hproper slots exterior x y hcalls | |
| cases pDown with | |
| | refl => exact False.elim ((Nat.lt_irrefl _) hproper) | |
| | step i pDown => | |
| funext direction | |
| have hvalues : | |
| ((fun terminal => y.1 (.inl (direction, terminal))) : | |
| LowerCutScalar.Values F2 repeats (.step i pDown) (LeafDomain slots)) = | |
| (fun terminal => x.1 (.inl (direction, terminal))) := by | |
| funext terminal | |
| apply hcalls (.inl (direction, terminal)) | |
| intro row h | |
| cases h | |
| simp only [buckets, rootBuckets] | |
| rw [hvalues] | |
| | step i pUp ih => | |
| intro m pDown hproper slots exterior x y hcalls | |
| apply ih pDown hproper (childSlots slots i) exterior.2.1 | |
| (fun call => x.1 (.inr call), x.2) (fun call => y.1 (.inr call), y.2) | |
| intro call hcall | |
| apply hcalls (.inr call) | |
| intro row h | |
| exact hcall row (Sum.inr.inj h) | |
| theorem buckets_splitTape (rows repeats : Nat → Nat) (pUp : Path branch n k) : | |
| ∀ {m : Nat} (pDown : Path branch k (m + 1)) (hproper : m + 1 < k) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (tape : WholeCutSampler.Tape rows repeats (pUp.append pDown) slots), | |
| buckets rows repeats pUp pDown hproper slots | |
| (splitTape rows repeats (pUp.append pDown) slots tape).1 | |
| (CutChildGrouping.assemble (cutSlots (pUp.append pDown) slots) rows | |
| (splitTape rows repeats (pUp.append pDown) slots tape).2) = | |
| WholeCutSubtree.rootBuckets rows repeats pDown hproper (subtreeSlots pUp slots) | |
| (WholeCutSubtree.extract rows repeats pUp pDown slots tape) := by | |
| induction pUp with | |
| | refl k => | |
| intro m pDown hproper slots tape | |
| cases pDown with | |
| | refl => exact False.elim ((Nat.lt_irrefl _) hproper) | |
| | step i pDown => | |
| funext direction | |
| exact LowerCutScalar.reconstruct_splitTape F2 repeats (.step i pDown) | |
| (LeafDomain slots) (tape.1 direction) | |
| | step i pUp ih => | |
| intro m pDown hproper slots tape | |
| exact ih pDown hproper (childSlots slots i) tape.2.1 | |
| def knownBuckets (rows repeats : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) (upper : Nodes branch n) | |
| (pDown : Path branch (Nodes.height upper) (m + 1)) | |
| (hproper : m + 1 < Nodes.height upper) | |
| (W : Submodule F2 (OwnInputReference.UpperVector rows upper)) | |
| (exterior : Exterior rows repeats ((Nodes.path upper).append pDown) slots) | |
| (assembled : LowerCutCallSplit.Assembled rows repeats ((Nodes.path upper).append pDown) slots) : | |
| HiddenBucketBias.VisibleDirection W → OwnInputReference.UpperSpace slots upper := | |
| fun v => buckets rows repeats (Nodes.path upper) pDown hproper slots exterior assembled v.val | |
| def knownBucketsOfComplement (rows repeats : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) (upper : Nodes branch n) | |
| (pDown : Path branch (Nodes.height upper) (m + 1)) | |
| (hproper : m + 1 < Nodes.height upper) | |
| (W : Submodule F2 (OwnInputReference.UpperVector rows upper)) | |
| (exterior : Exterior rows repeats ((Nodes.path upper).append pDown) slots) | |
| (complement : LowerCutCallSplit.Complement rows repeats ((Nodes.path upper).append pDown) slots) : | |
| HiddenBucketBias.VisibleDirection W → OwnInputReference.UpperSpace slots upper := | |
| knownBuckets rows repeats slots upper pDown hproper W exterior | |
| (LowerCutCallSplit.merge rows repeats ((Nodes.path upper).append pDown) slots complement 0) | |
| theorem knownBuckets_merge (rows repeats : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) (upper : Nodes branch n) | |
| (pDown : Path branch (Nodes.height upper) (m + 1)) | |
| (hproper : m + 1 < Nodes.height upper) | |
| (W : Submodule F2 (OwnInputReference.UpperVector rows upper)) | |
| (exterior : Exterior rows repeats ((Nodes.path upper).append pDown) slots) | |
| (complement : LowerCutCallSplit.Complement rows repeats ((Nodes.path upper).append pDown) slots) | |
| (fresh : LowerCutCallSplit.NativeRows rows ((Nodes.path upper).append pDown) slots) : | |
| knownBuckets rows repeats slots upper pDown hproper W exterior | |
| (LowerCutCallSplit.merge rows repeats ((Nodes.path upper).append pDown) slots complement fresh) = | |
| knownBucketsOfComplement rows repeats slots upper pDown hproper W exterior complement := by | |
| have h := buckets_congr_otherCalls rows repeats (Nodes.path upper) pDown hproper slots exterior | |
| (LowerCutCallSplit.merge rows repeats ((Nodes.path upper).append pDown) slots complement 0) | |
| (LowerCutCallSplit.merge rows repeats ((Nodes.path upper).append pDown) slots complement fresh) | |
| (fun call hcall => | |
| (LowerCutCallSplit.merge_otherCall rows repeats ((Nodes.path upper).append pDown) slots | |
| complement fresh ⟨call, hcall⟩).trans | |
| (LowerCutCallSplit.merge_otherCall rows repeats ((Nodes.path upper).append pDown) slots | |
| complement 0 ⟨call, hcall⟩).symm) | |
| funext v | |
| exact congrFun h v.val | |
| theorem knownBuckets_eq_complement (rows repeats : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) (upper : Nodes branch n) | |
| (pDown : Path branch (Nodes.height upper) (m + 1)) | |
| (hproper : m + 1 < Nodes.height upper) | |
| (W : Submodule F2 (OwnInputReference.UpperVector rows upper)) | |
| (exterior : Exterior rows repeats ((Nodes.path upper).append pDown) slots) | |
| (assembled : LowerCutCallSplit.Assembled rows repeats ((Nodes.path upper).append pDown) slots) : | |
| knownBuckets rows repeats slots upper pDown hproper W exterior assembled = | |
| knownBucketsOfComplement rows repeats slots upper pDown hproper W exterior | |
| (LowerCutCallSplit.complement rows repeats ((Nodes.path upper).append pDown) slots assembled) := by | |
| have h := buckets_congr_otherCalls rows repeats (Nodes.path upper) pDown hproper slots exterior | |
| (LowerCutCallSplit.merge rows repeats ((Nodes.path upper).append pDown) slots | |
| (LowerCutCallSplit.complement rows repeats ((Nodes.path upper).append pDown) slots assembled) 0) | |
| assembled (fun call hcall => | |
| (LowerCutCallSplit.merge_otherCall rows repeats ((Nodes.path upper).append pDown) slots | |
| (LowerCutCallSplit.complement rows repeats ((Nodes.path upper).append pDown) slots assembled) | |
| 0 ⟨call, hcall⟩).symm) | |
| funext v | |
| exact congrFun h v.val | |
| theorem knownBucketsAtNode_eq_complement (rows repeats : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) (upper : Nodes branch n) | |
| (pDown : Path branch (Nodes.height upper) (m + 1)) | |
| (hproper : m + 1 < Nodes.height upper) | |
| (W : Submodule F2 (OwnInputReference.UpperVector rows upper)) | |
| (tape : WholeCutSampler.Tape rows repeats ((Nodes.path upper).append pDown) slots) : | |
| WholeCutSubtree.knownBucketsAtNode rows repeats slots upper pDown hproper W tape = | |
| knownBucketsOfComplement rows repeats slots upper pDown hproper W | |
| (splitTape rows repeats ((Nodes.path upper).append pDown) slots tape).1 | |
| (LowerCutCallSplit.complement rows repeats ((Nodes.path upper).append pDown) slots | |
| (CutChildGrouping.assemble (cutSlots ((Nodes.path upper).append pDown) slots) rows | |
| (splitTape rows repeats ((Nodes.path upper).append pDown) slots tape).2)) := by | |
| have hreader := buckets_splitTape rows repeats (Nodes.path upper) pDown hproper slots tape | |
| have hknown : WholeCutSubtree.knownBucketsAtNode rows repeats slots upper pDown hproper W tape = | |
| knownBuckets rows repeats slots upper pDown hproper W | |
| (splitTape rows repeats ((Nodes.path upper).append pDown) slots tape).1 | |
| (CutChildGrouping.assemble (cutSlots ((Nodes.path upper).append pDown) slots) rows | |
| (splitTape rows repeats ((Nodes.path upper).append pDown) slots tape).2) := by | |
| funext v | |
| exact (congrFun hreader v.val).symm | |
| exact hknown.trans (knownBuckets_eq_complement rows repeats slots upper pDown hproper W _ _) | |
| theorem knownBucketsAtNode_join (rows repeats : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) (upper : Nodes branch n) | |
| (pDown : Path branch (Nodes.height upper) (m + 1)) | |
| (hproper : m + 1 < Nodes.height upper) | |
| (W : Submodule F2 (OwnInputReference.UpperVector rows upper)) | |
| (exterior : Exterior rows repeats ((Nodes.path upper).append pDown) slots) | |
| (raw : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats ((Nodes.path upper).append pDown)) | |
| (cutSlots ((Nodes.path upper).append pDown) slots) rows) : | |
| WholeCutSubtree.knownBucketsAtNode rows repeats slots upper pDown hproper W | |
| ((splitTape rows repeats ((Nodes.path upper).append pDown) slots).symm (exterior, raw)) = | |
| knownBucketsOfComplement rows repeats slots upper pDown hproper W exterior | |
| (LowerCutCallSplit.complement rows repeats ((Nodes.path upper).append pDown) slots | |
| (CutChildGrouping.assemble (cutSlots ((Nodes.path upper).append pDown) slots) rows raw)) := by | |
| have h := knownBucketsAtNode_eq_complement rows repeats slots upper pDown hproper W | |
| ((splitTape rows repeats ((Nodes.path upper).append pDown) slots).symm (exterior, raw)) | |
| rw [Equiv.apply_symm_apply] at h | |
| exact h | |
| theorem knownBucketsAtNode_erase_eq (rows repeats : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) (upper : Nodes branch n) | |
| (pDown : Path branch (Nodes.height upper) (m + 1)) | |
| (hproper : m + 1 < Nodes.height upper) (clean : Fin (branch m) → Prop) | |
| (W : Submodule F2 (OwnInputReference.UpperVector rows upper)) | |
| (tape : WholeCutSampler.Tape rows repeats ((Nodes.path upper).append pDown) slots) | |
| (hzero : WholeCutZero.ZeroAtClean rows repeats ((Nodes.path upper).append pDown) slots clean tape) : | |
| WholeReplaySubtree.knownBucketsAtNode rows repeats slots upper pDown hproper clean W | |
| (WholeCutReplay.erase rows repeats ((Nodes.path upper).append pDown) slots clean tape) = | |
| knownBucketsOfComplement rows repeats slots upper pDown hproper W | |
| (splitTape rows repeats ((Nodes.path upper).append pDown) slots tape).1 | |
| (LowerCutCallSplit.complement rows repeats ((Nodes.path upper).append pDown) slots | |
| (CutChildGrouping.assemble (cutSlots ((Nodes.path upper).append pDown) slots) rows | |
| (splitTape rows repeats ((Nodes.path upper).append pDown) slots tape).2)) := | |
| (WholeCutSubtree.knownBucketsAtNode_erase rows repeats slots upper pDown hproper clean W tape hzero).trans | |
| (knownBucketsAtNode_eq_complement rows repeats slots upper pDown hproper W tape) | |
| theorem knownBucketsAtNode_assemble_splitTape (rows repeats : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) (upper : Nodes branch n) | |
| (pDown : Path branch (Nodes.height upper) (m + 1)) | |
| (hproper : m + 1 < Nodes.height upper) (clean : Fin (branch m) → Prop) | |
| (W : Submodule F2 (OwnInputReference.UpperVector rows upper)) | |
| (tape : WholeCutSampler.Tape rows repeats ((Nodes.path upper).append pDown) slots) | |
| (hzero : WholeCutZero.ZeroAtClean rows repeats ((Nodes.path upper).append pDown) slots clean tape) : | |
| WholeReplaySubtree.knownBucketsAtNode rows repeats slots upper pDown hproper clean W | |
| (WholeCutReplayGrouping.assemble rows repeats ((Nodes.path upper).append pDown) slots clean | |
| (splitTape rows repeats ((Nodes.path upper).append pDown) slots tape).1 | |
| (fun child => (splitTape rows repeats ((Nodes.path upper).append pDown) slots tape).2 child.val)) = | |
| knownBucketsOfComplement rows repeats slots upper pDown hproper W | |
| (splitTape rows repeats ((Nodes.path upper).append pDown) slots tape).1 | |
| (LowerCutCallSplit.complement rows repeats ((Nodes.path upper).append pDown) slots | |
| (CutChildGrouping.assemble (cutSlots ((Nodes.path upper).append pDown) slots) rows | |
| (splitTape rows repeats ((Nodes.path upper).append pDown) slots tape).2)) := by | |
| exact (congrArg (WholeReplaySubtree.knownBucketsAtNode rows repeats slots upper pDown hproper clean W) | |
| (WholeCutReplayGrouping.assemble_splitTape rows repeats ((Nodes.path upper).append pDown) | |
| slots clean tape)).trans | |
| (knownBucketsAtNode_erase_eq rows repeats slots upper pDown hproper clean W tape hzero) | |
| theorem knownBucketsAtNode_assemble (rows repeats : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) (upper : Nodes branch n) | |
| (pDown : Path branch (Nodes.height upper) (m + 1)) | |
| (hproper : m + 1 < Nodes.height upper) (clean : Fin (branch m) → Prop) | |
| (W : Submodule F2 (OwnInputReference.UpperVector rows upper)) | |
| (exterior : Exterior rows repeats ((Nodes.path upper).append pDown) slots) | |
| (raw : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats ((Nodes.path upper).append pDown)) | |
| (cutSlots ((Nodes.path upper).append pDown) slots) rows) | |
| (hzero : WholeCutZero.ZeroAtClean rows repeats ((Nodes.path upper).append pDown) slots clean | |
| ((splitTape rows repeats ((Nodes.path upper).append pDown) slots).symm (exterior, raw))) : | |
| WholeReplaySubtree.knownBucketsAtNode rows repeats slots upper pDown hproper clean W | |
| (WholeCutReplayGrouping.assemble rows repeats ((Nodes.path upper).append pDown) slots clean | |
| exterior (fun child => raw child.val)) = | |
| knownBucketsOfComplement rows repeats slots upper pDown hproper W exterior | |
| (LowerCutCallSplit.complement rows repeats ((Nodes.path upper).append pDown) slots | |
| (CutChildGrouping.assemble (cutSlots ((Nodes.path upper).append pDown) slots) rows raw)) := by | |
| have h := knownBucketsAtNode_assemble_splitTape rows repeats slots upper pDown hproper clean W | |
| ((splitTape rows repeats ((Nodes.path upper).append pDown) slots).symm (exterior, raw)) hzero | |
| rw [Equiv.apply_symm_apply] at h | |
| exact h | |
| end | |
| end PerfectCompleteness.LowerCutBucketLocality | |
| end | |
| end OAI | |