openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Construction /StoppedBucketRows.lean
| import OAI.Computability.PerfectCompleteness.Decoding.SelectedArrayProjection | |
| import OAI.Computability.PerfectCompleteness.Sampling.WholeArrayBucketsLaw | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.StoppedBucketRows | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open WholeArraySubtreeSplit SelectedArrayReplacement | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| variable {branch : Nat → Nat} {N n m t : Nat} | |
| theorem selectedRows_evaluate (rows repeats : Nat → Nat) | |
| (p : Path branch N (n + 1)) (chosen : Fin (branch n)) (q : Path branch n m) | |
| (slots : Slots branch N → Fin t → MixedSupport.Slot) | |
| (tape : WholeArraySampler.Tape rows repeats (p.append (.step chosen q)) slots) : | |
| selectedRows rows p slots | |
| (WholeArraySampler.evaluate rows repeats (p.append (.step chosen q)) slots tape) = | |
| BucketSampler.evaluate (rows (n + 1)) | |
| (RecursiveSampler.evaluate F2 repeats (.step chosen q) | |
| (LeafDomain (WholeCutGrouping.cutSlots p slots))) | |
| (WholeArrayBucketsLaw.split rows repeats p chosen q slots tape).1 := by | |
| exact congrArg | |
| (fun arrays : Arrays (subtreeSlots p slots) rows => arrays (.inl ())) | |
| (restrictArrays_evaluate rows repeats p (.step chosen q) slots tape) | |
| theorem pullback_bucket_evaluate {Ω : Type*} (width : Nat) | |
| {left right : Slots branch n → Fin t → MixedSupport.Slot} | |
| (projection : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (eval : Ω → H right) (tape : BucketSampler.Tape width Ω) : | |
| (fun row => HPullback projection (BucketSampler.evaluate width eval tape row)) = | |
| BucketSampler.evaluate width id (fun direction => HPullback projection (eval (tape direction))) := by | |
| funext row | |
| simp only [BucketSampler.evaluate, map_sum, map_smul, id_eq] | |
| theorem selectedRows_pullback_evaluate (rows repeats : Nat → Nat) | |
| (p : Path branch N (n + 1)) (chosen : Fin (branch n)) (q : Path branch n m) | |
| (left right : Slots branch N → Fin t → MixedSupport.Slot) | |
| (projection : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (tape : WholeArraySampler.Tape rows repeats (p.append (.step chosen q)) right) : | |
| selectedRows rows p left | |
| (ChildBlockProjection.arraysPullback rows projection | |
| (WholeArraySampler.evaluate rows repeats (p.append (.step chosen q)) right tape)) = | |
| BucketSampler.evaluate (rows (n + 1)) id | |
| (fun direction => HPullback (CutGroupedProjection.cutProjection p projection) | |
| (RecursiveSampler.evaluate F2 repeats (.step chosen q) | |
| (LeafDomain (WholeCutGrouping.cutSlots p right)) | |
| ((WholeArrayBucketsLaw.split rows repeats p chosen q right tape).1 direction))) := by | |
| rw [SelectedArrayProjection.selectedRows_pullback] | |
| simp only [selectedRows_evaluate rows repeats p chosen q right tape] | |
| exact pullback_bucket_evaluate (rows (n + 1)) | |
| (CutGroupedProjection.cutProjection p projection) | |
| (RecursiveSampler.evaluate F2 repeats (.step chosen q) | |
| (LeafDomain (WholeCutGrouping.cutSlots p right))) | |
| (WholeArrayBucketsLaw.split rows repeats p chosen q right tape).1 | |
| end | |
| end PerfectCompleteness.StoppedBucketRows | |
| end | |
| end OAI | |