openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /LowerCutPairBackground.lean
| import OAI.Computability.PerfectCompleteness.Algebra.HierarchicalMatrixTable | |
| import OAI.Computability.PerfectCompleteness.Decoding.LowerCutNodeCoordinates | |
| import OAI.Computability.PerfectCompleteness.Decoding.LowerCutPairLemmas | |
| import OAI.Computability.PerfectCompleteness.Decoding.LowerCutRowsLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.LowerCutPairBackground | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open WholeCutGrouping | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped Classical | |
| abbrev F2 := ZMod 2 | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n m t : Nat} | |
| (rows repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| abbrev Complement := LowerCutCallSplit.Complement rows repeats p slots | |
| abbrev NativeRows := LowerCutPair.NativeRows rows p slots | |
| abbrev NativeMatrix := LowerCutPair.NativeMatrix rows p slots | |
| abbrev Observation := Complement rows repeats p slots × | |
| (NativeRows rows p slots × H (cutSlots p slots)) | |
| def splitObservation : | |
| (CutChildGrouping.Assembled (C := LowerCutPair.Calls rows repeats p) | |
| (cutSlots p slots) rows × H (cutSlots p slots)) →ₗ[F2] | |
| Observation rows repeats p slots where | |
| toFun z := | |
| ((LowerCutCallSplit.splitAssembled rows repeats p slots z.1).1, | |
| ((LowerCutCallSplit.splitAssembled rows repeats p slots z.1).2, z.2)) | |
| map_add' _ _ := rfl | |
| map_smul' _ _ := rfl | |
| theorem splitObservation_surjective : | |
| Function.Surjective (splitObservation rows repeats p slots) := by | |
| rintro ⟨c, S, h⟩ | |
| obtain ⟨record, hrecord⟩ := | |
| LowerCutCallSplit.splitAssembled_surjective rows repeats p slots (c, S) | |
| refine ⟨(record, h), ?_⟩ | |
| change ((LowerCutCallSplit.splitAssembled rows repeats p slots record).1, | |
| ((LowerCutCallSplit.splitAssembled rows repeats p slots record).2, h)) = (c, S, h) | |
| rw [hrecord] | |
| def observe (raw : LowerCutPair.Raw rows repeats p slots) : | |
| Observation rows repeats p slots := | |
| splitObservation rows repeats p slots (CutCallExtension.observe (cutSlots p slots) rows raw) | |
| theorem observe_law : | |
| (CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p)) | |
| (cutSlots p slots) rows).pushforward (observe rows repeats p slots) = | |
| (FiniteDistribution.uniform (Complement rows repeats p slots)).product | |
| ((FiniteDistribution.uniform (NativeRows rows p slots)).product | |
| (FiniteDistribution.uniform (H (cutSlots p slots)))) := by | |
| change (CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p)) | |
| (cutSlots p slots) rows).pushforward | |
| (fun raw => splitObservation rows repeats p slots | |
| (CutCallExtension.observe (cutSlots p slots) rows raw)) = _ | |
| rw [← FiniteDistribution.pushforward_comp, CutCallExtension.observe_law, | |
| WholeCutSampler.uniform_product, | |
| UniformLinearImage.uniform_pushforward_linearMap | |
| (splitObservation rows repeats p slots) (splitObservation_surjective rows repeats p slots)] | |
| rw [WholeCutSampler.uniform_product, WholeCutSampler.uniform_product] | |
| @[simp] theorem observe_rows (exterior : Exterior rows repeats p slots) | |
| (raw : LowerCutPair.Raw rows repeats p slots) : | |
| (observe rows repeats p slots raw).2.1 = | |
| SelectedArrayReplacement.selectedRows rows p slots | |
| (LowerCutPair.first rows repeats p slots exterior raw) := by | |
| exact (LowerCutRows.selectedRows_reconstruct_eq rows repeats p slots exterior _).symm | |
| @[simp] theorem observe_fresh (raw : LowerCutPair.Raw rows repeats p slots) : | |
| (observe rows repeats p slots raw).2.2 = LowerCutPair.fresh rows repeats p slots raw := rfl | |
| def rowPair (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) | |
| (raw : LowerCutPair.Raw rows repeats p slots) : | |
| Complement rows repeats p slots × (NativeRows rows p slots × NativeRows rows p slots) := | |
| ((observe rows repeats p slots raw).1, | |
| LowerCutPair.selectedPair rows repeats p slots exterior a raw) | |
| theorem rowPair_law (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) : | |
| (CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p)) | |
| (cutSlots p slots) rows).pushforward (rowPair rows repeats p slots exterior a) = | |
| (FiniteDistribution.uniform (Complement rows repeats p slots)).product | |
| (LowerDirectionFiber.fiberPairLaw (H := H (cutSlots p slots)) a) := by | |
| have h := congrArg | |
| (fun μ : FiniteDistribution (Observation rows repeats p slots) => μ.pushforward | |
| (fun z => (z.1, (z.2.1, LowerDirectionFiber.replace a z.2.1 z.2.2)))) | |
| (observe_law rows repeats p slots) | |
| rw [FiniteDistribution.pushforward_comp] at h | |
| have hp := FiniteDistribution.product_pushforward | |
| (FiniteDistribution.uniform (Complement rows repeats p slots)) | |
| ((FiniteDistribution.uniform (NativeRows rows p slots)).product | |
| (FiniteDistribution.uniform (H (cutSlots p slots)))) | |
| (id : Complement rows repeats p slots → Complement rows repeats p slots) | |
| (fun z => (z.1, LowerDirectionFiber.replace a z.1 z.2)) | |
| simp only [FiniteDistribution.pushforward_id, id_eq, LowerDirectionFiber.two_copy_law] at hp | |
| have hfinal := h.trans hp | |
| change (CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p)) | |
| (cutSlots p slots) rows).pushforward (fun raw => | |
| ((observe rows repeats p slots raw).1, | |
| (SelectedArrayReplacement.selectedRows rows p slots | |
| (LowerCutPair.first rows repeats p slots exterior raw), | |
| SelectedArrayReplacement.selectedRows rows p slots | |
| (LowerCutPair.second rows repeats p slots exterior a raw)))) = _ | |
| simpa only [LowerCutPair.selected_second, | |
| observe_rows rows repeats p slots exterior, observe_fresh] using hfinal | |
| attribute [local instance] UniqueGamesTheorem.Appendix.RankLevelFilter.linearMapFintype | |
| def matrixPair (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) | |
| (raw : LowerCutPair.Raw rows repeats p slots) : | |
| Complement rows repeats p slots × (NativeMatrix rows p slots × NativeMatrix rows p slots) := | |
| ((observe rows repeats p slots raw).1, | |
| LowerCutPair.matrixPair rows repeats p slots exterior a raw) | |
| theorem matrixPair_law (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) : | |
| (CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p)) | |
| (cutSlots p slots) rows).pushforward (matrixPair rows repeats p slots exterior a) = | |
| (FiniteDistribution.uniform (Complement rows repeats p slots)).product | |
| (((FiniteDistribution.uniform (NativeMatrix rows p slots)).product | |
| (FiniteDistribution.uniform (H (cutSlots p slots)))).pushforward | |
| (fun z => (z.1, EvaluationMatrix.shift (H (cutSlots p slots)) z.1 a.val z.2))) := by | |
| have h := congrArg | |
| (fun μ : FiniteDistribution | |
| (Complement rows repeats p slots × (NativeRows rows p slots × NativeRows rows p slots)) => | |
| μ.pushforward (fun z => (z.1, | |
| (EvaluationMatrix.ofRows (H (cutSlots p slots)) z.2.1, | |
| EvaluationMatrix.ofRows (H (cutSlots p slots)) z.2.2)))) | |
| (rowPair_law rows repeats p slots exterior a) | |
| rw [FiniteDistribution.pushforward_comp] at h | |
| have hp := FiniteDistribution.product_pushforward | |
| (FiniteDistribution.uniform (Complement rows repeats p slots)) | |
| (LowerDirectionFiber.fiberPairLaw (H := H (cutSlots p slots)) a) | |
| (id : Complement rows repeats p slots → Complement rows repeats p slots) | |
| (fun z => (EvaluationMatrix.ofRows (H (cutSlots p slots)) z.1, | |
| EvaluationMatrix.ofRows (H (cutSlots p slots)) z.2)) | |
| simp only [FiniteDistribution.pushforward_id, id_eq, LowerDirectionFiber.matrix_pair_law] at hp | |
| exact h.trans hp | |
| def backgroundArrays (exterior : Exterior rows repeats p slots) | |
| (c : Complement rows repeats p slots) : Arrays slots rows := | |
| LowerCutAssembled.reconstruct rows repeats p slots exterior | |
| (LowerCutCallSplit.merge rows repeats p slots c 0) | |
| def installRows (exterior : Exterior rows repeats p slots) | |
| (c : Complement rows repeats p slots) (S : NativeRows rows p slots) : Arrays slots rows := | |
| SelectedArrayReplacement.replace rows p slots | |
| (backgroundArrays rows repeats p slots exterior c) S | |
| theorem reconstruct_merge (exterior : Exterior rows repeats p slots) | |
| (c : Complement rows repeats p slots) (S : NativeRows rows p slots) : | |
| LowerCutAssembled.reconstruct rows repeats p slots exterior | |
| (LowerCutCallSplit.merge rows repeats p slots c S) = | |
| installRows rows repeats p slots exterior c S := by | |
| have h := LowerCutAssembled.reconstruct_congr_replace rows repeats p slots exterior | |
| (LowerCutCallSplit.merge rows repeats p slots c 0) | |
| (LowerCutCallSplit.merge rows repeats p slots c S) | |
| (fun call hcall => | |
| (LowerCutCallSplit.merge_otherCall rows repeats p slots c S ⟨call, hcall⟩).trans | |
| (LowerCutCallSplit.merge_otherCall rows repeats p slots c 0 ⟨call, hcall⟩).symm) rfl | |
| rw [LowerCutCallSplit.selectRows_merge] at h | |
| exact h | |
| theorem first_eq_installRows (exterior : Exterior rows repeats p slots) | |
| (raw : LowerCutPair.Raw rows repeats p slots) : | |
| LowerCutPair.first rows repeats p slots exterior raw = | |
| installRows rows repeats p slots exterior (observe rows repeats p slots raw).1 | |
| (SelectedArrayReplacement.selectedRows rows p slots | |
| (LowerCutPair.first rows repeats p slots exterior raw)) := by | |
| rw [← observe_rows rows repeats p slots exterior] | |
| change LowerCutRows.reconstruct rows repeats p slots exterior | |
| (CutCallExtension.original (cutSlots p slots) rows raw) = | |
| installRows rows repeats p slots exterior | |
| (LowerCutCallSplit.splitAssembled rows repeats p slots | |
| (CutChildGrouping.assemble (cutSlots p slots) rows | |
| (CutCallExtension.original (cutSlots p slots) rows raw))).1 | |
| (LowerCutCallSplit.splitAssembled rows repeats p slots | |
| (CutChildGrouping.assemble (cutSlots p slots) rows | |
| (CutCallExtension.original (cutSlots p slots) rows raw))).2 | |
| rw [← LowerCutAssembled.reconstruct_assemble, ← reconstruct_merge, | |
| LowerCutCallSplit.merge_split] | |
| private theorem replace_replace (arrays : Arrays slots rows) | |
| (S T : NativeRows rows p slots) : | |
| SelectedArrayReplacement.replace rows p slots | |
| (SelectedArrayReplacement.replace rows p slots arrays S) T = | |
| SelectedArrayReplacement.replace rows p slots arrays T := by | |
| simp only [LowerCutNodeCoordinates.replace_eq_update] | |
| funext node | |
| by_cases hnode : node = WholeArrayInteriorExterior.upperNode p | |
| · subst node | |
| simp only [Function.update_self] | |
| · simp only [Function.update_of_ne hnode] | |
| theorem second_eq_installRows (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) | |
| (raw : LowerCutPair.Raw rows repeats p slots) : | |
| LowerCutPair.second rows repeats p slots exterior a raw = | |
| installRows rows repeats p slots exterior (observe rows repeats p slots raw).1 | |
| (SelectedArrayReplacement.selectedRows rows p slots | |
| (LowerCutPair.second rows repeats p slots exterior a raw)) := by | |
| rw [LowerCutPair.selected_second] | |
| exact (congrArg | |
| (fun arrays => SelectedArrayReplacement.replace rows p slots arrays | |
| (LowerDirectionFiber.replace a | |
| (SelectedArrayReplacement.selectedRows rows p slots | |
| (LowerCutPair.first rows repeats p slots exterior raw)) | |
| (LowerCutPair.fresh rows repeats p slots raw))) | |
| (first_eq_installRows rows repeats p slots exterior raw)).trans | |
| (replace_replace rows p slots _ _ _) | |
| def installMatrix (exterior : Exterior rows repeats p slots) | |
| (c : Complement rows repeats p slots) (X : NativeMatrix rows p slots) : Arrays slots rows := | |
| installRows rows repeats p slots exterior c (EvaluationMatrix.rows (H (cutSlots p slots)) X) | |
| theorem arraysPair_eq (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) | |
| (raw : LowerCutPair.Raw rows repeats p slots) : | |
| LowerCutPair.arraysPair rows repeats p slots exterior a raw = | |
| (installMatrix rows repeats p slots exterior (matrixPair rows repeats p slots exterior a raw).1 | |
| (matrixPair rows repeats p slots exterior a raw).2.1, | |
| installMatrix rows repeats p slots exterior (matrixPair rows repeats p slots exterior a raw).1 | |
| (matrixPair rows repeats p slots exterior a raw).2.2) := by | |
| simp only [matrixPair, LowerCutPair.matrixPair, LowerCutPair.selectedPair, | |
| installMatrix, EvaluationMatrix.rows_ofRows] | |
| exact Prod.ext (first_eq_installRows rows repeats p slots exterior raw) | |
| (second_eq_installRows rows repeats p slots exterior a raw) | |
| theorem arraysPair_law (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) : | |
| (CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p)) | |
| (cutSlots p slots) rows).pushforward (LowerCutPair.arraysPair rows repeats p slots exterior a) = | |
| ((FiniteDistribution.uniform (Complement rows repeats p slots)).product | |
| (((FiniteDistribution.uniform (NativeMatrix rows p slots)).product | |
| (FiniteDistribution.uniform (H (cutSlots p slots)))).pushforward | |
| (fun z => (z.1, EvaluationMatrix.shift (H (cutSlots p slots)) z.1 a.val z.2)))).pushforward | |
| (fun z => (installMatrix rows repeats p slots exterior z.1 z.2.1, | |
| installMatrix rows repeats p slots exterior z.1 z.2.2)) := by | |
| have h := congrArg | |
| (fun μ : FiniteDistribution | |
| (Complement rows repeats p slots × (NativeMatrix rows p slots × NativeMatrix rows p slots)) => | |
| μ.pushforward (fun z => (installMatrix rows repeats p slots exterior z.1 z.2.1, | |
| installMatrix rows repeats p slots exterior z.1 z.2.2))) | |
| (matrixPair_law rows repeats p slots exterior a) | |
| rw [FiniteDistribution.pushforward_comp] at h | |
| simpa only [← arraysPair_eq] using h | |
| theorem installMatrix_eq_update (exterior : Exterior rows repeats p slots) | |
| (c : Complement rows repeats p slots) (X : NativeMatrix rows p slots) : | |
| installMatrix rows repeats p slots exterior c X = | |
| Function.update (backgroundArrays rows repeats p slots exterior c) | |
| (WholeArrayInteriorExterior.upperNode p) | |
| (LowerCutNodeCoordinates.rowEquiv rows p slots | |
| (EvaluationMatrix.rows (H (cutSlots p slots)) X)) := | |
| LowerCutNodeCoordinates.replace_eq_update rows p slots _ _ | |
| theorem installMatrix_outside (exterior : Exterior rows repeats p slots) | |
| (c : Complement rows repeats p slots) (X : NativeMatrix rows p slots) | |
| (node : Nodes branch n) (hne : node ≠ WholeArrayInteriorExterior.upperNode p) : | |
| installMatrix rows repeats p slots exterior c X node = | |
| backgroundArrays rows repeats p slots exterior c node := | |
| SelectedArrayReplacement.replace_outside rows p slots _ _ node hne | |
| theorem backgroundOf_installMatrix (exterior : Exterior rows repeats p slots) | |
| (c : Complement rows repeats p slots) (X : NativeMatrix rows p slots) : | |
| HierarchicalMatrixTable.backgroundOf slots (WholeArrayInteriorExterior.upperNode p) | |
| (installMatrix rows repeats p slots exterior c X) = | |
| HierarchicalMatrixTable.backgroundOf slots (WholeArrayInteriorExterior.upperNode p) | |
| (backgroundArrays rows repeats p slots exterior c) := by | |
| funext node | |
| exact installMatrix_outside rows repeats p slots exterior c X node.val node.property | |
| theorem upperMatrix_installMatrix (exterior : Exterior rows repeats p slots) | |
| (c : Complement rows repeats p slots) (X : NativeMatrix rows p slots) | |
| (upper : Nodes branch n) (hne : upper ≠ WholeArrayInteriorExterior.upperNode p) : | |
| NodeEmbedding.matrix (installMatrix rows repeats p slots exterior c X) upper = | |
| NodeEmbedding.matrix (backgroundArrays rows repeats p slots exterior c) upper := by | |
| unfold NodeEmbedding.matrix NodeEmbedding.embeddedRows | |
| rw [installMatrix_outside rows repeats p slots exterior c X upper hne] | |
| end | |
| end PerfectCompleteness.LowerCutPairBackground | |
| end | |
| end OAI | |