openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /FinishedWitness.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Arithmetic.MatrixMapDecidable | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.FinalState | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialZeroFinish | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ZeroFinish | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.StageCFinish | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.Terminal | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldFinishedWitness | |
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily | |
| open AllFieldTerminalStatisticLaws AllFieldTerminalRates | |
| open scoped BigOperators Classical | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| private def localMapWithFintypes {F : Type*} [Field F] | |
| {X Y Z X' Y' Z' : Type} [newX : Fintype X] [newY : Fintype Y] [newZ : Fintype Z] | |
| {oldX : Fintype X} {oldY : Fintype Y} {oldZ : Fintype Z} | |
| {T : Tensor F X Y Z} {Q : Tensor F X' Y' Z'} | |
| (M : @LocalMap F _ X Y Z X' Y' Z' oldX oldY oldZ T Q) : | |
| @LocalMap F _ X Y Z X' Y' Z' newX newY newZ T Q := by | |
| cases Subsingleton.elim oldX newX | |
| cases Subsingleton.elim oldY newY | |
| cases Subsingleton.elim oldZ newZ | |
| exact M | |
| private def matrixProductMap (F : Type*) [Field F] | |
| (A B C D E G : Type) [Fintype A] [Fintype B] [Fintype C] | |
| [Fintype D] [Fintype E] [Fintype G] : | |
| LocalMap (Tensor.product (Tensor.matrixCoefficients (K := F) A B C) | |
| (Tensor.matrixCoefficients D E G)) | |
| (Tensor.matrixCoefficients (A × D) (B × E) (C × G)) := by | |
| apply LocalMap.ofExists | |
| refine ⟨_, _, _, (Tensor.pullback_eq_restrict | |
| (fun x : (A × D) × (B × E) => ((x.1.1, x.2.1), (x.1.2, x.2.2))) | |
| (fun y : (B × E) × (C × G) => ((y.1.1, y.2.1), (y.1.2, y.2.2))) | |
| (fun z : (C × G) × (A × D) => ((z.1.1, z.2.1), (z.1.2, z.2.2))) _).symm.trans ?_⟩ | |
| funext x y z | |
| exact Tensor.matrixCoefficients_product x y z | |
| private def combineMatrixMaps {F : Type*} [Field F] | |
| {X Y Z U V W A B C D E G : Type} | |
| [Fintype X] [Fintype Y] [Fintype Z] [Fintype U] [Fintype V] [Fintype W] | |
| [Fintype A] [Fintype B] [Fintype C] [Fintype D] [Fintype E] [Fintype G] | |
| {T : Tensor F X Y Z} {Q : Tensor F U V W} | |
| (first : LocalMap T (Tensor.matrixCoefficients A B C)) | |
| (second : LocalMap Q (Tensor.matrixCoefficients D E G)) : | |
| LocalMap (Tensor.product T Q) | |
| (Tensor.matrixCoefficients (A × D) (B × E) (C × G)) := | |
| (first.product second).comp (matrixProductMap F A B C D E G) | |
| variable {K : ℕ} (allocation : Allocation) (m : ℕ) | |
| abbrev ZeroRows := ∀ h : TerminalZero K, AllFieldZeroFinish.Row allocation m h | |
| abbrev ZeroMiddles := ∀ h : TerminalZero K, AllFieldZeroFinish.Middle allocation m h | |
| abbrev ZeroColumns := ∀ h : TerminalZero K, AllFieldZeroFinish.Column allocation m h | |
| abbrev Rows := AllFieldInitialZeroFinish.Rows (K := K) allocation (terminalDilation K m) × | |
| (ZeroRows (K := K) allocation m × | |
| AllFieldStageCFinish.Rows (K := K) allocation (terminalDilation K m)) | |
| abbrev Middles := AllFieldInitialZeroFinish.Middles (K := K) allocation (terminalDilation K m) × | |
| (ZeroMiddles (K := K) allocation m × | |
| AllFieldStageCFinish.Middles (K := K) allocation (terminalDilation K m)) | |
| abbrev Columns := AllFieldInitialZeroFinish.Columns (K := K) allocation (terminalDilation K m) × | |
| (ZeroColumns (K := K) allocation m × | |
| AllFieldStageCFinish.Columns (K := K) allocation (terminalDilation K m)) | |
| local instance initialRowsFintype : | |
| Fintype (AllFieldInitialZeroFinish.Rows (K := K) allocation (terminalDilation K m)) := by | |
| letI : DecidableEq (AllFieldInitialZeroFinish.Index K) := Classical.decEq _ | |
| exact inferInstanceAs (Fintype (∀ h : AllFieldInitialZeroFinish.Index K, | |
| AllFieldInitialZeroFinish.RowIndex allocation (terminalDilation K m) h)) | |
| local instance initialMiddlesFintype : | |
| Fintype (AllFieldInitialZeroFinish.Middles (K := K) allocation (terminalDilation K m)) := by | |
| letI : DecidableEq (AllFieldInitialZeroFinish.Index K) := Classical.decEq _ | |
| exact inferInstanceAs (Fintype (∀ h : AllFieldInitialZeroFinish.Index K, | |
| AllFieldInitialZeroFinish.MiddleIndex allocation (terminalDilation K m) h)) | |
| local instance initialColumnsFintype : | |
| Fintype (AllFieldInitialZeroFinish.Columns (K := K) allocation (terminalDilation K m)) := by | |
| letI : DecidableEq (AllFieldInitialZeroFinish.Index K) := Classical.decEq _ | |
| exact inferInstanceAs (Fintype (∀ h : AllFieldInitialZeroFinish.Index K, | |
| AllFieldInitialZeroFinish.ColumnIndex allocation (terminalDilation K m) h)) | |
| local instance stageCRowsFintype : | |
| Fintype (AllFieldStageCFinish.Rows (K := K) allocation (terminalDilation K m)) := by | |
| exact inferInstanceAs (Fintype (∀ h : CWStageCProducts.PopulationHistory K, | |
| CWStageCProducts.RowWords (CWStageCProducts.populationSide h) | |
| (CWStageCProducts.populationAtomCounts allocation (terminalDilation K m) h))) | |
| local instance stageCMiddlesFintype : | |
| Fintype (AllFieldStageCFinish.Middles (K := K) allocation (terminalDilation K m)) := by | |
| exact inferInstanceAs (Fintype (∀ h : CWStageCProducts.PopulationHistory K, | |
| CWStageCProducts.MiddleWords (CWStageCProducts.populationSide h) | |
| (CWStageCProducts.populationAtomCounts allocation (terminalDilation K m) h))) | |
| local instance stageCColumnsFintype : | |
| Fintype (AllFieldStageCFinish.Columns (K := K) allocation (terminalDilation K m)) := by | |
| exact inferInstanceAs (Fintype (∀ h : CWStageCProducts.PopulationHistory K, | |
| CWStageCProducts.ColumnWords (CWStageCProducts.populationSide h) | |
| (CWStageCProducts.populationAtomCounts allocation (terminalDilation K m) h))) | |
| local instance rowsFintype : Fintype (Rows (K := K) allocation m) := | |
| inferInstanceAs (Fintype | |
| (AllFieldInitialZeroFinish.Rows (K := K) allocation (terminalDilation K m) × | |
| (ZeroRows (K := K) allocation m × | |
| AllFieldStageCFinish.Rows (K := K) allocation (terminalDilation K m)))) | |
| local instance middlesFintype : Fintype (Middles (K := K) allocation m) := | |
| inferInstanceAs (Fintype | |
| (AllFieldInitialZeroFinish.Middles (K := K) allocation (terminalDilation K m) × | |
| (ZeroMiddles (K := K) allocation m × | |
| AllFieldStageCFinish.Middles (K := K) allocation (terminalDilation K m)))) | |
| local instance columnsFintype : Fintype (Columns (K := K) allocation m) := | |
| inferInstanceAs (Fintype | |
| (AllFieldInitialZeroFinish.Columns (K := K) allocation (terminalDilation K m) × | |
| (ZeroColumns (K := K) allocation m × | |
| AllFieldStageCFinish.Columns (K := K) allocation (terminalDilation K m)))) | |
| local instance initialRawFintype : | |
| Fintype (AllFieldFinalState.InitialWords (K := K) allocation (terminalDilation K m)) := | |
| inferInstanceAs (Fintype (∀ h : AllFieldFinalState.InitialIndex K, | |
| HistoryWord allocation (terminalDilation K m) (.initial h.1.val, h.2))) | |
| local instance zeroRawFintype : | |
| Fintype (AllFieldFinalState.ZeroWords (K := K) allocation (terminalDilation K m)) := | |
| inferInstanceAs (Fintype (∀ h : TerminalZero K, | |
| HistoryWord allocation (terminalDilation K m) (AllFieldTerminalStatisticLaws.history h))) | |
| local instance stageCRawFintype : | |
| Fintype (AllFieldFinalState.CWords (K := K) allocation (terminalDilation K m)) := | |
| inferInstanceAs (Fintype (∀ h : AllFieldFinalState.CIndex K, | |
| HistoryWord allocation (terminalDilation K m) (.afterC h.1, h.2))) | |
| theorem zero_volume : | |
| Fintype.card (ZeroRows (K := K) allocation m) * | |
| Fintype.card (ZeroMiddles (K := K) allocation m) * | |
| Fintype.card (ZeroColumns (K := K) allocation m) = zeroVolume (K := K) allocation m := by | |
| simp only [ZeroRows, ZeroMiddles, ZeroColumns, Fintype.card_pi, zeroVolume] | |
| rw [← Finset.prod_mul_distrib, ← Finset.prod_mul_distrib] | |
| exact Finset.prod_congr rfl fun h _ => AllFieldZeroFinish.volume allocation m h | |
| theorem volume_eq : | |
| Fintype.card (Rows (K := K) allocation m) * | |
| Fintype.card (Middles (K := K) allocation m) * | |
| Fintype.card (Columns (K := K) allocation m) = terminalVolume (K := K) allocation m := by | |
| simp only [Rows, Middles, Columns, Fintype.card_prod] | |
| calc | |
| _ = (Fintype.card (AllFieldInitialZeroFinish.Rows (K := K) allocation (terminalDilation K m)) * | |
| Fintype.card (AllFieldInitialZeroFinish.Middles (K := K) allocation (terminalDilation K m)) * | |
| Fintype.card (AllFieldInitialZeroFinish.Columns (K := K) allocation (terminalDilation K m))) * | |
| (Fintype.card (ZeroRows (K := K) allocation m) * | |
| Fintype.card (ZeroMiddles (K := K) allocation m) * | |
| Fintype.card (ZeroColumns (K := K) allocation m)) * | |
| (Fintype.card (AllFieldStageCFinish.Rows (K := K) allocation (terminalDilation K m)) * | |
| Fintype.card (AllFieldStageCFinish.Middles (K := K) allocation (terminalDilation K m)) * | |
| Fintype.card (AllFieldStageCFinish.Columns (K := K) allocation (terminalDilation K m))) := by ring | |
| _ = _ := by rw [AllFieldInitialZeroFinish.volume_eq, zero_volume, | |
| AllFieldStageCFinish.volume_eq_populationVolume]; rfl | |
| theorem dimensions_pos : | |
| 0 < Fintype.card (Rows (K := K) allocation m) ∧ | |
| 0 < Fintype.card (Middles (K := K) allocation m) ∧ | |
| 0 < Fintype.card (Columns (K := K) allocation m) := by | |
| have hp := terminalVolume_pos (K := K) allocation m | |
| rw [← volume_eq allocation m] at hp | |
| have hab := Nat.pos_of_mul_pos_right hp | |
| exact ⟨Nat.pos_of_mul_pos_right hab, Nat.pos_of_mul_pos_left hab, | |
| Nat.pos_of_mul_pos_left hp⟩ | |
| def zeroMap (F : Type*) [Field F] {ε : ℝ} (hm : 0 < m) (hε : 0 ≤ ε) : | |
| LocalMap (AllFieldFinalState.zeroTensor (K := K) allocation (terminalDilation K m) F ε) | |
| (Tensor.matrixCoefficients (ZeroRows (K := K) allocation m) | |
| (ZeroMiddles (K := K) allocation m) (ZeroColumns (K := K) allocation m)) := by | |
| let result := AllFieldTerminal.terminalProductMap | |
| (fun h : TerminalZero K => historyTensor F allocation (terminalDilation K m) ε | |
| (AllFieldTerminalStatisticLaws.history h)) | |
| (fun h => AllFieldZeroFinish.Row allocation m h) | |
| (fun h => AllFieldZeroFinish.Middle allocation m h) | |
| (fun h => AllFieldZeroFinish.Column allocation m h) | |
| (fun h => LocalMap.redecideMatrix | |
| (AllFieldZeroFinish.localMap F allocation hm hε h) _ _ _) | |
| let aligned := localMapWithFintypes (F := F) | |
| (X := AllFieldFinalState.ZeroWords (K := K) allocation (terminalDilation K m)) | |
| (Y := AllFieldFinalState.ZeroWords (K := K) allocation (terminalDilation K m)) | |
| (Z := AllFieldFinalState.ZeroWords (K := K) allocation (terminalDilation K m)) | |
| (newX := inferInstance) (newY := inferInstance) (newZ := inferInstance) result | |
| exact LocalMap.redecideMatrix aligned _ _ _ | |
| def terminalMap (F : Type*) [Field F] {ε : ℝ} (hm : 0 < m) (hε : 0 ≤ ε) : | |
| LocalMap (stateTensor F (K := K) allocation (terminalDilation K m) ε (K + 2)) | |
| (Tensor.matrixCoefficients (Rows (K := K) allocation m) | |
| (Middles (K := K) allocation m) (Columns (K := K) allocation m)) := by | |
| let initialAligned := localMapWithFintypes (F := F) | |
| (X := AllFieldInitialZeroFinish.RawWords (K := K) allocation (terminalDilation K m)) | |
| (Y := AllFieldInitialZeroFinish.RawWords (K := K) allocation (terminalDilation K m)) | |
| (Z := AllFieldInitialZeroFinish.RawWords (K := K) allocation (terminalDilation K m)) | |
| (newX := inferInstance) (newY := inferInstance) (newZ := inferInstance) | |
| (AllFieldInitialZeroFinish.finishMap (K := K) allocation (terminalDilation K m) F ε) | |
| let initial := LocalMap.redecideMatrix initialAligned | |
| (Classical.decEq _) (Classical.decEq _) (Classical.decEq _) | |
| let zeros := LocalMap.redecideMatrix (zeroMap (K := K) allocation m F hm hε) | |
| (Classical.decEq _) (Classical.decEq _) (Classical.decEq _) | |
| let interior := LocalMap.redecideMatrix | |
| (AllFieldStageCFinish.finishMap (K := K) allocation (terminalDilation K m) F ε) | |
| (Classical.decEq _) (Classical.decEq _) (Classical.decEq _) | |
| let second := LocalMap.redecideMatrix | |
| (combineMatrixMaps zeros interior) | |
| (Classical.decEq _) (Classical.decEq _) (Classical.decEq _) | |
| let third := combineMatrixMaps initial second | |
| exact LocalMap.redecideMatrix | |
| ((AllFieldFinalState.map (K := K) allocation (terminalDilation K m) F ε).comp third) _ _ _ | |
| variable {F : Type*} [Field F] {ε : ℝ} {I : Type} [Fintype I] | |
| def execution (hm : 0 < m) (hε : 0 ≤ ε) | |
| (E : Execution (cwSource F K (terminalLength allocation K m)) | |
| (Tensor.directSum (fun _ : I => | |
| stateTensor F (K := K) allocation (terminalDilation K m) ε (K + 2)))) : | |
| Execution (cwSource F K (terminalLength allocation K m)) | |
| (AllFieldSource.matrixTarget F (Fintype.card (Rows (K := K) allocation m)) | |
| (Fintype.card (Middles (K := K) allocation m)) | |
| (Fintype.card (Columns (K := K) allocation m)) (Fintype.card I)) := | |
| (E.restrict _ ((terminalMap allocation m F hm hε).parallelCopies I)).restrict _ | |
| (LocalMap.ofExists (TerminalRelabel.directSum_matrix_restrict_fin (K := F) | |
| I (Rows (K := K) allocation m) (Middles (K := K) allocation m) | |
| (Columns (K := K) allocation m))) | |
| def witness (hm : 0 < m) (hε : 0 ≤ ε) | |
| (E : Execution (cwSource F K (terminalLength allocation K m)) | |
| (Tensor.directSum (fun _ : I => | |
| stateTensor F (K := K) allocation (terminalDilation K m) ε (K + 2)))) | |
| (hI : 0 < Fintype.card I) : AllFieldWitness F := | |
| (execution allocation m hm hε E).toWitness | |
| (dimensions_pos allocation m).1 (dimensions_pos allocation m).2.1 | |
| (dimensions_pos allocation m).2.2 hI | |
| @[simp] theorem witness_volume (hm : 0 < m) (hε : 0 ≤ ε) | |
| (E : Execution (cwSource F K (terminalLength allocation K m)) | |
| (Tensor.directSum (fun _ : I => | |
| stateTensor F (K := K) allocation (terminalDilation K m) ε (K + 2)))) | |
| (hI : 0 < Fintype.card I) : | |
| (witness allocation m hm hε E hI).volume = terminalVolume (K := K) allocation m := | |
| volume_eq allocation m | |
| @[simp] theorem witness_multiplicity (hm : 0 < m) (hε : 0 ≤ ε) | |
| (E : Execution (cwSource F K (terminalLength allocation K m)) | |
| (Tensor.directSum (fun _ : I => | |
| stateTensor F (K := K) allocation (terminalDilation K m) ε (K + 2)))) | |
| (hI : 0 < Fintype.card I) : | |
| (witness allocation m hm hε E hI).multiplicity = Fintype.card I := rfl | |
| @[simp] theorem witness_rankBound (hm : 0 < m) (hε : 0 ≤ ε) | |
| (E : Execution (cwSource F K (terminalLength allocation K m)) | |
| (Tensor.directSum (fun _ : I => | |
| stateTensor F (K := K) allocation (terminalDilation K m) ε (K + 2)))) | |
| (hI : 0 < Fintype.card I) : | |
| (witness allocation m hm hε E hI).rankBound = | |
| AllFieldSource.rankBudget E.Copies K (terminalLength allocation K m) := rfl | |
| end MatrixMultiplication.AllFieldFinishedWitness | |
| end | |
| end OAI | |