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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
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