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import OAI.Computability.PerfectCompleteness.Decoding.ChildAssemblyProjection
import OAI.Computability.PerfectCompleteness.Decoding.LowerCutNodeCoordinates
import OAI.Computability.PerfectCompleteness.Decoding.LowerCutRowsLemmas
import OAI.Computability.PerfectCompleteness.Decoding.LowerDirectionFiber
import OAI.Computability.PerfectCompleteness.Foundations.CutCallExtension
import OAI.Computability.PerfectCompleteness.Foundations.MultiplicationFormInjectiveLemmas
import OAI.Computability.PerfectCompleteness.Machines.LowerCutDecoderInput
import OAI.Computability.PerfectCompleteness.Machines.OwnInputReferenceLemmas
namespace OAI
section
namespace PerfectCompleteness.LowerCutPair
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 Calls := WholeCutCalls.Index rows repeats p
abbrev Raw := CutCallExtension.Extended (C := Calls rows repeats p) (cutSlots p slots) rows
abbrev NativeRows := Fin (rows (m + 1)) → H (cutSlots p slots)
abbrev NativeMatrix := Module.Dual F2 (H (cutSlots p slots)) →ₗ[F2] (Fin (rows (m + 1)) → F2)
def first (exterior : Exterior rows repeats p slots) (raw : Raw rows repeats p slots) :
Arrays slots rows :=
LowerCutRows.reconstruct rows repeats p slots exterior
(CutCallExtension.original (cutSlots p slots) rows raw)
def fresh (raw : Raw rows repeats p slots) : H (cutSlots p slots) :=
CutCallExtension.fresh (cutSlots p slots) rows raw
def second (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) :
Arrays slots rows :=
SelectedArrayReplacement.replace rows p slots (first rows repeats p slots exterior raw)
(LowerDirectionFiber.replace a
(SelectedArrayReplacement.selectedRows rows p slots (first rows repeats p slots exterior raw))
(fresh rows repeats p slots raw))
def arraysPair (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) :
Arrays slots rows × Arrays slots rows :=
(first rows repeats p slots exterior raw, second rows repeats p slots exterior a raw)
@[simp] theorem selected_second (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) :
SelectedArrayReplacement.selectedRows rows p slots (second rows repeats p slots exterior a raw) =
LowerDirectionFiber.replace a
(SelectedArrayReplacement.selectedRows rows p slots (first rows repeats p slots exterior raw))
(fresh rows repeats p slots raw) :=
SelectedArrayReplacement.selectedRows_replace rows p slots _ _
theorem second_outside (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots)
(node : Nodes branch n) (hne : node ≠ WholeArrayInteriorExterior.upperNode p) :
second rows repeats p slots exterior a raw node =
first rows repeats p slots exterior raw node :=
SelectedArrayReplacement.replace_outside rows p slots _ _ node hne
theorem selected_quotient (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) :
LowerDirectionFiber.quotientQuery (H (cutSlots p slots)) a
(SelectedArrayReplacement.selectedRows rows p slots
(second rows repeats p slots exterior a raw)) =
LowerDirectionFiber.quotientQuery (H (cutSlots p slots)) a
(SelectedArrayReplacement.selectedRows rows p slots
(first rows repeats p slots exterior raw)) := by
rw [selected_second]
exact LowerDirectionFiber.quotientQuery_replace _ _ _ _
def readRowsFresh (exterior : Exterior rows repeats p slots) (raw : Raw rows repeats p slots) :
NativeRows rows p slots × H (cutSlots p slots) :=
(SelectedArrayReplacement.selectedRows rows p slots (first rows repeats p slots exterior raw),
fresh rows repeats p slots raw)
def selectPair :
(CutChildGrouping.Assembled (C := Calls rows repeats p) (cutSlots p slots) rows ×
H (cutSlots p slots)) →ₗ[F2] (NativeRows rows p slots × H (cutSlots p slots)) :=
(LowerCutRows.selectRows rows repeats p slots).prodMap LinearMap.id
theorem selectPair_surjective : Function.Surjective (selectPair rows repeats p slots) := by
rintro ⟨S, h⟩
obtain ⟨record, hrecord⟩ := LowerCutRows.selectRows_surjective rows repeats p slots S
exact ⟨(record, h), Prod.ext hrecord rfl⟩
@[simp] theorem readRowsFresh_eq (exterior : Exterior rows repeats p slots)
(raw : Raw rows repeats p slots) :
readRowsFresh rows repeats p slots exterior raw =
selectPair rows repeats p slots
(CutCallExtension.observe (cutSlots p slots) rows raw) := by
apply Prod.ext
· exact LowerCutRows.selectedRows_reconstruct_eq rows repeats p slots exterior _
· rfl
theorem readRowsFresh_law (exterior : Exterior rows repeats p slots) :
(CutChildGrouping.rawLaw (C := Option (Calls rows repeats p)) (cutSlots p slots) rows).pushforward
(readRowsFresh rows repeats p slots exterior) =
(FiniteDistribution.uniform (NativeRows rows p slots)).product
(FiniteDistribution.uniform (H (cutSlots p slots))) := by
change (CutChildGrouping.rawLaw (C := Option (Calls rows repeats p)) (cutSlots p slots) rows).pushforward
(fun raw => readRowsFresh rows repeats p slots exterior raw) = _
simp only [readRowsFresh_eq]
rw [← FiniteDistribution.pushforward_comp, CutCallExtension.observe_law,
WholeCutSampler.uniform_product,
UniformLinearImage.uniform_pushforward_linearMap
(selectPair rows repeats p slots) (selectPair_surjective rows repeats p slots)]
exact WholeCutSampler.uniform_product.symm
def selectedPair (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) :
NativeRows rows p slots × NativeRows rows p slots :=
(SelectedArrayReplacement.selectedRows rows p slots (first rows repeats p slots exterior raw),
SelectedArrayReplacement.selectedRows rows p slots (second rows repeats p slots exterior a raw))
theorem selectedPair_law (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) :
(CutChildGrouping.rawLaw (C := Option (Calls rows repeats p)) (cutSlots p slots) rows).pushforward
(selectedPair rows repeats p slots exterior a) =
LowerDirectionFiber.fiberPairLaw (H := H (cutSlots p slots)) a := by
have h := congrArg
(fun μ : FiniteDistribution (NativeRows rows p slots × H (cutSlots p slots)) =>
μ.pushforward (fun z => (z.1, LowerDirectionFiber.replace a z.1 z.2)))
(readRowsFresh_law rows repeats p slots exterior)
rw [FiniteDistribution.pushforward_comp, LowerDirectionFiber.two_copy_law] at h
change (CutChildGrouping.rawLaw (C := Option (Calls rows repeats p)) (cutSlots p slots) rows).pushforward
(fun raw =>
(SelectedArrayReplacement.selectedRows rows p slots (first rows repeats p slots exterior raw),
SelectedArrayReplacement.selectedRows rows p slots (second rows repeats p slots exterior a raw))) = _
simpa only [readRowsFresh, selected_second] using h
def matrixPair (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) :
NativeMatrix rows p slots × NativeMatrix rows p slots :=
let z := selectedPair rows repeats p slots exterior a raw
(EvaluationMatrix.ofRows (H (cutSlots p slots)) z.1,
EvaluationMatrix.ofRows (H (cutSlots p slots)) z.2)
local instance nativeMatrixFintype : Fintype (NativeMatrix rows p slots) :=
Fintype.ofEquiv (NativeRows rows p slots) (EvaluationMatrix.rowsEquiv (H (cutSlots p slots))).symm
theorem matrixPair_law (exterior : Exterior rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) :
(CutChildGrouping.rawLaw (C := Option (Calls rows repeats p)) (cutSlots p slots) rows).pushforward
(matrixPair rows repeats p slots exterior a) =
((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 (NativeRows rows p slots × NativeRows rows p slots) =>
μ.pushforward (fun z => (EvaluationMatrix.ofRows (H (cutSlots p slots)) z.1,
EvaluationMatrix.ofRows (H (cutSlots p slots)) z.2)))
(selectedPair_law rows repeats p slots exterior a)
rw [FiniteDistribution.pushforward_comp, LowerDirectionFiber.matrix_pair_law] at h
exact h
end
end PerfectCompleteness.LowerCutPair
end
section
namespace PerfectCompleteness.LowerCutDecoderForm
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open WholeCutGrouping
variable {branch rows repeats : Nat → Nat} {n height t : Nat}
(path : Path branch n (height + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(upper : Nodes branch n) (lowerLevel : Nat)
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(hnode : WholeArrayInteriorExterior.upperNode path =
HierarchicalLeftDecoder.LowerNode upper lowerLevel d)
def spaceEquiv : H (cutSlots path slots) ≃ₗ[ZMod 2]
HierarchicalDecoderTables.LowerH slots upper lowerLevel d :=
(LowerCutNodeCoordinates.scalarEquiv path slots).trans (by
rw [hnode])
def form (answer : HierarchicalDecoderTables.Answer slots upper lowerLevel d) :
ChildBilinearCollisionTransfer.ParentForm (cutSlots path slots) where
toFun f := (OddListExtraction.multiplicationForm
(HierarchicalDecoderTables.LowerH slots upper lowerLevel d) answer
(spaceEquiv path slots upper lowerLevel d hnode f)).comp
(spaceEquiv path slots upper lowerLevel d hnode).toLinearMap
map_add' f g := by
ext h
simp only [LinearMap.comp_apply, map_add, LinearMap.add_apply]
map_smul' c f := by
ext h
simp only [LinearMap.comp_apply, map_smul,
LinearMap.smul_apply, RingHom.id_apply]
@[simp] theorem form_apply
(answer : HierarchicalDecoderTables.Answer slots upper lowerLevel d)
(f g : H (cutSlots path slots)) :
form path slots upper lowerLevel d hnode answer f g =
OddListExtraction.multiplicationForm
(HierarchicalDecoderTables.LowerH slots upper lowerLevel d) answer
(spaceEquiv path slots upper lowerLevel d hnode f)
(spaceEquiv path slots upper lowerLevel d hnode g) := rfl
theorem form_injective :
Function.Injective (form path slots upper lowerLevel d hnode) := by
intro answer answer' hform
apply MultiplicationFormInjective.multiplicationForm_injective
(HierarchicalDecoderTables.LowerH slots upper lowerLevel d)
ext f g
obtain ⟨f', rfl⟩ := (spaceEquiv path slots upper lowerLevel d hnode).surjective f
obtain ⟨g', rfl⟩ := (spaceEquiv path slots upper lowerLevel d hnode).surjective g
exact congrArg (fun F : ChildBilinearCollisionTransfer.ParentForm
(cutSlots path slots) => F f' g') hform
variable (hbranch : ∀ k < n, 0 < branch k) {adviceRows : Nat}
(A : ManyGoodRows.RowMap (Block rows upper) adviceRows)
(table : HierarchicalAllDecoderTables.Input (rows := rows)
slots upper lowerLevel adviceRows → HierarchicalAllDecoderTables.UpperAnswer slots upper)
(cutoff : Nat)
def answer (arrays : Arrays slots rows) :
Option (HierarchicalDecoderTables.Answer slots upper lowerLevel d) :=
HierarchicalDecoderTables.decode slots upper lowerLevel hbranch d cutoff
(table (LowerCutDecoderInput.actualInput slots upper lowerLevel arrays A))
def readForm (arrays : Arrays slots rows) :
Option (ChildBilinearCollisionTransfer.ParentForm (cutSlots path slots)) :=
(answer slots upper lowerLevel d hbranch A table cutoff arrays).map
(form path slots upper lowerLevel d hnode)
def first (exterior : Exterior rows repeats path slots)
(raw : LowerCutPair.Raw rows repeats path slots) :
Option (ChildBilinearCollisionTransfer.ParentForm (cutSlots path slots)) :=
readForm path slots upper lowerLevel d hnode hbranch A table cutoff
(LowerCutPair.first rows repeats path slots exterior raw)
def second (exterior : Exterior rows repeats path slots)
(direction : BucketSampler.Direction (rows (height + 1)))
(raw : LowerCutPair.Raw rows repeats path slots) :
Option (ChildBilinearCollisionTransfer.ParentForm (cutSlots path slots)) :=
readForm path slots upper lowerLevel d hnode hbranch A table cutoff
(LowerCutPair.second rows repeats path slots exterior direction raw)
theorem fullCollision_readForm (arrays arrays' : Arrays slots rows) :
BilinearCollisionTransfer.fullCollision
(readForm path slots upper lowerLevel d hnode hbranch A table cutoff arrays)
(readForm path slots upper lowerLevel d hnode hbranch A table cutoff arrays') =
DecoderTableCoupling.definedEqual
(answer slots upper lowerLevel d hbranch A table cutoff arrays)
(answer slots upper lowerLevel d hbranch A table cutoff arrays') :=
MultiplicationFormInjective.definedEqual_map_of_injective
(form path slots upper lowerLevel d hnode)
(form_injective path slots upper lowerLevel d hnode) _ _
theorem fullCollision_pair (exterior : Exterior rows repeats path slots)
(direction : BucketSampler.Direction (rows (height + 1)))
(raw : LowerCutPair.Raw rows repeats path slots) :
BilinearCollisionTransfer.fullCollision
(first path slots upper lowerLevel d hnode hbranch A table cutoff exterior raw)
(second path slots upper lowerLevel d hnode hbranch A table cutoff exterior direction raw) =
TwoResponseCollision.collision
(answer slots upper lowerLevel d hbranch A table cutoff)
(LowerCutPair.arraysPair rows repeats path slots exterior direction raw) :=
fullCollision_readForm path slots upper lowerLevel d hnode hbranch A table cutoff _ _
end
end PerfectCompleteness.LowerCutDecoderForm
end
end OAI
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