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import OAI.Computability.PerfectCompleteness.Reduction.FixedRows
namespace OAI
section
namespace PerfectCompleteness.HierarchicalLowerInverse
noncomputable section
open scoped Classical
open TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Fourier.MatrixLevelBridge
open UniqueGamesTheorem.Appendix.RankLevelFilter
attribute [local instance] linearMapFintype
variable {branch rows : Nat → Nat} {n t : Nat}
(slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot)
(upper : Nodes branch n) (lowerLevel : Nat)
(background : HierarchicalMatrixTable.Background (rows := rows) slots upper)
local instance lowerHFintype (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) :
Fintype (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) := Fintype.ofFinite _
abbrev Answer (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) :=
HierarchicalDecoderTables.Answer slots upper lowerLevel d
local instance answerFintype (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) :
Fintype (Answer slots upper lowerLevel d) :=
HierarchicalAllDecoderTables.answerFintype slots upper lowerLevel d
local instance upperDualFintype : Fintype (HierarchicalAllDecoderTables.UpperAnswer slots upper) :=
HierarchicalAllDecoderTables.upperDualFintype slots upper
def matrixTable {r : Nat}
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper)
(A : ManyGoodRows.RowMap (Block rows upper) r)
(table : HierarchicalAllDecoderTables.Input (rows := rows) slots upper lowerLevel r →
Option (Answer slots upper lowerLevel d)) :
HierarchicalLowerMatrixInput.Matrix (rows := rows) slots upper lowerLevel d →
Option (Answer slots upper lowerLevel d) :=
fun X => table (HierarchicalLowerMatrixInput.input slots upper lowerLevel
background d upperMatrix A X)
variable (hbranch : ∀ k < n, 0 < branch k)
theorem matrixTable_tested {r s : Nat}
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper)
(A : ManyGoodRows.RowMap (Block rows upper) r)
(table : HierarchicalAllDecoderTables.Input (rows := rows) slots upper lowerLevel r →
Option (Answer slots upper lowerLevel d))
(hvalid : DecoderTableAdmissibility.Admissible
(HierarchicalAllDecoderTables.RowValid slots upper lowerLevel hbranch r d s) table)
(X : HierarchicalLowerMatrixInput.Matrix (rows := rows) slots upper lowerLevel d)
(z : Answer slots upper lowerLevel d)
(hz : matrixTable slots upper lowerLevel background d upperMatrix A table X = some z) :
(LowerAffineSliceRank.testedGram
(FunctionSpaceLowerRank.matrixForm
(HierarchicalDecoderTables.LowerH slots upper lowerLevel d)
(OddListExtraction.multiplicationForm
(HierarchicalDecoderTables.LowerH slots upper lowerLevel d) z)) X).rank ≤ 1 := by
have h := hvalid
(HierarchicalLowerMatrixInput.input slots upper lowerLevel background d upperMatrix A X)
z hz
have ht := h.2.1
simpa only [HierarchicalLowerMatrixInput.input_testedMatrix] using ht
theorem matrixTable_high_rank {r s : Nat}
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper)
(A : ManyGoodRows.RowMap (Block rows upper) r)
(table : HierarchicalAllDecoderTables.Input (rows := rows) slots upper lowerLevel r →
Option (Answer slots upper lowerLevel d))
(hvalid : DecoderTableAdmissibility.Admissible
(HierarchicalAllDecoderTables.RowValid slots upper lowerLevel hbranch r d s) table)
(X : HierarchicalLowerMatrixInput.Matrix (rows := rows) slots upper lowerLevel d)
(z : Answer slots upper lowerLevel d)
(hz : matrixTable slots upper lowerLevel background d upperMatrix A table X = some z) :
s < Module.finrank F2 (LinearMap.range
(FunctionSpaceLowerRank.matrixForm
(HierarchicalDecoderTables.LowerH slots upper lowerLevel d)
(OddListExtraction.multiplicationForm
(HierarchicalDecoderTables.LowerH slots upper lowerLevel d) z))) := by
rw [FunctionSpaceLowerRank.matrixForm_rank]
exact (hvalid
(HierarchicalLowerMatrixInput.input slots upper lowerLevel background d upperMatrix A X)
z hz).2.2
theorem matrixTable_agreement_lt {r s m adviceRows : Nat}
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper)
(A : ManyGoodRows.RowMap (Block rows upper) adviceRows)
(table : HierarchicalAllDecoderTables.Input (rows := rows) slots upper lowerLevel adviceRows →
Option (Answer slots upper lowerLevel d))
(hvalid : DecoderTableAdmissibility.Admissible
(HierarchicalAllDecoderTables.RowValid slots upper lowerLevel hbranch adviceRows d s) table)
(ρ η : ℝ) (hm : 0 < m) (hρ : 0 < ρ) (hρ1 : ρ < 1)
(hconstants : levelCutoffConstant r ρ + node (r + 1) < η)
(hrows : 2 * m + r ≤ rows (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d)))
(herror : (2 : ℝ) ^ m / (2 : ℝ) ^ (s - 2 * r) +
(2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) ≤ ρ) :
PartialTableInverse.nonzeroAgreement
(matrixTable slots upper lowerLevel background d upperMatrix A table) < η := by
let f := matrixTable slots upper lowerLevel background d upperMatrix A table
let form := fun z : Answer slots upper lowerLevel d =>
FunctionSpaceLowerRank.matrixForm
(HierarchicalDecoderTables.LowerH slots upper lowerLevel d)
(OddListExtraction.multiplicationForm
(HierarchicalDecoderTables.LowerH slots upper lowerLevel d) z)
have tested : ∀ X z, f X = some z → (LowerAffineSliceRank.testedGram (form z) X).rank ≤ 1 :=
fun X z hz => matrixTable_tested slots upper lowerLevel background hbranch
d upperMatrix A table hvalid X z hz
have high : ∀ X z, f X = some z → s < Module.finrank F2 (LinearMap.range (form z)) :=
fun X z hz => matrixTable_high_rank slots upper lowerLevel background hbranch
d upperMatrix A table hvalid X z hz
have h := LowerAffineSliceRank.high_rank_agreement_lt f form tested r s ρ η
hm hρ hρ1 hconstants hrows herror
have herase : MatrixErasure.erase f (fun X => ∃ z, f X = some z ∧
Module.finrank F2 (LinearMap.range (form z)) ≤ s) = f := by
funext X
apply MatrixErasure.erase_eq_of_not
rintro ⟨z, hz, hrank⟩
exact (not_le_of_gt (high X z hz)) hrank
rw [herase] at h
exact h
variable (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t))
(useful : (bg : HierarchicalMatrixTable.Background (rows := rows) slots upper) →
HierarchicalFrozenTables.QuotientMatrix slots upper lowerLevel bg → Prop)
include hbranch σ useful
theorem supported_matrixTable_agreement_lt {r s m adviceRows : Nat}
(decoderDensity : ℝ) (hdecoderDensity : 0 < decoderDensity)
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper)
(A : ManyGoodRows.RowMap (Block rows upper) adviceRows)
(table : HierarchicalAllDecoderTables.Input (rows := rows) slots upper lowerLevel adviceRows →
Option (Answer slots upper lowerLevel d))
(hsupport : (HierarchicalAllDecoderTables.tableLaw slots upper lowerLevel hbranch σ useful
adviceRows decoderDensity d s).weight table ≠ 0)
(ρ η : ℝ) (hm : 0 < m) (hρ : 0 < ρ) (hρ1 : ρ < 1)
(hconstants : levelCutoffConstant r ρ + node (r + 1) < η)
(hrows : 2 * m + r ≤ rows (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d)))
(herror : (2 : ℝ) ^ m / (2 : ℝ) ^ (s - 2 * r) +
(2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) ≤ ρ) :
PartialTableInverse.nonzeroAgreement
(matrixTable slots upper lowerLevel background d upperMatrix A table) < η := by
exact matrixTable_agreement_lt slots upper lowerLevel background hbranch
d upperMatrix A table
(HierarchicalAllDecoderTables.supported_table_valid slots upper lowerLevel hbranch σ useful
adviceRows decoderDensity hdecoderDensity d s table hsupport)
ρ η hm hρ hρ1 hconstants hrows herror
theorem mean_matrixTable_agreement_le {r s m adviceRows : Nat}
(decoderDensity : ℝ) (hdecoderDensity : 0 < decoderDensity)
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper)
(A : ManyGoodRows.RowMap (Block rows upper) adviceRows)
(ρ η : ℝ) (hm : 0 < m) (hρ : 0 < ρ) (hρ1 : ρ < 1)
(hconstants : levelCutoffConstant r ρ + node (r + 1) < η)
(hrows : 2 * m + r ≤ rows (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d)))
(herror : (2 : ℝ) ^ m / (2 : ℝ) ^ (s - 2 * r) +
(2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) ≤ ρ) :
(HierarchicalAllDecoderTables.tableLaw slots upper lowerLevel hbranch σ useful
adviceRows decoderDensity d s).expectation
(fun table => PartialTableInverse.nonzeroAgreement
(matrixTable slots upper lowerLevel background d upperMatrix A table)) ≤ η := by
apply DecoderTableAdmissibility.expectation_le_of_support
intro table hsupport
exact (supported_matrixTable_agreement_lt slots upper lowerLevel background hbranch
σ useful decoderDensity hdecoderDensity d upperMatrix A table hsupport
ρ η hm hρ hρ1 hconstants hrows herror).le
theorem supported_upper_matrixTable_agreement_lt {r s m adviceRows : Nat}
(decoderDensity : ℝ) (hdecoderDensity : 0 < decoderDensity)
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper)
(A : ManyGoodRows.RowMap (Block rows upper) adviceRows)
(table : HierarchicalAllDecoderTables.Input (rows := rows) slots upper lowerLevel adviceRows →
HierarchicalAllDecoderTables.UpperAnswer slots upper)
(hsupport : (HierarchicalAllDecoderTables.upperTableLaw slots upper lowerLevel σ useful
adviceRows decoderDensity).weight table ≠ 0)
(ρ η : ℝ) (hm : 0 < m) (hρ : 0 < ρ) (hρ1 : ρ < 1)
(hconstants : levelCutoffConstant r ρ + node (r + 1) < η)
(hrows : 2 * m + r ≤ rows (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d)))
(herror : (2 : ℝ) ^ m / (2 : ℝ) ^ (s - 2 * r) +
(2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) ≤ ρ) :
PartialTableInverse.nonzeroAgreement
(matrixTable slots upper lowerLevel background d upperMatrix A
(HierarchicalAllDecoderTables.decodeTable slots upper lowerLevel hbranch adviceRows d s table)) < η := by
exact matrixTable_agreement_lt slots upper lowerLevel background hbranch
d upperMatrix A
(HierarchicalAllDecoderTables.decodeTable slots upper lowerLevel hbranch adviceRows d s table)
(HierarchicalAllDecoderTables.decoded_supported_table_valid slots upper lowerLevel hbranch
σ useful adviceRows decoderDensity hdecoderDensity table hsupport d s)
ρ η hm hρ hρ1 hconstants hrows herror
end
end PerfectCompleteness.HierarchicalLowerInverse
end
section
namespace PerfectCompleteness.HierarchicalFixedLowerInverse
noncomputable section
open scoped Classical
open TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
attribute [local instance] UniqueGamesTheorem.Appendix.RankLevelFilter.linearMapFintype
variable {δ : ℚ} (plan : FixedRows.Plan δ) (hδ : 0 < δ)
{branch : Nat → Nat} {t : Nat}
(slots : RecursiveSpaces.Slots branch plan.depth → Fin t → MixedSupport.Slot)
(upper : Nodes branch plan.depth) (lowerLevel : Nat)
(background : HierarchicalMatrixTable.Background
(rows := FixedRows.rows plan) slots upper)
local instance lowerHFintype (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) :
Fintype (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) := Fintype.ofFinite _
local instance upperDualFintype : Fintype (HierarchicalAllDecoderTables.UpperAnswer slots upper) :=
HierarchicalAllDecoderTables.upperDualFintype slots upper
variable (hbranch : ∀ k < plan.depth, 0 < branch k)
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch plan.depth t))
(useful : (bg : HierarchicalMatrixTable.Background
(rows := FixedRows.rows plan) slots upper) →
HierarchicalFrozenTables.QuotientMatrix slots upper lowerLevel bg → Prop)
include hbranch σ useful
theorem supported_upper_matrixTable_agreement_lt {adviceRows : Nat}
(decoderDensity : ℝ) (hdecoderDensity : 0 < decoderDensity)
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(upperMatrix : HierarchicalMatrixTable.Matrix
(rows := FixedRows.rows plan) slots upper)
(A : ManyGoodRows.RowMap (Block (FixedRows.rows plan) upper) adviceRows)
(table : HierarchicalAllDecoderTables.Input
(rows := FixedRows.rows plan) slots upper lowerLevel adviceRows →
HierarchicalAllDecoderTables.UpperAnswer slots upper)
(hsupport : (HierarchicalAllDecoderTables.upperTableLaw slots upper lowerLevel σ useful
adviceRows decoderDensity).weight table ≠ 0) :
PartialTableInverse.nonzeroAgreement
(HierarchicalLowerInverse.matrixTable slots upper lowerLevel background d upperMatrix A
(HierarchicalAllDecoderTables.decodeTable slots upper lowerLevel hbranch adviceRows d
(FixedRows.cutoff plan hδ) table)) <
UpperParameterScalars.gamma (InitialParameters.useful δ) plan.density
(InitialParameters.inverse δ) plan.order
(FixedRows.rows plan (Nodes.height upper)) ^ 2 / 8 := by
let lower := DescendingRows.lowerChoice (InitialParameters.useful_pos hδ)
plan.density_pos (InitialParameters.inverse_pos hδ) plan.order
(FixedRows.rows plan (Nodes.height upper))
have hheight : Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d) =
lowerLevel :=
(RelativeDescendantProducts.relativeNode_height upper d.val.val).trans d.property
have hproper : lowerLevel < Nodes.height upper := by
rw [← hheight]
exact (RelativeDescendantProducts.descendantIndex_below upper d.val).height_lt
have hminimum : lower.minimumRows ≤ FixedRows.rows plan lowerLevel :=
FixedRows.lower_row_requirement plan hδ hproper (Nodes.height_le upper)
have hrows : 2 * lower.crossSize + lower.order ≤
FixedRows.rows plan (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d)) := by
rw [hheight]
exact lower.row_bound _ hminimum
have hcutoff : lower.cutoff ≤ FixedRows.cutoff plan hδ := by
simpa only [lower, FixedRows.rows, FixedRows.cutoff] using
DescendingRows.tree_cutoff_le_common (InitialParameters.useful_pos hδ)
plan.density_pos (InitialParameters.inverse_pos hδ) plan.order plan.depth
plan.rootFirstRows (Nat.zero_lt_of_lt hproper) (Nodes.height_le upper)
exact HierarchicalLowerInverse.supported_upper_matrixTable_agreement_lt
(r := lower.order) (s := FixedRows.cutoff plan hδ) (m := lower.crossSize)
slots upper lowerLevel background hbranch σ useful decoderDensity hdecoderDensity
d upperMatrix A table hsupport lower.density _ lower.cross_pos lower.density_pos
lower.density_lt_one lower.spectral hrows (lower.error_bound _ hcutoff).le
end
end PerfectCompleteness.HierarchicalFixedLowerInverse
end
end OAI
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