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2f887db | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 | import OAI.Computability.PerfectCompleteness.Algebra.FunctionSpaceLowerRank
import OAI.Computability.PerfectCompleteness.Decoding.LeftDecoderKnownRows
import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalFrozenTablesLemmas
namespace OAI
section
namespace PerfectCompleteness.HierarchicalLeftDecoder
noncomputable section
open scoped Classical
open TreeSourceSpaces HierarchicalArrays PointwiseSpaces
open UniqueGamesTheorem.Foundations.Games
attribute [local instance] UniqueGamesTheorem.Appendix.RankLevelFilter.linearMapFintype
variable {branch rows : Nat → Nat} {n t : Nat}
(slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot)
(upper : Nodes branch n) (lowerLevel : Nat)
abbrev LowerNode (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) :=
(HierarchicalFrozenTables.lowerIndex upper lowerLevel d).val
local instance rowSpaceFintype (node : Nodes branch n) :
Fintype (NodeEmbedding.RowSpace slots node) := Fintype.ofFinite _
local instance nodeSpaceFintype (node : Nodes branch n) :
Fintype (NodeEmbedding.NodeH slots node) := Fintype.ofFinite _
local instance upperDualFintype : Fintype (Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) :=
LeftDecoder.dualFintype (V := NodeEmbedding.RowSpace slots upper)
variable (hbranch : ∀ k < n, 0 < branch k)
def squareEmbedding (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) :
squareSpace (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) →ₗ[F2]
NodeEmbedding.RowSpace slots upper where
toFun f := ⟨PointwiseSpaces.pullback F2
(NodeEmbedding.numberedRestriction slots (LowerNode upper lowerLevel d)) f.val, by
apply RelativeDescendantProducts.squareSpace_le_rowSpace slots hbranch
(RelativeDescendantProducts.descendantIndex_below upper d.val)
rw [NodeEmbedding.squareSpace_eq]
exact Submodule.mem_map_of_mem f.property⟩
map_add' _ _ := Subtype.ext rfl
map_smul' _ _ := Subtype.ext rfl
@[simp] theorem squareEmbedding_apply
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(f : squareSpace (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)))
(x : NodeEmbedding.NumberedDomain slots) :
(squareEmbedding slots upper lowerLevel hbranch d f).val x =
f.val (NodeEmbedding.numberedRestriction slots (LowerNode upper lowerLevel d) x) := rfl
def output (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) :
Module.Dual F2 (squareSpace (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d))) :=
q.comp (squareEmbedding slots upper lowerLevel hbranch d)
def lowerOne (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) :
squareSpace (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) :=
⟨1, TreeSourceSpaces.one_mem_squareSpace
(nodeSlots slots (LowerNode upper lowerLevel d))
(fun k hk => hbranch k (Nat.lt_of_lt_of_le hk
(Nodes.height_le (LowerNode upper lowerLevel d))))⟩
variable (background : HierarchicalMatrixTable.Background (rows := rows) slots upper)
def testedMatrix (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) :
Module.Dual F2 (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d)) →ₗ[F2]
Block rows (LowerNode upper lowerLevel d) :=
EvaluationMatrix.ofRows (NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d))
(background (HierarchicalFrozenTables.lowerIndex upper lowerLevel d))
def oneKnown : HierarchicalFrozenTables.knownRows slots upper lowerLevel background :=
⟨⟨1, RelativeDescendantProducts.one_mem_rowSpace slots hbranch upper⟩,
HierarchicalKnownRows.one_mem_knownSpace slots upper background
(HierarchicalFrozenTables.lowerIndex upper lowerLevel)⟩
def productKnown (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel)
(i j : Fin (rows (Nodes.height (LowerNode upper lowerLevel d)))) :
HierarchicalFrozenTables.knownRows slots upper lowerLevel background :=
⟨squareEmbedding slots upper lowerLevel hbranch d
(OddListExtraction.productElement
(NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d))
(background (HierarchicalFrozenTables.lowerIndex upper lowerLevel d) i)
(background (HierarchicalFrozenTables.lowerIndex upper lowerLevel d) j)),
HierarchicalKnownRows.product_mem_knownSpace slots upper background
(HierarchicalFrozenTables.lowerIndex upper lowerLevel) d i j⟩
theorem output_normalized_of_known_realizer
(q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper))
(x : NodeEmbedding.NumberedDomain slots)
(hknown : ∀ b : HierarchicalFrozenTables.knownRows slots upper lowerLevel background,
q (b : NodeEmbedding.RowSpace slots upper) = b.val.val x)
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) :
output slots upper lowerLevel hbranch d q (lowerOne slots upper lowerLevel hbranch d) = 1 := by
exact hknown (oneKnown slots upper lowerLevel hbranch background)
theorem tested_rank_le_one_of_known_realizer
(q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper))
(x : NodeEmbedding.NumberedDomain slots)
(hknown : ∀ b : HierarchicalFrozenTables.knownRows slots upper lowerLevel background,
q (b : NodeEmbedding.RowSpace slots upper) = b.val.val x)
(d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) :
(LowerAffineSliceRank.testedGram
(FunctionSpaceLowerRank.matrixForm
(NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d))
(OddListExtraction.multiplicationForm
(NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d))
(output slots upper lowerLevel hbranch d q)))
(testedMatrix slots upper lowerLevel background d)).rank ≤ 1 := by
let v := HierarchicalKnownRows.lower slots upper background
(HierarchicalFrozenTables.lowerIndex upper lowerLevel) x d
have hgram : LowerAffineSliceRank.testedGram
(FunctionSpaceLowerRank.matrixForm
(NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d))
(OddListExtraction.multiplicationForm
(NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d))
(output slots upper lowerLevel hbranch d q)))
(testedMatrix slots upper lowerLevel background d) = fun i j => v i * v j := by
rw [FunctionSpaceLowerRank.testedGram_multiplicationForm]
simp only [testedMatrix, EvaluationMatrix.rows_ofRows]
funext i j
exact hknown (productKnown slots upper lowerLevel hbranch background d i j)
rw [hgram]
exact Matrix.rank_vecMulVec_le v v
def Valid (q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) : Prop :=
∀ d : HierarchicalFrozenTables.LowerNodes upper lowerLevel,
output slots upper lowerLevel hbranch d q (lowerOne slots upper lowerLevel hbranch d) = 1 ∧
(LowerAffineSliceRank.testedGram
(FunctionSpaceLowerRank.matrixForm
(NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d))
(OddListExtraction.multiplicationForm
(NodeEmbedding.NodeH slots (LowerNode upper lowerLevel d))
(output slots upper lowerLevel hbranch d q)))
(testedMatrix slots upper lowerLevel background d)).rank ≤ 1
theorem valid_of_known_realizer
(q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper))
(hreal : ∃ x : NodeEmbedding.NumberedDomain slots,
∀ b : HierarchicalFrozenTables.knownRows slots upper lowerLevel background,
q (b : NodeEmbedding.RowSpace slots upper) = b.val.val x) :
Valid slots upper lowerLevel hbranch background q := by
obtain ⟨x, hx⟩ := hreal
intro d
exact ⟨output_normalized_of_known_realizer slots upper lowerLevel hbranch background q x hx d,
tested_rank_le_one_of_known_realizer slots upper lowerLevel hbranch background q x hx d⟩
variable (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t))
(useful : HierarchicalFrozenTables.QuotientMatrix slots upper lowerLevel background → Prop)
def adviceLaw (r : Nat) (ρ : ℝ) (A : ManyGoodRows.RowMap (Block rows upper) r)
(U : Module.Dual F2 (NodeEmbedding.RowSpace slots upper ⧸
HierarchicalFrozenTables.knownRows slots upper lowerLevel background) →ₗ[F2]
(Fin r → F2)) : FiniteDistribution (Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) :=
LeftDecoder.adviceLaw (TreeCanonical.numberedSlots slots) (NodeEmbedding.RowSpace slots upper)
(HierarchicalMatrixTable.other slots upper background) σ
(HierarchicalFrozenTables.knownRows slots upper lowerLevel background)
(HierarchicalFrozenTables.sectionMap slots upper lowerLevel background)
(HierarchicalFrozenTables.sectionMap_spec slots upper lowerLevel background) useful r ρ A U
theorem adviceLaw_valid (r : Nat) (ρ : ℝ) (hρ : 0 < ρ)
(A : ManyGoodRows.RowMap (Block rows upper) r)
(U : Module.Dual F2 (NodeEmbedding.RowSpace slots upper ⧸
HierarchicalFrozenTables.knownRows slots upper lowerLevel background) →ₗ[F2]
(Fin r → F2)) :
(adviceLaw slots upper lowerLevel background σ useful r ρ A U).probability
(fun q => decide (Valid slots upper lowerLevel hbranch background q)) = 1 := by
unfold adviceLaw
apply LeftDecoderKnownRows.adviceLaw_probability_eq_one_of_known_realizer
(TreeCanonical.numberedSlots slots) (NodeEmbedding.RowSpace slots upper)
(HierarchicalMatrixTable.other slots upper background) σ
(HierarchicalFrozenTables.knownRows slots upper lowerLevel background)
(HierarchicalFrozenTables.sectionMap slots upper lowerLevel background)
(HierarchicalFrozenTables.sectionMap_spec slots upper lowerLevel background) useful
(HierarchicalFrozenTables.determined slots upper lowerLevel background) r ρ hρ A U
intro q hq
exact decide_eq_true (valid_of_known_realizer slots upper lowerLevel hbranch background q hq)
variable {r : Nat} {ρ : ℝ} {A : ManyGoodRows.RowMap (Block rows upper) r}
{U : Module.Dual F2 (NodeEmbedding.RowSpace slots upper ⧸
HierarchicalFrozenTables.knownRows slots upper lowerLevel background) →ₗ[F2]
(Fin r → F2)}
theorem candidate_valid
(w : DenseWitnessRealization.RepresentativeWitness
(TreeCanonical.numberedSlots slots) (NodeEmbedding.RowSpace slots upper)
(HierarchicalMatrixTable.other slots upper background) σ
(HierarchicalFrozenTables.knownRows slots upper lowerLevel background)
(HierarchicalFrozenTables.sectionMap slots upper lowerLevel background) useful
(r := r) (ρ := ρ) (A := A) (U := U))
(hρ : 0 < ρ)
(q : PulledColumnSpace.Candidates
(HierarchicalFrozenTables.knownRows slots upper lowerLevel background) w.columnSpace
(LeftDecoder.center (TreeCanonical.numberedSlots slots) (NodeEmbedding.RowSpace slots upper)
(HierarchicalMatrixTable.other slots upper background) σ
(HierarchicalFrozenTables.knownRows slots upper lowerLevel background)
(HierarchicalFrozenTables.sectionMap slots upper lowerLevel background)
(HierarchicalFrozenTables.sectionMap_spec slots upper lowerLevel background) useful w)) :
Valid slots upper lowerLevel hbranch background q.val := by
obtain ⟨x, _, hx⟩ := LeftDecoder.exists_candidate_realizer
(TreeCanonical.numberedSlots slots) (NodeEmbedding.RowSpace slots upper)
(HierarchicalMatrixTable.other slots upper background) σ
(HierarchicalFrozenTables.knownRows slots upper lowerLevel background)
(HierarchicalFrozenTables.sectionMap slots upper lowerLevel background)
(HierarchicalFrozenTables.sectionMap_spec slots upper lowerLevel background) useful
(HierarchicalFrozenTables.determined slots upper lowerLevel background) w hρ
exact valid_of_known_realizer slots upper lowerLevel hbranch background q.val ⟨x, hx q⟩
end
end PerfectCompleteness.HierarchicalLeftDecoder
end
end OAI
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