File size: 7,912 Bytes
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 | import OAI.Computability.PerfectCompleteness.Algebra.CanonicalMatrixTable
import OAI.Computability.PerfectCompleteness.Construction.NodeEmbedding
namespace OAI
section
namespace PerfectCompleteness.HierarchicalMatrixTable
noncomputable section
open scoped Classical
open TreeSourceSpaces HierarchicalArrays
variable {branch rows : Nat → Nat} {n t : Nat}
(slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot)
(node : Nodes branch n)
abbrev Background := (j : {j : Nodes branch n // j ≠ node}) →
Fin (rows (Nodes.height j.val)) → NodeEmbedding.NodeH slots j.val
abbrev SideOutput : Type :=
BlockQuotient.OtherBlocks (fun j : Nodes branch n => Block rows j) node × PUnit.{1}
abbrev Matrix := Module.Dual F2 (NodeEmbedding.RowSpace slots node) →ₗ[F2] Block rows node
def backgroundOf (arrays : Arrays slots rows) : Background (rows := rows) slots node :=
fun j => arrays j.val
def nativeRows (X : Matrix (rows := rows) slots node) :
Fin (rows (Nodes.height node)) → NodeEmbedding.NodeH slots node :=
fun i => (NodeEmbedding.equiv slots node).symm
(EvaluationMatrix.rows (NodeEmbedding.RowSpace slots node) X i)
def assemble (background : Background (rows := rows) slots node)
(X : Matrix (rows := rows) slots node) : Arrays slots rows :=
fun j => if h : j = node then h.symm ▸ nativeRows slots node X else background ⟨j, h⟩
@[simp] theorem assemble_selected (background : Background (rows := rows) slots node)
(X : Matrix (rows := rows) slots node) :
assemble slots node background X node = nativeRows slots node X := by
simp [assemble]
@[simp] theorem assemble_other (background : Background (rows := rows) slots node)
(X : Matrix (rows := rows) slots node) (j : {j : Nodes branch n // j ≠ node}) :
assemble slots node background X j.val = background j := by
simp only [assemble, dite_eq_right j.property]
theorem embedded_assembled_rows (background : Background (rows := rows) slots node)
(X : Matrix (rows := rows) slots node) :
NodeEmbedding.embeddedRows (assemble slots node background X) node =
EvaluationMatrix.rows (NodeEmbedding.RowSpace slots node) X := by
funext i
change NodeEmbedding.embed slots node ((assemble slots node background X) node i) = _
rw [assemble_selected]
exact (NodeEmbedding.equiv slots node).apply_symm_apply _
theorem matrix_assemble (background : Background (rows := rows) slots node)
(X : Matrix (rows := rows) slots node) :
NodeEmbedding.matrix (assemble slots node background X) node = X := by
unfold NodeEmbedding.matrix
rw [embedded_assembled_rows, EvaluationMatrix.ofRows_rows]
theorem nativeRows_matrix (arrays : Arrays slots rows) :
nativeRows slots node (NodeEmbedding.matrix arrays node) = arrays node := by
funext i
change (NodeEmbedding.equiv slots node).symm
(EvaluationMatrix.rows (NodeEmbedding.RowSpace slots node)
(EvaluationMatrix.ofRows (NodeEmbedding.RowSpace slots node)
(NodeEmbedding.embeddedRows arrays node)) i) = arrays node i
rw [EvaluationMatrix.rows_ofRows]
exact (NodeEmbedding.equiv slots node).symm_apply_apply _
theorem assemble_original (arrays : Arrays slots rows) :
assemble slots node (backgroundOf slots node arrays) (NodeEmbedding.matrix arrays node) = arrays := by
funext j
by_cases hj : j = node
· subst j
rw [assemble_selected, nativeRows_matrix]
· simp only [assemble, dite_eq_right hj, backgroundOf]
def other (background : Background (rows := rows) slots node)
(x : NodeEmbedding.NumberedDomain slots) : SideOutput (rows := rows) node :=
(fun j i => (background j i).val (NodeEmbedding.numberedRestriction slots j.val x), PUnit.unit)
def displayed (background : Background (rows := rows) slots node)
(X : Matrix (rows := rows) slots node) :
NodeEmbedding.NumberedDomain slots → (Output branch n rows : Type) :=
TreeCanonical.numberedFunction slots (fullJoint (assemble slots node background X))
theorem displayed_original (arrays : Arrays slots rows) :
displayed slots node (backgroundOf slots node arrays) (NodeEmbedding.matrix arrays node) =
TreeCanonical.numberedFunction slots (fullJoint arrays) := by
unfold displayed
rw [assemble_original]
theorem query_eq_split_displayed (background : Background (rows := rows) slots node)
(X : Matrix (rows := rows) slots node) :
CanonicalMatrixTable.query (TreeCanonical.numberedSlots slots)
(NodeEmbedding.RowSpace slots node) (other slots node background) X =
BlockQuotient.splitOutput (fun j : Nodes branch n => Block rows j) PUnit.{1} node ∘
displayed slots node background X := by
funext x
apply Prod.ext
· change EvaluationMatrix.evaluate (NodeEmbedding.RowSpace slots node) X x =
(displayed slots node background X x).1 node
have he := NodeEmbedding.matrix_evaluate (assemble slots node background X) node x
rw [matrix_assemble] at he
exact he
· apply Prod.ext
· funext j i
change (background j i).val (NodeEmbedding.numberedRestriction slots j.val x) =
((assemble slots node background X) j.val i).val
(NodeEmbedding.numberedRestriction slots j.val x)
rw [assemble_other]
· rfl
variable (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t))
theorem response_eq_split_response (background : Background (rows := rows) slots node)
(X : Matrix (rows := rows) slots node) :
CanonicalMatrixTable.response (TreeCanonical.numberedSlots slots)
(NodeEmbedding.RowSpace slots node) (other slots node background) σ X =
BlockQuotient.splitOutput (fun j : Nodes branch n => Block rows j) PUnit.{1} node
(KeyStrategy.response σ .left (TreeCanonical.numberedSlots slots)
(displayed slots node background X)) := by
unfold CanonicalMatrixTable.response
rw [query_eq_split_displayed]
exact KeyStrategy.response_postcomp_injective σ .left (TreeCanonical.numberedSlots slots)
(displayed slots node background X)
(BlockQuotient.splitOutput (fun j : Nodes branch n => Block rows j) PUnit.{1} node)
(BlockQuotient.splitOutput (fun j : Nodes branch n => Block rows j) PUnit.{1} node).injective
theorem rightQuery_eq (background : Background (rows := rows) slots node)
(X : Matrix (rows := rows) slots node) (a : Block rows node) :
CanonicalMatrixTable.rightQuery (TreeCanonical.numberedSlots slots)
(NodeEmbedding.RowSpace slots node) (other slots node background) a X =
BlockQuotient.projectBlock node a ∘ displayed slots node background X := by
unfold CanonicalMatrixTable.rightQuery
rw [query_eq_split_displayed]
rfl
theorem accepts_iff_canonical (background : Background (rows := rows) slots node)
(X : Matrix (rows := rows) slots node) (a : Block rows node) :
CanonicalMatrixTable.Accepts (TreeCanonical.numberedSlots slots)
(NodeEmbedding.RowSpace slots node) (other slots node background) σ a X ↔
CanonicalEdges.coarsen (TreeCanonical.numberedSlots slots)
(displayed slots node background X) (BlockQuotient.projectBlock node a)
(KeyStrategy.label σ .left (TreeCanonical.numberedSlots slots)
(displayed slots node background X)) =
KeyStrategy.label σ .right (TreeCanonical.numberedSlots slots)
(BlockQuotient.projectBlock node a ∘ displayed slots node background X) := by
have hresponse := congrArg (DirectionQuotient.projectWithSide a)
(response_eq_split_response slots node σ background X)
have hright := congrArg
(KeyStrategy.response σ .right (TreeCanonical.numberedSlots slots))
(rightQuery_eq slots node background X a)
rw [CanonicalMatrixTable.accepts_iff, CanonicalEdges.coarsen_eq_iff]
constructor
· intro h
exact hresponse.symm.trans (h.trans hright)
· intro h
exact hresponse.trans (h.trans hright.symm)
end
end PerfectCompleteness.HierarchicalMatrixTable
end
end OAI
|