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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 | import OAI.Computability.PerfectCompleteness.Algebra.HierarchicalMatrixTable
import OAI.Computability.PerfectCompleteness.Foundations.QuarterBalanceLemmas
import OAI.Computability.PerfectCompleteness.Sampling.BucketUniformLemmas
import OAI.Computability.PerfectCompleteness.Sampling.WholeArraySampler
namespace OAI
section
namespace PerfectCompleteness.HierarchicalArrayLaw
noncomputable section
open scoped Classical
open TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
variable {branch : Nat → Nat} {n t : Nat}
(slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot)
(rows : Nat → Nat)
abbrev NativeRows (node : Nodes branch n) :=
Fin (rows (Nodes.height node)) → NodeEmbedding.NodeH slots node
def law (μ : (node : Nodes branch n) → FiniteDistribution (NativeRows slots rows node)) :
FiniteDistribution (Arrays slots rows) :=
FiniteProduct.law μ
def otherLaw (μ : (node : Nodes branch n) → FiniteDistribution (NativeRows slots rows node))
(node : Nodes branch n) :
FiniteDistribution (HierarchicalMatrixTable.Background (rows := rows) slots node) :=
FiniteProductSplit.otherLaw μ node
def nativeToMatrix (node : Nodes branch n) (v : NativeRows slots rows node) :
HierarchicalMatrixTable.Matrix (rows := rows) slots node :=
EvaluationMatrix.ofRows (NodeEmbedding.RowSpace slots node)
(NodeEmbedding.rowsEquiv slots node (rows (Nodes.height node)) v)
@[simp] theorem nativeToMatrix_selected (arrays : Arrays slots rows) (node : Nodes branch n) :
nativeToMatrix slots rows node (arrays node) = NodeEmbedding.matrix arrays node := rfl
private theorem pushforward_id {A : Type*} [Fintype A] (μ : FiniteDistribution A) :
μ.pushforward (fun a => a) = μ := by
apply FiniteDistribution.eq_of_weight_eq
intro a
simp [FiniteDistribution.pushforward]
theorem matrix_background_law
(μ : (node : Nodes branch n) → FiniteDistribution (NativeRows slots rows node))
(node : Nodes branch n)
[Fintype (HierarchicalMatrixTable.Matrix (rows := rows) slots node)] :
(law slots rows μ).pushforward (fun arrays =>
(NodeEmbedding.matrix arrays node, HierarchicalMatrixTable.backgroundOf slots node arrays)) =
((μ node).pushforward (nativeToMatrix slots rows node)).product
(otherLaw slots rows μ node) := by
have h := congrArg
(fun ν : FiniteDistribution (NativeRows slots rows node ×
HierarchicalMatrixTable.Background (rows := rows) slots node) =>
ν.pushforward (fun p => (nativeToMatrix slots rows node p.1, p.2)))
(FiniteProductSplit.split_law μ node)
rw [FiniteDistribution.pushforward_comp] at h
have hp := FiniteDistribution.product_pushforward
(μ node) (FiniteProductSplit.otherLaw μ node) (nativeToMatrix slots rows node)
(id : HierarchicalMatrixTable.Background (rows := rows) slots node →
HierarchicalMatrixTable.Background (rows := rows) slots node)
have hi : (FiniteProductSplit.otherLaw μ node).pushforward
(id : HierarchicalMatrixTable.Background (rows := rows) slots node →
HierarchicalMatrixTable.Background (rows := rows) slots node) =
FiniteProductSplit.otherLaw μ node := pushforward_id (FiniteProductSplit.otherLaw μ node)
rw [hi] at hp
exact h.trans (by simpa only [id_eq, otherLaw] using hp)
theorem uniform_nativeToMatrix (node : Nodes branch n)
[Fintype (HierarchicalMatrixTable.Matrix (rows := rows) slots node)] :
(FiniteDistribution.uniform (NativeRows slots rows node)).pushforward
(nativeToMatrix slots rows node) =
FiniteDistribution.uniform (HierarchicalMatrixTable.Matrix (rows := rows) slots node) := by
let : Fintype (NodeEmbedding.RowSpace slots node) := Fintype.ofFinite _
calc
_ = ((FiniteDistribution.uniform (NativeRows slots rows node)).pushforward
(NodeEmbedding.rowsEquiv slots node (rows (Nodes.height node)))).pushforward
(EvaluationMatrix.ofRows (NodeEmbedding.RowSpace slots node)) :=
(FiniteDistribution.pushforward_comp _ _ _).symm
_ = (FiniteDistribution.uniform
(Fin (rows (Nodes.height node)) → NodeEmbedding.RowSpace slots node)).pushforward
(EvaluationMatrix.ofRows (NodeEmbedding.RowSpace slots node)) := by
rw [NodeEmbedding.uniform_rowsEquiv]
_ = _ := UniformLinearImage.uniform_pushforward_linearMap
(BucketUniform.ofRowsLinear (NodeEmbedding.RowSpace slots node) (rows (Nodes.height node)))
(BucketUniform.ofRowsLinear_surjective
(NodeEmbedding.RowSpace slots node) (rows (Nodes.height node)))
theorem uniform_matrix_background_law
(μ : (node : Nodes branch n) → FiniteDistribution (NativeRows slots rows node))
(node : Nodes branch n)
[Fintype (HierarchicalMatrixTable.Matrix (rows := rows) slots node)]
(hselected : μ node = FiniteDistribution.uniform (NativeRows slots rows node)) :
(law slots rows μ).pushforward (fun arrays =>
(NodeEmbedding.matrix arrays node, HierarchicalMatrixTable.backgroundOf slots node arrays)) =
(FiniteDistribution.uniform
(HierarchicalMatrixTable.Matrix (rows := rows) slots node)).product
(otherLaw slots rows μ node) := by
rw [matrix_background_law, hselected, uniform_nativeToMatrix]
theorem matrix_coarse_background_law
(μ : (node : Nodes branch n) → FiniteDistribution (NativeRows slots rows node))
(node : Nodes branch n)
[Fintype (HierarchicalMatrixTable.Matrix (rows := rows) slots node)]
{B : Type*} [Fintype B]
(record : HierarchicalMatrixTable.Background (rows := rows) slots node → B) :
(law slots rows μ).pushforward (fun arrays =>
(NodeEmbedding.matrix arrays node, record (HierarchicalMatrixTable.backgroundOf slots node arrays))) =
((μ node).pushforward (nativeToMatrix slots rows node)).product
((otherLaw slots rows μ node).pushforward record) := by
have h := congrArg
(fun ν : FiniteDistribution
(HierarchicalMatrixTable.Matrix (rows := rows) slots node ×
HierarchicalMatrixTable.Background (rows := rows) slots node) =>
ν.pushforward (fun p => (p.1, record p.2)))
(matrix_background_law slots rows μ node)
rw [FiniteDistribution.pushforward_comp] at h
have hp := FiniteDistribution.product_pushforward
((μ node).pushforward (nativeToMatrix slots rows node)) (otherLaw slots rows μ node)
(id : HierarchicalMatrixTable.Matrix (rows := rows) slots node →
HierarchicalMatrixTable.Matrix (rows := rows) slots node) record
have hi : ((μ node).pushforward (nativeToMatrix slots rows node)).pushforward
(id : HierarchicalMatrixTable.Matrix (rows := rows) slots node →
HierarchicalMatrixTable.Matrix (rows := rows) slots node) =
(μ node).pushforward (nativeToMatrix slots rows node) :=
pushforward_id ((μ node).pushforward (nativeToMatrix slots rows node))
rw [hi] at hp
exact h.trans (by simpa only [id_eq] using hp)
theorem uniform_matrix_coarse_background_law
(μ : (node : Nodes branch n) → FiniteDistribution (NativeRows slots rows node))
(node : Nodes branch n)
[Fintype (HierarchicalMatrixTable.Matrix (rows := rows) slots node)]
(hselected : μ node = FiniteDistribution.uniform (NativeRows slots rows node))
{B : Type*} [Fintype B]
(record : HierarchicalMatrixTable.Background (rows := rows) slots node → B) :
(law slots rows μ).pushforward (fun arrays =>
(NodeEmbedding.matrix arrays node, record (HierarchicalMatrixTable.backgroundOf slots node arrays))) =
(FiniteDistribution.uniform
(HierarchicalMatrixTable.Matrix (rows := rows) slots node)).product
((otherLaw slots rows μ node).pushforward record) := by
rw [matrix_coarse_background_law, hselected, uniform_nativeToMatrix]
end
end PerfectCompleteness.HierarchicalArrayLaw
end
end OAI
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