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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 | import OAI.Computability.PerfectCompleteness.Construction.NodeEmbedding
import OAI.Computability.PerfectCompleteness.Decoding.DecoderSourcePullback
namespace OAI
section
namespace PerfectCompleteness.ProjectedNodeEmbedding
noncomputable section
open RecursiveSpaces TreeSourceSpaces HierarchicalArrays
open ChildBlockProjection (nodeProjection squarePullback)
variable {branch : Nat → Nat} {n t : Nat}
{slots projected : Slots branch n → Fin t → MixedSupport.Slot}
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(upper : Nodes branch n)
def nativePullback :
NodeEmbedding.NodeH projected upper →ₗ[F2] NodeEmbedding.NodeH slots upper :=
HPullback (nodeProjection p upper)
@[simp] theorem nativePullback_apply
(f : NodeEmbedding.NodeH projected upper) (x : Domain (nodeSlots slots upper)) :
(nativePullback p upper f).val x =
f.val (sourceProjection (nodeProjection p upper) x) := rfl
theorem nativePullback_injective : Function.Injective (nativePullback p upper) :=
HPullback_injective (nodeProjection p upper)
def intoOriginal :
NodeEmbedding.NodeH projected upper →ₗ[F2] NodeEmbedding.RowSpace slots upper :=
(NodeEmbedding.embed slots upper).comp (nativePullback p upper)
theorem intoOriginal_injective : Function.Injective (intoOriginal p upper) :=
(NodeEmbedding.embed_injective slots upper).comp (nativePullback_injective p upper)
@[simp] theorem intoOriginal_apply
(f : NodeEmbedding.NodeH projected upper) (x : NodeEmbedding.NumberedDomain slots) :
(intoOriginal p upper f).val x =
f.val (sourceProjection (nodeProjection p upper)
(NodeEmbedding.numberedRestriction slots upper x)) := rfl
theorem numberedRestriction_projection (x : NodeEmbedding.NumberedDomain slots) :
sourceProjection (nodeProjection p upper) (NodeEmbedding.numberedRestriction slots upper x) =
NodeEmbedding.numberedRestriction projected upper
(MixedSupport.projectionMap (TreeCanonical.numberedProjection p) x) := by
change restrictNode projected upper
(sourceProjection p ((TreeCanonical.assignmentEquiv slots).symm x)) = _
exact congrArg (restrictNode projected upper)
(TreeCanonical.assignmentEquiv_symm_projection p x)
theorem intoOriginal_apply_projected
(f : NodeEmbedding.NodeH projected upper) (x : NodeEmbedding.NumberedDomain slots) :
(intoOriginal p upper f).val x =
(NodeEmbedding.embed projected upper f).val
(MixedSupport.projectionMap (TreeCanonical.numberedProjection p) x) := by
exact congrArg f.val (numberedRestriction_projection p upper x)
theorem intoOriginal_val (f : NodeEmbedding.NodeH projected upper) :
(intoOriginal p upper f).val =
PointwiseSpaces.pullback F2
(MixedSupport.projectionMap (TreeCanonical.numberedProjection p))
(NodeEmbedding.embed projected upper f).val := by
funext x
exact intoOriginal_apply_projected p upper f x
theorem intoOriginal_at_assignment
(f : NodeEmbedding.NodeH projected upper) (x : Domain slots) :
(intoOriginal p upper f).val (TreeCanonical.assignmentEquiv slots x) =
f.val (restrictNode projected upper (sourceProjection p x)) := by
change f.val (sourceProjection (nodeProjection p upper)
(restrictNode slots upper ((TreeCanonical.assignmentEquiv slots).symm
(TreeCanonical.assignmentEquiv slots x)))) = _
rw [Equiv.symm_apply_apply]
rfl
theorem dualMap_eq_upperTarget
(q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) :
(intoOriginal p upper).dualMap q = DecoderSourcePullback.upperTarget p upper q := by
apply LinearMap.ext
intro f
rfl
theorem evaluation_dualMap (x : NodeEmbedding.NumberedDomain slots) :
(intoOriginal p upper).dualMap
(EvaluationMatrix.evaluation (NodeEmbedding.RowSpace slots upper) x) =
EvaluationMatrix.evaluation (NodeEmbedding.NodeH projected upper)
(NodeEmbedding.numberedRestriction projected upper
(MixedSupport.projectionMap (TreeCanonical.numberedProjection p) x)) := by
apply LinearMap.ext
intro f
exact congrArg f.val (numberedRestriction_projection p upper x)
theorem intoOriginal_mul (f g : NodeEmbedding.NodeH projected upper) :
(intoOriginal p upper f).val * (intoOriginal p upper g).val =
PointwiseSpaces.pullback F2
(sourceProjection (nodeProjection p upper) ∘ NodeEmbedding.numberedRestriction slots upper)
(f.val * g.val) := rfl
def intoOriginalSquare :
PointwiseSpaces.squareSpace (NodeEmbedding.NodeH projected upper) →ₗ[F2]
PointwiseSpaces.squareSpace (NodeEmbedding.RowSpace slots upper) where
toFun f := ⟨PointwiseSpaces.pullback F2 (NodeEmbedding.numberedRestriction slots upper)
(squarePullback (nodeProjection p upper) f).val, by
rw [NodeEmbedding.squareSpace_eq]
exact Submodule.mem_map_of_mem (squarePullback (nodeProjection p upper) f).property⟩
map_add' f g := by apply Subtype.ext; rfl
map_smul' c f := by apply Subtype.ext; rfl
@[simp] theorem intoOriginalSquare_apply
(f : PointwiseSpaces.squareSpace (NodeEmbedding.NodeH projected upper))
(x : NodeEmbedding.NumberedDomain slots) :
(intoOriginalSquare p upper f).val x =
f.val (sourceProjection (nodeProjection p upper)
(NodeEmbedding.numberedRestriction slots upper x)) := rfl
theorem intoOriginalSquare_product (f g : NodeEmbedding.NodeH projected upper) :
intoOriginalSquare p upper
(OddListExtraction.productElement (NodeEmbedding.NodeH projected upper) f g) =
OddListExtraction.productElement (NodeEmbedding.RowSpace slots upper)
(intoOriginal p upper f) (intoOriginal p upper g) := by
apply Subtype.ext
rfl
end
end PerfectCompleteness.ProjectedNodeEmbedding
end
end OAI
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