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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 | import OAI.Computability.PerfectCompleteness.Construction.CutChildGrouping
import OAI.Computability.PerfectCompleteness.Decoding.ChildBlockProjection
namespace OAI
section
namespace PerfectCompleteness.ChildAssemblyProjection
open RecursiveSpaces TreeSourceSpaces HierarchicalArrays PointwiseSpaces
open UniqueGamesTheorem.Foundations.Games
open scoped BigOperators Classical
noncomputable section
variable {branch : Nat → Nat} {n t : Nat}
{slots projected : Slots branch (n + 1) → Fin t → MixedSupport.Slot}
def childProjection
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(i : Fin (branch n)) :
∀ s k, MixedSupport.Projection (childSlots slots i s k) (childSlots projected i s k) :=
fun s k => p (i, s) k
theorem restrictChild_projection
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(i : Fin (branch n)) (x : Domain slots) :
sourceProjection (childProjection p i) (restrictChild slots i x) =
restrictChild projected i (sourceProjection p x) := rfl
theorem recursiveSum_pullback
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(f : (i : Fin (branch n)) → squareSpace (H (childSlots projected i))) :
UniformChildSum.recursiveSum (LeafDomain slots)
(fun i => ChildBlockProjection.squarePullback (childProjection p i) (f i)) =
HPullback p (UniformChildSum.recursiveSum (LeafDomain projected) f) := by
apply Subtype.ext
change (∑ i, PointwiseSpaces.pullback F2 (childRestriction (LeafDomain slots) i)
(ChildBlockProjection.squarePullback (childProjection p i) (f i)).val) =
PointwiseSpaces.pullback F2 (sourceProjection p)
(∑ i, PointwiseSpaces.pullback F2 (childRestriction (LeafDomain projected) i) (f i).val)
rw [map_sum]
apply Finset.sum_congr rfl
intro i _
rfl
variable {C : Type*} [Fintype C]
def childPullback (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(i : Fin (branch n)) :
CutChildGrouping.Child (C := C) projected rows i →ₗ[F2]
CutChildGrouping.Child (C := C) slots rows i where
toFun block :=
(fun call => ChildBlockProjection.squarePullback (childProjection p i) (block.1 call),
ChildBlockProjection.arraysPullback rows (childProjection p i) block.2)
map_add' a b := by
apply Prod.ext
· funext call
exact map_add (ChildBlockProjection.squarePullback (childProjection p i))
(a.1 call) (b.1 call)
· exact map_add (ChildBlockProjection.arraysPullback rows (childProjection p i)) a.2 b.2
map_smul' c a := by
apply Prod.ext
· funext call
exact map_smul (ChildBlockProjection.squarePullback (childProjection p i)) c (a.1 call)
· exact map_smul (ChildBlockProjection.arraysPullback rows (childProjection p i)) c a.2
theorem childPullback_fin (calls : Nat) (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(i : Fin (branch n)) :
childPullback (C := Fin calls) rows p i =
ChildBlockProjection.rawPullback calls rows (childProjection p i) := rfl
def rawPullback (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) :
CutChildGrouping.Raw (C := C) projected rows →ₗ[F2]
CutChildGrouping.Raw (C := C) slots rows where
toFun block i := childPullback rows p i (block i)
map_add' a b := by
funext i
exact map_add (childPullback rows p i) (a i) (b i)
map_smul' c a := by
funext i
exact map_smul (childPullback rows p i) c (a i)
def assembledPullback (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) :
CutChildGrouping.Assembled (C := C) projected rows →ₗ[F2]
CutChildGrouping.Assembled (C := C) slots rows where
toFun z := (fun call => HPullback p (z.1 call),
fun i => ChildBlockProjection.arraysPullback rows (childProjection p i) (z.2 i))
map_add' a b := by
apply Prod.ext
· funext call
exact map_add (HPullback p) (a.1 call) (b.1 call)
· funext i
exact map_add (ChildBlockProjection.arraysPullback rows (childProjection p i)) (a.2 i) (b.2 i)
map_smul' c a := by
apply Prod.ext
· funext call
exact map_smul (HPullback p) c (a.1 call)
· funext i
exact map_smul (ChildBlockProjection.arraysPullback rows (childProjection p i)) c (a.2 i)
omit [Fintype C] in
theorem assemble_pullback (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(block : CutChildGrouping.Raw (C := C) projected rows) :
CutChildGrouping.assemble slots rows (rawPullback rows p block) =
assembledPullback rows p (CutChildGrouping.assemble projected rows block) := by
apply Prod.ext
· funext call
exact recursiveSum_pullback p (fun i => (block i).1 call)
· rfl
omit [Fintype C] in
theorem assemble_comp_pullback (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) :
(CutChildGrouping.assemble (C := C) slots rows).comp (rawPullback rows p) =
(assembledPullback rows p).comp (CutChildGrouping.assemble projected rows) := by
apply LinearMap.ext
intro block
exact assemble_pullback rows p block
def evaluate {rows : Nat → Nat}
(z : CutChildGrouping.Assembled (C := C) slots rows) (x : Domain slots) :
(C → F2) × (Fin (branch n) → Output branch n rows) :=
(fun call => (z.1 call).val x,
fun i => fullJoint (z.2 i) (restrictChild slots i x))
omit [Fintype C] in
theorem evaluate_pullback (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(z : CutChildGrouping.Assembled (C := C) projected rows) (x : Domain slots) :
evaluate (assembledPullback rows p z) x = evaluate z (sourceProjection p x) := rfl
omit [Fintype C] in
theorem evaluate_assemble_pullback (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(block : CutChildGrouping.Raw (C := C) projected rows) (x : Domain slots) :
evaluate (CutChildGrouping.assemble slots rows (rawPullback rows p block)) x =
evaluate (CutChildGrouping.assemble projected rows block) (sourceProjection p x) := by
rw [assemble_pullback]
exact evaluate_pullback rows p _ x
theorem pushforward_assemble (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(μ : FiniteDistribution (CutChildGrouping.Raw (C := C) projected rows)) :
(μ.pushforward (rawPullback rows p)).pushforward (CutChildGrouping.assemble slots rows) =
(μ.pushforward (CutChildGrouping.assemble projected rows)).pushforward
(assembledPullback rows p) := by
rw [FiniteDistribution.pushforward_comp, FiniteDistribution.pushforward_comp]
congr 1
funext block
exact assemble_pullback rows p block
theorem uniform_projected_assembly (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) :
((CutChildGrouping.rawLaw (C := C) projected rows).pushforward
(rawPullback rows p)).pushforward (CutChildGrouping.assemble slots rows) =
(FiniteDistribution.uniform (CutChildGrouping.Assembled (C := C) projected rows)).pushforward
(assembledPullback rows p) := by
rw [pushforward_assemble, CutChildGrouping.assemble_law]
end
end PerfectCompleteness.ChildAssemblyProjection
end
end OAI
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