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import OAI.Computability.PerfectCompleteness.Decoding.LowerCutPairBackground
import OAI.Computability.PerfectCompleteness.Decoding.LowerCutPairLemmas
namespace OAI
section
namespace PerfectCompleteness.LowerCutFiberDisintegration
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open WholeCutGrouping
open UniqueGamesTheorem.Foundations.Games
variable {branch : Nat → Nat} {n m t : Nat}
(rows repeats : Nat → Nat) (path : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(a : BucketSampler.Direction (rows (m + 1)))
abbrev Scalar := H (cutSlots path slots)
abbrev Quotient := LowerDirectionFiber.Complement a (Scalar path slots)
abbrev Background := LowerCutPairBackground.Complement rows repeats path slots ×
Quotient rows path slots a
abbrev Coordinates := Background rows repeats path slots a ×
(Scalar path slots × Scalar path slots)
def coordinateEquiv : LowerCutPairBackground.Observation rows repeats path slots ≃
Coordinates rows repeats path slots a where
toFun z := ((z.1, LowerDirectionFiber.complement a z.2.1),
(z.2.1 (LowerDirectionFiber.pivot a), z.2.2))
invFun z := (z.1.1, (LowerDirectionFiber.restore a z.1.2 z.2.1, z.2.2))
left_inv z := by
change (z.1, (LowerDirectionFiber.restore a
(LowerDirectionFiber.complement a z.2.1) (z.2.1 (LowerDirectionFiber.pivot a)),
z.2.2)) = z
exact Prod.ext rfl (Prod.ext (LowerDirectionFiber.restore_complement a z.2.1) rfl)
right_inv z := by
apply Prod.ext
· exact Prod.ext rfl (LowerDirectionFiber.complement_restore a z.1.2 z.2.1)
· exact Prod.ext (LowerDirectionFiber.restore_pivot a z.1.2 z.2.1) rfl
@[simp] theorem coordinateEquiv_apply (z : LowerCutPairBackground.Observation rows repeats path slots) :
coordinateEquiv rows repeats path slots a z =
((z.1, LowerDirectionFiber.complement a z.2.1),
(z.2.1 (LowerDirectionFiber.pivot a), z.2.2)) := rfl
def coordinates (raw : LowerCutPair.Raw rows repeats path slots) :
Coordinates rows repeats path slots a :=
coordinateEquiv rows repeats path slots a
(LowerCutPairBackground.observe rows repeats path slots raw)
def law : FiniteDistribution (Coordinates rows repeats path slots a) :=
((FiniteDistribution.uniform (LowerCutPairBackground.Complement rows repeats path slots)).product
(FiniteDistribution.uniform (Quotient rows path slots a))).product
((FiniteDistribution.uniform (Scalar path slots)).product
(FiniteDistribution.uniform (Scalar path slots)))
theorem coordinates_law :
(CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats path))
(cutSlots path slots) rows).pushforward (coordinates rows repeats path slots a) =
law rows repeats path slots a := by
change (CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats path))
(cutSlots path slots) rows).pushforward
(fun raw => coordinateEquiv rows repeats path slots a
(LowerCutPairBackground.observe rows repeats path slots raw)) = _
rw [← FiniteDistribution.pushforward_comp,
LowerCutPairBackground.observe_law]
have h := UniformConditioning.uniform_transport
(coordinateEquiv rows repeats path slots a)
rw [FiniteDistribution.transport_eq_pushforward] at h
simpa only [law, WholeCutSampler.uniform_product] using h
def arraysAt (exterior : Exterior rows repeats path slots)
(background : Background rows repeats path slots a) (fresh : Scalar path slots) :
Arrays slots rows :=
LowerCutPairBackground.installRows rows repeats path slots exterior background.1
(LowerDirectionFiber.restore a background.2 fresh)
theorem first_eq_arraysAt (exterior : Exterior rows repeats path slots)
(raw : LowerCutPair.Raw rows repeats path slots) :
LowerCutPair.first rows repeats path slots exterior raw =
arraysAt rows repeats path slots a exterior
(coordinates rows repeats path slots a raw).1
(coordinates rows repeats path slots a raw).2.1 := by
simp only [arraysAt, coordinates, coordinateEquiv_apply, LowerDirectionFiber.restore_complement]
rw [LowerCutPairBackground.observe_rows rows repeats path slots exterior]
exact LowerCutPairBackground.first_eq_installRows rows repeats path slots exterior raw
theorem second_eq_arraysAt (exterior : Exterior rows repeats path slots)
(raw : LowerCutPair.Raw rows repeats path slots) :
LowerCutPair.second rows repeats path slots exterior a raw =
arraysAt rows repeats path slots a exterior
(coordinates rows repeats path slots a raw).1
(coordinates rows repeats path slots a raw).2.2 := by
rw [LowerCutPairBackground.second_eq_installRows,
LowerCutPair.selected_second]
simp only [arraysAt, coordinates, coordinateEquiv_apply,
LowerDirectionFiber.replace, LowerCutPairBackground.observe_rows rows repeats path slots exterior,
LowerCutPairBackground.observe_fresh]
def pair (exterior : Exterior rows repeats path slots)
(z : Coordinates rows repeats path slots a) : Arrays slots rows × Arrays slots rows :=
(arraysAt rows repeats path slots a exterior z.1 z.2.1,
arraysAt rows repeats path slots a exterior z.1 z.2.2)
theorem pair_coordinates (exterior : Exterior rows repeats path slots)
(raw : LowerCutPair.Raw rows repeats path slots) :
pair rows repeats path slots a exterior (coordinates rows repeats path slots a raw) =
LowerCutPair.arraysPair rows repeats path slots exterior a raw :=
Prod.ext (first_eq_arraysAt rows repeats path slots a exterior raw).symm
(second_eq_arraysAt rows repeats path slots a exterior raw).symm
theorem arraysPair_law (exterior : Exterior rows repeats path slots) :
(CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats path))
(cutSlots path slots) rows).pushforward
(LowerCutPair.arraysPair rows repeats path slots exterior a) =
(law rows repeats path slots a).pushforward (pair rows repeats path slots a exterior) := by
have h := congrArg
(fun μ : FiniteDistribution (Coordinates rows repeats path slots a) =>
μ.pushforward (pair rows repeats path slots a exterior))
(coordinates_law rows repeats path slots a)
rw [FiniteDistribution.pushforward_comp] at h
simpa only [pair_coordinates] using h
theorem probability_pair (exterior : Exterior rows repeats path slots)
(event : Arrays slots rows × Arrays slots rows → Bool) :
(CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats path))
(cutSlots path slots) rows).probability
(fun raw => event (LowerCutPair.arraysPair rows repeats path slots exterior a raw)) =
(law rows repeats path slots a).probability
(fun z => event (pair rows repeats path slots a exterior z)) := by
rw [← FiniteDistribution.probability_pushforward,
arraysPair_law, FiniteDistribution.probability_pushforward]
end
end PerfectCompleteness.LowerCutFiberDisintegration
end
section
namespace PerfectCompleteness.LowerCutProjection
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
open scoped Classical
noncomputable section
section Linear
variable {H K : Type*} [AddCommGroup H] [AddCommGroup K]
[Module F2 H] [Module F2 K] {ℓ : Nat}
theorem linearMap_replace (f : H →ₗ[F2] K) (a : BucketSampler.Direction ℓ)
(S : Fin ℓ → H) (fresh : H) :
(fun i => f (LowerDirectionFiber.replace a S fresh i)) =
LowerDirectionFiber.replace a (fun i => f (S i)) (f fresh) := by
rw [LowerDirectionFiber.replace_eq, LowerDirectionFiber.replace_eq]
funext i
simp only [Pi.add_apply, BucketSampler.rankOne, map_add, map_smul, map_sub]
end Linear
section Local
variable {branch : Nat → Nat} {n t : Nat} {C : Type*} [Fintype C]
{slots projected : Slots branch (n + 1) → Fin t → MixedSupport.Slot}
omit [Fintype C] in
theorem original_pullback (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(raw : CutCallExtension.Extended (C := C) projected rows) :
CutCallExtension.original slots rows
(ChildAssemblyProjection.rawPullback (C := Option C) rows p raw) =
ChildAssemblyProjection.rawPullback (C := C) rows p
(CutCallExtension.original projected rows raw) := rfl
omit [Fintype C] in
theorem fresh_pullback (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(raw : CutCallExtension.Extended (C := C) projected rows) :
CutCallExtension.fresh slots rows
(ChildAssemblyProjection.rawPullback (C := Option C) rows p raw) =
HPullback p (CutCallExtension.fresh projected rows raw) :=
ChildAssemblyProjection.recursiveSum_pullback p (fun i => (raw i).1 none)
def observePullback (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) :
(CutChildGrouping.Assembled (C := C) projected rows × H projected) →ₗ[F2]
(CutChildGrouping.Assembled (C := C) slots rows × H slots) :=
(ChildAssemblyProjection.assembledPullback (C := C) rows p).prodMap (HPullback p)
omit [Fintype C] in
theorem observe_pullback (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(raw : CutCallExtension.Extended (C := C) projected rows) :
CutCallExtension.observe slots rows
(ChildAssemblyProjection.rawPullback (C := Option C) rows p raw) =
observePullback rows p (CutCallExtension.observe projected rows raw) := by
apply Prod.ext
· change CutChildGrouping.assemble slots rows
(CutCallExtension.original slots rows
(ChildAssemblyProjection.rawPullback (C := Option C) rows p raw)) =
ChildAssemblyProjection.assembledPullback rows p
(CutChildGrouping.assemble projected rows (CutCallExtension.original projected rows raw))
rw [original_pullback]
exact ChildAssemblyProjection.assemble_pullback rows p _
· exact fresh_pullback rows p raw
theorem pushforward_observe (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(μ : FiniteDistribution (CutCallExtension.Extended (C := C) projected rows)) :
(μ.pushforward (ChildAssemblyProjection.rawPullback (C := Option C) rows p)).pushforward
(CutCallExtension.observe slots rows) =
(μ.pushforward (CutCallExtension.observe projected rows)).pushforward
(observePullback rows p) := by
rw [FiniteDistribution.pushforward_comp, FiniteDistribution.pushforward_comp]
congr 1
funext raw
exact observe_pullback rows p raw
theorem uniform_projected_observe (rows : Nat → Nat)
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) :
((CutChildGrouping.rawLaw (C := Option C) projected rows).pushforward
(ChildAssemblyProjection.rawPullback rows p)).pushforward
(CutCallExtension.observe slots rows) =
((FiniteDistribution.uniform (CutChildGrouping.Assembled (C := C) projected rows)).product
(FiniteDistribution.uniform (H projected))).pushforward (observePullback rows p) := by
rw [pushforward_observe, CutCallExtension.observe_law]
theorem HPullback_replace {ℓ : Nat}
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(a : BucketSampler.Direction ℓ) (S : Fin ℓ → H projected) (fresh : H projected) :
(fun i => HPullback p (LowerDirectionFiber.replace a S fresh i)) =
LowerDirectionFiber.replace a (fun i => HPullback p (S i)) (HPullback p fresh) :=
linearMap_replace (HPullback p) a S fresh
end Local
section Selected
variable {branch : Nat → Nat} {n m t : Nat}
(rows repeats : Nat → Nat) (path : Path branch n (m + 1))
{slots projected : Slots branch n → Fin t → MixedSupport.Slot}
theorem firstRows_pullback
(p : ∀ s k, MixedSupport.Projection
(WholeCutGrouping.cutSlots path slots s k) (WholeCutGrouping.cutSlots path projected s k))
(exterior : WholeCutGrouping.Exterior rows repeats path slots)
(projectedExterior : WholeCutGrouping.Exterior rows repeats path projected)
(raw : LowerCutPair.Raw rows repeats path projected) :
SelectedArrayReplacement.selectedRows rows path slots
(LowerCutPair.first rows repeats path slots exterior
(ChildAssemblyProjection.rawPullback rows p raw)) =
(fun i => HPullback p
(SelectedArrayReplacement.selectedRows rows path projected
(LowerCutPair.first rows repeats path projected projectedExterior raw) i)) := by
simp only [LowerCutPair.first, LowerCutRows.selectedRows_reconstruct_eq]
rw [original_pullback]
exact congrArg (LowerCutRows.selectRows rows repeats path slots)
(ChildAssemblyProjection.assemble_pullback rows p
(CutCallExtension.original (WholeCutGrouping.cutSlots path projected) rows raw))
theorem selectedPair_pullback
(p : ∀ s k, MixedSupport.Projection
(WholeCutGrouping.cutSlots path slots s k) (WholeCutGrouping.cutSlots path projected s k))
(exterior : WholeCutGrouping.Exterior rows repeats path slots)
(projectedExterior : WholeCutGrouping.Exterior rows repeats path projected)
(a : BucketSampler.Direction (rows (m + 1)))
(raw : LowerCutPair.Raw rows repeats path projected) :
LowerCutPair.selectedPair rows repeats path slots exterior a
(ChildAssemblyProjection.rawPullback rows p raw) =
((fun i => HPullback p
((LowerCutPair.selectedPair rows repeats path projected projectedExterior a raw).1 i)),
(fun i => HPullback p
((LowerCutPair.selectedPair rows repeats path projected projectedExterior a raw).2 i))) := by
simp only [LowerCutPair.selectedPair, LowerCutPair.selected_second]
apply Prod.ext
· exact firstRows_pullback rows repeats path p exterior projectedExterior raw
· rw [firstRows_pullback rows repeats path p exterior projectedExterior raw]
simp only [LowerCutPair.fresh]
rw [fresh_pullback]
exact (HPullback_replace p a _ _).symm
end Selected
end
end PerfectCompleteness.LowerCutProjection
end
section
namespace PerfectCompleteness.LowerCutSingleFiberDisintegration
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open WholeCutGrouping
open UniqueGamesTheorem.Foundations.Games
variable {branch : Nat → Nat} {n m t : Nat}
(rows repeats : Nat → Nat) (path : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(a : BucketSampler.Direction (rows (m + 1)))
abbrev Raw := CutChildGrouping.Raw (C := LowerCutPair.Calls rows repeats path)
(cutSlots path slots) rows
abbrev Coordinates := LowerCutFiberDisintegration.Background rows repeats path slots a ×
H (cutSlots path slots)
def coordinateEquiv :
(LowerCutCallSplit.Complement rows repeats path slots ×
LowerCutCallSplit.NativeRows rows path slots) ≃
Coordinates rows repeats path slots a where
toFun z := ((z.1, LowerDirectionFiber.complement a z.2),
z.2 (LowerDirectionFiber.pivot a))
invFun z := (z.1.1, LowerDirectionFiber.restore a z.1.2 z.2)
left_inv z := Prod.ext rfl (LowerDirectionFiber.restore_complement a z.2)
right_inv z := Prod.ext
(Prod.ext rfl (LowerDirectionFiber.complement_restore a z.1.2 z.2))
(LowerDirectionFiber.restore_pivot a z.1.2 z.2)
@[simp] theorem coordinateEquiv_apply
(z : LowerCutCallSplit.Complement rows repeats path slots ×
LowerCutCallSplit.NativeRows rows path slots) :
coordinateEquiv rows repeats path slots a z =
((z.1, LowerDirectionFiber.complement a z.2),
z.2 (LowerDirectionFiber.pivot a)) := rfl
def coordinates (raw : Raw rows repeats path slots) :
Coordinates rows repeats path slots a :=
coordinateEquiv rows repeats path slots a
(LowerCutCallSplit.splitAssembled rows repeats path slots
(CutChildGrouping.assemble (cutSlots path slots) rows raw))
@[simp] theorem coordinates_complement (raw : Raw rows repeats path slots) :
(coordinates rows repeats path slots a raw).1.1 =
LowerCutCallSplit.complement rows repeats path slots
(CutChildGrouping.assemble (cutSlots path slots) rows raw) := rfl
theorem coordinates_law :
(CutChildGrouping.rawLaw (C := LowerCutPair.Calls rows repeats path)
(cutSlots path slots) rows).pushforward (coordinates rows repeats path slots a) =
((FiniteDistribution.uniform
(LowerCutPairBackground.Complement rows repeats path slots)).product
(FiniteDistribution.uniform
(LowerCutFiberDisintegration.Quotient rows path slots a))).product
(FiniteDistribution.uniform (H (cutSlots path slots))) := by
have hsplit :
(CutChildGrouping.rawLaw (C := LowerCutPair.Calls rows repeats path)
(cutSlots path slots) rows).pushforward (fun raw =>
LowerCutCallSplit.splitAssembled rows repeats path slots
(CutChildGrouping.assemble (cutSlots path slots) rows raw)) =
FiniteDistribution.uniform
(LowerCutCallSplit.Complement rows repeats path slots ×
LowerCutCallSplit.NativeRows rows path slots) := by
rw [← FiniteDistribution.pushforward_comp, CutChildGrouping.assemble_law]
exact UniformLinearImage.uniform_pushforward_linearMap
(LowerCutCallSplit.splitAssembled rows repeats path slots)
(LowerCutCallSplit.splitAssembled_surjective rows repeats path slots)
have h := congrArg
(fun μ : FiniteDistribution
(LowerCutCallSplit.Complement rows repeats path slots ×
LowerCutCallSplit.NativeRows rows path slots) =>
μ.pushforward (coordinateEquiv rows repeats path slots a)) hsplit
rw [FiniteDistribution.pushforward_comp] at h
have hcoord := UniformConditioning.uniform_transport
(coordinateEquiv rows repeats path slots a)
rw [FiniteDistribution.transport_eq_pushforward] at hcoord
have hfinal := h.trans hcoord
change (CutChildGrouping.rawLaw (C := LowerCutPair.Calls rows repeats path)
(cutSlots path slots) rows).pushforward (fun raw =>
coordinateEquiv rows repeats path slots a
(LowerCutCallSplit.splitAssembled rows repeats path slots
(CutChildGrouping.assemble (cutSlots path slots) rows raw))) = _
calc
_ = FiniteDistribution.uniform (Coordinates rows repeats path slots a) := hfinal
_ = _ := by
apply FiniteDistribution.eq_of_weight_eq
intro z
simp only [FiniteDistribution.uniform, FiniteDistribution.product,
Coordinates, LowerCutFiberDisintegration.Background,
Fintype.card_prod, Nat.cast_mul, one_div_mul_one_div]
theorem reconstruct_eq_arraysAt (exterior : Exterior rows repeats path slots)
(raw : Raw rows repeats path slots) :
LowerCutRows.reconstruct rows repeats path slots exterior raw =
LowerCutFiberDisintegration.arraysAt rows repeats path slots a exterior
(coordinates rows repeats path slots a raw).1
(coordinates rows repeats path slots a raw).2 := by
simp only [LowerCutFiberDisintegration.arraysAt, coordinates,
coordinateEquiv_apply, LowerDirectionFiber.restore_complement]
rw [← LowerCutAssembled.reconstruct_assemble,
← LowerCutPairBackground.reconstruct_merge, LowerCutCallSplit.merge_split]
theorem expectation_fiber (exterior : Exterior rows repeats path slots)
(f : LowerCutFiberDisintegration.Background rows repeats path slots a →
Arrays slots rows → ℝ) :
(CutChildGrouping.rawLaw (C := LowerCutPair.Calls rows repeats path)
(cutSlots path slots) rows).expectation (fun raw =>
f (coordinates rows repeats path slots a raw).1
(LowerCutRows.reconstruct rows repeats path slots exterior raw)) =
(((FiniteDistribution.uniform
(LowerCutPairBackground.Complement rows repeats path slots)).product
(FiniteDistribution.uniform
(LowerCutFiberDisintegration.Quotient rows path slots a))).product
(FiniteDistribution.uniform (H (cutSlots path slots)))).expectation
(fun z => f z.1
(LowerCutFiberDisintegration.arraysAt rows repeats path slots a exterior z.1 z.2)) := by
have h := congrArg
(fun μ : FiniteDistribution (Coordinates rows repeats path slots a) =>
μ.expectation (fun z => f z.1
(LowerCutFiberDisintegration.arraysAt rows repeats path slots a exterior z.1 z.2)))
(coordinates_law rows repeats path slots a)
rw [FiniteDistribution.expectation_pushforward] at h
simpa only [← reconstruct_eq_arraysAt rows repeats path slots a exterior] using h
theorem probability_fiber (exterior : Exterior rows repeats path slots)
(event : LowerCutFiberDisintegration.Background rows repeats path slots a →
Arrays slots rows → Bool) :
(CutChildGrouping.rawLaw (C := LowerCutPair.Calls rows repeats path)
(cutSlots path slots) rows).probability (fun raw =>
event (coordinates rows repeats path slots a raw).1
(LowerCutRows.reconstruct rows repeats path slots exterior raw)) =
(((FiniteDistribution.uniform
(LowerCutPairBackground.Complement rows repeats path slots)).product
(FiniteDistribution.uniform
(LowerCutFiberDisintegration.Quotient rows path slots a))).product
(FiniteDistribution.uniform (H (cutSlots path slots)))).probability
(fun z => event z.1
(LowerCutFiberDisintegration.arraysAt rows repeats path slots a exterior z.1 z.2)) := by
have h := congrArg
(fun μ : FiniteDistribution (Coordinates rows repeats path slots a) =>
μ.probability (fun z => event z.1
(LowerCutFiberDisintegration.arraysAt rows repeats path slots a exterior z.1 z.2)))
(coordinates_law rows repeats path slots a)
rw [FiniteDistribution.probability_pushforward] at h
simpa only [← reconstruct_eq_arraysAt rows repeats path slots a exterior] using h
end
end PerfectCompleteness.LowerCutSingleFiberDisintegration
end
end OAI
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