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import OAI.Computability.PerfectCompleteness.Decoding.LowerCutPairLemmas
import OAI.Computability.PerfectCompleteness.Foundations.CutNativeForms
import OAI.Computability.PerfectCompleteness.Repetition.CleanPhysicalReplay
import OAI.Computability.PerfectCompleteness.Sampling.SourceChildMarkedLaw
namespace OAI
section
namespace PerfectCompleteness.LowerCutScalarProjection
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open WholeArraySubtreeSplit WholeArrayInteriorExterior
open scoped Classical
noncomputable section
variable {branch : Nat → Nat} {n m t : Nat}
theorem scalarEquiv_HPullback :
{n m : Nat} → (p : Path branch n (m + 1)) →
(slots projected : Slots branch n → Fin t → MixedSupport.Slot) →
(q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) →
(f : H (subtreeSlots p projected)) →
LowerCutNodeCoordinates.scalarEquiv p slots
(HPullback (fun s k => q (p.slotEmbedding s) k) f) =
HPullback (ChildBlockProjection.nodeProjection q (upperNode p))
(LowerCutNodeCoordinates.scalarEquiv p projected f)
| _, _, .refl _, _, _, _, _ => rfl
| _, _, .step i p, slots, projected, q, f =>
scalarEquiv_HPullback p (childSlots slots i) (childSlots projected i)
(fun s k => q (i, s) k) f
theorem projectionCast_apply
{a b c d : Slots branch m → Fin t → MixedSupport.Slot}
(hab : a = b) (hcd : c = d)
(q : ∀ s k, MixedSupport.Projection (a s k) (c s k))
(s : Slots branch m) (k : Fin t) :
CutNativeForms.projectionCast hab hcd q s k =
CutProjectionAssembly.castProjection
(congrFun (congrFun hab s) k) (congrFun (congrFun hcd s) k) (q s k) := by
cases hab
cases hcd
rfl
theorem projectionCast_fill (p : Path branch n (m + 1))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(left right : Slots branch (m + 1) → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) :
CutNativeForms.projectionCast
(CleanPhysicalReplay.cutSlots_fill p outside left)
(CleanPhysicalReplay.cutSlots_fill p outside right)
(fun s k => CutProjectionAssembly.fillProjection p outside left right q
(p.slotEmbedding s) k) = q := by
funext s k
rw [projectionCast_apply]
exact CutProjectionAssembly.fillProjection_at_cut_cast p outside left right q s k
theorem inverse_spaceCast_HPullback
{a b c d : Slots branch m → Fin t → MixedSupport.Slot}
(hab : a = b) (hcd : c = d)
(q : ∀ s k, MixedSupport.Projection (a s k) (c s k)) (f : H d) :
CutNativeForms.spaceCast hab.symm
(HPullback (CutNativeForms.projectionCast hab hcd q) f) =
HPullback q (CutNativeForms.spaceCast hcd.symm f) := by
cases hab
cases hcd
rfl
theorem fill_spaceCast_HPullback (p : Path branch n (m + 1))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(left right : Slots branch (m + 1) → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) (f : H right) :
CutNativeForms.spaceCast (CleanPhysicalReplay.cutSlots_fill p outside left).symm
(HPullback q f) =
HPullback
(fun s k => CutProjectionAssembly.fillProjection p outside left right q
(p.slotEmbedding s) k)
(CutNativeForms.spaceCast
(CleanPhysicalReplay.cutSlots_fill p outside right).symm f) := by
have h := inverse_spaceCast_HPullback
(CleanPhysicalReplay.cutSlots_fill p outside left)
(CleanPhysicalReplay.cutSlots_fill p outside right)
(fun s k => CutProjectionAssembly.fillProjection p outside left right q
(p.slotEmbedding s) k) f
rw [projectionCast_fill] at h
exact h
theorem scalarEquiv_fill_HPullback (p : Path branch n (m + 1))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(left right : Slots branch (m + 1) → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) (f : H right) :
LowerCutNodeCoordinates.scalarEquiv p (CutSlotAssembly.fill p outside left)
(CutNativeForms.spaceCast (CleanPhysicalReplay.cutSlots_fill p outside left).symm
(HPullback q f)) =
HPullback (ChildBlockProjection.nodeProjection
(CutProjectionAssembly.fillProjection p outside left right q) (upperNode p))
(LowerCutNodeCoordinates.scalarEquiv p (CutSlotAssembly.fill p outside right)
(CutNativeForms.spaceCast
(CleanPhysicalReplay.cutSlots_fill p outside right).symm f)) := by
rw [fill_spaceCast_HPullback]
exact scalarEquiv_HPullback p _ _ _ _
def nodeEquiv (slots : Slots branch n → Fin t → MixedSupport.Slot)
{a b : Nodes branch n} (hab : a = b) :
H (nodeSlots slots a) ≃ₗ[F2] H (nodeSlots slots b) := by
rw [hab]
theorem nodeEquiv_HPullback
(slots projected : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
{a b : Nodes branch n} (hab : a = b) (f : H (nodeSlots projected a)) :
nodeEquiv slots hab (HPullback (ChildBlockProjection.nodeProjection q a) f) =
HPullback (ChildBlockProjection.nodeProjection q b) (nodeEquiv projected hab f) := by
cases hab
rfl
theorem spaceEquiv_HPullback (p : Path branch n (m + 1))
(slots projected : Slots branch n → Fin t → MixedSupport.Slot)
(upper : Nodes branch n) (level : Nat)
(d : HierarchicalFrozenTables.LowerNodes upper level)
(hnode : upperNode p = HierarchicalLeftDecoder.LowerNode upper level d)
(q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k))
(f : H (subtreeSlots p projected)) :
LowerCutDecoderForm.spaceEquiv p slots upper level d hnode
(HPullback (fun s k => q (p.slotEmbedding s) k) f) =
HPullback (ChildBlockProjection.nodeProjection q
(HierarchicalLeftDecoder.LowerNode upper level d))
(LowerCutDecoderForm.spaceEquiv p projected upper level d hnode f) := by
change nodeEquiv slots hnode
(LowerCutNodeCoordinates.scalarEquiv p slots
(HPullback (fun s k => q (p.slotEmbedding s) k) f)) =
HPullback (ChildBlockProjection.nodeProjection q
(HierarchicalLeftDecoder.LowerNode upper level d))
(nodeEquiv projected hnode (LowerCutNodeCoordinates.scalarEquiv p projected f))
rw [scalarEquiv_HPullback, nodeEquiv_HPullback]
theorem spaceEquiv_fill_HPullback (p : Path branch n (m + 1))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(left right : Slots branch (m + 1) → Fin t → MixedSupport.Slot)
(upper : Nodes branch n) (level : Nat)
(d : HierarchicalFrozenTables.LowerNodes upper level)
(hnode : upperNode p = HierarchicalLeftDecoder.LowerNode upper level d)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) (f : H right) :
LowerCutDecoderForm.spaceEquiv p (CutSlotAssembly.fill p outside left)
upper level d hnode
(CutNativeForms.spaceCast (CleanPhysicalReplay.cutSlots_fill p outside left).symm
(HPullback q f)) =
HPullback (ChildBlockProjection.nodeProjection
(CutProjectionAssembly.fillProjection p outside left right q)
(HierarchicalLeftDecoder.LowerNode upper level d))
(LowerCutDecoderForm.spaceEquiv p (CutSlotAssembly.fill p outside right)
upper level d hnode
(CutNativeForms.spaceCast
(CleanPhysicalReplay.cutSlots_fill p outside right).symm f)) := by
rw [fill_spaceCast_HPullback]
exact spaceEquiv_HPullback p _ _ upper level d hnode _ _
end
end PerfectCompleteness.LowerCutScalarProjection
end
section
namespace PerfectCompleteness.SourceQuestionLowerForms
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
section Casts
variable {branch : Nat → Nat} {height t : Nat} {C : Type*} [Fintype C]
(rows : Nat → Nat)
{source target : Slots branch (height + 1) → Fin t → MixedSupport.Slot}
def rawCast (h : source = target) :
CutChildGrouping.Raw (C := C) source rows ≃ CutChildGrouping.Raw (C := C) target rows := by
cases h
exact Equiv.refl _
theorem rawCast_law (h : source = target) :
(CutChildGrouping.rawLaw (C := C) source rows).pushforward (rawCast rows h) =
CutChildGrouping.rawLaw (C := C) target rows := by
cases h
exact FiniteDistribution.pushforward_id _
theorem formCast_injective (h : source = target) :
Function.Injective (CutNativeForms.formCast h) := by
cases h
exact Function.injective_id
theorem fullCollision_formCast (h : source = target)
(F G : Option (ChildBilinearCollisionTransfer.ParentForm source)) :
BilinearCollisionTransfer.fullCollision (F.map (CutNativeForms.formCast h))
(G.map (CutNativeForms.formCast h)) =
BilinearCollisionTransfer.fullCollision F G :=
MultiplicationFormInjective.definedEqual_map_of_injective
(CutNativeForms.formCast h) (formCast_injective h) F G
end Casts
variable {branch rows repeats : Nat → Nat} {n height t v m : Nat}
(path : Path branch n (height + 1))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(placeholder : Slots branch (height + 1) → Fin t → MixedSupport.Slot)
(clauses : Fin m → SourceClause.NormalizedClause v)
abbrev Questions := SourceChildMarkedLaw.Questions
(branch := branch) (n := height) (t := t) (m := m)
abbrev Calls := Option (WholeCutCalls.Index rows repeats path)
def slots (q : Questions (branch := branch) (height := height) (t := t) (m := m)) :
Slots branch n → Fin t → MixedSupport.Slot :=
CutSlotAssembly.fill path outside (SourceChildMarkedLaw.questionSlots clauses q)
abbrev Raw (q : Questions (branch := branch) (height := height) (t := t) (m := m)) :=
CutChildGrouping.Raw (C := Calls (rows := rows) (repeats := repeats) path)
(SourceChildMarkedLaw.questionSlots clauses q) rows
abbrev Form (q : Questions (branch := branch) (height := height) (t := t) (m := m)) :=
ChildBilinearCollisionTransfer.ParentForm (SourceChildMarkedLaw.questionSlots clauses q)
theorem cutSlots_eq (q : Questions (branch := branch) (height := height) (t := t) (m := m)) :
WholeCutGrouping.cutSlots path (slots path outside clauses q) =
SourceChildMarkedLaw.questionSlots clauses q :=
CleanPhysicalReplay.cutSlots_fill path outside _
def rawAtCut (q : Questions (branch := branch) (height := height) (t := t) (m := m))
(raw : Raw (rows := rows) (repeats := repeats) path clauses q) :
LowerCutPair.Raw rows repeats path (slots path outside clauses q) :=
rawCast rows (cutSlots_eq path outside clauses q).symm raw
theorem rawAtCut_law
(q : Questions (branch := branch) (height := height) (t := t) (m := m)) :
(CutChildGrouping.rawLaw (C := Calls (rows := rows) (repeats := repeats) path)
(SourceChildMarkedLaw.questionSlots clauses q) rows).pushforward
(rawAtCut (rows := rows) (repeats := repeats) path outside clauses q) =
CutChildGrouping.rawLaw (C := Calls (rows := rows) (repeats := repeats) path)
(WholeCutGrouping.cutSlots path (slots path outside clauses q)) rows :=
rawCast_law rows (cutSlots_eq path outside clauses q).symm
variable (upper : Nodes branch n) (level : Nat)
(d : HierarchicalFrozenTables.LowerNodes upper level)
(hnode : WholeArrayInteriorExterior.upperNode path =
HierarchicalLeftDecoder.LowerNode upper level d)
(hbranch : ∀ k < n, 0 < branch k)
{adviceRows : Nat} (A : ManyGoodRows.RowMap (Block rows upper) adviceRows)
(exterior : CleanPhysicalReplay.Exterior rows repeats path outside placeholder)
abbrev UpperTable (q : Questions (branch := branch) (height := height) (t := t) (m := m)) :=
HierarchicalAllDecoderTables.Input (rows := rows)
(slots path outside clauses q) upper level adviceRows →
HierarchicalAllDecoderTables.UpperAnswer (slots path outside clauses q) upper
variable (tables : (q : Questions (branch := branch) (height := height) (t := t) (m := m)) →
UpperTable (rows := rows) path outside clauses upper level (adviceRows := adviceRows) q)
(cutoff : Nat)
def first (q : Questions (branch := branch) (height := height) (t := t) (m := m))
(raw : Raw (rows := rows) (repeats := repeats) path clauses q) : Option (Form clauses q) :=
(LowerCutDecoderForm.first path (slots path outside clauses q) upper level d hnode hbranch A
(tables q) cutoff
(CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder
(SourceChildMarkedLaw.questionSlots clauses q) exterior)
(rawAtCut path outside clauses q raw)).map
(CutNativeForms.formCast (cutSlots_eq path outside clauses q))
def second (direction : BucketSampler.Direction (rows (height + 1)))
(q : Questions (branch := branch) (height := height) (t := t) (m := m))
(raw : Raw (rows := rows) (repeats := repeats) path clauses q) : Option (Form clauses q) :=
(LowerCutDecoderForm.second path (slots path outside clauses q) upper level d hnode hbranch A
(tables q) cutoff
(CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder
(SourceChildMarkedLaw.questionSlots clauses q) exterior)
direction (rawAtCut path outside clauses q raw)).map
(CutNativeForms.formCast (cutSlots_eq path outside clauses q))
theorem fullCollision_probability
(direction : BucketSampler.Direction (rows (height + 1)))
(q : Questions (branch := branch) (height := height) (t := t) (m := m)) :
(CutChildGrouping.rawLaw (C := Calls (rows := rows) (repeats := repeats) path)
(SourceChildMarkedLaw.questionSlots clauses q) rows).probability
(fun raw => BilinearCollisionTransfer.fullCollision
(first path outside placeholder clauses upper level d hnode hbranch A exterior tables cutoff q raw)
(second path outside placeholder clauses upper level d hnode hbranch A exterior tables cutoff
direction q raw)) =
(CutChildGrouping.rawLaw (C := Calls (rows := rows) (repeats := repeats) path)
(WholeCutGrouping.cutSlots path (slots path outside clauses q)) rows).probability
(fun raw => BilinearCollisionTransfer.fullCollision
(LowerCutDecoderForm.first path (slots path outside clauses q) upper level d hnode hbranch A
(tables q) cutoff (CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder
(SourceChildMarkedLaw.questionSlots clauses q) exterior) raw)
(LowerCutDecoderForm.second path (slots path outside clauses q) upper level d hnode hbranch A
(tables q) cutoff (CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder
(SourceChildMarkedLaw.questionSlots clauses q) exterior) direction raw)) := by
have h := congrArg (fun μ : FiniteDistribution
(CutChildGrouping.Raw (C := Calls (rows := rows) (repeats := repeats) path)
(WholeCutGrouping.cutSlots path (slots path outside clauses q)) rows) => μ.probability
(fun raw => BilinearCollisionTransfer.fullCollision
(LowerCutDecoderForm.first path (slots path outside clauses q) upper level d hnode hbranch A
(tables q) cutoff (CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder
(SourceChildMarkedLaw.questionSlots clauses q) exterior) raw)
(LowerCutDecoderForm.second path (slots path outside clauses q) upper level d hnode hbranch A
(tables q) cutoff (CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder
(SourceChildMarkedLaw.questionSlots clauses q) exterior) direction raw)))
(rawAtCut_law (rows := rows) (repeats := repeats) path outside clauses q)
rw [FiniteDistribution.probability_pushforward] at h
simpa only [first, second, fullCollision_formCast] using h
end
end PerfectCompleteness.SourceQuestionLowerForms
end
end OAI
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