File size: 2,798 Bytes
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 | import OAI.Computability.PerfectCompleteness.Decoding.ProjectedNativeCollision
import OAI.Computability.PerfectCompleteness.Foundations.CutNativeForms
import OAI.Computability.PerfectCompleteness.Sampling.SourceChildMarkedLaw
namespace OAI
section
namespace PerfectCompleteness.SourceQuestionUnmarkedRange
open scoped Classical
open RecursiveSpaces TreeSourceSpaces SourceChildKernel
noncomputable section
variable {branch : Nat → Nat} {height t v m : Nat}
private theorem unmarked_range_cast
{left native right : Slots branch (height + 1) → Fin t → MixedSupport.Slot}
(h : left = native)
(p : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(mask : Fin (branch height) → Bool)
(hin : ChildUnmarkedSpace.space left mask ≤ LinearMap.range (HPullback p)) :
ChildUnmarkedSpace.space native mask ≤
LinearMap.range (HPullback (CutNativeForms.projectionCast h rfl p)) := by
cases h
exact hin
variable (rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch height) → Slots branch height)
(q : SourceChildMarkedLaw.Questions (branch := branch) (n := height) (t := t) (m := m))
(choices : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := height) (t := t))
theorem leftSlots_eq :
parentLeftSlots clauses designated (SourceQuestionKernelJoint.sources designated q choices) =
SourceChildMarkedLaw.questionSlots clauses q :=
SourceQuestionPositionSplit.parentLeftSlots_join clauses designated q _
variable {C : Type*} [Fintype C]
(raw : (i : Fin (branch height)) → Raw (C := C) (t := t) rows clauses designated i)
def projection : ∀ s k, MixedSupport.Projection
(SourceChildMarkedLaw.questionSlots clauses q s k)
(parentRightSlots rows clauses designated
(SourceQuestionKernelJoint.sources designated q choices) raw s k) :=
CutNativeForms.projectionCast (leftSlots_eq clauses designated q choices) rfl
(parentProjection rows clauses designated
(SourceQuestionKernelJoint.sources designated q choices) raw)
theorem unmarked_le_range :
ChildUnmarkedSpace.space (SourceChildMarkedLaw.questionSlots clauses q)
(fun i => rawProjected rows clauses designated i (raw i)) ≤
LinearMap.range (HPullback (projection rows clauses designated q choices raw)) := by
exact unmarked_range_cast (leftSlots_eq clauses designated q choices)
(parentProjection rows clauses designated
(SourceQuestionKernelJoint.sources designated q choices) raw)
(fun i => rawProjected rows clauses designated i (raw i))
(ProjectedNativeCollision.parent_unmarked_le_range clauses designated
(SourceQuestionKernelJoint.sources designated q choices) raw)
end
end PerfectCompleteness.SourceQuestionUnmarkedRange
end
end OAI
|