File size: 5,359 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 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 | import OAI.Computability.PerfectCompleteness.Construction.SourceQuestionReconstruction
import OAI.Computability.PerfectCompleteness.Foundations.CutSlotAssembly
namespace OAI
section
namespace PerfectCompleteness.CleanEndpointSlots
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces
open SourceQuestionReconstruction
variable {branch : Nat → Nat} {n h t v m : Nat} [NeZero m]
(clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch h) → Slots branch h)
(clean : Fin (branch h) → Prop)
(visible : Visible (E := SourceTuple m t) designated clean)
def leftInside (own : {i : Fin (branch h) // clean i} → Fin t → Fin m) :
Slots branch (h + 1) → Fin t → MixedSupport.Slot :=
fun s k => SourceKeys.slot clauses
(reconstructLeft designated clean visible own s.1 s.2 k)
def rightInside (projected : Fin (branch h) → Bool)
(own : {i : Fin (branch h) // clean i} → Fin t → Fin v) :
Slots branch (h + 1) → Fin t → MixedSupport.Slot :=
fun s k => SourceKeys.slot clauses
(reconstructRight designated clean clauses projected visible own s.1 s.2 k)
omit [NeZero m] in
theorem leftInside_nonclean
(own other : {i : Fin (branch h) // clean i} → Fin t → Fin m)
(i : Fin (branch h)) (hi : ¬ clean i) (s : Slots branch h) :
leftInside clauses designated clean visible own (i, s) =
leftInside clauses designated clean visible other (i, s) := by
funext k
have hn : ¬ (clean i ∧ s = designated i) := fun h => hi h.1
simp only [leftInside, reconstructLeft, reconstruct, dite_eq_right hn]
omit [NeZero m] in
theorem rightInside_nonclean (projected : Fin (branch h) → Bool)
(own other : {i : Fin (branch h) // clean i} → Fin t → Fin v)
(i : Fin (branch h)) (hi : ¬ clean i) (s : Slots branch h) :
rightInside clauses designated clean visible projected own (i, s) =
rightInside clauses designated clean visible projected other (i, s) := by
funext k
have hn : ¬ (clean i ∧ s = designated i) := fun h => hi h.1
simp only [rightInside, reconstructRight, reconstruct, dite_eq_right hn]
def leftReference : Slots branch (h + 1) → Fin t → MixedSupport.Slot :=
leftInside clauses designated clean visible (fun _ _ => 0)
def rightReference (projected : Fin (branch h) → Bool) :
Slots branch (h + 1) → Fin t → MixedSupport.Slot :=
rightInside clauses designated clean visible projected
(fun _ _ => (clauses 0).variable 0)
variable (rows repeats : Nat → Nat) (p : Path branch n (h + 1))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
def leftSlots (own : {i : Fin (branch h) // clean i} → Fin t → Fin m) :=
CutSlotAssembly.fill p outside (leftInside clauses designated clean visible own)
def rightSlots (projected : Fin (branch h) → Bool)
(own : {i : Fin (branch h) // clean i} → Fin t → Fin v) :=
CutSlotAssembly.fill p outside (rightInside clauses designated clean visible projected own)
omit [NeZero m] in
theorem leftSlots_kept_eq
(own other : {i : Fin (branch h) // clean i} → Fin t → Fin m)
(s : Slots branch n) (hs : CutSamplerKeyLocality.keptLeaf p clean s) :
leftSlots clauses designated clean visible p outside own s =
leftSlots clauses designated clean visible p outside other s :=
CutSlotAssembly.fill_kept_eq p outside _ _ clean
(leftInside_nonclean clauses designated clean visible own other) s hs
omit [NeZero m] in
theorem rightSlots_kept_eq (projected : Fin (branch h) → Bool)
(own other : {i : Fin (branch h) // clean i} → Fin t → Fin v)
(s : Slots branch n) (hs : CutSamplerKeyLocality.keptLeaf p clean s) :
rightSlots clauses designated clean visible p outside projected own s =
rightSlots clauses designated clean visible p outside projected other s :=
CutSlotAssembly.fill_kept_eq p outside _ _ clean
(rightInside_nonclean clauses designated clean visible projected own other) s hs
def leftReplay
(record : WholeCutReplay.Tape rows repeats p
(CutSlotAssembly.fill p outside (leftReference clauses designated clean visible)) clean)
(own : {i : Fin (branch h) // clean i} → Fin t → Fin m) :
WholeCutReplay.Tape rows repeats p
(leftSlots clauses designated clean visible p outside own) clean :=
CutSlotAssembly.transport rows repeats p outside
(leftReference clauses designated clean visible)
(leftInside clauses designated clean visible own) clean
(leftInside_nonclean clauses designated clean visible (fun _ _ => 0) own) record
def rightReplay (projected : Fin (branch h) → Bool)
(record : WholeCutReplay.Tape rows repeats p
(CutSlotAssembly.fill p outside
(rightReference clauses designated clean visible projected)) clean)
(own : {i : Fin (branch h) // clean i} → Fin t → Fin v) :
WholeCutReplay.Tape rows repeats p
(rightSlots clauses designated clean visible p outside projected own) clean :=
CutSlotAssembly.transport rows repeats p outside
(rightReference clauses designated clean visible projected)
(rightInside clauses designated clean visible projected own) clean
(rightInside_nonclean clauses designated clean visible projected
(fun _ _ => (clauses 0).variable 0) own) record
end
end PerfectCompleteness.CleanEndpointSlots
end
end OAI
|