File size: 3,982 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 | import OAI.Computability.PerfectCompleteness.Foundations.CanonicalEdgesLemmas
namespace OAI
namespace PerfectCompleteness.SourceQuestionReconstruction
open SourceClause SourceKeys
noncomputable section
variable {I E Q : Type*} {Leaf : I → Type*}
(designated : (i : I) → Leaf i) (clean : I → Prop) [DecidablePred clean]
[∀ i, DecidableEq (Leaf i)]
abbrev Visible := (i : I) → {s : Leaf i // ¬(clean i ∧ s = designated i)} → E
def reveal (x : (i : I) → Leaf i → E) : Visible (E := E) designated clean :=
fun i s => x i s.1
def reconstruct (endpoint : I → E → Q)
(visible : Visible (E := E) designated clean) (own : {i : I // clean i} → Q) :
(i : I) → Leaf i → Q :=
fun i s => if h : clean i ∧ s = designated i then own ⟨i, h.1⟩
else endpoint i (visible i ⟨s, h⟩)
theorem reconstruct_eq (endpoint : I → E → Q) (x : (i : I) → Leaf i → E) :
reconstruct designated clean endpoint (reveal designated clean x)
(fun i => endpoint i.1 (x i.1 (designated i.1))) =
fun i s => endpoint i (x i s) := by
funext i s
by_cases h : clean i ∧ s = designated i
· simp [reconstruct, h]
· simp [reconstruct, reveal, h]
variable {v m t : Nat}
(clauses : Fin m → NormalizedClause v)
abbrev SourceTuple (m t : Nat) := Fin t → Occurrence m
def leftTuple (e : SourceTuple m t) : Fin t → Endpoint v m :=
fun k => .clause (e k).1
def rightTuple (projected : Bool) (e : SourceTuple m t) : Fin t → Endpoint v m :=
fun k => if projected then .variable (occurrenceVariable clauses (e k)) else .clause (e k).1
def reconstructLeft
(visible : Visible (E := SourceTuple m t) designated clean)
(own : {i : I // clean i} → Fin t → Fin m) :
(i : I) → Leaf i → Fin t → Endpoint v m :=
reconstruct designated clean (fun _ => leftTuple) visible
(fun i k => .clause (own i k))
def reconstructRight (projected : I → Bool)
(visible : Visible (E := SourceTuple m t) designated clean)
(own : {i : I // clean i} → Fin t → Fin v) :
(i : I) → Leaf i → Fin t → Endpoint v m :=
reconstruct designated clean (fun i => rightTuple clauses (projected i)) visible
(fun i k => .variable (own i k))
theorem reconstructLeft_eq (x : (i : I) → Leaf i → SourceTuple m t) :
reconstructLeft designated clean (reveal designated clean x)
(fun i k => (x i.1 (designated i.1) k).1) =
fun i s => leftTuple (v := v) (x i s) := by
exact reconstruct_eq designated clean (fun _ => leftTuple) x
theorem reconstructRight_eq (projected : I → Bool)
(clean_projected : ∀ i, clean i → projected i = true)
(x : (i : I) → Leaf i → SourceTuple m t) :
reconstructRight designated clean clauses projected (reveal designated clean x)
(fun i k => occurrenceVariable clauses (x i.1 (designated i.1) k)) =
fun i s => rightTuple clauses (projected i) (x i s) := by
have hown : (fun (i : {i : I // clean i}) k =>
Endpoint.variable (m := m) (occurrenceVariable clauses (x i.1 (designated i.1) k))) =
(fun (i : {i : I // clean i}) =>
rightTuple clauses (projected i.1) (x i.1 (designated i.1))) := by
funext i k
simp only [rightTuple, clean_projected i.1 i.2, ite_true]
unfold reconstructRight
rw [hown]
exact reconstruct_eq designated clean (fun i => rightTuple clauses (projected i)) x
theorem reconstructRight_slots (projected : I → Bool)
(clean_projected : ∀ i, clean i → projected i = true)
(x : (i : I) → Leaf i → SourceTuple m t) :
(fun i s k => slot clauses
(reconstructRight designated clean clauses projected (reveal designated clean x)
(fun i k => occurrenceVariable clauses (x i.1 (designated i.1) k)) i s k)) =
(fun i s k => slot clauses (rightTuple clauses (projected i) (x i s) k)) := by
rw [reconstructRight_eq designated clean clauses projected clean_projected x]
end
end PerfectCompleteness.SourceQuestionReconstruction
end OAI
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