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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Computability.PerfectCompleteness.Decoding.CutProjectionAssembly
import OAI.Computability.PerfectCompleteness.Foundations.SourceOddListsLemmas
import OAI.Computability.PerfectCompleteness.Repetition.CleanEndpointSlots
namespace OAI
section
namespace PerfectCompleteness.CleanEndpointProjection
open scoped Classical
open RecursiveSpaces MixedSupport SourceQuestionReconstruction
noncomputable section
private theorem reconstruct_designated {I E Q : Type*} {Leaf : I → Type*}
(designated : (i : I) → Leaf i) (clean : I → Prop) [DecidablePred clean]
[∀ i, DecidableEq (Leaf i)] (endpoint : I → E → Q)
(visible : Visible (E := E) designated clean) (own : {i : I // clean i} → Q)
(i : {i : I // clean i}) :
reconstruct designated clean endpoint visible own i.val (designated i.val) = own i := by
exact dite_eq_left ⟨i.property, rfl⟩
private theorem dite_heq_of_pos {α β : Type} {P : Prop} [Decidable P]
(f : P → α) (g : ¬ P → α) (b : β) (hP : P)
(h : HEq (f hP) b) : HEq (dite P f g) b := by
rw [dite_eq_left hP]
exact h
variable {v m t : Nat} (clauses : Fin m → SourceClause.NormalizedClause v)
def sourceProjection (projected : Bool) (e : SourceOddLists.Occurrences m t) :
∀ k, Projection (SourceAnswerEquiv.leftSlots clauses (SourceOddLists.left e) k)
(SourceKeys.slot clauses (rightTuple clauses projected e k)) :=
match projected with
| false => fun k => Projection.keep
(SourceAnswerEquiv.leftSlots clauses (SourceOddLists.left e) k)
| true => SourceAnswerEquiv.slotProjection clauses e
theorem sourceProjection_of_true (projected : Bool)
(e : SourceOddLists.Occurrences m t) (k : Fin t) (hp : projected = true) :
HEq (sourceProjection clauses projected e k)
(SourceAnswerEquiv.slotProjection clauses e k) := by
subst projected
rfl
variable {branch : Nat → Nat} {h : Nat}
(designated : Fin (branch h) → Slots branch h)
(clean : Fin (branch h) → Prop)
(visible : Visible (E := SourceTuple m t) designated clean)
theorem leftInside_designated
(own : {i : Fin (branch h) // clean i} → Fin t → Fin m)
(i : {i : Fin (branch h) // clean i}) :
CleanEndpointSlots.leftInside clauses designated clean visible own
(i.val, designated i.val) =
SourceAnswerEquiv.leftSlots clauses (own i) := by
funext k
exact congrArg
(fun question : Fin t → SourceKeys.Endpoint v m => SourceKeys.slot clauses (question k))
(reconstruct_designated designated clean (fun _ => leftTuple (v := v)) visible
(fun j k => SourceKeys.Endpoint.clause (v := v) (own j k)) i)
variable (projected : Fin (branch h) → Bool)
theorem rightInside_designated
(own : {i : Fin (branch h) // clean i} → Fin t → Fin v)
(i : {i : Fin (branch h) // clean i}) :
CleanEndpointSlots.rightInside clauses designated clean visible projected own
(i.val, designated i.val) =
SourceAnswerEquiv.rightSlots clauses (own i) := by
funext k
exact congrArg
(fun question : Fin t → SourceKeys.Endpoint v m => SourceKeys.slot clauses (question k))
(reconstruct_designated designated clean (fun j => rightTuple clauses (projected j)) visible
(fun j k => SourceKeys.Endpoint.variable (m := m) (own j k)) i)
def insideProjection (cleanProjected : ∀ i, clean i → projected i = true)
(e : {i : Fin (branch h) // clean i} → SourceOddLists.Occurrences m t) :
∀ s k, Projection
(CleanEndpointSlots.leftInside clauses designated clean visible
(fun i => SourceOddLists.left (e i)) s k)
(CleanEndpointSlots.rightInside clauses designated clean visible projected
(fun i => SourceOddLists.right clauses (e i)) s k) :=
fun s k => if hs : clean s.1 ∧ s.2 = designated s.1 then
CutProjectionAssembly.castProjection
(by simp only [SourceAnswerEquiv.leftSlots, SourceOddLists.left,
CleanEndpointSlots.leftInside, reconstructLeft, reconstruct, dite_eq_left hs])
(by simp only [rightTuple, cleanProjected s.1 hs.1, ite_true,
CleanEndpointSlots.rightInside, reconstructRight, reconstruct, dite_eq_left hs,
SourceOddLists.right])
(sourceProjection clauses (projected s.1) (e ⟨s.1, hs.1⟩) k)
else
CutProjectionAssembly.castProjection
(by simp only [SourceAnswerEquiv.leftSlots, SourceOddLists.left,
CleanEndpointSlots.leftInside, reconstructLeft, reconstruct, dite_eq_right hs, leftTuple])
(by simp only [CleanEndpointSlots.rightInside, reconstructRight, reconstruct, dite_eq_right hs])
(sourceProjection clauses (projected s.1) (visible s.1 ⟨s.2, hs⟩) k)
theorem insideProjection_designated
(cleanProjected : ∀ i, clean i → projected i = true)
(e : {i : Fin (branch h) // clean i} → SourceOddLists.Occurrences m t)
(i : {i : Fin (branch h) // clean i}) (k : Fin t) :
HEq (insideProjection clauses designated clean visible projected cleanProjected e
(i.val, designated i.val) k)
(SourceAnswerEquiv.slotProjection clauses (e i) k) := by
refine @dite_heq_of_pos _ _
(clean i.val ∧ designated i.val = designated i.val) _ _ _
(SourceAnswerEquiv.slotProjection clauses (e i) k) ⟨i.property, rfl⟩ ?_
exact (CutProjectionAssembly.castProjection_heq _ _ _).trans
(sourceProjection_of_true clauses (projected i.val) (e i) k
(cleanProjected i.val i.property))
end
end PerfectCompleteness.CleanEndpointProjection
end
end OAI
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