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import OAI.Computability.VertexCover.Reduction.RejectionCount
namespace OAI
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namespace VertexCover.ClauseProjection
open UniqueGames.Foundations.Target
def Consistent (F : Formula) (i : Fin F.clauses.length) (P : Pattern) : Prop :=
∀ k l : Fin 3, (literalAt F i k).variableIndex = (literalAt F i l).variableIndex →
flipSign (literalAt F i k).positive (P.val k) =
flipSign (literalAt F i l).positive (P.val l)
instance (F : Formula) (i : Fin F.clauses.length) (P : Pattern) :
Decidable (Consistent F i P) := inferInstanceAs (Decidable (∀ _k _l, _ → _))
theorem consistent_truth (F : Formula) (a : Fin F.«variables» → Bool)
(i : Fin F.clauses.length) (hp : ∃ k, clauseValue F a i k = true) :
Consistent F i ⟨clauseValue F a i, hp⟩ := by
intro k l hkl
change flipSign _ (flipSign _ _) = flipSign _ (flipSign _ _)
simp only [flipSign_flipSign]
rw [hkl]
def defaultAssignment (F : Formula) (i : Fin F.clauses.length)
(v : Fin F.«variables») : Bool :=
if v = (literalAt F i 0).variableIndex then (literalAt F i 0).positive else false
theorem default_true (F : Formula) (i : Fin F.clauses.length) :
clauseValue F (defaultAssignment F i) i 0 = true := by
simp only [clauseValue, Literal.eval, defaultAssignment, ↓reduceIte]
cases (literalAt F i 0).positive <;> rfl
def defaultPattern (F : Formula) (i : Fin F.clauses.length) : Pattern :=
⟨clauseValue F (defaultAssignment F i) i, ⟨0, default_true F i⟩⟩
def normalizePattern (F : Formula) (i : Fin F.clauses.length) (P : Pattern) : Pattern :=
if Consistent F i P then P else defaultPattern F i
theorem normalize_consistent (F : Formula) (i : Fin F.clauses.length) (P : Pattern) :
Consistent F i (normalizePattern F i P) := by
unfold normalizePattern
split
· assumption
· exact consistent_truth F (defaultAssignment F i) i _
theorem normalize_truth (F : Formula) (a : Fin F.«variables» → Bool)
(i : Fin F.clauses.length) (hp : ∃ k, clauseValue F a i k = true) :
normalizePattern F i ⟨clauseValue F a i, hp⟩ = ⟨clauseValue F a i, hp⟩ := by
exact ite_eq_left (consistent_truth F a i hp)
noncomputable def normalizeLabel (F : Formula) (i : Fin F.clauses.length) (l : Fin 7) : Fin 7 :=
patternEquiv (normalizePattern F i (patternEquiv.symm l))
noncomputable def simpleGame (F : Formula) (hne : F.clauses ≠ []) : LabelCover :=
{ game F hne with projection := fun e label =>
(game F hne).projection e (normalizeLabel F ((game F hne).left e) label) }
noncomputable def decodedLabeling (F : Formula) (hne : F.clauses ≠ [])
(A : (simpleGame F hne).Labeling) : (game F hne).Labeling :=
(fun i => normalizeLabel F i (A.1 i), A.2)
theorem simple_satisfies_iff (F : Formula) (hne : F.clauses ≠ [])
(A : (simpleGame F hne).Labeling) (e : Fin (simpleGame F hne).M) :
(simpleGame F hne).Satisfies A e ↔ (game F hne).Satisfies (decodedLabeling F hne A) e :=
Iff.rfl
theorem simple_perfect_complete (F : Formula) (hne : F.clauses ≠ []) (hF : F.Satisfiable) :
∃ A : (simpleGame F hne).Labeling, ∀ e, (simpleGame F hne).Satisfies A e := by
obtain ⟨a, ha⟩ := hF
let A : (simpleGame F hne).Labeling := honestLabeling F hne a ha
refine ⟨A, ?_⟩
intro e
change (game F hne).Satisfies (decodedLabeling F hne A) e
have hdecode : decodedLabeling F hne A = honestLabeling F hne a ha := by
apply Prod.ext
· change (fun i : Fin F.clauses.length => normalizeLabel F i
(patternEquiv ⟨clauseValue F a i, _⟩)) =
(fun i : Fin F.clauses.length => patternEquiv ⟨clauseValue F a i, _⟩)
funext i
simp only [normalizeLabel, Equiv.symm_apply_apply, normalize_truth]
· rfl
rw [hdecode]
exact honest_satisfies F hne a ha e
theorem simple_rejection_count (F : Formula) (hne : F.clauses ≠ [])
(A : (simpleGame F hne).Labeling) :
(unsatisfiedClauses F (assignmentOf F hne (decodedLabeling F hne A))).card ≤
(Finset.univ.filter (fun e => ¬(simpleGame F hne).Satisfies A e)).card :=
rejection_count F hne (decodedLabeling F hne A)
theorem simple_parallel (F : Formula) (hne : F.clauses ≠ [])
(e e' : Fin (simpleGame F hne).M)
(hl : (simpleGame F hne).left e = (simpleGame F hne).left e')
(hr : (simpleGame F hne).right e = (simpleGame F hne).right e') :
(simpleGame F hne).projection e = (simpleGame F hne).projection e' := by
funext label
change (finProdFinEquiv.symm e).1 = (finProdFinEquiv.symm e').1 at hl
change (literalAt F (finProdFinEquiv.symm e).1 (finProdFinEquiv.symm e).2).variableIndex =
(literalAt F (finProdFinEquiv.symm e').1 (finProdFinEquiv.symm e').2).variableIndex at hr
simp only [simpleGame, game, normalizeLabel, Equiv.symm_apply_apply]
rw [hl] at hr ⊢
congr 1
exact normalize_consistent F (finProdFinEquiv.symm e').1 (patternEquiv.symm label) _ _ hr
end VertexCover.ClauseProjection
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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