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import OAI.Computability.VertexCover.PCP.TableGapReduction
import OAI.Computability.VertexCover.Reduction.SimpleParallel
namespace OAI
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namespace VertexCover.LabelCover
theorem value_le_of_counts (Φ : LabelCover) (σ : ℝ)
(h : ∀ A : Φ.Labeling, (Φ.satisfiedCount A : ℝ) ≤ σ * Φ.M) : Φ.value ≤ σ := by
classical
obtain ⟨A, _, hA⟩ := Finset.exists_mem_eq_sup (Finset.univ : Finset Φ.Labeling)
⟨Φ.defaultLabeling, Finset.mem_univ _⟩ Φ.satisfiedCount
apply (div_le_iff₀ (Nat.cast_pos.mpr Φ.M_pos)).mpr
change ((Finset.univ.sup Φ.satisfiedCount : ℕ) : ℝ) ≤ _
rw [hA]
exact h A
def ParallelCoherent (Φ : LabelCover) : Prop :=
∀ e e', Φ.left e = Φ.left e' → Φ.right e = Φ.right e' → Φ.projection e = Φ.projection e'
end VertexCover.LabelCover
namespace VertexCover.ClauseProjection
open UniqueGames.Foundations
open Target
theorem failureCount_eq_sum (F : Formula) (a : Fin F.«variables» → Bool)
(indices : List (Fin F.clauses.length)) :
PCP.failureCount F a indices =
(indices.map (fun i => if (F.clauses.get i).eval a = false then 1 else 0)).sum := by
induction indices with
| nil => rfl
| cons i rest ih =>
simp only [PCP.failureCount, List.map_cons, List.sum_cons, ih]
congr 1
unfold PCP.clauseFailure PCP.clauseAt
change (if (F.clauses.get i).eval a = true then 0 else 1) = _
cases (F.clauses.get i).eval a <;> rfl
theorem failureCount_all (F : Formula) (a : Fin F.«variables» → Bool) :
PCP.failureCount F a (PCP.allIndices F) = (unsatisfiedClauses F a).card := by
rw [failureCount_eq_sum]
simp only [PCP.allIndices, List.finRange, List.map_ofFn, List.sum_ofFn, Function.comp_apply]
exact (Finset.card_filter _ _).symm
theorem simple_value_le (F : Formula) (η : ℚ) (hgap : Hastad.SourceGap.ClauseGap F η) :
(simpleGame F hgap.1).value ≤ 1 - (η : ℝ) / 3 := by
classical
apply LabelCover.value_le_of_counts
intro A
have hbad := hgap.2 (assignmentOf F hgap.1 (decodedLabeling F hgap.1 A))
rw [failureCount_all] at hbad
have hreject := simple_rejection_count F hgap.1 A
have hrejectR := Nat.cast_le (α := ℝ) |>.mpr hreject
have hpart := Finset.card_filter_add_card_filter_not ((simpleGame F hgap.1).Satisfies A)
(s := Finset.univ)
have hpartR : ((simpleGame F hgap.1).satisfiedCount A : ℝ) +
((Finset.univ.filter (fun e => ¬(simpleGame F hgap.1).Satisfies A e)).card : ℝ) =
(F.clauses.length : ℝ) * 3 := by
simp only [Finset.card_univ, Fintype.card_fin] at hpart
change (simpleGame F hgap.1).satisfiedCount A + _ = F.clauses.length * 3 at hpart
exact_mod_cast hpart
change ((simpleGame F hgap.1).satisfiedCount A : ℝ) ≤
(1 - (η : ℝ) / 3) * ((F.clauses.length * 3 : ℕ) : ℝ)
push_cast
nlinarith
noncomputable def fixedBaseTable : PCP.RoundTables.BaseTable :=
PCP.ExpanderTables.exists_base_table.choose
theorem fixedBaseTable_certificate :
PCP.SpectralReturn.SpectralCertificate (PCP.ExpanderTables.graph fixedBaseTable) (1/100 : ℝ) :=
PCP.ExpanderTables.exists_base_table.choose_spec
noncomputable def baseReduction (F : Formula) : LabelCover :=
simpleGame (PCP.TableIteration.gapMap fixedBaseTable F)
(PCP.TableIteration.gapMap_nonempty fixedBaseTable F)
theorem baseReduction_complete (F : Formula) (hF : F.Satisfiable) :
∃ A : (baseReduction F).Labeling, ∀ e, (baseReduction F).Satisfies A e :=
simple_perfect_complete _ _ (PCP.TableIteration.gapMap_completeness fixedBaseTable F hF)
theorem baseReduction_sound (F : Formula) (hF : ¬F.Satisfiable) :
(baseReduction F).value ≤ 1 - (PCP.PCPIteration.finalClauseGap : ℝ) / 3 :=
simple_value_le _ _ (PCP.TableGapReduction.gapMap_clauseGap fixedBaseTable
fixedBaseTable_certificate F hF)
theorem baseReduction_coherent (F : Formula) : (baseReduction F).ParallelCoherent :=
simple_parallel _ _
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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