File size: 2,865 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 | import OAI.Computability.PerfectCompleteness.Reduction.FixedTarget
namespace OAI
namespace PerfectCompleteness.FixedTargetValue
open UniqueGamesTheorem.Foundations Target
open FixedParameters FixedRows FixedPreliminaryGame
open scoped Classical
noncomputable section
private theorem packed_value_le (formula : Formula) [NeZero formula.clauses.length]
(nonempty : formula.clauses ≠ [])
(distinct : ∀ clause ∈ formula.clauses, ∀ i j : Fin 3,
clause[i].variableIndex = clause[j].variableIndex → i = j)
(branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat)
(hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k)
(hrows : ∀ k, 0 < rows (k + 1)) (δ : ℚ) (hδ : 0 < δ) :
(NormalizedTarget.construct (branch := branch) (n := n) (t := t)
rows repeats hn hbranch hrows δ ⟨formula, nonempty, distinct⟩).value ≤
PreliminaryOutput.value
(fun index : Fin formula.clauses.length =>
({ clause := PCP.clauseAt formula index
distinct := fun i j same => distinct _ (List.getElem_mem index.isLt) i j same } :
SourceClause.NormalizedClause formula.variables))
branch n t rows repeats hn hbranch hrows + (δ : ℝ) / 3 := by
have clauses_eq :
NormalizedSourceInput.clauses ⟨formula, nonempty, distinct⟩ =
(fun index : Fin formula.clauses.length =>
({ clause := PCP.clauseAt formula index
distinct := fun i j same => distinct _ (List.getElem_mem index.isLt) i j same } :
SourceClause.NormalizedClause formula.variables)) := by
funext index
rfl
have native :
(NormalizedTarget.construct (branch := branch) (n := n) (t := t)
rows repeats hn hbranch hrows δ ⟨formula, nonempty, distinct⟩).value ≤
PreliminaryOutput.value (NormalizedSourceInput.clauses ⟨formula, nonempty, distinct⟩)
branch n t rows repeats hn hbranch hrows + (δ : ℝ) / 3 := by
simpa only [PreliminaryOutput.value] using
(NormalizedTarget.value_le (branch := branch) (n := n) (t := t)
rows repeats hn hbranch hrows δ ⟨formula, nonempty, distinct⟩ hδ)
rw [clauses_eq] at native
exact native
variable {δ : ℚ} {hδ : 0 < δ} (p : Parameters δ hδ)
theorem value_le (input : List Bool) :
(FixedTarget.construct p input).value ≤
FixedPreliminaryGame.value p input + (δ : ℝ) / 3 :=
packed_value_le (PCPSource.normalizedFormula (BinaryLanguage.totalRename input))
(PCPSource.normalizedFormula_nonempty (BinaryLanguage.totalRename input))
(PCPSource.normalizedFormula_distinct (BinaryLanguage.totalRename input))
(branch p) p.plan.depth (sourceLength p.plan hδ) (rows p.plan) (repeats p.plan)
(depth_pos p) (fun k _ => branch_pos p k) (fun k => rows_pos p (k + 1)) δ hδ
end
end PerfectCompleteness.FixedTargetValue
end OAI
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