File size: 3,422 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 | import Mathlib
import OAI.Computability.MinUncut.Encoding.Binary
import OAI.Computability.MinUncut.Graphs.DemandOrder
namespace OAI
section
noncomputable section
namespace MinUncut.Outer.Parameters
open MinUncut.Inner MinUncut.SourceBridge
open MinUncutGames MinUncutGames.Reduction MinUncutGames.Foundations
def sourceError {J : ℕ} (P : Parameters J) : ℚ := min (1/2) (P.o.b4/P.o.t)
lemma sourceError_pos {J : ℕ} (P : Parameters J) : 0<P.sourceError := by
apply lt_min (by norm_num)
exact div_pos P.ho.2.1 (by exact_mod_cast P.t_pos)
lemma sourceError_small {J : ℕ} (P : Parameters J) : P.sourceError≤1/2 := min_le_left _ _
lemma sourceError_budget {J : ℕ} (P : Parameters J) : (P.o.t:ℝ)*(P.sourceError:ℝ)≤P.o.b4 := by
have hh := min_le_right (1/2:ℚ) (P.o.b4/P.o.t)
have ht : (0:ℚ)<P.o.t := by exact_mod_cast P.t_pos
have hh' : (P.o.t:ℚ)*P.sourceError≤P.o.b4 := by
rw [mul_comm]
exact (le_div_iff₀ ht).mp hh
exact_mod_cast hh'
def baseTable : PCP.RoundTables.BaseTable := Classical.choose PCP.ExpanderTables.exists_base_table
lemma baseTable_certificate :
PCP.SpectralReturn.SpectralCertificate (PCP.ExpanderTables.graph baseTable) (1/100:ℝ) :=
Classical.choose_spec PCP.ExpanderTables.exists_base_table
def sourceReduction {J : ℕ} (P : Parameters J) : SourceBridge.BinaryReduction P.sourceError :=
SourceBridge.binaryReduction {
reduce := MinUncutGames.Outer.HastadSource.output baseTable P.sourceError P.sourceError_pos
completeness := MinUncutGames.Outer.HastadSource.complete baseTable P.sourceError P.sourceError_pos
soundness := MinUncutGames.Outer.HastadSource.sound baseTable baseTable_certificate P.sourceError P.sourceError_pos
computation := MinUncutGames.Outer.HastadSource.computation baseTable P.sourceError P.sourceError_pos
finiteAlphabet := MinUncutGames.Outer.HastadSource.finiteAlphabet baseTable P.sourceError P.sourceError_pos }
def sourceGraph {J : ℕ} (P : Parameters J) (F : BinaryFormula.Formula) : MinUncut.Output :=
P.graph (equations (P.sourceReduction.reduce F))
lemma sourceGraph_yes {J : ℕ} (P : Parameters J) (F : BinaryFormula.Formula) (hF : F.Satisfiable) :
(P.sourceGraph F).opt≤(P.sourceGraph F).threshold := by
obtain ⟨s,hs⟩ := binaryReduction_complete P.sourceReduction F hF P.o.t P.o.b4 P.sourceError_budget
exact P.graph_yes _ s hs
lemma sourceGraph_no {K J : ℕ} (hK : 2≤K) (hKJ : 8*K≤J) (P : Parameters J)
(F : BinaryFormula.Formula) (hF : ¬F.Satisfiable) :
K*(P.sourceGraph F).threshold<(P.sourceGraph F).opt :=
P.graph_no hK hKJ _ (binaryReduction_sound P.sourceError_small P.sourceReduction F hF)
end MinUncut.Outer.Parameters
namespace MinUncut
open Outer
def parameters (K : ℕ) : Parameters (10*(max K 2)) := Parameters.select _ (by omega)
def semanticReduction (K : ℕ) (F : MinUncutGames.BinaryFormula.Formula) : Output :=
(parameters K).sourceGraph F
lemma semanticReduction_yes {K : ℕ} (F : MinUncutGames.BinaryFormula.Formula) (hF : F.Satisfiable) :
(semanticReduction K F).opt≤(semanticReduction K F).threshold :=
(parameters K).sourceGraph_yes F hF
lemma semanticReduction_no {K : ℕ} (hK : 2≤K) (F : MinUncutGames.BinaryFormula.Formula) (hF : ¬F.Satisfiable) :
K*(semanticReduction K F).threshold<(semanticReduction K F).opt :=
(parameters K).sourceGraph_no hK (by omega) F hF
end MinUncut
end
end
end OAI
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