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ea76aac | 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 | import Mathlib
import OAI.Computability.MinUncut.Estimates.Bridge
import OAI.Computability.MinUncut.Machines.MachineBinaryRenameMachine
namespace OAI
section
noncomputable section
namespace MinUncut.SourceBridge
open MinUncutGames MinUncutGames.Reduction MinUncutGames.Foundations
open MinUncutGames.Foundations.Complexity
open Outer MinUncut.Inner
structure BinaryReduction (ξ : ℚ) where
reduce : BinaryFormula.Formula → SourceEncoding.Input
complete : ∀ F, F.Satisfiable → ∃ bits, 1-ξ≤Outer.HastadSource.success (reduce F) bits
sound : ∀ F, ¬F.Satisfiable → ∀ bits, Outer.HastadSource.success (reduce F) bits≤(1+ξ)/2
computation : Turing.TM2ComputableInPolyTime BinaryEncoding.formulaBits SourceEncoding.inputBits reduce
finiteAlphabet : MachineFiniteAlphabet.FiniteAlphabet computation.tm
def binaryReduction {ξ : ℚ} (R : Outer.HastadSource.Reduction ξ) : BinaryReduction ξ where
reduce F := R.reduce (BinaryOccurrenceRename.renamed F)
complete F hF := R.completeness _ ((BinaryOccurrenceRename.renamed_satisfiable_iff F).mpr hF)
sound F hF := R.soundness _ (fun h => hF ((BinaryOccurrenceRename.renamed_satisfiable_iff F).mp h))
computation := MachineSequential.composeBits BinaryRenameMachine.computableInPolyTime R.computation
finiteAlphabet := MachineFiniteAlphabet.composeBits _ _
BinaryRenameMachine.computation_finiteAlphabet R.finiteAlphabet
lemma binaryReduction_exists {ξ : ℚ} (hξ : 0<ξ) : Nonempty (BinaryReduction ξ) := by
obtain ⟨R⟩ := Outer.HastadSource.exists_reduction ξ hξ
exact ⟨binaryReduction R⟩
lemma binaryReduction_sound {ξ : ℚ} (hξ : ξ≤1/2) (R : BinaryReduction ξ)
(F : BinaryFormula.Formula) (hF : ¬F.Satisfiable) (s : Fin (R.reduce F).«variables» → F₂) :
equationFraction (equations (R.reduce F)) s≤3/4 := by
rw [fraction_eq_assignment]
have hh : (Outer.HastadSource.success (R.reduce F) (fun x => fieldBool (s x)):ℝ)≤
((1+ξ)/2:ℚ) := Rat.cast_le.mpr (R.sound F hF _)
have hξ' : (ξ:ℝ)≤1/2 := by
simpa only [Rat.cast_div,Rat.cast_one,Rat.cast_ofNat] using
(Rat.cast_le.mpr hξ : (ξ:ℝ)≤((1/2:ℚ):ℝ))
push_cast at hh
linarith
lemma binaryReduction_complete {ξ : ℚ} (R : BinaryReduction ξ)
(F : BinaryFormula.Formula) (hF : F.Satisfiable) (t : ℕ) (b : ℝ)
(hξ : (t:ℝ)*(ξ:ℝ)≤b) :
∃ s : Fin (R.reduce F).«variables» → F₂,
(t:ℝ)*(1-equationFraction (equations (R.reduce F)) s)≤b := by
obtain ⟨bits,hb⟩ := R.complete F hF
refine ⟨fun x => boolField (bits x),?_⟩
rw [fraction_eq]
have hb' : 1-(ξ:ℝ)≤(Outer.HastadSource.success (R.reduce F) bits:ℝ) := by exact_mod_cast hb
exact (mul_le_mul_of_nonneg_left (by linarith :
1-(Outer.HastadSource.success (R.reduce F) bits:ℝ)≤(ξ:ℝ)) (Nat.cast_nonneg t)).trans hξ
end MinUncut.SourceBridge
end
end
end OAI
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