import OAI.Computability.UniqueGames.Games.ProductSoundness import OAI.Computability.UniqueGames.Machines.MachineProductRuntime import OAI.Computability.UniqueGames.Machines.MachineSubdivisionBodySpecLemmas import OAI.Computability.UniqueGames.Machines.MachineUniformRun import OAI.Computability.UniqueGames.Reduction.ParameterSelection namespace OAI /-! The concrete final composition, once the matrix reduction has been proved. This constructor copies the actual occurrence list, subdivides every occurrence into four edges, and runs the full tuple-product machine. Its hypotheses are the matrix reduction's actual value bounds and finite-machine computation. It is an intermediate composition theorem, not the unconditional endpoint. -/ namespace UniqueGamesTheorem.FinishReduction open UniqueGamesTheorem.Foundations Target open UniqueGamesTheorem.Foundations.Complexity open UniqueGamesTheorem.Integration open Explicit noncomputable section /-- Compose the proved matrix reduction with the explicit three-stage finish. All parameters, machines, alphabets, and time polynomials precede the input. The original raw binary input codec is retained by sequential composition. -/ def of_matrix_reduction {ε δ : ℝ} (P : ParameterSelection.OuterParameters ε δ) {q s : ℕ} (hq : 2 ≤ q) (hs : 1 ≤ s) (coordinates : Fin q ≃ BinaryLinear.Vector s) (matrix : List Bool → Instance q) (translations : ∀ input, TranslationTarget.IsTranslationInstance coordinates (matrix input)) (computation : Turing.TM2ComputableInPolyTime (id : List Bool → List Bool) gameBits matrix) (finiteAlphabet : MachineFiniteAlphabet.FiniteAlphabet computation.tm) (d : ℚ) (hd : d ≤ P.epsilon0 / P.repetitions) (complete : ∀ input, BinaryLanguage.language input → 1 - (d : ℝ) ≤ InstanceValue.value (matrix input)) (sound : ∀ input, ¬BinaryLanguage.language input → InstanceValue.value (matrix input) ≤ 99 / 100) : MachineOutputContract.BinaryGapReduction ε δ := by classical have hqpos : 0 < q := by omega have hC : 0 < P.copies := by have h := P.copies_large; omega letI : Nonempty (Fin q) := ⟨⟨0, hqpos⟩⟩ let rounded := fun input => UniformTarget.construct P.copies hC (matrix input) have roundedTranslations : ∀ input, TranslationTarget.IsTranslationInstance coordinates (rounded input) := fun input => UniformTarget.translations P.copies hC (matrix input) coordinates (translations input) let roundedMachine := MachineSequential.composeBits (f := matrix) (g := UniformTarget.construct P.copies hC) computation (MachineUniformRun.computation (q := q) P.copies hC) have roundedFinite : MachineFiniteAlphabet.FiniteAlphabet roundedMachine.tm := MachineFiniteAlphabet.composeBits (f := matrix) (g := UniformTarget.construct P.copies hC) computation (MachineUniformRun.computation (q := q) P.copies hC) finiteAlphabet (MachineUniformRun.finiteAlphabet (q := q) P.copies hC) let base := fun input => SubdivisionTarget.subdivide (rounded input) let basePresentation := fun input => SubdivisionTarget.simpleBipartite (rounded input) let baseMachine := MachineSubdivisionCompose.translationComputableInPolyTime coordinates roundedMachine roundedTranslations have baseFinite : MachineFiniteAlphabet.FiniteAlphabet baseMachine.tm := MachineSubdivisionCompose.finiteAlphabet roundedMachine (fun input => MachineSubdivisionCorrespondence.translations_involutive coordinates (rounded input) (roundedTranslations input)) roundedFinite let output := fun input => ProductPaddedOutput.output (base input) (basePresentation input) P.repetitions P.repetitions_pos let outputMachine := MachineProductRuntime.compose baseMachine basePresentation P.repetitions P.repetitions_pos have hqout : 2 ≤ q ^ P.repetitions := hq.trans (Nat.le_self_pow P.repetitions_pos.ne' q) have hqoutpos : 0 < q ^ P.repetitions := by omega have baseValue (input : List Bool) : InstanceValue.value (base input) = 1 - (1 - InstanceValue.value (matrix input)) / 4 := by dsimp [base, rounded] rw [SubdivisionTarget.value_eq _ hqpos, UniformTarget.value P.copies hC (matrix input) hqpos] have hτ : (0 : ℝ) < (P.pStar : ℝ) := by exact_mod_cast P.pStar_pos have hτsmall : (P.pStar : ℝ) ≤ 1 / 200 := by simpa only [ParameterSelection.OuterParameters.pStar, Rat.cast_div, Rat.cast_ofNat, Rat.cast_one] using ((Rat.cast_le (K := ℝ)).mpr P.reciprocal_small) have hτbudget : (P.pStar : ℝ) ≤ (P.epsilon0 : ℝ) / P.repetitions := by exact_mod_cast P.reciprocal_budget have hdbudget : (d : ℝ) ≤ (P.epsilon0 : ℝ) / P.repetitions := by exact_mod_cast hd have hε0 : (0 : ℝ) ≤ (P.epsilon0 : ℝ) := by exact_mod_cast P.epsilon0_pos.le have roundingError (input : List Bool) : |InstanceValue.value (base input) - (1 - (1 - InstanceValue.value (matrix input)) / 4)| ≤ (P.pStar : ℝ) / 4 := by rw [baseValue input, sub_self, abs_zero] positivity refine { alphabet := q ^ P.repetitions alphabetAtLeastTwo := hqout dimension := s * P.repetitions dimensionPositive := by have hpos := Nat.mul_pos (show 0 < s by omega) P.repetitions_pos omega coordinates := ProductPaddedOutput.finalCoordinates coordinates P.repetitions construct := output simpleBipartite := fun input => ProductPaddedOutput.simpleBipartite (base input) (basePresentation input) P.repetitions P.repetitions_pos translations := fun input => ProductPaddedOutput.output_translations (base input) (basePresentation input) coordinates (SubdivisionTarget.subdivide_translation coordinates (rounded input) (roundedTranslations input)) P.repetitions P.repetitions_pos computation := outputMachine finiteAlphabet := MachineProductRuntime.compose_finiteAlphabet baseMachine basePresentation P.repetitions P.repetitions_pos baseFinite completeness := ?_ soundness := ?_ } · intro input yes apply (InstanceValue.exists_rate_ge_iff (output input) hqoutpos (1 - ε)).mpr have hbase := FinishBounds.subdivision_completeness _ _ (d : ℝ) (P.pStar : ℝ) (complete input yes) (roundingError input) have hproduct := ProductPaddedOutput.output_completeness (base input) (basePresentation input) P.repetitions P.repetitions_pos (error := 1 - InstanceValue.value (base input)) (by linarith) exact FinishBounds.final_completeness _ _ (d : ℝ) (P.pStar : ℝ) (P.epsilon0 : ℝ) ε P.repetitions P.repetitions_pos hbase hproduct hdbudget hτbudget hε0 P.epsilon0_le · intro input no apply (InstanceValue.forall_rate_le_iff (output input) hqoutpos δ).mpr have hbase := FinishBounds.subdivision_soundness _ _ (P.pStar : ℝ) (sound input no) hτsmall (roundingError input) exact (ProductPaddedOutput.output_soundness (base input) (basePresentation input) P.repetitions P.repetitions_pos P.soundness_rate hbase).trans P.delta0_le end end UniqueGamesTheorem.FinishReduction end OAI