| import OAI.Probability.SATComputability.EffectiveTrialValues |
| import OAI.Probability.SATComputability.UpperCertificates |
|
|
| namespace OAI |
|
|
| namespace FixedClauseThreshold.Computability |
|
|
| open DilutedSpinGlass Nat.Partrec PeriodicLattice.CertifiedReal |
| open scoped NNReal |
|
|
| local instance compiledTrialsRatPrimcodable : Primcodable ℚ := |
| PeriodicLattice.RecursiveArithmetic.ratPrimcodable |
|
|
| noncomputable def decodeTrial (j : ℕ) : TrialData := |
| (Encodable.decode j).getD (0,0,.zero,[]) |
|
|
| @[fun_prop] theorem decodeTrial_computable : Computable decodeTrial := by |
| unfold decodeTrial |
| fun_prop |
|
|
| theorem decodeTrial_encode (t : TrialData) : decodeTrial (Encodable.encode t) = t := by |
| simp [decodeTrial] |
|
|
| noncomputable def trialDataValid (t : TrialData) : Bool := |
| decide (0 < t.1 ∧ 0 < t.2.1) && exponentCheck t.2.2.2 |
|
|
| @[fun_prop] theorem trialDataValid_computable : Computable trialDataValid := by |
| unfold trialDataValid |
| fun_prop |
|
|
| theorem trialDataValid_iff (t : TrialData) : trialDataValid t = true ↔ |
| 0 < t.1 ∧ 0 < t.2.1 ∧ exponentCheck t.2.2.2 = true := by |
| simp only [trialDataValid, Bool.and_eq_true, decide_eq_true_eq, and_assoc] |
|
|
| private noncomputable def trialApproximation : TrialData × ℕ → ℚ := |
| Classical.choose trialValue_effective |
|
|
| private theorem trialApproximation_computable : Computable trialApproximation := |
| (Classical.choose_spec trialValue_effective).1 |
|
|
| private theorem trialApproximation_error (t : TrialData) (s : ℕ) : |
| |trialValue t - (trialApproximation (t,s) : ℝ)| ≤ error s := |
| (Classical.choose_spec trialValue_effective).2 t s |
|
|
| noncomputable def satTrialEvaluation : TrialEvaluation where |
| density j := (decodeTrial j).1 |
| valid j := trialDataValid (decodeTrial j) |
| value j := trialValue (decodeTrial j) |
| approximate p := trialApproximation (decodeTrial p.1, 2^p.2) |
| density_computable := by fun_prop |
| valid_computable := by fun_prop |
| approximate_computable := trialApproximation_computable.comp (by fun_prop) |
| density_positive j hj := ((trialDataValid_iff _).mp hj).1 |
| approximation_error j s _ := by |
| rw [abs_sub_comm] |
| apply (trialApproximation_error _ _).trans |
| simp only [error, tolerance, Rat.cast_inv, Rat.cast_pow, Rat.cast_ofNat] |
| rw [one_div] |
| apply inv_anti₀ (by positivity) |
| push_cast |
| exact le_add_of_nonneg_right (by norm_num) |
|
|
| theorem satTrialEvaluation_valid {j : ℕ} (hj : satTrialEvaluation.valid j = true) : |
| 0 < (decodeTrial j).1 ∧ 0 < (decodeTrial j).2.1 ∧ |
| exponentCheck (decodeTrial j).2.2.2 = true := |
| (trialDataValid_iff _).mp hj |
|
|
| theorem satTrialEvaluation_penalty {j : ℕ} (hj : satTrialEvaluation.valid j = true) : |
| 0 < (decodeTrial j).2.1 := (satTrialEvaluation_valid hj).2.1 |
|
|
| theorem satTrialEvaluation_bound {j : ℕ} (hj : satTrialEvaluation.valid j = true) : |
| satPressure (trialDensityNN (decodeTrial j)) ((decodeTrial j).2.1 : ℝ) ≤ |
| satTrialEvaluation.value j := by |
| have ht := satTrialEvaluation_valid hj |
| change _ ≤ trialValue (decodeTrial j) |
| rw [trialValue_eq_functional _ ht.2.2] |
| apply satPressure_le_rational_trial |
| · change (0 : ℝ) < (max 0 (decodeTrial j).1 : ℚ) |
| exact_mod_cast lt_max_of_lt_right ht.1 |
| · exact_mod_cast ht.2.1 |
| · exact (exponentCheck_iff _).mp ht.2.2 |
|
|
| end FixedClauseThreshold.Computability |
|
|
| end OAI |
|
|
