File size: 3,302 Bytes
a2ee19a | 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 80 81 82 83 84 85 86 | 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
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