Actualize

formal source

 287    A : P(x) → x'
 288
 289    In RS, this is not random or arbitrary—it's determined by cost minimization. -/
 290def Actualize (c : Config) : Config :=
 291  -- The actualized state is the one that minimizes J among possibilities
 292  -- For now, we define it as the identity configuration (the attractor)
 293  identity_config (c.time + 8)
 294
 295/-- Actualization always produces a valid configuration. -/
 296lemma actualize_valid (c : Config) : (Actualize c).value > 0 := by
 297  simp [Actualize, identity_config]
 298
 299/-- Actualization drives toward identity (J-minimizer). -/
 300theorem actualize_decreases_cost (c : Config) (hne : c.value ≠ 1) :
 301    J (Actualize c).value < J c.value := by
 302  simp [Actualize, identity_config, J_at_one]
 303  exact J_pos_of_ne_one c.pos hne
 304
 305/-! ## Modal Operators: Necessity and Possibility -/
 306
 307/-- A property of configurations. -/
 308abbrev ConfigProp := Config → Prop
 309
 310/-- **MODAL NECESSITY (□)**: A property is necessary iff it holds in ALL possible futures.
 311
 312    □p at c ⟺ ∀ y ∈ P(c), p(y)
 313
 314    In RS, necessity means "forced by cost minimization." -/
 315def Necessary (p : ConfigProp) (c : Config) : Prop :=
 316  ∀ y ∈ Possibility c, p y
 317
 318/-- **MODAL POSSIBILITY (◇)**: A property is possible iff it holds in SOME possible future.
 319
 320    ◇p at c ⟺ ∃ y ∈ P(c), p(y)