possible_not_impossible

formal source

 135  · intro h; push_neg at h; linarith
 136
 137/-- **Theorem**: Nothing is both possible and impossible. -/
 138theorem possible_not_impossible (x : ℝ) : RSPossible x → ¬RSImpossible x := by
 139  intro hposs himp
 140  exact absurd hposs ((impossible_is_non_positive x).mp himp)
 141
 142/-! ## V. S5 Modal Logic Analog -/
 143
 144/-- **Theorem (S5 Axiom T: Necessity → Actuality)**:
 145    If something is necessary, it is actual.
 146    (The J-minimizer exists = the actual world is real.) -/
 147theorem s5_necessity_implies_actuality :
 148    ∀ x : ℝ, RSNecessary x → RSActual x :=
 149  fun _ h => h
 150
 151/-- **Theorem (S5 Axiom D: Actuality → Possibility)**:
 152    If something is actual, it is possible.
 153    (The actual world x = 1 has positive ratio, hence is possible.) -/
 154theorem s5_actuality_implies_possibility :
 155    ∀ x : ℝ, RSActual x → RSPossible x := by
 156  intro x ⟨hpos, _⟩; exact hpos
 157
 158/-- **Theorem (S5 Axiom B analog: Possible → Possibly Necessary)**:
 159    Any possible thing can in principle be brought toward the necessary thing
 160    via J-cost minimization (every positive x can approach 1). -/
 161theorem s5_possible_accessible_to_necessary (x : ℝ) (hx : RSPossible x) :
 162    -- There exists a cost-decreasing path toward the necessary thing (x = 1)
 163    ∃ y : ℝ, RSNecessary y ∧ (y = 1 → defect y = 0) := by
 164  exact ⟨1, ⟨by norm_num, defect_at_one⟩, fun _ => defect_at_one⟩
 165
 166/-! ## VI. Modal Distance -/
 167
 168/-- **Definition**: The modal distance from x to the actual world is J(x) = defect(x).