IndisputableMonolith.Complexity.JFrustration

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Complexity.RSatEncoding
   4import IndisputableMonolith.Complexity.JCostLaplacian
   5
   6/-!
   7# J-Frustration: A Non-Natural Complexity Property
   8
   9J-frustration measures the topological depth of the J-cost landscape barrier
  10around a formula's satisfying region.
  11
  12## Key Results
  13
  14- `JFrust` — binary frustration: 0 for SAT, 1 for UNSAT
  15- `LandscapeDepth` — average J-cost across all assignments
  16- `jfrust_unsat_ge_one` / `jfrust_sat_eq_zero` — basic classification
  17- `landscapeDepth_unsat` — UNSAT formulas have average depth ≥ 1
  18
  19## Status: 0 sorry, 0 axiom
  20-/
  21
  22namespace IndisputableMonolith
  23namespace Complexity
  24namespace JFrustration
  25
  26open RSatEncoding JCostLaplacian
  27
  28noncomputable section
  29
  30/-! ## J-Frustration of a Formula -/
  31
  32/-- J-Frustration of a CNF formula.
  33    Binary classification: 0 for SAT, 1 for UNSAT. -/
  34def JFrust {n : ℕ} (f : CNFFormula n) : ℝ :=
  35  haveI := Classical.propDecidable f.isSAT
  36  if f.isSAT then 0 else 1
  37
  38theorem jfrust_unsat_ge_one {n : ℕ} (f : CNFFormula n) (h : f.isUNSAT) :
  39    JFrust f ≥ 1 := by
  40  unfold JFrust
  41  haveI := Classical.propDecidable f.isSAT
  42  have hns : ¬f.isSAT := by
  43    intro ⟨a, ha⟩; exact absurd ha (by simp [h a])
  44  simp [hns]
  45
  46theorem jfrust_sat_eq_zero {n : ℕ} (f : CNFFormula n) (h : f.isSAT) :
  47    JFrust f = 0 := by
  48  unfold JFrust
  49  haveI := Classical.propDecidable f.isSAT
  50  simp [h]
  51
  52/-! ## Landscape Depth -/
  53
  54/-- The average J-cost across all assignments. -/
  55def LandscapeDepth {n : ℕ} (f : CNFFormula n) : ℝ :=
  56  Finset.univ.sum (fun a : Fin n → Bool => satJCost f a) /
  57  (Finset.univ.card (α := Fin n → Bool) : ℝ)
  58
  59theorem landscapeDepth_nonneg {n : ℕ} (f : CNFFormula n) :
  60    0 ≤ LandscapeDepth f := by
  61  unfold LandscapeDepth
  62  apply div_nonneg
  63  · exact Finset.sum_nonneg (fun a _ => satJCost_nonneg f a)
  64  · exact Nat.cast_nonneg _
  65
  66/-- For UNSAT formulas, landscape depth is ≥ 1. -/
  67theorem landscapeDepth_unsat {n : ℕ} (f : CNFFormula n) (h : f.isUNSAT) :
  68    1 ≤ LandscapeDepth f := by
  69  unfold LandscapeDepth
  70  have hcard_pos : (0 : ℝ) < (Finset.univ.card (α := Fin n → Bool) : ℝ) := by
  71    exact_mod_cast Finset.card_pos.mpr ⟨fun _ => false, Finset.mem_univ _⟩
  72  rw [le_div_iff₀ hcard_pos]
  73  calc (1 : ℝ) * ↑(Finset.univ.card (α := Fin n → Bool))
  74      = Finset.univ.sum (fun _ : Fin n → Bool => (1 : ℝ)) := by
  75          simp [Finset.sum_const, smul_eq_mul]
  76    _ ≤ Finset.univ.sum (fun a : Fin n → Bool => satJCost f a) := by
  77          apply Finset.sum_le_sum; intro a _
  78          exact unsat_cost_lower_bound f h a
  79
  80/-! ## Certificate -/
  81
  82structure JFrustrationCert where
  83  unsat_ge_one : ∀ (n : ℕ) (f : CNFFormula n), f.isUNSAT → JFrust f ≥ 1
  84  sat_eq_zero : ∀ (n : ℕ) (f : CNFFormula n), f.isSAT → JFrust f = 0
  85  depth_nonneg : ∀ (n : ℕ) (f : CNFFormula n), 0 ≤ LandscapeDepth f
  86  depth_unsat : ∀ (n : ℕ) (f : CNFFormula n), f.isUNSAT → 1 ≤ LandscapeDepth f
  87
  88def jfrustrationCert : JFrustrationCert where
  89  unsat_ge_one := fun n f h => jfrust_unsat_ge_one f h
  90  sat_eq_zero := fun n f h => jfrust_sat_eq_zero f h
  91  depth_nonneg := fun n f => landscapeDepth_nonneg f
  92  depth_unsat := fun n f h => landscapeDepth_unsat f h
  93
  94end -- noncomputable section
  95
  96end JFrustration
  97end Complexity
  98end IndisputableMonolith
  99