lean/OAI/Combinatorics/IndependentSets/Model.lean

import Mathlib

namespace OAI

namespace LargeIndependentSets

structure Literal where
  name : ℕ
  positive : Bool
  deriving DecidableEq

structure Formula where
  clauses : List (List Literal)
  width : ∀ C ∈ clauses, C.length ≤ 3

def Formula.Satisfiable (φ : Formula) : Prop :=
  ∃ assignment : ℕ → Bool, ∀ C ∈ φ.clauses,
    ∃ l ∈ C, assignment l.name = l.positive

/-- Binary framing: `true,b` for a payload bit and `false` as terminator. -/
def frame : List Bool → List Bool
  | [] => [false]
  | b :: bs => true :: b :: frame bs

def nameBits (n : ℕ) : List Bool := frame n.bits

def literalBits (l : Literal) : List Bool := l.positive :: nameBits l.name

def clauseBits (C : List Literal) : List Bool :=
  nameBits C.length ++ C.flatMap literalBits

def formulaBits (φ : Formula) : List Bool :=
  nameBits φ.clauses.length ++ φ.clauses.flatMap clauseBits

/-- A nonempty, finite, simple, undirected, unweighted graph. -/
structure Graph where
  vertices : ℕ
  nonempty : 0 < vertices
  adj : Fin vertices → Fin vertices → Bool
  symm : ∀ u v, adj u v = adj v u
  loopless : ∀ u, adj u u = false

def Graph.Independent (G : Graph) (S : Finset (Fin G.vertices)) : Prop :=
  ∀ u ∈ S, ∀ v ∈ S, u ≠ v → G.adj u v = false

instance (G : Graph) (S : Finset (Fin G.vertices)) : Decidable (G.Independent S) :=
  inferInstanceAs (Decidable (∀ u ∈ S, ∀ v ∈ S, u ≠ v → G.adj u v = false))

def Graph.independenceNumber (G : Graph) : ℕ :=
  ((Finset.univ : Finset (Fin G.vertices)).powerset.filter G.Independent).sup Finset.card

def Graph.ThreeColorable (G : Graph) : Prop :=
  ∃ color : Fin G.vertices → Fin 3,
    ∀ u v, G.adj u v = true → color u ≠ color v

/-- Full explicit adjacency matrix, row-major, with binary vertex count. -/
def graphBits (G : Graph) : List Bool :=
  nameBits G.vertices ++ (List.finRange G.vertices).flatMap
    (fun u => (List.finRange G.vertices).map (G.adj u))

open scoped Classical
noncomputable section

/-- Polynomial-time graph reduction with a real independence-ratio threshold. -/
structure RealGraphReduction (δ : ℝ) where
  reduce : List Bool → Graph
  computation : Turing.TM2ComputableInPolyTime (id : List Bool → List Bool) graphBits reduce
  finiteAlphabets : ∀ k : computation.tm.K, Finite (computation.tm.Γ k)
  outputBound : Polynomial ℕ
  outputSize : ∀ b, (graphBits (reduce b)).length ≤ outputBound.eval b.length
  completeness : ∀ φ : Formula, φ.Satisfiable → (reduce (formulaBits φ)).ThreeColorable
  soundness : ∀ φ : Formula, ¬φ.Satisfiable →
    ((reduce (formulaBits φ)).independenceNumber : ℝ) <
      δ * (reduce (formulaBits φ)).vertices

end
end LargeIndependentSets

end OAI