lean/OAI/Optimization/Tropical/TruffetModel.lean

import Mathlib

namespace OAI

noncomputable section
namespace TruffetV4Counterexample

abbrev Coeff := WithBot ℝ

abbrev Form := Fin 3 → Coeff

abbrev Var := Fin 2

abbrev RowIndex := Fin 2

def column (j : Var) : Fin 3 := ⟨j.val, Nat.lt_trans j.isLt (by decide)⟩

def nullForm : Form := fun _ => ⊥

structure Row where
  lhs : Form
  rhs : Form

structure State where
  rows : RowIndex → Row
  cost : Form
  remaining : Var → Prop

def wellFormed (s : State) : Prop := ∀ j, ¬ s.remaining j →
  s.cost (column j) = ⊥ ∧ ∀ i,
    (s.rows i).lhs (column j) = ⊥ ∧ (s.rows i).rhs (column j) = ⊥

def eval (f : Form) (w : Fin 3 → ℝ) : Coeff :=
  max (f 0 + (w 0 : Coeff)) (max (f 1 + (w 1 : Coeff)) (f 2 + (w 2 : Coeff)))

def feasible (s : State) (w : Fin 3 → ℝ) : Prop :=
  ∀ i, eval (s.rows i).rhs w ≤ eval (s.rows i).lhs w

def lower (s : State) (i : RowIndex) (j : Var) : Prop :=
  (s.rows i).lhs (column j) ≠ ⊥ ∧
    (s.rows i).rhs (column j) < (s.rows i).lhs (column j)

def upper (s : State) (i : RowIndex) (j : Var) : Prop :=
  (s.rows i).rhs (column j) ≠ ⊥ ∧
    (s.rows i).lhs (column j) < (s.rows i).rhs (column j)

def dominating (s : State) (j : Var) : Prop :=
  s.remaining j ∧ (∃ i, lower s i j) ∧ ∃ i, upper s i j

def eligible (s : State) (i : RowIndex) (j : Var) : Prop :=
  s.remaining j ∧ lower s i j ∧ ((∃ k, dominating s k) → dominating s j)

def bounded (s : State) : Prop :=
  ∀ j, s.remaining j → (∃ i, lower s i j) ∨ ∃ i, upper s i j

def stop (s : State) : Prop := ∀ i, (s.rows i).rhs 2 ≤ (s.rows i).lhs 2

def switch (s : State) : Prop :=
  (∀ j, s.remaining j → s.cost (column j) = ⊥) ∧ ¬ stop s ∧
  (∃ j, s.remaining j) ∧
  (¬ ∃ i j, s.remaining j ∧ lower s i j) ∧
  ∃ i j, s.remaining j ∧ upper s i j

def failure (s : State) : Prop := ¬ stop s ∧
  ((¬ ∃ j, s.remaining j) ∨
   ((¬ ∃ i j, s.remaining j ∧ lower s i j) ∧
    (¬ ∃ i j, s.remaining j ∧ upper s i j)))

def lowerForm (s : State) (i : RowIndex) (j : Var) : Form := fun k =>
  if k = column j then ⊥ else
    (s.rows i).rhs k + ((-WithBot.unbotD 0 ((s.rows i).lhs (column j)) : ℝ) : Coeff)

def substitute (f : Form) (j : Var) (g : Form) : Form := fun k =>
  if k = column j then ⊥ else max (f k) (f (column j) + g k)

def cleanup (r : Row) : Row := by
  classical
  exact if ∀ k, r.rhs k ≤ r.lhs k then ⟨nullForm, nullForm⟩ else r

def advance (s : State) (i : RowIndex) (j : Var) : State where
  rows := fun r => cleanup ⟨substitute (s.rows r).lhs j (lowerForm s i j),
    substitute (s.rows r).rhs j (lowerForm s i j)⟩
  cost := substitute s.cost j (lowerForm s i j)
  remaining := fun k => s.remaining k ∧ k ≠ j

def ForcedSingletonStep (s s' : State) (i : RowIndex) (j : Var) : Prop :=
  wellFormed s ∧ wellFormed s' ∧
  bounded s ∧ ¬ stop s ∧ ¬ switch s ∧ ¬ failure s ∧
  (∀ r k, eligible s r k ↔ r = i ∧ k = j) ∧
  (∃ k, (substitute s.cost j (lowerForm s i j)) k ≠ ⊥) ∧
  s' = advance s i j

def initial : State where
  rows := ![⟨![(0 : Coeff), 0, ⊥], ![⊥, ⊥, (0 : Coeff)]⟩,
    ⟨![(0 : Coeff), ⊥, ⊥], ![⊥, (1 : Coeff), ((-2 : ℝ) : Coeff)]⟩]
  cost := ![(0 : Coeff), ⊥, ⊥]
  remaining := fun _ => True

def middle : State where
  rows := ![⟨nullForm, nullForm⟩,
    ⟨![(0 : Coeff), ⊥, ⊥], ![⊥, ⊥, (1 : Coeff)]⟩]
  cost := ![(0 : Coeff), ⊥, ⊥]
  remaining := fun j => j = 0

def finalState : State where
  rows := fun _ => ⟨nullForm, nullForm⟩
  cost := ![⊥, ⊥, (1 : Coeff)]
  remaining := fun _ => False

def crossRowRestriction (s : State) : Prop :=
  ∀ i r : RowIndex, i ≠ r → ∀ α : ℝ, ∃ w : Fin 3 → ℝ,
    eval (s.rows i).lhs w < (α : Coeff) + eval (s.rows r).rhs w

def output : Fin 3 → ℝ := ![1, 0, 0]

def better : Fin 3 → ℝ := ![0, -1, 0]

end TruffetV4Counterexample
end

end OAI