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180bc03 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | 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
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