private def ackermann_fuel''._unary : (n : Nat) ×' (m : Nat) ×' (fuel : Nat) ×' g n m < fuel → Nat :=
WellFounded.Nat.fix
  (fun x => PSigma.casesOn x fun n m => PSigma.casesOn m fun m fuel => PSigma.casesOn fuel fun fuel h => fuel)
  fun _x a =>
  PSigma.casesOn (motive := fun _x =>
    ((y : (n : Nat) ×' (m : Nat) ×' (fuel : Nat) ×' g n m < fuel) →
        InvImage (fun x1 x2 => x1 < x2)
            (fun x => PSigma.casesOn x fun n m => PSigma.casesOn m fun m fuel => PSigma.casesOn fuel fun fuel h => fuel)
            y _x →
          Nat) →
      Nat)
    _x
    (fun n m a =>
      PSigma.casesOn (motive := fun m =>
        ((y : (n : Nat) ×' (m : Nat) ×' (fuel : Nat) ×' g n m < fuel) →
            InvImage (fun x1 x2 => x1 < x2)
                (fun x =>
                  PSigma.casesOn x fun n m => PSigma.casesOn m fun m fuel => PSigma.casesOn fuel fun fuel h => fuel)
                y ⟨n, m⟩ →
              Nat) →
          Nat)
        m
        (fun m fuel a =>
          PSigma.casesOn (motive := fun fuel =>
            ((y : (n : Nat) ×' (m : Nat) ×' (fuel : Nat) ×' g n m < fuel) →
                InvImage (fun x1 x2 => x1 < x2)
                    (fun x =>
                      PSigma.casesOn x fun n m => PSigma.casesOn m fun m fuel => PSigma.casesOn fuel fun fuel h => fuel)
                    y ⟨n, ⟨m, fuel⟩⟩ →
                  Nat) →
              Nat)
            fuel
            (fun fuel h a =>
              (match (motive :=
                  (x x_1 x_2 : Nat) →
                    (x_3 : g x x_1 < x_2) →
                      ((y : (n : Nat) ×' (m : Nat) ×' (fuel : Nat) ×' g n m < fuel) →
                          InvImage (fun x1 x2 => x1 < x2)
                              (fun x =>
                                PSigma.casesOn x fun n m =>
                                  PSigma.casesOn m fun m fuel => PSigma.casesOn fuel fun fuel h => fuel)
                              y ⟨x, ⟨x_1, ⟨x_2, x_3⟩⟩⟩ →
                            Nat) →
                        Nat)
                  n, m, fuel, h with
                | 0, m, x, x_1 => fun x => m + 1
                | n.succ, 0, f, h => fun x => x ⟨n, ⟨1, ⟨f - 1, sorry⟩⟩⟩ (id ((fun f => sorry) f))
                | n.succ, m.succ, f, h => fun x =>
                  x ⟨n, ⟨x ⟨n + 1, ⟨m, ⟨f - 1, sorry⟩⟩⟩ (id ((fun f => sorry) f)), ⟨f - 1, sorry⟩⟩⟩
                    (id ((fun f => sorry) f)))
                a)
            a)
        a)
    a