SGH与FGH对照：修订间差异

←上一编辑

可视化wikitext

2025年8月25日 (一) 13:28的最新版本

本条目展示 SGH 与 FGH 的对照，使用 Veblen 函数和 MOCF。本条目分析来自梅天狸。

Part 1

$g_{ε_{0} \times 2} (n) = g_{ε_{0} + ε_{0} [n]} (n) = (n ↑ ↑ n) \times 2$
$g_{ε_{0} \times ω} (n) = g_{ε_{0} \times n} (n) = (n ↑ ↑ n) \times n$
$g_{ε_{0} \times ω^{2}} (n) = g_{ε_{0} \times ω \times n} (n) = (n ↑ ↑ n) \times n^{2}$
$g_{ε_{0} \times ω^{ω}} (n) = g_{ε_{0} \times ω^{n}} (n) = (n ↑ ↑ n) \times n^{n}$
$g_{ε_{0} \times ω^{ω^{ω}}} (n) = (n ↑ ↑ n) \times (n ↑ ↑ 3)$
$g_{ε_{0}^{2}} (n) = g_{ε_{0} \times ε_{0} [n]} (n) = (n ↑ ↑ n)^{2}$
$g_{ε_{0}^{3}} (n) = (n ↑ ↑ n)^{3}$
$g_{ε_{0}^{ω}} (n) = g_{ε_{0}^{n}} (n) = (n ↑ ↑ n)^{n}$
$g_{ε_{0}^{ω^{ω}}} (n) = (n ↑ ↑ n)^{n^{n}}$
$g_{ε_{0}^{ε_{0}}} (n) = (n ↑ ↑ n)^{n ↑ ↑ n} = (n ↑ ↑ n) ↑ ↑ 2$
$g_{ε_{0}^{ε_{0}^{ε_{0}}}} (n) = (n ↑ ↑ n) ↑ ↑ 3$
$g_{ε_{1}} (n) = (n ↑ ↑ n) ↑ ↑ n \approx n ↑ ↑ (n \times 2)$
$g_{ε_{2}} (n) = ((n ↑ ↑ n) ↑ ↑ n) ↑ ↑ n \approx n ↑ ↑ (n \times 3)$
$n ↑ ↑ (n \times 3) \sim ε_{2}$
$n ↑ ↑ (n \times 3 + 1) \sim ε_{2}^{ε_{2}}$
$n ↑ ↑ (n \times 3 + 2) \sim ε_{2}^{ε_{2}^{ε_{2}}}$
$n ↑ ↑ (n \times 4) \sim ε_{3}$
$n ↑ ↑ (n \times 5) \sim ε_{4}$
$n ↑ ↑ (n^{2}) \sim ε_{ω}$
$n ↑ ↑ (n^{2} + 1) \sim ε_{ω}^{ε_{ω}}$
$n ↑ ↑ (n^{2} + n) \sim ε_{ω + 1}$
$n ↑ ↑ (n^{2} \times 2) \sim ε_{ω \times 2}$
$n ↑ ↑ (n^{3}) \sim ε_{ω^{2}}$
$n ↑ ↑ (n^{n}) = n ↑ ↑ n ↑ ↑ 2 \sim ε_{ω^{ω}}$
$n ↑ ↑ (n^{n} + n) \sim ε_{ω^{ω} + 1}$
$n ↑ ↑ (n^{n + 1}) \sim ε_{ω^{ω + 1}}$
$n ↑ ↑ (n^{n^{2}}) \sim ε_{ω^{ω^{2}}}$
$n ↑ ↑ n ↑ ↑ 3 \sim ε_{ω^{ω^{ω}}}$
$n ↑ ↑ n ↑ ↑ n = n ↑ ↑ ↑ 3 \sim ε_{ε_{0}}$
$n ↑ ↑ (n ↑ ↑ n + n) \sim ε_{ε_{0} + 1}$
$n ↑ ↑ ((n ↑ ↑ n) \times 2) \sim ε_{ε_{0} \times 2}$
$n ↑ ↑ ((n ↑ ↑ n) ↑ ↑ 2) \approx n ↑ ↑ n ↑ ↑ (n + 1) \sim ε_{ε_{0}^{ε_{0}}}$
$n ↑ ↑ n ↑ ↑ (n \times 2) \sim ε_{ε_{1}}$
$n ↑ ↑ n ↑ ↑ (n^{2}) \sim ε_{ε_{ω}}$
$n ↑ ↑ n ↑ ↑ n ↑ ↑ n = n ↑ ↑ ↑ 4 \sim ε_{ε_{ε_{0}}}$
$n ↑ ↑ ↑ n \sim ζ_{0}$
$(n ↑ ↑ ↑ n) ↑ ↑ n \sim ε_{ζ_{0} + 1}$
$(n ↑ ↑ ↑ n) ↑ ↑ (n \times 2) \sim ε_{ζ_{0} + 2}$
$(n ↑ ↑ ↑ n) ↑ ↑ (n ↑ ↑ n) \sim ε_{ζ_{0} + ε_{0}}$
$(n ↑ ↑ ↑ n) ↑ ↑ (n ↑ ↑ ↑ n) \approx n ↑ ↑ ↑ (n + 1) \sim ε_{ζ_{0} \times 2}$
$(n ↑ ↑ ↑ n) ↑ ↑ (n ↑ ↑ ↑ n) ↑ ↑ n \sim ε_{ε_{ζ_{0} + 1}}$
$(n ↑ ↑ ↑ n) ↑ ↑ (n ↑ ↑ ↑ n) ↑ ↑ (n ↑ ↑ ↑ n) \sim ε_{ε_{ζ_{0} \times 2}}$
$(n ↑ ↑ ↑ n) ↑ ↑ ↑ n \approx n ↑ ↑ ↑ (n \times 2) \sim ζ_{1}$
$(n ↑ ↑ ↑ (n \times 2)) ↑ ↑ n \sim ε_{ζ_{1} + 1}$
$(n ↑ ↑ ↑ (n \times 2)) ↑ ↑ (n ↑ ↑ ↑ n) \sim ε_{ζ_{1} + ζ_{0}}$
$(n ↑ ↑ ↑ (n \times 2)) ↑ ↑ (n ↑ ↑ ↑ (n \times 2)) \sim ε_{ζ_{1} \times 2}$
$(n ↑ ↑ ↑ (n \times 2)) ↑ ↑ ↑ n \approx n ↑ ↑ ↑ (n \times 3) \sim ζ_{2}$
$n ↑ ↑ ↑ (n^{2}) \sim ζ_{ω}$
$n ↑ ↑ ↑ (n^{n}) \sim ζ_{ω^{ω}}$
$n ↑ ↑ ↑ (n ↑ ↑ n) \sim ζ_{ε_{0}}$
$n ↑ ↑ ↑ n ↑ ↑ ↑ n \sim ζ_{ζ_{0}}$
$n ↑ ↑ ↑ ↑ n \sim η_{0}$
$(n ↑ ↑ ↑ ↑ n) ↑ ↑ ↑ n \sim ζ_{η_{0} + 1}$
$n ↑ ↑ ↑ ↑ (n \times 2) \sim η_{1}$
$n ↑^{5} 3 \sim η_{η_{0}}$
$n ↑^{5} n \sim φ (4, 0)$
$n ↑^{5} (n \times 2) \sim φ (4, 1)$
$n ↑^{6} n \sim φ (5, 0)$
$n ↑^{n} n \sim φ (ω, 0)$

Part 2

$f_{3} (f_{ω} (n)) \approx (n ↑^{n} n) ↑ ↑ n \sim ψ (Ω^{ω} + 1)$
$f_{3}^{2} (f_{ω} (n)) \approx (n ↑^{n} n) ↑ ↑ (n \times 2) \sim ψ (Ω^{ω} + 2)$
$f_{3}^{n} (f_{ω} (n)) \approx (n ↑^{n} n) ↑ ↑ (n^{2}) \sim ψ (Ω^{ω} + ω)$
$f_{3}^{f_{3} (n)} (f_{ω} (n)) \approx (n ↑^{n} n) ↑ ↑ (n ↑ ↑ n) \sim ψ (Ω^{ω} + ψ (0))$
$f_{4} (f_{ω} (n)) = f_{3}^{f_{ω} (n)} (f_{ω} (n)) \approx (n ↑^{n} n) ↑ ↑ ↑ 2 \sim ψ (Ω^{ω} + ψ (Ω^{ω}))$
$f_{3} (f_{4} (f_{ω} (n))) \approx ((n ↑^{n} n) ↑ ↑ ↑ 2) ↑ ↑ n \sim ψ (Ω^{ω} + ψ (Ω^{ω}) + 1)$
$f_{3}^{f_{ω} (n)} (f_{4} (f_{ω} (n))) \approx ((n ↑^{n} n) ↑ ↑ ↑ 2) ↑ ↑ (n ↑^{n} n) \sim ψ (Ω^{ω} + ψ (Ω^{ω}) \times 2)$
$f_{3}^{f_{ω} (n) \times 2} (f_{4} (f_{ω} (n))) \sim ψ (Ω^{ω} + ψ (Ω^{ω}) \times 3)$
$f_{3}^{f_{3} (f_{ω} (n))} (f_{4} (f_{ω} (n))) \sim ψ (Ω^{ω} + ψ (Ω^{ω} + 1))$
$f_{3}^{f_{3}^{2} (f_{ω} (n))} (f_{4} (f_{ω} (n))) \sim ψ (Ω^{ω} + ψ (Ω^{ω} + 2))$
$f_{4}^{2} (f_{ω} (n)) = f_{3}^{f_{3}^{f_{ω} (n)} (f_{ω} (n))} (f_{4} (f_{ω} (n))) \sim ψ (Ω^{ω} + ψ (Ω^{ω} + ψ (Ω^{ω})))$
$f_{3} (f_{4}^{2} (f_{ω} (n))) \sim ψ (Ω^{ω} + ψ (Ω^{ω} + ψ (Ω^{ω})) + 1)$
$f_{3}^{f_{4} (f_{ω} (n))} (f_{4}^{2} (f_{ω} (n))) \sim ψ (Ω^{ω} + ψ (Ω^{ω} + ψ (Ω^{ω})) \times 2)$
$f_{3}^{f_{3} (f_{4} (f_{ω} (n)))} (f_{4}^{2} (f_{ω} (n))) \sim ψ (Ω^{ω} + ψ (Ω^{ω} + ψ (Ω^{ω}) + 1))$
$f_{3}^{f_{3}^{f_{ω} (n)} (f_{4} (f_{ω} (n)))} (f_{4}^{2} (f_{ω} (n))) \sim ψ (Ω^{ω} + ψ (Ω^{ω} + ψ (Ω^{ω}) \times 2))$
$f_{4}^{3} (f_{ω} (n)) \sim ψ (Ω^{ω} + ψ (Ω^{ω} + ψ (Ω^{ω} + ψ (Ω^{ω}))))$
$f_{4}^{n} (f_{ω} (n)) \approx (n ↑^{n} n) ↑ ↑ ↑ n \sim ψ (Ω^{ω} + Ω)$
$f_{4}^{n + 1} (f_{ω} (n)) \sim ψ (Ω^{ω} + Ω + ψ (Ω^{ω} + Ω))$
$f_{4}^{n + 2} (f_{ω} (n)) \sim ψ (Ω^{ω} + Ω + ψ (Ω^{ω} + Ω + ψ (Ω^{ω} + Ω)))$
$f_{4}^{n \times 2} (f_{ω} (n)) \sim ψ (Ω^{ω} + Ω \times 2)$
$f_{4}^{n^{2}} (f_{ω} (n)) \sim ψ (Ω^{ω} + Ω \times ω)$
$f_{4}^{f_{3} (n)} (f_{ω} (n)) \sim ψ (Ω^{ω} + Ω \times ψ (0))$
$f_{5} (f_{ω} (n)) = f_{4}^{f_{ω} (n)} (f_{ω} (n)) \sim ψ (Ω^{ω} + Ω \times ψ (Ω^{ω}))$
$f_{3} (f_{5} (f_{ω} (n))) \sim ψ (Ω^{ω} + Ω \times ψ (Ω^{ω}) + 1)$
$f_{3}^{f_{ω} (n)} (f_{5} (f_{ω} (n))) \sim ψ (Ω^{ω} + Ω \times ψ (Ω^{ω}) + ψ (Ω^{ω}))$
$f_{4} (f_{5} (f_{ω} (n))) \sim ψ (Ω^{ω} + Ω \times ψ (Ω^{ω}) + ψ (Ω^{ω} + Ω \times ψ (Ω^{ω})))$
$f_{4}^{n} (f_{5} (f_{ω} (n))) \sim ψ (Ω^{ω} + Ω \times ψ (Ω^{ω}) + Ω)$
$f_{4}^{f_{ω} (n)} (f_{5} (f_{ω} (n))) \sim ψ (Ω^{ω} + Ω \times ψ (Ω^{ω}) \times 2)$
$f_{4}^{f_{ω} (n) \times 2} (f_{5} (f_{ω} (n))) \sim ψ (Ω^{ω} + Ω \times ψ (Ω^{ω}) \times 3)$
$f_{4}^{f_{3} (f_{ω} (n))} (f_{5} (f_{ω} (n))) \sim ψ (Ω^{ω} + Ω \times ψ (Ω^{ω} + 1))$
$f_{4}^{f_{4} (f_{ω} (n))} (f_{5} (f_{ω} (n))) \sim ψ (Ω^{ω} + Ω \times ψ (Ω^{ω} + Ω))$
$f_{5}^{2} (f_{ω} (n)) = f_{4}^{f_{5} (f_{ω} (n))} (f_{5} (f_{ω} (n))) \sim ψ (Ω^{ω} + Ω \times ψ (Ω^{ω} + Ω \times ψ (Ω^{ω})))$
$f_{5}^{n} (f_{ω} (n)) \approx (n ↑^{n} n) ↑ ↑ ↑ ↑ n \sim ψ (Ω^{ω} + Ω^{2})$
$f_{5}^{n \times 2} (f_{ω} (n)) \sim ψ (Ω^{ω} + Ω^{2} \times 2)$
$f_{6} (f_{ω} (n)) \sim ψ (Ω^{ω} + Ω^{2} \times ψ (Ω^{ω}))$
$f_{6}^{n} (f_{ω} (n)) \sim ψ (Ω^{ω} + Ω^{3})$
$f_{7}^{n} (f_{ω} (n)) \sim ψ (Ω^{ω} + Ω^{4})$
$f_{n} (f_{ω} (n)) \approx (n ↑^{n} n) ↑^{n} n \approx n ↑^{n} (2 n) \sim ψ (Ω^{ω} \times 2)$
$f_{n}^{2} (f_{ω} (n)) \approx n ↑^{n} (3 n) \sim ψ (Ω^{ω} \times 3)$
$f_{n}^{n} (f_{ω} (n)) \approx n ↑^{n} (n^{2}) \sim ψ (Ω^{ω} \times ω)$
$f_{n}^{f_{3} (n)} (f_{ω} (n)) \approx n ↑^{n} n ↑ ↑ n \sim ψ (Ω^{ω} \times ψ (0))$
$f_{n + 1} (f_{ω} (n)) = f_{n}^{f_{ω} (n)} (f_{ω} (n)) \approx n ↑^{n} n ↑^{n} n \sim ψ (Ω^{ω} \times ψ (Ω^{ω}))$
$f_{n} (f_{n + 1} (f_{ω} (n))) \sim ψ (Ω^{ω} \times ψ (Ω^{ω}) + Ω^{ω})$
$f_{n}^{f_{ω} (n)} (f_{n + 1} (f_{ω} (n))) \sim ψ (Ω^{ω} \times ψ (Ω^{ω}) \times 2)$
$f_{n}^{f_{3} (f_{ω} (n))} (f_{n + 1} (f_{ω} (n))) \sim ψ (Ω^{ω} \times ψ (Ω^{ω} + 1))$
$f_{n}^{f_{n} (f_{ω} (n))} (f_{n + 1} (f_{ω} (n))) \sim ψ (Ω^{ω} \times ψ (Ω^{ω} \times 2))$
$f_{n + 1}^{2} (f_{ω} (n)) \approx n ↑^{n} n ↑^{n} n ↑^{n} n \sim ψ (Ω^{ω} \times ψ (Ω^{ω} \times ψ (Ω^{ω})))$
$f_{n + 1}^{n} (f_{ω} (n)) \approx n ↑^{n + 1} n \sim ψ (Ω^{ω + 1})$
$f_{n} (f_{n + 1}^{n} (f_{ω} (n))) \sim ψ (Ω^{ω + 1} + Ω^{ω})$
$f_{n + 1}^{n + 1} (f_{ω} (n)) = f_{n}^{f_{n + 1}^{n} (f_{ω} (n))} (f_{n + 1}^{n} (f_{ω} (n))) \sim ψ (Ω^{ω + 1} + Ω^{ω} \times ψ (Ω^{ω + 1}))$
$f_{n + 1}^{n \times 2} (f_{ω} (n)) \approx n ↑^{n + 1} (2 n) \sim ψ (Ω^{ω + 1} \times 2)$
$f_{n + 2} (f_{ω} (n)) \sim ψ (Ω^{ω + 1} \times ψ (Ω^{ω}))$
$f_{n} (f_{n + 2} (f_{ω} (n))) \sim ψ (Ω^{ω + 1} \times ψ (Ω^{ω}) + Ω^{ω})$
$f_{n}^{f_{n + 1}^{n} (f_{ω} (n))} (f_{n + 2} (f_{ω} (n))) \sim ψ (Ω^{ω + 1} \times ψ (Ω^{ω}) + Ω^{ω} \times ψ (Ω^{ω + 1}))$
$f_{n + 1} (f_{n + 2} (f_{ω} (n))) \sim ψ (Ω^{ω + 1} \times ψ (Ω^{ω}) + Ω^{ω} \times ψ (Ω^{ω + 1} \times ψ (Ω^{ω})))$
$f_{n + 1}^{n} (f_{n + 2} (f_{ω} (n))) \sim ψ (Ω^{ω + 1} \times ψ (Ω^{ω}) + Ω^{ω + 1})$
$f_{n + 1}^{f_{ω} (n)} (f_{n + 2} (f_{ω} (n))) \sim ψ (Ω^{ω + 1} \times ψ (Ω^{ω}) \times 2)$
$f_{n + 1}^{f_{n + 1} (f_{ω} (n))} (f_{n + 2} (f_{ω} (n))) \sim ψ (Ω^{ω + 1} \times ψ (Ω^{ω} \times ψ (Ω^{ω})))$
$f_{n + 1}^{f_{n + 1}^{n} (f_{ω} (n))} (f_{n + 2} (f_{ω} (n))) \sim ψ (Ω^{ω + 1} \times ψ (Ω^{ω + 1}))$
$f_{n + 2}^{2} (f_{ω} (n)) \sim ψ (Ω^{ω + 1} \times ψ (Ω^{ω + 1} \times ψ (Ω^{ω})))$
$f_{n + 2}^{n} (f_{ω} (n)) \sim ψ (Ω^{ω + 2})$
$f_{n + 2}^{n + 1} (f_{ω} (n)) \sim ψ (Ω^{ω + 2} + Ω^{ω + 1} \times ψ (Ω^{ω}))$
$f_{n + 2}^{n \times 2} (f_{ω} (n)) \sim ψ (Ω^{ω + 2} \times 2)$
$f_{n + 3} (f_{ω} (n)) \sim ψ (Ω^{ω + 2} \times ψ (Ω^{ω}))$
$f_{n + 3}^{2} (f_{ω} (n)) \sim ψ (Ω^{ω + 2} \times ψ (Ω^{ω + 2} \times ψ (Ω^{ω})))$
$f_{n + 3}^{n} (f_{ω} (n)) \sim ψ (Ω^{ω + 3})$
$f_{n \times 2} (f_{ω} (n)) \sim ψ (Ω^{ω \times 2})$
$f_{n \times 2 + 1} (f_{ω} (n)) \sim ψ (Ω^{ω \times 2} \times ψ (Ω^{ω}))$
$f_{n \times 2 + 1}^{n} (f_{ω} (n)) \sim ψ (Ω^{ω \times 2 + 1})$
$f_{n \times 3} (f_{ω} (n)) \sim ψ (Ω^{ω \times 3})$
$f_{n^{2}} (f_{ω} (n)) \sim ψ (Ω^{ω^{2}})$
$f_{f_{3} (n)} (f_{ω} (n)) \approx n ↑^{n ↑ ↑ n} n \sim ψ (Ω^{ψ (0)})$
$f_{f_{4} (n)} (f_{ω} (n)) \approx n ↑^{n ↑ ↑ ↑ n} n \sim ψ (Ω^{ψ (Ω)})$
$f_{ω}^{2} (n) = f_{f_{ω} (n)} (f_{ω} (n)) \approx n ↑^{n ↑^{n} n} n \sim ψ (Ω^{ψ (Ω^{ω})})$
$f_{n} (f_{ω}^{2} (n)) \sim ψ (Ω^{ψ (Ω^{ω})} + Ω^{ω})$
$f_{n + 1}^{n} (f_{ω}^{2} (n)) \sim ψ (Ω^{ψ (Ω^{ω})} + Ω^{ω + 1})$
$f_{f_{ω} (n)} (f_{ω}^{2} (n)) \sim ψ (Ω^{ψ (Ω^{ω})} \times 2)$
$f_{f_{ω} (n)}^{n} (f_{ω}^{2} (n)) \sim ψ (Ω^{ψ (Ω^{ω})} \times ω)$
$f_{f_{ω} (n) + 1} (f_{ω}^{2} (n)) \sim ψ (Ω^{ψ (Ω^{ω})} \times ψ (Ω^{ψ (Ω^{ω})}))$
$f_{f_{ω} (n) + 1}^{n} (f_{ω}^{2} (n)) \sim ψ (Ω^{ψ (Ω^{ω}) + 1})$
$f_{f_{ω} (n) + n} (f_{ω}^{2} (n)) \sim ψ (Ω^{ψ (Ω^{ω}) + ω})$
$f_{f_{ω} (n) \times 2} (f_{ω}^{2} (n)) \sim ψ (Ω^{ψ (Ω^{ω}) \times 2})$
$f_{f_{3} (f_{ω} (n))} (f_{ω}^{2} (n)) \sim ψ (Ω^{ψ (Ω^{ω} + 1)})$
$f_{f_{n + 1}^{n} (f_{ω} (n))} (f_{ω}^{2} (n)) \sim ψ (Ω^{ψ (Ω^{ω + 1})})$
$f_{ω}^{3} (n) = f_{f_{f_{ω} (n)} (f_{ω} (n))} (f_{ω}^{2} (n)) \sim ψ (Ω^{ψ (Ω^{ψ (Ω^{ω})})})$
$f_{ω + 1} (n) = f_{ω}^{n} (n) \sim ψ (Ω^{Ω}) = φ (1, 0, 0)$

Part 3

$f_{3} (f_{ω + 1} (n)) \sim ψ (Ω^{Ω} + 1)$
$f_{4} (f_{ω + 1} (n)) = f_{3}^{f_{ω + 1} (n)} (f_{ω + 1} (n)) \sim ψ (Ω^{Ω} + ψ (Ω^{Ω}))$
$f_{4}^{n} (f_{ω + 1} (n)) \sim ψ (Ω^{Ω} + Ω)$
$f_{5}^{n} (f_{ω + 1} (n)) \sim ψ (Ω^{Ω} + Ω^{2})$
$f_{n} (f_{ω + 1} (n)) \sim ψ (Ω^{Ω} + Ω^{ω})$
$f_{n + 1}^{n} (f_{ω + 1} (n)) \sim ψ (Ω^{Ω} + Ω^{ω + 1})$
$f_{f_{3} (n)} (f_{ω + 1} (n)) \sim ψ (Ω^{Ω} + Ω^{ψ (0)})$
$f_{f_{ω} (n)} (f_{ω + 1} (n)) \sim ψ (Ω^{Ω} + Ω^{ψ (Ω^{ω})})$
$f_{ω} (f_{ω + 1} (n)) = f_{f_{ω + 1} (n)} (f_{ω + 1} (n)) \sim ψ (Ω^{Ω} + Ω^{ψ (Ω^{Ω})}) = φ (φ (1, 0, 0), 1)$
$f_{f_{ω + 1} (n)} (f_{ω} (f_{ω + 1} (n))) \sim ψ (Ω^{Ω} + Ω^{ψ (Ω^{Ω})} \times 2)$
$f_{f_{ω + 1} (n) + 1}^{n} (f_{ω} (f_{ω + 1} (n))) \sim ψ (Ω^{Ω} + Ω^{ψ (Ω^{Ω}) + 1})$
$f_{f_{3} (f_{ω + 1} (n))} (f_{ω} (f_{ω + 1} (n))) \sim ψ (Ω^{Ω} + Ω^{ψ (Ω^{Ω} + 1)})$
$f_{f_{n} (f_{ω + 1} (n))} (f_{ω} (f_{ω + 1} (n))) \sim ψ (Ω^{Ω} + Ω^{ψ (Ω^{Ω} + Ω^{ω})})$
$f_{ω}^{2} (f_{ω + 1} (n)) = f_{f_{ω} (f_{ω + 1} (n))} (f_{ω} (f_{ω + 1} (n))) \sim ψ (Ω^{Ω} + Ω^{ψ (Ω^{Ω} + Ω^{ψ (Ω^{Ω})})})$
$f_{ω}^{n} (f_{ω + 1} (n)) \sim ψ (Ω^{Ω} \times 2) = φ (1, 0, 1)$
$f_{ω}^{n \times 2} (f_{ω + 1} (n)) \sim ψ (Ω^{Ω} \times 3)$
$f_{ω + 1}^{2} (n) = f_{ω}^{f_{ω + 1} (n)} (f_{ω + 1} (n)) \sim ψ (Ω^{Ω} \times ψ (Ω^{Ω}))$
$f_{ω}^{n} (f_{ω + 1}^{2} (n)) \sim ψ (Ω^{Ω} \times ψ (Ω^{Ω}) + Ω^{Ω})$
$f_{ω}^{f_{ω + 1} (n)} (f_{ω + 1}^{2} (n)) \sim ψ (Ω^{Ω} \times ψ (Ω^{Ω}) \times 2)$
$f_{ω}^{f_{n} (f_{ω + 1} (n))} (f_{ω + 1}^{2} (n)) \sim ψ (Ω^{Ω} \times ψ (Ω^{Ω} + Ω^{ω}))$
$f_{ω}^{f_{ω} (f_{ω + 1} (n))} (f_{ω + 1}^{2} (n)) \sim ψ (Ω^{Ω} \times ψ (Ω^{Ω} + Ω^{ψ (Ω^{Ω})}))$
$f_{ω}^{f_{ω}^{n} (f_{ω + 1} (n))} (f_{ω + 1}^{2} (n)) \sim ψ (Ω^{Ω} \times ψ (Ω^{Ω} \times 2))$
$f_{ω + 1}^{3} (n) \sim ψ (Ω^{Ω} \times ψ (Ω^{Ω} \times ψ (Ω^{Ω})))$
$f_{ω + 2} (n) = f_{ω + 1}^{n} (n) \sim ψ (Ω^{Ω + 1}) = φ (1, 1, 0)$
$f_{ω} (f_{ω + 2} (n)) \sim ψ (Ω^{Ω + 1} + Ω^{ψ (Ω^{Ω + 1})})$
$f_{ω}^{n} (f_{ω + 2} (n)) \sim ψ (Ω^{Ω + 1} + Ω^{Ω})$
$f_{ω}^{n \times 2} (f_{ω + 2} (n)) \sim ψ (Ω^{Ω + 1} + Ω^{Ω} \times 2)$
$f_{ω + 1} (f_{ω + 2} (n)) = f_{ω}^{f_{ω + 2} (n)} (f_{ω + 2} (n)) \sim ψ (Ω^{Ω + 1} + Ω^{Ω} \times ψ (Ω^{Ω + 1}))$
$f_{ω}^{f_{ω + 2} (n)} (f_{ω + 1} (f_{ω + 2} (n))) \sim ψ (Ω^{Ω + 1} + Ω^{Ω} \times ψ (Ω^{Ω + 1}) \times 2)$
$f_{ω}^{f_{ω}^{n} (f_{ω + 2} (n))} (f_{ω + 1} (f_{ω + 2} (n))) \sim ψ (Ω^{Ω + 1} + Ω^{Ω} \times ψ (Ω^{Ω + 1} + Ω^{Ω}))$
$f_{ω + 1}^{2} (f_{ω + 2} (n)) \sim ψ (Ω^{Ω + 1} + Ω^{Ω} \times ψ (Ω^{Ω + 1} + Ω^{Ω} \times ψ (Ω^{Ω + 1})))$
$f_{ω + 1}^{n} (f_{ω + 2} (n)) \sim ψ (Ω^{Ω + 1} \times 2)$
$f_{ω + 2}^{2} (n) \sim ψ (Ω^{Ω + 1} \times ψ (Ω^{Ω + 1}))$
$f_{ω + 3} (n) = f_{ω + 2}^{n} (n) \sim ψ (Ω^{Ω + 2})$
$f_{ω + 4} (n) \sim ψ (Ω^{Ω + 3})$
$f_{ω \times 2} (n) = f_{ω + n} (n) \sim ψ (Ω^{Ω + ω})$
$f_{ω} (f_{ω \times 2} (n)) \sim ψ (Ω^{Ω + ω} + Ω^{ψ (Ω^{Ω + ω})})$
$f_{ω}^{n} (f_{ω \times 2} (n)) \sim ψ (Ω^{Ω + ω} + Ω^{Ω})$
$f_{ω + 1} (f_{ω \times 2} (n)) \sim ψ (Ω^{Ω + ω} + Ω^{Ω} \times ψ (Ω^{Ω + ω}))$
$f_{ω + 1}^{n} (f_{ω \times 2} (n)) \sim ψ (Ω^{Ω + ω} + Ω^{Ω + 1})$
$f_{ω + 2}^{n} (f_{ω \times 2} (n)) \sim ψ (Ω^{Ω + ω} + Ω^{Ω + 2})$
$f_{ω + n} (f_{ω \times 2} (n)) \sim ψ (Ω^{Ω + ω} \times 2)$
$f_{ω + n + 1} (f_{ω \times 2} (n)) \sim ψ (Ω^{Ω + ω} \times ψ (Ω^{Ω + ω}))$
$f_{ω + n + 1}^{n} (f_{ω \times 2} (n)) \sim ψ (Ω^{Ω + ω + 1})$
$f_{ω + n \times 2} (f_{ω \times 2} (n)) \sim ψ (Ω^{Ω + ω \times 2})$
$f_{ω + f_{ω + 1} (n)} (f_{ω \times 2} (n)) \sim ψ (Ω^{Ω + ψ (Ω^{Ω})})$
$f_{ω + f_{ω + 2} (n)} (f_{ω \times 2} (n)) \sim ψ (Ω^{Ω + ψ (Ω^{Ω + 1})})$
$f_{ω \times 2}^{2} (n) = f_{ω + f_{ω + n} (n)} (f_{ω \times 2} (n)) \sim ψ (Ω^{Ω + ψ (Ω^{Ω + ω})})$
$f_{ω \times 2}^{3} (n) \sim ψ (Ω^{Ω + ψ (Ω^{Ω + ψ (Ω^{Ω + ω})})})$
$f_{ω \times 2 + 1} (n) \sim ψ (Ω^{Ω \times 2})$
$f_{ω}^{n} (f_{ω \times 2 + 1} (n)) \sim ψ (Ω^{Ω \times 2} + Ω^{Ω})$
$f_{ω \times 2}^{n} (f_{ω \times 2 + 1} (n)) \sim ψ (Ω^{Ω \times 2} \times 2)$
$f_{ω \times 2 + 1}^{2} (n) \sim ψ (Ω^{Ω \times 2} \times ψ (Ω^{Ω \times 2}))$
$f_{ω \times 2 + 2} (n) \sim ψ (Ω^{Ω \times 2 + 1})$
$f_{ω \times 3} (n) \sim ψ (Ω^{Ω \times 2 + ω})$
$f_{ω \times 3}^{2} (n) \sim ψ (Ω^{Ω \times 2 + ψ (Ω^{Ω \times 2 + ω})})$
$f_{ω \times 3 + 1} (n) \sim ψ (Ω^{Ω \times 3})$
$f_{ω \times 4} (n) \sim ψ (Ω^{Ω \times 3 + ω})$
$f_{ω \times 4 + 1} (n) \sim ψ (Ω^{Ω \times 4})$
$f_{ω^{2}} (n) \sim ψ (Ω^{Ω \times ω}) = φ (ω, 0, 0)$
$f_{ω}^{n} (f_{ω^{2}} (n)) \sim ψ (Ω^{Ω \times ω} + Ω^{Ω})$
$f_{ω \times 2}^{n} (f_{ω^{2}} (n)) \sim ψ (Ω^{Ω \times ω} + Ω^{Ω \times 2})$
$f_{ω \times n} (f_{ω^{2}} (n)) \sim ψ (Ω^{Ω \times ω} \times 2)$
$f_{ω \times n + 1} (f_{ω^{2}} (n)) \sim ψ (Ω^{Ω \times ω} \times ψ (Ω^{Ω \times ω}))$
$f_{ω \times n + 1}^{n} (f_{ω^{2}} (n)) \sim ψ (Ω^{Ω \times ω + 1})$
$f_{ω \times n + 2}^{n} (f_{ω^{2}} (n)) \sim ψ (Ω^{Ω \times ω + 2})$
$f_{ω \times n + n} (f_{ω^{2}} (n)) \sim ψ (Ω^{Ω \times ω + ω})$
$f_{ω \times n + f_{ω^{2}} (n)} (f_{ω^{2}} (n)) \sim ψ (Ω^{Ω \times ω + ψ (Ω^{Ω \times ω})})$
$f_{ω \times n + ω} (f_{ω^{2}} (n)) \sim ψ (Ω^{Ω \times ω + Ω})$
$f_{ω \times n + ω \times 2} (f_{ω^{2}} (n)) \sim ψ (Ω^{Ω \times ω + Ω \times 2})$
$f_{ω \times n \times 2} (f_{ω^{2}} (n)) \sim ψ (Ω^{Ω \times ω \times 2})$
$f_{ω^{2}}^{2} (n) = f_{ω \times f_{ω^{2}} (n)} (f_{ω^{2}} (n)) \sim ψ (Ω^{Ω \times ψ (Ω^{Ω \times ω})})$
$f_{ω^{2}}^{3} (n) \sim ψ (Ω^{Ω \times ψ (Ω^{Ω \times ψ (Ω^{Ω \times ω})})})$
$f_{ω^{2} + 1} (n) \sim ψ (Ω^{Ω^{2}}) = φ (1, 0, 0, 0)$
$f_{ω}^{n} (f_{ω^{2} + 1} (n)) \sim ψ (Ω^{Ω^{2}} + Ω^{Ω})$
$f_{ω \times 2}^{n} (f_{ω^{2} + 1} (n)) \sim ψ (Ω^{Ω^{2}} + Ω^{Ω \times 2})$
$f_{ω \times n} (f_{ω^{2} + 1} (n)) \sim ψ (Ω^{Ω^{2}} + Ω^{Ω \times ω})$
$f_{ω^{2}} (f_{ω^{2} + 1} (n)) = f_{ω \times f_{ω^{2} + 1} (n)} (f_{ω^{2} + 1} (n)) \sim ψ (Ω^{Ω^{2}} + Ω^{Ω \times ψ (Ω^{Ω^{2}})}) = φ (φ (1, 0, 0, 0), 0, 1)$
$f_{ω^{2}}^{2} (f_{ω^{2} + 1} (n)) \sim ψ (Ω^{Ω^{2}} + Ω^{Ω \times ψ (Ω^{Ω^{2}} + Ω^{Ω \times ψ (Ω^{Ω^{2}})})})$
$f_{ω^{2}}^{n} (f_{ω^{2} + 1} (n)) \sim ψ (Ω^{Ω^{2}} \times 2)$
$f_{ω^{2} + 1}^{2} (n) \sim ψ (Ω^{Ω^{2}} \times ψ (Ω^{Ω^{2}}))$
$f_{ω^{2} + 2} (n) \sim ψ (Ω^{Ω^{2} + 1})$
$f_{ω^{2} + ω} (n) \sim ψ (Ω^{Ω^{2} + ω})$
$f_{ω^{2} + ω + 1} (n) \sim ψ (Ω^{Ω^{2} + Ω})$
$f_{ω^{2} + ω + 2} (n) \sim ψ (Ω^{Ω^{2} + Ω + 1})$
$f_{ω^{2} + ω \times 2 + 1} (n) \sim ψ (Ω^{Ω^{2} + Ω \times 2})$
$f_{ω^{2} \times 2} (n) \sim ψ (Ω^{Ω^{2} + Ω \times ω})$
$f_{ω^{2} \times 2 + 1} (n) \sim ψ (Ω^{Ω^{2} \times 2})$
$f_{ω^{2} \times 3 + 1} (n) \sim ψ (Ω^{Ω^{2} \times 3})$
$f_{ω^{3}} (n) \sim ψ (Ω^{Ω^{2} \times ω})$
$f_{ω^{2} \times n} (f_{ω^{3}} (n)) \sim ψ (Ω^{Ω^{2} \times ω} \times 2)$
$f_{ω^{2} \times n + 1}^{n} (f_{ω^{3}} (n)) \sim ψ (Ω^{Ω^{2} \times ω + 1})$
$f_{ω^{2} \times n + n} (f_{ω^{3}} (n)) \sim ψ (Ω^{Ω^{2} \times ω + ω})$
$f_{ω^{2} \times n + ω} (f_{ω^{3}} (n)) \sim ψ (Ω^{Ω^{2} \times ω + ψ (Ω^{Ω^{2} \times ω})})$
$f_{ω^{2} \times n + ω}^{n} (f_{ω^{3}} (n)) \sim ψ (Ω^{Ω^{2} \times ω + Ω})$
$f_{ω^{2} \times n + ω \times n} (f_{ω^{3}} (n)) \sim ψ (Ω^{Ω^{2} \times ω + Ω \times ω})$
$f_{ω^{2} \times n + ω^{2}}^{n} (f_{ω^{3}} (n)) \sim ψ (Ω^{Ω^{2} \times ω + Ω^{2}})$
$f_{ω^{2} \times n \times 2} (f_{ω^{3}} (n)) \sim ψ (Ω^{Ω^{2} \times ω \times 2})$
$f_{ω^{3}}^{2} (n) \sim ψ (Ω^{Ω^{2} \times ψ (Ω^{Ω^{2}})})$
$f_{ω^{3} + 1} (n) \sim ψ (Ω^{Ω^{3}})$
$f_{ω^{3} + 2} (n) \sim ψ (Ω^{Ω^{3} + 1})$
$f_{ω^{3} + ω + 1} (n) \sim ψ (Ω^{Ω^{3} + Ω})$
$f_{ω^{3} + ω^{2} + 1} (n) \sim ψ (Ω^{Ω^{3} + Ω^{2}})$
$f_{ω^{3} \times 2 + 1} (n) \sim ψ (Ω^{Ω^{3} \times 2})$
$f_{ω^{4}} (n) \sim ψ (Ω^{Ω^{3} \times ω})$
$f_{ω^{4} + 1} (n) \sim ψ (Ω^{Ω^{4}})$
$f_{\omega^\omega}(n)\sim\psi(\Omega^{\Omega^\omega})=\varphi(1@\omega)$

Part 4

$f_{ω^{ω}} (n) \sim ψ (Ω^{Ω^{ω}})$
$f_{3} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} + 1)$
$f_{4}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} + Ω)$
$f_{5}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} + Ω^{2})$
$f_{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} + Ω^{ω})$
$f_{ω} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} + Ω^{ψ (Ω^{Ω^{ω}})})$
$f_{ω}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} + Ω^{Ω})$
$f_{ω + 1}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} + Ω^{Ω + 1})$
$f_{ω + n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} + Ω^{Ω + ω})$
$f_{ω \times 2}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} + Ω^{Ω \times 2})$
$f_{ω \times n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} + Ω^{Ω \times ω})$
$f_{ω^{2}}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} + Ω^{Ω^{2}})$
$f_{ω^{3}}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} + Ω^{Ω^{3}})$
$f_{ω^{n}} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} \times 2)$
$f_{ω^{n}}^{2} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} \times 3)$
$f_{ω^{n}}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} \times ω)$
$f_{ω^{n} + 1} (f_{ω^{ω}} (n)) = f_{ω^{n}}^{f_{ω^{ω}} (n)} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω}} \times ψ (Ω^{Ω^{ω}}))$
$f_{ω^{n} + 1}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω} + 1})$
$f_{ω^{n} + 2}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω} + 2})$
$f_{ω^{n} + n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω} + ω})$
$f_{ω^{n} + ω}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω} + Ω})$
$f_{ω^{n} + ω \times 2}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω} + Ω \times 2})$
$f_{ω^{n} + ω^{2}}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω} + Ω^{2}})$
$f_{ω^{n} \times 2} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω} \times 2})$
$f_{ω^{n} \times 3} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω} \times 3})$
$f_{ω^{n} \times n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω} \times ω})$
$f_{ω^{n + 1}} (f_{ω^{ω}} (n)) = f_{ω^{n} \times f_{ω^{ω}} (n)} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω + 1}})$
$f_{ω^{n}} (f_{ω^{n + 1}} (f_{ω^{ω}} (n))) \sim ψ (Ω^{Ω^{ω + 1}} + Ω^{Ω^{ω}})$
$f_{ω^{n} + 1}^{n} (f_{ω^{n + 1}} (f_{ω^{ω}} (n))) \sim ψ (Ω^{Ω^{ω + 1}} + Ω^{Ω^{ω} + 1})$
$f_{ω^{n} \times n} (f_{ω^{n + 1}} (f_{ω^{ω}} (n))) \sim ψ (Ω^{Ω^{ω + 1}} + Ω^{Ω^{ω} \times ω})$
$f_{ω^{n + 1}}^{2} (f_{ω^{ω}} (n)) = f_{ω^{n} \times f_{ω^{n + 1}} (f_{ω^{ω}} (n))} (f_{ω^{n + 1}} (f_{ω^{ω}} (n))) \sim ψ (Ω^{Ω^{ω + 1}} \times 2)$
$f_{ω^{n + 1}}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω + 1}} \times ω)$
$f_{ω^{n + 1} + 1}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω + 1} + 1})$
$f_{ω^{n + 1} + ω^{n}} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω + 1} + Ω^{ω}})$
$f_{ω^{n + 1} \times 2}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω + 1} \times 2})$
$f_{ω^{n + 1} \times n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω + 1} \times ω})$
$f_{ω^{n + 2}}^{n} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω + 2}})$
$f_{ω^{n \times 2}} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω \times 2}})$
$f_{ω^{n^{2}}} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ω^{2}}})$
$f_{ω^{f_{3} (n)}} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ψ (0)}})$
$f_{ω^{ω}}^{2} (n) = f_{ω^{f_{ω^{ω}} (n)}} (f_{ω^{ω}} (n)) \sim ψ (Ω^{Ω^{ψ (Ω^{Ω^{ω}})}})$
$f_{ω^{ω}}^{3} (n) \sim ψ (Ω^{Ω^{ψ (Ω^{Ω^{ψ (Ω^{Ω^{ω}})}})}})$
$f_{ω^{ω} + 1} (n) = f_{ω^{ω}}^{n} (n) \sim ψ (Ω^{Ω^{Ω}})$
$f_{ω^{ω} + 2} (n) \sim ψ (Ω^{Ω^{Ω} + 1})$
$f_{ω^{ω} + 3} (n) \sim ψ (Ω^{Ω^{Ω} + 2})$
$f_{ω^{ω} + ω} (n) \sim ψ (Ω^{Ω^{Ω} + ω})$
$f_{ω^{ω} + ω + 1} (n) \sim ψ (Ω^{Ω^{Ω} + Ω})$
$f_{ω^{ω} + ω \times 2} (n) \sim ψ (Ω^{Ω^{Ω} + Ω + ω})$
$f_{ω^{ω} + ω \times 2 + 1} (n) \sim ψ (Ω^{Ω^{Ω} + Ω \times 2})$
$f_{ω^{ω} + ω^{2}} (n) \sim ψ (Ω^{Ω^{Ω} + Ω \times ω})$
$f_{ω^{ω} + ω^{2} + 1} (n) \sim ψ (Ω^{Ω^{Ω} + Ω^{2}})$
$f_{ω^{ω} + ω^{3} + 1} (n) \sim ψ (Ω^{Ω^{Ω} + Ω^{3}})$
$f_{ω^{ω} \times 2} (n) \sim ψ (Ω^{Ω^{Ω} + Ω^{ω}})$
$f_{ω^{ω} \times 2 + 1} (n) \sim ψ (Ω^{Ω^{Ω} \times 2})$
$f_{ω^{ω} \times 3 + 1} (n) \sim ψ (Ω^{Ω^{Ω} \times 3})$
$f_{ω^{ω + 1}} (n) \sim ψ (Ω^{Ω^{Ω} \times ω})$
$f_{ω^{ω + 1} + 1} (n) \sim ψ (Ω^{Ω^{Ω + 1}})$
$f_{ω^{ω + 1} + ω + 1} (n) \sim ψ (Ω^{Ω^{Ω + 1} + Ω})$
$f_{ω^{ω + 1} + ω^{ω} + 1} (n) \sim ψ (Ω^{Ω^{Ω + 1} + Ω^{Ω}})$
$f_{ω^{ω + 1} \times 2 + 1} (n) \sim ψ (Ω^{Ω^{Ω + 1} \times 2})$
$f_{ω^{ω + 2}} (n) \sim ψ (Ω^{Ω^{Ω + 1} \times ω})$
$f_{ω^{ω + 2} + 1} (n) \sim ψ (Ω^{Ω^{Ω + 2}})$
$f_{ω^{ω \times 2}} (n) \sim ψ (Ω^{Ω^{Ω + ω}})$
$f_{ω^{ω \times 2} + 1} (n) \sim ψ (Ω^{Ω^{Ω \times 2}})$
$f_{ω^{ω \times 3} + 1} (n) \sim ψ (Ω^{Ω^{Ω \times 3}})$
$f_{ω^{ω^{2}}} (n) \sim ψ (Ω^{Ω^{Ω \times ω}})$
$f_{ω^{ω^{2}} + 1} (n) \sim ψ (Ω^{Ω^{Ω^{2}}})$
$f_{ω^{ω^{2}} \times 2 + 1} (n) \sim ψ (Ω^{Ω^{Ω^{2}} \times 2})$
$f_{ω^{ω^{2} + 1} + 1} (n) \sim ψ (Ω^{Ω^{Ω^{2} + 1}})$
$f_{ω^{ω^{2} + ω} + 1} (n) \sim ψ (Ω^{Ω^{Ω^{2} + Ω}})$
$f_{ω^{ω^{2} \times 2} + 1} (n) \sim ψ (Ω^{Ω^{Ω^{2} \times 2}})$
$f_{ω^{ω^{3}} + 1} (n) \sim ψ (Ω^{Ω^{Ω^{3}}})$
$f_{ω^{ω^{ω}}} (n) \sim ψ (Ω^{Ω^{Ω^{ω}}})$
$f_{ω^{ω^{ω}} + 1} (n) \sim ψ (Ω^{Ω^{Ω^{Ω}}})$
$f_{ω^{ω^{ω^{ω}}} + 1} (n) \sim ψ (Ω^{Ω^{Ω^{Ω^{Ω}}}})$
$f_{ψ (0)} (n) \sim ψ (ψ_{1} (0))$
$f_{3} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) + 1)$
$f_{4}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) + Ω)$
$f_{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) + Ω^{ω})$
$f_{ω} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) + Ω^{ψ (ψ_{1} (0))})$
$f_{ω}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) + Ω^{Ω})$
$f_{ω + 1}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) + Ω^{Ω + 1})$
$f_{ω \times 2}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) + Ω^{Ω \times 2})$
$f_{ω^{ω}}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) + Ω^{Ω^{Ω}})$
$f_{ω^{ω} + 1}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) + Ω^{Ω^{Ω} + 1})$
$f_{ω^{ω^{ω}}}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) + Ω^{Ω^{Ω^{Ω}}})$
$f_{ω^{ω^{ω^{ω}}}}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) + Ω^{Ω^{Ω^{Ω^{Ω}}}})$
$f_{ψ (0) [n]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) \times 2)$
$f_{ψ (0) [n]}^{2} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) \times 3)$
$f_{ψ (0) [n]}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) \times ω)$
$f_{ψ (0) [n] + 1} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) \times ψ (ψ_{1} (0)))$
$f_{ψ (0) [n] + 1}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) \times Ω)$
$f_{ψ (0) [n]} (f_{ψ (0) [n] + 1}^{n} (f_{ψ (0)} (n))) \sim ψ (ψ_{1} (0) \times Ω + ψ_{1} (0))$
$f_{ψ (0) [n] + 1}^{n + 1} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) \times Ω + ψ_{1} (0) \times ψ (ψ_{1} (0) \times Ω))$
$f_{ψ (0) [n] + 1}^{n \times 2} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) \times Ω \times 2)$
$f_{ψ (0) [n] + 2} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) \times Ω \times ψ (ψ_{1} (0)))$
$f_{ψ (0) [n] + 2}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) \times Ω^{2})$
$f_{ψ (0) [n] + n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) \times Ω^{ω})$
$f_{ψ (0) [n] + ω}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0) \times Ω^{Ω})$
$f_{ψ (0) [n] \times 2} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{2})$
$f_{ψ (0) [n] \times n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ω})$
$f_{ψ (0) [n] \times ω}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{Ω})$
$f_{ψ (0) [n] \times ω^{2}}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{Ω^{2}})$
$f_{ψ (0) [n] \times ω^{ω}}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{Ω^{Ω}})$
$f_{ψ (0) [n]^{2}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)})$
$f_{ψ (0) [n]^{2} + ψ (0) [n]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0) + 1})$
$f_{ψ (0) [n]^{2} \times 2} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0) \times 2})$
$f_{ψ (0) [n]^{2} \times n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0) \times ω})$
$f_{ψ (0) [n]^{2} \times ω}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0) \times Ω})$
$f_{ψ (0) [n]^{3}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{2}})$
$f_{ψ (0) [n]^{n}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ω}})$
$f_{ψ (0) [n]^{ω}}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{Ω}})$
$f_{ψ (0) [n]^{ψ (0) [n]}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)}})$
$f_{ψ (0) [n + 1]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)}})$
$f_{ψ (0) [n + 1] + ψ (0) [n]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)} + 1})$
$f_{ψ (0) [n + 1] + ψ (0) [n]^{2}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)} + ψ_{1} (0)})$
$f_{ψ (0) [n + 1] + ψ (0) [n]^{ω}}^{n} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)} + ψ_{1} (0)^{Ω}})$
$f_{ψ (0) [n + 1] \times 2} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)} \times 2})$
$f_{ψ (0) [n + 1] \times ψ (0) [n]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0) + 1}})$
$f_{ψ (0) [n + 1] \times ψ (0) [n]^{2}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0) + 2}})$
$f_{ψ (0) [n + 1]^{2}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0) \times 2}})$
$f_{ψ (0) [n + 1]^{ψ (0) [n]}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)^{2}}})$
$f_{ψ (0) [n + 1]^{ψ (0) [n] + 1}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)^{2} + ψ_{1} (0)}})$
$f_{ψ (0) [n + 1]^{ψ (0) [n] \times 2}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)^{2} \times 2}})$
$f_{ψ (0) [n + 1]^{ψ (0) [n]^{2}}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)^{3}}})$
$f_{ψ (0) [n + 2]} (f_{ψ (0)} (n)) = f_{ψ (0) [n + 1]^{ψ (0) [n + 1]}} (f_{ψ (0)} (n)) = f_{ψ (0) [n + 1]^{ψ (0) [n]^{ψ (0) [n]}}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)}}})$
$f_{ψ (0) [n + 3]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)}}}})$
$f_{ψ (0) [n \times 2]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (1))$
$f_{ψ (0) [n]} (f_{ψ (0) [n \times 2]} (f_{ψ (0)} (n))) \sim ψ (ψ_{1} (1) + ψ_{1} (0))$
$f_{ψ (0) [n \times 2]}^{2} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (1) \times 2)$
$f_{ψ (0) [n \times 2] + ψ (0) [n]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (1) \times ψ_{1} (0))$
$f_{ψ (0) [n \times 2] + ψ (0) [n + 1]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (1) \times ψ_{1} (0)^{ψ_{1} (0)^{ψ_{1} (0)}})$
$f_{ψ (0) [n \times 2] \times 2} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (1)^{2})$
$f_{ψ (0) [n \times 2] \times ψ (0) [n]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (1)^{ψ_{1} (0)})$
$f_{ψ (0) [n \times 2]^{2}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (1)^{ψ_{1} (1)})$
$f_{ψ (0) [n \times 2]^{3}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (1)^{ψ_{1} (1)^{2}})$
$f_{ψ (0) [n \times 2 + 1]} (f_{ψ (0)} (n)) = f_{ψ (0) [n \times 2]^{ψ (0) [n \times 2]}} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (1)^{ψ_{1} (1)^{ψ_{1} (1)}})$
$f_{ψ (0) [n \times 3]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (2))$
$f_{ψ (0) [n^{2}]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (ω))$
$f_{ψ (0) [n^{2} + n]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (ω + 1))$
$f_{ψ (0) [n^{3}]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (ω^{2}))$
$f_{ψ (0) [n^{n}]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (ω^{ω}))$
$f_{ψ (0) [f_{3} (n)]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (ψ (0)))$
$f_{ψ (0)}^{2} (n) = f_{ψ (0) [f_{ψ (0)} (n)]} (f_{ψ (0)} (n)) \sim ψ (ψ_{1} (ψ (ψ_{1} (0))))$
$f_{ψ (0) [n]} (f_{ψ (0)}^{2} (n)) \sim ψ (ψ_{1} (ψ (ψ_{1} (0))) \times ψ_{1} (0))$
$f_{ψ (0) [n \times 2]} (f_{ψ (0)}^{2} (n)) \sim ψ (ψ_{1} (ψ (ψ_{1} (0))) \times ψ_{1} (1))$
$f_{ψ (0) [f_{ψ (0)} (n)]} (f_{ψ (0)}^{2} (n)) \sim ψ (ψ_{1} (ψ (ψ_{1} (0)))^{2})$
$f_{ψ (0) [f_{ψ (0)} (n) + 1]} (f_{ψ (0)}^{2} (n)) \sim ψ (ψ_{1} (ψ (ψ_{1} (0)))^{ψ_{1} (ψ (ψ_{1} (0)))^{ψ_{1} (ψ (ψ_{1} (0)))}})$
$f_{ψ (0) [f_{ψ (0)} (n) + n]} (f_{ψ (0)}^{2} (n)) \sim ψ (ψ_{1} (ψ (ψ_{1} (0)) + 1))$
$f_{ψ (0) [f_{ψ (0)} (n) \times 2]} (f_{ψ (0)}^{2} (n)) \sim ψ (ψ_{1} (ψ (ψ_{1} (0)) \times 2))$
$f_{ψ (0) [f_{3} (f_{ψ (0)} (n))]} (f_{ψ (0)}^{2} (n)) \sim ψ (ψ_{1} (ψ (ψ_{1} (0) + 1)))$
$f_{ψ (0) [f_{ψ (0) [n]} (f_{ψ (0)} (n))]} (f_{ψ (0)}^{2} (n)) \sim ψ (ψ_{1} (ψ (ψ_{1} (0) \times 2)))$
$f_{ψ (0) [f_{ψ (0) [n \times 2]} (f_{ψ (0)} (n))]} (f_{ψ (0)}^{2} (n)) \sim ψ (ψ_{1} (ψ (ψ_{1} (1))))$
$f_{ψ (0) [f_{ψ (0) [n^{2}]} (f_{ψ (0)} (n))]} (f_{ψ (0)}^{2} (n)) \sim ψ (ψ_{1} (ψ (ψ_{1} (ω))))$
$f_{ψ (0)}^{3} (n) \sim ψ (ψ_{1} (ψ (ψ_{1} (ψ (ψ_{1} (0))))))$
$f_{ψ (0) + 1} (n) \sim ψ (ψ_{1} (Ω))$
$f_{ψ (0) [n]} (f_{ψ (0) + 1} (n)) \sim ψ (ψ_{1} (Ω) + ψ_{1} (0))$
$f_{ψ (0) [n \times 2]} (f_{ψ (0) + 1} (n)) \sim ψ (ψ_{1} (Ω) + ψ_{1} (1))$
$f_{ψ (0) [f_{ψ (0)} (n)]} (f_{ψ (0) + 1} (n)) \sim ψ (ψ_{1} (Ω) + ψ_{1} (ψ (ψ_{1} (0))))$
$f_{ψ (0)} (f_{ψ (0) + 1} (n)) = f_{ψ (0) [f_{ψ (0) + 1} (n)]} (f_{ψ (0) + 1} (n)) \sim ψ (ψ_{1} (Ω) + ψ_{1} (ψ (ψ_{1} (Ω))))$
$f_{ψ (0)}^{2} (f_{ψ (0) + 1} (n)) \sim ψ (ψ_{1} (Ω) + ψ_{1} (ψ (ψ_{1} (Ω) + ψ_{1} (ψ (ψ_{1} (Ω))))))$
$f_{ψ (0)}^{n} (f_{ψ (0) + 1} (n)) \sim ψ (ψ_{1} (Ω) \times 2)$
$f_{ψ (0)}^{n^{2}} (f_{ψ (0) + 1} (n)) \sim ψ (ψ_{1} (Ω) \times ω)$
$f_{ψ (0) + 1}^{2} (n) \sim ψ (ψ_{1} (Ω) \times ψ (ψ_{1} (Ω)))$
$f_{ψ (0) + 2} (n) \sim ψ (ψ_{1} (Ω) \times Ω)$
$f_{ψ (0) + 3} (n) \sim ψ (ψ_{1} (Ω) \times Ω^{2})$
$f_{ψ (0) + ω} (n) \sim ψ (ψ_{1} (Ω) \times Ω^{ω})$
$f_{ψ (0) + ω + 1} (n) \sim ψ (ψ_{1} (Ω) \times Ω^{Ω})$
$f_{ψ (0) + ω^{ω} + 1} (n) \sim ψ (ψ_{1} (Ω) \times Ω^{Ω^{Ω}})$
$f_{ψ (0) \times 2} (n) \sim ψ (ψ_{1} (Ω) \times ψ_{1} (0))$
$f_{ψ (0) \times 2 + 1} (n) \sim ψ (ψ_{1} (Ω)^{2})$
$f_{ψ (0) \times 3 + 1} (n) \sim ψ (ψ_{1} (Ω)^{3})$
$f_{ψ (0) \times ω} (n) \sim ψ (ψ_{1} (Ω)^{ω})$
$f_{ψ (0) \times ω + 1} (n) \sim ψ (ψ_{1} (Ω)^{Ω})$
$f_{ψ (0)^{2}} (n) \sim ψ (ψ_{1} (Ω)^{ψ_{1} (0)})$
$f_{ψ (0)^{2} + 1} (n) \sim ψ (ψ_{1} (Ω)^{ψ_{1} (Ω)})$
$f_{ψ (0)^{2} + ψ (0) + 1} (n) \sim ψ (ψ_{1} (Ω)^{ψ_{1} (Ω) + 1})$
$f_{ψ (0)^{2} \times 2 + 1} (n) \sim ψ (ψ_{1} (Ω)^{ψ_{1} (Ω) \times 2})$
$f_{ψ (0)^{3} + 1} (n) \sim ψ (ψ_{1} (Ω)^{ψ_{1} (Ω)^{2}})$
$f_{ψ (0)^{ω} + 1} (n) \sim ψ (ψ_{1} (Ω)^{ψ_{1} (Ω)^{Ω}})$
$f_{ψ (0)^{ψ (0)}} (n) \sim ψ (ψ_{1} (Ω)^{ψ_{1} (Ω)^{ψ_{1} (0)}})$
$f_{ψ (0)^{ψ (0)} + 1} (n) \sim ψ (ψ_{1} (Ω)^{ψ_{1} (Ω)^{ψ_{1} (Ω)}})$
$f_{ψ (1)} (n) \sim ψ (ψ_{1} (Ω + 1))$
$f_{ψ (0) [n]} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1) + ψ_{1} (0))$
$f_{ψ (0)}^{n} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1) + ψ_{1} (Ω))$
$f_{ψ (0)^{ψ (0)}}^{n} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1) + ψ_{1} (Ω)^{ψ_{1} (Ω)^{ψ_{1} (Ω)}})$
$f_{ψ (1) [n]} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1) \times 2)$
$f_{ψ (1) [n]}^{2} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1) \times 3)$
$f_{ψ (1) [n] + 1} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1) \times ψ (ψ_{1} (Ω + 1)))$
$f_{ψ (1) [n] + 1}^{n} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1) \times Ω)$
$f_{ψ (1) [n] + ψ (0) [n]} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1) \times ψ_{1} (0))$
$f_{ψ (1) [n] + ψ (0)}^{n} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1) \times ψ_{1} (Ω))$
$f_{ψ (1) [n] \times 2} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1)^{2})$
$f_{ψ (1) [n] \times ψ (0) [n]} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1)^{ψ_{1} (0)})$
$f_{ψ (1) [n]^{2}} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1)^{ψ_{1} (Ω + 1)})$
$f_{ψ (1) [n + 1]} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 1)^{ψ_{1} (Ω + 1)^{ψ_{1} (Ω + 1)}})$
$f_{ψ (1) [n \times 2]} (f_{ψ (1)} (n)) \sim ψ (ψ_{1} (Ω + 2))$
$f_{ψ (1)}^{2} (n) \sim ψ (ψ_{1} (Ω + ψ (ψ_{1} (Ω))))$
$f_{ψ (1) + 1} (n) \sim ψ (ψ_{1} (Ω \times 2))$
$f_{ψ (1) + 2} (n) \sim ψ (ψ_{1} (Ω \times 2) \times Ω)$
$f_{ψ (1) + ψ (0) + 1} (n) \sim ψ (ψ_{1} (Ω \times 2) \times ψ_{1} (Ω))$
$f_{ψ (1) \times 2 + 1} (n) \sim ψ (ψ_{1} (Ω \times 2)^{2})$
$f_{ψ (1) \times ψ (0) + 1} (n) \sim ψ (ψ_{1} (Ω \times 2)^{ψ_{1} (Ω)})$
$f_{ψ (1)^{2} + 1} (n) \sim ψ (ψ_{1} (Ω \times 2)^{ψ_{1} (Ω \times 2)})$
$f_{ψ (1)^{ψ (1)} + 1} (n) \sim ψ (ψ_{1} (Ω \times 2)^{ψ_{1} (Ω \times 2)^{ψ_{1} (Ω \times 2)}})$
$f_{ψ (2)} (n) \sim ψ (ψ_{1} (Ω \times 2 + 1))$
$f_{ψ (2) + 1} (n) \sim ψ (ψ_{1} (Ω \times 3))$
$f_{ψ (3)} (n) \sim ψ (ψ_{1} (Ω \times 3 + 1))$
$f_{ψ (4)} (n) \sim ψ (ψ_{1} (Ω \times 4 + 1))$
$f_{ψ (ω)} (n) \sim ψ (ψ_{1} (Ω \times ω))$
$f_{ψ (ω) + 1} (n) \sim ψ (ψ_{1} (Ω^{2}))$
$f_{ψ (ω) + 2} (n) \sim ψ (ψ_{1} (Ω^{2}) \times Ω)$
$f_{ψ (ω) \times 2 + 1} (n) \sim ψ (ψ_{1} (Ω^{2})^{2})$
$f_{ψ (ω)^{2} + 1} (n) \sim ψ (ψ_{1} (Ω^{2})^{ψ_{1} (Ω^{2})})$
$f_{ψ (ω + 1)} (n) \sim ψ (ψ_{1} (Ω^{2} + 1))$
$f_{ψ (ω + 1) + 1} (n) \sim ψ (ψ_{1} (Ω^{2} + Ω))$
$f_{ψ (ω + 2) + 1} (n) \sim ψ (ψ_{1} (Ω^{2} + Ω \times 2))$
$f_{ψ (ω \times 2) + 1} (n) \sim ψ (ψ_{1} (Ω^{2} \times 2))$
$f_{ψ (ω^{2}) + 1} (n) \sim ψ (ψ_{1} (Ω^{3}))$
$f_{ψ (ω^{ω})} (n) \sim ψ (ψ_{1} (Ω^{ω}))$
$f_{ψ (ω^{ω}) + 1} (n) \sim ψ (ψ_{1} (Ω^{Ω}))$
$f_{ψ (ω^{ω^{ω}}) + 1} (n) \sim ψ (ψ_{1} (Ω^{Ω^{Ω}}))$
$f_{ψ (ψ (0))} (n) \sim ψ (ψ_{1} (ψ_{1} (0)))$
$f_{ψ (ψ (0)) + 1} (n) \sim ψ (ψ_{1} (ψ_{1} (Ω)))$
$f_{ψ (ψ (1)) + 1} (n) \sim ψ (ψ_{1} (ψ_{1} (Ω \times 2)))$
$f_{ψ (ψ (ψ (0))) + 1} (n) \sim ψ (ψ_{1} (ψ_{1} (ψ_{1} (Ω))))$
$f_{ψ (Ω)} (n) \sim ψ (Ω_{2})$
$f_{3} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + 1)$
$f_{ω}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + Ω^{Ω})$
$f_{ψ (0) [n]} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (0))$
$f_{ψ (0)}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω))$
$f_{ψ (1) [n]} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω + 1))$
$f_{ψ (1)}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω \times 2))$
$f_{ψ (ω)}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω^{2}))$
$f_{ψ (ψ (0))}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (ψ_{1} (Ω)))$
$f_{ψ (ψ (ψ (0)))}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (ψ_{1} (ψ_{1} (Ω))))$
$f_{ψ (Ω) [n]}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2}) \times ω)$
$f_{ψ (Ω) [n] + 1}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2}) \times Ω)$
$f_{ψ (Ω) [n] \times 2} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2})^{2})$
$f_{ψ (Ω) [n] \times ω}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2})^{Ω})$
$f_{ψ (Ω) [n] \times ψ (0) + 1} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2})^{ψ_{1} (Ω)})$
$f_{ψ (Ω) [n]^{2}} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2})^{ψ_{1} (Ω_{2})})$
$f_{ψ (Ω) [n]^{ψ (Ω) [n]}} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2})^{ψ_{1} (Ω_{2})^{ψ_{1} (Ω_{2})}})$
$f_{ψ (ψ (Ω) [n - 1] + 1) [n]} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2} + 1))$
$f_{ψ (ψ (Ω) [n - 1] + 1)}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2} + Ω))$
$f_{ψ (ψ (Ω) [n - 1] + n)} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2} + Ω \times ω))$
$f_{ψ (ψ (Ω) [n - 1] + ω)}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2} + Ω^{2}))$
$f_{ψ (ψ (Ω) [n - 1] + ψ (0))}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2} + ψ_{1} (Ω)))$
$f_{ψ (ψ (Ω) [n - 1] + ψ (ψ (0)))}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2} + ψ_{1} (ψ_{1} (Ω))))$
$f_{ψ (Ω) [n + 1]} (f_{ψ (Ω)} (n)) = f_{ψ (ψ (Ω) [n])} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2} + ψ_{1} (Ω_{2})))$
$f_{ψ (ψ (Ω) [n] + 1)} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2} + ψ_{1} (Ω_{2}) + 1))$
$f_{ψ (ψ (Ω) [n] \times 2)} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2} + ψ_{1} (Ω_{2}) \times 2))$
$f_{ψ (ψ (Ω) [n]^{2})} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2} + ψ_{1} (Ω_{2})^{2}))$
$f_{ψ (ψ (ψ (Ω) [n - 1] + 1))} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2} + ψ_{1} (Ω_{2} + 1)))$
$f_{ψ (Ω) [n + 2]} (f_{ψ (Ω)} (n)) = f_{ψ (ψ (ψ (Ω) [n]))} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} + ψ_{1} (Ω_{2} + ψ_{1} (Ω_{2} + ψ_{1} (Ω_{2}))))$
$f_{ψ (Ω) [n \times 2]} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times 2)$
$f_{ψ (Ω) [n]} (f_{ψ (Ω) [n \times 2]} (f_{ψ (Ω)} (n))) \sim ψ (Ω_{2} \times 2 + ψ_{1} (Ω_{2}))$
$f_{ψ (Ω) [n \times 2]}^{2} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times 2 + ψ_{1} (Ω_{2} \times 2))$
$f_{ψ (Ω) [n \times 2] + 1}^{n} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times 2 + ψ_{1} (Ω_{2} \times 2) \times Ω)$
$f_{ψ (Ω) [n \times 2] \times 2} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times 2 + ψ_{1} (Ω_{2} \times 2)^{2})$
$f_{ψ (Ω) [n \times 2]^{2}} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times 2 + ψ_{1} (Ω_{2} \times 2)^{ψ_{!} (Ω_{2} \times 2)})$
$f_{ψ (ψ (Ω) [n \times 2 - 1] + 1)} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times 2 + ψ_{1} (Ω_{2} \times 2 + 1))$
$f_{ψ (Ω) [n \times 2 + 1]} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times 2 + ψ_{1} (Ω_{2} \times 2 + ψ_{1} (Ω_{2} \times 2)))$
$f_{ψ (Ω) [n \times 3]} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times 3)$
$f_{ψ (Ω) [n^{2}]} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times ω)$
$f_{ψ (Ω) [n^{2} + 1]} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times ω + ψ_{1} (Ω_{2} \times ω + ψ_{1} (Ω_{2} \times ω)))$
$f_{ψ (Ω) [n^{2} + n]} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times ω + Ω_{2})$
$f_{ψ (Ω) [n^{3}]} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times ω^{2})$
$f_{ψ (Ω) [f_{3} (n)]} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times ψ (0))$
$f_{ψ (Ω)}^{2} (n) = f_{ψ (Ω) [f_{ψ (Ω)} (n)]} (f_{ψ (Ω)} (n)) \sim ψ (Ω_{2} \times ψ (Ω_{2}))$
$f_{ψ (Ω)}^{3} (n) \sim ψ (Ω_{2} \times ψ (Ω_{2} \times ψ (Ω_{2})))$
$f_{ψ (Ω) + 1} (n) = f_{ψ (Ω)}^{n} (n) \sim ψ (Ω_{2} \times Ω)$
$f_{ψ (Ω)} (f_{ψ (Ω) + 1} (n)) \sim ψ (Ω_{2} \times Ω + Ω_{2} \times ψ (Ω_{2} \times Ω))$
$f_{ψ (Ω)}^{n} (f_{ψ (Ω) + 1} (n)) \sim ψ (Ω_{2} \times Ω \times 2)$
$f_{ψ (Ω) + 1}^{2} (n) \sim ψ (Ω_{2} \times Ω \times ψ (Ω_{2} \times Ω))$
$f_{ψ (Ω) + 2} (n) \sim ψ (Ω_{2} \times Ω^{2})$
$f_{ψ (Ω) + ω} (n) \sim ψ (Ω_{2} \times Ω^{ω})$
$f_{ψ (Ω) + ω + 1} (n) \sim ψ (Ω_{2} \times Ω^{Ω})$
$f_{ψ (Ω) + ω^{ω} + 1} (n) \sim ψ (Ω_{2} \times Ω^{Ω^{Ω}})$
$f_{ψ (Ω) + ψ (0) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω))$
$f_{ψ (Ω) + ψ (1) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω \times 2))$
$f_{ψ (Ω) + ψ (ψ (0)) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (ψ_{1} (Ω)))$
$f_{ψ (Ω) \times 2} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2}))$
$f_{ψ (Ω)}^{n} (f_{ψ (Ω) \times 2} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2}) + Ω_{2} \times Ω)$
$f_{ψ (Ω) + ψ (0)}^{n} (f_{ψ (Ω) \times 2} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2}) + Ω_{2} \times ψ_{1} (Ω))$
$f_{ψ (Ω) + ψ (Ω) [n]} (f_{ψ (Ω) \times 2} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2}) \times 2)$
$f_{ψ (Ω) + ψ (Ω) [n]}^{2} (f_{ψ (Ω) \times 2} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2}) \times 3)$
$f_{ψ (Ω) + ψ (Ω) [n] + 1} (f_{ψ (Ω) \times 2} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2}) \times ψ (Ω_{2} \times ψ_{1} (Ω_{2})))$
$f_{ψ (Ω) + ψ (Ω) [n] + 1}^{n} (f_{ψ (Ω) \times 2} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2}) \times Ω)$
$f_{ψ (Ω) + ψ (Ω) [n] \times 2} (f_{ψ (Ω) \times 2} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2})^{2})$
$f_{ψ (Ω) + ψ (ψ (Ω) [n - 1] + 1) [n]} (f_{ψ (Ω) \times 2} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} + 1))$
$f_{ψ (Ω) + ψ (Ω) [n + 1]} (f_{ψ (Ω) \times 2} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} + ψ_{1} (Ω_{2})))$
$f_{ψ (Ω) + ψ (Ω) [n \times 2]} (f_{ψ (Ω) \times 2} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times 2))$
$f_{ψ (Ω) \times 2}^{2} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times ψ (Ω_{2} \times ψ_{1} (Ω_{2}))))$
$f_{ψ (Ω) \times 2 + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω))$
$f_{ψ (Ω) \times 2} (f_{ψ (Ω) \times 2 + 1} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω) + Ω_{2} \times ψ_{1} (Ω_{2} \times ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω))))$
$f_{ψ (Ω) \times 2}^{n} (f_{ψ (Ω) \times 2 + 1} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω) \times 2)$
$f_{ψ (Ω) \times 2 + 1}^{2} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω) \times ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω)))$
$f_{ψ (Ω) \times 2 + 2} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω) \times Ω)$
$f_{ψ (Ω) \times 3} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω) \times ψ_{1} (Ω_{2}))$
$f_{ψ (Ω) \times 3 + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω)^{2})$
$f_{ψ (Ω) \times ω + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω)^{Ω})$
$f_{ψ (Ω) \times ω + ψ (Ω) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω)^{Ω + 1})$
$f_{ψ (Ω) \times ψ (0)} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω)^{ψ_{1} (0)})$
$f_{ψ (Ω)^{2}} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω)^{ψ_{1} (Ω_{2})})$
$f_{ψ (Ω)^{2} + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω)^{ψ_{1} (Ω_{2} \times Ω)})$
$f_{ψ (Ω)^{ψ (Ω)} + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω)^{ψ_{1} (Ω_{2} \times Ω)^{ψ_{1} (Ω_{2} \times Ω)}})$
$f_{ψ (Ω + 1)} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + 1))$
$f_{ψ (Ω + 1) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + Ω))$
$f_{ψ (Ω + 2)} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + Ω + 1))$
$f_{ψ (Ω + 2) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + Ω \times 2))$
$f_{ψ (Ω + ω) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + Ω^{2}))$
$f_{ψ (Ω + ψ (0))} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + ψ_{1} (0)))$
$f_{ψ (Ω + ψ (0)) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + ψ_{1} (Ω)))$
$f_{ψ (Ω + ψ (Ω))} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + ψ_{1} (Ω_{2})))$
$f_{ψ (Ω + ψ (Ω)) [n \times 2]} (f_{ψ (Ω + ψ (Ω))} (n)) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + ψ_{1} (Ω_{2} \times 2)))$
$f_{ψ (Ω + ψ (Ω)) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + ψ_{1} (Ω_{2} \times Ω)))$
$f_{ψ (Ω + ψ (Ω) + 1)} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + ψ_{1} (Ω_{2} \times Ω) + 1))$
$f_{ψ (Ω + ψ (Ω + 1))} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + ψ_{1} (Ω_{2} \times Ω + 1)))$
$f_{ψ (Ω + ψ (Ω + ψ (Ω))) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + ψ_{1} (Ω_{2} \times Ω + ψ_{1} (Ω_{2} \times Ω))))$
$f_{ψ (Ω \times 2)} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + Ω_{2}))$
$f_{ψ (Ω \times 2) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω \times 2))$
$f_{ψ (Ω \times 2) \times 2 + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω \times 2)^{2})$
$f_{ψ (Ω \times 2 + 1)} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω \times 2 + 1))$
$f_{ψ (Ω \times 2 + 1) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω \times 2 + Ω))$
$f_{ψ (Ω \times 2 + ψ (Ω \times 2)) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω \times 2 + ψ_{1} (Ω_{2} \times Ω \times 2)))$
$f_{ψ (Ω \times 3)} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω \times 2 + Ω_{2}))$
$f_{ψ (Ω \times 3) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω \times 3))$
$f_{ψ (Ω \times ω)} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω \times ω))$
$f_{ψ (Ω \times ω) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω^{2}))$
$f_{ψ (Ω \times ω + 1)} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω^{2} + 1))$
$f_{ψ (Ω \times ω + Ω) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω^{2} + Ω_{2} \times Ω))$
$f_{ψ (Ω \times ω \times 2) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω^{2} \times 2))$
$f_{ψ (Ω \times ω^{2}) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω^{3}))$
$f_{ψ (Ω \times ω^{ω}) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω^{Ω}))$
$f_{ψ (Ω \times ψ (0))} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times ψ_{1} (0)))$
$f_{ψ (Ω \times ψ (0)) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times ψ_{1} (Ω)))$
$f_{ψ (Ω \times ψ (1))} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times ψ_{1} (Ω + 1)))$
$f_{ψ (Ω \times ψ (Ω))} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times ψ_{1} (Ω_{2})))$
$f_{ψ (Ω \times ψ (Ω)) + 1} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω)))$
$f_{ψ (Ω \times ψ (Ω + 1))} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω + 1)))$
$f_{ψ (Ω \times ψ (Ω \times 2))} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times ψ_{1} (Ω_{2} \times Ω \times 2)))$
$f_{ψ (Ω \times ψ (Ω \times ψ (0)))} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times ψ_{1} (Ω_{2} \times ψ_{1} (0))))$
$f_{ψ (Ω \times ψ (Ω \times ψ (Ω)))} (n) \sim ψ (Ω_{2} \times ψ_{1} (Ω_{2} \times ψ_{1} (Ω_{2} \times ψ_{1} (Ω_{2}))))$
$f_{ψ (Ω^{2})} (n) \sim ψ (Ω_{2}^{2})$
$f_{ψ (Ω^{2}) [n]} (f_{ψ (Ω^{2})} (n)) \sim ψ (Ω_{2}^{2} + ψ_{1} (Ω_{2}^{2}))$
$f_{ψ (Ω \times ψ (Ω^{2}) [n - 1] + 1)} (f_{ψ (Ω^{2})} (n)) \sim ψ (Ω_{2}^{2} + ψ_{1} (Ω_{2}^{2} + 1))$
$ψ (Ω^{2}) [n] = ψ (Ω \times ψ (Ω^{2}) [n - 1])$
$f_{ψ (Ω \times ψ (Ω^{2}) [n - 1] + Ω) [n]} (f_{ψ (Ω^{2})} (n)) \sim ψ (Ω_{2}^{2} + Ω_{2})$
$f_{ψ (Ω \times ψ (Ω^{2}) [n - 1] + Ω)} (f_{ψ (Ω^{2})} (n)) \sim ψ (Ω_{2}^{2} + Ω_{2} \times Ω)$
$f_{ψ (Ω \times ψ (Ω^{2}) [n - 1] + Ω \times ψ (Ω))} (f_{ψ (Ω^{2})} (n)) \sim ψ (Ω_{2}^{2} + Ω_{2} \times ψ_{1} (Ω_{2}))$
$f_{ψ (Ω^{2}) [n + 1]} (f_{ψ (Ω^{2})} (n)) \sim ψ (Ω_{2}^{2} + Ω_{2} \times ψ_{1} (Ω_{2}^{2}))$
$f_{ψ (Ω^{2}) [n \times 2]} (f_{ψ (Ω^{2})} (n)) \sim ψ (Ω_{2}^{2} \times 2)$
$f_{ψ (Ω^{2}) + 1} (n) \sim ψ (Ω_{2}^{2} \times Ω)$
$f_{ψ (Ω^{2}) + 2} (n) \sim ψ (Ω_{2}^{2} \times Ω^{2})$
$f_{ψ (Ω^{2}) + ψ (0)} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (0))$
$f_{ψ (Ω^{2}) + ψ (Ω)} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}))$
$f_{ψ (Ω^{2}) + ψ (Ω \times ψ (Ω))} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2} \times ψ_{1} (Ω_{2})))$
$f_{ψ (Ω^{2}) \times 2} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2}))$
$f_{ψ (Ω^{2}) \times 2 + 1} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω))$
$f_{ψ (Ω^{2})^{2} + 1} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω)^{ψ_{1} (Ω_{2}^{2} \times Ω)})$
$f_{ψ (Ω^{2} + 1)} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω + 1))$
$f_{ψ (Ω^{2} + 1) + 1} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω + Ω))$
$f_{ψ (Ω^{2} + ψ (Ω^{2})) + 1} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω + ψ_{1} (Ω_{2}^{2} \times Ω)))$
$f_{ψ (Ω^{2} + Ω)} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω + Ω_{2}))$
$f_{ψ (Ω^{2} + Ω) + 1} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω + Ω_{2} \times Ω))$
$f_{ψ (Ω^{2} + Ω \times 2) + 1} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω + Ω_{2} \times Ω \times 2))$
$f_{ψ (Ω^{2} + Ω \times ψ (0)) + 1} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω + Ω_{2} \times ψ_{1} (Ω)))$
$f_{ψ (Ω^{2} + Ω \times ψ (Ω^{2})) + 1} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω + Ω_{2} \times ψ_{1} (Ω_{2}^{2} \times Ω)))$
$f_{ψ (Ω^{2} \times 2)} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω \times 2))$
$f_{ψ (Ω^{2} \times ω) + 1} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω^{2}))$
$f_{ψ (Ω^{2} \times ψ (0)) + 1} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times ψ_{1} (0)))$
$f_{ψ (Ω^{2} \times ψ (Ω^{2}))} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2})))$
$f_{ψ (Ω^{2} \times ψ (Ω^{2})) + 1} (n) \sim ψ (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times ψ_{1} (Ω_{2}^{2} \times Ω)))$
$f_{ψ (Ω^{3})} (n) \sim ψ (Ω_{2}^{3})$
$f_{ψ (Ω^{4})} (n) \sim ψ (Ω_{2}^{4})$
$f_{ψ (Ω^{ω})} (n) \sim ψ (Ω_{2}^{ω})$
$f_{ψ (Ω^{ω}) + 1} (n) \sim ψ (Ω_{2}^{Ω})$
$f_{ψ (Ω^{ω}) + 2} (n) \sim ψ (Ω_{2}^{Ω} \times Ω)$
$f_{ψ (Ω^{ω}) \times 2} (n) \sim ψ (Ω_{2}^{Ω} \times ψ_{1} (Ω_{2}^{ω}))$
$f_{ψ (Ω^{ω}) \times 2 + 1} (n) \sim ψ (Ω_{2}^{Ω} \times ψ_{1} (Ω_{2}^{Ω}))$
$f_{ψ (Ω^{ω}) \times 2 + 2} (n) \sim ψ (Ω_{2}^{Ω} \times ψ_{1} (Ω_{2}^{Ω}) \times Ω)$
$f_{ψ (Ω^{ω})^{2} + 1} (n) \sim ψ (Ω_{2}^{Ω} \times ψ_{1} (Ω_{2}^{Ω})^{ψ_{1} (Ω_{2}^{Ω})})$
$f_{ψ (Ω^{ω} + 1)} (n) \sim ψ (Ω_{2}^{Ω} \times ψ_{1} (Ω_{2}^{Ω} + 1))$
$f_{ψ (Ω^{ω} + ψ (Ω^{ω})) + 1} (n) \sim ψ (Ω_{2}^{Ω} \times ψ_{1} (Ω_{2}^{Ω} + ψ_{1} (Ω_{2}^{Ω})))$
$f_{ψ (Ω^{ω} + Ω) + 1} (n) \sim ψ (Ω_{2}^{Ω} \times ψ_{1} (Ω_{2}^{Ω} + Ω_{2} \times Ω))$
$f_{ψ (Ω^{ω} \times 2)} (n) \sim ψ (Ω_{2}^{Ω} \times ψ_{1} (Ω_{2}^{Ω} + Ω_{2}^{ω}))$
$f_{ψ (Ω^{ω} \times 2) + 1} (n) \sim ψ (Ω_{2}^{Ω} \times ψ_{1} (Ω_{2}^{Ω} \times 2))$
$f_{ψ (Ω^{ω} \times ψ (0)) + 1} (n) \sim ψ (Ω_{2}^{Ω} \times ψ_{1} (Ω_{2}^{Ω} \times ψ_{1} (Ω)))$
$f_{ψ (Ω^{ω} \times ψ (Ω^{ω})) + 1} (n) \sim ψ (Ω_{2}^{Ω} \times ψ_{1} (Ω_{2}^{Ω} \times ψ_{1} (Ω^{Ω})))$
$f_{ψ (Ω^{ω + 1})} (n) \sim ψ (Ω_{2}^{Ω + 1})$
$f_{ψ (Ω^{ω \times 2}) + 1} (n) \sim ψ (Ω_{2}^{Ω \times 2})$
$f_{ψ (Ω^{ω^{2}}) + 1} (n) \sim ψ (Ω_{2}^{Ω^{2}})$
$f_{ψ (Ω^{ω^{ω}}) + 1} (n) \sim ψ (Ω_{2}^{Ω^{Ω}})$
$f_{ψ (Ω^{ψ (0)})} (n) \sim ψ (Ω_{2}^{ψ_{1} (0)})$
$f_{ψ (Ω^{ψ (0)}) + 1} (n) \sim ψ (Ω_{2}^{ψ_{1} (Ω)})$
$f_{ψ (Ω^{ψ (Ω)})} (n) \sim ψ (Ω_{2}^{ψ_{1} (Ω_{2})})$
$f_{ψ (Ω^{ψ (Ω^{2})})} (n) \sim ψ (Ω_{2}^{ψ_{1} (Ω_{2}^{2})})$
$f_{ψ (Ω^{ψ (Ω^{ψ (Ω)})})} (n) \sim ψ (Ω_{2}^{ψ_{1} (Ω_{2}^{ψ_{1} (Ω_{2})})})$
$f_{ψ (Ω^{Ω})} (n) \sim ψ (Ω_{2}^{Ω_{2}})$
$f_{ψ (Ω^{Ω}) + 1} (n) \sim ψ (Ω_{2}^{Ω_{2}} \times Ω)$
$f_{ψ (Ω^{Ω}) \times 2} (n) \sim ψ (Ω_{2}^{Ω_{2}} \times ψ_{1} (Ω_{2}^{Ω_{2}}))$
$f_{ψ (Ω^{Ω} + 1) + 1} (n) \sim ψ (Ω_{2}^{Ω_{2}} \times ψ_{1} (Ω_{2}^{Ω_{2}} \times Ω + Ω))$
$f_{ψ (Ω^{Ω} + Ω) + 1} (n) \sim ψ (Ω_{2}^{Ω_{2}} \times ψ_{1} (Ω_{2}^{Ω_{2}} \times Ω + Ω_{2} \times Ω))$
$f_{ψ (Ω^{Ω} + Ω^{ψ (Ω^{Ω})})} (n) \sim ψ (Ω_{2}^{Ω_{2}} \times ψ_{1} (Ω_{2}^{Ω_{2}} \times Ω + Ω_{2}^{ψ_{1} (Ω_{2}^{Ω_{2}})}))$
$f_{ψ (Ω^{Ω} \times 2)} (n) \sim ψ (Ω_{2}^{Ω_{2}} \times ψ_{1} (Ω_{2}^{Ω_{2}} \times Ω \times 2))$
$f_{ψ (Ω^{Ω} \times ψ (0))} (n) \sim ψ (Ω_{2}^{Ω_{2}} \times ψ_{1} (Ω_{2}^{Ω_{2}} \times ψ_{1} (0)))$
$f_{ψ (Ω^{Ω + 1})} (n) \sim ψ (Ω_{2}^{Ω_{2} + 1})$
$f_{ψ (Ω^{Ω + 2})} (n) \sim ψ (Ω_{2}^{Ω_{2} + 2})$
$f_{ψ (Ω^{Ω + ω}) + 1} (n) \sim ψ (Ω_{2}^{Ω_{2} + Ω})$
$f_{ψ (Ω^{Ω + ψ (Ω^{Ω})})} (n) \sim ψ (Ω_{2}^{Ω_{2} + ψ_{1} (Ω_{2}^{Ω_{2}})})$
$f_{ψ (Ω^{Ω \times 2})} (n) \sim ψ (Ω_{2}^{Ω_{2} \times 2})$
$f_{ψ (Ω^{Ω^{2}})} (n) \sim ψ (Ω_{2}^{Ω_{2}^{2}})$
$f_{ψ (Ω^{Ω^{ω}})} (n) \sim ψ (Ω_{2}^{Ω_{2}^{ω}})$
$f_{ψ (Ω^{Ω^{ω}}) + 1} (n) \sim ψ (Ω_{2}^{Ω_{2}^{Ω}})$
$f_{ψ (Ω^{Ω^{Ω}})} (n) \sim ψ (Ω_{2}^{Ω_{2}^{Ω_{2}}})$
$f_{ψ (ψ_{1} (0))} (n) \sim ψ (ψ_{2} (0))$
$f_{ψ (ψ_{1} (0) \times Ω)} (n) \sim ψ (ψ_{2} (0) \times Ω_{2})$
$f_{ψ (ψ_{1} (0)^{2})} (n) \sim ψ (ψ_{2} (0)^{2})$
$f_{ψ (ψ_{1} (0)^{Ω})} (n) \sim ψ (ψ_{2} (0)^{Ω_{2}})$
$f_{ψ (ψ_{1} (0)^{ψ_{1} (0)})} (n) \sim ψ (ψ_{2} (0)^{ψ_{2} (0)})$
$f_{ψ (ψ_{1} (1))} (n) \sim ψ (ψ_{2} (1))$
$f_{ψ (ψ_{1} (ω)) + 1} (n) \sim ψ (ψ_{2} (Ω))$
$f_{ψ (ψ_{1} (ψ (0)))} (n) \sim ψ (ψ_{2} (ψ_{1} (0)))$
$f_{ψ (ψ_{1} (ψ (Ω)))} (n) \sim ψ (ψ_{2} (ψ_{1} (Ω_{2})))$
$f_{ψ (ψ_{1} (ψ (Ω^{Ω})))} (n) \sim ψ (ψ_{2} (ψ_{1} (Ω_{2}^{Ω_{2}})))$
$f_{ψ (ψ_{1} (ψ (ψ_{1} (0))))} (n) \sim ψ (ψ_{2} (ψ_{1} (ψ_{2} (0))))$
$f_{ψ (ψ_{1} (Ω))} (n) \sim ψ (ψ_{2} (Ω_{2}))$
$f_{ψ (ψ_{1} (ψ_{1} (0)))} (n) \sim ψ (ψ_{2} (ψ_{2} (0)))$
$f_{ψ (Ω_{2})} (n) \sim ψ (Ω_{3})$
$f_{ψ (Ω_{2}) [n]} (f_{ψ (Ω_{2})} (n)) \sim ψ (Ω_{3} + ψ_{2} (Ω_{3}))$
$f_{ψ (Ω_{2}) [n + 1]} (f_{ψ (Ω_{2})} (n)) \sim ψ (Ω_{3} + ψ_{2} (Ω_{3} + ψ_{2} (Ω_{3})))$
$f_{ψ (Ω_{2}) [n \times 2]} (f_{ψ (Ω_{2})} (n)) \sim ψ (Ω_{3} \times 2)$
$f_{ψ (Ω_{2}) + 1} (n) \sim ψ (Ω_{3} \times Ω)$
$f_{ψ (Ω_{2}) + ψ (0)} (n) \sim ψ (Ω_{3} \times ψ_{1} (0))$
$f_{ψ (Ω_{2}) + ψ (Ω)} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{2}))$
$f_{ψ (Ω_{2}) \times 2} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{3}))$
$f_{ψ (Ω_{2}) \times 2 + 1} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{3} \times Ω))$
$f_{ψ (Ω_{2}) \times 3 + 1} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{3} \times Ω)^{2})$
$f_{ψ (Ω_{2})^{2} + 1} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{3} \times Ω)^{ψ_{1} (Ω_{3} \times Ω)})$
$f_{ψ (Ω_{2} + 1)} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{3} \times Ω + 1))$
$f_{ψ (Ω_{2} + Ω)} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{3} \times Ω + Ω_{2}))$
$f_{ψ (Ω_{2} + ψ_{1} (Ω_{2}))} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{3} \times Ω + ψ_{2} (Ω_{3})))$
$f_{ψ (Ω_{2} + ψ_{1} (Ω_{2})) + 1} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{3} \times Ω + ψ_{2} (Ω_{3} \times Ω)))$
$f_{ψ (Ω_{2} + ψ_{1} (Ω_{2} + 1))} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{3} \times Ω + ψ_{2} (Ω_{3} \times Ω + 1)))$
$f_{ψ (Ω_{2} \times 2)} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{3} \times Ω + Ω_{3}))$
$f_{ψ (Ω_{2} \times 2) + 1} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{3} \times Ω \times 2))$
$f_{ψ (Ω_{2} \times ψ (Ω_{2}))} (n) \sim ψ (Ω_{3} \times ψ_{1} (Ω_{3} \times ψ_{1} (Ω_{3})))$
$f_{ψ (Ω_{2} \times Ω)} (n) \sim ψ (Ω_{3} \times Ω_{2})$
$f_{ψ (Ω_{2} \times ψ_{1} (0))} (n) \sim ψ (Ω_{3} \times ψ_{2} (0))$
$f_{ψ (Ω_{2} \times ψ_{1} (Ω_{2}))} (n) \sim ψ (Ω_{3} \times ψ_{2} (Ω_{3}))$
$f_{ψ (Ω_{2}^{2})} (n) \sim ψ (Ω_{3}^{2})$
$f_{ψ (Ω_{2}^{Ω_{2}})} (n) \sim ψ (Ω_{3}^{Ω_{3}})$
$f_{ψ (ψ_{2} (0))} (n) \sim ψ (ψ_{3} (0))$
$f_{ψ (Ω_{3})} (n) \sim ψ (Ω_{4})$
$f_{ψ (Ω_{4})} (n) \sim ψ (Ω_{5})$
$f_{ψ (Ω_{ω})} \sim ψ (Ω_{ω}) = BO$

@@ 第1行： / 第1行： @@
-本条目展示[[SGH]]与[[FGH]]的对照
+本条目展示 [[SGH]] 与 [[FGH]] 的对照，使用 [[Veblen 函数]]和 [[序数坍缩函数#MOCF|MOCF]]。本条目分析来自梅天狸。
-<math>g_{\varepsilon_0\times2}(n)=g_{\varepsilon_0+\varepsilon_0[n]}(n)=(n\uparrow\uparrow n)\times2</math>
+=== Part 1 ===
-<nowiki>\begin{align} & g_{\varepsilon_0\times\omega}(n)=g_{\varepsilon_0\times n}(n)=(n\uparrow\uparrow n)\times n\\ &g_{\varepsilon_0\times\omega^2}(n)=g_{\varepsilon_0\times\omega\times n}(n)=(n\uparrow\uparrow n)\times n^2\\ &g_{\varepsilon_0\times\omega^\omega}(n)=g_{\varepsilon_0\times\omega^n}(n)=(n\uparrow\uparrow n)\times n^n\\ &g_{\varepsilon_0\times\omega^{\omega^\omega}}(n)=(n\uparrow\uparrow n)\times (n\uparrow\uparrow3)\\ &g_{\varepsilon_0^2}(n)=g_{\varepsilon_0\times\varepsilon_0[n]}(n)=(n\uparrow\uparrow n)^2\\ &g_{\varepsilon_0^3}(n)=(n\uparrow\uparrow n)^3\\ &g_{\varepsilon_0^\omega}(n)=g_{\varepsilon_0^n}(n)=(n\uparrow\uparrow n)^n\\ &g_{\varepsilon_0^{\omega^\omega}}(n)=(n\uparrow\uparrow n)^{n^n}\\ &g_{\varepsilon_0^{\varepsilon_0}}(n)=(n\uparrow\uparrow n)^{n\uparrow\uparrow n}=(n\uparrow\uparrow n)\uparrow\uparrow2\\ &g_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}(n)=(n\uparrow\uparrow n)\uparrow\uparrow3\\ &g_{\varepsilon_1}(n)=(n\uparrow\uparrow n)\uparrow\uparrow n\approx n\uparrow\uparrow(n\times2)\\ &g_{\varepsilon_2}(n)=((n\uparrow\uparrow n)\uparrow\uparrow n)\uparrow\uparrow n\approx n\uparrow\uparrow(n\times3)\\ &n\uparrow\uparrow(n\times3)\sim\varepsilon_2\\ &n\uparrow\uparrow(n\times3+1)\sim\varepsilon_2^{\varepsilon_2}\\ &n\uparrow\uparrow(n\times3+2)\sim\varepsilon_2^{\varepsilon_2^{\varepsilon_2}}\\ &n\uparrow\uparrow(n\times4)\sim\varepsilon_3\\ &n\uparrow\uparrow(n\times5)\sim\varepsilon_4\\ &n\uparrow\uparrow(n^2)\sim\varepsilon_\omega\\ &n\uparrow\uparrow(n^2+1)\sim\varepsilon_\omega^{\varepsilon_\omega}\\&n\uparrow\uparrow(n^2+n)\sim\varepsilon_{\omega+1}\\&n\uparrow\uparrow(n^2\times2)\sim\varepsilon_{\omega\times2}\\&n\uparrow\uparrow(n^3)\sim\varepsilon_{\omega^2}\\&n\uparrow\uparrow(n^n)=n\uparrow\uparrow n\uparrow\uparrow2\sim\varepsilon_{\omega^\omega}\\&n\uparrow\uparrow(n^n+n)\sim\varepsilon_{\omega^\omega+1}\\&</nowiki><nowiki>n\uparrow\uparrow(n^{n+1})\sim\varepsilon_{\omega^{\omega+1}}\\&</nowiki><nowiki>n\uparrow\uparrow(n^{n^2})\sim\varepsilon_{\omega^{\omega^2}}\\&</nowiki><nowiki>n\uparrow\uparrow n\uparrow\uparrow3\sim\varepsilon_{\omega^{\omega^\omega}}\\&</nowiki>n\uparrow\uparrow n\uparrow\uparrow n=n\uparrow\uparrow\uparrow 3\sim\varepsilon_{\varepsilon_0}\\&n\uparrow\uparrow (n\uparrow\uparrow n+n)\sim\varepsilon_{\varepsilon_0+1}\\&n\uparrow\uparrow ((n\uparrow\uparrow n)\times2)\sim\varepsilon_{\varepsilon_0\times2}\\&<nowiki>n\uparrow\uparrow ((n\uparrow\uparrow n)\uparrow\uparrow 2)\approx n\uparrow\uparrow n\uparrow\uparrow(n+1)\sim\varepsilon_{\varepsilon_0^{\varepsilon_0}}\\&</nowiki>n\uparrow\uparrow n\uparrow\uparrow(n\times2)\sim\varepsilon_{\varepsilon_1}\\&n\uparrow\uparrow n\uparrow\uparrow(n^2)\sim\varepsilon_{\varepsilon_\omega}\\&<nowiki>n\uparrow\uparrow n\uparrow\uparrow n\uparrow\uparrow n=n\uparrow\uparrow\uparrow4\sim\varepsilon_{\varepsilon_{\varepsilon_0}}\\&</nowiki>n\uparrow\uparrow\uparrow n\sim\zeta_0\\&(n\uparrow\uparrow\uparrow n)\uparrow\uparrow n\sim\varepsilon_{\zeta_0+1}\\&(n\uparrow\uparrow\uparrow n)\uparrow\uparrow (n\times2)\sim\varepsilon_{\zeta_0+2}\\&(n\uparrow\uparrow\uparrow n)\uparrow\uparrow (n\uparrow\uparrow n)\sim\varepsilon_{\zeta_0+\varepsilon_0}\\&(n\uparrow\uparrow\uparrow n)\uparrow\uparrow (n\uparrow\uparrow\uparrow n)\approx n\uparrow\uparrow\uparrow(n+1)\sim\varepsilon_{\zeta_0\times2}\\&<nowiki>(n\uparrow\uparrow\uparrow n)\uparrow\uparrow (n\uparrow\uparrow\uparrow n)\uparrow\uparrow n\sim\varepsilon_{\varepsilon_{\zeta_0+1}}\\&</nowiki><nowiki>(n\uparrow\uparrow\uparrow n)\uparrow\uparrow (n\uparrow\uparrow\uparrow n)\uparrow\uparrow (n\uparrow\uparrow\uparrow n)\sim\varepsilon_{\varepsilon_{\zeta_0\times2}}\\&</nowiki>(n\uparrow\uparrow\uparrow n)\uparrow\uparrow \uparrow n\approx n\uparrow\uparrow\uparrow(n\times2)\sim\zeta_1\\&(n\uparrow\uparrow\uparrow(n\times2))\uparrow\uparrow n\sim\varepsilon_{\zeta_1+1}\\&(n\uparrow\uparrow\uparrow(n\times2))\uparrow\uparrow (n\uparrow\uparrow\uparrow n)\sim\varepsilon_{\zeta_1+\zeta_0}\\&(n\uparrow\uparrow\uparrow(n\times2))\uparrow\uparrow (n\uparrow\uparrow\uparrow(n\times2))\sim\varepsilon_{\zeta_1\times2}\\&(n\uparrow\uparrow\uparrow(n\times2))\uparrow\uparrow \uparrow n\approx n\uparrow\uparrow\uparrow(n\times3)\sim\zeta_2\\&n\uparrow\uparrow\uparrow(n^2)\sim\zeta_\omega\\&n\uparrow\uparrow\uparrow(n^n)\sim\zeta_{\omega^\omega}\\&n\uparrow\uparrow\uparrow(n\uparrow\uparrow n)\sim\zeta_{\varepsilon_0}\\&n\uparrow\uparrow\uparrow n\uparrow\uparrow\uparrow n \sim\zeta_{\zeta_0}\\&n\uparrow\uparrow\uparrow\uparrow n \sim\eta_0\\&(n\uparrow\uparrow\uparrow\uparrow n)\uparrow\uparrow\uparrow n \sim\zeta_{\eta_0+1}\\&n\uparrow\uparrow\uparrow\uparrow (n\times2) \sim\eta_1\\&n\uparrow^53 \sim\eta_{\eta_0}\\&n\uparrow^5n \sim\varphi(4,0)\\&n\uparrow^5(n\times2) \sim\varphi(4,1)\\&n\uparrow^6 n \sim\varphi(5,0)\\&n\uparrow^n n \sim\varphi(\omega,0)\\ \end{align}
+* <math>g_{\varepsilon_0\times2}(n)=g_{\varepsilon_0+\varepsilon_0[n]}(n)=(n\uparrow\uparrow n)\times2</math>
+* <math>g_{\varepsilon_0\times\omega}(n)=g_{\varepsilon_0\times n}(n)=(n\uparrow\uparrow n)\times n</math>
+* <math>g_{\varepsilon_0\times\omega^2}(n)=g_{\varepsilon_0\times\omega\times n}(n)=(n\uparrow\uparrow n)\times n^2</math>
+* <math>g_{\varepsilon_0\times\omega^\omega}(n)=g_{\varepsilon_0\times\omega^n}(n)=(n\uparrow\uparrow n)\times n^n</math>
+* <math>g_{\varepsilon_0\times\omega^{\omega^\omega}}(n)=(n\uparrow\uparrow n)\times (n\uparrow\uparrow3)</math>
+* <math>g_{\varepsilon_0^2}(n)=g_{\varepsilon_0\times\varepsilon_0[n]}(n)=(n\uparrow\uparrow n)^2</math>
+* <math>g_{\varepsilon_0^3}(n)=(n\uparrow\uparrow n)^3</math>
+* <math>g_{\varepsilon_0^\omega}(n)=g_{\varepsilon_0^n}(n)=(n\uparrow\uparrow n)^n</math>
+* <math>g_{\varepsilon_0^{\omega^\omega}}(n)=(n\uparrow\uparrow n)^{n^n}</math>
+* <math>g_{\varepsilon_0^{\varepsilon_0}}(n)=(n\uparrow\uparrow n)^{n\uparrow\uparrow n}=(n\uparrow\uparrow n)\uparrow\uparrow2</math>
+* <math>g_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}(n)=(n\uparrow\uparrow n)\uparrow\uparrow3</math>
+* <math>g_{\varepsilon_1}(n)=(n\uparrow\uparrow n)\uparrow\uparrow n\approx n\uparrow\uparrow(n\times2)</math>
+* <math>g_{\varepsilon_2}(n)=((n\uparrow\uparrow n)\uparrow\uparrow n)\uparrow\uparrow n\approx n\uparrow\uparrow(n\times3)</math>
+* <math>n\uparrow\uparrow(n\times3)\sim\varepsilon_2</math>
+* <math>n\uparrow\uparrow(n\times3+1)\sim\varepsilon_2^{\varepsilon_2}</math>
+* <math>n\uparrow\uparrow(n\times3+2)\sim\varepsilon_2^{\varepsilon_2^{\varepsilon_2}}</math>
+* <math>n\uparrow\uparrow(n\times4)\sim\varepsilon_3</math>
+* <math>n\uparrow\uparrow(n\times5)\sim\varepsilon_4</math>
+* <math>n\uparrow\uparrow(n^2)\sim\varepsilon_\omega</math>
+* <math>n\uparrow\uparrow(n^2+1)\sim\varepsilon_\omega^{\varepsilon_\omega}</math>
+* <math>n\uparrow\uparrow(n^2+n)\sim\varepsilon_{\omega+1}</math>
+* <math>n\uparrow\uparrow(n^2\times2)\sim\varepsilon_{\omega\times2}</math>
+* <math>n\uparrow\uparrow(n^3)\sim\varepsilon_{\omega^2}</math>
+* <math>n\uparrow\uparrow(n^n)=n\uparrow\uparrow n\uparrow\uparrow2\sim\varepsilon_{\omega^\omega}</math>
+* <math>n\uparrow\uparrow(n^n+n)\sim\varepsilon_{\omega^\omega+1}</math>
+* <math>n\uparrow\uparrow(n^{n+1})\sim\varepsilon_{\omega^{\omega+1}}</math>
+* <math>n\uparrow\uparrow(n^{n^2})\sim\varepsilon_{\omega^{\omega^2}}</math>
+* <math>n\uparrow\uparrow n\uparrow\uparrow3\sim\varepsilon_{\omega^{\omega^\omega}}</math>
+* <math>n\uparrow\uparrow n\uparrow\uparrow n=n\uparrow\uparrow\uparrow 3\sim\varepsilon_{\varepsilon_0}</math>
+* <math>n\uparrow\uparrow (n\uparrow\uparrow n+n)\sim\varepsilon_{\varepsilon_0+1}</math>
+* <math>n\uparrow\uparrow ((n\uparrow\uparrow n)\times2)\sim\varepsilon_{\varepsilon_0\times2}</math>
+* <math>n\uparrow\uparrow ((n\uparrow\uparrow n)\uparrow\uparrow 2)\approx n\uparrow\uparrow n\uparrow\uparrow(n+1)\sim\varepsilon_{\varepsilon_0^{\varepsilon_0}}</math>
+* <math>n\uparrow\uparrow n\uparrow\uparrow(n\times2)\sim\varepsilon_{\varepsilon_1}</math>
+* <math>n\uparrow\uparrow n\uparrow\uparrow(n^2)\sim\varepsilon_{\varepsilon_\omega}</math>
+* <math>n\uparrow\uparrow n\uparrow\uparrow n\uparrow\uparrow n=n\uparrow\uparrow\uparrow4\sim\varepsilon_{\varepsilon_{\varepsilon_0}}</math>
+* <math>n\uparrow\uparrow\uparrow n\sim\zeta_0</math>
+* <math>(n\uparrow\uparrow\uparrow n)\uparrow\uparrow n\sim\varepsilon_{\zeta_0+1}</math>
+* <math>(n\uparrow\uparrow\uparrow n)\uparrow\uparrow (n\times2)\sim\varepsilon_{\zeta_0+2}</math>
+* <math>(n\uparrow\uparrow\uparrow n)\uparrow\uparrow (n\uparrow\uparrow n)\sim\varepsilon_{\zeta_0+\varepsilon_0}</math>
+* <math>(n\uparrow\uparrow\uparrow n)\uparrow\uparrow (n\uparrow\uparrow\uparrow n)\approx n\uparrow\uparrow\uparrow(n+1)\sim\varepsilon_{\zeta_0\times2}</math>
+* <math>(n\uparrow\uparrow\uparrow n)\uparrow\uparrow (n\uparrow\uparrow\uparrow n)\uparrow\uparrow n\sim\varepsilon_{\varepsilon_{\zeta_0+1}}</math>
+* <math>(n\uparrow\uparrow\uparrow n)\uparrow\uparrow (n\uparrow\uparrow\uparrow n)\uparrow\uparrow (n\uparrow\uparrow\uparrow n)\sim\varepsilon_{\varepsilon_{\zeta_0\times2}}</math>
+* <math>(n\uparrow\uparrow\uparrow n)\uparrow\uparrow \uparrow n\approx n\uparrow\uparrow\uparrow(n\times2)\sim\zeta_1</math>
+* <math>(n\uparrow\uparrow\uparrow(n\times2))\uparrow\uparrow n\sim\varepsilon_{\zeta_1+1}</math>
+* <math>(n\uparrow\uparrow\uparrow(n\times2))\uparrow\uparrow (n\uparrow\uparrow\uparrow n)\sim\varepsilon_{\zeta_1+\zeta_0}</math>
+* <math>(n\uparrow\uparrow\uparrow(n\times2))\uparrow\uparrow (n\uparrow\uparrow\uparrow(n\times2))\sim\varepsilon_{\zeta_1\times2}</math>
+* <math>(n\uparrow\uparrow\uparrow(n\times2))\uparrow\uparrow \uparrow n\approx n\uparrow\uparrow\uparrow(n\times3)\sim\zeta_2</math>
+* <math>n\uparrow\uparrow\uparrow(n^2)\sim\zeta_\omega</math>
+* <math>n\uparrow\uparrow\uparrow(n^n)\sim\zeta_{\omega^\omega}</math>
+* <math>n\uparrow\uparrow\uparrow(n\uparrow\uparrow n)\sim\zeta_{\varepsilon_0}</math>
+* <math>n\uparrow\uparrow\uparrow n\uparrow\uparrow\uparrow n \sim\zeta_{\zeta_0}</math>
+* <math>n\uparrow\uparrow\uparrow\uparrow n \sim\eta_0</math>
+* <math>(n\uparrow\uparrow\uparrow\uparrow n)\uparrow\uparrow\uparrow n \sim\zeta_{\eta_0+1}</math>
+* <math>n\uparrow\uparrow\uparrow\uparrow (n\times2) \sim\eta_1</math>
+* <math>n\uparrow^53 \sim\eta_{\eta_0}</math>
+* <math>n\uparrow^5n \sim\varphi(4,0)</math>
+* <math>n\uparrow^5(n\times2) \sim\varphi(4,1)</math>
+* <math>n\uparrow^6 n \sim\varphi(5,0)</math>
+* <math>n\uparrow^n n \sim\varphi(\omega,0)</math>
+=== Part 2 ===
-\begin{align} &f_3(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow n\sim\psi(\Omega^\omega+1)\\&f_3^2(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow (n\times2)\sim\psi(\Omega^\omega+2)\\&f_3^n(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow (n^2)\sim\psi(\Omega^\omega+\omega)\\&f_3^{f_3(n)}(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow (n\uparrow\uparrow n)\sim\psi(\Omega^\omega+\psi(0))\\&f_4(f_\omega(n))=f_3^{f_\omega(n)}(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow\uparrow2\sim\psi(\Omega^\omega+\psi(\Omega^\omega))\\&f_3(f_4(f_\omega(n)))\approx ((n\uparrow^nn)\uparrow\uparrow\uparrow2)\uparrow\uparrow n\sim\psi(\Omega^\omega+\psi(\Omega^\omega)+1)\\&f_3^{f_{\omega}(n)}(f_4(f_\omega(n)))\approx ((n\uparrow^nn)\uparrow\uparrow\uparrow2)\uparrow\uparrow (n\uparrow^nn)\sim\psi(\Omega^\omega+\psi(\Omega^\omega)\times2)\\&f_3^{f_{\omega}(n)\times2}(f_4(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega)\times3)\\&f_3^{f_3(f_{\omega}(n))}(f_4(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+1))\\&f_3^{f_3^2(f_{\omega}(n))}(f_4(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+2))\\&f_4^2(f_\omega(n))=f_3^{f_3^{f_\omega(n)}(f_{\omega}(n))}(f_4(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega)))\\&f_3(f_4^2(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega))+1)\\&f_3^{f_4(f_\omega(n))}(f_4^2(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega))\times2)\\&f_3^{f_3(f_4(f_\omega(n)))}(f_4^2(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega)+1))\\&f_3^{f_3^{f_\omega(n)}(f_4(f_\omega(n)))}(f_4^2(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega)\times2))\\&f_4^3(f_\omega(n))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega))))\\&f_4^n(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow\uparrow n\sim\psi(\Omega^\omega+\Omega)\\&f_4^{n+1}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega+\psi(\Omega^\omega+\Omega))\\&f_4^{n+2}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega+\psi(\Omega^\omega+\Omega+\psi(\Omega^\omega+\Omega)))\\&f_4^{n\times2}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega\times2)\\&f_4^{n^2}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega\times\omega)\\&f_4^{f_3(n)}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega\times\psi(0))\\&f_5(f_\omega(n))=f_4^{f_\omega(n)}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega))\\&f_3(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)+1)\\&f_3^{f_\omega(n)}(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)+\psi(\Omega^\omega))\\&f_4(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)+\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)))\\&f_4^n(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)+\Omega)\\&f_4^{f_\omega(n)}(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)\times2)\\&f_4^{f_\omega(n)\times2}(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)\times3)\\&f_4^{f_3(f_\omega(n))}(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega+1))\\&f_4^{f_4(f_\omega(n))}(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega+\Omega))\\&f_5^2(f_\omega(n))=f_4^{f_5(f_\omega(n))}(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)))\\&f_5^n(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow\uparrow\uparrow n\sim\psi(\Omega^\omega+\Omega^2)\\&f_5^{n\times2}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega^2\times2)\\&f_6(f_\omega(n))\sim\psi(\Omega^\omega+\Omega^2\times\psi(\Omega^\omega))\\&f_6^n(f_\omega(n))\sim\psi(\Omega^\omega+\Omega^3)\\&f_7^n(f_\omega(n))\sim\psi(\Omega^\omega+\Omega^4)\\&f_n(f_\omega(n))\approx (n\uparrow^nn)\uparrow^nn\approx n\uparrow^n(2n)\sim\psi(\Omega^\omega\times2)\\&f_n^2(f_\omega(n))\approx n\uparrow^n(3n)\sim\psi(\Omega^\omega\times3)\\&f_n^n(f_\omega(n))\approx n\uparrow^n(n^2)\sim\psi(\Omega^\omega\times\omega)\\&f_n^{f_3(n)}(f_\omega(n))\approx n\uparrow^nn\uparrow\uparrow n\sim\psi(\Omega^\omega\times\psi(0))\\&f_{n+1}(f_\omega(n))=f_n^{f_\omega(n)}(f_\omega(n))\approx n\uparrow^nn\uparrow^nn\sim\psi(\Omega^\omega\times\psi(\Omega^\omega))\\&f_n(f_{n+1}(f_\omega(n)))\sim\psi(\Omega^\omega\times\psi(\Omega^\omega)+\Omega^\omega)\\&f_n^{f_\omega(n)}(f_{n+1}(f_\omega(n)))\sim\psi(\Omega^\omega\times\psi(\Omega^\omega)\times2)\\&f_n^{f_3(f_\omega(n))}(f_{n+1}(f_\omega(n)))\sim\psi(\Omega^\omega\times\psi(\Omega^\omega+1))\\&f_n^{f_n(f_\omega(n))}(f_{n+1}(f_\omega(n)))\sim\psi(\Omega^\omega\times\psi(\Omega^\omega\times2))\\&f_{n+1}^2(f_\omega(n))\approx n\uparrow^nn\uparrow^nn\uparrow^nn\sim\psi(\Omega^\omega\times\psi(\Omega^\omega\times\psi(\Omega^\omega)))\\&f_{n+1}^n(f_\omega(n))\approx n\uparrow^{n+1}n\sim\psi(\Omega^{\omega+1})\\&f_n(f_{n+1}^n(f_\omega(n)))\sim\psi(\Omega^{\omega+1}+\Omega^\omega)\\&f_{n+1}^{n+1}(f_\omega(n))=f_n^{f_{n+1}^n(f_\omega(n))}(f_{n+1}^n(f_\omega(n)))\sim\psi(\Omega^{\omega+1}+\Omega^\omega\times\psi(\Omega^{\omega+1}))\\&f_{n+1}^{n\times2}(f_\omega(n))\approx n\uparrow^{n+1}(2n)\sim\psi(\Omega^{\omega+1}\times2)\\&f_{n+2}(f_\omega(n))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega))\\&f_n(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)+\Omega^\omega)\\&f_n^{f_{n+1}^n(f_\omega(n))}(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)+\Omega^\omega\times\psi(\Omega^{\omega+1}))\\&f_{n+1}(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)+\Omega^\omega\times\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)))\\&f_{n+1}^n(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)+\Omega^{\omega+1})\\&f_{n+1}^{f_\omega(n)}(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)\times2)\\&f_{n+1}^{f_{n+1}(f_\omega(n))}(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega\times\psi(\Omega^\omega)))\\&f_{n+1}^{f_{n+1}^n(f_\omega(n))}(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^{\omega+1}))\\&f_{n+2}^2(f_\omega(n))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)))\\&f_{n+2}^n(f_\omega(n))\sim\psi(\Omega^{\omega+2})\\&f_{n+2}^{n+1}(f_\omega(n))\sim\psi(\Omega^{\omega+2}+\Omega^{\omega+1}\times\psi(\Omega^\omega))\\&f_{n+2}^{n\times2}(f_\omega(n))\sim\psi(\Omega^{\omega+2}\times2)\\&f_{n+3}(f_\omega(n))\sim\psi(\Omega^{\omega+2}\times\psi(\Omega^\omega))\\&f_{n+3}^2(f_\omega(n))\sim\psi(\Omega^{\omega+2}\times\psi(\Omega^{\omega+2}\times\psi(\Omega^\omega)))\\&f_{n+3}^n(f_\omega(n))\sim\psi(\Omega^{\omega+3})\\&f_{n\times2}(f_\omega(n))\sim\psi(\Omega^{\omega\times2})\\&f_{n\times2+1}(f_\omega(n))\sim\psi(\Omega^{\omega\times2}\times\psi(\Omega^{\omega}))\\&f_{n\times2+1}^n(f_\omega(n))\sim\psi(\Omega^{\omega\times2+1})\\&f_{n\times3}(f_\omega(n))\sim\psi(\Omega^{\omega\times3})\\&f_{n^2}(f_\omega(n))\sim\psi(\Omega^{\omega^2})\\&f_{f_3(n)}(f_\omega(n))\approx n\uparrow^{n\uparrow\uparrow n}n\sim\psi(\Omega^{\psi(0)})\\&f_{f_4(n)}(f_\omega(n))\approx n\uparrow^{n\uparrow\uparrow\uparrow n}n\sim\psi(\Omega^{\psi(\Omega)})\\&f_\omega^2(n)=f_{f_\omega(n)}(f_\omega(n))\approx n\uparrow^{n\uparrow^n n}n\sim\psi(\Omega^{\psi(\Omega^\omega)})\\&f_n(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)}+\Omega^\omega)\\&f_{n+1}^n(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)}+\Omega^{\omega+1})\\&f_{f_\omega(n)}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)}\times2)\\&f_{f_\omega(n)}^n(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)}\times\omega)\\&f_{f_\omega(n)+1}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)}\times\psi(\Omega^{\psi(\Omega^\omega)}))\\&f_{f_\omega(n)+1}^n(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)+1})\\&f_{f_\omega(n)+n}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)+\omega})\\&f_{f_\omega(n)\times2}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)\times2})\\&f_{f_3(f_\omega(n))}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega+1)})\\&f_{f_{n+1}^n(f_\omega(n))}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^{\omega+1})})\\&f_\omega^3(n)=f_{f_{f_\omega(n)}(f_\omega(n))}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^{\psi(\Omega^\omega)})})\\&f_{\omega+1}(n)=f_\omega^n(n)\sim\psi(\Omega^\Omega)=\varphi(1,0,0)\\ \end{align}
+* <math>f_3(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow n\sim\psi(\Omega^\omega+1)</math>
+* <math>f_3^2(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow (n\times2)\sim\psi(\Omega^\omega+2)</math>
+* <math>f_3^n(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow (n^2)\sim\psi(\Omega^\omega+\omega)</math>
+* <math>f_3^{f_3(n)}(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow (n\uparrow\uparrow n)\sim\psi(\Omega^\omega+\psi(0))</math>
+* <math>f_4(f_\omega(n))=f_3^{f_\omega(n)}(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow\uparrow2\sim\psi(\Omega^\omega+\psi(\Omega^\omega))</math>
+* <math>f_3(f_4(f_\omega(n)))\approx ((n\uparrow^nn)\uparrow\uparrow\uparrow2)\uparrow\uparrow n\sim\psi(\Omega^\omega+\psi(\Omega^\omega)+1)</math>
+* <math>f_3^{f_{\omega}(n)}(f_4(f_\omega(n)))\approx ((n\uparrow^nn)\uparrow\uparrow\uparrow2)\uparrow\uparrow (n\uparrow^nn)\sim\psi(\Omega^\omega+\psi(\Omega^\omega)\times2)</math>
+* <math>f_3^{f_{\omega}(n)\times2}(f_4(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega)\times3)</math>
+* <math>f_3^{f_3(f_{\omega}(n))}(f_4(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+1))</math>
+* <math>f_3^{f_3^2(f_{\omega}(n))}(f_4(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+2))</math>
+* <math>f_4^2(f_\omega(n))=f_3^{f_3^{f_\omega(n)}(f_{\omega}(n))}(f_4(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega)))</math>
+* <math>f_3(f_4^2(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega))+1)</math>
+* <math>f_3^{f_4(f_\omega(n))}(f_4^2(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega))\times2)</math>
+* <math>f_3^{f_3(f_4(f_\omega(n)))}(f_4^2(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega)+1))</math>
+* <math>f_3^{f_3^{f_\omega(n)}(f_4(f_\omega(n)))}(f_4^2(f_\omega(n)))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega)\times2))</math>
+* <math>f_4^3(f_\omega(n))\sim\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega+\psi(\Omega^\omega))))</math>
+* <math>f_4^n(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow\uparrow n\sim\psi(\Omega^\omega+\Omega)</math>
+* <math>f_4^{n+1}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega+\psi(\Omega^\omega+\Omega))</math>
+* <math>f_4^{n+2}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega+\psi(\Omega^\omega+\Omega+\psi(\Omega^\omega+\Omega)))</math>
+* <math>f_4^{n\times2}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega\times2)</math>
+* <math>f_4^{n^2}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega\times\omega)</math>
+* <math>f_4^{f_3(n)}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega\times\psi(0))</math>
+* <math>f_5(f_\omega(n))=f_4^{f_\omega(n)}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega))</math>
+* <math>f_3(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)+1)</math>
+* <math>f_3^{f_\omega(n)}(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)+\psi(\Omega^\omega))</math>
+* <math>f_4(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)+\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)))</math>
+* <math>f_4^n(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)+\Omega)</math>
+* <math>f_4^{f_\omega(n)}(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)\times2)</math>
+* <math>f_4^{f_\omega(n)\times2}(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)\times3)</math>
+* <math>f_4^{f_3(f_\omega(n))}(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega+1))</math>
+* <math>f_4^{f_4(f_\omega(n))}(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega+\Omega))</math>
+* <math>f_5^2(f_\omega(n))=f_4^{f_5(f_\omega(n))}(f_5(f_\omega(n)))\sim\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega+\Omega\times\psi(\Omega^\omega)))</math>
+* <math>f_5^n(f_\omega(n))\approx (n\uparrow^nn)\uparrow\uparrow\uparrow\uparrow n\sim\psi(\Omega^\omega+\Omega^2)</math>
+* <math>f_5^{n\times2}(f_\omega(n))\sim\psi(\Omega^\omega+\Omega^2\times2)</math>
+* <math>f_6(f_\omega(n))\sim\psi(\Omega^\omega+\Omega^2\times\psi(\Omega^\omega))</math>
+* <math>f_6^n(f_\omega(n))\sim\psi(\Omega^\omega+\Omega^3)</math>
+* <math>f_7^n(f_\omega(n))\sim\psi(\Omega^\omega+\Omega^4)</math>
+* <math>f_n(f_\omega(n))\approx (n\uparrow^nn)\uparrow^nn\approx n\uparrow^n(2n)\sim\psi(\Omega^\omega\times2)</math>
+* <math>f_n^2(f_\omega(n))\approx n\uparrow^n(3n)\sim\psi(\Omega^\omega\times3)</math>
+* <math>f_n^n(f_\omega(n))\approx n\uparrow^n(n^2)\sim\psi(\Omega^\omega\times\omega)</math>
+* <math>f_n^{f_3(n)}(f_\omega(n))\approx n\uparrow^nn\uparrow\uparrow n\sim\psi(\Omega^\omega\times\psi(0))</math>
+* <math>f_{n+1}(f_\omega(n))=f_n^{f_\omega(n)}(f_\omega(n))\approx n\uparrow^nn\uparrow^nn\sim\psi(\Omega^\omega\times\psi(\Omega^\omega))</math>
+* <math>f_n(f_{n+1}(f_\omega(n)))\sim\psi(\Omega^\omega\times\psi(\Omega^\omega)+\Omega^\omega)</math>
+* <math>f_n^{f_\omega(n)}(f_{n+1}(f_\omega(n)))\sim\psi(\Omega^\omega\times\psi(\Omega^\omega)\times2)</math>
+* <math>f_n^{f_3(f_\omega(n))}(f_{n+1}(f_\omega(n)))\sim\psi(\Omega^\omega\times\psi(\Omega^\omega+1))</math>
+* <math>f_n^{f_n(f_\omega(n))}(f_{n+1}(f_\omega(n)))\sim\psi(\Omega^\omega\times\psi(\Omega^\omega\times2))</math>
+* <math>f_{n+1}^2(f_\omega(n))\approx n\uparrow^nn\uparrow^nn\uparrow^nn\sim\psi(\Omega^\omega\times\psi(\Omega^\omega\times\psi(\Omega^\omega)))</math>
+* <math>f_{n+1}^n(f_\omega(n))\approx n\uparrow^{n+1}n\sim\psi(\Omega^{\omega+1})</math>
+* <math>f_n(f_{n+1}^n(f_\omega(n)))\sim\psi(\Omega^{\omega+1}+\Omega^\omega)</math>
+* <math>f_{n+1}^{n+1}(f_\omega(n))=f_n^{f_{n+1}^n(f_\omega(n))}(f_{n+1}^n(f_\omega(n)))\sim\psi(\Omega^{\omega+1}+\Omega^\omega\times\psi(\Omega^{\omega+1}))</math>
+* <math>f_{n+1}^{n\times2}(f_\omega(n))\approx n\uparrow^{n+1}(2n)\sim\psi(\Omega^{\omega+1}\times2)</math>
+* <math>f_{n+2}(f_\omega(n))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega))</math>
+* <math>f_n(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)+\Omega^\omega)</math>
+* <math>f_n^{f_{n+1}^n(f_\omega(n))}(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)+\Omega^\omega\times\psi(\Omega^{\omega+1}))</math>
+* <math>f_{n+1}(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)+\Omega^\omega\times\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)))</math>
+* <math>f_{n+1}^n(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)+\Omega^{\omega+1})</math>
+* <math>f_{n+1}^{f_\omega(n)}(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)\times2)</math>
+* <math>f_{n+1}^{f_{n+1}(f_\omega(n))}(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega\times\psi(\Omega^\omega)))</math>
+* <math>f_{n+1}^{f_{n+1}^n(f_\omega(n))}(f_{n+2}(f_\omega(n)))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^{\omega+1}))</math>
+* <math>f_{n+2}^2(f_\omega(n))\sim\psi(\Omega^{\omega+1}\times\psi(\Omega^{\omega+1}\times\psi(\Omega^\omega)))</math>
+* <math>f_{n+2}^n(f_\omega(n))\sim\psi(\Omega^{\omega+2})</math>
+* <math>f_{n+2}^{n+1}(f_\omega(n))\sim\psi(\Omega^{\omega+2}+\Omega^{\omega+1}\times\psi(\Omega^\omega))</math>
+* <math>f_{n+2}^{n\times2}(f_\omega(n))\sim\psi(\Omega^{\omega+2}\times2)</math>
+* <math>f_{n+3}(f_\omega(n))\sim\psi(\Omega^{\omega+2}\times\psi(\Omega^\omega))</math>
+* <math>f_{n+3}^2(f_\omega(n))\sim\psi(\Omega^{\omega+2}\times\psi(\Omega^{\omega+2}\times\psi(\Omega^\omega)))</math>
+* <math>f_{n+3}^n(f_\omega(n))\sim\psi(\Omega^{\omega+3})</math>
+* <math>f_{n\times2}(f_\omega(n))\sim\psi(\Omega^{\omega\times2})</math>
+* <math>f_{n\times2+1}(f_\omega(n))\sim\psi(\Omega^{\omega\times2}\times\psi(\Omega^{\omega}))</math>
+* <math>f_{n\times2+1}^n(f_\omega(n))\sim\psi(\Omega^{\omega\times2+1})</math>
+* <math>f_{n\times3}(f_\omega(n))\sim\psi(\Omega^{\omega\times3})</math>
+* <math>f_{n^2}(f_\omega(n))\sim\psi(\Omega^{\omega^2})</math>
+* <math>f_{f_3(n)}(f_\omega(n))\approx n\uparrow^{n\uparrow\uparrow n}n\sim\psi(\Omega^{\psi(0)})</math>
+* <math>f_{f_4(n)}(f_\omega(n))\approx n\uparrow^{n\uparrow\uparrow\uparrow n}n\sim\psi(\Omega^{\psi(\Omega)})</math>
+* <math>f_\omega^2(n)=f_{f_\omega(n)}(f_\omega(n))\approx n\uparrow^{n\uparrow^n n}n\sim\psi(\Omega^{\psi(\Omega^\omega)})</math>
+* <math>f_n(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)}+\Omega^\omega)</math>
+* <math>f_{n+1}^n(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)}+\Omega^{\omega+1})</math>
+* <math>f_{f_\omega(n)}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)}\times2)</math>
+* <math>f_{f_\omega(n)}^n(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)}\times\omega)</math>
+* <math>f_{f_\omega(n)+1}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)}\times\psi(\Omega^{\psi(\Omega^\omega)}))</math>
+* <math>f_{f_\omega(n)+1}^n(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)+1})</math>
+* <math>f_{f_\omega(n)+n}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)+\omega})</math>
+* <math>f_{f_\omega(n)\times2}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega)\times2})</math>
+* <math>f_{f_3(f_\omega(n))}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^\omega+1)})</math>
+* <math>f_{f_{n+1}^n(f_\omega(n))}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^{\omega+1})})</math>
+* <math>f_\omega^3(n)=f_{f_{f_\omega(n)}(f_\omega(n))}(f_\omega^2(n))\sim\psi(\Omega^{\psi(\Omega^{\psi(\Omega^\omega)})})</math>
+* <math>f_{\omega+1}(n)=f_\omega^n(n)\sim\psi(\Omega^\Omega)=\varphi(1,0,0)</math>
-<
+=== Part 3 ===
+* <math>f_3(f_{\omega+1}(n))\sim\psi(\Omega^\Omega+1)</math>
+* <math>f_4(f_{\omega+1}(n))=f_3^{f_{\omega+1}(n)}(f_{\omega+1}(n))\sim\psi(\Omega^\Omega+\psi(\Omega^\Omega))</math>
+* <math>f_4^n(f_{\omega+1}(n))\sim\psi(\Omega^\Omega+\Omega)</math>
+* <math>f_5^n(f_{\omega+1}(n))\sim\psi(\Omega^\Omega+\Omega^2)</math>
+* <math>f_n(f_{\omega+1}(n))\sim\psi(\Omega^\Omega+\Omega^\omega)</math>
+* <math>f_{n+1}^n(f_{\omega+1}(n))\sim\psi(\Omega^\Omega+\Omega^{\omega+1})</math>
+* <math>f_{f_3(n)}(f_{\omega+1}(n))\sim\psi(\Omega^\Omega+\Omega^{\psi(0)})</math>
+* <math>f_{f_\omega(n)}(f_{\omega+1}(n))\sim\psi(\Omega^\Omega+\Omega^{\psi(\Omega^\omega)})</math>
+* <math>f_\omega(f_{\omega+1}(n))=f_{f_{\omega+1}(n)}(f_{\omega+1}(n))\sim\psi(\Omega^\Omega+\Omega^{\psi(\Omega^\Omega)})=\varphi(\varphi(1,0,0),1)</math>
+* <math>f_{f_{\omega+1}(n)}(f_{\omega}(f_{\omega+1}(n)))\sim\psi(\Omega^\Omega+\Omega^{\psi(\Omega^\Omega)}\times2)</math>
+* <math>f_{f_{\omega+1}(n)+1}^n(