ČN

emnv oewi wvis view rvi  IE #Iq f9bq87B8ne0qnp WIP (could also be called ChN)

Č notation

$$\check{C}(a)$$ = b a-ated to c+c a-ated to b

b = a a-ated to a a-1ated to a a-2ated to a...a+a

c = a+a*a^a...a a-ated to a

$$\check{C}(a) \approx f_{\omega}^3(a)$$

$$\check{C}([a]) = \check{C}^a(a)$$iterated a times

$$\check{C}([a]) \approx f_{\omega2}(a)$$

$$\check{C}([a][0]) = \check{C}^a([a])$$iterated a times

$$\check{C}([a][0]) \approx f_{\omega3}(a)$$

$$\check{C}([a][b+1]) = \check{C}^a([a][b])$$iterated a times

$$\check{C}([a][a]) \approx f_{\omega^2}(a)$$

Now, define A as nesting on a a times. Also define ! as a [a]s and $ as a+1 [a]s.

$$\check{C}($) = \check{C}(![A])$$

$$\check{C}([a]/[0]) = \check{C}(!)$$

$$\check{C}([a]/[0]) \approx f_{\omega^\omega}(a)$$

Define the same rules for before the /.

$$\check{C}([a]/[b+1]) = \check{C}(!/[b])$$

$$\check{C}([a]/[a]) \approx f_{\omega^{\omega+1}}(a)$$

There isn't any twist to the rules here.

$$\check{C}([a]//[0]) = \check{C}([a]/[a]/[a]\dots[a]/[a]/[a])$$

Define $$!_2$$as a /s and $$$_2$$as a+1 /s.

$$\check{C}([a]$_2[0]) = \check{C}([a]!_2[a]!_2[a]\dots[a]!_2[a]!_2[a])$$a times

$$\check{C}([a]/_2[0]) = \check{C}([a]!_2[a])$$

$$\check{C}([a]/_2[0]) \approx f_{\omega^{\omega+2}}(a)$$

Define $$!_{2~a}$$as a $$/_a$$s.

$$\check{C}([a]/_{b+1}[0]) = \check{C}([a]!_{2~b}[a])$$

Note: you  can  have things like $$/_{a+1}/_a$$.

$$\check{C}([a]/_{a}[0]) \approx f_{\omega^{\omega2}}(a)$$

$$\check{C}([a]/_{[0]}[0]) = \check{C}([a]/_{A}[0])$$

$$\check{C}([a]/_{[0]}[0]) \approx f_{\omega^{\omega2}+1}(a)$$

Use the same rules.

Define $$A_2$$as like A but it uses the notation to nest.

$$\check{C}([a]\backslash[0]) = \check{C}([a]/_{A_2}[0])$$

$$\check{C}([a]\backslash[0]) \approx f_{\varepsilon_0}(a)$$

$$B_0 = /, B_1 = \backslash, etc.$$

$$\check{C}([a]\lozenge[0]) = \check{C}([a]B_a[0])$$

$$\check{C}([a]\lozenge[0]) \approx f_{\varepsilon_\omega}(a)$$