RMI: Phi Growing Hierarchy

Consult the sheet or consult this.

1. Φf_n(x) = x[x]x = x[x+1]2

2. Φf_α(x) = Φf_α[x](x)

3. Φf_(n,0)(x) = Φf_n(x)

4. Φf_(n,m)(x) = Φf_(Φf_(n,m-1))()

5. Φf_(n...o,0)(x) = Φf_(n...o)

6. Φf_(n...o,m)(x) = Φf_(n...Φf_(n...o,m-1),0)(x)

7. Φf_(<n,m>) = Φf_(0,0...m...0,0)(x)    (a <#,#> is an optaparameter and yes, optaparameters can use ordinals)

8. Φf_(<10,1>)(x) = Φf_(0,0,0,0,0,0,0,0,0,1)(x)

9. Φf_(<5,1><10,1>)(x) = Φf_(0,0,0,0,1,0,0,0,0,1)(x)

10. Φf_(<n,m,l>)(x) = Φf_(<n[1],m[1]>...<n[l],m[l]>)(x)

11. Φf_(<n,m,l,o>)(x) = Φf_(<n[o],m[o],l[o]>...<n[o],m[o],l[o]>)(x)