Veblen Function primed

In googology, Veblen function is one of the most popular that defines ordinal fundamental sequence. Here I want to redefine it into something more powerful.

1- Ordinal Arithmetic using Hyper Operators

1. For positive integer a and Ordinal O:
1. O《1》a = O+a
2. O《2》a = O*a
3. O《3》a = O↑a
4. O《4》a = O↑↑a
5. O《5》a = O↑↑↑a
6. And so on.
2. For positive integer b and ordinal P:
1. O《b+1》c+1 = (O《b+1》c)《b》O
2. O《b+1》P+c+1 = (O《b+1》P+c)《b》O
3. Other rules shall follow standard ordinal arithmetic and up arrow's rules.

2- One-variable

1. For non-negative integer c: φ'(c) = c+1.
2. φ'(0,0)
1. = ... φ'(φ'(φ'(0))) ...
2. = ω

3- Two-variable Veblen Function primed

1. The function is right associative.
2. φ'(0,c+1)
1. = φ'(0,c)《ω》ω
2. =｛φ'(0,c)《1》1, φ'(0,c)《2》2, φ'(0,c)《3》3, ...｝
3. =｛φ'(0,c)+1, φ'(0,c)*2, φ'(0,c)↑3, ...｝
3. φ'(1,0) = ... φ'(0,φ'(0,φ'(0,0))) ...
4. φ'(1,c+1) = φ'(1,c)《ω》ω.
5. φ'(1,O+c+1) = φ'(1,O+c)《ω》ω.
6. φ'(2,0) = ... φ'(1,φ'(1,φ'(1,0))) ...
7. Obviously the rest shall follow rule 2.1 to 2.5; and this shall continue for φ'(3,0), φ'(4,0) and so on.

4- SVO primed

1. For ending with array of 0's:
1. φ'(1,0,0) = ... φ'(φ'(φ'(1,0),0),0) ...
2. φ'(1,0,0,0) = ... φ'(φ'(φ'(1,0,0),0,0),0,0) ...
3. And so on.
2. For array of non-negative integers #:
1. φ'(#,c+1,0) = ... φ'(#,c,φ'(#,c,φ'(#,c,0))) ...
2. φ'(#,c+1,0,0) = ... φ'(#,c,φ'(#,c,φ'(#,c,0,0),0),0) ...
3. And so on.
3. φ'(#,c+1) = φ'(#,c)《ω》ω.
4. φ'(#,O+c+1) = φ'(#,O+c)《ω》ω.
5. This shall continue for the whole SVO'.

5- LVO primed

Thanks to p-bot and gamesfan for giving me the idea of LVO. Here I used @ to indicate number of 0's in the array after 1.

1. SVO'
   1. = φ'(1@φ'(0,0))
   2. =｛φ'(1@1), φ'(1@2). φ'(1@3), ...｝
   3. =｛φ'(1,0), φ'(1,0,0), φ'(1,0,0,0), ...｝
2. LVO'
   1. = ...φ'(1@φ'(1@φ'(0,0)))...
   2. =｛φ'(0,0), φ'(1@φ'(0,0)), φ'(1@φ'(1@φ'(0,0))), ...｝
3. φ'(1@O+c+1) = ...φ'(φ'(φ'(1@O+c)@O+c)@O+c)... , where O denotes an intermediate ordinal.

Note: This definition is also good for original SVO and LVO.