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# General Edit

## Non-confusing definition Edit

First, TON, short for Taranovsky's ordinal notation, is divided into an infinite number of systems, namely the 1st system, the 2nd system, the 3rd system, etc. Expressions in TON look like this: C(α,β). The nth system uses only a few symbols, namely "C", "(", ")", ",", "0", "Ωn". Note that unlike in other notations, Ωn is not the nth uncountable. It isn't even an ordinal, though it could be treated as an ordinal fixed point. It's just a symbol.

The easiest thing about TON is comparing the size of two ordinals. To do this, we use the postfix form. To compare ordinals this way, first delete all of the "("s, the ")"s, and the ","s. Then reverse the string. Now, compare them in alphabetical order, where the "alphabet" here is C0Ωn. For example, to confirm that C(C(0,0),C(C(0,C(0,0)),0))=ω2+ω is smaller than C(C(0,C(0,C(0,C(0,0)))),0)=ω4, we write the postfix forms:

000C0CC00CC

000C0C0C0CC

Now we see that the second is bigger.

Next, we define the standard form of an ordinal. Note that "standard" is an understatement, since we only treat ordinals in standard form as real ordinals.

Now, of course, 0 and Ωn are ordinals. Also, obviously, for C(α,β) to be an ordinal, α and β have to be ordinals.

The second rule is that if the β equals C(γ,δ), γ≥α. Thus, if a ordinal is written as C(α1,C(α2,...C(αk-1,C(αk,0 or Ω_n))...)), α1≤α2≤...≤αk-1≤αk. This rule makes things like C(C(0,0),C(0,0)) non-standard.

The third rule is the hardest to understand. First, we have to understand what is the "subterm". Basically, the subterm of some expression is a part of it. Formally this could be expressed as:

• η is a subterm of η;
• if η=C(α,β), then the subterms of α and β are the subterms of η.

Now we define "n-built from below from":

• α is 0-built from below from <β means that α<β;
• α is (k+1)-built from below from <β means that: for every subterm γ of α,
• γ≤α, or
• there exists a δ "between" γ and α (i.e. γ is a subterm of δ, which is a subterm of α) such that δ is k-built from below from β.

Finally, we can express the third rule! It is: α is n-built from below from <C(α,β).

Now, we've defined the standard form! But there's one more thing before we can fully define TON. This one is easy:

C(α1,C(α2,...C(αk-1,C(αkn))...))=Ωnαkαk-1+...+ωα2α1

We can combine all of the systems into a notation like this: Ωk=C(Ωk+1,0).

That's it!

## Explanation Edit

Now, you might say: "You haven't actually defined the notation. You only gave the definition of the standard form!" Well, this is the full definition. Remember that all ordinals can be expressed in TON. Therefore, there is a one-to-one correspondence:

TON ordinals (the real ones, not things like C(0,Ω1)=Ω1+1) <--> Normal (countable) ordinals (up to the limit of TON)

Since we can compare the size of TON ordinals, we can "pack" them to correspond them to the normal ordinals.

Think of it this way. TON is like an infinite library. Each book in the TON library has a title with the alphabet "C0Ωn", which is the postfix form of a TON ordinal. The library only accepts standard books. The books are ordered in their alphabetical order. Now, if the library does all of this, then the books will each correspond to its order in the library, its ordinal. No reordering of books is possible, nor will any new book fit in.

# Analysis Edit

To make things simple, Ω is for Ωn here.

## Before First System Edit

Since there is no specific system for this, pretend that this is just the 1st system.

Before I start the table, notice that for expressions without Ωs, any ordinal α will be larger than all of its subterms, so the third rule of the standard form doesn't matter.

Simple form Standard form Postfix form Normal ordinals
0 0 0 0
C(0,0) C(0,0) 00C 1
C(0,1) C(0,C(0,0)) 00C0C 2
C(0,2) C(0,C(0,C(0,0))) 00C0C0C 3
C(1,0) C(C(0,0),0) 000CC ω
C(0,ω) C(0,C(C(0,0),0)) 000CC0C ω+1
C(0,ω+1) C(0,C(0,C(C(0,0),0))) 000CC0C0C ω+2
C(1,ω) C(C(0,0),C(C(0,0),0)) 000CC00CC ω2
C(0,ω2) C(0,C(C(0,0),C(C(0,0),0))) 000CC00CC0C ω2+1
C(1,ω2) C(C(0,0),C(C(0,0),C(C(0,0),0))) 000CC00CC00CC ω3
C(1,ω3) C(C(0,0),C(C(0,0),C(C(0,0),C(C(0,0),0)))) 000CC00CC00CC00CC ω4
C(2,0) C(C(0,C(0,0)),0) 000C0CC ω2
C(0,ω2) C(0,C(C(0,C(0,0)),0)) 000C0CC0C ω2+1
C(1,ω2) C(C(0,0),C(C(0,C(0,0)),0)) 000C0CC00CC ω2
C(2,ω2) C(C(0,C(0,0)),C(C(0,C(0,0)),0)) 000C0CC00C0CC ω22
C(3,0) C(C(0,C(0,C(0,0))),0) 000C0C0CC ω3
C(3,ω3) C(C(0,C(0,C(0,0))),C(C(0,C(0,C(0,0))),0)) 000C0C0CC00C0C0CC ω32
C(4,0) C(C(0,C(0,C(0,C(0,0)))),0) 000C0C0C0CC ω4
C(5,0) C(C(0,C(0,C(0,C(0,C(0,0))))),0) 000C0C0C0C0CC ω5
C(ω,0) C(C(C(0,0),0),0) 0000CCC ωω
C(0,ωω) C(0,C(C(C(0,0),0),0)) 0000CCC0C ωω+1
C(1,ωω) C(C(0,0),C(C(C(0,0),0),0)) 0000CCC00CC ωω
C(ω,ωω) C(C(C(0,0),0),C(C(C(0,0),0),0)) 0000CCC000CCC ωω2
C(ω+1,0) C(C(0,C(C(0,0),0)),0) 0000CC0CC ωω+1
C(ω+2,0) C(C(0,C(0,C(C(0,0),0))),0) 0000CC0C0CC ωω+2
C(ω2,0) C(C(C(0,0),C(C(0,0),0)),0) 0000CC00CCC ωω2
C(ω2,0) C(C(C(0,C(0,0)),0),0) 0000C0CCC ωω2
C(ω3,0) C(C(C(0,C(0,C(0,0))),0),0) 0000C0C0CCC ωω3
C(ωω,0) C(C(C(C(0,0),0),0),0) 00000CCCC ωωω
C(ωωω,0) C(C(C(C(C(0,0),0),0),0),0) 000000CCCCC ωωωω

Turns out, if α<ε0, then C(α,β)=β+ωα. Actually, this addition principle works for larger α too, but you have to be careful about standardness.

## First SystemEdit

### 0th System??Edit

The 0th system is a joke system proposed by Hyp cos in this page's talk page. It can exist theoretically, but we never really use it when we are talking about TON. It just uses "0-built from below from <", and satisfies Ω0=C(Ω1,0). Therefore, Ω00. The 0th system extends from ε0 to ε1.

Simple form Standard form Postfix form Normal ordinals
C(Ω,0) C(Ω,0) 0ΩC ψ(0)
C(0,ψ(0)) C(0,C(Ω,0)) 0ΩC0C ψ(0)+1
C(1,ψ(0)) C(C(0,0),C(Ω,0)) 0ΩC00CC ψ(0)+ω
C(ω,ψ(0)) C(C(C(0,0),0),C(Ω,0)) 0ΩC000CCC ψ(0)+ωω
C(ψ(0),ψ(0)) C(C(Ω,0),C(Ω,0)) 0ΩC0ΩCC ψ(0)2
C(ψ(0),ψ(0)2) C(C(Ω,0),C(C(Ω,0),C(Ω,0))) 0ΩC0ΩCC0ΩCC ψ(0)3
C(ψ(0)+1,ψ(0)) C(C(0,C(Ω,0)),C(Ω,0)) 0ΩC0ΩC0CC ψ(0)ω
C(ψ(0)+2,ψ(0)) C(C(0,C(0,C(Ω,0))),C(Ω,0)) 0ΩC0ΩC0C0CC ψ(0)ω2
C(ψ(0)+ω,ψ(0)) C(C(C(0,0),C(Ω,0)),C(Ω,0)) 0ΩC0ΩC00CCC ψ(0)ωω
C(ψ(0)+ωω,ψ(0) C(C(C(C(0,0),0),C(Ω,0)),C(Ω,0)) 0ΩC0ΩC000CCCC ψ(0)ωωω
C(ψ(0)2,ψ(0)) C(C(C(Ω,0),C(Ω,0)),C(Ω,0)) 0ΩC0ΩC0ΩCCC ψ(0)2
C(ψ(0)3,ψ(0)) C(C(C(Ω,0),C(C(Ω,0),C(Ω,0))),C(Ω,0)) 0ΩC0ΩC0ΩCC0ΩCCC ψ(0)3
C(ψ(0)ω,ψ(0)) C(C(C(0,C(Ω,0)),C(Ω,0)),C(Ω,0)) 0ΩC0ΩC0ΩC0CCC ψ(0)ω
C(ψ(0)ωω,ψ(0)) C(C(C(C(0,0),C(Ω,0)),C(Ω,0)),C(Ω,0)) 0ΩC0ΩC0ΩC00CCCC ψ(0)ωω
C(ψ(0)2,ψ(0)) C(C(C(C(Ω,0),C(Ω,0)),C(Ω,0)),C(Ω,0)) 0ΩC0ΩC0ΩC0ΩCCCC ψ(0)ψ(0)
C(ψ(0)3,ψ(0)) C(C(C(C(Ω,0),C(C(Ω,0),C(Ω,0))),C(Ω,0)),C(Ω,0)) 0ΩC0ΩC0ΩC0ΩCC0ΩCCCC ψ(0)ψ(0)2
C(ψ(0)ω,ψ(0)) C(C(C(C(0,C(Ω,0)),C(Ω,0)),C(Ω,0)),C(Ω,0)) 0ΩC0ΩC0ΩC0ΩC0CCCC ψ(0)ψ(0)ω
C(ψ(0)ψ(0),ψ(0)) C(C(C(C(C(Ω,0),C(Ω,0)),C(Ω,0)),C(Ω,0)),C(Ω,0)) 0ΩC0ΩC0ΩC0ΩC0ΩCCCCC ψ(0)ψ(0)ψ(0)
C(ψ(0)ψ(0)ψ(0),ψ(0)) C(C(C(C(C(C(Ω,0),C(Ω,0)),C(Ω,0)),C(Ω,0)),C(Ω,0)),C(Ω,0)) 0ΩC0ΩC0ΩC0ΩC0ΩC0ΩCCCCCC ψ(0)ψ(0)ψ(0)ψ(0)
C(Ω,ψ(0)) C(Ω,C(Ω,0)) 0ΩCΩC ψ(1)
C(ψ(1),ψ(1)) C(C(Ω,C(Ω,0)),C(Ω,C(Ω,0))) 0ΩCΩC0ΩCΩCC ψ(1)2
C(ψ(1)+1,ψ(1)) C(0,C(Ω,C(Ω,0))),C(Ω,C(Ω,0))) 0ΩCΩC0ΩCΩC0CC ψ(1)ω
C(ψ(1)+ψ(0),ψ(1)) C(C(C(Ω,0),C(Ω,C(Ω,0))),C(Ω,C(Ω,0))) 0ΩCΩC0ΩCΩC0ΩCCC ψ(1)ψ(0)
C(ψ(1)2,ψ(1)) C(C(C(Ω,C(Ω,0)),C(Ω,C(Ω,0))),C(Ω,C(Ω,0))) 0ΩCΩC0ΩCΩC0ΩCΩCCC ψ(1)2
C(ψ(1)ω,ψ(1)) C(C(C(0,C(Ω,C(Ω,0))),C(Ω,C(Ω,0))),C(Ω,C(Ω,0))) 0ΩCΩC0ΩCΩC0ΩCΩC0CCC ψ(1)ω
C(ψ(1)2,ψ(1)) C(C(C(C(Ω,C(Ω,0)),C(Ω,C(Ω,0))),C(Ω,C(Ω,0))),C(Ω,C(Ω,0))) 0ΩCΩC0ΩCΩC0ΩCΩC0ΩCΩCCCC ψ(1)ψ(1)
C(Ω,ψ(1)) C(Ω,C(Ω,C(Ω,0))) 0ΩCΩCΩC ψ(2)
C(Ω,ψ(2)) C(Ω,C(Ω,C(Ω,C(Ω,0)))) 0ΩCΩCΩCΩC ψ(3)
C(Ω+1,0) C(C(0,Ω),0) 0Ω0CC ψ(ω)
C(ψ(ω)+1,ψ(ω)) C(C(0,C(C(0,Ω),0)),C(C(0,Ω),0)) 0Ω0CC0Ω0CC0CC ψ(ω)ω
C(ψ(ω)ω,ψ(ω)) C(C(C(0,C(C(0,Ω),0)),C(C(0,Ω),0)),C(C(0,Ω),0)) 0Ω0CC0Ω0CC0Ω0CC0CCC ψ(ω)ω
C(Ω,ψ(ω)) C(Ω,C(C(0,Ω),0)) 0Ω0CCΩC ψ(ω+1)
C(Ω,ψ(ω+1)) C(Ω,C(Ω,C(C(0,Ω),0))) 0Ω0CCΩCΩC ψ(ω+2)
C(Ω+1,ψ(ω)) C(C(0,Ω),C(C(0,Ω),0)) 0Ω0CCΩ0CC ψ(ω2)
C(Ω+1,ψ(ω2)) C(C(0,Ω),C(C(0,Ω),C(C(0,Ω),0))) 0Ω0CCΩ0CCΩ0CC ψ(ω3)
C(Ω+2,0) C(C(0,C(0,Ω)),0) 0Ω0C0CC ψ(ω2)
C(Ω+ω,0) C(C(C(0,0),Ω),0) 0Ω00CCC ψ(ωω)
C(Ω+ωω,0) C(C(C(C(0,0),0),Ω),0) 0Ω000CCCC ψ(ωωω)
C(Ω+ψ(0),0) C(C(C(Ω,0),Ω),0) 0Ω0ΩCCC ψ(ψ(0))
C(Ω+ψ(1),0) C(C(C(Ω,C(Ω,0)),Ω),0) 0Ω0ΩCΩCCC ψ(ψ(1))
C(Ω+ψ(ω),0) C(C(C(C(0,Ω),0),Ω),0) 0Ω0Ω0CCCC ψ(ψ(ω))
C(Ω+ψ(ψ(0)),0) C(C(C(C(C(Ω,0),Ω),0),Ω),0) 0Ω0Ω0ΩCCCCC ψ(ψ(ψ(0)))
C(Ω+ψ(ψ(ψ(0))),0) C(C(C(C(C(C(C(Ω,0),Ω),0),Ω),0),Ω),0) 0Ω0Ω0Ω0ΩCCCCCCC ψ(ψ(ψ(ψ(0))))
C(Ω2,0) C(C(Ω,Ω),0) 0ΩΩCC ψ(Ω)
C(Ω2,ψ(Ω)) C(C(Ω,Ω),C(C(Ω,Ω),0)) 0ΩΩCCΩΩCC ψ(Ω2)
C(Ω3,0) C(C(Ω,C(Ω,Ω)),0) 0ΩΩCΩCC ψ(Ω2)
C(Ωω,0) C(C(C(0,Ω),Ω),0) 0ΩΩ0CCC ψ(Ωω)
C(Ωψ(0),0) C(C(C(C(Ω,0),Ω),Ω),0) 0ΩΩ0ΩCCCC ψ(Ωψ(0))
C(Ωψ(Ω),0) C(C(C(C(C(Ω,Ω),0),Ω),Ω),0) 0ΩΩ0ΩΩCCCCC ψ(Ωψ(Ω))
C(Ωψ(Ωψ(Ω)),0) C(C(C(C(C(C(C(C(Ω,Ω),0),Ω),Ω),0),Ω),Ω),0) 0ΩΩ0ΩΩ0ΩΩCCCCCCCC ψ(Ωψ(Ωψ(Ω)))
C(Ω2,0) C(C(C(Ω,Ω),Ω),0) 0ΩΩΩCCC ψ(ΩΩ)
C(Ω2+1,0) C(C(0,C(C(Ω,Ω),Ω)),0) 0ΩΩΩCC0CC ψ(ΩΩω)
C(Ω2+ψ(ΩΩ),0) C(C(C(C(C(Ω,Ω),Ω),0),C(C(Ω,Ω),Ω)),0) 0ΩΩΩCC0ΩΩΩCCCCC ψ(ΩΩψ(ΩΩ))
C(Ω2+Ω,0) C(C(Ω,C(C(Ω,Ω),Ω)),0) 0ΩΩΩCCΩCC ψ(ΩΩ+1)
C(Ω2+Ωψ(ΩΩ),0) C(C(C(C(C(C(Ω,Ω),Ω),0),Ω),C(C(Ω,Ω),Ω)),0) 0ΩΩΩCCΩ0ΩΩΩCCCCCC ψ(ΩΩ+ψ(ΩΩ))
C(Ω22,0) C(C(C(Ω,Ω),C(C(Ω,Ω),Ω)),0) 0ΩΩΩCCΩΩCCC ψ(ΩΩ2)
C(Ω23,0) C(C(C(Ω,Ω),C(C(Ω,Ω),C(C(Ω,Ω),Ω))),0) 0ΩΩΩCCΩΩCCΩΩCCC ψ(ΩΩ3)
C(Ω2ω,0) C(C(C(0,C(Ω,Ω)),Ω),0) 0ΩΩΩC0CCC ψ(ΩΩω)
C(Ω2ψ(ΩΩ),0) C(C(C(C(C(C(Ω,Ω),Ω),0),C(Ω,Ω)),Ω),0) 0ΩΩΩC0ΩΩΩCCCCCC ψ(ΩΩψ(ΩΩ))
C(Ω2ψ(ΩΩψ(ΩΩ)),0) C(C(C(C(C(C(C(C(C(Ω,Ω),Ω),0),C(Ω,Ω)),Ω),0),C(Ω,Ω)),Ω),0) 0ΩΩΩC0ΩΩΩC0ΩΩΩCCCCCCCCC ψ(ΩΩψ(ΩΩψ(ΩΩ)))
C(Ω3,0) C(C(C(Ω,C(Ω,Ω)),Ω),0) 0ΩΩΩCΩCCC ψ(ΩΩ2)
C(Ω4,0) C(C(C(Ω,C(Ω,C(Ω,Ω))),Ω),0) 0ΩΩΩCΩCΩCCC ψ(ΩΩ3)
C(Ωω,0) C(C(C(C(0,Ω),Ω),Ω),0) 0ΩΩΩ0CCCC ψ(ΩΩω)
C(Ωψ(ΩΩω),0) C(C(C(C(C(C(C(C(0,Ω),Ω),Ω),0),Ω),Ω),Ω),0) 0ΩΩΩ0ΩΩΩ0CCCCCCCC ψ(ΩΩψ(ΩΩω))
C(ΩΩ,0) C(C(C(C(Ω,Ω),Ω),Ω),0) 0ΩΩΩΩCCCC ψ(ΩΩΩ)
C(ΩΩ2,0) C(C(C(C(Ω,C(Ω,Ω)),Ω),Ω),0) 0ΩΩΩΩCΩCCCC ψ(ΩΩΩ2)
C(ΩΩω,0) C(C(C(C(C(0,Ω),Ω),Ω),Ω),0) 0ΩΩΩΩ0CCCCC ψ(ΩΩΩω)
C(ΩΩΩ,0) C(C(C(C(C(Ω,Ω),Ω),Ω),Ω),0) 0ΩΩΩΩΩCCCCC ψ(ΩΩΩΩ)
C(ΩΩΩΩ,0) C(C(C(C(C(C(Ω,Ω),Ω),Ω),Ω),Ω),0) 0ΩΩΩΩΩΩCCCCCC ψ(ΩΩΩΩΩ)

New pattern: C(Ω,α)=ωα. Also, notice that Ω in the OCF works similarly to Ω1 in TON. Again, watch out for standardness!

## Second SystemEdit

In the 2nd system, C(Ω2,0) acts like Ω1.

### Up to C(C(Ω22,0),0)Edit

So, what's the limit if we replace every Ω1 with C(Ω2,0), 0Ω2C in the postfix form?

Simple form Standard form Postfix form Normal ordinals
C(Ω1,0) C(C(Ω,0),0) 00ΩCC ψ(0)
C(Ω12,0) C(C(C(Ω,0),C(Ω,0)),0) 00ΩC0ΩCCC ψ(Ω)
C(Ω12,0) C(C(C(C(Ω,0),C(Ω,0)),C(Ω,0)),0) 00ΩC0ΩC0ΩCCCC ψ(ΩΩ)
C(Ω1Ω1,0) C(C(C(C(C(Ω,0),C(Ω,0)),C(Ω,0)),C(Ω,0)),0) 00ΩC0ΩC0ΩC0ΩCCCCC ψ(ΩΩΩ)
C(C(Ω21),0) C(C(Ω,C(Ω,0)),0) 00ΩCΩCC ψ(ψΩ2(0))
C(C(Ω21),ψ(ψΩ2(0))) C(C(Ω,C(Ω,0)),C(C(Ω,C(Ω,0)),0)) 00ΩCΩCC0ΩCΩCC ψ(ψΩ2(0)2)
C(C(Ω21)+1,0) C(C(0,C(Ω,C(Ω,0))),0) 00ΩCΩC0CC ψ(ψΩ2(0)ω)
C(C(Ω21)+Ω1,0) C(C(C(Ω,0),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩCCC ψ(ψΩ2(0)Ω)
C(C(Ω21)+Ω12,0) C(C(C(C(Ω,0),C(Ω,0)),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩC0ΩCCCC ψ(ψΩ2(0)Ω2)
C(C(Ω21)+Ω12,0) C(C(C(C(C(Ω,0),C(Ω,0)),C(Ω,0)),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩC0ΩC0ΩCCCCC ψ(ψΩ2(0)ΩΩ)
C(C(Ω21)+Ω1Ω1,0) C(C(C(C(C(C(Ω,0),C(Ω,0)),C(Ω,0)),C(Ω,0)),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩC0ΩC0ΩC0ΩCCCCCC ψ(ψΩ2(0)ΩΩΩ)
C(C(Ω21)+εΩ1+1,0) C(C(C(C(Ω,C(Ω,0)),C(Ω,0)),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩC0ΩCΩCCCC ψ(ψΩ2(0)2)
C(C(Ω21)+εΩ1+12,0) C(C(C(C(Ω,C(Ω,0)),C(Ω,0)),C(C(C(Ω,C(Ω,0)),C(Ω,0)),C(Ω,C(Ω,0)))),0) 00ΩCΩC0ΩC0ΩCΩCCC0ΩC0ΩCΩCCCC ψ(ψΩ2(0)3)
C(C(Ω21)+εΩ1+1ω,0) C(C(C(0,C(C(Ω,C(Ω,0)),C(Ω,0))),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩC0ΩCΩCC0CCC ψ(ψΩ2(0)ω)
C(C(Ω21)+εΩ1+1Ω1,0) C(C(C(C(Ω,0),C(C(Ω,C(Ω,0)),C(Ω,0))),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩC0ΩCΩCC0ΩCCCC ψ(ψΩ2(0)Ω)
C(C(Ω21)+εΩ1+12,0) C(C(C(C(C(Ω,C(Ω,0)),C(Ω,0)),C(C(Ω,C(Ω,0)),C(Ω,0))),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩC0ΩCΩCC0ΩC0ΩCΩCCCCC ψ(ψΩ2(0)ψΩ2(0))
C(C(Ω21)+εΩ1+2,0) C(C(C(C(Ω,C(Ω,0)),C(C(Ω,C(Ω,0)),C(Ω,0))),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩC0ΩCΩCC0ΩCΩCCCC ψ(ψΩ2(1))
C(C(Ω21)+εΩ1+3,0) C(C(C(C(Ω,C(Ω,0)),C(C(Ω,C(Ω,0)),C(C(Ω,C(Ω,0)),C(Ω,0)))),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩC0ΩCΩCC0ΩCΩCC0ΩCΩCCCC ψ(ψΩ2(2))
C(C(Ω21)+εΩ1,0) C(C(C(C(0,C(Ω,C(Ω,0))),C(Ω,0)),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩC0ΩCΩC0CCCC ψ(ψΩ2(ω))
C(C(Ω21)+εεΩ1+1,0) C(C(C(C(C(C(Ω,C(Ω,0)),C(Ω,0)),C(Ω,C(Ω,0))),C(Ω,0)),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩC0ΩCΩC0ΩC0ΩCΩCCCCCC ψ(ψΩ2Ω2(0)))
C(C(Ω21)2,0) C(C(C(Ω,C(Ω,0)),C(Ω,C(Ω,0))),0) 00ΩCΩC0ΩCΩCCC ψ(Ω2)
C(C(Ω21)2+1,0) C(C(0,C(C(Ω,C(Ω,0)),C(Ω,C(Ω,0)))),0) 00ΩCΩC0ΩCΩCC0CC ψ(Ω2Ω22)ω)
C(C(Ω21)2+εΩ1+1,0) C(C(C(C(Ω,C(Ω,0)),C(Ω,0)),C(C(Ω,C(Ω,0)),C(Ω,C(Ω,0)))),0) 00ΩCΩC0ΩCΩCC0ΩC0ΩCΩCCCC ψ(Ω2Ω22Ω2(0))

testing 123
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