User:Hyp cos/Taranovsky's various ordinal notations/Built-from-below with Passthrough

Here is the definition of Built-from-below with Passthrough, a special system among Taranovsky's ordinal notations.

Syntax
The system uses one constant: 0, and one ternary function: C.
 * 1) \(0\) is term
 * 2) If \(a\), \(b\) and \(c\) are terms, then \(C_a(b,c)\) is a term

Comparison
0 is the smallest term. \[C_a(b,c)b\rightarrow\exists y\sqsupseteq x(y<C_a(b,c)\lor\) \((y\neq x\land d(y)<a\land\forall z(x\sqsubseteq z\sqsubseteq y\rightarrow z\ge y))))\) (quantifiers range over terms with their positions in the syntax tree of b; \(d(y)< a\) means that y in standard term is Cu(v,w) where u < a)