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I-notation (or |-notation) is a notation for large numbers by Denis Maksudov[1]. The creator stated that it reaches $$\psi_0(\Omega_{\omega})$$ with respect to Buchholz's function in fast-growing hierarchy without a proof.[2]

## Definition

The notation consists of a natural number, followed by the symbol "|" (vertical bar, possibly also writen as "I", a capital letter i), then a sequence of natural numbers and parentheses, which are in turn followed by subscripts consisting of natural numbers.

Let $$a$$ be a natural number. Here are the rules for expansion:

1. $$a|\ =\ a+1$$
2. $$a|b()\ =\ a\underbrace{|b|b\cdots |b|b}_{a\textrm{ copies of }|"}$$, where $$b$$ is the rest of the expression
3. $$a|b(c())_de\ =\ a|b\underbrace{(c)_d(c)_d\cdots (c)_d (c)_d}_{a\textrm{ copies of }(c)_d"}e$$, where $$d$$ is a natural number, $$b$$ and $$c$$ are parts of the expression that can include any parentheses with any subscript, and $$e$$ is a part of the expression that includes only closing parentheses with any subscript.
4. $$a\vert b(c()_{d+1}e)_fg=a\vert b(c(c(\cdots(c\underbrace{()_de)_d\cdots e)_de)_d}_{a\;d\textrm{'s}}e)_fg$$, where $$a$$, $$d$$, $$f$$ are natural numbers; $$b$$, $$c$$ can include left and right parentheses with any subscripts; $$g$$ can include right parentheses with any subscripts; $$e$$ can include only right parentheses with subscript $$d+1$$ or more; $$d+1>f$$

Note: if some pair of brackets is not enclosed in any other then subscript of this pair should be zero, for example $$(()_2()_2)_0(()_1()_0)_0()_0$$. We use $$(...)$$ as an abbreviation for $$(...)_0$$.