Definition:Bourbaki Assembly

Definition
The notion of assembly is an example of a collation used in.

An assembly is a succession of signs written one after another.

Certain signs which are not variables are allowed to be joined in pairs by links, as follows:
 * $\overbrace {\tau A \Box}^{} A'$

Concatenation
Let $\mathbf A$ and $\mathbf B$ denote assemblies.

Then $\mathbf {A B}$ denotes the assembly formed from writing the assembly $\mathbf B$ immediately to the right of assembly $\mathbf A$.

Extracting the Variable
Let $\mathbf A$ denote an assembly and let $x$ be a letter.

Then the assembly $\map {\tau_x} {\mathbf A}$ is constructed as follows:
 * $(1): \quad$ Form the assembly $\tau \mathbf A$
 * $(2): \quad$ Link each occurrence of $x$ wherever it appears in $\mathbf A$ to the $\tau$ written to the left of $\mathbf A$
 * $(3): \quad$ Replace $x$ by $\Box$ wherever it occurs in $\mathbf A$.

The assembly denoted $\map {\tau_x} {\mathbf A}$ therefore does not contain $x$.

Replacement
Let $\mathbf A$ and $\mathbf B$ denote assemblies.

We denote by:
 * $\paren {\mathbf B \mid x} \mathbf A$

the assembly obtained by replacing $x$, wherever it occurs, by $\mathbf B$.

This is read::
 * $\mathbf B$ replaces $x$ in $\mathbf A$.

If $x$ does not appear in $\mathbf A$, then $\paren {\mathbf B \mid x} \mathbf A$ is identical with $\mathbf A$.

In particular:
 * $\paren {\mathbf B \mid x} \map {\tau_x} {\mathbf A}$

is identical with $\map {\tau_x} {\mathbf A}$.