Shadow and Injection

Definition¹

For $n \in \mathbb{N}$ and $0 \le i \le n$, the following function $p_{i}$

$$ \begin{align*} p_{i} : &\left\{ 0, 1 \right\}^{n+1} \to \left\{ 0, 1 \right\}^{n} \\ & (a_{0}, \dots, a_{n}) \mapsto (a_{0}, \dots, a_{i-1}, a_{i+1}, \dots, a_{n}) \end{align*} $$

is called a projection. The following two functions $\imath_{i}$, $\jmath_{i}$

$$ \begin{align*} \imath : &\left\{ 0, 1 \right\}^{n} \to \left\{ 0, 1 \right\}^{n+1} \\ & (a_{0}, \dots, a_{n-1}) \mapsto (a_{0}, \dots, a_{i-1}, 0, a_{i+1}, \dots, a_{n-1}) \end{align*} $$

$$ \begin{align*} \jmath : &\left\{ 0, 1 \right\}^{n} \to \left\{ 0, 1 \right\}^{n+1} \\ & (a_{0}, \dots, a_{n-1}) \mapsto (a_{0}, \dots, a_{i-1}, 1, a_{i+1}, \dots, a_{n-1}) \end{align*} $$

are called injections.

Explanation

A projection is a mapping that deletes the $i$th truth value, and an injection is a mapping that pushes the $i$th truth value back and adds $0$ or $1$ in its place. Since these can actually be implemented by discarding or adding wires in circuits, it is assumed that they can be used without restrictions in proofs or theoretical developments.