Physlib.Relativity.Tensors.ComponentIdx.Product

Let $c$ and $c_1$ be sequences of tensor colors of lengths $n_1$ and $n_2$, respectively. Let $p = (p_0, \dots, p_{n_1-1})$ be a component index (multi-index) for $c$, and let $q = (q_0, \dots, q_{n_2-1})$ be a component index for $c_1$. The equivalence `ComponentIdx.prod` identifies the pair $(p, q)$ with a component index $b$ for the concatenated color sequence $c \mathbin{+\!+} c_1$. This theorem states that for any index $i \in \{0, \dots, n_2-1\}$, the value of the concatenated component index $b$ at the shifted position $n_1 + i$ is equal to $q_i$.

Let $c: \{0, \dots, n_1-1\} \to C$ and $c_1: \{0, \dots, n_2-1\} \to C$ be sequences of index colors. Let $p$ and $q$ be component indices (multi-indices) for $c$ and $c_1$, respectively. The product equivalence provides a map $\text{prod.symm}$ that takes the pair $(p, q)$ and returns a combined multi-index for the concatenated color sequence $c \mathbin{+\!+} c_1$. This theorem states that for any index $i \in \{0, \dots, n_1-1\}$, the value of this combined multi-index at position $i$ is equal to the $i$-th component of $p$.