\(i\)を階数、\(A:\mathcal {U}(i)\)を型、\(B:A\to \mathcal {U}(i)\)を型の族、\(C:\prod _{x:A}B(x)\to \mathcal {U}(i)\)を型の族とする。

• -関数\(\mathord {\textnormal {\textsf {assoc}}}(C):(\sum _{z:\sum _{x:A}B(x)}C(\mathord {\textnormal {\textsf {proj}}}_{1}(z),\mathord {\textnormal {\textsf {proj}}}_{2}(z)))\to (\sum _{x:A}\sum _{y:B(x)}C(x,y))\)を\(\lambda w.\mathord {\textnormal {\textsf {pair}}}(\mathord {\textnormal {\textsf {proj}}}_{1}(\mathord {\textnormal {\textsf {proj}}}_{1}(w)),\mathord {\textnormal {\textsf {pair}}}(\mathord {\textnormal {\textsf {proj}}}_{2}(\mathord {\textnormal {\textsf {proj}}}_{1}(w)),\mathord {\textnormal {\textsf {proj}}}_{2}(w)))\)と定義する。
• -関数\(\mathord {\textnormal {\textsf {assoc}}}^{-1}(C):(\sum _{x:A}\sum _{y:B(x)}C(x,y))\to (\sum _{z:\sum _{x:A}B(x)}C(\mathord {\textnormal {\textsf {proj}}}_{1}(z),\mathord {\textnormal {\textsf {proj}}}_{2}(z)))\)を\(\lambda w.\mathord {\textnormal {\textsf {pair}}}(\mathord {\textnormal {\textsf {pair}}}(\mathord {\textnormal {\textsf {proj}}}_{1}(w),\mathord {\textnormal {\textsf {proj}}}_{1}(\mathord {\textnormal {\textsf {proj}}}_{2}(w))),\mathord {\textnormal {\textsf {proj}}}_{2}(\mathord {\textnormal {\textsf {proj}}}_{2}(w)))\)と定義する。