Type theory with records

A record type is a set of fields. A field is a pair consisting of a label and a type. Within a record type, field labels are unique. The witness of a record type is a record. A record is a similar set of fields, but fields contain objects instead of types. The object in each field must be of the type declared in the corresponding field in the record type.^[3]

Basic type: ${\displaystyle {\begin{bmatrix}{\text{x}}:Ind\end{bmatrix}}}$

Object: ${\displaystyle {\begin{bmatrix}{\text{x}}=a\end{bmatrix}}}$

Ptype: ${\displaystyle \left[{\begin{array}{lll}{\text{x}}&:&Ind\\{\text{c}}_{\text{boy}}&:&boy({\text{x}})\\{\text{y}}&:&Ind\\{\text{c}}_{\text{dog}}&:&dog({\text{y}})\\{\text{c}}_{\text{hug}}&:&hug(x,y)\end{array}}\right]}$

Object: ${\displaystyle \left[{\begin{array}{lll}{\text{x}}&=&a\\{\text{c}}_{\text{boy}}&=&p_{1}\\{\text{y}}&=&b\\{\text{c}}_{\text{dog}}&=&p_{2}\\{\text{c}}_{\text{hug}}&=&p_{3}\end{array}}\right]}$

where ${\displaystyle a}$ and ${\displaystyle b}$ are individuals (type ${\displaystyle Ind}$ ), ${\displaystyle p_{1}}$ is proof that ${\displaystyle a}$ is a boy, etc.