Sum type
Type
| context | $X, Y, Z$ … type |
| rule | ${\large\frac{a\ :\ X}{\nu_{\mathcal l}(a)\ :\ X+Y}}(+\mathcal l \mathcal I)$ |
| rule | ${\large\frac{b\ :\ Y}{\nu_{\mathcal r}(b)\ :\ X+Y}}(+\mathcal r \mathcal I)$ |
| rule | ${\large\frac{\Gamma,\ x\ :\ X\ \vdash\ f\ :\ Z\hspace{1cm}\Gamma,\ y\ :\ Y\ \vdash\ g\ :\ Z\hspace{1cm}c\ :\ X+Y}{\Gamma\ \vdash\ \left[\mathrm{\bf if}\,c\ \text{matches} \ \nu_{\mathcal l}(x)\ \text{do} \ f(x),\ \mathrm{\bf elseif}\, c\ \text{matches} \ \nu_{\mathcal r}(y)\ \text{do} \ g(y)\right]\ :\ Z}}(+\mathcal E)$ |
Discussion
Notation
The matching-term is generally written
$\left[\nu_{\mathcal l}x.f(x),\ \nu_{\mathcal r}y.g(y)\right](c),$
which is shorter but less suggestive.
Logic
Note how the rules reflect the logical “or”.
Element of
