If you have been following the exercises you should have an almost complete encoding of Ola’s syntax and typing rules. Some extra exercises follow:
Add Type Annotations and Products/Pairs
Extend your mechanisation to include the following data structures and operations:
Type Annotations
Type annotations, ensuring that an expression has a specific type
e := (the t e) ...
g |- e : t
---- [ Annotations ]
g |- (the t e) : tProduct Types
Pairs with two values, together with primitives to access the first and second element;
t := (t,t) ...
e := (e,e) | fst e | snd e ...g |- l : t_{l}
g |- r : t_{r}
---- [ New Pair ]
g |- (l,r) : (t_{l}, t_{r})g |- e : (t_{l},t_{r})
---- [ First ]
g |- fst e : t_{l}g |- e : (t_{l},t_{r})
---- [ Second ]
g |- snd e : t_{r}Sum Types
Extend your mechanisation further to support sums:
t := <t,t> ...
e := Left e | Right e ...
s := match e { Left xref => s; Right xref => s }g |- e : t_{l}
---- [ Left ]
g |- e : <t_{l},t_{r}>g |- e : t_{r}
---- [ Right ]
g |- e : <t_{l},t_{r}>g |- e : <t_{l},t_{r}>
g, (lref, t_{l}) |- l : t
g, (rref, t_{r}) |- r : t
---- [ Match ]
g |- match e { Left lref => l; Right rref => r }