Section type_reification.
Inductive term :Type :=
Fun : term -> term -> term
| Prod : term -> term -> term
| Bool : term
| SET :term
| PROP :term
| TYPE :term
| Var : Type -> term.
Fixpoint interp (t:term) :=
match t with
Bool => bool
| SET => Set
| PROP => Prop
| TYPE => Type
| Fun a b => interp a -> interp b
| Prod a b => interp a * interp b
| Var x => x
end.
Record interp_pair :Type :=
{ repr:>term;
abs:>Type;
link: abs = interp repr }.

Lemma fun_interp :forall (a b:interp_pair), (a -> b) = interp (Fun a b).