How can I enumerate all simply typed lambda terms which have a specified type?

More precisely, suppose we have the simply typed lambda calculus augmented with numerals and iteration, as described in this answer. How can I enumerate all lambda terms of type N (natural number)?

For example, the first few lambda terms of type N are

zero
succ zero
succ (succ zero), K zero zero
succ (succ (succ zero)), K zero (suc zero), K (suc zero) zero, iter zero suc zero

and so on. How can I systematically continue this pattern, while ensuring that only well-typed terms are generated?

How can I enumerate all simply typed lambda terms which have a specified type?

More precisely, suppose we have the simply typed lambda calculus augmented with numerals and iteration, as described in this answer. How can I enumerate all lambda terms of type N (natural number)?

For example, the first few lambda terms of type N are

zero
succ zero
succ (succ zero), K zero zero
succ (succ (succ zero)), K zero (suc zero), K (suc zero) zero, iter zero suc zero

and so on. How can I systematically continue this pattern, while ensuring that only well-typed terms are generated?

How can I enumerate all simply typed lambda terms which have a specified type?

More precisely, supposed we have the simply typed lambda calculus augmented with numerals and iteration, as described in this answer. How can I enumerate all lambda terms of type ℕ (natural number)?

For example, the first few lambda terms of type ℕ are

zero
succ zero
succ (succ zero), K zero zero
succ (succ (succ zero)), K zero (suc zero), K (suc zero) zero, iter zero suc zero

and so on. How can I systematically continue this pattern, while ensuring that only well-typed terms are generated?

How can I enumerate all simply typed lambda terms which have a specified type?

More precisely, suppose we have the simply typed lambda calculus augmented with numerals and iteration, as described in this answer. How can I enumerate all lambda terms of type N (natural number)?

For example, the first few lambda terms of type N are

zero
succ zero
succ (succ zero), K zero zero
succ (succ (succ zero)), K zero (suc zero), K (suc zero) zero, iter zero suc zero

and so on. How can I systematically continue this pattern, while ensuring that only well-typed terms are generated?