System F i

Lambda encodings in type theory

“…Induction principle is not derivable in extensions of System F such as Calculus of Construcitons (CC) [20]. 3. Computing type from data is not possible with Church encoding.…”

“…Since F ω 's types do not have dependency on terms, P : Nat → * will get erased to P : * . It is known that one can also erase the implicit product [3]. The erasure of Nat will be ∀P : * .…”

“…We point out that 0 = 1 is derivable with this notion. Our main contribution is for problem (3). We adapt CC to support dependent elimination with Church or Parigot encodings, using a novel type construct called self types, ιx.T , to express dependency of a type on its subject.…”

“…Observe that the lambda term for the proof is Church numberal zero λs.λz.z 3. The lambda term for this proof is Church successor λn.λs.λz.s(n s z).…”