A:Type, f:(A(A + Top)), x:A, n:. (can-apply(f;x))  ((f^n+1(x)) ~ (f^n(do-apply(f;x)))) 

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ProofTree

Definitions	Type, t  T, Top, left + right, x:AB(x), , f(a), x:A. B(x), isl(x), b, P  Q, do-apply(f;x), can-apply(f;x), A  B, #$n, a < b, Void, False, A, {x:A| B(x)} , , x:A.B(x), f^n, i  j , s ~ t, -n, n+m, n - m, p-id(), f o g  , True, s = t, , b, , (i = j), x:A  B(x), P & Q, P  Q, Unit
Lemmas	eqtt to assert, eqff to assert, iff transitivity, assert of bnot, not functionality wrt iff, assert of eq int, eq int wf, bool wf, bnot wf, not wf, ge wf, nat properties, simple-primrec-add, le wf, assert wf, isl wf, nat wf, top wf