Term Rewriting System R:
[Z, X, Y, X1, X2]
afst(0, Z) -> nil
afst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
afst(X1, X2) -> fst(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
alen(nil) -> 0
alen(cons(X, Z)) -> s(len(Z))
alen(X) -> len(X)
mark(fst(X1, X2)) -> afst(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(len(X)) -> alen(mark(X))
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

AFST(s(X), cons(Y, Z)) -> MARK(Y)
AFROM(X) -> MARK(X)
AADD(0, X) -> MARK(X)
MARK(fst(X1, X2)) -> AFST(mark(X1), mark(X2))
MARK(fst(X1, X2)) -> MARK(X1)
MARK(fst(X1, X2)) -> MARK(X2)
MARK(from(X)) -> AFROM(mark(X))
MARK(from(X)) -> MARK(X)
MARK(add(X1, X2)) -> AADD(mark(X1), mark(X2))
MARK(add(X1, X2)) -> MARK(X1)
MARK(add(X1, X2)) -> MARK(X2)
MARK(len(X)) -> ALEN(mark(X))
MARK(len(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(len(X)) -> MARK(X)
MARK(add(X1, X2)) -> MARK(X2)
MARK(add(X1, X2)) -> MARK(X1)
AADD(0, X) -> MARK(X)
MARK(add(X1, X2)) -> AADD(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
AFROM(X) -> MARK(X)
MARK(from(X)) -> AFROM(mark(X))
MARK(fst(X1, X2)) -> MARK(X2)
MARK(fst(X1, X2)) -> MARK(X1)
MARK(fst(X1, X2)) -> AFST(mark(X1), mark(X2))
AFST(s(X), cons(Y, Z)) -> MARK(Y)


Rules:


afst(0, Z) -> nil
afst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
afst(X1, X2) -> fst(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
alen(nil) -> 0
alen(cons(X, Z)) -> s(len(Z))
alen(X) -> len(X)
mark(fst(X1, X2)) -> afst(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(len(X)) -> alen(mark(X))
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:03 minutes