Termination Proof using AProVE

Term Rewriting System R:
[Z, X, Y, X1, X2]
a_{_fst}(0, Z) -> nil
a_{_fst}(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
a_{_fst}(X1, X2) -> fst(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_add}(0, X) -> mark(X)
a_{_add}(s(X), Y) -> s(add(X, Y))
a_{_add}(X1, X2) -> add(X1, X2)
a_{_len}(nil) -> 0
a_{_len}(cons(X, Z)) -> s(len(Z))
a_{_len}(X) -> len(X)
mark(fst(X1, X2)) -> a_{_fst}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(add(X1, X2)) -> a_{_add}(mark(X1), mark(X2))
mark(len(X)) -> a_{_len}(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.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_FST}(s(X), cons(Y, Z)) -> MARK(Y)
A_{_FROM}(X) -> MARK(X)
A_{_ADD}(0, X) -> MARK(X)
MARK(fst(X1, X2)) -> A_{_FST}(mark(X1), mark(X2))
MARK(fst(X1, X2)) -> MARK(X1)
MARK(fst(X1, X2)) -> MARK(X2)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(from(X)) -> MARK(X)
MARK(add(X1, X2)) -> A_{_ADD}(mark(X1), mark(X2))
MARK(add(X1, X2)) -> MARK(X1)
MARK(add(X1, X2)) -> MARK(X2)
MARK(len(X)) -> A_{_LEN}(mark(X))
MARK(len(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)

Furthermore, R contains one SCC.

     ↳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)
A_{_ADD}(0, X) -> MARK(X)
MARK(add(X1, X2)) -> A_{_ADD}(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(fst(X1, X2)) -> MARK(X2)
MARK(fst(X1, X2)) -> MARK(X1)
MARK(fst(X1, X2)) -> A_{_FST}(mark(X1), mark(X2))
A_{_FST}(s(X), cons(Y, Z)) -> MARK(Y)

Rules:

a_{_fst}(0, Z) -> nil
a_{_fst}(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
a_{_fst}(X1, X2) -> fst(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_add}(0, X) -> mark(X)
a_{_add}(s(X), Y) -> s(add(X, Y))
a_{_add}(X1, X2) -> add(X1, X2)
a_{_len}(nil) -> 0
a_{_len}(cons(X, Z)) -> s(len(Z))
a_{_len}(X) -> len(X)
mark(fst(X1, X2)) -> a_{_fst}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(add(X1, X2)) -> a_{_add}(mark(X1), mark(X2))
mark(len(X)) -> a_{_len}(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