Termination Proof using AProVE

Term Rewriting System R:
[N, X, Y, X1, X2, XS]
a_{_fib}(N) -> a_{_sel}(mark(N), a_{_fib1}(s(0), s(0)))
a_{_fib}(X) -> fib(X)
a_{_fib1}(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
a_{_fib1}(X1, X2) -> fib1(X1, X2)
a_{_add}(0, X) -> mark(X)
a_{_add}(s(X), Y) -> s(a_{_add}(mark(X), mark(Y)))
a_{_add}(X1, X2) -> add(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(fib(X)) -> a_{_fib}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(fib1(X1, X2)) -> a_{_fib1}(mark(X1), mark(X2))
mark(add(X1, X2)) -> a_{_add}(mark(X1), mark(X2))
mark(s(X)) -> s(mark(X))
mark(0) -> 0
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_{_FIB}(N) -> A_{_SEL}(mark(N), a_{_fib1}(s(0), s(0)))
A_{_FIB}(N) -> MARK(N)
A_{_FIB}(N) -> A_{_FIB1}(s(0), s(0))
A_{_FIB1}(X, Y) -> MARK(X)
A_{_ADD}(0, X) -> MARK(X)
A_{_ADD}(s(X), Y) -> A_{_ADD}(mark(X), mark(Y))
A_{_ADD}(s(X), Y) -> MARK(X)
A_{_ADD}(s(X), Y) -> MARK(Y)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), mark(XS))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
MARK(fib(X)) -> A_{_FIB}(mark(X))
MARK(fib(X)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(fib1(X1, X2)) -> A_{_FIB1}(mark(X1), mark(X2))
MARK(fib1(X1, X2)) -> MARK(X1)
MARK(fib1(X1, X2)) -> MARK(X2)
MARK(add(X1, X2)) -> A_{_ADD}(mark(X1), mark(X2))
MARK(add(X1, X2)) -> MARK(X1)
MARK(add(X1, X2)) -> MARK(X2)
MARK(s(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:

A_{_FIB}(N) -> A_{_FIB1}(s(0), s(0))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
A_{_ADD}(s(X), Y) -> MARK(Y)
A_{_ADD}(s(X), Y) -> MARK(X)
A_{_ADD}(s(X), Y) -> A_{_ADD}(mark(X), mark(Y))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(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(fib1(X1, X2)) -> MARK(X2)
MARK(fib1(X1, X2)) -> MARK(X1)
A_{_FIB1}(X, Y) -> MARK(X)
MARK(fib1(X1, X2)) -> A_{_FIB1}(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), mark(XS))
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(fib(X)) -> MARK(X)
A_{_FIB}(N) -> MARK(N)
MARK(fib(X)) -> A_{_FIB}(mark(X))
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
A_{_FIB}(N) -> A_{_SEL}(mark(N), a_{_fib1}(s(0), s(0)))

Rules:

a_{_fib}(N) -> a_{_sel}(mark(N), a_{_fib1}(s(0), s(0)))
a_{_fib}(X) -> fib(X)
a_{_fib1}(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
a_{_fib1}(X1, X2) -> fib1(X1, X2)
a_{_add}(0, X) -> mark(X)
a_{_add}(s(X), Y) -> s(a_{_add}(mark(X), mark(Y)))
a_{_add}(X1, X2) -> add(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(fib(X)) -> a_{_fib}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(fib1(X1, X2)) -> a_{_fib1}(mark(X1), mark(X2))
mark(add(X1, X2)) -> a_{_add}(mark(X1), mark(X2))
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(cons(X1, X2)) -> cons(mark(X1), X2)

Strategy:

innermost

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