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)

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

         ↳Narrowing Transformation

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)

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

A_{_FIB}(N) -> A_{_SEL}(mark(N), a_{_fib1}(s(0), s(0)))

nine new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

A_{_FIB}(N) -> A_{_SEL}(mark(N), cons(mark(s(0)), fib1(s(0), add(s(0), s(0)))))
A_{_FIB}(0) -> A_{_SEL}(0, a_{_fib1}(s(0), s(0)))
A_{_FIB}(s(X')) -> A_{_SEL}(s(mark(X')), a_{_fib1}(s(0), s(0)))
A_{_FIB}(add(X1', X2')) -> A_{_SEL}(a_{_add}(mark(X1'), mark(X2')), a_{_fib1}(s(0), s(0)))
A_{_FIB}(fib1(X1', X2')) -> A_{_SEL}(a_{_fib1}(mark(X1'), mark(X2')), a_{_fib1}(s(0), s(0)))
A_{_FIB}(sel(X1', X2')) -> A_{_SEL}(a_{_sel}(mark(X1'), mark(X2')), a_{_fib1}(s(0), s(0)))
A_{_FIB}(fib(X')) -> A_{_SEL}(a_{_fib}(mark(X')), a_{_fib1}(s(0), s(0)))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), 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)
MARK(fib1(X1, X2)) -> A_{_FIB1}(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
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_{_FIB1}(X, Y) -> MARK(X)
A_{_FIB}(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)

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

A_{_ADD}(s(X), Y) -> A_{_ADD}(mark(X), mark(Y))

14 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Narrowing Transformation

Dependency Pairs:

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)

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), mark(XS))

14 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Narrowing Transformation

Dependency Pairs:

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)

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))

14 new Dependency Pairs are created:

MARK(sel(fib(X'), X2)) -> A_{_SEL}(a_{_fib}(mark(X')), mark(X2))
MARK(sel(sel(X1'', X2''), X2)) -> A_{_SEL}(a_{_sel}(mark(X1''), mark(X2'')), mark(X2))
MARK(sel(fib1(X1'', X2''), X2)) -> A_{_SEL}(a_{_fib1}(mark(X1''), mark(X2'')), mark(X2))
MARK(sel(add(X1'', X2''), X2)) -> A_{_SEL}(a_{_add}(mark(X1''), mark(X2'')), mark(X2))
MARK(sel(s(X'), X2)) -> A_{_SEL}(s(mark(X')), mark(X2))
MARK(sel(0, X2)) -> A_{_SEL}(0, mark(X2))
MARK(sel(cons(X1'', X2''), X2)) -> A_{_SEL}(cons(mark(X1''), X2''), mark(X2))
MARK(sel(X1, fib(X'))) -> A_{_SEL}(mark(X1), a_{_fib}(mark(X')))
MARK(sel(X1, sel(X1'', X2''))) -> A_{_SEL}(mark(X1), a_{_sel}(mark(X1''), mark(X2'')))
MARK(sel(X1, fib1(X1'', X2''))) -> A_{_SEL}(mark(X1), a_{_fib1}(mark(X1''), mark(X2'')))
MARK(sel(X1, add(X1'', X2''))) -> A_{_SEL}(mark(X1), a_{_add}(mark(X1''), mark(X2'')))
MARK(sel(X1, s(X'))) -> A_{_SEL}(mark(X1), s(mark(X')))
MARK(sel(X1, 0)) -> A_{_SEL}(mark(X1), 0)
MARK(sel(X1, cons(X1'', X2''))) -> A_{_SEL}(mark(X1), cons(mark(X1''), X2''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

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

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(add(X1, X2)) -> A_{_ADD}(mark(X1), mark(X2))

14 new Dependency Pairs are created:

MARK(add(fib(X'), X2)) -> A_{_ADD}(a_{_fib}(mark(X')), mark(X2))
MARK(add(sel(X1'', X2''), X2)) -> A_{_ADD}(a_{_sel}(mark(X1''), mark(X2'')), mark(X2))
MARK(add(fib1(X1'', X2''), X2)) -> A_{_ADD}(a_{_fib1}(mark(X1''), mark(X2'')), mark(X2))
MARK(add(add(X1'', X2''), X2)) -> A_{_ADD}(a_{_add}(mark(X1''), mark(X2'')), mark(X2))
MARK(add(s(X'), X2)) -> A_{_ADD}(s(mark(X')), mark(X2))
MARK(add(0, X2)) -> A_{_ADD}(0, mark(X2))
MARK(add(cons(X1'', X2''), X2)) -> A_{_ADD}(cons(mark(X1''), X2''), mark(X2))
MARK(add(X1, fib(X'))) -> A_{_ADD}(mark(X1), a_{_fib}(mark(X')))
MARK(add(X1, sel(X1'', X2''))) -> A_{_ADD}(mark(X1), a_{_sel}(mark(X1''), mark(X2'')))
MARK(add(X1, fib1(X1'', X2''))) -> A_{_ADD}(mark(X1), a_{_fib1}(mark(X1''), mark(X2'')))
MARK(add(X1, add(X1'', X2''))) -> A_{_ADD}(mark(X1), a_{_add}(mark(X1''), mark(X2'')))
MARK(add(X1, s(X'))) -> A_{_ADD}(mark(X1), s(mark(X')))
MARK(add(X1, 0)) -> A_{_ADD}(mark(X1), 0)
MARK(add(X1, cons(X1'', X2''))) -> A_{_ADD}(mark(X1), cons(mark(X1''), X2''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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

The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes