Termination Proof using AProVE

Term Rewriting System R:
[N, X, Y, X1, X2, XS]
fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, n_{_fib1}(Y, n_{_add}(X, Y)))
fib1(X1, X2) -> n_{_fib1}(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
add(X1, X2) -> n_{_add}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_fib1}(X1, X2)) -> fib1(activate(X1), activate(X2))
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

FIB(N) -> SEL(N, fib1(s(0), s(0)))
FIB(N) -> FIB1(s(0), s(0))
ADD(s(X), Y) -> ADD(X, Y)
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(n_{_fib1}(X1, X2)) -> FIB1(activate(X1), activate(X2))
ACTIVATE(n_{_fib1}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_fib1}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(activate(X1), activate(X2))
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X2)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

Dependency Pair:

ADD(s(X), Y) -> ADD(X, Y)

Rules:

fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, n_{_fib1}(Y, n_{_add}(X, Y)))
fib1(X1, X2) -> n_{_fib1}(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
add(X1, X2) -> n_{_add}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_fib1}(X1, X2)) -> fib1(activate(X1), activate(X2))
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ADD(s(X), Y) -> ADD(X, Y)

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

ADD(x₁, x₂) -> ADD(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 4

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

Dependency Pair:

Rules:

fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, n_{_fib1}(Y, n_{_add}(X, Y)))
fib1(X1, X2) -> n_{_fib1}(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
add(X1, X2) -> n_{_add}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_fib1}(X1, X2)) -> fib1(activate(X1), activate(X2))
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_fib1}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_fib1}(X1, X2)) -> ACTIVATE(X1)

Rules:

fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, n_{_fib1}(Y, n_{_add}(X, Y)))
fib1(X1, X2) -> n_{_fib1}(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
add(X1, X2) -> n_{_add}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_fib1}(X1, X2)) -> fib1(activate(X1), activate(X2))
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_fib1}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_fib1}(X1, X2)) -> ACTIVATE(X1)

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

ACTIVATE(x₁) -> ACTIVATE(x₁)
n_{_fib1}(x₁, x₂) -> n_{_fib1}(x₁, x₂)
n_{_add}(x₁, x₂) -> n_{_add}(x₁, x₂)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 5

             ↳Dependency Graph

       →DP Problem 3

         ↳Nar

Dependency Pair:

Rules:

fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, n_{_fib1}(Y, n_{_add}(X, Y)))
fib1(X1, X2) -> n_{_fib1}(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
add(X1, X2) -> n_{_add}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_fib1}(X1, X2)) -> fib1(activate(X1), activate(X2))
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Narrowing Transformation

Dependency Pair:

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))

Rules:

fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, n_{_fib1}(Y, n_{_add}(X, Y)))
fib1(X1, X2) -> n_{_fib1}(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
add(X1, X2) -> n_{_add}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_fib1}(X1, X2)) -> fib1(activate(X1), activate(X2))
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(X) -> X

Strategy:

innermost

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

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))

three new Dependency Pairs are created:

SEL(s(N), cons(X, n_{_fib1}(X1', X2'))) -> SEL(N, fib1(activate(X1'), activate(X2')))
SEL(s(N), cons(X, n_{_add}(X1', X2'))) -> SEL(N, add(activate(X1'), activate(X2')))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Forward Instantiation Transformation

Dependency Pairs:

SEL(s(N), cons(X, XS')) -> SEL(N, XS')
SEL(s(N), cons(X, n_{_add}(X1', X2'))) -> SEL(N, add(activate(X1'), activate(X2')))
SEL(s(N), cons(X, n_{_fib1}(X1', X2'))) -> SEL(N, fib1(activate(X1'), activate(X2')))

Rules:

fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, n_{_fib1}(Y, n_{_add}(X, Y)))
fib1(X1, X2) -> n_{_fib1}(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
add(X1, X2) -> n_{_add}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_fib1}(X1, X2)) -> fib1(activate(X1), activate(X2))
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(X) -> X

Strategy:

innermost

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

SEL(s(N), cons(X, XS')) -> SEL(N, XS')

three new Dependency Pairs are created:

SEL(s(s(N'')), cons(X, cons(X'', n_{_fib1}(X1''', X2''')))) -> SEL(s(N''), cons(X'', n_{_fib1}(X1''', X2''')))
SEL(s(s(N'')), cons(X, cons(X'', n_{_add}(X1''', X2''')))) -> SEL(s(N''), cons(X'', n_{_add}(X1''', X2''')))
SEL(s(s(N'')), cons(X, cons(X'', XS'''))) -> SEL(s(N''), cons(X'', XS'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳FwdInst

...

               →DP Problem 7

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

SEL(s(s(N'')), cons(X, cons(X'', XS'''))) -> SEL(s(N''), cons(X'', XS'''))
SEL(s(s(N'')), cons(X, cons(X'', n_{_add}(X1''', X2''')))) -> SEL(s(N''), cons(X'', n_{_add}(X1''', X2''')))
SEL(s(s(N'')), cons(X, cons(X'', n_{_fib1}(X1''', X2''')))) -> SEL(s(N''), cons(X'', n_{_fib1}(X1''', X2''')))
SEL(s(N), cons(X, n_{_fib1}(X1', X2'))) -> SEL(N, fib1(activate(X1'), activate(X2')))
SEL(s(N), cons(X, n_{_add}(X1', X2'))) -> SEL(N, add(activate(X1'), activate(X2')))

Rules:

fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, n_{_fib1}(Y, n_{_add}(X, Y)))
fib1(X1, X2) -> n_{_fib1}(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
add(X1, X2) -> n_{_add}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_fib1}(X1, X2)) -> fib1(activate(X1), activate(X2))
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(X) -> X

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:09 minutes