Termination Proof using AProVE

Term Rewriting System R:
[N, X, Y, X1, X2, Z]
terms(N) -> cons(recip(sqr(N)), n_{_terms}(s(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(n_{_add}(sqr(activate(X)), dbl(activate(X))))
dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
add(0, X) -> X
add(s(X), Y) -> s(n_{_add}(activate(X), Y))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

TERMS(N) -> SQR(N)
TERMS(N) -> S(N)
SQR(s(X)) -> S(n_{_add}(sqr(activate(X)), dbl(activate(X))))
SQR(s(X)) -> SQR(activate(X))
SQR(s(X)) -> ACTIVATE(X)
SQR(s(X)) -> DBL(activate(X))
DBL(s(X)) -> S(n_{_s}(n_{_dbl}(activate(X))))
DBL(s(X)) -> ACTIVATE(X)
ADD(s(X), Y) -> S(n_{_add}(activate(X), Y))
ADD(s(X), Y) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_terms}(X)) -> TERMS(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(X1, X2)
ACTIVATE(n_{_s}(X)) -> S(X)
ACTIVATE(n_{_dbl}(X)) -> DBL(X)
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

SQR(s(X)) -> DBL(activate(X))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(X)) -> DBL(X)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(X1, X2)
ACTIVATE(n_{_terms}(X)) -> TERMS(X)
SQR(s(X)) -> ACTIVATE(X)
SQR(s(X)) -> SQR(activate(X))
TERMS(N) -> SQR(N)

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(s(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(n_{_add}(sqr(activate(X)), dbl(activate(X))))
dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
add(0, X) -> X
add(s(X), Y) -> s(n_{_add}(activate(X), Y))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(X) -> X

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

SQR(s(X)) -> SQR(activate(X))

six new Dependency Pairs are created:

SQR(s(n_{_terms}(X''))) -> SQR(terms(X''))
SQR(s(n_{_add}(X1', X2'))) -> SQR(add(X1', X2'))
SQR(s(n_{_s}(X''))) -> SQR(s(X''))
SQR(s(n_{_dbl}(X''))) -> SQR(dbl(X''))
SQR(s(n_{_first}(X1', X2'))) -> SQR(first(X1', X2'))
SQR(s(X'')) -> SQR(X'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

SQR(s(X'')) -> SQR(X'')
SQR(s(n_{_first}(X1', X2'))) -> SQR(first(X1', X2'))
SQR(s(n_{_dbl}(X''))) -> SQR(dbl(X''))
SQR(s(n_{_s}(X''))) -> SQR(s(X''))
SQR(s(n_{_add}(X1', X2'))) -> SQR(add(X1', X2'))
SQR(s(n_{_terms}(X''))) -> SQR(terms(X''))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)
ACTIVATE(n_{_dbl}(X)) -> DBL(X)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(X1, X2)
SQR(s(X)) -> ACTIVATE(X)
TERMS(N) -> SQR(N)
ACTIVATE(n_{_terms}(X)) -> TERMS(X)
DBL(s(X)) -> ACTIVATE(X)
SQR(s(X)) -> DBL(activate(X))

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(s(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(n_{_add}(sqr(activate(X)), dbl(activate(X))))
dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
add(0, X) -> X
add(s(X), Y) -> s(n_{_add}(activate(X), Y))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(X) -> X

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

SQR(s(X)) -> DBL(activate(X))

six new Dependency Pairs are created:

SQR(s(n_{_terms}(X''))) -> DBL(terms(X''))
SQR(s(n_{_add}(X1', X2'))) -> DBL(add(X1', X2'))
SQR(s(n_{_s}(X''))) -> DBL(s(X''))
SQR(s(n_{_dbl}(X''))) -> DBL(dbl(X''))
SQR(s(n_{_first}(X1', X2'))) -> DBL(first(X1', X2'))
SQR(s(X'')) -> DBL(X'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

SQR(s(X'')) -> DBL(X'')
SQR(s(n_{_first}(X1', X2'))) -> DBL(first(X1', X2'))
SQR(s(n_{_dbl}(X''))) -> DBL(dbl(X''))
SQR(s(n_{_s}(X''))) -> DBL(s(X''))
SQR(s(n_{_add}(X1', X2'))) -> DBL(add(X1', X2'))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)
ACTIVATE(n_{_dbl}(X)) -> DBL(X)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(X1, X2)
DBL(s(X)) -> ACTIVATE(X)
SQR(s(n_{_terms}(X''))) -> DBL(terms(X''))
SQR(s(n_{_first}(X1', X2'))) -> SQR(first(X1', X2'))
SQR(s(n_{_dbl}(X''))) -> SQR(dbl(X''))
SQR(s(n_{_s}(X''))) -> SQR(s(X''))
SQR(s(n_{_add}(X1', X2'))) -> SQR(add(X1', X2'))
SQR(s(n_{_terms}(X''))) -> SQR(terms(X''))
TERMS(N) -> SQR(N)
ACTIVATE(n_{_terms}(X)) -> TERMS(X)
SQR(s(X)) -> ACTIVATE(X)
SQR(s(X'')) -> SQR(X'')

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(s(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(n_{_add}(sqr(activate(X)), dbl(activate(X))))
dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
add(0, X) -> X
add(s(X), Y) -> s(n_{_add}(activate(X), Y))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(X) -> X

Termination of R could not be shown.
Duration:
0:01 minutes