Termination Proof using AProVE

Term Rewriting System R:
[N, X, Y, Z, X1, X2]
terms(N) -> cons(recip(sqr(N)), n_{_terms}(s(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
activate(n_{_terms}(X)) -> terms(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)
SQR(s(X)) -> ADD(sqr(X), dbl(X))
SQR(s(X)) -> SQR(X)
SQR(s(X)) -> DBL(X)
DBL(s(X)) -> DBL(X)
ADD(s(X), Y) -> ADD(X, Y)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
HALF(s(s(X))) -> HALF(X)
ACTIVATE(n_{_terms}(X)) -> TERMS(X)
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)

Furthermore, R contains five SCCs.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

Dependency Pair:

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

Rules:

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

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{first, nil, n_{_first}} > {terms, activate} > n_{_terms}
{first, nil, n_{_first}} > {terms, activate} > cons
{first, nil, n_{_first}} > {terms, activate} > recip
{first, nil, n_{_first}} > {terms, activate} > sqr > add > {half, s}
{first, nil, n_{_first}} > {terms, activate} > sqr > dbl > {half, s}

resulting in one new DP problem.
Used Argument Filtering System:

ADD(x₁, x₂) -> ADD(x₁, x₂)
s(x₁) -> s(x₁)
terms(x₁) -> terms(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)
recip(x₁) -> recip(x₁)
n_{_terms}(x₁) -> n_{_terms}(x₁)
sqr(x₁) -> sqr(x₁)
add(x₁, x₂) -> add(x₁, x₂)
dbl(x₁) -> dbl(x₁)
first(x₁, x₂) -> first(x₁, x₂)
n_{_first}(x₁, x₂) -> n_{_first}(x₁, x₂)
activate(x₁) -> activate(x₁)
half(x₁) -> half(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 6

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

Dependency Pair:

Rules:

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

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

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

Dependency Pair:

DBL(s(X)) -> DBL(X)

Rules:

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

The following dependency pair can be strictly oriented:

DBL(s(X)) -> DBL(X)

The following rules can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{0, nil}
{first, n_{_first}} > {terms, activate} > n_{_terms}
{first, n_{_first}} > {terms, activate} > {dbl, sqr} > add > s
{first, n_{_first}} > {terms, activate} > cons
{first, n_{_first}} > {terms, activate} > recip
half > s

resulting in one new DP problem.
Used Argument Filtering System:

DBL(x₁) -> DBL(x₁)
s(x₁) -> s(x₁)
terms(x₁) -> terms(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)
recip(x₁) -> recip(x₁)
n_{_terms}(x₁) -> n_{_terms}(x₁)
sqr(x₁) -> sqr(x₁)
add(x₁, x₂) -> add(x₁, x₂)
dbl(x₁) -> dbl(x₁)
first(x₁, x₂) -> first(x₁, x₂)
n_{_first}(x₁, x₂) -> n_{_first}(x₁, x₂)
activate(x₁) -> activate(x₁)
half(x₁) -> half(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 7

             ↳Dependency Graph

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

Dependency Pair:

HALF(s(s(X))) -> HALF(X)

Rules:

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

The following dependency pair can be strictly oriented:

HALF(s(s(X))) -> HALF(X)

The following rules can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

0 > nil
{first, n_{_first}} > activate > terms > n_{_terms} > nil
{first, n_{_first}} > activate > terms > cons > nil
{first, n_{_first}} > activate > terms > recip > nil
{first, n_{_first}} > activate > terms > sqr > add > {half, s} > nil
{first, n_{_first}} > activate > terms > sqr > dbl > {half, s} > nil
HALF > nil

resulting in one new DP problem.
Used Argument Filtering System:

HALF(x₁) -> HALF(x₁)
s(x₁) -> s(x₁)
terms(x₁) -> terms(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)
recip(x₁) -> recip(x₁)
n_{_terms}(x₁) -> n_{_terms}(x₁)
sqr(x₁) -> sqr(x₁)
add(x₁, x₂) -> add(x₁, x₂)
dbl(x₁) -> dbl(x₁)
first(x₁, x₂) -> first(x₁, x₂)
n_{_first}(x₁, x₂) -> n_{_first}(x₁, x₂)
activate(x₁) -> activate(x₁)
half(x₁) -> half(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

           →DP Problem 8

             ↳Dependency Graph

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Argument Filtering and Ordering

       →DP Problem 5

         ↳AFS

Dependency Pair:

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

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

SQR > nil
0 > nil
{first, n_{_first}} > {terms, activate} > n_{_terms} > nil
{first, n_{_first}} > {terms, activate} > {dbl, sqr} > add > s > half > nil
{first, n_{_first}} > {terms, activate} > cons > nil
{first, n_{_first}} > {terms, activate} > recip > nil

resulting in one new DP problem.
Used Argument Filtering System:

SQR(x₁) -> SQR(x₁)
s(x₁) -> s(x₁)
terms(x₁) -> terms(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)
recip(x₁) -> recip(x₁)
n_{_terms}(x₁) -> n_{_terms}(x₁)
sqr(x₁) -> sqr(x₁)
add(x₁, x₂) -> add(x₁, x₂)
dbl(x₁) -> dbl(x₁)
first(x₁, x₂) -> first(x₁, x₂)
n_{_first}(x₁, x₂) -> n_{_first}(x₁, x₂)
activate(x₁) -> activate(x₁)
half(x₁) -> half(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

           →DP Problem 9

             ↳Dependency Graph

       →DP Problem 5

         ↳AFS

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳Argument Filtering and Ordering

Dependency Pairs:

ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

Rules:

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

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

The following rules can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

0 > nil
{first, FIRST, n_{_first}} > activate > terms > n_{_terms} > nil
{first, FIRST, n_{_first}} > activate > terms > cons > nil
{first, FIRST, n_{_first}} > activate > terms > recip > nil
{first, FIRST, n_{_first}} > activate > terms > sqr > add > {half, s} > nil
{first, FIRST, n_{_first}} > activate > terms > sqr > dbl > {half, s} > nil
{first, FIRST, n_{_first}} > ACTIVATE > nil

resulting in one new DP problem.
Used Argument Filtering System:

ACTIVATE(x₁) -> ACTIVATE(x₁)
FIRST(x₁, x₂) -> FIRST(x₁, x₂)
n_{_first}(x₁, x₂) -> n_{_first}(x₁, x₂)
s(x₁) -> s(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)
terms(x₁) -> terms(x₁)
recip(x₁) -> recip(x₁)
n_{_terms}(x₁) -> n_{_terms}(x₁)
sqr(x₁) -> sqr(x₁)
add(x₁, x₂) -> add(x₁, x₂)
dbl(x₁) -> dbl(x₁)
first(x₁, x₂) -> first(x₁, x₂)
activate(x₁) -> activate(x₁)
half(x₁) -> half(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

           →DP Problem 10

             ↳Dependency Graph

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:02 minutes