Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

Dependency Pairs:

ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_dbl}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_sqr}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__terms(x₁)) = x₁

POL(n__sqr(x₁)) = x₁

POL(ACTIVATE(x₁)) = x₁

POL(n__add(x₁, x₂)) = x₁ + x₂

POL(n__dbl(x₁)) = x₁

POL(n__first(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Polynomial Ordering

Dependency Pairs:

ACTIVATE(n_{_dbl}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_sqr}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_dbl}(X)) -> ACTIVATE(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__terms(x₁)) = x₁

POL(n__sqr(x₁)) = x₁

POL(ACTIVATE(x₁)) = x₁

POL(n__add(x₁, x₂)) = x₁ + x₂

POL(n__dbl(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Polo

...

               →DP Problem 3

                 ↳Polynomial Ordering

Dependency Pairs:

ACTIVATE(n_{_sqr}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_sqr}(X)) -> ACTIVATE(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__terms(x₁)) = x₁

POL(n__sqr(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = x₁

POL(n__add(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Polo

...

               →DP Problem 4

                 ↳Polynomial Ordering

Dependency Pairs:

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

Rules:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__terms(x₁)) = x₁

POL(ACTIVATE(x₁)) = x₁

POL(n__add(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Polo

...

               →DP Problem 5

                 ↳Polynomial Ordering

Dependency Pair:

ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__terms(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Polo

...

               →DP Problem 6

                 ↳Dependency Graph

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes

POL(n__terms(x₁))	= x₁
POL(n__sqr(x₁))	= x₁
POL(ACTIVATE(x₁))	= x₁
POL(n__add(x₁, x₂))	= x₁ + x₂
POL(n__dbl(x₁))	= x₁
POL(n__first(x₁, x₂))	= 1 + x₁ + x₂

POL(n__terms(x₁))	= 1 + x₁
POL(ACTIVATE(x₁))	= x₁