Termination Proof using AProVE

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

     ↳DPs

       →DP Problem 1

         ↳Size-Change Principle

       →DP Problem 2

         ↳SCP

       →DP Problem 3

         ↳SCP

       →DP Problem 4

         ↳SCP

       →DP Problem 5

         ↳Neg POLO

Dependency Pair:

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

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_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
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

We number the DPs as follows:

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

and get the following Size-Change Graph(s):

{1}	,	{1}
1	>	1
2	=	2

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	>	1
2	=	2

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳SCP

       →DP Problem 2

         ↳Size-Change Principle

       →DP Problem 3

         ↳SCP

       →DP Problem 4

         ↳SCP

       →DP Problem 5

         ↳Neg POLO

Dependency Pair:

DBL(s(X)) -> DBL(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(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
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

We number the DPs as follows:

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

and get the following Size-Change Graph(s):

{1}	,	{1}
1	>	1

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	>	1

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳SCP

       →DP Problem 2

         ↳SCP

       →DP Problem 3

         ↳Size-Change Principle

       →DP Problem 4

         ↳SCP

       →DP Problem 5

         ↳Neg POLO

Dependency Pair:

HALF(s(s(X))) -> HALF(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(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
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

We number the DPs as follows:

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

and get the following Size-Change Graph(s):

{1}	,	{1}
1	>	1

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	>	1

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳SCP

       →DP Problem 2

         ↳SCP

       →DP Problem 3

         ↳SCP

       →DP Problem 4

         ↳Size-Change Principle

       →DP Problem 5

         ↳Neg POLO

Dependency Pair:

SQR(s(X)) -> SQR(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(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
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

We number the DPs as follows:

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

and get the following Size-Change Graph(s):

{1}	,	{1}
1	>	1

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	>	1

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳SCP

       →DP Problem 2

         ↳SCP

       →DP Problem 3

         ↳SCP

       →DP Problem 4

         ↳SCP

       →DP Problem 5

         ↳Negative Polynomial Order

Dependency Pairs:

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

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_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
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

The following Dependency Pairs can be strictly oriented using the given order.

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

Moreover, the following usable rules (regarding the implicit AFS) are oriented.

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

Used ordering:
Polynomial Order with Interpretation:

POL( ACTIVATE(x₁) ) = x₁

POL( n_{_first}(x₁, x₂) ) = x₁ + x₂ + 1

POL( FIRST(x₁, x₂) ) = x₂

POL( activate(x₁) ) = x₁

POL( n_{_terms}(x₁) ) = x₁

POL( n_{_s}(x₁) ) = x₁

POL( cons(x₁, x₂) ) = x₂

POL( terms(x₁) ) = x₁

POL( s(x₁) ) = x₁

POL( first(x₁, x₂) ) = x₁ + x₂ + 1

POL( 0 ) = 0

POL( nil ) = 0

POL( sqr(x₁) ) = 0

POL( add(x₁, x₂) ) = x₂

POL( dbl(x₁) ) = 0

This results in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳SCP

       →DP Problem 2

         ↳SCP

       →DP Problem 3

         ↳SCP

       →DP Problem 4

         ↳SCP

       →DP Problem 5

         ↳Neg POLO

           →DP Problem 6

             ↳Dependency Graph

Dependency Pairs:

ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_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
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

Using the Dependency Graph the DP problem was split into 1 DP problems.

     ↳DPs

       →DP Problem 1

         ↳SCP

       →DP Problem 2

         ↳SCP

       →DP Problem 3

         ↳SCP

       →DP Problem 4

         ↳SCP

       →DP Problem 5

         ↳Neg POLO

           →DP Problem 6

             ↳DGraph

...

               →DP Problem 7

                 ↳Size-Change Principle

Dependency Pairs:

ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_s}(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(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
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

We number the DPs as follows:

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

and get the following Size-Change Graph(s):

{1, 2}	,	{1, 2}
1	>	1

which lead(s) to this/these maximal multigraph(s):

{1, 2}	,	{1, 2}
1	>	1

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

n_{_terms}(x₁) -> n_{_terms}(x₁)
n_{_s}(x₁) -> n_{_s}(x₁)

We obtain no new DP problems.

Termination of R successfully shown.
Duration:
0:01 minutes