Termination Proof using AProVE

Term Rewriting System R:
[X, Y, X1, X2, Z]
and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(activate(X1), X2)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

AND(true, X) -> ACTIVATE(X)
IF(true, X, Y) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
ADD(0, X) -> ACTIVATE(X)
ADD(s(X), Y) -> S(n_{_add}(activate(X), activate(Y)))
ADD(s(X), Y) -> ACTIVATE(X)
ADD(s(X), Y) -> ACTIVATE(Y)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Y)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(activate(X1), X2)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_s}(X)) -> S(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

Dependency Pairs:

FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ADD(0, X) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(activate(X1), X2)

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(activate(X1), X2)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_from}(X)) -> FROM(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(from(x₁)) = 0

POL(FROM(x₁)) = x₁

POL(activate(x₁)) = 0

POL(n__s(x₁)) = 0

POL(ACTIVATE(x₁)) = x₁

POL(ADD(x₁, x₂)) = x₂

POL(add(x₁, x₂)) = 0

POL(n__from(x₁)) = 1 + x₁

POL(first(x₁, x₂)) = 0

POL(0) = 0

POL(cons(x₁, x₂)) = 0

POL(nil) = 0

POL(s(x₁)) = 0

POL(n__add(x₁, x₂)) = x₁ + x₂

POL(n__first(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Dependency Graph

Dependency Pairs:

FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ADD(0, X) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(activate(X1), X2)

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(activate(X1), X2)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

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

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳DGraph

...

               →DP Problem 3

                 ↳Polynomial Ordering

Dependency Pairs:

ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ADD(0, X) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(activate(X1), X2)

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(activate(X1), X2)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

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(from(x₁)) = 0

POL(activate(x₁)) = 0

POL(n__s(x₁)) = 0

POL(ACTIVATE(x₁)) = x₁

POL(ADD(x₁, x₂)) = x₂

POL(add(x₁, x₂)) = 0

POL(n__from(x₁)) = 0

POL(first(x₁, x₂)) = 0

POL(0) = 0

POL(cons(x₁, x₂)) = 0

POL(nil) = 0

POL(s(x₁)) = 0

POL(n__add(x₁, x₂)) = 1 + x₁ + x₂

POL(n__first(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳DGraph

...

               →DP Problem 4

                 ↳Dependency Graph

Dependency Pairs:

ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ADD(0, X) -> ACTIVATE(X)

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(activate(X1), X2)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

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

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳DGraph

...

               →DP Problem 5

                 ↳Polynomial Ordering

Dependency Pairs:

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

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(activate(X1), X2)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

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(ACTIVATE(x₁)) = x₁

POL(n__first(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳DGraph

...

               →DP Problem 6

                 ↳Dependency Graph

Dependency Pair:

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(activate(X1), X2)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(X)
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(from(x₁))	= 0
POL(FROM(x₁))	= x₁
POL(activate(x₁))	= 0
POL(n__s(x₁))	= 0
POL(ACTIVATE(x₁))	= x₁
POL(ADD(x₁, x₂))	= x₂
POL(add(x₁, x₂))	= 0
POL(n__from(x₁))	= 1 + x₁
POL(first(x₁, x₂))	= 0
POL(0)	= 0
POL(cons(x₁, x₂))	= 0
POL(nil)	= 0
POL(s(x₁))	= 0
POL(n__add(x₁, x₂))	= x₁ + x₂
POL(n__first(x₁, x₂))	= x₁ + x₂