Term Rewriting System R:
[X, Y, Z, X1, X2]
2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

2ND(cons(X, ncons(Y, Z))) -> ACTIVATE(Y)
FROM(X) -> CONS(X, nfrom(ns(X)))
ACTIVATE(ncons(X1, X2)) -> CONS(activate(X1), X2)
ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfrom(X)) -> FROM(activate(X))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Polynomial Ordering

Dependency Pairs:

ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)

Rules:

2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(ns(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__from(x1)) =  x1 POL(n__cons(x1, x2)) =  x1 POL(n__s(x1)) =  1 + x1 POL(ACTIVATE(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polynomial Ordering

Dependency Pairs:

ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)

Rules:

2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(nfrom(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__from(x1)) =  1 + x1 POL(n__cons(x1, x2)) =  x1 POL(ACTIVATE(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
...
→DP Problem 3
Polynomial Ordering

Dependency Pair:

ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)

Rules:

2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(ncons(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__cons(x1, x2)) =  1 + x1 POL(ACTIVATE(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
...
→DP Problem 4
Dependency Graph

Dependency Pair:

Rules:

2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(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