Termination Proof using AProVE

Term Rewriting System R:
[X, XS, N]
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

AFTER(s(N), cons(X, XS)) -> AFTER(N, activate(XS))
AFTER(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(n_{_from}(X)) -> FROM(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pair:

AFTER(s(N), cons(X, XS)) -> AFTER(N, activate(XS))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

AFTER(s(N), cons(X, XS)) -> AFTER(N, activate(XS))

two new Dependency Pairs are created:

AFTER(s(N), cons(X, n_{_from}(X''))) -> AFTER(N, from(X''))
AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')
AFTER(s(N), cons(X, n_{_from}(X''))) -> AFTER(N, from(X''))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

AFTER(s(N), cons(X, n_{_from}(X''))) -> AFTER(N, from(X''))

two new Dependency Pairs are created:

AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, cons(X''', n_{_from}(s(X'''))))
AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, n_{_from}(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, cons(X''', n_{_from}(s(X'''))))
AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')

two new Dependency Pairs are created:

AFTER(s(s(N'')), cons(X, cons(X'', XS'''))) -> AFTER(s(N''), cons(X'', XS'''))
AFTER(s(s(N'')), cons(X, cons(X'', n_{_from}(X''''')))) -> AFTER(s(N''), cons(X'', n_{_from}(X''''')))

The transformation is resulting in two new DP problems:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Argument Filtering and Ordering

Dependency Pair:

AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, cons(X''', n_{_from}(s(X'''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, cons(X''', n_{_from}(s(X'''))))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

AFTER > n_{_from}
AFTER > cons
AFTER > s

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

AFTER(x₁, x₂) -> AFTER(x₁, x₂)
s(x₁) -> s(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)
n_{_from}(x₁) -> n_{_from}(x₁)

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Dependency Graph

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Argument Filtering and Ordering

Dependency Pair:

AFTER(s(s(N'')), cons(X, cons(X'', XS'''))) -> AFTER(s(N''), cons(X'', XS'''))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

AFTER(s(s(N'')), cons(X, cons(X'', XS'''))) -> AFTER(s(N''), cons(X'', XS'''))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

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

AFTER(x₁, x₂) -> AFTER(x₁, x₂)
s(x₁) -> s(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)

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