Termination Proof using AProVE

Term Rewriting System R:
[YS, X, XS, X1, X2, Y, L]
app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, n_{_app}(activate(XS), YS))
app(X1, X2) -> n_{_app}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n_{_nil})), n_{_zWadr}(activate(XS), activate(YS)))
zWadr(X1, X2) -> n_{_zWadr}(X1, X2)
prefix(L) -> cons(nil, n_{_zWadr}(L, prefix(L)))
nil -> n_{_nil}
activate(n_{_app}(X1, X2)) -> app(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_nil}) -> nil
activate(n_{_zWadr}(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(cons(X, XS), YS) -> ACTIVATE(XS)
ZWADR(cons(X, XS), cons(Y, YS)) -> APP(Y, cons(X, n_{_nil}))
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
PREFIX(L) -> NIL
PREFIX(L) -> PREFIX(L)
ACTIVATE(n_{_app}(X1, X2)) -> APP(X1, X2)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_nil}) -> NIL
ACTIVATE(n_{_zWadr}(X1, X2)) -> ZWADR(X1, X2)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ZWADR(cons(X, XS), cons(Y, YS)) -> APP(Y, cons(X, n_{_nil}))
ACTIVATE(n_{_zWadr}(X1, X2)) -> ZWADR(X1, X2)
ACTIVATE(n_{_app}(X1, X2)) -> APP(X1, X2)
APP(cons(X, XS), YS) -> ACTIVATE(XS)

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, n_{_app}(activate(XS), YS))
app(X1, X2) -> n_{_app}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n_{_nil})), n_{_zWadr}(activate(XS), activate(YS)))
zWadr(X1, X2) -> n_{_zWadr}(X1, X2)
prefix(L) -> cons(nil, n_{_zWadr}(L, prefix(L)))
nil -> n_{_nil}
activate(n_{_app}(X1, X2)) -> app(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_nil}) -> nil
activate(n_{_zWadr}(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X

Strategy:

innermost

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

APP(cons(X, XS), YS) -> ACTIVATE(XS)

two new Dependency Pairs are created:

APP(cons(X, n_{_app}(X1'', X2'')), YS) -> ACTIVATE(n_{_app}(X1'', X2''))
APP(cons(X, n_{_zWadr}(X1'', X2'')), YS) -> ACTIVATE(n_{_zWadr}(X1'', X2''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ZWADR(cons(X, XS), cons(Y, YS)) -> APP(Y, cons(X, n_{_nil}))
ACTIVATE(n_{_zWadr}(X1, X2)) -> ZWADR(X1, X2)
APP(cons(X, n_{_zWadr}(X1'', X2'')), YS) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
APP(cons(X, n_{_app}(X1'', X2'')), YS) -> ACTIVATE(n_{_app}(X1'', X2''))
ACTIVATE(n_{_app}(X1, X2)) -> APP(X1, X2)
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, n_{_app}(activate(XS), YS))
app(X1, X2) -> n_{_app}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n_{_nil})), n_{_zWadr}(activate(XS), activate(YS)))
zWadr(X1, X2) -> n_{_zWadr}(X1, X2)
prefix(L) -> cons(nil, n_{_zWadr}(L, prefix(L)))
nil -> n_{_nil}
activate(n_{_app}(X1, X2)) -> app(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_nil}) -> nil
activate(n_{_zWadr}(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X

Strategy:

innermost

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

ZWADR(cons(X, XS), cons(Y, YS)) -> APP(Y, cons(X, n_{_nil}))

two new Dependency Pairs are created:

ZWADR(cons(X0, XS), cons(cons(X'', n_{_app}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_app}(X1'''', X2'''')), cons(X0, n_{_nil}))
ZWADR(cons(X0, XS), cons(cons(X'', n_{_zWadr}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(X0, n_{_nil}))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳FwdInst

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

ZWADR(cons(X0, XS), cons(cons(X'', n_{_zWadr}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(X0, n_{_nil}))
ZWADR(cons(X0, XS), cons(cons(X'', n_{_app}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_app}(X1'''', X2'''')), cons(X0, n_{_nil}))
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWadr}(X1, X2)) -> ZWADR(X1, X2)
APP(cons(X, n_{_zWadr}(X1'', X2'')), YS) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
APP(cons(X, n_{_app}(X1'', X2'')), YS) -> ACTIVATE(n_{_app}(X1'', X2''))
ACTIVATE(n_{_app}(X1, X2)) -> APP(X1, X2)
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, n_{_app}(activate(XS), YS))
app(X1, X2) -> n_{_app}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n_{_nil})), n_{_zWadr}(activate(XS), activate(YS)))
zWadr(X1, X2) -> n_{_zWadr}(X1, X2)
prefix(L) -> cons(nil, n_{_zWadr}(L, prefix(L)))
nil -> n_{_nil}
activate(n_{_app}(X1, X2)) -> app(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_nil}) -> nil
activate(n_{_zWadr}(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X

Strategy:

innermost

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

ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)

two new Dependency Pairs are created:

ZWADR(cons(X, n_{_app}(X1'', X2'')), cons(Y, YS)) -> ACTIVATE(n_{_app}(X1'', X2''))
ZWADR(cons(X, n_{_zWadr}(X1'', X2'')), cons(Y, YS)) -> ACTIVATE(n_{_zWadr}(X1'', X2''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳FwdInst

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

ZWADR(cons(X, n_{_zWadr}(X1'', X2'')), cons(Y, YS)) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
ZWADR(cons(X, n_{_app}(X1'', X2'')), cons(Y, YS)) -> ACTIVATE(n_{_app}(X1'', X2''))
ZWADR(cons(X0, XS), cons(cons(X'', n_{_app}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_app}(X1'''', X2'''')), cons(X0, n_{_nil}))
APP(cons(X, n_{_app}(X1'', X2'')), YS) -> ACTIVATE(n_{_app}(X1'', X2''))
ACTIVATE(n_{_app}(X1, X2)) -> APP(X1, X2)
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWadr}(X1, X2)) -> ZWADR(X1, X2)
APP(cons(X, n_{_zWadr}(X1'', X2'')), YS) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
ZWADR(cons(X0, XS), cons(cons(X'', n_{_zWadr}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(X0, n_{_nil}))

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, n_{_app}(activate(XS), YS))
app(X1, X2) -> n_{_app}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n_{_nil})), n_{_zWadr}(activate(XS), activate(YS)))
zWadr(X1, X2) -> n_{_zWadr}(X1, X2)
prefix(L) -> cons(nil, n_{_zWadr}(L, prefix(L)))
nil -> n_{_nil}
activate(n_{_app}(X1, X2)) -> app(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_nil}) -> nil
activate(n_{_zWadr}(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X

Strategy:

innermost

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

ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)

two new Dependency Pairs are created:

ZWADR(cons(X, XS), cons(Y, n_{_app}(X1'', X2''))) -> ACTIVATE(n_{_app}(X1'', X2''))
ZWADR(cons(X, XS), cons(Y, n_{_zWadr}(X1'', X2''))) -> ACTIVATE(n_{_zWadr}(X1'', X2''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳FwdInst

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

ZWADR(cons(X, XS), cons(Y, n_{_zWadr}(X1'', X2''))) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
ZWADR(cons(X, XS), cons(Y, n_{_app}(X1'', X2''))) -> ACTIVATE(n_{_app}(X1'', X2''))
ZWADR(cons(X, n_{_app}(X1'', X2'')), cons(Y, YS)) -> ACTIVATE(n_{_app}(X1'', X2''))
ZWADR(cons(X0, XS), cons(cons(X'', n_{_zWadr}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(X0, n_{_nil}))
APP(cons(X, n_{_zWadr}(X1'', X2'')), YS) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
ACTIVATE(n_{_app}(X1, X2)) -> APP(X1, X2)
APP(cons(X, n_{_app}(X1'', X2'')), YS) -> ACTIVATE(n_{_app}(X1'', X2''))
ZWADR(cons(X0, XS), cons(cons(X'', n_{_app}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_app}(X1'''', X2'''')), cons(X0, n_{_nil}))
ACTIVATE(n_{_zWadr}(X1, X2)) -> ZWADR(X1, X2)
ZWADR(cons(X, n_{_zWadr}(X1'', X2'')), cons(Y, YS)) -> ACTIVATE(n_{_zWadr}(X1'', X2''))

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, n_{_app}(activate(XS), YS))
app(X1, X2) -> n_{_app}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n_{_nil})), n_{_zWadr}(activate(XS), activate(YS)))
zWadr(X1, X2) -> n_{_zWadr}(X1, X2)
prefix(L) -> cons(nil, n_{_zWadr}(L, prefix(L)))
nil -> n_{_nil}
activate(n_{_app}(X1, X2)) -> app(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_nil}) -> nil
activate(n_{_zWadr}(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(n_{_app}(X1, X2)) -> APP(X1, X2)

two new Dependency Pairs are created:

ACTIVATE(n_{_app}(cons(X'', n_{_app}(X1'''', X2'''')), X2')) -> APP(cons(X'', n_{_app}(X1'''', X2'''')), X2')
ACTIVATE(n_{_app}(cons(X'', n_{_zWadr}(X1'''', X2'''')), X2')) -> APP(cons(X'', n_{_zWadr}(X1'''', X2'''')), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳FwdInst

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

ZWADR(cons(X, XS), cons(Y, n_{_app}(X1'', X2''))) -> ACTIVATE(n_{_app}(X1'', X2''))
ZWADR(cons(X, n_{_zWadr}(X1'', X2'')), cons(Y, YS)) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
ZWADR(cons(X, n_{_app}(X1'', X2'')), cons(Y, YS)) -> ACTIVATE(n_{_app}(X1'', X2''))
ZWADR(cons(X0, XS), cons(cons(X'', n_{_zWadr}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(X0, n_{_nil}))
APP(cons(X, n_{_zWadr}(X1'', X2'')), YS) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
ACTIVATE(n_{_app}(cons(X'', n_{_zWadr}(X1'''', X2'''')), X2')) -> APP(cons(X'', n_{_zWadr}(X1'''', X2'''')), X2')
ACTIVATE(n_{_app}(cons(X'', n_{_app}(X1'''', X2'''')), X2')) -> APP(cons(X'', n_{_app}(X1'''', X2'''')), X2')
APP(cons(X, n_{_app}(X1'', X2'')), YS) -> ACTIVATE(n_{_app}(X1'', X2''))
ZWADR(cons(X0, XS), cons(cons(X'', n_{_app}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_app}(X1'''', X2'''')), cons(X0, n_{_nil}))
ACTIVATE(n_{_zWadr}(X1, X2)) -> ZWADR(X1, X2)
ZWADR(cons(X, XS), cons(Y, n_{_zWadr}(X1'', X2''))) -> ACTIVATE(n_{_zWadr}(X1'', X2''))

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, n_{_app}(activate(XS), YS))
app(X1, X2) -> n_{_app}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n_{_nil})), n_{_zWadr}(activate(XS), activate(YS)))
zWadr(X1, X2) -> n_{_zWadr}(X1, X2)
prefix(L) -> cons(nil, n_{_zWadr}(L, prefix(L)))
nil -> n_{_nil}
activate(n_{_app}(X1, X2)) -> app(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_nil}) -> nil
activate(n_{_zWadr}(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(n_{_zWadr}(X1, X2)) -> ZWADR(X1, X2)

six new Dependency Pairs are created:

ACTIVATE(n_{_zWadr}(cons(X0'', XS''), cons(cons(X'''', n_{_app}(X1'''''', X2'''''')), YS''))) -> ZWADR(cons(X0'', XS''), cons(cons(X'''', n_{_app}(X1'''''', X2'''''')), YS''))
ACTIVATE(n_{_zWadr}(cons(X0'', XS''), cons(cons(X'''', n_{_zWadr}(X1'''''', X2'''''')), YS''))) -> ZWADR(cons(X0'', XS''), cons(cons(X'''', n_{_zWadr}(X1'''''', X2'''''')), YS''))
ACTIVATE(n_{_zWadr}(cons(X'', n_{_app}(X1'''', X2'''')), cons(Y'', YS''))) -> ZWADR(cons(X'', n_{_app}(X1'''', X2'''')), cons(Y'', YS''))
ACTIVATE(n_{_zWadr}(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(Y'', YS''))) -> ZWADR(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(Y'', YS''))
ACTIVATE(n_{_zWadr}(cons(X'', XS''), cons(Y'', n_{_app}(X1'''', X2'''')))) -> ZWADR(cons(X'', XS''), cons(Y'', n_{_app}(X1'''', X2'''')))
ACTIVATE(n_{_zWadr}(cons(X'', XS''), cons(Y'', n_{_zWadr}(X1'''', X2'''')))) -> ZWADR(cons(X'', XS''), cons(Y'', n_{_zWadr}(X1'''', X2'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
ACTIVATE(n_{_zWadr}(cons(X'', XS''), cons(Y'', n_{_zWadr}(X1'''', X2'''')))) -> ZWADR(cons(X'', XS''), cons(Y'', n_{_zWadr}(X1'''', X2'''')))
ACTIVATE(n_{_zWadr}(cons(X'', XS''), cons(Y'', n_{_app}(X1'''', X2'''')))) -> ZWADR(cons(X'', XS''), cons(Y'', n_{_app}(X1'''', X2'''')))
ACTIVATE(n_{_zWadr}(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(Y'', YS''))) -> ZWADR(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(Y'', YS''))
ACTIVATE(n_{_zWadr}(cons(X'', n_{_app}(X1'''', X2'''')), cons(Y'', YS''))) -> ZWADR(cons(X'', n_{_app}(X1'''', X2'''')), cons(Y'', YS''))
ZWADR(cons(X, XS), cons(Y, n_{_zWadr}(X1'', X2''))) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
ZWADR(cons(X0, XS), cons(cons(X'', n_{_zWadr}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(X0, n_{_nil}))
ACTIVATE(n_{_zWadr}(cons(X0'', XS''), cons(cons(X'''', n_{_zWadr}(X1'''''', X2'''''')), YS''))) -> ZWADR(cons(X0'', XS''), cons(cons(X'''', n_{_zWadr}(X1'''''', X2'''''')), YS''))
ZWADR(cons(X, n_{_zWadr}(X1'', X2'')), cons(Y, YS)) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
ZWADR(cons(X, n_{_app}(X1'', X2'')), cons(Y, YS)) -> ACTIVATE(n_{_app}(X1'', X2''))
ZWADR(cons(X0, XS), cons(cons(X'', n_{_app}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_app}(X1'''', X2'''')), cons(X0, n_{_nil}))
ACTIVATE(n_{_zWadr}(cons(X0'', XS''), cons(cons(X'''', n_{_app}(X1'''''', X2'''''')), YS''))) -> ZWADR(cons(X0'', XS''), cons(cons(X'''', n_{_app}(X1'''''', X2'''''')), YS''))
APP(cons(X, n_{_zWadr}(X1'', X2'')), YS) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
ACTIVATE(n_{_app}(cons(X'', n_{_zWadr}(X1'''', X2'''')), X2')) -> APP(cons(X'', n_{_zWadr}(X1'''', X2'''')), X2')
APP(cons(X, n_{_app}(X1'', X2'')), YS) -> ACTIVATE(n_{_app}(X1'', X2''))
ACTIVATE(n_{_app}(cons(X'', n_{_app}(X1'''', X2'''')), X2')) -> APP(cons(X'', n_{_app}(X1'''', X2'''')), X2')
ZWADR(cons(X, XS), cons(Y, n_{_app}(X1'', X2''))) -> ACTIVATE(n_{_app}(X1'', X2''))

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, n_{_app}(activate(XS), YS))
app(X1, X2) -> n_{_app}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n_{_nil})), n_{_zWadr}(activate(XS), activate(YS)))
zWadr(X1, X2) -> n_{_zWadr}(X1, X2)
prefix(L) -> cons(nil, n_{_zWadr}(L, prefix(L)))
nil -> n_{_nil}
activate(n_{_app}(X1, X2)) -> app(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_nil}) -> nil
activate(n_{_zWadr}(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X

Strategy:
innermost
Dependency Pair:
PREFIX(L) -> PREFIX(L)

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, n_{_app}(activate(XS), YS))
app(X1, X2) -> n_{_app}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n_{_nil})), n_{_zWadr}(activate(XS), activate(YS)))
zWadr(X1, X2) -> n_{_zWadr}(X1, X2)
prefix(L) -> cons(nil, n_{_zWadr}(L, prefix(L)))
nil -> n_{_nil}
activate(n_{_app}(X1, X2)) -> app(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_nil}) -> nil
activate(n_{_zWadr}(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
ACTIVATE(n_{_zWadr}(cons(X'', XS''), cons(Y'', n_{_zWadr}(X1'''', X2'''')))) -> ZWADR(cons(X'', XS''), cons(Y'', n_{_zWadr}(X1'''', X2'''')))
ACTIVATE(n_{_zWadr}(cons(X'', XS''), cons(Y'', n_{_app}(X1'''', X2'''')))) -> ZWADR(cons(X'', XS''), cons(Y'', n_{_app}(X1'''', X2'''')))
ACTIVATE(n_{_zWadr}(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(Y'', YS''))) -> ZWADR(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(Y'', YS''))
ACTIVATE(n_{_zWadr}(cons(X'', n_{_app}(X1'''', X2'''')), cons(Y'', YS''))) -> ZWADR(cons(X'', n_{_app}(X1'''', X2'''')), cons(Y'', YS''))
ZWADR(cons(X, XS), cons(Y, n_{_zWadr}(X1'', X2''))) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
ZWADR(cons(X0, XS), cons(cons(X'', n_{_zWadr}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_zWadr}(X1'''', X2'''')), cons(X0, n_{_nil}))
ACTIVATE(n_{_zWadr}(cons(X0'', XS''), cons(cons(X'''', n_{_zWadr}(X1'''''', X2'''''')), YS''))) -> ZWADR(cons(X0'', XS''), cons(cons(X'''', n_{_zWadr}(X1'''''', X2'''''')), YS''))
ZWADR(cons(X, n_{_zWadr}(X1'', X2'')), cons(Y, YS)) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
ZWADR(cons(X, n_{_app}(X1'', X2'')), cons(Y, YS)) -> ACTIVATE(n_{_app}(X1'', X2''))
ZWADR(cons(X0, XS), cons(cons(X'', n_{_app}(X1'''', X2'''')), YS)) -> APP(cons(X'', n_{_app}(X1'''', X2'''')), cons(X0, n_{_nil}))
ACTIVATE(n_{_zWadr}(cons(X0'', XS''), cons(cons(X'''', n_{_app}(X1'''''', X2'''''')), YS''))) -> ZWADR(cons(X0'', XS''), cons(cons(X'''', n_{_app}(X1'''''', X2'''''')), YS''))
APP(cons(X, n_{_zWadr}(X1'', X2'')), YS) -> ACTIVATE(n_{_zWadr}(X1'', X2''))
ACTIVATE(n_{_app}(cons(X'', n_{_zWadr}(X1'''', X2'''')), X2')) -> APP(cons(X'', n_{_zWadr}(X1'''', X2'''')), X2')
APP(cons(X, n_{_app}(X1'', X2'')), YS) -> ACTIVATE(n_{_app}(X1'', X2''))
ACTIVATE(n_{_app}(cons(X'', n_{_app}(X1'''', X2'''')), X2')) -> APP(cons(X'', n_{_app}(X1'''', X2'''')), X2')
ZWADR(cons(X, XS), cons(Y, n_{_app}(X1'', X2''))) -> ACTIVATE(n_{_app}(X1'', X2''))

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, n_{_app}(activate(XS), YS))
app(X1, X2) -> n_{_app}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n_{_nil})), n_{_zWadr}(activate(XS), activate(YS)))
zWadr(X1, X2) -> n_{_zWadr}(X1, X2)
prefix(L) -> cons(nil, n_{_zWadr}(L, prefix(L)))
nil -> n_{_nil}
activate(n_{_app}(X1, X2)) -> app(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_nil}) -> nil
activate(n_{_zWadr}(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X

Strategy:
innermost
Dependency Pair:
PREFIX(L) -> PREFIX(L)

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, n_{_app}(activate(XS), YS))
app(X1, X2) -> n_{_app}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n_{_nil})), n_{_zWadr}(activate(XS), activate(YS)))
zWadr(X1, X2) -> n_{_zWadr}(X1, X2)
prefix(L) -> cons(nil, n_{_zWadr}(L, prefix(L)))
nil -> n_{_nil}
activate(n_{_app}(X1, X2)) -> app(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_nil}) -> nil
activate(n_{_zWadr}(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X

Strategy:
innermost

Innermost Termination of R could not be shown.
Duration:
0:06 minutes