Termination Proof using AProVE

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

FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)
ACTIVATE(n_{_from}(X)) -> FROM(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
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

FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

one new Dependency Pair is created:

FIRST(s(X), cons(Y, n_{_first}(X1'', X2''))) -> ACTIVATE(n_{_first}(X1'', X2''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳Forward Instantiation Transformation

Dependency Pairs:

FIRST(s(X), cons(Y, n_{_first}(X1'', X2''))) -> ACTIVATE(n_{_first}(X1'', X2''))
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
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

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

one new Dependency Pair is created:

ACTIVATE(n_{_first}(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))) -> FIRST(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_first}(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))) -> FIRST(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))
FIRST(s(X), cons(Y, n_{_first}(X1'', X2''))) -> ACTIVATE(n_{_first}(X1'', X2''))

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
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

FIRST(s(X), cons(Y, n_{_first}(X1'', X2''))) -> ACTIVATE(n_{_first}(X1'', X2''))

one new Dependency Pair is created:

FIRST(s(X), cons(Y, n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

FIRST(s(X), cons(Y, n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))
ACTIVATE(n_{_first}(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))) -> FIRST(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
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

ACTIVATE(n_{_first}(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))) -> FIRST(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))

one new Dependency Pair is created:

ACTIVATE(n_{_first}(s(X'''), cons(Y''', n_{_first}(s(X''''''), cons(Y'''''', n_{_first}(X1'''''''', X2'''''''')))))) -> FIRST(s(X'''), cons(Y''', n_{_first}(s(X''''''), cons(Y'''''', n_{_first}(X1'''''''', X2'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 5

                 ↳Argument Filtering and Ordering

Dependency Pairs:

ACTIVATE(n_{_first}(s(X'''), cons(Y''', n_{_first}(s(X''''''), cons(Y'''''', n_{_first}(X1'''''''', X2'''''''')))))) -> FIRST(s(X'''), cons(Y''', n_{_first}(s(X''''''), cons(Y'''''', n_{_first}(X1'''''''', X2'''''''')))))
FIRST(s(X), cons(Y, n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_first}(s(X'''), cons(Y''', n_{_first}(s(X''''''), cons(Y'''''', n_{_first}(X1'''''''', X2'''''''')))))) -> FIRST(s(X'''), cons(Y''', n_{_first}(s(X''''''), cons(Y'''''', n_{_first}(X1'''''''', X2'''''''')))))
FIRST(s(X), cons(Y, n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))

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

{n_{_first}, FIRST} > ACTIVATE

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

FIRST(x₁, x₂) -> FIRST(x₁, x₂)
ACTIVATE(x₁) -> ACTIVATE(x₁)
s(x₁) -> s(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)
n_{_first}(x₁, x₂) -> n_{_first}(x₁, x₂)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 6

                 ↳Dependency Graph

Dependency Pair:

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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