Term Rewriting System R:
[a, x, k, d]
f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(a, empty) -> G(a, empty)
F(a, cons(x, k)) -> F(cons(x, a), k)
G(cons(x, k), d) -> G(k, cons(x, d))

Furthermore, R contains two SCCs.


   R
DPs
       →DP Problem 1
Instantiation Transformation
       →DP Problem 2
Inst


Dependency Pair:

G(cons(x, k), d) -> G(k, cons(x, d))


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))


Strategy:

innermost




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

G(cons(x, k), d) -> G(k, cons(x, d))
one new Dependency Pair is created:

G(cons(x0, k''), cons(x'', d'')) -> G(k'', cons(x0, cons(x'', d'')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Inst
           →DP Problem 3
Instantiation Transformation
       →DP Problem 2
Inst


Dependency Pair:

G(cons(x0, k''), cons(x'', d'')) -> G(k'', cons(x0, cons(x'', d'')))


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))


Strategy:

innermost




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

G(cons(x0, k''), cons(x'', d'')) -> G(k'', cons(x0, cons(x'', d'')))
one new Dependency Pair is created:

G(cons(x0'', k''''), cons(x'''', cons(x''''', d''''))) -> G(k'''', cons(x0'', cons(x'''', cons(x''''', d''''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Inst
           →DP Problem 3
Inst
             ...
               →DP Problem 4
Argument Filtering and Ordering
       →DP Problem 2
Inst


Dependency Pair:

G(cons(x0'', k''''), cons(x'''', cons(x''''', d''''))) -> G(k'''', cons(x0'', cons(x'''', cons(x''''', d''''))))


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))


Strategy:

innermost




The following dependency pair can be strictly oriented:

G(cons(x0'', k''''), cons(x'''', cons(x''''', d''''))) -> G(k'''', cons(x0'', cons(x'''', cons(x''''', d''''))))


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:
{cons, G}

resulting in one new DP problem.
Used Argument Filtering System:
G(x1, x2) -> G(x1, x2)
cons(x1, x2) -> cons(x1, x2)


   R
DPs
       →DP Problem 1
Inst
           →DP Problem 3
Inst
             ...
               →DP Problem 5
Dependency Graph
       →DP Problem 2
Inst


Dependency Pair:


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Inst
       →DP Problem 2
Instantiation Transformation


Dependency Pair:

F(a, cons(x, k)) -> F(cons(x, a), k)


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))


Strategy:

innermost




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

F(a, cons(x, k)) -> F(cons(x, a), k)
one new Dependency Pair is created:

F(cons(x'', a''), cons(x0, k'')) -> F(cons(x0, cons(x'', a'')), k'')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Inst
       →DP Problem 2
Inst
           →DP Problem 6
Instantiation Transformation


Dependency Pair:

F(cons(x'', a''), cons(x0, k'')) -> F(cons(x0, cons(x'', a'')), k'')


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))


Strategy:

innermost




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

F(cons(x'', a''), cons(x0, k'')) -> F(cons(x0, cons(x'', a'')), k'')
one new Dependency Pair is created:

F(cons(x'''', cons(x''''', a'''')), cons(x0'', k'''')) -> F(cons(x0'', cons(x'''', cons(x''''', a''''))), k'''')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Inst
       →DP Problem 2
Inst
           →DP Problem 6
Inst
             ...
               →DP Problem 7
Argument Filtering and Ordering


Dependency Pair:

F(cons(x'''', cons(x''''', a'''')), cons(x0'', k'''')) -> F(cons(x0'', cons(x'''', cons(x''''', a''''))), k'''')


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(cons(x'''', cons(x''''', a'''')), cons(x0'', k'''')) -> F(cons(x0'', cons(x'''', cons(x''''', a''''))), k'''')


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:
F(x1, x2) -> x2
cons(x1, x2) -> cons(x1, x2)


   R
DPs
       →DP Problem 1
Inst
       →DP Problem 2
Inst
           →DP Problem 6
Inst
             ...
               →DP Problem 8
Dependency Graph


Dependency Pair:


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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