Term Rewriting System R:
[X]
active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(c) -> F(g(c))
ACTIVE(c) -> G(c)
PROPER(f(X)) -> F(proper(X))
PROPER(f(X)) -> PROPER(X)
PROPER(g(X)) -> G(proper(X))
PROPER(g(X)) -> PROPER(X)
F(ok(X)) -> F(X)
G(ok(X)) -> G(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains four SCCs.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
Nar


Dependency Pair:

F(ok(X)) -> F(X)


Rules:


active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(ok(X)) -> F(X)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 5
Dependency Graph
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
Nar


Dependency Pair:


Rules:


active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Argument Filtering and Ordering
       →DP Problem 3
AFS
       →DP Problem 4
Nar


Dependency Pair:

G(ok(X)) -> G(X)


Rules:


active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

G(ok(X)) -> G(X)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
G(x1) -> G(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
           →DP Problem 6
Dependency Graph
       →DP Problem 3
AFS
       →DP Problem 4
Nar


Dependency Pair:


Rules:


active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
Argument Filtering and Ordering
       →DP Problem 4
Nar


Dependency Pairs:

PROPER(g(X)) -> PROPER(X)
PROPER(f(X)) -> PROPER(X)


Rules:


active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

PROPER(g(X)) -> PROPER(X)
PROPER(f(X)) -> PROPER(X)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
PROPER(x1) -> PROPER(x1)
f(x1) -> f(x1)
g(x1) -> g(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
           →DP Problem 7
Dependency Graph
       →DP Problem 4
Nar


Dependency Pair:


Rules:


active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
Narrowing Transformation


Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))


Rules:


active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

TOP(mark(X)) -> TOP(proper(X))
three new Dependency Pairs are created:

TOP(mark(c)) -> TOP(ok(c))
TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
Nar
           →DP Problem 8
Narrowing Transformation


Dependency Pairs:

TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(mark(c)) -> TOP(ok(c))
TOP(ok(X)) -> TOP(active(X))


Rules:


active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




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

TOP(ok(X)) -> TOP(active(X))
two new Dependency Pairs are created:

TOP(ok(c)) -> TOP(mark(f(g(c))))
TOP(ok(f(g(X'')))) -> TOP(mark(g(X'')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
Nar
           →DP Problem 8
Nar
             ...
               →DP Problem 9
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

TOP(ok(f(g(X'')))) -> TOP(mark(g(X'')))
TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(ok(c)) -> TOP(mark(f(g(c))))
TOP(mark(c)) -> TOP(ok(c))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))


Rules:


active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes