Term Rewriting System R:
[X, Y, Z, X1, X2]
active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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(f(X)) -> CONS(X, f(g(X)))
ACTIVE(f(X)) -> F(g(X))
ACTIVE(f(X)) -> G(X)
ACTIVE(g(0)) -> S(0)
ACTIVE(g(s(X))) -> S(s(g(X)))
ACTIVE(g(s(X))) -> S(g(X))
ACTIVE(g(s(X))) -> G(X)
ACTIVE(sel(s(X), cons(Y, Z))) -> SEL(X, Z)
ACTIVE(f(X)) -> F(active(X))
ACTIVE(f(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(g(X)) -> G(active(X))
ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(sel(X1, X2)) -> SEL(active(X1), X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> SEL(X1, active(X2))
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
F(mark(X)) -> F(X)
F(ok(X)) -> F(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
G(mark(X)) -> G(X)
G(ok(X)) -> G(X)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
SEL(mark(X1), X2) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
PROPER(f(X)) -> F(proper(X))
PROPER(f(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(g(X)) -> G(proper(X))
PROPER(g(X)) -> PROPER(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(sel(X1, X2)) -> SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
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 eight SCCs.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Nar


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

F(ok(X)) -> F(X)
F(mark(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)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 9
Dependency Graph
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Nar


Dependency Pair:


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Nar


Dependency Pairs:

S(ok(X)) -> S(X)
S(mark(X)) -> S(X)


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

S(ok(X)) -> S(X)
S(mark(X)) -> S(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:
S(x1) -> S(x1)
ok(x1) -> ok(x1)
mark(x1) -> mark(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
           →DP Problem 10
Dependency Graph
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Nar


Dependency Pair:


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Nar


Dependency Pairs:

SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
SEL(mark(X1), X2) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
SEL(mark(X1), X2) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)


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:
SEL(x1, x2) -> SEL(x1, x2)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
           →DP Problem 11
Dependency Graph
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Nar


Dependency Pair:


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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
Argument Filtering and Ordering
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Nar


Dependency Pairs:

CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)


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:
CONS(x1, x2) -> CONS(x1, x2)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
           →DP Problem 12
Dependency Graph
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Nar


Dependency Pair:


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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
AFS
       →DP Problem 5
Argument Filtering and Ordering
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Nar


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

G(ok(X)) -> G(X)
G(mark(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)
mark(x1) -> mark(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
           →DP Problem 13
Dependency Graph
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Nar


Dependency Pair:


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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
AFS
       →DP Problem 5
AFS
       →DP Problem 6
Argument Filtering and Ordering
       →DP Problem 7
AFS
       →DP Problem 8
Nar


Dependency Pairs:

ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(f(X)) -> ACTIVE(X)


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(f(X)) -> ACTIVE(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:
ACTIVE(x1) -> ACTIVE(x1)
s(x1) -> s(x1)
sel(x1, x2) -> sel(x1, x2)
cons(x1, x2) -> cons(x1, x2)
g(x1) -> g(x1)
f(x1) -> f(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
           →DP Problem 14
Dependency Graph
       →DP Problem 7
AFS
       →DP Problem 8
Nar


Dependency Pair:


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
Argument Filtering and Ordering
       →DP Problem 8
Nar


Dependency Pairs:

PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(g(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(f(X)) -> PROPER(X)


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(g(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
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)
sel(x1, x2) -> sel(x1, x2)
cons(x1, x2) -> cons(x1, x2)
s(x1) -> s(x1)
g(x1) -> g(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
           →DP Problem 15
Dependency Graph
       →DP Problem 8
Nar


Dependency Pair:


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Narrowing Transformation


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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))
six new Dependency Pairs are created:

TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))

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
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Nar
           →DP Problem 16
Narrowing Transformation


Dependency Pairs:

TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(ok(X)) -> TOP(active(X))


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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))
11 new Dependency Pairs are created:

TOP(ok(f(X''))) -> TOP(mark(cons(X'', f(g(X'')))))
TOP(ok(g(0))) -> TOP(mark(s(0)))
TOP(ok(g(s(X'')))) -> TOP(mark(s(s(g(X'')))))
TOP(ok(sel(0, cons(X'', Y')))) -> TOP(mark(X''))
TOP(ok(sel(s(X''), cons(Y', Z')))) -> TOP(mark(sel(X'', Z')))
TOP(ok(f(X''))) -> TOP(f(active(X'')))
TOP(ok(cons(X1', X2'))) -> TOP(cons(active(X1'), X2'))
TOP(ok(g(X''))) -> TOP(g(active(X'')))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(sel(X1', X2'))) -> TOP(sel(active(X1'), X2'))
TOP(ok(sel(X1', X2'))) -> TOP(sel(X1', active(X2')))

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
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Nar
           →DP Problem 16
Nar
             ...
               →DP Problem 17
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

TOP(ok(sel(X1', X2'))) -> TOP(sel(X1', active(X2')))
TOP(ok(sel(X1', X2'))) -> TOP(sel(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(g(X''))) -> TOP(g(active(X'')))
TOP(ok(cons(X1', X2'))) -> TOP(cons(active(X1'), X2'))
TOP(ok(f(X''))) -> TOP(f(active(X'')))
TOP(ok(sel(s(X''), cons(Y', Z')))) -> TOP(mark(sel(X'', Z')))
TOP(ok(sel(0, cons(X'', Y')))) -> TOP(mark(X''))
TOP(ok(g(s(X'')))) -> TOP(mark(s(s(g(X'')))))
TOP(ok(g(0))) -> TOP(mark(s(0)))
TOP(ok(f(X''))) -> TOP(mark(cons(X'', f(g(X'')))))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:56 minutes