Term Rewriting System R:
[X, Y, X1, X2]
active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

ACTIVE(f(g(X), Y)) -> F(X, f(g(X), Y))
ACTIVE(f(X1, X2)) -> F(active(X1), X2)
ACTIVE(f(X1, X2)) -> ACTIVE(X1)
ACTIVE(g(X)) -> G(active(X))
ACTIVE(g(X)) -> ACTIVE(X)
F(mark(X1), X2) -> F(X1, X2)
F(ok(X1), ok(X2)) -> F(X1, X2)
G(mark(X)) -> G(X)
G(ok(X)) -> G(X)
PROPER(f(X1, X2)) -> F(proper(X1), proper(X2))
PROPER(f(X1, X2)) -> PROPER(X1)
PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(g(X)) -> G(proper(X))
PROPER(g(X)) -> PROPER(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 five 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

Dependency Pairs:

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

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pair can be strictly oriented:

F(ok(X1), ok(X2)) -> F(X1, X2)

The following rules can be oriented:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(top(x1)) =  x1 POL(active(x1)) =  x1 POL(proper(x1)) =  x1 POL(g(x1)) =  x1 POL(mark(x1)) =  x1 POL(ok(x1)) =  1 + x1 POL(F(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
ok(x1) -> ok(x1)
mark(x1) -> mark(x1)
active(x1) -> active(x1)
f(x1, x2) -> x1
g(x1) -> g(x1)
proper(x1) -> proper(x1)
top(x1) -> top(x1)

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

Dependency Pair:

F(mark(X1), X2) -> F(X1, X2)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pair can be strictly oriented:

F(mark(X1), X2) -> F(X1, X2)

The following rules can be oriented:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(top(x1)) =  x1 POL(active(x1)) =  1 + x1 POL(proper(x1)) =  x1 POL(g(x1)) =  x1 POL(mark(x1)) =  1 + x1 POL(ok(x1)) =  1 + x1 POL(F(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
mark(x1) -> mark(x1)
active(x1) -> active(x1)
f(x1, x2) -> x1
g(x1) -> g(x1)
ok(x1) -> ok(x1)
proper(x1) -> proper(x1)
top(x1) -> top(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 6
AFS
...
→DP Problem 7
Dependency Graph
→DP Problem 2
AFS
→DP Problem 3
AFS
→DP Problem 4
AFS
→DP Problem 5
AFS

Dependency Pair:

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

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

Dependency Pairs:

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

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(top(x1)) =  x1 POL(active(x1)) =  x1 POL(proper(x1)) =  x1 POL(g(x1)) =  x1 POL(G(x1)) =  x1 POL(mark(x1)) =  x1 POL(ok(x1)) =  1 + x1

resulting in one new DP problem.
Used Argument Filtering System:
G(x1) -> G(x1)
ok(x1) -> ok(x1)
mark(x1) -> mark(x1)
active(x1) -> active(x1)
f(x1, x2) -> x1
g(x1) -> g(x1)
proper(x1) -> proper(x1)
top(x1) -> top(x1)

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

Dependency Pair:

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

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(top(x1)) =  x1 POL(active(x1)) =  1 + x1 POL(proper(x1)) =  x1 POL(g(x1)) =  x1 POL(G(x1)) =  x1 POL(mark(x1)) =  1 + x1 POL(ok(x1)) =  1 + x1

resulting in one new DP problem.
Used Argument Filtering System:
G(x1) -> G(x1)
mark(x1) -> mark(x1)
active(x1) -> active(x1)
f(x1, x2) -> x1
g(x1) -> g(x1)
ok(x1) -> ok(x1)
proper(x1) -> proper(x1)
top(x1) -> top(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 8
AFS
...
→DP Problem 9
Dependency Graph
→DP Problem 3
AFS
→DP Problem 4
AFS
→DP Problem 5
AFS

Dependency Pair:

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

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

Dependency Pairs:

ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(f(X1, X2)) -> ACTIVE(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pair can be strictly oriented:

ACTIVE(g(X)) -> ACTIVE(X)

The following rules can be oriented:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(top) =  0 POL(ACTIVE(x1)) =  1 + x1 POL(active(x1)) =  x1 POL(proper(x1)) =  x1 POL(g(x1)) =  1 + x1 POL(mark) =  0 POL(ok(x1)) =  x1 POL(f(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVE(x1) -> ACTIVE(x1)
g(x1) -> g(x1)
f(x1, x2) -> f(x1, x2)
active(x1) -> active(x1)
mark(x1) -> mark
ok(x1) -> ok(x1)
proper(x1) -> proper(x1)
top(x1) -> top

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 3
AFS
→DP Problem 10
Argument Filtering and Ordering
→DP Problem 4
AFS
→DP Problem 5
AFS

Dependency Pair:

ACTIVE(f(X1, X2)) -> ACTIVE(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pair can be strictly oriented:

ACTIVE(f(X1, X2)) -> ACTIVE(X1)

The following rules can be oriented:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(top) =  0 POL(ACTIVE(x1)) =  1 + x1 POL(active(x1)) =  x1 POL(proper(x1)) =  x1 POL(g(x1)) =  x1 POL(mark) =  0 POL(ok(x1)) =  x1 POL(f(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVE(x1) -> ACTIVE(x1)
f(x1, x2) -> f(x1, x2)
active(x1) -> active(x1)
mark(x1) -> mark
g(x1) -> g(x1)
ok(x1) -> ok(x1)
proper(x1) -> proper(x1)
top(x1) -> top

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 3
AFS
→DP Problem 10
AFS
...
→DP Problem 11
Dependency Graph
→DP Problem 4
AFS
→DP Problem 5
AFS

Dependency Pair:

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

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

Dependency Pairs:

PROPER(g(X)) -> PROPER(X)
PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(f(X1, X2)) -> PROPER(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(f(X1, X2)) -> PROPER(X1)

The following rules can be oriented:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(top) =  0 POL(active(x1)) =  x1 POL(proper(x1)) =  x1 POL(g(x1)) =  x1 POL(PROPER(x1)) =  1 + x1 POL(mark) =  0 POL(ok(x1)) =  x1 POL(f(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
PROPER(x1) -> PROPER(x1)
f(x1, x2) -> f(x1, x2)
g(x1) -> g(x1)
active(x1) -> active(x1)
mark(x1) -> mark
ok(x1) -> ok(x1)
proper(x1) -> proper(x1)
top(x1) -> top

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 3
AFS
→DP Problem 4
AFS
→DP Problem 12
Argument Filtering and Ordering
→DP Problem 5
AFS

Dependency Pair:

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

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(top(x1)) =  x1 POL(active(x1)) =  x1 POL(proper(x1)) =  x1 POL(g(x1)) =  1 + x1 POL(PROPER(x1)) =  1 + x1 POL(mark(x1)) =  x1 POL(ok(x1)) =  x1

resulting in one new DP problem.
Used Argument Filtering System:
PROPER(x1) -> PROPER(x1)
g(x1) -> g(x1)
active(x1) -> active(x1)
mark(x1) -> mark(x1)
f(x1, x2) -> x1
ok(x1) -> ok(x1)
proper(x1) -> proper(x1)
top(x1) -> top(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 3
AFS
→DP Problem 4
AFS
→DP Problem 12
AFS
...
→DP Problem 13
Dependency Graph
→DP Problem 5
AFS

Dependency Pair:

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

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

Dependency Pairs:

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

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(top(x1)) =  x1 POL(proper(x1)) =  x1 POL(active(x1)) =  1 + x1 POL(g(x1)) =  x1 POL(mark(x1)) =  1 + x1 POL(TOP(x1)) =  x1 POL(ok(x1)) =  1 + x1

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
mark(x1) -> mark(x1)
proper(x1) -> proper(x1)
ok(x1) -> ok(x1)
active(x1) -> active(x1)
f(x1, x2) -> x1
g(x1) -> g(x1)
top(x1) -> top(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 14
Argument Filtering and Ordering

Dependency Pair:

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

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(top) =  0 POL(active(x1)) =  x1 POL(proper(x1)) =  x1 POL(g(x1)) =  x1 POL(mark) =  0 POL(TOP(x1)) =  x1 POL(ok(x1)) =  1 + x1 POL(f(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
ok(x1) -> ok(x1)
active(x1) -> active(x1)
f(x1, x2) -> f(x1, x2)
mark(x1) -> mark
g(x1) -> g(x1)
proper(x1) -> proper(x1)
top(x1) -> top

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 3
AFS
→DP Problem 4
AFS
→DP Problem 5
AFS
→DP Problem 14
AFS
...
→DP Problem 15
Dependency Graph

Dependency Pair:

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Using the Dependency Graph resulted in no new DP problems.

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