Term Rewriting System R:
[X]
active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(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(g(X)) -> H(X)
ACTIVE(h(d)) -> G(c)
PROPER(g(X)) -> G(proper(X))
PROPER(g(X)) -> PROPER(X)
PROPER(h(X)) -> H(proper(X))
PROPER(h(X)) -> PROPER(X)
G(ok(X)) -> G(X)
H(ok(X)) -> H(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
Polynomial Ordering
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Remaining

Dependency Pair:

H(ok(X)) -> H(X)

Rules:

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

The following dependency pair can be strictly oriented:

H(ok(X)) -> H(X)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(H(x1)) =  x1 POL(ok(x1)) =  1 + x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 5
Dependency Graph
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Remaining

Dependency Pair:

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(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
Polo
→DP Problem 2
Polynomial Ordering
→DP Problem 3
Polo
→DP Problem 4
Remaining

Dependency Pair:

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

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(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)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(G(x1)) =  x1 POL(ok(x1)) =  1 + x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 6
Dependency Graph
→DP Problem 3
Polo
→DP Problem 4
Remaining

Dependency Pair:

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(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
Polo
→DP Problem 2
Polo
→DP Problem 3
Polynomial Ordering
→DP Problem 4
Remaining

Dependency Pairs:

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

Rules:

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

The following dependency pair can be strictly oriented:

PROPER(h(X)) -> PROPER(X)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  x1 POL(PROPER(x1)) =  x1 POL(h(x1)) =  1 + x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 7
Polynomial Ordering
→DP Problem 4
Remaining

Dependency Pair:

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

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(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)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(PROPER(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 7
Polo
...
→DP Problem 8
Dependency Graph
→DP Problem 4
Remaining

Dependency Pair:

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(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
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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

Rules:

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

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