Term Rewriting System R:
[x]
rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

REC(rec(x)) -> SENT(rec(x))
REC(sent(x)) -> SENT(rec(x))
REC(sent(x)) -> REC(x)
REC(no(x)) -> SENT(rec(x))
REC(no(x)) -> REC(x)
REC(bot) -> SENT(bot)
REC(up(x)) -> REC(x)
SENT(up(x)) -> SENT(x)
NO(up(x)) -> NO(x)
TOP(rec(up(x))) -> TOP(check(rec(x)))
TOP(rec(up(x))) -> CHECK(rec(x))
TOP(rec(up(x))) -> REC(x)
TOP(sent(up(x))) -> TOP(check(rec(x)))
TOP(sent(up(x))) -> CHECK(rec(x))
TOP(sent(up(x))) -> REC(x)
TOP(no(up(x))) -> TOP(check(rec(x)))
TOP(no(up(x))) -> CHECK(rec(x))
TOP(no(up(x))) -> REC(x)
CHECK(up(x)) -> CHECK(x)
CHECK(sent(x)) -> SENT(check(x))
CHECK(sent(x)) -> CHECK(x)
CHECK(rec(x)) -> REC(check(x))
CHECK(rec(x)) -> CHECK(x)
CHECK(no(x)) -> NO(check(x))
CHECK(no(x)) -> CHECK(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 Pair:

SENT(up(x)) -> SENT(x)

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

The following dependency pair can be strictly oriented:

SENT(up(x)) -> SENT(x)

There are no usable rules 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:
SENT(x1) -> SENT(x1)
up(x1) -> up(x1)

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

Dependency Pair:

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(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 Pair:

NO(up(x)) -> NO(x)

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

The following dependency pair can be strictly oriented:

NO(up(x)) -> NO(x)

There are no usable rules 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:
NO(x1) -> NO(x1)
up(x1) -> up(x1)

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

Dependency Pair:

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(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:

REC(up(x)) -> REC(x)
REC(no(x)) -> REC(x)
REC(sent(x)) -> REC(x)

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

The following dependency pairs can be strictly oriented:

REC(up(x)) -> REC(x)
REC(no(x)) -> REC(x)
REC(sent(x)) -> REC(x)

There are no usable rules 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:
REC(x1) -> REC(x1)
no(x1) -> no(x1)
sent(x1) -> sent(x1)
up(x1) -> up(x1)

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

Dependency Pair:

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(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:

CHECK(no(x)) -> CHECK(x)
CHECK(rec(x)) -> CHECK(x)
CHECK(sent(x)) -> CHECK(x)
CHECK(up(x)) -> CHECK(x)

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

The following dependency pairs can be strictly oriented:

CHECK(no(x)) -> CHECK(x)
CHECK(rec(x)) -> CHECK(x)
CHECK(sent(x)) -> CHECK(x)
CHECK(up(x)) -> CHECK(x)

There are no usable rules 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:
CHECK(x1) -> CHECK(x1)
up(x1) -> up(x1)
no(x1) -> no(x1)
sent(x1) -> sent(x1)
rec(x1) -> rec(x1)

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

Dependency Pair:

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(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(no(up(x))) -> TOP(check(rec(x)))
TOP(sent(up(x))) -> TOP(check(rec(x)))
TOP(rec(up(x))) -> TOP(check(rec(x)))

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

The following dependency pairs can be strictly oriented:

TOP(no(up(x))) -> TOP(check(rec(x)))
TOP(rec(up(x))) -> TOP(check(rec(x)))

The following usable rules w.r.t. to the AFS can be oriented:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
no > {rec, up}

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
no(x1) -> no(x1)
check(x1) -> x1
up(x1) -> up(x1)
rec(x1) -> rec(x1)
sent(x1) -> 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 10
Remaining Obligation(s)

The following remains to be proven:
Dependency Pair:

TOP(sent(up(x))) -> TOP(check(rec(x)))

Rules:

rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
rec(bot) -> up(sent(bot))
rec(up(x)) -> up(rec(x))
sent(up(x)) -> up(sent(x))
no(up(x)) -> up(no(x))
top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(x)))
check(up(x)) -> up(check(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

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