Termination Proof using AProVE

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.

     ↳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.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

The following remains to be proven:

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)
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)
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)
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)
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)

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

The following remains to be proven:

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)
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)
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)
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)
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)

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

The following remains to be proven:

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)
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)
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)
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)
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)

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

The following remains to be proven:

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)
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)
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)
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)
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)

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

The following remains to be proven:

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)
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)
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)
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)
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)

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