YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 13 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 2 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 2 ms]
(8) QTRS
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   c(c(c(a(x1)))) -> d(d(x1))
   d(b(x1)) -> c(c(x1))
   c(x1) -> a(a(a(a(x1))))
   d(x1) -> b(b(b(b(x1))))
   b(d(x1)) -> c(c(x1))
   a(c(c(c(x1)))) -> d(d(x1))

Q is empty.

----------------------------------------

(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(c(c(c(x1)))) -> d(d(x1))
   b(d(x1)) -> c(c(x1))
   c(x1) -> a(a(a(a(x1))))
   d(x1) -> b(b(b(b(x1))))
   d(b(x1)) -> c(c(x1))
   c(c(c(a(x1)))) -> d(d(x1))

Q is empty.

----------------------------------------

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(a(x_1)) = 3 + x_1
   POL(b(x_1)) = 5 + x_1
   POL(c(x_1)) = 13 + x_1
   POL(d(x_1)) = 21 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   c(x1) -> a(a(a(a(x1))))
   d(x1) -> b(b(b(b(x1))))




----------------------------------------

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(c(c(c(x1)))) -> d(d(x1))
   b(d(x1)) -> c(c(x1))
   d(b(x1)) -> c(c(x1))
   c(c(c(a(x1)))) -> d(d(x1))

Q is empty.

----------------------------------------

(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(a(x_1)) = x_1
   POL(b(x_1)) = 1 + x_1
   POL(c(x_1)) = x_1
   POL(d(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   b(d(x1)) -> c(c(x1))
   d(b(x1)) -> c(c(x1))




----------------------------------------

(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(c(c(c(x1)))) -> d(d(x1))
   c(c(c(a(x1)))) -> d(d(x1))

Q is empty.

----------------------------------------

(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(a(x_1)) = 1 + x_1
   POL(c(x_1)) = x_1
   POL(d(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a(c(c(c(x1)))) -> d(d(x1))
   c(c(c(a(x1)))) -> d(d(x1))




----------------------------------------

(8)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

----------------------------------------

(9) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(10)
YES