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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QTRSRRRProof [EQUIVALENT, 61 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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

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

   +(a, b) -> +(b, a)
   +(a, +(b, z)) -> +(b, +(a, z))
   +(+(x, y), z) -> +(x, +(y, z))
   f(a, y) -> a
   f(b, y) -> b
   f(+(x, y), z) -> +(f(x, z), f(y, z))

Q is empty.

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

(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
f_2 > [+_2, a] > b


Status:
+_2: [1,2]
a: multiset status
b: multiset status
f_2: multiset status

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   +(a, b) -> +(b, a)
   +(a, +(b, z)) -> +(b, +(a, z))
   +(+(x, y), z) -> +(x, +(y, z))
   f(a, y) -> a
   f(b, y) -> b
   f(+(x, y), z) -> +(f(x, z), f(y, z))




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

(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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

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

(4)
YES