YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given ETRS could be proven:

(0) ETRS
(1) RRRPoloETRSProof [EQUIVALENT, 138 ms]
(2) ETRS
(3) RRRPoloETRSProof [EQUIVALENT, 0 ms]
(4) ETRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

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

   plus(0, y) -> y
   plus(s(x), 0) -> s(x)
   plus(s(x), s(y)) -> s(plus(s(x), plus(y, 0)))

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))


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

(1) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   plus(0, y) -> y
   plus(s(x), 0) -> s(x)
   plus(s(x), s(y)) -> s(plus(s(x), plus(y, 0)))

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

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

   plus(s(x), s(y)) -> s(plus(s(x), plus(y, 0)))
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(plus(x_1, x_2)) = x_1 + x_1*x_2 + x_2
   POL(s(x_1)) = 1 + x_1




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

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

   plus(0, y) -> y
   plus(s(x), 0) -> s(x)

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))


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

(3) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   plus(0, y) -> y
   plus(s(x), 0) -> s(x)

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

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

   plus(0, y) -> y
   plus(s(x), 0) -> s(x)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
   POL(s(x_1)) = 3 + x_1^2




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

(4)
Obligation:
Equational rewrite system:
R is empty.
The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))


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

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

(6)
YES