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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 155 ms]
(2) ETRS
(3) RRRPoloETRSProof [EQUIVALENT, 241 ms]
(4) ETRS
(5) RRRPoloETRSProof [EQUIVALENT, 207 ms]
(6) ETRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

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

   0(S) -> S
   plus(x, S) -> x
   plus(0(x), 0(y)) -> 0(plus(x, y))
   plus(0(x), 1(y)) -> 1(plus(x, y))
   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))
   times(x, S) -> S
   times(x, 0(y)) -> 0(times(x, y))
   times(x, 1(y)) -> plus(x, 0(times(x, y)))

The set E consists of the following equations:

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


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

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

   0(S) -> S
   plus(x, S) -> x
   plus(0(x), 0(y)) -> 0(plus(x, y))
   plus(0(x), 1(y)) -> 1(plus(x, y))
   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))
   times(x, S) -> S
   times(x, 0(y)) -> 0(times(x, y))
   times(x, 1(y)) -> plus(x, 0(times(x, y)))

The set E consists of the following equations:

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

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

   0(S) -> S
   plus(0(x), 0(y)) -> 0(plus(x, y))
   plus(0(x), 1(y)) -> 1(plus(x, y))
   times(x, S) -> S
   times(x, 0(y)) -> 0(times(x, y))
   times(x, 1(y)) -> plus(x, 0(times(x, y)))
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = 2 + x_1
   POL(1(x_1)) = 2 + x_1
   POL(S) = 0
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(times(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2




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

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

   plus(x, S) -> x
   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))

The set E consists of the following equations:

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


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

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

   plus(x, S) -> x
   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))

The set E consists of the following equations:

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

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

   plus(x, S) -> x
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = x_1
   POL(1(x_1)) = 2 + x_1
   POL(S) = 2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(times(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2




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

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

   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))

The set E consists of the following equations:

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


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

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

   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))

The set E consists of the following equations:

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

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

   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = x_1
   POL(1(x_1)) = 2 + x_1
   POL(S) = 0
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(times(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2




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

(6)
Obligation:
Equational rewrite system:
R is empty.
The set E consists of the following equations:

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


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

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

(8)
YES