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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given ETRS could be proven:

(0) ETRS
(1) RRRPoloETRSProof [EQUIVALENT, 157 ms]
(2) ETRS
(3) RRRPoloETRSProof [EQUIVALENT, 24 ms]
(4) ETRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

   zero(S) -> S
   plus(x, S) -> x
   plus(zero(x), zero(y)) -> zero(plus(x, y))
   plus(zero(x), un(y)) -> un(plus(x, y))
   plus(un(x), un(y)) -> zero(plus(x, plus(y, un(S))))
   times(x, S) -> S
   times(x, times(S, z)) -> times(S, z)
   times(x, zero(y)) -> zero(times(x, y))
   times(x, times(zero(y), z)) -> times(zero(times(x, y)), z)
   times(x, un(y)) -> plus(x, zero(times(x, y)))
   times(x, times(un(y), z)) -> times(plus(x, zero(times(x, y))), z)

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))


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

   zero(S) -> S
   plus(x, S) -> x
   plus(zero(x), zero(y)) -> zero(plus(x, y))
   plus(zero(x), un(y)) -> un(plus(x, y))
   plus(un(x), un(y)) -> zero(plus(x, plus(y, un(S))))
   times(x, S) -> S
   times(x, times(S, z)) -> times(S, z)
   times(x, zero(y)) -> zero(times(x, y))
   times(x, times(zero(y), z)) -> times(zero(times(x, y)), z)
   times(x, un(y)) -> plus(x, zero(times(x, y)))
   times(x, times(un(y), z)) -> times(plus(x, zero(times(x, y))), z)

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(un(x), un(y)) -> zero(plus(x, plus(y, un(S))))
   times(x, un(y)) -> plus(x, zero(times(x, y)))
   times(x, times(un(y), z)) -> times(plus(x, zero(times(x, y))), z)
Used ordering:
Polynomial interpretation [POLO]:

   POL(S) = 0
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(times(x_1, x_2)) = x_1 + x_1*x_2 + x_2
   POL(un(x_1)) = 2 + x_1
   POL(zero(x_1)) = x_1




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

   zero(S) -> S
   plus(x, S) -> x
   plus(zero(x), zero(y)) -> zero(plus(x, y))
   plus(zero(x), un(y)) -> un(plus(x, y))
   times(x, S) -> S
   times(x, times(S, z)) -> times(S, z)
   times(x, zero(y)) -> zero(times(x, y))
   times(x, times(zero(y), z)) -> times(zero(times(x, y)), z)

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))


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

   zero(S) -> S
   plus(x, S) -> x
   plus(zero(x), zero(y)) -> zero(plus(x, y))
   plus(zero(x), un(y)) -> un(plus(x, y))
   times(x, S) -> S
   times(x, times(S, z)) -> times(S, z)
   times(x, zero(y)) -> zero(times(x, y))
   times(x, times(zero(y), z)) -> times(zero(times(x, y)), z)

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:

   zero(S) -> S
   plus(x, S) -> x
   plus(zero(x), zero(y)) -> zero(plus(x, y))
   plus(zero(x), un(y)) -> un(plus(x, y))
   times(x, S) -> S
   times(x, times(S, z)) -> times(S, z)
   times(x, zero(y)) -> zero(times(x, y))
   times(x, times(zero(y), z)) -> times(zero(times(x, y)), z)
Used ordering:
Polynomial interpretation [POLO]:

   POL(S) = 3
   POL(plus(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2
   POL(times(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2
   POL(un(x_1)) = 2 + 2*x_1
   POL(zero(x_1)) = 2 + x_1




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(4)
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))


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(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES