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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 128 ms]
(2) ETRS
(3) RRRPoloETRSProof [EQUIVALENT, 20 ms]
(4) ETRS
(5) RRRPoloETRSProof [EQUIVALENT, 31 ms]
(6) ETRS
(7) RRRPoloETRSProof [EQUIVALENT, 0 ms]
(8) ETRS
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

   f(plus(x, y)) -> plus(f(x), y)
   plus(g(x), y) -> g(plus(x, y))
   plus(f(a), g(b)) -> plus(f(b), g(a))
   h(a, b) -> h(b, a)
   h(a, g(g(a))) -> h(g(a), f(a))
   h(g(a), a) -> h(a, g(b))
   h(g(a), b) -> h(a, g(a))

The set E consists of the following equations:

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

   f(plus(x, y)) -> plus(f(x), y)
   plus(g(x), y) -> g(plus(x, y))
   plus(f(a), g(b)) -> plus(f(b), g(a))
   h(a, b) -> h(b, a)
   h(a, g(g(a))) -> h(g(a), f(a))
   h(g(a), a) -> h(a, g(b))
   h(g(a), b) -> h(a, g(a))

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:

   f(plus(x, y)) -> plus(f(x), y)
   plus(g(x), y) -> g(plus(x, y))
   h(a, g(g(a))) -> h(g(a), f(a))
Used ordering:
Polynomial interpretation [POLO]:

   POL(a) = 0
   POL(b) = 0
   POL(f(x_1)) = 2 + 3*x_1
   POL(g(x_1)) = 3 + x_1
   POL(h(x_1, x_2)) = x_1 + x_2
   POL(plus(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2




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

   plus(f(a), g(b)) -> plus(f(b), g(a))
   h(a, b) -> h(b, a)
   h(g(a), a) -> h(a, g(b))
   h(g(a), b) -> h(a, g(a))

The set E consists of the following equations:

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

   plus(f(a), g(b)) -> plus(f(b), g(a))
   h(a, b) -> h(b, a)
   h(g(a), a) -> h(a, g(b))
   h(g(a), b) -> h(a, g(a))

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:

   h(a, b) -> h(b, a)
   h(g(a), a) -> h(a, g(b))
Used ordering:
Polynomial interpretation [POLO]:

   POL(a) = 2
   POL(b) = 0
   POL(f(x_1)) = 2*x_1
   POL(g(x_1)) = x_1^2
   POL(h(x_1, x_2)) = 2*x_1 + x_2
   POL(plus(x_1, x_2)) = 3 + x_1 + x_2




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

   plus(f(a), g(b)) -> plus(f(b), g(a))
   h(g(a), b) -> h(a, g(a))

The set E consists of the following equations:

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


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

   plus(f(a), g(b)) -> plus(f(b), g(a))
   h(g(a), b) -> h(a, g(a))

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(f(a), g(b)) -> plus(f(b), g(a))
Used ordering:
Polynomial interpretation [POLO]:

   POL(a) = 2
   POL(b) = 0
   POL(f(x_1)) = 3*x_1
   POL(g(x_1)) = 2*x_1
   POL(h(x_1, x_2)) = 2*x_1 + x_2
   POL(plus(x_1, x_2)) = 3 + x_1 + x_2




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

   h(g(a), b) -> h(a, g(a))

The set E consists of the following equations:

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


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

   h(g(a), b) -> h(a, g(a))

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:

   h(g(a), b) -> h(a, g(a))
Used ordering:
Polynomial interpretation [POLO]:

   POL(a) = 0
   POL(b) = 1
   POL(g(x_1)) = x_1^2
   POL(h(x_1, x_2)) = 2*x_1 + x_2
   POL(plus(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2




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(8)
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))


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(9) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(10)
YES