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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 155 ms]
(2) ETRS
(3) RRRPoloETRSProof [EQUIVALENT, 31 ms]
(4) ETRS
(5) RRRPoloETRSProof [EQUIVALENT, 0 ms]
(6) ETRS
(7) RRRPoloETRSProof [EQUIVALENT, 0 ms]
(8) ETRS
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

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

   0(#) -> #
   +(#, x) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 1(y)) -> 0(+(1(#), +(x, y)))
   *(#, x) -> #
   *(0(x), y) -> 0(*(x, y))
   *(1(x), y) -> +(0(*(x, y)), y)
   U(empty, b) -> b
   sum(empty) -> 0(#)
   sum(singl(x)) -> x
   sum(U(x, y)) -> +(sum(x), sum(y))
   prod(empty) -> 1(#)
   prod(singl(x)) -> x
   prod(U(x, y)) -> *(prod(x), prod(y))

The set E consists of the following equations:

   *(x, y) == *(y, x)
   +(x, y) == +(y, x)
   U(x, y) == U(y, x)
   *(*(x, y), z) == *(x, *(y, z))
   +(+(x, y), z) == +(x, +(y, z))
   U(U(x, y), z) == U(x, U(y, z))


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

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

   0(#) -> #
   +(#, x) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 1(y)) -> 0(+(1(#), +(x, y)))
   *(#, x) -> #
   *(0(x), y) -> 0(*(x, y))
   *(1(x), y) -> +(0(*(x, y)), y)
   U(empty, b) -> b
   sum(empty) -> 0(#)
   sum(singl(x)) -> x
   sum(U(x, y)) -> +(sum(x), sum(y))
   prod(empty) -> 1(#)
   prod(singl(x)) -> x
   prod(U(x, y)) -> *(prod(x), prod(y))

The set E consists of the following equations:

   *(x, y) == *(y, x)
   +(x, y) == +(y, x)
   U(x, y) == U(y, x)
   *(*(x, y), z) == *(x, *(y, z))
   +(+(x, y), z) == +(x, +(y, z))
   U(U(x, y), z) == U(x, U(y, z))

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

   +(#, x) -> x
   *(#, x) -> #
   *(1(x), y) -> +(0(*(x, y)), y)
   U(empty, b) -> b
   sum(singl(x)) -> x
   sum(U(x, y)) -> +(sum(x), sum(y))
   prod(empty) -> 1(#)
   prod(singl(x)) -> x
   prod(U(x, y)) -> *(prod(x), prod(y))
Used ordering:
Polynomial interpretation [POLO]:

   POL(#) = 3
   POL(*(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2
   POL(+(x_1, x_2)) = x_1 + x_2
   POL(0(x_1)) = x_1
   POL(1(x_1)) = 3 + x_1
   POL(U(x_1, x_2)) = 2 + 3*x_1 + 3*x_1*x_2 + 3*x_2
   POL(empty) = 3
   POL(prod(x_1)) = x_1^2
   POL(singl(x_1)) = 2 + x_1 + 3*x_1^2
   POL(sum(x_1)) = x_1




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

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

   0(#) -> #
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 1(y)) -> 0(+(1(#), +(x, y)))
   *(0(x), y) -> 0(*(x, y))
   sum(empty) -> 0(#)

The set E consists of the following equations:

   *(x, y) == *(y, x)
   +(x, y) == +(y, x)
   U(x, y) == U(y, x)
   *(*(x, y), z) == *(x, *(y, z))
   +(+(x, y), z) == +(x, +(y, z))
   U(U(x, y), z) == U(x, U(y, z))


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

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

   0(#) -> #
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 1(y)) -> 0(+(1(#), +(x, y)))
   *(0(x), y) -> 0(*(x, y))
   sum(empty) -> 0(#)

The set E consists of the following equations:

   *(x, y) == *(y, x)
   +(x, y) == +(y, x)
   U(x, y) == U(y, x)
   *(*(x, y), z) == *(x, *(y, z))
   +(+(x, y), z) == +(x, +(y, z))
   U(U(x, y), z) == U(x, U(y, z))

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

   +(1(x), 1(y)) -> 0(+(1(#), +(x, y)))
   sum(empty) -> 0(#)
Used ordering:
Polynomial interpretation [POLO]:

   POL(#) = 0
   POL(*(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2
   POL(+(x_1, x_2)) = x_1 + 3*x_1*x_2 + x_2
   POL(0(x_1)) = x_1
   POL(1(x_1)) = 1 + 2*x_1
   POL(U(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2
   POL(empty) = 3
   POL(sum(x_1)) = 3 + 3*x_1




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

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

   0(#) -> #
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   *(0(x), y) -> 0(*(x, y))

The set E consists of the following equations:

   *(x, y) == *(y, x)
   +(x, y) == +(y, x)
   U(x, y) == U(y, x)
   *(*(x, y), z) == *(x, *(y, z))
   +(+(x, y), z) == +(x, +(y, z))
   U(U(x, y), z) == U(x, U(y, z))


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

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

   0(#) -> #
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   *(0(x), y) -> 0(*(x, y))

The set E consists of the following equations:

   *(x, y) == *(y, x)
   +(x, y) == +(y, x)
   U(x, y) == U(y, x)
   *(*(x, y), z) == *(x, *(y, z))
   +(+(x, y), z) == +(x, +(y, z))
   U(U(x, y), z) == U(x, U(y, z))

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

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

   POL(#) = 2
   POL(*(x_1, x_2)) = x_1 + 2*x_1*x_2 + x_2
   POL(+(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2
   POL(0(x_1)) = 2 + 3*x_1
   POL(1(x_1)) = 2*x_1
   POL(U(x_1, x_2)) = 2 + 3*x_1 + 3*x_1*x_2 + 3*x_2




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

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

   *(0(x), y) -> 0(*(x, y))

The set E consists of the following equations:

   *(x, y) == *(y, x)
   +(x, y) == +(y, x)
   U(x, y) == U(y, x)
   *(*(x, y), z) == *(x, *(y, z))
   +(+(x, y), z) == +(x, +(y, z))
   U(U(x, y), z) == U(x, U(y, z))


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

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

   *(0(x), y) -> 0(*(x, y))

The set E consists of the following equations:

   *(x, y) == *(y, x)
   +(x, y) == +(y, x)
   U(x, y) == U(y, x)
   *(*(x, y), z) == *(x, *(y, z))
   +(+(x, y), z) == +(x, +(y, z))
   U(U(x, y), z) == U(x, U(y, z))

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

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

   POL(*(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2
   POL(+(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2
   POL(0(x_1)) = 2 + x_1
   POL(U(x_1, x_2)) = 2 + 3*x_1 + 3*x_1*x_2 + 3*x_2




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

(8)
Obligation:
Equational rewrite system:
R is empty.
The set E consists of the following equations:

   *(x, y) == *(y, x)
   +(x, y) == +(y, x)
   U(x, y) == U(y, x)
   *(*(x, y), z) == *(x, *(y, z))
   +(+(x, y), z) == +(x, +(y, z))
   U(U(x, y), z) == U(x, U(y, z))


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

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

(10)
YES