4.24/1.96	YES
4.24/1.97	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.24/1.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.24/1.97	
4.24/1.97	
4.24/1.97	Termination of the given ETRS could be proven:
4.24/1.97	
4.24/1.97	(0) ETRS
4.24/1.97	(1) RRRPoloETRSProof [EQUIVALENT, 151 ms]
4.24/1.97	(2) ETRS
4.24/1.97	(3) RRRPoloETRSProof [EQUIVALENT, 32 ms]
4.24/1.97	(4) ETRS
4.24/1.97	(5) RRRPoloETRSProof [EQUIVALENT, 0 ms]
4.24/1.97	(6) ETRS
4.24/1.97	(7) RRRPoloETRSProof [EQUIVALENT, 0 ms]
4.24/1.97	(8) ETRS
4.24/1.97	(9) RisEmptyProof [EQUIVALENT, 0 ms]
4.24/1.97	(10) YES
4.24/1.97	
4.24/1.97	
4.24/1.97	----------------------------------------
4.24/1.97	
4.24/1.97	(0)
4.24/1.97	Obligation:
4.24/1.97	Equational rewrite system:
4.24/1.97	The TRS R consists of the following rules:
4.24/1.97	
4.24/1.97	   0(#) -> #
4.24/1.97	   +(#, x) -> x
4.24/1.97	   +(0(x), 0(y)) -> 0(+(x, y))
4.24/1.97	   +(0(x), 1(y)) -> 1(+(x, y))
4.24/1.97	   +(1(x), 1(y)) -> 0(+(1(#), +(x, y)))
4.24/1.97	   *(#, x) -> #
4.24/1.97	   *(0(x), y) -> 0(*(x, y))
4.24/1.97	   *(1(x), y) -> +(0(*(x, y)), y)
4.24/1.97	   U(empty, b) -> b
4.24/1.97	   sum(empty) -> 0(#)
4.24/1.97	   sum(singl(x)) -> x
4.24/1.97	   sum(U(x, y)) -> +(sum(x), sum(y))
4.24/1.97	   prod(empty) -> 1(#)
4.24/1.97	   prod(singl(x)) -> x
4.24/1.97	   prod(U(x, y)) -> *(prod(x), prod(y))
4.24/1.97	
4.24/1.97	The set E consists of the following equations:
4.24/1.97	
4.24/1.97	   *(x, y) == *(y, x)
4.24/1.97	   +(x, y) == +(y, x)
4.24/1.97	   U(x, y) == U(y, x)
4.24/1.97	   *(*(x, y), z) == *(x, *(y, z))
4.24/1.97	   +(+(x, y), z) == +(x, +(y, z))
4.24/1.97	   U(U(x, y), z) == U(x, U(y, z))
4.24/1.97	
4.24/1.97	
4.24/1.97	----------------------------------------
4.24/1.97	
4.24/1.97	(1) RRRPoloETRSProof (EQUIVALENT)
4.24/1.97	The following E TRS is given: Equational rewrite system:
4.24/1.97	The TRS R consists of the following rules:
4.24/1.97	
4.24/1.97	   0(#) -> #
4.24/1.97	   +(#, x) -> x
4.24/1.97	   +(0(x), 0(y)) -> 0(+(x, y))
4.24/1.97	   +(0(x), 1(y)) -> 1(+(x, y))
4.24/1.97	   +(1(x), 1(y)) -> 0(+(1(#), +(x, y)))
4.24/1.97	   *(#, x) -> #
4.24/1.97	   *(0(x), y) -> 0(*(x, y))
4.24/1.97	   *(1(x), y) -> +(0(*(x, y)), y)
4.24/1.97	   U(empty, b) -> b
4.24/1.97	   sum(empty) -> 0(#)
4.24/1.97	   sum(singl(x)) -> x
4.24/1.97	   sum(U(x, y)) -> +(sum(x), sum(y))
4.24/1.97	   prod(empty) -> 1(#)
4.24/1.97	   prod(singl(x)) -> x
4.24/1.97	   prod(U(x, y)) -> *(prod(x), prod(y))
4.24/1.97	
4.24/1.97	The set E consists of the following equations:
4.24/1.97	
4.24/1.97	   *(x, y) == *(y, x)
4.24/1.97	   +(x, y) == +(y, x)
4.24/1.97	   U(x, y) == U(y, x)
4.24/1.97	   *(*(x, y), z) == *(x, *(y, z))
4.24/1.97	   +(+(x, y), z) == +(x, +(y, z))
4.24/1.97	   U(U(x, y), z) == U(x, U(y, z))
4.24/1.97	
4.24/1.97	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
4.24/1.97	
4.24/1.97	   +(#, x) -> x
4.24/1.97	   *(#, x) -> #
4.24/1.97	   *(1(x), y) -> +(0(*(x, y)), y)
4.24/1.97	   U(empty, b) -> b
4.24/1.97	   sum(singl(x)) -> x
4.24/1.97	   sum(U(x, y)) -> +(sum(x), sum(y))
4.24/1.97	   prod(empty) -> 1(#)
4.24/1.97	   prod(singl(x)) -> x
4.24/1.97	   prod(U(x, y)) -> *(prod(x), prod(y))
4.24/1.97	Used ordering:
4.24/1.97	Polynomial interpretation [POLO]:
4.24/1.97	
4.24/1.97	   POL(#) = 3
4.24/1.97	   POL(*(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2
4.24/1.97	   POL(+(x_1, x_2)) = x_1 + x_2
4.24/1.97	   POL(0(x_1)) = x_1
4.24/1.97	   POL(1(x_1)) = 3 + x_1
4.24/1.97	   POL(U(x_1, x_2)) = 2 + 3*x_1 + 3*x_1*x_2 + 3*x_2
4.24/1.97	   POL(empty) = 3
4.24/1.97	   POL(prod(x_1)) = x_1^2
4.24/1.97	   POL(singl(x_1)) = 2 + x_1 + 3*x_1^2
4.24/1.97	   POL(sum(x_1)) = x_1
4.24/1.97	
4.24/1.97	
4.24/1.97	
4.24/1.97	
4.24/1.97	----------------------------------------
4.24/1.97	
4.24/1.97	(2)
4.24/1.97	Obligation:
4.24/1.97	Equational rewrite system:
4.24/1.97	The TRS R consists of the following rules:
4.24/1.97	
4.24/1.97	   0(#) -> #
4.24/1.97	   +(0(x), 0(y)) -> 0(+(x, y))
4.24/1.97	   +(0(x), 1(y)) -> 1(+(x, y))
4.24/1.97	   +(1(x), 1(y)) -> 0(+(1(#), +(x, y)))
4.24/1.97	   *(0(x), y) -> 0(*(x, y))
4.24/1.97	   sum(empty) -> 0(#)
4.24/1.97	
4.24/1.97	The set E consists of the following equations:
4.24/1.97	
4.24/1.97	   *(x, y) == *(y, x)
4.24/1.97	   +(x, y) == +(y, x)
4.24/1.97	   U(x, y) == U(y, x)
4.24/1.97	   *(*(x, y), z) == *(x, *(y, z))
4.24/1.97	   +(+(x, y), z) == +(x, +(y, z))
4.24/1.97	   U(U(x, y), z) == U(x, U(y, z))
4.24/1.97	
4.24/1.97	
4.24/1.97	----------------------------------------
4.24/1.97	
4.24/1.97	(3) RRRPoloETRSProof (EQUIVALENT)
4.24/1.97	The following E TRS is given: Equational rewrite system:
4.24/1.97	The TRS R consists of the following rules:
4.24/1.97	
4.24/1.97	   0(#) -> #
4.24/1.97	   +(0(x), 0(y)) -> 0(+(x, y))
4.24/1.97	   +(0(x), 1(y)) -> 1(+(x, y))
4.24/1.97	   +(1(x), 1(y)) -> 0(+(1(#), +(x, y)))
4.24/1.97	   *(0(x), y) -> 0(*(x, y))
4.24/1.97	   sum(empty) -> 0(#)
4.24/1.97	
4.24/1.97	The set E consists of the following equations:
4.24/1.97	
4.24/1.97	   *(x, y) == *(y, x)
4.24/1.97	   +(x, y) == +(y, x)
4.24/1.97	   U(x, y) == U(y, x)
4.24/1.97	   *(*(x, y), z) == *(x, *(y, z))
4.24/1.97	   +(+(x, y), z) == +(x, +(y, z))
4.24/1.97	   U(U(x, y), z) == U(x, U(y, z))
4.24/1.97	
4.24/1.97	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
4.24/1.97	
4.24/1.97	   +(1(x), 1(y)) -> 0(+(1(#), +(x, y)))
4.24/1.97	   sum(empty) -> 0(#)
4.24/1.97	Used ordering:
4.24/1.97	Polynomial interpretation [POLO]:
4.24/1.97	
4.24/1.97	   POL(#) = 0
4.24/1.97	   POL(*(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2
4.24/1.97	   POL(+(x_1, x_2)) = x_1 + 3*x_1*x_2 + x_2
4.24/1.97	   POL(0(x_1)) = x_1
4.24/1.97	   POL(1(x_1)) = 1 + 2*x_1
4.24/1.97	   POL(U(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2
4.24/1.97	   POL(empty) = 3
4.24/1.97	   POL(sum(x_1)) = 3 + 3*x_1
4.24/1.97	
4.24/1.97	
4.24/1.97	
4.24/1.97	
4.24/1.97	----------------------------------------
4.24/1.97	
4.24/1.97	(4)
4.24/1.97	Obligation:
4.24/1.97	Equational rewrite system:
4.24/1.97	The TRS R consists of the following rules:
4.24/1.97	
4.24/1.97	   0(#) -> #
4.24/1.97	   +(0(x), 0(y)) -> 0(+(x, y))
4.24/1.97	   +(0(x), 1(y)) -> 1(+(x, y))
4.24/1.97	   *(0(x), y) -> 0(*(x, y))
4.24/1.97	
4.24/1.97	The set E consists of the following equations:
4.24/1.97	
4.24/1.97	   *(x, y) == *(y, x)
4.24/1.97	   +(x, y) == +(y, x)
4.24/1.97	   U(x, y) == U(y, x)
4.24/1.97	   *(*(x, y), z) == *(x, *(y, z))
4.24/1.97	   +(+(x, y), z) == +(x, +(y, z))
4.24/1.97	   U(U(x, y), z) == U(x, U(y, z))
4.24/1.97	
4.24/1.97	
4.24/1.97	----------------------------------------
4.24/1.97	
4.24/1.97	(5) RRRPoloETRSProof (EQUIVALENT)
4.24/1.97	The following E TRS is given: Equational rewrite system:
4.24/1.97	The TRS R consists of the following rules:
4.24/1.97	
4.24/1.97	   0(#) -> #
4.24/1.97	   +(0(x), 0(y)) -> 0(+(x, y))
4.24/1.97	   +(0(x), 1(y)) -> 1(+(x, y))
4.24/1.97	   *(0(x), y) -> 0(*(x, y))
4.24/1.97	
4.24/1.97	The set E consists of the following equations:
4.24/1.97	
4.24/1.97	   *(x, y) == *(y, x)
4.24/1.97	   +(x, y) == +(y, x)
4.24/1.97	   U(x, y) == U(y, x)
4.24/1.97	   *(*(x, y), z) == *(x, *(y, z))
4.24/1.97	   +(+(x, y), z) == +(x, +(y, z))
4.24/1.97	   U(U(x, y), z) == U(x, U(y, z))
4.24/1.97	
4.24/1.97	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
4.24/1.97	
4.24/1.97	   0(#) -> #
4.24/1.97	   +(0(x), 0(y)) -> 0(+(x, y))
4.24/1.97	   +(0(x), 1(y)) -> 1(+(x, y))
4.24/1.97	Used ordering:
4.24/1.97	Polynomial interpretation [POLO]:
4.24/1.97	
4.24/1.97	   POL(#) = 2
4.24/1.97	   POL(*(x_1, x_2)) = x_1 + 2*x_1*x_2 + x_2
4.24/1.97	   POL(+(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2
4.24/1.97	   POL(0(x_1)) = 2 + 3*x_1
4.24/1.97	   POL(1(x_1)) = 2*x_1
4.24/1.97	   POL(U(x_1, x_2)) = 2 + 3*x_1 + 3*x_1*x_2 + 3*x_2
4.24/1.97	
4.24/1.97	
4.24/1.97	
4.24/1.97	
4.24/1.97	----------------------------------------
4.24/1.97	
4.24/1.97	(6)
4.24/1.97	Obligation:
4.24/1.97	Equational rewrite system:
4.24/1.97	The TRS R consists of the following rules:
4.24/1.97	
4.24/1.97	   *(0(x), y) -> 0(*(x, y))
4.24/1.97	
4.24/1.97	The set E consists of the following equations:
4.24/1.97	
4.24/1.97	   *(x, y) == *(y, x)
4.24/1.97	   +(x, y) == +(y, x)
4.24/1.97	   U(x, y) == U(y, x)
4.24/1.97	   *(*(x, y), z) == *(x, *(y, z))
4.24/1.97	   +(+(x, y), z) == +(x, +(y, z))
4.24/1.97	   U(U(x, y), z) == U(x, U(y, z))
4.24/1.97	
4.24/1.97	
4.24/1.97	----------------------------------------
4.24/1.97	
4.24/1.97	(7) RRRPoloETRSProof (EQUIVALENT)
4.24/1.97	The following E TRS is given: Equational rewrite system:
4.24/1.97	The TRS R consists of the following rules:
4.24/1.97	
4.24/1.97	   *(0(x), y) -> 0(*(x, y))
4.24/1.97	
4.24/1.97	The set E consists of the following equations:
4.24/1.97	
4.24/1.97	   *(x, y) == *(y, x)
4.24/1.97	   +(x, y) == +(y, x)
4.24/1.97	   U(x, y) == U(y, x)
4.24/1.97	   *(*(x, y), z) == *(x, *(y, z))
4.24/1.97	   +(+(x, y), z) == +(x, +(y, z))
4.24/1.97	   U(U(x, y), z) == U(x, U(y, z))
4.24/1.97	
4.24/1.97	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
4.24/1.97	
4.24/1.97	   *(0(x), y) -> 0(*(x, y))
4.24/1.97	Used ordering:
4.24/1.97	Polynomial interpretation [POLO]:
4.24/1.97	
4.24/1.97	   POL(*(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2
4.24/1.97	   POL(+(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2
4.24/1.97	   POL(0(x_1)) = 2 + x_1
4.24/1.97	   POL(U(x_1, x_2)) = 2 + 3*x_1 + 3*x_1*x_2 + 3*x_2
4.24/1.97	
4.24/1.97	
4.24/1.97	
4.24/1.97	
4.24/1.97	----------------------------------------
4.24/1.97	
4.24/1.97	(8)
4.24/1.97	Obligation:
4.24/1.97	Equational rewrite system:
4.24/1.97	R is empty.
4.24/1.97	The set E consists of the following equations:
4.24/1.97	
4.24/1.97	   *(x, y) == *(y, x)
4.24/1.97	   +(x, y) == +(y, x)
4.24/1.97	   U(x, y) == U(y, x)
4.24/1.97	   *(*(x, y), z) == *(x, *(y, z))
4.24/1.97	   +(+(x, y), z) == +(x, +(y, z))
4.24/1.97	   U(U(x, y), z) == U(x, U(y, z))
4.24/1.97	
4.24/1.97	
4.24/1.97	----------------------------------------
4.24/1.97	
4.24/1.97	(9) RisEmptyProof (EQUIVALENT)
4.24/1.97	The TRS R is empty. Hence, termination is trivially proven.
4.24/1.97	----------------------------------------
4.24/1.97	
4.24/1.97	(10)
4.24/1.97	YES
4.28/1.99	EOF