28.69/8.26	YES
29.09/8.38	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
29.09/8.38	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
29.09/8.38	
29.09/8.38	
29.09/8.38	Termination w.r.t. Q of the given QTRS could be proven:
29.09/8.38	
29.09/8.38	(0) QTRS
29.09/8.38	(1) QTRSRRRProof [EQUIVALENT, 66 ms]
29.09/8.38	(2) QTRS
29.09/8.38	(3) DependencyPairsProof [EQUIVALENT, 23 ms]
29.09/8.38	(4) QDP
29.09/8.38	(5) QDPOrderProof [EQUIVALENT, 15 ms]
29.09/8.38	(6) QDP
29.09/8.38	(7) MRRProof [EQUIVALENT, 1 ms]
29.09/8.38	(8) QDP
29.09/8.38	(9) PisEmptyProof [EQUIVALENT, 0 ms]
29.09/8.38	(10) YES
29.09/8.38	
29.09/8.38	
29.09/8.38	----------------------------------------
29.09/8.38	
29.09/8.38	(0)
29.09/8.38	Obligation:
29.09/8.38	Q restricted rewrite system:
29.09/8.38	The TRS R consists of the following rules:
29.09/8.38	
29.09/8.38	   0(0(*(*(x1)))) -> *(*(1(1(x1))))
29.09/8.38	   1(1(*(*(x1)))) -> 0(0(#(#(x1))))
29.09/8.38	   #(#(0(0(x1)))) -> 0(0(#(#(x1))))
29.09/8.38	   #(#(1(1(x1)))) -> 1(1(#(#(x1))))
29.09/8.38	   #(#($($(x1)))) -> *(*($($(x1))))
29.09/8.38	   #(#(#(#(x1)))) -> #(#(x1))
29.09/8.38	   #(#(*(*(x1)))) -> *(*(x1))
29.09/8.38	
29.09/8.38	Q is empty.
29.09/8.38	
29.09/8.38	----------------------------------------
29.09/8.38	
29.09/8.38	(1) QTRSRRRProof (EQUIVALENT)
29.09/8.38	Used ordering:
29.09/8.38	Polynomial interpretation [POLO]:
29.09/8.38	
29.09/8.38	   POL(#(x_1)) = 1 + x_1
29.09/8.38	   POL($(x_1)) = x_1
29.09/8.38	   POL(*(x_1)) = 1 + x_1
29.09/8.38	   POL(0(x_1)) = x_1
29.09/8.38	   POL(1(x_1)) = x_1
29.09/8.38	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
29.09/8.38	
29.09/8.38	   #(#(#(#(x1)))) -> #(#(x1))
29.09/8.38	   #(#(*(*(x1)))) -> *(*(x1))
29.09/8.38	
29.09/8.38	
29.09/8.38	
29.09/8.38	
29.09/8.38	----------------------------------------
29.09/8.38	
29.09/8.38	(2)
29.09/8.38	Obligation:
29.09/8.38	Q restricted rewrite system:
29.09/8.38	The TRS R consists of the following rules:
29.09/8.38	
29.09/8.38	   0(0(*(*(x1)))) -> *(*(1(1(x1))))
29.09/8.38	   1(1(*(*(x1)))) -> 0(0(#(#(x1))))
29.09/8.38	   #(#(0(0(x1)))) -> 0(0(#(#(x1))))
29.09/8.38	   #(#(1(1(x1)))) -> 1(1(#(#(x1))))
29.09/8.38	   #(#($($(x1)))) -> *(*($($(x1))))
29.09/8.38	
29.09/8.38	Q is empty.
29.09/8.38	
29.09/8.38	----------------------------------------
29.09/8.38	
29.09/8.38	(3) DependencyPairsProof (EQUIVALENT)
29.09/8.38	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
29.09/8.38	----------------------------------------
29.09/8.38	
29.09/8.38	(4)
29.09/8.38	Obligation:
29.09/8.38	Q DP problem:
29.09/8.38	The TRS P consists of the following rules:
29.09/8.38	
29.09/8.38	   0^1(0(*(*(x1)))) -> 1^1(1(x1))
29.09/8.38	   0^1(0(*(*(x1)))) -> 1^1(x1)
29.09/8.38	   1^1(1(*(*(x1)))) -> 0^1(0(#(#(x1))))
29.09/8.38	   1^1(1(*(*(x1)))) -> 0^1(#(#(x1)))
29.09/8.38	   1^1(1(*(*(x1)))) -> #^1(#(x1))
29.09/8.38	   1^1(1(*(*(x1)))) -> #^1(x1)
29.09/8.38	   #^1(#(0(0(x1)))) -> 0^1(0(#(#(x1))))
29.09/8.38	   #^1(#(0(0(x1)))) -> 0^1(#(#(x1)))
29.09/8.38	   #^1(#(0(0(x1)))) -> #^1(#(x1))
29.09/8.38	   #^1(#(0(0(x1)))) -> #^1(x1)
29.09/8.38	   #^1(#(1(1(x1)))) -> 1^1(1(#(#(x1))))
29.09/8.38	   #^1(#(1(1(x1)))) -> 1^1(#(#(x1)))
29.09/8.38	   #^1(#(1(1(x1)))) -> #^1(#(x1))
29.09/8.38	   #^1(#(1(1(x1)))) -> #^1(x1)
29.09/8.38	
29.09/8.38	The TRS R consists of the following rules:
29.09/8.38	
29.09/8.38	   0(0(*(*(x1)))) -> *(*(1(1(x1))))
29.09/8.38	   1(1(*(*(x1)))) -> 0(0(#(#(x1))))
29.09/8.38	   #(#(0(0(x1)))) -> 0(0(#(#(x1))))
29.09/8.38	   #(#(1(1(x1)))) -> 1(1(#(#(x1))))
29.09/8.38	   #(#($($(x1)))) -> *(*($($(x1))))
29.09/8.38	
29.09/8.38	Q is empty.
29.09/8.38	We have to consider all minimal (P,Q,R)-chains.
29.09/8.38	----------------------------------------
29.09/8.38	
29.09/8.38	(5) QDPOrderProof (EQUIVALENT)
29.09/8.38	We use the reduction pair processor [LPAR04,JAR06].
29.09/8.38	
29.09/8.38	
29.09/8.38	The following pairs can be oriented strictly and are deleted.
29.09/8.38	
29.09/8.38	   0^1(0(*(*(x1)))) -> 1^1(x1)
29.09/8.38	   1^1(1(*(*(x1)))) -> 0^1(#(#(x1)))
29.09/8.38	   1^1(1(*(*(x1)))) -> #^1(#(x1))
29.09/8.38	   1^1(1(*(*(x1)))) -> #^1(x1)
29.09/8.38	   #^1(#(0(0(x1)))) -> 0^1(0(#(#(x1))))
29.09/8.38	   #^1(#(0(0(x1)))) -> 0^1(#(#(x1)))
29.09/8.38	   #^1(#(0(0(x1)))) -> #^1(#(x1))
29.09/8.38	   #^1(#(0(0(x1)))) -> #^1(x1)
29.09/8.38	   #^1(#(1(1(x1)))) -> 1^1(1(#(#(x1))))
29.09/8.38	   #^1(#(1(1(x1)))) -> 1^1(#(#(x1)))
29.09/8.38	   #^1(#(1(1(x1)))) -> #^1(#(x1))
29.09/8.38	   #^1(#(1(1(x1)))) -> #^1(x1)
29.09/8.38	The remaining pairs can at least be oriented weakly.
29.09/8.38	Used ordering:  Polynomial interpretation [POLO]:
29.09/8.38	
29.09/8.38	   POL(#(x_1)) = x_1
29.09/8.38	   POL(#^1(x_1)) = x_1
29.09/8.38	   POL($(x_1)) = x_1
29.09/8.38	   POL(*(x_1)) = x_1
29.09/8.38	   POL(0(x_1)) = 1 + x_1
29.09/8.38	   POL(0^1(x_1)) = x_1
29.09/8.38	   POL(1(x_1)) = 1 + x_1
29.09/8.38	   POL(1^1(x_1)) = x_1
29.09/8.38	
29.09/8.38	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.09/8.38	
29.09/8.38	   1(1(*(*(x1)))) -> 0(0(#(#(x1))))
29.09/8.38	   #(#(0(0(x1)))) -> 0(0(#(#(x1))))
29.09/8.38	   #(#(1(1(x1)))) -> 1(1(#(#(x1))))
29.09/8.38	   #(#($($(x1)))) -> *(*($($(x1))))
29.09/8.38	   0(0(*(*(x1)))) -> *(*(1(1(x1))))
29.09/8.38	
29.09/8.38	
29.09/8.38	----------------------------------------
29.09/8.38	
29.09/8.38	(6)
29.09/8.38	Obligation:
29.09/8.38	Q DP problem:
29.09/8.38	The TRS P consists of the following rules:
29.09/8.38	
29.09/8.38	   0^1(0(*(*(x1)))) -> 1^1(1(x1))
29.09/8.38	   1^1(1(*(*(x1)))) -> 0^1(0(#(#(x1))))
29.09/8.38	
29.09/8.38	The TRS R consists of the following rules:
29.09/8.38	
29.09/8.38	   0(0(*(*(x1)))) -> *(*(1(1(x1))))
29.09/8.38	   1(1(*(*(x1)))) -> 0(0(#(#(x1))))
29.09/8.38	   #(#(0(0(x1)))) -> 0(0(#(#(x1))))
29.09/8.38	   #(#(1(1(x1)))) -> 1(1(#(#(x1))))
29.09/8.38	   #(#($($(x1)))) -> *(*($($(x1))))
29.09/8.38	
29.09/8.38	Q is empty.
29.09/8.38	We have to consider all minimal (P,Q,R)-chains.
29.09/8.38	----------------------------------------
29.09/8.38	
29.09/8.38	(7) MRRProof (EQUIVALENT)
29.09/8.38	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
29.09/8.38	
29.09/8.38	Strictly oriented dependency pairs:
29.09/8.38	
29.09/8.38	   0^1(0(*(*(x1)))) -> 1^1(1(x1))
29.09/8.38	   1^1(1(*(*(x1)))) -> 0^1(0(#(#(x1))))
29.09/8.38	
29.09/8.38	
29.09/8.38	Used ordering: Polynomial interpretation [POLO]:
29.09/8.38	
29.09/8.38	   POL(#(x_1)) = 1 + 2*x_1
29.09/8.38	   POL($(x_1)) = 2*x_1
29.09/8.38	   POL(*(x_1)) = 1 + 2*x_1
29.09/8.38	   POL(0(x_1)) = x_1
29.09/8.38	   POL(0^1(x_1)) = 2*x_1
29.09/8.38	   POL(1(x_1)) = x_1
29.09/8.38	   POL(1^1(x_1)) = 3 + 2*x_1
29.09/8.38	
29.09/8.38	
29.09/8.38	----------------------------------------
29.09/8.38	
29.09/8.38	(8)
29.09/8.38	Obligation:
29.09/8.38	Q DP problem:
29.09/8.38	P is empty.
29.09/8.38	The TRS R consists of the following rules:
29.09/8.38	
29.09/8.38	   0(0(*(*(x1)))) -> *(*(1(1(x1))))
29.09/8.38	   1(1(*(*(x1)))) -> 0(0(#(#(x1))))
29.09/8.38	   #(#(0(0(x1)))) -> 0(0(#(#(x1))))
29.09/8.38	   #(#(1(1(x1)))) -> 1(1(#(#(x1))))
29.09/8.38	   #(#($($(x1)))) -> *(*($($(x1))))
29.09/8.38	
29.09/8.38	Q is empty.
29.09/8.38	We have to consider all minimal (P,Q,R)-chains.
29.09/8.38	----------------------------------------
29.09/8.38	
29.09/8.38	(9) PisEmptyProof (EQUIVALENT)
29.09/8.38	The TRS P is empty. Hence, there is no (P,Q,R) chain.
29.09/8.38	----------------------------------------
29.09/8.38	
29.09/8.38	(10)
29.09/8.38	YES
29.40/8.47	EOF