3.51/1.77	YES
3.51/1.78	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.51/1.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.51/1.78	
3.51/1.78	
3.51/1.78	Termination w.r.t. Q of the given QTRS could be proven:
3.51/1.78	
3.51/1.78	(0) QTRS
3.51/1.78	(1) QTRSRRRProof [EQUIVALENT, 24 ms]
3.51/1.78	(2) QTRS
3.51/1.78	(3) RisEmptyProof [EQUIVALENT, 0 ms]
3.51/1.78	(4) YES
3.51/1.78	
3.51/1.78	
3.51/1.78	----------------------------------------
3.51/1.78	
3.51/1.78	(0)
3.51/1.78	Obligation:
3.51/1.78	Q restricted rewrite system:
3.51/1.78	The TRS R consists of the following rules:
3.51/1.78	
3.51/1.78	   append(@l1, @l2) -> append#1(@l1, @l2)
3.51/1.78	   append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2))
3.51/1.78	   append#1(nil, @l2) -> @l2
3.51/1.78	   appendAll(@l) -> appendAll#1(@l)
3.51/1.78	   appendAll#1(::(@l1, @ls)) -> append(@l1, appendAll(@ls))
3.51/1.78	   appendAll#1(nil) -> nil
3.51/1.78	   appendAll2(@l) -> appendAll2#1(@l)
3.51/1.78	   appendAll2#1(::(@l1, @ls)) -> append(appendAll(@l1), appendAll2(@ls))
3.51/1.78	   appendAll2#1(nil) -> nil
3.51/1.78	   appendAll3(@l) -> appendAll3#1(@l)
3.51/1.78	   appendAll3#1(::(@l1, @ls)) -> append(appendAll2(@l1), appendAll3(@ls))
3.51/1.78	   appendAll3#1(nil) -> nil
3.51/1.78	
3.51/1.78	The set Q consists of the following terms:
3.51/1.78	
3.51/1.78	   append(x0, x1)
3.51/1.78	   append#1(::(x0, x1), x2)
3.51/1.78	   append#1(nil, x0)
3.51/1.78	   appendAll(x0)
3.51/1.78	   appendAll#1(::(x0, x1))
3.51/1.78	   appendAll#1(nil)
3.51/1.78	   appendAll2(x0)
3.51/1.78	   appendAll2#1(::(x0, x1))
3.51/1.78	   appendAll2#1(nil)
3.51/1.78	   appendAll3(x0)
3.51/1.78	   appendAll3#1(::(x0, x1))
3.51/1.78	   appendAll3#1(nil)
3.51/1.78	
3.51/1.78	
3.51/1.78	----------------------------------------
3.51/1.78	
3.51/1.78	(1) QTRSRRRProof (EQUIVALENT)
3.51/1.78	Used ordering:
3.51/1.78	Knuth-Bendix order [KBO] with precedence:appendAll3#1_1 > appendAll3_1 > appendAll2#1_1 > append_2 > append#1_2 > appendAll2_1 > ::_2 > appendAll_1 > appendAll#1_1 > nil
3.51/1.78	
3.51/1.78	and weight map:
3.51/1.78	
3.51/1.78	   nil=1
3.51/1.78	   appendAll_1=2
3.51/1.78	   appendAll#1_1=2
3.51/1.78	   appendAll2_1=2
3.51/1.78	   appendAll2#1_1=1
3.51/1.78	   appendAll3_1=1
3.51/1.78	   appendAll3#1_1=0
3.51/1.78	   append_2=0
3.51/1.78	   append#1_2=0
3.51/1.78	   ::_2=3
3.51/1.78	
3.51/1.78	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.51/1.78	
3.51/1.78	   append(@l1, @l2) -> append#1(@l1, @l2)
3.51/1.78	   append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2))
3.51/1.78	   append#1(nil, @l2) -> @l2
3.51/1.78	   appendAll(@l) -> appendAll#1(@l)
3.51/1.78	   appendAll#1(::(@l1, @ls)) -> append(@l1, appendAll(@ls))
3.51/1.78	   appendAll#1(nil) -> nil
3.51/1.78	   appendAll2(@l) -> appendAll2#1(@l)
3.51/1.78	   appendAll2#1(::(@l1, @ls)) -> append(appendAll(@l1), appendAll2(@ls))
3.51/1.78	   appendAll2#1(nil) -> nil
3.51/1.78	   appendAll3(@l) -> appendAll3#1(@l)
3.51/1.78	   appendAll3#1(::(@l1, @ls)) -> append(appendAll2(@l1), appendAll3(@ls))
3.51/1.78	   appendAll3#1(nil) -> nil
3.51/1.78	
3.51/1.78	
3.51/1.78	
3.51/1.78	
3.51/1.78	----------------------------------------
3.51/1.78	
3.51/1.78	(2)
3.51/1.78	Obligation:
3.51/1.78	Q restricted rewrite system:
3.51/1.78	R is empty.
3.51/1.78	The set Q consists of the following terms:
3.51/1.78	
3.51/1.78	   append(x0, x1)
3.51/1.78	   append#1(::(x0, x1), x2)
3.51/1.78	   append#1(nil, x0)
3.51/1.78	   appendAll(x0)
3.51/1.78	   appendAll#1(::(x0, x1))
3.51/1.78	   appendAll#1(nil)
3.51/1.78	   appendAll2(x0)
3.51/1.78	   appendAll2#1(::(x0, x1))
3.51/1.78	   appendAll2#1(nil)
3.51/1.78	   appendAll3(x0)
3.51/1.78	   appendAll3#1(::(x0, x1))
3.51/1.78	   appendAll3#1(nil)
3.51/1.78	
3.51/1.78	
3.51/1.78	----------------------------------------
3.51/1.78	
3.51/1.78	(3) RisEmptyProof (EQUIVALENT)
3.51/1.78	The TRS R is empty. Hence, termination is trivially proven.
3.51/1.78	----------------------------------------
3.51/1.78	
3.51/1.78	(4)
3.51/1.78	YES
3.51/1.81	EOF