3.64/1.78	YES
3.64/1.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.64/1.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.64/1.79	
3.64/1.79	
3.64/1.79	Termination w.r.t. Q of the given QTRS could be proven:
3.64/1.79	
3.64/1.79	(0) QTRS
3.64/1.79	(1) QTRSRRRProof [EQUIVALENT, 118 ms]
3.64/1.79	(2) QTRS
3.64/1.79	(3) QTRSRRRProof [EQUIVALENT, 0 ms]
3.64/1.79	(4) QTRS
3.64/1.79	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.64/1.79	(6) YES
3.64/1.79	
3.64/1.79	
3.64/1.79	----------------------------------------
3.64/1.79	
3.64/1.79	(0)
3.64/1.79	Obligation:
3.64/1.79	Q restricted rewrite system:
3.64/1.79	The TRS R consists of the following rules:
3.64/1.79	
3.64/1.79	   append(@l1, @l2) -> append#1(@l1, @l2)
3.64/1.79	   append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2))
3.64/1.79	   append#1(nil, @l2) -> @l2
3.64/1.79	   subtrees(@t) -> subtrees#1(@t)
3.64/1.79	   subtrees#1(leaf) -> nil
3.64/1.79	   subtrees#1(node(@x, @t1, @t2)) -> subtrees#2(subtrees(@t1), @t1, @t2, @x)
3.64/1.79	   subtrees#2(@l1, @t1, @t2, @x) -> subtrees#3(subtrees(@t2), @l1, @t1, @t2, @x)
3.64/1.79	   subtrees#3(@l2, @l1, @t1, @t2, @x) -> ::(node(@x, @t1, @t2), append(@l1, @l2))
3.64/1.79	
3.64/1.79	The set Q consists of the following terms:
3.64/1.79	
3.64/1.79	   append(x0, x1)
3.64/1.79	   append#1(::(x0, x1), x2)
3.64/1.79	   append#1(nil, x0)
3.64/1.79	   subtrees(x0)
3.64/1.79	   subtrees#1(leaf)
3.64/1.79	   subtrees#1(node(x0, x1, x2))
3.64/1.79	   subtrees#2(x0, x1, x2, x3)
3.64/1.79	   subtrees#3(x0, x1, x2, x3, x4)
3.64/1.79	
3.64/1.79	
3.64/1.79	----------------------------------------
3.64/1.79	
3.64/1.79	(1) QTRSRRRProof (EQUIVALENT)
3.64/1.79	Used ordering:
3.64/1.79	Quasi precedence:
3.64/1.79	[nil, subtrees_1, subtrees#1_1, subtrees#2_4] > [node_3, subtrees#3_5] > [append_2, append#1_2] > ::_2
3.64/1.79	
3.64/1.79	
3.64/1.79	Status:
3.64/1.79	append_2: multiset status
3.64/1.79	append#1_2: multiset status
3.64/1.79	::_2: multiset status
3.64/1.79	nil: multiset status
3.64/1.79	subtrees_1: [1]
3.64/1.79	subtrees#1_1: [1]
3.64/1.79	leaf: multiset status
3.64/1.79	node_3: multiset status
3.64/1.79	subtrees#2_4: [3,1,2,4]
3.64/1.79	subtrees#3_5: multiset status
3.64/1.79	
3.64/1.79	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.64/1.79	
3.64/1.79	   append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2))
3.64/1.79	   append#1(nil, @l2) -> @l2
3.64/1.79	   subtrees#1(leaf) -> nil
3.64/1.79	   subtrees#1(node(@x, @t1, @t2)) -> subtrees#2(subtrees(@t1), @t1, @t2, @x)
3.64/1.79	   subtrees#2(@l1, @t1, @t2, @x) -> subtrees#3(subtrees(@t2), @l1, @t1, @t2, @x)
3.64/1.79	   subtrees#3(@l2, @l1, @t1, @t2, @x) -> ::(node(@x, @t1, @t2), append(@l1, @l2))
3.64/1.79	
3.64/1.79	
3.64/1.79	
3.64/1.79	
3.64/1.79	----------------------------------------
3.64/1.79	
3.64/1.79	(2)
3.64/1.79	Obligation:
3.64/1.79	Q restricted rewrite system:
3.64/1.79	The TRS R consists of the following rules:
3.64/1.79	
3.64/1.79	   append(@l1, @l2) -> append#1(@l1, @l2)
3.64/1.79	   subtrees(@t) -> subtrees#1(@t)
3.64/1.79	
3.64/1.79	The set Q consists of the following terms:
3.64/1.79	
3.64/1.79	   append(x0, x1)
3.64/1.79	   append#1(::(x0, x1), x2)
3.64/1.79	   append#1(nil, x0)
3.64/1.79	   subtrees(x0)
3.64/1.79	   subtrees#1(leaf)
3.64/1.79	   subtrees#1(node(x0, x1, x2))
3.64/1.79	   subtrees#2(x0, x1, x2, x3)
3.64/1.79	   subtrees#3(x0, x1, x2, x3, x4)
3.64/1.79	
3.64/1.79	
3.64/1.79	----------------------------------------
3.64/1.79	
3.64/1.79	(3) QTRSRRRProof (EQUIVALENT)
3.64/1.79	Used ordering:
3.64/1.79	Knuth-Bendix order [KBO] with precedence:subtrees_1 > subtrees#1_1 > append_2 > append#1_2
3.64/1.79	
3.64/1.79	and weight map:
3.64/1.79	
3.64/1.79	   subtrees_1=1
3.64/1.79	   subtrees#1_1=1
3.64/1.79	   append_2=0
3.64/1.79	   append#1_2=0
3.64/1.79	
3.64/1.79	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.64/1.79	
3.64/1.79	   append(@l1, @l2) -> append#1(@l1, @l2)
3.64/1.79	   subtrees(@t) -> subtrees#1(@t)
3.64/1.79	
3.64/1.79	
3.64/1.79	
3.64/1.79	
3.64/1.79	----------------------------------------
3.64/1.79	
3.64/1.79	(4)
3.64/1.79	Obligation:
3.64/1.79	Q restricted rewrite system:
3.64/1.79	R is empty.
3.64/1.79	The set Q consists of the following terms:
3.64/1.79	
3.64/1.79	   append(x0, x1)
3.64/1.79	   append#1(::(x0, x1), x2)
3.64/1.79	   append#1(nil, x0)
3.64/1.79	   subtrees(x0)
3.64/1.79	   subtrees#1(leaf)
3.64/1.79	   subtrees#1(node(x0, x1, x2))
3.64/1.79	   subtrees#2(x0, x1, x2, x3)
3.64/1.79	   subtrees#3(x0, x1, x2, x3, x4)
3.64/1.79	
3.64/1.79	
3.64/1.79	----------------------------------------
3.64/1.79	
3.64/1.79	(5) RisEmptyProof (EQUIVALENT)
3.64/1.79	The TRS R is empty. Hence, termination is trivially proven.
3.64/1.79	----------------------------------------
3.64/1.79	
3.64/1.79	(6)
3.64/1.79	YES
3.64/1.80	EOF