6.68/2.50	WORST_CASE(NON_POLY, ?)
6.68/2.51	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.68/2.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.68/2.51	
6.68/2.51	
6.68/2.51	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).
6.68/2.51	
6.68/2.51	(0) CpxRelTRS
6.68/2.51	(1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 184 ms]
6.68/2.51	(2) CpxRelTRS
6.68/2.51	(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
6.68/2.51	(4) TRS for Loop Detection
6.68/2.51	(5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
6.68/2.51	(6) BEST
6.68/2.51	    (7) proven lower bound
6.68/2.51	        (8) LowerBoundPropagationProof [FINISHED, 0 ms]
6.68/2.51	        (9) BOUNDS(n^1, INF)
6.68/2.51	    (10) TRS for Loop Detection
6.68/2.51	        (11) InfiniteLowerBoundProof [FINISHED, 549 ms]
6.68/2.51	        (12) BOUNDS(INF, INF)
6.68/2.51	
6.68/2.51	
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(0)
6.68/2.51	Obligation:
6.68/2.51	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).
6.68/2.51	
6.68/2.51	
6.68/2.51	The TRS R consists of the following rules:
6.68/2.51	
6.68/2.51	   lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs)
6.68/2.51	   g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   lt0(x, Nil) -> False
6.68/2.51	   g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))
6.68/2.51	   f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)))
6.68/2.51	   number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil))
6.68/2.51	
6.68/2.51	The (relative) TRS S consists of the following rules:
6.68/2.51	
6.68/2.51	   g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y))
6.68/2.51	   g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs)
6.68/2.51	   f[Ite][False][Ite](False, Cons(x, xs), y) -> xs
6.68/2.51	   f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs
6.68/2.51	   f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y)
6.68/2.51	   f[Ite][False][Ite](True, x, y) -> x
6.68/2.51	
6.68/2.51	Rewrite Strategy: INNERMOST
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
6.68/2.51	proved termination of relative rules
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(2)
6.68/2.51	Obligation:
6.68/2.51	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).
6.68/2.51	
6.68/2.51	
6.68/2.51	The TRS R consists of the following rules:
6.68/2.51	
6.68/2.51	   lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs)
6.68/2.51	   g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   lt0(x, Nil) -> False
6.68/2.51	   g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))
6.68/2.51	   f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)))
6.68/2.51	   number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil))
6.68/2.51	
6.68/2.51	The (relative) TRS S consists of the following rules:
6.68/2.51	
6.68/2.51	   g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y))
6.68/2.51	   g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs)
6.68/2.51	   f[Ite][False][Ite](False, Cons(x, xs), y) -> xs
6.68/2.51	   f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs
6.68/2.51	   f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y)
6.68/2.51	   f[Ite][False][Ite](True, x, y) -> x
6.68/2.51	
6.68/2.51	Rewrite Strategy: INNERMOST
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
6.68/2.51	Transformed a relative TRS into a decreasing-loop problem.
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(4)
6.68/2.51	Obligation:
6.68/2.51	Analyzing the following TRS for decreasing loops:
6.68/2.51	
6.68/2.51	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).
6.68/2.51	
6.68/2.51	
6.68/2.51	The TRS R consists of the following rules:
6.68/2.51	
6.68/2.51	   lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs)
6.68/2.51	   g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   lt0(x, Nil) -> False
6.68/2.51	   g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))
6.68/2.51	   f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)))
6.68/2.51	   number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil))
6.68/2.51	
6.68/2.51	The (relative) TRS S consists of the following rules:
6.68/2.51	
6.68/2.51	   g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y))
6.68/2.51	   g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs)
6.68/2.51	   f[Ite][False][Ite](False, Cons(x, xs), y) -> xs
6.68/2.51	   f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs
6.68/2.51	   f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y)
6.68/2.51	   f[Ite][False][Ite](True, x, y) -> x
6.68/2.51	
6.68/2.51	Rewrite Strategy: INNERMOST
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(5) DecreasingLoopProof (LOWER BOUND(ID))
6.68/2.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
6.68/2.51	
6.68/2.51	The rewrite sequence
6.68/2.51	
6.68/2.51	lt0(Cons(x', xs'), Cons(x, xs)) ->^+ lt0(xs', xs)
6.68/2.51	
6.68/2.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
6.68/2.51	
6.68/2.51	The pumping substitution is [xs' / Cons(x', xs'), xs / Cons(x, xs)].
6.68/2.51	
6.68/2.51	The result substitution is [ ].
6.68/2.51	
6.68/2.51	
6.68/2.51	
6.68/2.51	
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(6)
6.68/2.51	Complex Obligation (BEST)
6.68/2.51	
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(7)
6.68/2.51	Obligation:
6.68/2.51	Proved the lower bound n^1 for the following obligation:
6.68/2.51	
6.68/2.51	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).
6.68/2.51	
6.68/2.51	
6.68/2.51	The TRS R consists of the following rules:
6.68/2.51	
6.68/2.51	   lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs)
6.68/2.51	   g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   lt0(x, Nil) -> False
6.68/2.51	   g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))
6.68/2.51	   f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)))
6.68/2.51	   number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil))
6.68/2.51	
6.68/2.51	The (relative) TRS S consists of the following rules:
6.68/2.51	
6.68/2.51	   g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y))
6.68/2.51	   g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs)
6.68/2.51	   f[Ite][False][Ite](False, Cons(x, xs), y) -> xs
6.68/2.51	   f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs
6.68/2.51	   f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y)
6.68/2.51	   f[Ite][False][Ite](True, x, y) -> x
6.68/2.51	
6.68/2.51	Rewrite Strategy: INNERMOST
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(8) LowerBoundPropagationProof (FINISHED)
6.68/2.51	Propagated lower bound.
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(9)
6.68/2.51	BOUNDS(n^1, INF)
6.68/2.51	
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(10)
6.68/2.51	Obligation:
6.68/2.51	Analyzing the following TRS for decreasing loops:
6.68/2.51	
6.68/2.51	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).
6.68/2.51	
6.68/2.51	
6.68/2.51	The TRS R consists of the following rules:
6.68/2.51	
6.68/2.51	   lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs)
6.68/2.51	   g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   lt0(x, Nil) -> False
6.68/2.51	   g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))
6.68/2.51	   f(x, Cons(x', xs)) -> f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)))
6.68/2.51	   number42 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
6.68/2.51	   goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil))
6.68/2.51	
6.68/2.51	The (relative) TRS S consists of the following rules:
6.68/2.51	
6.68/2.51	   g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y))
6.68/2.51	   g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs)
6.68/2.51	   f[Ite][False][Ite](False, Cons(x, xs), y) -> xs
6.68/2.51	   f[Ite][False][Ite](True, x', Cons(x, xs)) -> xs
6.68/2.51	   f[Ite][False][Ite](False, x, y) -> Cons(Cons(Nil, Nil), y)
6.68/2.51	   f[Ite][False][Ite](True, x, y) -> x
6.68/2.51	
6.68/2.51	Rewrite Strategy: INNERMOST
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(11) InfiniteLowerBoundProof (FINISHED)
6.68/2.51	The following loop proves infinite runtime complexity:
6.68/2.51	
6.68/2.51	The rewrite sequence
6.68/2.51	
6.68/2.51	f(Cons(x'1_0, xs'2_0), Cons(x', xs)) ->^+ f(Cons(Cons(Nil, Nil), Cons(x', xs)), Cons(Cons(Nil, Nil), Cons(x', xs)))
6.68/2.51	
6.68/2.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
6.68/2.51	
6.68/2.51	The pumping substitution is [ ].
6.68/2.51	
6.68/2.51	The result substitution is [x'1_0 / Cons(Nil, Nil), xs'2_0 / Cons(x', xs), x' / Cons(Nil, Nil), xs / Cons(x', xs)].
6.68/2.51	
6.68/2.51	
6.68/2.51	
6.68/2.51	
6.68/2.51	----------------------------------------
6.68/2.51	
6.68/2.51	(12)
6.68/2.51	BOUNDS(INF, INF)
6.83/2.55	EOF