374.47/291.50	WORST_CASE(Omega(n^1), ?)
374.47/291.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
374.47/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
374.47/291.51	
374.47/291.51	
374.47/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
374.47/291.51	
374.47/291.51	(0) CpxTRS
374.47/291.51	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
374.47/291.51	(2) TRS for Loop Detection
374.47/291.51	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
374.47/291.51	(4) BEST
374.47/291.51	    (5) proven lower bound
374.47/291.51	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
374.47/291.51	        (7) BOUNDS(n^1, INF)
374.47/291.51	    (8) TRS for Loop Detection
374.47/291.51	
374.47/291.51	
374.47/291.51	----------------------------------------
374.47/291.51	
374.47/291.51	(0)
374.47/291.51	Obligation:
374.47/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
374.47/291.51	
374.47/291.51	
374.47/291.51	The TRS R consists of the following rules:
374.47/291.51	
374.47/291.51	   a(lambda(x), y) -> lambda(a(x, p(1, a(y, t))))
374.47/291.51	   a(p(x, y), z) -> p(a(x, z), a(y, z))
374.47/291.51	   a(a(x, y), z) -> a(x, a(y, z))
374.47/291.51	   a(id, x) -> x
374.47/291.51	   a(1, id) -> 1
374.47/291.51	   a(t, id) -> t
374.47/291.51	   a(1, p(x, y)) -> x
374.47/291.51	   a(t, p(x, y)) -> y
374.47/291.51	
374.47/291.51	S is empty.
374.47/291.51	Rewrite Strategy: FULL
374.47/291.51	----------------------------------------
374.47/291.51	
374.47/291.51	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
374.47/291.51	Transformed a relative TRS into a decreasing-loop problem.
374.47/291.51	----------------------------------------
374.47/291.51	
374.47/291.51	(2)
374.47/291.51	Obligation:
374.47/291.51	Analyzing the following TRS for decreasing loops:
374.47/291.51	
374.47/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
374.47/291.51	
374.47/291.51	
374.47/291.51	The TRS R consists of the following rules:
374.47/291.51	
374.47/291.51	   a(lambda(x), y) -> lambda(a(x, p(1, a(y, t))))
374.47/291.51	   a(p(x, y), z) -> p(a(x, z), a(y, z))
374.47/291.51	   a(a(x, y), z) -> a(x, a(y, z))
374.47/291.51	   a(id, x) -> x
374.47/291.51	   a(1, id) -> 1
374.47/291.51	   a(t, id) -> t
374.47/291.51	   a(1, p(x, y)) -> x
374.47/291.51	   a(t, p(x, y)) -> y
374.47/291.51	
374.47/291.51	S is empty.
374.47/291.51	Rewrite Strategy: FULL
374.47/291.51	----------------------------------------
374.47/291.51	
374.47/291.51	(3) DecreasingLoopProof (LOWER BOUND(ID))
374.47/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
374.47/291.51	
374.47/291.51	The rewrite sequence
374.47/291.51	
374.47/291.51	a(p(x, y), z) ->^+ p(a(x, z), a(y, z))
374.47/291.51	
374.47/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
374.47/291.51	
374.47/291.51	The pumping substitution is [x / p(x, y)].
374.47/291.51	
374.47/291.51	The result substitution is [ ].
374.47/291.51	
374.47/291.51	
374.47/291.51	
374.47/291.51	
374.47/291.51	----------------------------------------
374.47/291.51	
374.47/291.51	(4)
374.47/291.51	Complex Obligation (BEST)
374.47/291.51	
374.47/291.51	----------------------------------------
374.47/291.51	
374.47/291.51	(5)
374.47/291.51	Obligation:
374.47/291.51	Proved the lower bound n^1 for the following obligation:
374.47/291.51	
374.47/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
374.47/291.51	
374.47/291.51	
374.47/291.51	The TRS R consists of the following rules:
374.47/291.51	
374.47/291.51	   a(lambda(x), y) -> lambda(a(x, p(1, a(y, t))))
374.47/291.51	   a(p(x, y), z) -> p(a(x, z), a(y, z))
374.47/291.51	   a(a(x, y), z) -> a(x, a(y, z))
374.47/291.51	   a(id, x) -> x
374.47/291.51	   a(1, id) -> 1
374.47/291.51	   a(t, id) -> t
374.47/291.51	   a(1, p(x, y)) -> x
374.47/291.51	   a(t, p(x, y)) -> y
374.47/291.51	
374.47/291.51	S is empty.
374.47/291.51	Rewrite Strategy: FULL
374.47/291.51	----------------------------------------
374.47/291.51	
374.47/291.51	(6) LowerBoundPropagationProof (FINISHED)
374.47/291.51	Propagated lower bound.
374.47/291.51	----------------------------------------
374.47/291.51	
374.47/291.51	(7)
374.47/291.51	BOUNDS(n^1, INF)
374.47/291.51	
374.47/291.51	----------------------------------------
374.47/291.51	
374.47/291.51	(8)
374.47/291.51	Obligation:
374.47/291.51	Analyzing the following TRS for decreasing loops:
374.47/291.51	
374.47/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
374.47/291.51	
374.47/291.51	
374.47/291.51	The TRS R consists of the following rules:
374.47/291.51	
374.47/291.51	   a(lambda(x), y) -> lambda(a(x, p(1, a(y, t))))
374.47/291.51	   a(p(x, y), z) -> p(a(x, z), a(y, z))
374.47/291.51	   a(a(x, y), z) -> a(x, a(y, z))
374.47/291.51	   a(id, x) -> x
374.47/291.51	   a(1, id) -> 1
374.47/291.51	   a(t, id) -> t
374.47/291.51	   a(1, p(x, y)) -> x
374.47/291.51	   a(t, p(x, y)) -> y
374.47/291.51	
374.47/291.51	S is empty.
374.47/291.51	Rewrite Strategy: FULL
374.57/291.56	EOF