3.59/1.76	WORST_CASE(NON_POLY, ?)
3.59/1.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.59/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.59/1.77	
3.59/1.77	
3.59/1.77	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.77	
3.59/1.77	(0) CpxTRS
3.59/1.77	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.59/1.77	(2) TRS for Loop Detection
3.59/1.77	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.59/1.77	(4) BEST
3.59/1.77	    (5) proven lower bound
3.59/1.77	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.59/1.77	        (7) BOUNDS(n^1, INF)
3.59/1.77	    (8) TRS for Loop Detection
3.59/1.77	        (9) DecreasingLoopProof [FINISHED, 48 ms]
3.59/1.77	        (10) BOUNDS(EXP, INF)
3.59/1.77	
3.59/1.77	
3.59/1.77	----------------------------------------
3.59/1.77	
3.59/1.77	(0)
3.59/1.77	Obligation:
3.59/1.77	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.77	
3.59/1.77	
3.59/1.77	The TRS R consists of the following rules:
3.59/1.77	
3.59/1.77	   plus(s(s(x)), y) -> s(plus(x, s(y)))
3.59/1.77	   plus(x, s(s(y))) -> s(plus(s(x), y))
3.59/1.77	   plus(s(0), y) -> s(y)
3.59/1.77	   plus(0, y) -> y
3.59/1.77	   ack(0, y) -> s(y)
3.59/1.77	   ack(s(x), 0) -> ack(x, s(0))
3.59/1.77	   ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
3.59/1.77	
3.59/1.77	S is empty.
3.59/1.77	Rewrite Strategy: FULL
3.59/1.77	----------------------------------------
3.59/1.77	
3.59/1.77	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.59/1.77	Transformed a relative TRS into a decreasing-loop problem.
3.59/1.77	----------------------------------------
3.59/1.77	
3.59/1.77	(2)
3.59/1.77	Obligation:
3.59/1.77	Analyzing the following TRS for decreasing loops:
3.59/1.77	
3.59/1.77	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.77	
3.59/1.77	
3.59/1.77	The TRS R consists of the following rules:
3.59/1.77	
3.59/1.77	   plus(s(s(x)), y) -> s(plus(x, s(y)))
3.59/1.77	   plus(x, s(s(y))) -> s(plus(s(x), y))
3.59/1.77	   plus(s(0), y) -> s(y)
3.59/1.77	   plus(0, y) -> y
3.59/1.77	   ack(0, y) -> s(y)
3.59/1.77	   ack(s(x), 0) -> ack(x, s(0))
3.59/1.77	   ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
3.59/1.77	
3.59/1.77	S is empty.
3.59/1.77	Rewrite Strategy: FULL
3.59/1.77	----------------------------------------
3.59/1.77	
3.59/1.77	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.59/1.77	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.59/1.77	
3.59/1.77	The rewrite sequence
3.59/1.77	
3.59/1.77	plus(x, s(s(y))) ->^+ s(plus(s(x), y))
3.59/1.77	
3.59/1.77	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.59/1.77	
3.59/1.77	The pumping substitution is [y / s(s(y))].
3.59/1.77	
3.59/1.77	The result substitution is [x / s(x)].
3.59/1.77	
3.59/1.77	
3.59/1.77	
3.59/1.77	
3.59/1.77	----------------------------------------
3.59/1.77	
3.59/1.77	(4)
3.59/1.77	Complex Obligation (BEST)
3.59/1.77	
3.59/1.77	----------------------------------------
3.59/1.77	
3.59/1.77	(5)
3.59/1.77	Obligation:
3.59/1.77	Proved the lower bound n^1 for the following obligation:
3.59/1.77	
3.59/1.77	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.77	
3.59/1.77	
3.59/1.77	The TRS R consists of the following rules:
3.59/1.77	
3.59/1.77	   plus(s(s(x)), y) -> s(plus(x, s(y)))
3.59/1.77	   plus(x, s(s(y))) -> s(plus(s(x), y))
3.59/1.77	   plus(s(0), y) -> s(y)
3.59/1.77	   plus(0, y) -> y
3.59/1.77	   ack(0, y) -> s(y)
3.59/1.77	   ack(s(x), 0) -> ack(x, s(0))
3.59/1.77	   ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
3.59/1.77	
3.59/1.77	S is empty.
3.59/1.77	Rewrite Strategy: FULL
3.59/1.77	----------------------------------------
3.59/1.77	
3.59/1.77	(6) LowerBoundPropagationProof (FINISHED)
3.59/1.77	Propagated lower bound.
3.59/1.77	----------------------------------------
3.59/1.77	
3.59/1.77	(7)
3.59/1.77	BOUNDS(n^1, INF)
3.59/1.77	
3.59/1.77	----------------------------------------
3.59/1.77	
3.59/1.77	(8)
3.59/1.77	Obligation:
3.59/1.77	Analyzing the following TRS for decreasing loops:
3.59/1.77	
3.59/1.77	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.77	
3.59/1.77	
3.59/1.77	The TRS R consists of the following rules:
3.59/1.77	
3.59/1.77	   plus(s(s(x)), y) -> s(plus(x, s(y)))
3.59/1.77	   plus(x, s(s(y))) -> s(plus(s(x), y))
3.59/1.77	   plus(s(0), y) -> s(y)
3.59/1.77	   plus(0, y) -> y
3.59/1.77	   ack(0, y) -> s(y)
3.59/1.77	   ack(s(x), 0) -> ack(x, s(0))
3.59/1.77	   ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
3.59/1.77	
3.59/1.77	S is empty.
3.59/1.77	Rewrite Strategy: FULL
3.59/1.77	----------------------------------------
3.59/1.77	
3.59/1.77	(9) DecreasingLoopProof (FINISHED)
3.59/1.77	The following loop(s) give(s) rise to the lower bound EXP:
3.59/1.77	
3.59/1.77	The rewrite sequence
3.59/1.77	
3.59/1.77	ack(s(s(x1_1)), s(s(s(x1_0)))) ->^+ ack(x1_1, plus(plus(x1_0, s(ack(s(s(x1_1)), s(s(x1_0))))), ack(s(x1_1), plus(x1_0, s(ack(s(s(x1_1)), s(s(x1_0))))))))
3.59/1.77	
3.59/1.77	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,1,0].
3.59/1.77	
3.59/1.77	The pumping substitution is [x1_0 / s(x1_0)].
3.59/1.77	
3.59/1.77	The result substitution is [ ].
3.59/1.77	
3.59/1.77	
3.59/1.77	
3.59/1.77	The rewrite sequence
3.59/1.77	
3.59/1.77	ack(s(s(x1_1)), s(s(s(x1_0)))) ->^+ ack(x1_1, plus(plus(x1_0, s(ack(s(s(x1_1)), s(s(x1_0))))), ack(s(x1_1), plus(x1_0, s(ack(s(s(x1_1)), s(s(x1_0))))))))
3.59/1.77	
3.59/1.77	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1,1,1,0].
3.59/1.77	
3.59/1.77	The pumping substitution is [x1_0 / s(x1_0)].
3.59/1.77	
3.59/1.77	The result substitution is [ ].
3.59/1.77	
3.59/1.77	
3.59/1.77	
3.59/1.77	
3.59/1.77	----------------------------------------
3.59/1.77	
3.59/1.77	(10)
3.59/1.77	BOUNDS(EXP, INF)
3.86/1.80	EOF