3.98/1.87	WORST_CASE(NON_POLY, ?)
3.98/1.88	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.98/1.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.98/1.88	
3.98/1.88	
3.98/1.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.98/1.88	
3.98/1.88	(0) CpxTRS
3.98/1.88	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.98/1.88	(2) TRS for Loop Detection
3.98/1.88	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.98/1.88	(4) BEST
3.98/1.88	    (5) proven lower bound
3.98/1.88	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.98/1.88	        (7) BOUNDS(n^1, INF)
3.98/1.88	    (8) TRS for Loop Detection
3.98/1.88	        (9) InfiniteLowerBoundProof [FINISHED, 79 ms]
3.98/1.88	        (10) BOUNDS(INF, INF)
3.98/1.88	
3.98/1.88	
3.98/1.88	----------------------------------------
3.98/1.88	
3.98/1.88	(0)
3.98/1.88	Obligation:
3.98/1.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.98/1.88	
3.98/1.88	
3.98/1.88	The TRS R consists of the following rules:
3.98/1.88	
3.98/1.88	   f(x, h(y)) -> h(f(f(h(a), y), x))
3.98/1.88	
3.98/1.88	S is empty.
3.98/1.88	Rewrite Strategy: FULL
3.98/1.88	----------------------------------------
3.98/1.88	
3.98/1.88	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.98/1.88	Transformed a relative TRS into a decreasing-loop problem.
3.98/1.88	----------------------------------------
3.98/1.88	
3.98/1.88	(2)
3.98/1.88	Obligation:
3.98/1.88	Analyzing the following TRS for decreasing loops:
3.98/1.88	
3.98/1.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.98/1.88	
3.98/1.88	
3.98/1.88	The TRS R consists of the following rules:
3.98/1.88	
3.98/1.88	   f(x, h(y)) -> h(f(f(h(a), y), x))
3.98/1.88	
3.98/1.88	S is empty.
3.98/1.88	Rewrite Strategy: FULL
3.98/1.88	----------------------------------------
3.98/1.88	
3.98/1.88	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.98/1.88	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.98/1.88	
3.98/1.88	The rewrite sequence
3.98/1.88	
3.98/1.88	f(x, h(y)) ->^+ h(f(f(h(a), y), x))
3.98/1.88	
3.98/1.88	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.98/1.88	
3.98/1.88	The pumping substitution is [y / h(y)].
3.98/1.88	
3.98/1.88	The result substitution is [x / h(a)].
3.98/1.88	
3.98/1.88	
3.98/1.88	
3.98/1.88	
3.98/1.88	----------------------------------------
3.98/1.88	
3.98/1.88	(4)
3.98/1.88	Complex Obligation (BEST)
3.98/1.88	
3.98/1.88	----------------------------------------
3.98/1.88	
3.98/1.88	(5)
3.98/1.88	Obligation:
3.98/1.88	Proved the lower bound n^1 for the following obligation:
3.98/1.88	
3.98/1.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.98/1.88	
3.98/1.88	
3.98/1.88	The TRS R consists of the following rules:
3.98/1.88	
3.98/1.88	   f(x, h(y)) -> h(f(f(h(a), y), x))
3.98/1.88	
3.98/1.88	S is empty.
3.98/1.88	Rewrite Strategy: FULL
3.98/1.88	----------------------------------------
3.98/1.88	
3.98/1.88	(6) LowerBoundPropagationProof (FINISHED)
3.98/1.88	Propagated lower bound.
3.98/1.88	----------------------------------------
3.98/1.88	
3.98/1.88	(7)
3.98/1.88	BOUNDS(n^1, INF)
3.98/1.88	
3.98/1.88	----------------------------------------
3.98/1.88	
3.98/1.88	(8)
3.98/1.88	Obligation:
3.98/1.88	Analyzing the following TRS for decreasing loops:
3.98/1.88	
3.98/1.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.98/1.88	
3.98/1.88	
3.98/1.88	The TRS R consists of the following rules:
3.98/1.88	
3.98/1.88	   f(x, h(y)) -> h(f(f(h(a), y), x))
3.98/1.88	
3.98/1.88	S is empty.
3.98/1.88	Rewrite Strategy: FULL
3.98/1.88	----------------------------------------
3.98/1.88	
3.98/1.88	(9) InfiniteLowerBoundProof (FINISHED)
3.98/1.88	The following loop proves infinite runtime complexity:
3.98/1.88	
3.98/1.88	The rewrite sequence
3.98/1.88	
3.98/1.88	f(h(h(y2_2)), h(h(y2_0))) ->^+ h(h(f(h(h(f(f(h(a), a), f(h(a), y2_2)))), h(h(f(f(h(a), a), f(h(a), y2_0)))))))
3.98/1.88	
3.98/1.88	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.98/1.88	
3.98/1.88	The pumping substitution is [ ].
3.98/1.88	
3.98/1.88	The result substitution is [y2_2 / f(f(h(a), a), f(h(a), y2_2)), y2_0 / f(f(h(a), a), f(h(a), y2_0))].
3.98/1.88	
3.98/1.88	
3.98/1.88	
3.98/1.88	
3.98/1.88	----------------------------------------
3.98/1.88	
3.98/1.88	(10)
3.98/1.88	BOUNDS(INF, INF)
4.26/1.92	EOF