3.90/1.82	WORST_CASE(NON_POLY, ?)
4.28/1.82	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
4.28/1.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.28/1.82	
4.28/1.82	
4.28/1.82	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.28/1.82	
4.28/1.82	(0) CpxTRS
4.28/1.82	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.28/1.82	(2) TRS for Loop Detection
4.28/1.82	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.28/1.82	(4) BEST
4.28/1.82	    (5) proven lower bound
4.28/1.82	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
4.28/1.82	        (7) BOUNDS(n^1, INF)
4.28/1.82	    (8) TRS for Loop Detection
4.28/1.82	        (9) InfiniteLowerBoundProof [FINISHED, 87 ms]
4.28/1.82	        (10) BOUNDS(INF, INF)
4.28/1.82	
4.28/1.82	
4.28/1.82	----------------------------------------
4.28/1.82	
4.28/1.82	(0)
4.28/1.82	Obligation:
4.28/1.82	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.28/1.82	
4.28/1.82	
4.28/1.82	The TRS R consists of the following rules:
4.28/1.82	
4.28/1.82	   f(h(x), y) -> h(f(y, f(x, h(a))))
4.28/1.82	
4.28/1.82	S is empty.
4.28/1.82	Rewrite Strategy: FULL
4.28/1.82	----------------------------------------
4.28/1.82	
4.28/1.82	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.28/1.82	Transformed a relative TRS into a decreasing-loop problem.
4.28/1.82	----------------------------------------
4.28/1.82	
4.28/1.82	(2)
4.28/1.82	Obligation:
4.28/1.82	Analyzing the following TRS for decreasing loops:
4.28/1.82	
4.28/1.82	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.28/1.82	
4.28/1.82	
4.28/1.82	The TRS R consists of the following rules:
4.28/1.82	
4.28/1.82	   f(h(x), y) -> h(f(y, f(x, h(a))))
4.28/1.82	
4.28/1.82	S is empty.
4.28/1.82	Rewrite Strategy: FULL
4.28/1.82	----------------------------------------
4.28/1.82	
4.28/1.82	(3) DecreasingLoopProof (LOWER BOUND(ID))
4.28/1.82	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.28/1.82	
4.28/1.82	The rewrite sequence
4.28/1.82	
4.28/1.82	f(h(x), y) ->^+ h(f(y, f(x, h(a))))
4.28/1.82	
4.28/1.82	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
4.28/1.82	
4.28/1.82	The pumping substitution is [x / h(x)].
4.28/1.82	
4.28/1.82	The result substitution is [y / h(a)].
4.28/1.82	
4.28/1.82	
4.28/1.82	
4.28/1.82	
4.28/1.82	----------------------------------------
4.28/1.82	
4.28/1.82	(4)
4.28/1.82	Complex Obligation (BEST)
4.28/1.82	
4.28/1.82	----------------------------------------
4.28/1.82	
4.28/1.82	(5)
4.28/1.82	Obligation:
4.28/1.82	Proved the lower bound n^1 for the following obligation:
4.28/1.82	
4.28/1.82	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.28/1.82	
4.28/1.82	
4.28/1.82	The TRS R consists of the following rules:
4.28/1.82	
4.28/1.82	   f(h(x), y) -> h(f(y, f(x, h(a))))
4.28/1.82	
4.28/1.82	S is empty.
4.28/1.82	Rewrite Strategy: FULL
4.28/1.82	----------------------------------------
4.28/1.82	
4.28/1.82	(6) LowerBoundPropagationProof (FINISHED)
4.28/1.82	Propagated lower bound.
4.28/1.82	----------------------------------------
4.28/1.82	
4.28/1.82	(7)
4.28/1.82	BOUNDS(n^1, INF)
4.28/1.82	
4.28/1.82	----------------------------------------
4.28/1.82	
4.28/1.82	(8)
4.28/1.82	Obligation:
4.28/1.82	Analyzing the following TRS for decreasing loops:
4.28/1.82	
4.28/1.82	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.28/1.82	
4.28/1.82	
4.28/1.82	The TRS R consists of the following rules:
4.28/1.82	
4.28/1.82	   f(h(x), y) -> h(f(y, f(x, h(a))))
4.28/1.82	
4.28/1.82	S is empty.
4.28/1.82	Rewrite Strategy: FULL
4.28/1.82	----------------------------------------
4.28/1.82	
4.28/1.82	(9) InfiniteLowerBoundProof (FINISHED)
4.28/1.82	The following loop proves infinite runtime complexity:
4.28/1.82	
4.28/1.82	The rewrite sequence
4.28/1.82	
4.28/1.82	f(h(h(x1_0)), h(h(x1_2))) ->^+ h(h(f(h(h(f(f(x1_0, h(a)), f(a, h(a))))), h(h(f(f(x1_2, h(a)), f(a, h(a))))))))
4.28/1.82	
4.28/1.82	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
4.28/1.82	
4.28/1.82	The pumping substitution is [ ].
4.28/1.82	
4.28/1.82	The result substitution is [x1_0 / f(f(x1_0, h(a)), f(a, h(a))), x1_2 / f(f(x1_2, h(a)), f(a, h(a)))].
4.28/1.82	
4.28/1.82	
4.28/1.82	
4.28/1.82	
4.28/1.82	----------------------------------------
4.28/1.82	
4.28/1.82	(10)
4.28/1.82	BOUNDS(INF, INF)
4.28/1.86	EOF