4.49/2.16	WORST_CASE(NON_POLY, ?)
4.49/2.16	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.49/2.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.49/2.16	
4.49/2.16	
4.49/2.16	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.49/2.16	
4.49/2.16	(0) CpxTRS
4.49/2.16	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.49/2.16	(2) TRS for Loop Detection
4.49/2.16	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.49/2.16	(4) BEST
4.49/2.16	    (5) proven lower bound
4.49/2.16	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
4.49/2.16	        (7) BOUNDS(n^1, INF)
4.49/2.16	    (8) TRS for Loop Detection
4.49/2.16	        (9) InfiniteLowerBoundProof [FINISHED, 156 ms]
4.49/2.16	        (10) BOUNDS(INF, INF)
4.49/2.16	
4.49/2.16	
4.49/2.16	----------------------------------------
4.49/2.16	
4.49/2.16	(0)
4.49/2.16	Obligation:
4.49/2.16	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.49/2.16	
4.49/2.16	
4.49/2.16	The TRS R consists of the following rules:
4.49/2.16	
4.49/2.16	   f(h(x), y) -> h(f(y, f(x, h(f(a, a)))))
4.49/2.16	
4.49/2.16	S is empty.
4.49/2.16	Rewrite Strategy: FULL
4.49/2.16	----------------------------------------
4.49/2.16	
4.49/2.16	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.49/2.16	Transformed a relative TRS into a decreasing-loop problem.
4.49/2.16	----------------------------------------
4.49/2.16	
4.49/2.16	(2)
4.49/2.16	Obligation:
4.49/2.16	Analyzing the following TRS for decreasing loops:
4.49/2.16	
4.49/2.16	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.49/2.16	
4.49/2.16	
4.49/2.16	The TRS R consists of the following rules:
4.49/2.16	
4.49/2.16	   f(h(x), y) -> h(f(y, f(x, h(f(a, a)))))
4.49/2.16	
4.49/2.16	S is empty.
4.49/2.16	Rewrite Strategy: FULL
4.49/2.16	----------------------------------------
4.49/2.16	
4.49/2.16	(3) DecreasingLoopProof (LOWER BOUND(ID))
4.49/2.16	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.49/2.16	
4.49/2.16	The rewrite sequence
4.49/2.16	
4.49/2.16	f(h(x), y) ->^+ h(f(y, f(x, h(f(a, a)))))
4.49/2.16	
4.49/2.16	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
4.49/2.16	
4.49/2.16	The pumping substitution is [x / h(x)].
4.49/2.16	
4.49/2.16	The result substitution is [y / h(f(a, a))].
4.49/2.16	
4.49/2.16	
4.49/2.16	
4.49/2.16	
4.49/2.16	----------------------------------------
4.49/2.16	
4.49/2.16	(4)
4.49/2.16	Complex Obligation (BEST)
4.49/2.16	
4.49/2.16	----------------------------------------
4.49/2.16	
4.49/2.16	(5)
4.49/2.16	Obligation:
4.49/2.16	Proved the lower bound n^1 for the following obligation:
4.49/2.16	
4.49/2.16	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.49/2.16	
4.49/2.16	
4.49/2.16	The TRS R consists of the following rules:
4.49/2.16	
4.49/2.16	   f(h(x), y) -> h(f(y, f(x, h(f(a, a)))))
4.49/2.16	
4.49/2.16	S is empty.
4.49/2.16	Rewrite Strategy: FULL
4.49/2.16	----------------------------------------
4.49/2.16	
4.49/2.16	(6) LowerBoundPropagationProof (FINISHED)
4.49/2.16	Propagated lower bound.
4.49/2.16	----------------------------------------
4.49/2.16	
4.49/2.16	(7)
4.49/2.16	BOUNDS(n^1, INF)
4.49/2.16	
4.49/2.16	----------------------------------------
4.49/2.16	
4.49/2.16	(8)
4.49/2.16	Obligation:
4.49/2.16	Analyzing the following TRS for decreasing loops:
4.49/2.16	
4.49/2.16	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.49/2.16	
4.49/2.16	
4.49/2.16	The TRS R consists of the following rules:
4.49/2.16	
4.49/2.16	   f(h(x), y) -> h(f(y, f(x, h(f(a, a)))))
4.49/2.16	
4.49/2.16	S is empty.
4.49/2.16	Rewrite Strategy: FULL
4.49/2.16	----------------------------------------
4.49/2.16	
4.49/2.16	(9) InfiniteLowerBoundProof (FINISHED)
4.49/2.16	The following loop proves infinite runtime complexity:
4.49/2.16	
4.49/2.16	The rewrite sequence
4.49/2.16	
4.49/2.16	f(h(h(x1_0)), h(h(x1_2))) ->^+ h(h(f(h(h(f(f(x1_0, h(f(a, a))), f(f(a, a), h(f(a, a)))))), h(h(f(f(x1_2, h(f(a, a))), f(f(a, a), h(f(a, a)))))))))
4.49/2.16	
4.49/2.16	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
4.49/2.16	
4.49/2.16	The pumping substitution is [ ].
4.49/2.16	
4.49/2.16	The result substitution is [x1_0 / f(f(x1_0, h(f(a, a))), f(f(a, a), h(f(a, a)))), x1_2 / f(f(x1_2, h(f(a, a))), f(f(a, a), h(f(a, a))))].
4.49/2.16	
4.49/2.16	
4.49/2.16	
4.49/2.16	
4.49/2.16	----------------------------------------
4.49/2.16	
4.49/2.16	(10)
4.49/2.16	BOUNDS(INF, INF)
4.49/2.20	EOF