3.33/1.61	WORST_CASE(Omega(n^1), O(n^1))
3.48/1.61	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.48/1.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.48/1.61	
3.48/1.61	
3.48/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.48/1.61	
3.48/1.61	(0) CpxTRS
3.48/1.61	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.48/1.61	(2) CpxTRS
3.48/1.61	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
3.48/1.61	    (4) BOUNDS(1, n^1)
3.48/1.61	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.48/1.61	(6) TRS for Loop Detection
3.48/1.61	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.48/1.61	    (8) BEST
3.48/1.61	        (9) proven lower bound
3.48/1.61	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.48/1.61	            (11) BOUNDS(n^1, INF)
3.48/1.61	        (12) TRS for Loop Detection
3.48/1.61	
3.48/1.61	
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(0)
3.48/1.61	Obligation:
3.48/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.48/1.61	
3.48/1.61	
3.48/1.61	The TRS R consists of the following rules:
3.48/1.61	
3.48/1.61	   f(f(X)) -> c(n__f(n__g(n__f(X))))
3.48/1.61	   c(X) -> d(activate(X))
3.48/1.61	   h(X) -> c(n__d(X))
3.48/1.61	   f(X) -> n__f(X)
3.48/1.61	   g(X) -> n__g(X)
3.48/1.61	   d(X) -> n__d(X)
3.48/1.61	   activate(n__f(X)) -> f(activate(X))
3.48/1.61	   activate(n__g(X)) -> g(X)
3.48/1.61	   activate(n__d(X)) -> d(X)
3.48/1.61	   activate(X) -> X
3.48/1.61	
3.48/1.61	S is empty.
3.48/1.61	Rewrite Strategy: FULL
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.48/1.61	transformed relative TRS to TRS
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(2)
3.48/1.61	Obligation:
3.48/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.48/1.61	
3.48/1.61	
3.48/1.61	The TRS R consists of the following rules:
3.48/1.61	
3.48/1.61	   f(f(X)) -> c(n__f(n__g(n__f(X))))
3.48/1.61	   c(X) -> d(activate(X))
3.48/1.61	   h(X) -> c(n__d(X))
3.48/1.61	   f(X) -> n__f(X)
3.48/1.61	   g(X) -> n__g(X)
3.48/1.61	   d(X) -> n__d(X)
3.48/1.61	   activate(n__f(X)) -> f(activate(X))
3.48/1.61	   activate(n__g(X)) -> g(X)
3.48/1.61	   activate(n__d(X)) -> d(X)
3.48/1.61	   activate(X) -> X
3.48/1.61	
3.48/1.61	S is empty.
3.48/1.61	Rewrite Strategy: FULL
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(3) CpxTrsMatchBoundsProof (FINISHED)
3.48/1.61	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. 
3.48/1.61	The certificate found is represented by the following graph.
3.48/1.61	
3.48/1.61	"[31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]
3.48/1.61	{(31,32,[f_1|0, c_1|0, h_1|0, g_1|0, d_1|0, activate_1|0, n__f_1|1, n__g_1|1, n__d_1|1, g_1|1, d_1|1, n__g_1|2, n__d_1|2]), (31,33,[d_1|1, n__d_1|2]), (31,34,[c_1|1]), (31,35,[f_1|1, n__f_1|2]), (31,36,[d_1|2, n__d_1|3]), (31,37,[c_1|2]), (31,40,[d_1|3, n__d_1|4]), (32,32,[n__f_1|0, n__g_1|0, n__d_1|0]), (33,32,[activate_1|1, n__f_1|1, g_1|1, n__g_1|1, d_1|1, n__d_1|1, n__g_1|2, n__d_1|2]), (33,35,[f_1|1, n__f_1|2]), (33,37,[c_1|2]), (33,40,[d_1|3, n__d_1|4]), (34,32,[n__d_1|1]), (35,32,[activate_1|1, n__f_1|1, g_1|1, n__g_1|1, d_1|1, n__d_1|1, n__g_1|2, n__d_1|2]), (35,35,[f_1|1, n__f_1|2]), (35,37,[c_1|2]), (35,40,[d_1|3, n__d_1|4]), (36,34,[activate_1|2]), (36,32,[d_1|2, n__d_1|2, n__d_1|3]), (37,38,[n__f_1|2]), (38,39,[n__g_1|2]), (39,35,[n__f_1|2]), (40,37,[activate_1|3]), (40,41,[f_1|3, n__f_1|4]), (40,38,[n__f_1|3]), (41,38,[activate_1|3]), (41,39,[g_1|3, n__g_1|3, n__g_1|4])}"
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(4)
3.48/1.61	BOUNDS(1, n^1)
3.48/1.61	
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.48/1.61	Transformed a relative TRS into a decreasing-loop problem.
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(6)
3.48/1.61	Obligation:
3.48/1.61	Analyzing the following TRS for decreasing loops:
3.48/1.61	
3.48/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.48/1.61	
3.48/1.61	
3.48/1.61	The TRS R consists of the following rules:
3.48/1.61	
3.48/1.61	   f(f(X)) -> c(n__f(n__g(n__f(X))))
3.48/1.61	   c(X) -> d(activate(X))
3.48/1.61	   h(X) -> c(n__d(X))
3.48/1.61	   f(X) -> n__f(X)
3.48/1.61	   g(X) -> n__g(X)
3.48/1.61	   d(X) -> n__d(X)
3.48/1.61	   activate(n__f(X)) -> f(activate(X))
3.48/1.61	   activate(n__g(X)) -> g(X)
3.48/1.61	   activate(n__d(X)) -> d(X)
3.48/1.61	   activate(X) -> X
3.48/1.61	
3.48/1.61	S is empty.
3.48/1.61	Rewrite Strategy: FULL
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.48/1.61	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.48/1.61	
3.48/1.61	The rewrite sequence
3.48/1.61	
3.48/1.61	activate(n__f(X)) ->^+ f(activate(X))
3.48/1.61	
3.48/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.48/1.61	
3.48/1.61	The pumping substitution is [X / n__f(X)].
3.48/1.61	
3.48/1.61	The result substitution is [ ].
3.48/1.61	
3.48/1.61	
3.48/1.61	
3.48/1.61	
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(8)
3.48/1.61	Complex Obligation (BEST)
3.48/1.61	
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(9)
3.48/1.61	Obligation:
3.48/1.61	Proved the lower bound n^1 for the following obligation:
3.48/1.61	
3.48/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.48/1.61	
3.48/1.61	
3.48/1.61	The TRS R consists of the following rules:
3.48/1.61	
3.48/1.61	   f(f(X)) -> c(n__f(n__g(n__f(X))))
3.48/1.61	   c(X) -> d(activate(X))
3.48/1.61	   h(X) -> c(n__d(X))
3.48/1.61	   f(X) -> n__f(X)
3.48/1.61	   g(X) -> n__g(X)
3.48/1.61	   d(X) -> n__d(X)
3.48/1.61	   activate(n__f(X)) -> f(activate(X))
3.48/1.61	   activate(n__g(X)) -> g(X)
3.48/1.61	   activate(n__d(X)) -> d(X)
3.48/1.61	   activate(X) -> X
3.48/1.61	
3.48/1.61	S is empty.
3.48/1.61	Rewrite Strategy: FULL
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(10) LowerBoundPropagationProof (FINISHED)
3.48/1.61	Propagated lower bound.
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(11)
3.48/1.61	BOUNDS(n^1, INF)
3.48/1.61	
3.48/1.61	----------------------------------------
3.48/1.61	
3.48/1.61	(12)
3.48/1.61	Obligation:
3.48/1.61	Analyzing the following TRS for decreasing loops:
3.48/1.61	
3.48/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.48/1.61	
3.48/1.61	
3.48/1.61	The TRS R consists of the following rules:
3.48/1.61	
3.48/1.61	   f(f(X)) -> c(n__f(n__g(n__f(X))))
3.48/1.61	   c(X) -> d(activate(X))
3.48/1.61	   h(X) -> c(n__d(X))
3.48/1.61	   f(X) -> n__f(X)
3.48/1.61	   g(X) -> n__g(X)
3.48/1.61	   d(X) -> n__d(X)
3.48/1.61	   activate(n__f(X)) -> f(activate(X))
3.48/1.61	   activate(n__g(X)) -> g(X)
3.48/1.61	   activate(n__d(X)) -> d(X)
3.48/1.61	   activate(X) -> X
3.48/1.61	
3.48/1.61	S is empty.
3.48/1.61	Rewrite Strategy: FULL
3.48/1.64	EOF