7.38/2.70	WORST_CASE(Omega(n^1), O(n^1))
7.38/2.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
7.38/2.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.38/2.70	
7.38/2.70	
7.38/2.70	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.38/2.70	
7.38/2.70	(0) CpxTRS
7.38/2.70	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
7.38/2.70	(2) CpxTRS
7.38/2.70	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
7.38/2.70	    (4) BOUNDS(1, n^1)
7.38/2.70	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
7.38/2.70	(6) TRS for Loop Detection
7.38/2.70	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
7.38/2.70	    (8) BEST
7.38/2.70	        (9) proven lower bound
7.38/2.70	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
7.38/2.70	            (11) BOUNDS(n^1, INF)
7.38/2.70	        (12) TRS for Loop Detection
7.38/2.70	
7.38/2.70	
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(0)
7.38/2.70	Obligation:
7.38/2.70	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.38/2.70	
7.38/2.70	
7.38/2.70	The TRS R consists of the following rules:
7.38/2.70	
7.38/2.70	   f(X) -> g(n__h(n__f(X)))
7.38/2.70	   h(X) -> n__h(X)
7.38/2.70	   f(X) -> n__f(X)
7.38/2.70	   activate(n__h(X)) -> h(activate(X))
7.38/2.70	   activate(n__f(X)) -> f(activate(X))
7.38/2.70	   activate(X) -> X
7.38/2.70	
7.38/2.70	S is empty.
7.38/2.70	Rewrite Strategy: INNERMOST
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(1) RelTrsToTrsProof (UPPER BOUND(ID))
7.38/2.70	transformed relative TRS to TRS
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(2)
7.38/2.70	Obligation:
7.38/2.70	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
7.38/2.70	
7.38/2.70	
7.38/2.70	The TRS R consists of the following rules:
7.38/2.70	
7.38/2.70	   f(X) -> g(n__h(n__f(X)))
7.38/2.70	   h(X) -> n__h(X)
7.38/2.70	   f(X) -> n__f(X)
7.38/2.70	   activate(n__h(X)) -> h(activate(X))
7.38/2.70	   activate(n__f(X)) -> f(activate(X))
7.38/2.70	   activate(X) -> X
7.38/2.70	
7.38/2.70	S is empty.
7.38/2.70	Rewrite Strategy: INNERMOST
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(3) CpxTrsMatchBoundsProof (FINISHED)
7.38/2.70	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
7.38/2.70	The certificate found is represented by the following graph.
7.38/2.70	
7.38/2.70	"[1, 2, 3, 4, 5, 6, 7, 8]
7.38/2.70	{(1,2,[f_1|0, h_1|0, activate_1|0, n__f_1|1, n__h_1|1, g_1|1]), (1,3,[g_1|1]), (1,5,[h_1|1, n__h_1|2]), (1,6,[f_1|1, n__f_1|2]), (1,7,[g_1|2]), (2,2,[g_1|0, n__h_1|0, n__f_1|0]), (3,4,[n__h_1|1]), (4,2,[n__f_1|1]), (5,2,[activate_1|1, n__h_1|1, n__f_1|1, g_1|1]), (5,5,[h_1|1, n__h_1|2]), (5,6,[f_1|1, n__f_1|2]), (5,7,[g_1|2]), (6,2,[activate_1|1, n__h_1|1, n__f_1|1, g_1|1]), (6,5,[h_1|1, n__h_1|2]), (6,6,[f_1|1, n__f_1|2]), (6,7,[g_1|2]), (7,8,[n__h_1|2]), (8,6,[n__f_1|2])}"
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(4)
7.38/2.70	BOUNDS(1, n^1)
7.38/2.70	
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
7.38/2.70	Transformed a relative TRS into a decreasing-loop problem.
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(6)
7.38/2.70	Obligation:
7.38/2.70	Analyzing the following TRS for decreasing loops:
7.38/2.70	
7.38/2.70	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.38/2.70	
7.38/2.70	
7.38/2.70	The TRS R consists of the following rules:
7.38/2.70	
7.38/2.70	   f(X) -> g(n__h(n__f(X)))
7.38/2.70	   h(X) -> n__h(X)
7.38/2.70	   f(X) -> n__f(X)
7.38/2.70	   activate(n__h(X)) -> h(activate(X))
7.38/2.70	   activate(n__f(X)) -> f(activate(X))
7.38/2.70	   activate(X) -> X
7.38/2.70	
7.38/2.70	S is empty.
7.38/2.70	Rewrite Strategy: INNERMOST
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(7) DecreasingLoopProof (LOWER BOUND(ID))
7.38/2.70	The following loop(s) give(s) rise to the lower bound Omega(n^1):
7.38/2.70	
7.38/2.70	The rewrite sequence
7.38/2.70	
7.38/2.70	activate(n__f(X)) ->^+ f(activate(X))
7.38/2.70	
7.38/2.70	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
7.38/2.70	
7.38/2.70	The pumping substitution is [X / n__f(X)].
7.38/2.70	
7.38/2.70	The result substitution is [ ].
7.38/2.70	
7.38/2.70	
7.38/2.70	
7.38/2.70	
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(8)
7.38/2.70	Complex Obligation (BEST)
7.38/2.70	
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(9)
7.38/2.70	Obligation:
7.38/2.70	Proved the lower bound n^1 for the following obligation:
7.38/2.70	
7.38/2.70	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.38/2.70	
7.38/2.70	
7.38/2.70	The TRS R consists of the following rules:
7.38/2.70	
7.38/2.70	   f(X) -> g(n__h(n__f(X)))
7.38/2.70	   h(X) -> n__h(X)
7.38/2.70	   f(X) -> n__f(X)
7.38/2.70	   activate(n__h(X)) -> h(activate(X))
7.38/2.70	   activate(n__f(X)) -> f(activate(X))
7.38/2.70	   activate(X) -> X
7.38/2.70	
7.38/2.70	S is empty.
7.38/2.70	Rewrite Strategy: INNERMOST
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(10) LowerBoundPropagationProof (FINISHED)
7.38/2.70	Propagated lower bound.
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(11)
7.38/2.70	BOUNDS(n^1, INF)
7.38/2.70	
7.38/2.70	----------------------------------------
7.38/2.70	
7.38/2.70	(12)
7.38/2.70	Obligation:
7.38/2.70	Analyzing the following TRS for decreasing loops:
7.38/2.70	
7.38/2.70	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.38/2.70	
7.38/2.70	
7.38/2.70	The TRS R consists of the following rules:
7.38/2.70	
7.38/2.70	   f(X) -> g(n__h(n__f(X)))
7.38/2.70	   h(X) -> n__h(X)
7.38/2.70	   f(X) -> n__f(X)
7.38/2.70	   activate(n__h(X)) -> h(activate(X))
7.38/2.70	   activate(n__f(X)) -> f(activate(X))
7.38/2.70	   activate(X) -> X
7.38/2.70	
7.38/2.70	S is empty.
7.38/2.70	Rewrite Strategy: INNERMOST
7.67/2.74	EOF