9.21/3.11	WORST_CASE(Omega(n^1), O(n^1))
9.21/3.12	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
9.21/3.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.21/3.12	
9.21/3.12	
9.21/3.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
9.21/3.12	
9.21/3.12	(0) CpxTRS
9.21/3.12	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
9.21/3.12	(2) CpxTRS
9.21/3.12	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
9.21/3.12	    (4) BOUNDS(1, n^1)
9.21/3.12	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
9.21/3.12	(6) TRS for Loop Detection
9.21/3.12	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
9.21/3.12	    (8) BEST
9.21/3.12	        (9) proven lower bound
9.21/3.12	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
9.21/3.12	            (11) BOUNDS(n^1, INF)
9.21/3.12	        (12) TRS for Loop Detection
9.21/3.12	
9.21/3.12	
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(0)
9.21/3.12	Obligation:
9.21/3.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
9.21/3.12	
9.21/3.12	
9.21/3.12	The TRS R consists of the following rules:
9.21/3.12	
9.21/3.12	   f(a) -> f(c(a))
9.21/3.12	   f(c(X)) -> X
9.21/3.12	   f(c(a)) -> f(d(b))
9.21/3.12	   f(a) -> f(d(a))
9.21/3.12	   f(d(X)) -> X
9.21/3.12	   f(c(b)) -> f(d(a))
9.21/3.12	   e(g(X)) -> e(X)
9.21/3.12	
9.21/3.12	S is empty.
9.21/3.12	Rewrite Strategy: INNERMOST
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(1) RelTrsToTrsProof (UPPER BOUND(ID))
9.21/3.12	transformed relative TRS to TRS
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(2)
9.21/3.12	Obligation:
9.21/3.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
9.21/3.12	
9.21/3.12	
9.21/3.12	The TRS R consists of the following rules:
9.21/3.12	
9.21/3.12	   f(a) -> f(c(a))
9.21/3.12	   f(c(X)) -> X
9.21/3.12	   f(c(a)) -> f(d(b))
9.21/3.12	   f(a) -> f(d(a))
9.21/3.12	   f(d(X)) -> X
9.21/3.12	   f(c(b)) -> f(d(a))
9.21/3.12	   e(g(X)) -> e(X)
9.21/3.12	
9.21/3.12	S is empty.
9.21/3.12	Rewrite Strategy: INNERMOST
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(3) CpxTrsMatchBoundsProof (FINISHED)
9.21/3.12	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. 
9.21/3.12	The certificate found is represented by the following graph.
9.21/3.12	
9.21/3.12	"[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
9.21/3.12	{(1,2,[f_1|0, e_1|0, a|1, c_1|1, d_1|1, b|1, g_1|1, e_1|1, a|2, b|2, b|3]), (1,3,[f_1|1]), (1,5,[f_1|1]), (1,7,[f_1|1]), (1,9,[f_1|2]), (2,2,[a|0, c_1|0, d_1|0, b|0, g_1|0]), (3,4,[c_1|1]), (4,2,[a|1]), (5,6,[d_1|1]), (6,2,[a|1]), (7,8,[d_1|1]), (8,2,[b|1]), (9,10,[d_1|2]), (10,2,[b|2])}"
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(4)
9.21/3.12	BOUNDS(1, n^1)
9.21/3.12	
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
9.21/3.12	Transformed a relative TRS into a decreasing-loop problem.
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(6)
9.21/3.12	Obligation:
9.21/3.12	Analyzing the following TRS for decreasing loops:
9.21/3.12	
9.21/3.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
9.21/3.12	
9.21/3.12	
9.21/3.12	The TRS R consists of the following rules:
9.21/3.12	
9.21/3.12	   f(a) -> f(c(a))
9.21/3.12	   f(c(X)) -> X
9.21/3.12	   f(c(a)) -> f(d(b))
9.21/3.12	   f(a) -> f(d(a))
9.21/3.12	   f(d(X)) -> X
9.21/3.12	   f(c(b)) -> f(d(a))
9.21/3.12	   e(g(X)) -> e(X)
9.21/3.12	
9.21/3.12	S is empty.
9.21/3.12	Rewrite Strategy: INNERMOST
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(7) DecreasingLoopProof (LOWER BOUND(ID))
9.21/3.12	The following loop(s) give(s) rise to the lower bound Omega(n^1):
9.21/3.12	
9.21/3.12	The rewrite sequence
9.21/3.12	
9.21/3.12	e(g(X)) ->^+ e(X)
9.21/3.12	
9.21/3.12	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
9.21/3.12	
9.21/3.12	The pumping substitution is [X / g(X)].
9.21/3.12	
9.21/3.12	The result substitution is [ ].
9.21/3.12	
9.21/3.12	
9.21/3.12	
9.21/3.12	
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(8)
9.21/3.12	Complex Obligation (BEST)
9.21/3.12	
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(9)
9.21/3.12	Obligation:
9.21/3.12	Proved the lower bound n^1 for the following obligation:
9.21/3.12	
9.21/3.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
9.21/3.12	
9.21/3.12	
9.21/3.12	The TRS R consists of the following rules:
9.21/3.12	
9.21/3.12	   f(a) -> f(c(a))
9.21/3.12	   f(c(X)) -> X
9.21/3.12	   f(c(a)) -> f(d(b))
9.21/3.12	   f(a) -> f(d(a))
9.21/3.12	   f(d(X)) -> X
9.21/3.12	   f(c(b)) -> f(d(a))
9.21/3.12	   e(g(X)) -> e(X)
9.21/3.12	
9.21/3.12	S is empty.
9.21/3.12	Rewrite Strategy: INNERMOST
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(10) LowerBoundPropagationProof (FINISHED)
9.21/3.12	Propagated lower bound.
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(11)
9.21/3.12	BOUNDS(n^1, INF)
9.21/3.12	
9.21/3.12	----------------------------------------
9.21/3.12	
9.21/3.12	(12)
9.21/3.12	Obligation:
9.21/3.12	Analyzing the following TRS for decreasing loops:
9.21/3.12	
9.21/3.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
9.21/3.12	
9.21/3.12	
9.21/3.12	The TRS R consists of the following rules:
9.21/3.12	
9.21/3.12	   f(a) -> f(c(a))
9.21/3.12	   f(c(X)) -> X
9.21/3.12	   f(c(a)) -> f(d(b))
9.21/3.12	   f(a) -> f(d(a))
9.21/3.12	   f(d(X)) -> X
9.21/3.12	   f(c(b)) -> f(d(a))
9.21/3.12	   e(g(X)) -> e(X)
9.21/3.12	
9.21/3.12	S is empty.
9.21/3.12	Rewrite Strategy: INNERMOST
9.21/3.15	EOF