3.04/1.59	WORST_CASE(NON_POLY, ?)
3.04/1.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.04/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.04/1.60	
3.04/1.60	
3.04/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.04/1.60	
3.04/1.60	(0) CpxTRS
3.04/1.60	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.04/1.60	(2) TRS for Loop Detection
3.04/1.60	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.04/1.60	(4) BEST
3.04/1.60	    (5) proven lower bound
3.04/1.60	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.04/1.60	        (7) BOUNDS(n^1, INF)
3.04/1.60	    (8) TRS for Loop Detection
3.04/1.60	        (9) DecreasingLoopProof [FINISHED, 0 ms]
3.04/1.60	        (10) BOUNDS(EXP, INF)
3.04/1.60	
3.04/1.60	
3.04/1.60	----------------------------------------
3.04/1.60	
3.04/1.60	(0)
3.04/1.60	Obligation:
3.04/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.04/1.60	
3.04/1.60	
3.04/1.60	The TRS R consists of the following rules:
3.04/1.60	
3.04/1.60	   f(0) -> 0
3.04/1.60	   f(s(0)) -> s(0)
3.04/1.60	   f(s(s(x))) -> p(h(g(x)))
3.04/1.60	   g(0) -> pair(s(0), s(0))
3.04/1.60	   g(s(x)) -> h(g(x))
3.04/1.60	   h(x) -> pair(+(p(x), q(x)), p(x))
3.04/1.60	   p(pair(x, y)) -> x
3.04/1.60	   q(pair(x, y)) -> y
3.04/1.60	   +(x, 0) -> x
3.04/1.60	   +(x, s(y)) -> s(+(x, y))
3.04/1.60	   f(s(s(x))) -> +(p(g(x)), q(g(x)))
3.04/1.60	   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
3.04/1.60	
3.04/1.60	S is empty.
3.04/1.60	Rewrite Strategy: INNERMOST
3.04/1.60	----------------------------------------
3.04/1.60	
3.04/1.60	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.04/1.60	Transformed a relative TRS into a decreasing-loop problem.
3.04/1.60	----------------------------------------
3.04/1.60	
3.04/1.60	(2)
3.04/1.60	Obligation:
3.04/1.60	Analyzing the following TRS for decreasing loops:
3.04/1.60	
3.04/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.04/1.60	
3.04/1.60	
3.04/1.60	The TRS R consists of the following rules:
3.04/1.60	
3.04/1.60	   f(0) -> 0
3.04/1.60	   f(s(0)) -> s(0)
3.04/1.60	   f(s(s(x))) -> p(h(g(x)))
3.04/1.60	   g(0) -> pair(s(0), s(0))
3.04/1.60	   g(s(x)) -> h(g(x))
3.04/1.60	   h(x) -> pair(+(p(x), q(x)), p(x))
3.04/1.60	   p(pair(x, y)) -> x
3.04/1.60	   q(pair(x, y)) -> y
3.04/1.60	   +(x, 0) -> x
3.04/1.60	   +(x, s(y)) -> s(+(x, y))
3.04/1.60	   f(s(s(x))) -> +(p(g(x)), q(g(x)))
3.04/1.60	   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
3.04/1.60	
3.04/1.60	S is empty.
3.04/1.60	Rewrite Strategy: INNERMOST
3.04/1.60	----------------------------------------
3.04/1.60	
3.04/1.60	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.04/1.60	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.04/1.60	
3.04/1.60	The rewrite sequence
3.04/1.60	
3.04/1.60	+(x, s(y)) ->^+ s(+(x, y))
3.04/1.60	
3.04/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.04/1.60	
3.04/1.60	The pumping substitution is [y / s(y)].
3.04/1.60	
3.04/1.60	The result substitution is [ ].
3.04/1.60	
3.04/1.60	
3.04/1.60	
3.04/1.60	
3.04/1.60	----------------------------------------
3.04/1.60	
3.04/1.60	(4)
3.04/1.60	Complex Obligation (BEST)
3.04/1.60	
3.04/1.60	----------------------------------------
3.04/1.60	
3.04/1.60	(5)
3.04/1.60	Obligation:
3.04/1.60	Proved the lower bound n^1 for the following obligation:
3.04/1.60	
3.04/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.04/1.60	
3.04/1.60	
3.04/1.60	The TRS R consists of the following rules:
3.04/1.60	
3.04/1.60	   f(0) -> 0
3.04/1.60	   f(s(0)) -> s(0)
3.04/1.60	   f(s(s(x))) -> p(h(g(x)))
3.04/1.60	   g(0) -> pair(s(0), s(0))
3.04/1.60	   g(s(x)) -> h(g(x))
3.04/1.60	   h(x) -> pair(+(p(x), q(x)), p(x))
3.04/1.60	   p(pair(x, y)) -> x
3.04/1.60	   q(pair(x, y)) -> y
3.04/1.60	   +(x, 0) -> x
3.04/1.60	   +(x, s(y)) -> s(+(x, y))
3.04/1.60	   f(s(s(x))) -> +(p(g(x)), q(g(x)))
3.04/1.60	   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
3.04/1.60	
3.04/1.60	S is empty.
3.04/1.60	Rewrite Strategy: INNERMOST
3.04/1.60	----------------------------------------
3.04/1.60	
3.04/1.60	(6) LowerBoundPropagationProof (FINISHED)
3.04/1.60	Propagated lower bound.
3.04/1.60	----------------------------------------
3.04/1.60	
3.04/1.60	(7)
3.04/1.60	BOUNDS(n^1, INF)
3.04/1.60	
3.04/1.60	----------------------------------------
3.04/1.60	
3.04/1.60	(8)
3.04/1.60	Obligation:
3.04/1.60	Analyzing the following TRS for decreasing loops:
3.04/1.60	
3.04/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.04/1.60	
3.04/1.60	
3.04/1.60	The TRS R consists of the following rules:
3.04/1.60	
3.04/1.60	   f(0) -> 0
3.04/1.60	   f(s(0)) -> s(0)
3.04/1.60	   f(s(s(x))) -> p(h(g(x)))
3.04/1.60	   g(0) -> pair(s(0), s(0))
3.04/1.60	   g(s(x)) -> h(g(x))
3.04/1.60	   h(x) -> pair(+(p(x), q(x)), p(x))
3.04/1.60	   p(pair(x, y)) -> x
3.04/1.60	   q(pair(x, y)) -> y
3.04/1.60	   +(x, 0) -> x
3.04/1.60	   +(x, s(y)) -> s(+(x, y))
3.04/1.60	   f(s(s(x))) -> +(p(g(x)), q(g(x)))
3.04/1.60	   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
3.04/1.60	
3.04/1.60	S is empty.
3.04/1.60	Rewrite Strategy: INNERMOST
3.04/1.60	----------------------------------------
3.04/1.60	
3.04/1.60	(9) DecreasingLoopProof (FINISHED)
3.04/1.60	The following loop(s) give(s) rise to the lower bound EXP:
3.04/1.60	
3.04/1.60	The rewrite sequence
3.04/1.60	
3.04/1.60	g(s(x)) ->^+ pair(+(p(g(x)), q(g(x))), p(g(x)))
3.04/1.60	
3.04/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
3.04/1.60	
3.04/1.60	The pumping substitution is [x / s(x)].
3.04/1.60	
3.04/1.60	The result substitution is [ ].
3.04/1.60	
3.04/1.60	
3.04/1.60	
3.04/1.60	The rewrite sequence
3.04/1.60	
3.04/1.60	g(s(x)) ->^+ pair(+(p(g(x)), q(g(x))), p(g(x)))
3.04/1.60	
3.04/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0].
3.04/1.60	
3.04/1.60	The pumping substitution is [x / s(x)].
3.04/1.60	
3.04/1.60	The result substitution is [ ].
3.04/1.60	
3.04/1.60	
3.04/1.60	
3.04/1.60	
3.04/1.60	----------------------------------------
3.04/1.60	
3.04/1.60	(10)
3.04/1.60	BOUNDS(EXP, INF)
3.20/1.63	EOF