3.42/1.67	WORST_CASE(NON_POLY, ?)
3.42/1.68	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.42/1.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.42/1.68	
3.42/1.68	
3.42/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.42/1.68	
3.42/1.68	(0) CpxTRS
3.42/1.68	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.42/1.68	(2) TRS for Loop Detection
3.42/1.68	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.42/1.68	(4) BEST
3.42/1.68	    (5) proven lower bound
3.42/1.68	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.42/1.68	        (7) BOUNDS(n^1, INF)
3.42/1.68	    (8) TRS for Loop Detection
3.42/1.68	        (9) DecreasingLoopProof [FINISHED, 26 ms]
3.42/1.68	        (10) BOUNDS(EXP, INF)
3.42/1.68	
3.42/1.68	
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(0)
3.42/1.68	Obligation:
3.42/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.42/1.68	
3.42/1.68	
3.42/1.68	The TRS R consists of the following rules:
3.42/1.68	
3.42/1.68	   plus(0, x) -> x
3.42/1.68	   plus(s(x), y) -> s(plus(x, y))
3.42/1.68	   times(0, y) -> 0
3.42/1.68	   times(s(x), y) -> plus(y, times(x, y))
3.42/1.68	   exp(x, 0) -> s(0)
3.42/1.68	   exp(x, s(y)) -> times(x, exp(x, y))
3.42/1.68	   ge(x, 0) -> true
3.42/1.68	   ge(0, s(x)) -> false
3.42/1.68	   ge(s(x), s(y)) -> ge(x, y)
3.42/1.68	   tower(x, y) -> towerIter(0, x, y, s(0))
3.42/1.68	   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
3.42/1.68	   help(true, c, x, y, z) -> z
3.42/1.68	   help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
3.42/1.68	
3.42/1.68	S is empty.
3.42/1.68	Rewrite Strategy: FULL
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.42/1.68	Transformed a relative TRS into a decreasing-loop problem.
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(2)
3.42/1.68	Obligation:
3.42/1.68	Analyzing the following TRS for decreasing loops:
3.42/1.68	
3.42/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.42/1.68	
3.42/1.68	
3.42/1.68	The TRS R consists of the following rules:
3.42/1.68	
3.42/1.68	   plus(0, x) -> x
3.42/1.68	   plus(s(x), y) -> s(plus(x, y))
3.42/1.68	   times(0, y) -> 0
3.42/1.68	   times(s(x), y) -> plus(y, times(x, y))
3.42/1.68	   exp(x, 0) -> s(0)
3.42/1.68	   exp(x, s(y)) -> times(x, exp(x, y))
3.42/1.68	   ge(x, 0) -> true
3.42/1.68	   ge(0, s(x)) -> false
3.42/1.68	   ge(s(x), s(y)) -> ge(x, y)
3.42/1.68	   tower(x, y) -> towerIter(0, x, y, s(0))
3.42/1.68	   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
3.42/1.68	   help(true, c, x, y, z) -> z
3.42/1.68	   help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
3.42/1.68	
3.42/1.68	S is empty.
3.42/1.68	Rewrite Strategy: FULL
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.42/1.68	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.42/1.68	
3.42/1.68	The rewrite sequence
3.42/1.68	
3.42/1.68	exp(x, s(y)) ->^+ times(x, exp(x, y))
3.42/1.68	
3.42/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.42/1.68	
3.42/1.68	The pumping substitution is [y / s(y)].
3.42/1.68	
3.42/1.68	The result substitution is [ ].
3.42/1.68	
3.42/1.68	
3.42/1.68	
3.42/1.68	
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(4)
3.42/1.68	Complex Obligation (BEST)
3.42/1.68	
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(5)
3.42/1.68	Obligation:
3.42/1.68	Proved the lower bound n^1 for the following obligation:
3.42/1.68	
3.42/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.42/1.68	
3.42/1.68	
3.42/1.68	The TRS R consists of the following rules:
3.42/1.68	
3.42/1.68	   plus(0, x) -> x
3.42/1.68	   plus(s(x), y) -> s(plus(x, y))
3.42/1.68	   times(0, y) -> 0
3.42/1.68	   times(s(x), y) -> plus(y, times(x, y))
3.42/1.68	   exp(x, 0) -> s(0)
3.42/1.68	   exp(x, s(y)) -> times(x, exp(x, y))
3.42/1.68	   ge(x, 0) -> true
3.42/1.68	   ge(0, s(x)) -> false
3.42/1.68	   ge(s(x), s(y)) -> ge(x, y)
3.42/1.68	   tower(x, y) -> towerIter(0, x, y, s(0))
3.42/1.68	   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
3.42/1.68	   help(true, c, x, y, z) -> z
3.42/1.68	   help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
3.42/1.68	
3.42/1.68	S is empty.
3.42/1.68	Rewrite Strategy: FULL
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(6) LowerBoundPropagationProof (FINISHED)
3.42/1.68	Propagated lower bound.
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(7)
3.42/1.68	BOUNDS(n^1, INF)
3.42/1.68	
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(8)
3.42/1.68	Obligation:
3.42/1.68	Analyzing the following TRS for decreasing loops:
3.42/1.68	
3.42/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.42/1.68	
3.42/1.68	
3.42/1.68	The TRS R consists of the following rules:
3.42/1.68	
3.42/1.68	   plus(0, x) -> x
3.42/1.68	   plus(s(x), y) -> s(plus(x, y))
3.42/1.68	   times(0, y) -> 0
3.42/1.68	   times(s(x), y) -> plus(y, times(x, y))
3.42/1.68	   exp(x, 0) -> s(0)
3.42/1.68	   exp(x, s(y)) -> times(x, exp(x, y))
3.42/1.68	   ge(x, 0) -> true
3.42/1.68	   ge(0, s(x)) -> false
3.42/1.68	   ge(s(x), s(y)) -> ge(x, y)
3.42/1.68	   tower(x, y) -> towerIter(0, x, y, s(0))
3.42/1.68	   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
3.42/1.68	   help(true, c, x, y, z) -> z
3.42/1.68	   help(false, c, x, y, z) -> towerIter(s(c), x, y, exp(y, z))
3.42/1.68	
3.42/1.68	S is empty.
3.42/1.68	Rewrite Strategy: FULL
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(9) DecreasingLoopProof (FINISHED)
3.42/1.68	The following loop(s) give(s) rise to the lower bound EXP:
3.42/1.68	
3.42/1.68	The rewrite sequence
3.42/1.68	
3.42/1.68	exp(s(x1_0), s(y)) ->^+ plus(exp(s(x1_0), y), times(x1_0, exp(s(x1_0), y)))
3.42/1.68	
3.42/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.42/1.68	
3.42/1.68	The pumping substitution is [y / s(y)].
3.42/1.68	
3.42/1.68	The result substitution is [ ].
3.42/1.68	
3.42/1.68	
3.42/1.68	
3.42/1.68	The rewrite sequence
3.42/1.68	
3.42/1.68	exp(s(x1_0), s(y)) ->^+ plus(exp(s(x1_0), y), times(x1_0, exp(s(x1_0), y)))
3.42/1.68	
3.42/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
3.42/1.68	
3.42/1.68	The pumping substitution is [y / s(y)].
3.42/1.68	
3.42/1.68	The result substitution is [ ].
3.42/1.68	
3.42/1.68	
3.42/1.68	
3.42/1.68	
3.42/1.68	----------------------------------------
3.42/1.68	
3.42/1.68	(10)
3.42/1.68	BOUNDS(EXP, INF)
3.63/1.71	EOF