3.06/1.67	WORST_CASE(NON_POLY, ?)
3.06/1.68	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.06/1.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.06/1.68	
3.06/1.68	
3.06/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.06/1.68	
3.06/1.68	(0) CpxTRS
3.06/1.68	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.06/1.68	(2) TRS for Loop Detection
3.06/1.68	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.06/1.68	(4) BEST
3.06/1.68	    (5) proven lower bound
3.06/1.68	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.06/1.68	        (7) BOUNDS(n^1, INF)
3.06/1.68	    (8) TRS for Loop Detection
3.06/1.68	        (9) DecreasingLoopProof [FINISHED, 0 ms]
3.06/1.68	        (10) BOUNDS(EXP, INF)
3.06/1.68	
3.06/1.68	
3.06/1.68	----------------------------------------
3.06/1.68	
3.06/1.68	(0)
3.06/1.68	Obligation:
3.06/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.06/1.68	
3.06/1.68	
3.06/1.68	The TRS R consists of the following rules:
3.06/1.68	
3.06/1.68	   t(N) -> cs(r(q(N)), nt(ns(N)))
3.06/1.68	   q(0) -> 0
3.06/1.68	   q(s(X)) -> s(p(q(X), d(X)))
3.06/1.68	   d(0) -> 0
3.06/1.68	   d(s(X)) -> s(s(d(X)))
3.06/1.68	   p(0, X) -> X
3.06/1.68	   p(X, 0) -> X
3.06/1.68	   p(s(X), s(Y)) -> s(s(p(X, Y)))
3.06/1.68	   f(0, X) -> nil
3.06/1.68	   f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z)))
3.06/1.68	   t(X) -> nt(X)
3.06/1.68	   s(X) -> ns(X)
3.06/1.68	   f(X1, X2) -> nf(X1, X2)
3.06/1.68	   a(nt(X)) -> t(a(X))
3.06/1.68	   a(ns(X)) -> s(a(X))
3.06/1.68	   a(nf(X1, X2)) -> f(a(X1), a(X2))
3.06/1.68	   a(X) -> X
3.06/1.68	
3.06/1.68	S is empty.
3.06/1.68	Rewrite Strategy: FULL
3.06/1.68	----------------------------------------
3.06/1.68	
3.06/1.68	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.06/1.68	Transformed a relative TRS into a decreasing-loop problem.
3.06/1.68	----------------------------------------
3.06/1.68	
3.06/1.68	(2)
3.06/1.68	Obligation:
3.06/1.68	Analyzing the following TRS for decreasing loops:
3.06/1.68	
3.06/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.06/1.68	
3.06/1.68	
3.06/1.68	The TRS R consists of the following rules:
3.06/1.68	
3.06/1.68	   t(N) -> cs(r(q(N)), nt(ns(N)))
3.06/1.68	   q(0) -> 0
3.06/1.68	   q(s(X)) -> s(p(q(X), d(X)))
3.06/1.68	   d(0) -> 0
3.06/1.68	   d(s(X)) -> s(s(d(X)))
3.06/1.68	   p(0, X) -> X
3.06/1.68	   p(X, 0) -> X
3.06/1.68	   p(s(X), s(Y)) -> s(s(p(X, Y)))
3.06/1.68	   f(0, X) -> nil
3.06/1.68	   f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z)))
3.06/1.68	   t(X) -> nt(X)
3.06/1.68	   s(X) -> ns(X)
3.06/1.68	   f(X1, X2) -> nf(X1, X2)
3.06/1.68	   a(nt(X)) -> t(a(X))
3.06/1.68	   a(ns(X)) -> s(a(X))
3.06/1.68	   a(nf(X1, X2)) -> f(a(X1), a(X2))
3.06/1.68	   a(X) -> X
3.06/1.68	
3.06/1.68	S is empty.
3.06/1.68	Rewrite Strategy: FULL
3.06/1.68	----------------------------------------
3.06/1.68	
3.06/1.68	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.06/1.68	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.06/1.68	
3.06/1.68	The rewrite sequence
3.06/1.68	
3.06/1.68	a(nt(X)) ->^+ t(a(X))
3.06/1.68	
3.06/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.06/1.68	
3.06/1.68	The pumping substitution is [X / nt(X)].
3.06/1.68	
3.06/1.68	The result substitution is [ ].
3.06/1.68	
3.06/1.68	
3.06/1.68	
3.06/1.68	
3.06/1.68	----------------------------------------
3.06/1.68	
3.06/1.68	(4)
3.06/1.68	Complex Obligation (BEST)
3.06/1.68	
3.06/1.68	----------------------------------------
3.06/1.68	
3.06/1.68	(5)
3.06/1.68	Obligation:
3.06/1.68	Proved the lower bound n^1 for the following obligation:
3.06/1.68	
3.06/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.06/1.68	
3.06/1.68	
3.06/1.68	The TRS R consists of the following rules:
3.06/1.68	
3.06/1.68	   t(N) -> cs(r(q(N)), nt(ns(N)))
3.06/1.68	   q(0) -> 0
3.06/1.68	   q(s(X)) -> s(p(q(X), d(X)))
3.06/1.68	   d(0) -> 0
3.06/1.68	   d(s(X)) -> s(s(d(X)))
3.06/1.68	   p(0, X) -> X
3.06/1.68	   p(X, 0) -> X
3.06/1.68	   p(s(X), s(Y)) -> s(s(p(X, Y)))
3.06/1.68	   f(0, X) -> nil
3.06/1.68	   f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z)))
3.06/1.68	   t(X) -> nt(X)
3.06/1.68	   s(X) -> ns(X)
3.06/1.68	   f(X1, X2) -> nf(X1, X2)
3.06/1.68	   a(nt(X)) -> t(a(X))
3.06/1.68	   a(ns(X)) -> s(a(X))
3.06/1.68	   a(nf(X1, X2)) -> f(a(X1), a(X2))
3.06/1.68	   a(X) -> X
3.06/1.68	
3.06/1.68	S is empty.
3.06/1.68	Rewrite Strategy: FULL
3.06/1.68	----------------------------------------
3.06/1.68	
3.06/1.68	(6) LowerBoundPropagationProof (FINISHED)
3.06/1.68	Propagated lower bound.
3.06/1.68	----------------------------------------
3.06/1.68	
3.06/1.68	(7)
3.06/1.68	BOUNDS(n^1, INF)
3.06/1.68	
3.06/1.68	----------------------------------------
3.06/1.68	
3.06/1.68	(8)
3.06/1.68	Obligation:
3.06/1.68	Analyzing the following TRS for decreasing loops:
3.06/1.68	
3.06/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.06/1.68	
3.06/1.68	
3.06/1.68	The TRS R consists of the following rules:
3.06/1.68	
3.06/1.68	   t(N) -> cs(r(q(N)), nt(ns(N)))
3.06/1.68	   q(0) -> 0
3.06/1.68	   q(s(X)) -> s(p(q(X), d(X)))
3.06/1.68	   d(0) -> 0
3.06/1.68	   d(s(X)) -> s(s(d(X)))
3.06/1.68	   p(0, X) -> X
3.06/1.68	   p(X, 0) -> X
3.06/1.68	   p(s(X), s(Y)) -> s(s(p(X, Y)))
3.06/1.68	   f(0, X) -> nil
3.06/1.68	   f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z)))
3.06/1.68	   t(X) -> nt(X)
3.06/1.68	   s(X) -> ns(X)
3.06/1.68	   f(X1, X2) -> nf(X1, X2)
3.06/1.68	   a(nt(X)) -> t(a(X))
3.06/1.68	   a(ns(X)) -> s(a(X))
3.06/1.68	   a(nf(X1, X2)) -> f(a(X1), a(X2))
3.06/1.68	   a(X) -> X
3.06/1.68	
3.06/1.68	S is empty.
3.06/1.68	Rewrite Strategy: FULL
3.06/1.68	----------------------------------------
3.06/1.68	
3.06/1.68	(9) DecreasingLoopProof (FINISHED)
3.06/1.68	The following loop(s) give(s) rise to the lower bound EXP:
3.06/1.68	
3.06/1.68	The rewrite sequence
3.06/1.68	
3.06/1.68	a(nt(X)) ->^+ cs(r(q(a(X))), nt(ns(a(X))))
3.06/1.68	
3.06/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
3.06/1.68	
3.06/1.68	The pumping substitution is [X / nt(X)].
3.06/1.68	
3.06/1.68	The result substitution is [ ].
3.06/1.68	
3.06/1.68	
3.06/1.68	
3.06/1.68	The rewrite sequence
3.06/1.68	
3.06/1.68	a(nt(X)) ->^+ cs(r(q(a(X))), nt(ns(a(X))))
3.06/1.68	
3.06/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.06/1.68	
3.06/1.68	The pumping substitution is [X / nt(X)].
3.06/1.68	
3.06/1.68	The result substitution is [ ].
3.06/1.68	
3.06/1.68	
3.06/1.68	
3.06/1.68	
3.06/1.68	----------------------------------------
3.06/1.68	
3.06/1.68	(10)
3.06/1.68	BOUNDS(EXP, INF)
3.46/1.72	EOF