3.21/1.69	WORST_CASE(NON_POLY, ?)
3.41/1.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.41/1.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.41/1.70	
3.41/1.70	
3.41/1.70	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.70	
3.41/1.70	(0) CpxTRS
3.41/1.70	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.41/1.70	(2) TRS for Loop Detection
3.41/1.70	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.41/1.70	(4) BEST
3.41/1.70	    (5) proven lower bound
3.41/1.70	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.41/1.70	        (7) BOUNDS(n^1, INF)
3.41/1.70	    (8) TRS for Loop Detection
3.41/1.70	        (9) DecreasingLoopProof [FINISHED, 18 ms]
3.41/1.70	        (10) BOUNDS(EXP, INF)
3.41/1.70	
3.41/1.70	
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(0)
3.41/1.70	Obligation:
3.41/1.70	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.70	
3.41/1.70	
3.41/1.70	The TRS R consists of the following rules:
3.41/1.70	
3.41/1.70	   app(nil, k) -> k
3.41/1.70	   app(l, nil) -> l
3.41/1.70	   app(cons(x, l), k) -> cons(x, app(l, k))
3.41/1.70	   sum(cons(x, nil)) -> cons(x, nil)
3.41/1.70	   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
3.41/1.70	   a(h, h, x) -> s(x)
3.41/1.70	   a(x, s(y), h) -> a(x, y, s(h))
3.41/1.70	   a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
3.41/1.70	   a(s(x), h, z) -> a(x, z, z)
3.41/1.70	
3.41/1.70	S is empty.
3.41/1.70	Rewrite Strategy: FULL
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.41/1.70	Transformed a relative TRS into a decreasing-loop problem.
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(2)
3.41/1.70	Obligation:
3.41/1.70	Analyzing the following TRS for decreasing loops:
3.41/1.70	
3.41/1.70	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.70	
3.41/1.70	
3.41/1.70	The TRS R consists of the following rules:
3.41/1.70	
3.41/1.70	   app(nil, k) -> k
3.41/1.70	   app(l, nil) -> l
3.41/1.70	   app(cons(x, l), k) -> cons(x, app(l, k))
3.41/1.70	   sum(cons(x, nil)) -> cons(x, nil)
3.41/1.70	   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
3.41/1.70	   a(h, h, x) -> s(x)
3.41/1.70	   a(x, s(y), h) -> a(x, y, s(h))
3.41/1.70	   a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
3.41/1.70	   a(s(x), h, z) -> a(x, z, z)
3.41/1.70	
3.41/1.70	S is empty.
3.41/1.70	Rewrite Strategy: FULL
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.41/1.70	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.41/1.70	
3.41/1.70	The rewrite sequence
3.41/1.70	
3.41/1.70	app(cons(x, l), k) ->^+ cons(x, app(l, k))
3.41/1.70	
3.41/1.70	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.41/1.70	
3.41/1.70	The pumping substitution is [l / cons(x, l)].
3.41/1.70	
3.41/1.70	The result substitution is [ ].
3.41/1.70	
3.41/1.70	
3.41/1.70	
3.41/1.70	
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(4)
3.41/1.70	Complex Obligation (BEST)
3.41/1.70	
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(5)
3.41/1.70	Obligation:
3.41/1.70	Proved the lower bound n^1 for the following obligation:
3.41/1.70	
3.41/1.70	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.70	
3.41/1.70	
3.41/1.70	The TRS R consists of the following rules:
3.41/1.70	
3.41/1.70	   app(nil, k) -> k
3.41/1.70	   app(l, nil) -> l
3.41/1.70	   app(cons(x, l), k) -> cons(x, app(l, k))
3.41/1.70	   sum(cons(x, nil)) -> cons(x, nil)
3.41/1.70	   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
3.41/1.70	   a(h, h, x) -> s(x)
3.41/1.70	   a(x, s(y), h) -> a(x, y, s(h))
3.41/1.70	   a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
3.41/1.70	   a(s(x), h, z) -> a(x, z, z)
3.41/1.70	
3.41/1.70	S is empty.
3.41/1.70	Rewrite Strategy: FULL
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(6) LowerBoundPropagationProof (FINISHED)
3.41/1.70	Propagated lower bound.
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(7)
3.41/1.70	BOUNDS(n^1, INF)
3.41/1.70	
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(8)
3.41/1.70	Obligation:
3.41/1.70	Analyzing the following TRS for decreasing loops:
3.41/1.70	
3.41/1.70	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.70	
3.41/1.70	
3.41/1.70	The TRS R consists of the following rules:
3.41/1.70	
3.41/1.70	   app(nil, k) -> k
3.41/1.70	   app(l, nil) -> l
3.41/1.70	   app(cons(x, l), k) -> cons(x, app(l, k))
3.41/1.70	   sum(cons(x, nil)) -> cons(x, nil)
3.41/1.70	   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
3.41/1.70	   a(h, h, x) -> s(x)
3.41/1.70	   a(x, s(y), h) -> a(x, y, s(h))
3.41/1.70	   a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
3.41/1.70	   a(s(x), h, z) -> a(x, z, z)
3.41/1.70	
3.41/1.70	S is empty.
3.41/1.70	Rewrite Strategy: FULL
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(9) DecreasingLoopProof (FINISHED)
3.41/1.70	The following loop(s) give(s) rise to the lower bound EXP:
3.41/1.70	
3.41/1.70	The rewrite sequence
3.41/1.70	
3.41/1.70	a(s(x1_0), s(h), s(z)) ->^+ a(x1_0, a(s(x1_0), s(h), z), a(s(x1_0), s(h), z))
3.41/1.70	
3.41/1.70	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.41/1.70	
3.41/1.70	The pumping substitution is [z / s(z)].
3.41/1.70	
3.41/1.70	The result substitution is [ ].
3.41/1.70	
3.41/1.70	
3.41/1.70	
3.41/1.70	The rewrite sequence
3.41/1.70	
3.41/1.70	a(s(x1_0), s(h), s(z)) ->^+ a(x1_0, a(s(x1_0), s(h), z), a(s(x1_0), s(h), z))
3.41/1.70	
3.41/1.70	gives rise to a decreasing loop by considering the right hand sides subterm at position [2].
3.41/1.70	
3.41/1.70	The pumping substitution is [z / s(z)].
3.41/1.70	
3.41/1.70	The result substitution is [ ].
3.41/1.70	
3.41/1.70	
3.41/1.70	
3.41/1.70	
3.41/1.70	----------------------------------------
3.41/1.70	
3.41/1.70	(10)
3.41/1.70	BOUNDS(EXP, INF)
3.41/1.73	EOF