3.24/1.51	WORST_CASE(NON_POLY, ?)
3.24/1.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.24/1.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.24/1.52	
3.24/1.52	
3.24/1.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.52	
3.24/1.52	(0) CpxTRS
3.24/1.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.24/1.52	(2) TRS for Loop Detection
3.24/1.52	(3) DecreasingLoopProof [FINISHED, 0 ms]
3.24/1.52	(4) BOUNDS(EXP, INF)
3.24/1.52	
3.24/1.52	
3.24/1.52	----------------------------------------
3.24/1.52	
3.24/1.52	(0)
3.24/1.52	Obligation:
3.24/1.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.52	
3.24/1.52	
3.24/1.52	The TRS R consists of the following rules:
3.24/1.52	
3.24/1.52	   merge(nil, y) -> y
3.24/1.52	   merge(x, nil) -> x
3.24/1.52	   merge(.(x, y), .(u, v)) -> if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
3.24/1.52	   ++(nil, y) -> y
3.24/1.52	   ++(.(x, y), z) -> .(x, ++(y, z))
3.24/1.52	   if(true, x, y) -> x
3.24/1.52	   if(false, x, y) -> x
3.24/1.52	
3.24/1.52	S is empty.
3.24/1.52	Rewrite Strategy: INNERMOST
3.24/1.52	----------------------------------------
3.24/1.52	
3.24/1.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.24/1.52	Transformed a relative TRS into a decreasing-loop problem.
3.24/1.52	----------------------------------------
3.24/1.52	
3.24/1.52	(2)
3.24/1.52	Obligation:
3.24/1.52	Analyzing the following TRS for decreasing loops:
3.24/1.52	
3.24/1.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.52	
3.24/1.52	
3.24/1.52	The TRS R consists of the following rules:
3.24/1.52	
3.24/1.52	   merge(nil, y) -> y
3.24/1.52	   merge(x, nil) -> x
3.24/1.52	   merge(.(x, y), .(u, v)) -> if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
3.24/1.52	   ++(nil, y) -> y
3.24/1.52	   ++(.(x, y), z) -> .(x, ++(y, z))
3.24/1.52	   if(true, x, y) -> x
3.24/1.52	   if(false, x, y) -> x
3.24/1.52	
3.24/1.52	S is empty.
3.24/1.52	Rewrite Strategy: INNERMOST
3.24/1.52	----------------------------------------
3.24/1.52	
3.24/1.52	(3) DecreasingLoopProof (FINISHED)
3.24/1.52	The following loop(s) give(s) rise to the lower bound EXP:
3.24/1.52	
3.24/1.52	The rewrite sequence
3.24/1.52	
3.24/1.52	merge(.(x, y), .(u, v)) ->^+ if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
3.24/1.52	
3.24/1.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
3.24/1.52	
3.24/1.52	The pumping substitution is [y / .(x, y)].
3.24/1.52	
3.24/1.52	The result substitution is [ ].
3.24/1.52	
3.24/1.52	
3.24/1.52	
3.24/1.52	The rewrite sequence
3.24/1.52	
3.24/1.52	merge(.(x, y), .(u, v)) ->^+ if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
3.24/1.52	
3.24/1.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
3.24/1.52	
3.24/1.52	The pumping substitution is [v / .(u, v)].
3.24/1.52	
3.24/1.52	The result substitution is [ ].
3.24/1.52	
3.24/1.52	
3.24/1.52	
3.24/1.52	
3.24/1.52	----------------------------------------
3.24/1.52	
3.24/1.52	(4)
3.24/1.52	BOUNDS(EXP, INF)
3.29/1.62	EOF