3.26/1.64	WORST_CASE(NON_POLY, ?)
3.26/1.65	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.26/1.65	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.26/1.65	
3.26/1.65	
3.26/1.65	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.26/1.65	
3.26/1.65	(0) CpxTRS
3.26/1.65	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.26/1.65	(2) TRS for Loop Detection
3.26/1.65	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.26/1.65	(4) BEST
3.26/1.65	    (5) proven lower bound
3.26/1.65	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.26/1.65	        (7) BOUNDS(n^1, INF)
3.26/1.65	    (8) TRS for Loop Detection
3.26/1.65	        (9) DecreasingLoopProof [FINISHED, 0 ms]
3.26/1.65	        (10) BOUNDS(EXP, INF)
3.26/1.65	
3.26/1.65	
3.26/1.65	----------------------------------------
3.26/1.65	
3.26/1.65	(0)
3.26/1.65	Obligation:
3.26/1.65	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.26/1.65	
3.26/1.65	
3.26/1.65	The TRS R consists of the following rules:
3.26/1.65	
3.26/1.65	   msort(nil) -> nil
3.26/1.65	   msort(.(x, y)) -> .(min(x, y), msort(del(min(x, y), .(x, y))))
3.26/1.65	   min(x, nil) -> x
3.26/1.65	   min(x, .(y, z)) -> if(<=(x, y), min(x, z), min(y, z))
3.26/1.65	   del(x, nil) -> nil
3.26/1.65	   del(x, .(y, z)) -> if(=(x, y), z, .(y, del(x, z)))
3.26/1.65	
3.26/1.65	S is empty.
3.26/1.65	Rewrite Strategy: FULL
3.26/1.65	----------------------------------------
3.26/1.65	
3.26/1.65	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.26/1.65	Transformed a relative TRS into a decreasing-loop problem.
3.26/1.65	----------------------------------------
3.26/1.65	
3.26/1.65	(2)
3.26/1.65	Obligation:
3.26/1.65	Analyzing the following TRS for decreasing loops:
3.26/1.65	
3.26/1.65	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.26/1.65	
3.26/1.65	
3.26/1.65	The TRS R consists of the following rules:
3.26/1.65	
3.26/1.65	   msort(nil) -> nil
3.26/1.65	   msort(.(x, y)) -> .(min(x, y), msort(del(min(x, y), .(x, y))))
3.26/1.65	   min(x, nil) -> x
3.26/1.65	   min(x, .(y, z)) -> if(<=(x, y), min(x, z), min(y, z))
3.26/1.65	   del(x, nil) -> nil
3.26/1.65	   del(x, .(y, z)) -> if(=(x, y), z, .(y, del(x, z)))
3.26/1.65	
3.26/1.65	S is empty.
3.26/1.65	Rewrite Strategy: FULL
3.26/1.65	----------------------------------------
3.26/1.65	
3.26/1.65	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.26/1.65	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.26/1.65	
3.26/1.65	The rewrite sequence
3.26/1.65	
3.26/1.65	del(x, .(y, z)) ->^+ if(=(x, y), z, .(y, del(x, z)))
3.26/1.65	
3.26/1.65	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
3.26/1.65	
3.26/1.65	The pumping substitution is [z / .(y, z)].
3.26/1.65	
3.26/1.65	The result substitution is [ ].
3.26/1.65	
3.26/1.65	
3.26/1.65	
3.26/1.65	
3.26/1.65	----------------------------------------
3.26/1.65	
3.26/1.65	(4)
3.26/1.65	Complex Obligation (BEST)
3.26/1.65	
3.26/1.65	----------------------------------------
3.26/1.65	
3.26/1.65	(5)
3.26/1.65	Obligation:
3.26/1.65	Proved the lower bound n^1 for the following obligation:
3.26/1.65	
3.26/1.65	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.26/1.65	
3.26/1.65	
3.26/1.65	The TRS R consists of the following rules:
3.26/1.65	
3.26/1.65	   msort(nil) -> nil
3.26/1.65	   msort(.(x, y)) -> .(min(x, y), msort(del(min(x, y), .(x, y))))
3.26/1.65	   min(x, nil) -> x
3.26/1.65	   min(x, .(y, z)) -> if(<=(x, y), min(x, z), min(y, z))
3.26/1.65	   del(x, nil) -> nil
3.26/1.65	   del(x, .(y, z)) -> if(=(x, y), z, .(y, del(x, z)))
3.26/1.65	
3.26/1.65	S is empty.
3.26/1.65	Rewrite Strategy: FULL
3.26/1.65	----------------------------------------
3.26/1.65	
3.26/1.65	(6) LowerBoundPropagationProof (FINISHED)
3.26/1.65	Propagated lower bound.
3.26/1.65	----------------------------------------
3.26/1.65	
3.26/1.65	(7)
3.26/1.65	BOUNDS(n^1, INF)
3.26/1.65	
3.26/1.65	----------------------------------------
3.26/1.65	
3.26/1.65	(8)
3.26/1.65	Obligation:
3.26/1.65	Analyzing the following TRS for decreasing loops:
3.26/1.65	
3.26/1.65	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.26/1.65	
3.26/1.65	
3.26/1.65	The TRS R consists of the following rules:
3.26/1.65	
3.26/1.65	   msort(nil) -> nil
3.26/1.65	   msort(.(x, y)) -> .(min(x, y), msort(del(min(x, y), .(x, y))))
3.26/1.65	   min(x, nil) -> x
3.26/1.65	   min(x, .(y, z)) -> if(<=(x, y), min(x, z), min(y, z))
3.26/1.65	   del(x, nil) -> nil
3.26/1.65	   del(x, .(y, z)) -> if(=(x, y), z, .(y, del(x, z)))
3.26/1.65	
3.26/1.65	S is empty.
3.26/1.65	Rewrite Strategy: FULL
3.26/1.65	----------------------------------------
3.26/1.65	
3.26/1.65	(9) DecreasingLoopProof (FINISHED)
3.26/1.65	The following loop(s) give(s) rise to the lower bound EXP:
3.26/1.65	
3.26/1.65	The rewrite sequence
3.26/1.65	
3.26/1.65	min(x, .(y, z)) ->^+ if(<=(x, y), min(x, z), min(y, z))
3.26/1.65	
3.26/1.65	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.26/1.65	
3.26/1.65	The pumping substitution is [z / .(y, z)].
3.26/1.65	
3.26/1.65	The result substitution is [ ].
3.26/1.65	
3.26/1.65	
3.26/1.65	
3.26/1.65	The rewrite sequence
3.26/1.65	
3.26/1.65	min(x, .(y, z)) ->^+ if(<=(x, y), min(x, z), min(y, z))
3.26/1.65	
3.26/1.65	gives rise to a decreasing loop by considering the right hand sides subterm at position [2].
3.26/1.65	
3.26/1.65	The pumping substitution is [z / .(y, z)].
3.26/1.65	
3.26/1.65	The result substitution is [x / y].
3.26/1.65	
3.26/1.65	
3.26/1.65	
3.26/1.65	
3.26/1.65	----------------------------------------
3.26/1.65	
3.26/1.65	(10)
3.26/1.65	BOUNDS(EXP, INF)
3.39/1.68	EOF