2.90/1.72	WORST_CASE(NON_POLY, ?)
2.90/1.73	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
2.90/1.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.90/1.73	
2.90/1.73	
2.90/1.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
2.90/1.73	
2.90/1.73	(0) CpxTRS
2.90/1.73	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
2.90/1.73	(2) TRS for Loop Detection
2.90/1.73	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
2.90/1.73	(4) BEST
2.90/1.73	    (5) proven lower bound
2.90/1.73	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
2.90/1.73	        (7) BOUNDS(n^1, INF)
2.90/1.73	    (8) TRS for Loop Detection
2.90/1.73	        (9) DecreasingLoopProof [FINISHED, 0 ms]
2.90/1.73	        (10) BOUNDS(EXP, INF)
2.90/1.73	
2.90/1.73	
2.90/1.73	----------------------------------------
2.90/1.73	
2.90/1.73	(0)
2.90/1.73	Obligation:
2.90/1.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
2.90/1.73	
2.90/1.73	
2.90/1.73	The TRS R consists of the following rules:
2.90/1.73	
2.90/1.73	   msort(nil) -> nil
2.90/1.73	   msort(.(x, y)) -> .(min(x, y), msort(del(min(x, y), .(x, y))))
2.90/1.73	   min(x, nil) -> x
2.90/1.73	   min(x, .(y, z)) -> if(<=(x, y), min(x, z), min(y, z))
2.90/1.73	   del(x, nil) -> nil
2.90/1.73	   del(x, .(y, z)) -> if(=(x, y), z, .(y, del(x, z)))
2.90/1.73	
2.90/1.73	S is empty.
2.90/1.73	Rewrite Strategy: INNERMOST
2.90/1.73	----------------------------------------
2.90/1.73	
2.90/1.73	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
2.90/1.73	Transformed a relative TRS into a decreasing-loop problem.
2.90/1.73	----------------------------------------
2.90/1.73	
2.90/1.73	(2)
2.90/1.73	Obligation:
2.90/1.73	Analyzing the following TRS for decreasing loops:
2.90/1.73	
2.90/1.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
2.90/1.73	
2.90/1.73	
2.90/1.73	The TRS R consists of the following rules:
2.90/1.73	
2.90/1.73	   msort(nil) -> nil
2.90/1.73	   msort(.(x, y)) -> .(min(x, y), msort(del(min(x, y), .(x, y))))
2.90/1.73	   min(x, nil) -> x
2.90/1.73	   min(x, .(y, z)) -> if(<=(x, y), min(x, z), min(y, z))
2.90/1.73	   del(x, nil) -> nil
2.90/1.73	   del(x, .(y, z)) -> if(=(x, y), z, .(y, del(x, z)))
2.90/1.73	
2.90/1.73	S is empty.
2.90/1.73	Rewrite Strategy: INNERMOST
2.90/1.73	----------------------------------------
2.90/1.73	
2.90/1.73	(3) DecreasingLoopProof (LOWER BOUND(ID))
2.90/1.73	The following loop(s) give(s) rise to the lower bound Omega(n^1):
2.90/1.73	
2.90/1.73	The rewrite sequence
2.90/1.73	
2.90/1.73	del(x, .(y, z)) ->^+ if(=(x, y), z, .(y, del(x, z)))
2.90/1.73	
2.90/1.73	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
2.90/1.73	
2.90/1.73	The pumping substitution is [z / .(y, z)].
2.90/1.73	
2.90/1.73	The result substitution is [ ].
2.90/1.73	
2.90/1.73	
2.90/1.73	
2.90/1.73	
2.90/1.73	----------------------------------------
2.90/1.73	
2.90/1.73	(4)
2.90/1.73	Complex Obligation (BEST)
2.90/1.73	
2.90/1.73	----------------------------------------
2.90/1.73	
2.90/1.73	(5)
2.90/1.73	Obligation:
2.90/1.73	Proved the lower bound n^1 for the following obligation:
2.90/1.73	
2.90/1.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
2.90/1.73	
2.90/1.73	
2.90/1.73	The TRS R consists of the following rules:
2.90/1.73	
2.90/1.73	   msort(nil) -> nil
2.90/1.73	   msort(.(x, y)) -> .(min(x, y), msort(del(min(x, y), .(x, y))))
2.90/1.73	   min(x, nil) -> x
2.90/1.73	   min(x, .(y, z)) -> if(<=(x, y), min(x, z), min(y, z))
2.90/1.73	   del(x, nil) -> nil
2.90/1.73	   del(x, .(y, z)) -> if(=(x, y), z, .(y, del(x, z)))
2.90/1.73	
2.90/1.73	S is empty.
2.90/1.73	Rewrite Strategy: INNERMOST
2.90/1.73	----------------------------------------
2.90/1.73	
2.90/1.73	(6) LowerBoundPropagationProof (FINISHED)
2.90/1.73	Propagated lower bound.
2.90/1.73	----------------------------------------
2.90/1.73	
2.90/1.73	(7)
2.90/1.73	BOUNDS(n^1, INF)
2.90/1.73	
2.90/1.73	----------------------------------------
2.90/1.73	
2.90/1.73	(8)
2.90/1.73	Obligation:
2.90/1.73	Analyzing the following TRS for decreasing loops:
2.90/1.73	
2.90/1.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
2.90/1.73	
2.90/1.73	
2.90/1.73	The TRS R consists of the following rules:
2.90/1.73	
2.90/1.73	   msort(nil) -> nil
2.90/1.73	   msort(.(x, y)) -> .(min(x, y), msort(del(min(x, y), .(x, y))))
2.90/1.73	   min(x, nil) -> x
2.90/1.73	   min(x, .(y, z)) -> if(<=(x, y), min(x, z), min(y, z))
2.90/1.73	   del(x, nil) -> nil
2.90/1.73	   del(x, .(y, z)) -> if(=(x, y), z, .(y, del(x, z)))
2.90/1.73	
2.90/1.73	S is empty.
2.90/1.73	Rewrite Strategy: INNERMOST
2.90/1.73	----------------------------------------
2.90/1.73	
2.90/1.73	(9) DecreasingLoopProof (FINISHED)
2.90/1.73	The following loop(s) give(s) rise to the lower bound EXP:
2.90/1.73	
2.90/1.73	The rewrite sequence
2.90/1.73	
2.90/1.73	min(x, .(y, z)) ->^+ if(<=(x, y), min(x, z), min(y, z))
2.90/1.73	
2.90/1.73	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
2.90/1.73	
2.90/1.73	The pumping substitution is [z / .(y, z)].
2.90/1.73	
2.90/1.73	The result substitution is [ ].
2.90/1.73	
2.90/1.73	
2.90/1.73	
2.90/1.73	The rewrite sequence
2.90/1.73	
2.90/1.73	min(x, .(y, z)) ->^+ if(<=(x, y), min(x, z), min(y, z))
2.90/1.73	
2.90/1.73	gives rise to a decreasing loop by considering the right hand sides subterm at position [2].
2.90/1.73	
2.90/1.73	The pumping substitution is [z / .(y, z)].
2.90/1.73	
2.90/1.73	The result substitution is [x / y].
2.90/1.73	
2.90/1.73	
2.90/1.73	
2.90/1.73	
2.90/1.73	----------------------------------------
2.90/1.73	
2.90/1.73	(10)
2.90/1.73	BOUNDS(EXP, INF)
2.90/1.75	EOF