3.07/1.60	WORST_CASE(NON_POLY, ?)
3.07/1.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.07/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.07/1.60	
3.07/1.60	
3.07/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.07/1.60	
3.07/1.60	(0) CpxTRS
3.07/1.60	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.07/1.60	(2) TRS for Loop Detection
3.07/1.60	(3) DecreasingLoopProof [FINISHED, 0 ms]
3.07/1.60	(4) BOUNDS(EXP, INF)
3.07/1.60	
3.07/1.60	
3.07/1.60	----------------------------------------
3.07/1.60	
3.07/1.60	(0)
3.07/1.60	Obligation:
3.07/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.07/1.60	
3.07/1.60	
3.07/1.60	The TRS R consists of the following rules:
3.07/1.60	
3.07/1.60	   le(0, y) -> true
3.07/1.60	   le(s(x), 0) -> false
3.07/1.60	   le(s(x), s(y)) -> le(x, y)
3.07/1.60	   eq(0, 0) -> true
3.07/1.60	   eq(0, s(y)) -> false
3.07/1.60	   eq(s(x), 0) -> false
3.07/1.60	   eq(s(x), s(y)) -> eq(x, y)
3.07/1.60	   if(true, x, y) -> x
3.07/1.60	   if(false, x, y) -> y
3.07/1.60	   minsort(nil) -> nil
3.07/1.60	   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
3.07/1.60	   min(x, nil) -> x
3.07/1.60	   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
3.07/1.60	   del(x, nil) -> nil
3.07/1.60	   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))
3.07/1.60	
3.07/1.60	S is empty.
3.07/1.60	Rewrite Strategy: INNERMOST
3.07/1.60	----------------------------------------
3.07/1.60	
3.07/1.60	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.07/1.60	Transformed a relative TRS into a decreasing-loop problem.
3.07/1.60	----------------------------------------
3.07/1.60	
3.07/1.60	(2)
3.07/1.60	Obligation:
3.07/1.60	Analyzing the following TRS for decreasing loops:
3.07/1.60	
3.07/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.07/1.60	
3.07/1.60	
3.07/1.60	The TRS R consists of the following rules:
3.07/1.60	
3.07/1.60	   le(0, y) -> true
3.07/1.60	   le(s(x), 0) -> false
3.07/1.60	   le(s(x), s(y)) -> le(x, y)
3.07/1.60	   eq(0, 0) -> true
3.07/1.60	   eq(0, s(y)) -> false
3.07/1.60	   eq(s(x), 0) -> false
3.07/1.60	   eq(s(x), s(y)) -> eq(x, y)
3.07/1.60	   if(true, x, y) -> x
3.07/1.60	   if(false, x, y) -> y
3.07/1.60	   minsort(nil) -> nil
3.07/1.60	   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
3.07/1.60	   min(x, nil) -> x
3.07/1.60	   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
3.07/1.60	   del(x, nil) -> nil
3.07/1.60	   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))
3.07/1.60	
3.07/1.60	S is empty.
3.07/1.60	Rewrite Strategy: INNERMOST
3.07/1.60	----------------------------------------
3.07/1.60	
3.07/1.60	(3) DecreasingLoopProof (FINISHED)
3.07/1.60	The following loop(s) give(s) rise to the lower bound EXP:
3.07/1.60	
3.07/1.60	The rewrite sequence
3.07/1.60	
3.07/1.60	min(x, cons(y, z)) ->^+ if(le(x, y), min(x, z), min(y, z))
3.07/1.60	
3.07/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.07/1.60	
3.07/1.60	The pumping substitution is [z / cons(y, z)].
3.07/1.60	
3.07/1.60	The result substitution is [ ].
3.07/1.60	
3.07/1.60	
3.07/1.60	
3.07/1.60	The rewrite sequence
3.07/1.60	
3.07/1.60	min(x, cons(y, z)) ->^+ if(le(x, y), min(x, z), min(y, z))
3.07/1.60	
3.07/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [2].
3.07/1.60	
3.07/1.60	The pumping substitution is [z / cons(y, z)].
3.07/1.60	
3.07/1.60	The result substitution is [x / y].
3.07/1.60	
3.07/1.60	
3.07/1.60	
3.07/1.60	
3.07/1.60	----------------------------------------
3.07/1.60	
3.07/1.60	(4)
3.07/1.60	BOUNDS(EXP, INF)
3.16/1.64	EOF