3.24/1.65	WORST_CASE(NON_POLY, ?)
3.24/1.67	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.24/1.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.24/1.67	
3.24/1.67	
3.24/1.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.67	
3.24/1.67	(0) CpxTRS
3.24/1.67	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.24/1.67	(2) TRS for Loop Detection
3.24/1.67	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.24/1.67	(4) BEST
3.24/1.67	    (5) proven lower bound
3.24/1.67	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.24/1.67	        (7) BOUNDS(n^1, INF)
3.24/1.67	    (8) TRS for Loop Detection
3.24/1.67	        (9) DecreasingLoopProof [FINISHED, 0 ms]
3.24/1.67	        (10) BOUNDS(EXP, INF)
3.24/1.67	
3.24/1.67	
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(0)
3.24/1.67	Obligation:
3.24/1.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.67	
3.24/1.67	
3.24/1.67	The TRS R consists of the following rules:
3.24/1.67	
3.24/1.67	   sort(l) -> st(0, l)
3.24/1.67	   st(n, l) -> cond1(member(n, l), n, l)
3.24/1.67	   cond1(true, n, l) -> cons(n, st(s(n), l))
3.24/1.67	   cond1(false, n, l) -> cond2(gt(n, max(l)), n, l)
3.24/1.67	   cond2(true, n, l) -> nil
3.24/1.67	   cond2(false, n, l) -> st(s(n), l)
3.24/1.67	   member(n, nil) -> false
3.24/1.67	   member(n, cons(m, l)) -> or(equal(n, m), member(n, l))
3.24/1.67	   or(x, true) -> true
3.24/1.67	   or(x, false) -> x
3.24/1.67	   equal(0, 0) -> true
3.24/1.67	   equal(s(x), 0) -> false
3.24/1.67	   equal(0, s(y)) -> false
3.24/1.67	   equal(s(x), s(y)) -> equal(x, y)
3.24/1.67	   gt(0, v) -> false
3.24/1.67	   gt(s(u), 0) -> true
3.24/1.67	   gt(s(u), s(v)) -> gt(u, v)
3.24/1.67	   max(nil) -> 0
3.24/1.67	   max(cons(u, l)) -> if(gt(u, max(l)), u, max(l))
3.24/1.67	   if(true, u, v) -> u
3.24/1.67	   if(false, u, v) -> v
3.24/1.67	
3.24/1.67	S is empty.
3.24/1.67	Rewrite Strategy: INNERMOST
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.24/1.67	Transformed a relative TRS into a decreasing-loop problem.
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(2)
3.24/1.67	Obligation:
3.24/1.67	Analyzing the following TRS for decreasing loops:
3.24/1.67	
3.24/1.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.67	
3.24/1.67	
3.24/1.67	The TRS R consists of the following rules:
3.24/1.67	
3.24/1.67	   sort(l) -> st(0, l)
3.24/1.67	   st(n, l) -> cond1(member(n, l), n, l)
3.24/1.67	   cond1(true, n, l) -> cons(n, st(s(n), l))
3.24/1.67	   cond1(false, n, l) -> cond2(gt(n, max(l)), n, l)
3.24/1.67	   cond2(true, n, l) -> nil
3.24/1.67	   cond2(false, n, l) -> st(s(n), l)
3.24/1.67	   member(n, nil) -> false
3.24/1.67	   member(n, cons(m, l)) -> or(equal(n, m), member(n, l))
3.24/1.67	   or(x, true) -> true
3.24/1.67	   or(x, false) -> x
3.24/1.67	   equal(0, 0) -> true
3.24/1.67	   equal(s(x), 0) -> false
3.24/1.67	   equal(0, s(y)) -> false
3.24/1.67	   equal(s(x), s(y)) -> equal(x, y)
3.24/1.67	   gt(0, v) -> false
3.24/1.67	   gt(s(u), 0) -> true
3.24/1.67	   gt(s(u), s(v)) -> gt(u, v)
3.24/1.67	   max(nil) -> 0
3.24/1.67	   max(cons(u, l)) -> if(gt(u, max(l)), u, max(l))
3.24/1.67	   if(true, u, v) -> u
3.24/1.67	   if(false, u, v) -> v
3.24/1.67	
3.24/1.67	S is empty.
3.24/1.67	Rewrite Strategy: INNERMOST
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.24/1.67	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.24/1.67	
3.24/1.67	The rewrite sequence
3.24/1.67	
3.24/1.67	gt(s(u), s(v)) ->^+ gt(u, v)
3.24/1.67	
3.24/1.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.24/1.67	
3.24/1.67	The pumping substitution is [u / s(u), v / s(v)].
3.24/1.67	
3.24/1.67	The result substitution is [ ].
3.24/1.67	
3.24/1.67	
3.24/1.67	
3.24/1.67	
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(4)
3.24/1.67	Complex Obligation (BEST)
3.24/1.67	
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(5)
3.24/1.67	Obligation:
3.24/1.67	Proved the lower bound n^1 for the following obligation:
3.24/1.67	
3.24/1.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.67	
3.24/1.67	
3.24/1.67	The TRS R consists of the following rules:
3.24/1.67	
3.24/1.67	   sort(l) -> st(0, l)
3.24/1.67	   st(n, l) -> cond1(member(n, l), n, l)
3.24/1.67	   cond1(true, n, l) -> cons(n, st(s(n), l))
3.24/1.67	   cond1(false, n, l) -> cond2(gt(n, max(l)), n, l)
3.24/1.67	   cond2(true, n, l) -> nil
3.24/1.67	   cond2(false, n, l) -> st(s(n), l)
3.24/1.67	   member(n, nil) -> false
3.24/1.67	   member(n, cons(m, l)) -> or(equal(n, m), member(n, l))
3.24/1.67	   or(x, true) -> true
3.24/1.67	   or(x, false) -> x
3.24/1.67	   equal(0, 0) -> true
3.24/1.67	   equal(s(x), 0) -> false
3.24/1.67	   equal(0, s(y)) -> false
3.24/1.67	   equal(s(x), s(y)) -> equal(x, y)
3.24/1.67	   gt(0, v) -> false
3.24/1.67	   gt(s(u), 0) -> true
3.24/1.67	   gt(s(u), s(v)) -> gt(u, v)
3.24/1.67	   max(nil) -> 0
3.24/1.67	   max(cons(u, l)) -> if(gt(u, max(l)), u, max(l))
3.24/1.67	   if(true, u, v) -> u
3.24/1.67	   if(false, u, v) -> v
3.24/1.67	
3.24/1.67	S is empty.
3.24/1.67	Rewrite Strategy: INNERMOST
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(6) LowerBoundPropagationProof (FINISHED)
3.24/1.67	Propagated lower bound.
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(7)
3.24/1.67	BOUNDS(n^1, INF)
3.24/1.67	
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(8)
3.24/1.67	Obligation:
3.24/1.67	Analyzing the following TRS for decreasing loops:
3.24/1.67	
3.24/1.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.67	
3.24/1.67	
3.24/1.67	The TRS R consists of the following rules:
3.24/1.67	
3.24/1.67	   sort(l) -> st(0, l)
3.24/1.67	   st(n, l) -> cond1(member(n, l), n, l)
3.24/1.67	   cond1(true, n, l) -> cons(n, st(s(n), l))
3.24/1.67	   cond1(false, n, l) -> cond2(gt(n, max(l)), n, l)
3.24/1.67	   cond2(true, n, l) -> nil
3.24/1.67	   cond2(false, n, l) -> st(s(n), l)
3.24/1.67	   member(n, nil) -> false
3.24/1.67	   member(n, cons(m, l)) -> or(equal(n, m), member(n, l))
3.24/1.67	   or(x, true) -> true
3.24/1.67	   or(x, false) -> x
3.24/1.67	   equal(0, 0) -> true
3.24/1.67	   equal(s(x), 0) -> false
3.24/1.67	   equal(0, s(y)) -> false
3.24/1.67	   equal(s(x), s(y)) -> equal(x, y)
3.24/1.67	   gt(0, v) -> false
3.24/1.67	   gt(s(u), 0) -> true
3.24/1.67	   gt(s(u), s(v)) -> gt(u, v)
3.24/1.67	   max(nil) -> 0
3.24/1.67	   max(cons(u, l)) -> if(gt(u, max(l)), u, max(l))
3.24/1.67	   if(true, u, v) -> u
3.24/1.67	   if(false, u, v) -> v
3.24/1.67	
3.24/1.67	S is empty.
3.24/1.67	Rewrite Strategy: INNERMOST
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(9) DecreasingLoopProof (FINISHED)
3.24/1.67	The following loop(s) give(s) rise to the lower bound EXP:
3.24/1.67	
3.24/1.67	The rewrite sequence
3.24/1.67	
3.24/1.67	max(cons(u, l)) ->^+ if(gt(u, max(l)), u, max(l))
3.24/1.67	
3.24/1.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
3.24/1.67	
3.24/1.67	The pumping substitution is [l / cons(u, l)].
3.24/1.67	
3.24/1.67	The result substitution is [ ].
3.24/1.67	
3.24/1.67	
3.24/1.67	
3.24/1.67	The rewrite sequence
3.24/1.67	
3.24/1.67	max(cons(u, l)) ->^+ if(gt(u, max(l)), u, max(l))
3.24/1.67	
3.24/1.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [2].
3.24/1.67	
3.24/1.67	The pumping substitution is [l / cons(u, l)].
3.24/1.67	
3.24/1.67	The result substitution is [ ].
3.24/1.67	
3.24/1.67	
3.24/1.67	
3.24/1.67	
3.24/1.67	----------------------------------------
3.24/1.67	
3.24/1.67	(10)
3.24/1.67	BOUNDS(EXP, INF)
3.48/1.70	EOF