3.36/1.59	WORST_CASE(Omega(n^1), O(n^1))
3.36/1.60	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.36/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.36/1.60	
3.36/1.60	
3.36/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.60	
3.36/1.60	(0) CpxTRS
3.36/1.60	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.36/1.60	(2) CpxTRS
3.36/1.60	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 38 ms]
3.36/1.60	    (4) BOUNDS(1, n^1)
3.36/1.60	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.36/1.60	(6) TRS for Loop Detection
3.36/1.60	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.36/1.60	    (8) BEST
3.36/1.60	        (9) proven lower bound
3.36/1.60	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.36/1.60	            (11) BOUNDS(n^1, INF)
3.36/1.60	        (12) TRS for Loop Detection
3.36/1.60	
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(0)
3.36/1.60	Obligation:
3.36/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.60	
3.36/1.60	
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   addlist(Cons(x, xs'), Cons(S(0), xs)) -> Cons(S(x), addlist(xs', xs))
3.36/1.60	   addlist(Cons(S(0), xs'), Cons(x, xs)) -> Cons(S(x), addlist(xs', xs))
3.36/1.60	   addlist(Nil, ys) -> Nil
3.36/1.60	   notEmpty(Cons(x, xs)) -> True
3.36/1.60	   notEmpty(Nil) -> False
3.36/1.60	   goal(xs, ys) -> addlist(xs, ys)
3.36/1.60	
3.36/1.60	S is empty.
3.36/1.60	Rewrite Strategy: INNERMOST
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.36/1.60	transformed relative TRS to TRS
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(2)
3.36/1.60	Obligation:
3.36/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.36/1.60	
3.36/1.60	
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   addlist(Cons(x, xs'), Cons(S(0), xs)) -> Cons(S(x), addlist(xs', xs))
3.36/1.60	   addlist(Cons(S(0), xs'), Cons(x, xs)) -> Cons(S(x), addlist(xs', xs))
3.36/1.60	   addlist(Nil, ys) -> Nil
3.36/1.60	   notEmpty(Cons(x, xs)) -> True
3.36/1.60	   notEmpty(Nil) -> False
3.36/1.60	   goal(xs, ys) -> addlist(xs, ys)
3.36/1.60	
3.36/1.60	S is empty.
3.36/1.60	Rewrite Strategy: INNERMOST
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.36/1.60	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 
3.36/1.60	
3.36/1.60	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.36/1.60	final states : [1, 2, 3]
3.36/1.60	transitions: 
3.36/1.60	Cons0(0, 0) -> 0
3.36/1.60	S0(0) -> 0
3.36/1.60	00() -> 0
3.36/1.60	Nil0() -> 0
3.36/1.60	True0() -> 0
3.36/1.60	False0() -> 0
3.36/1.60	addlist0(0, 0) -> 1
3.36/1.60	notEmpty0(0) -> 2
3.36/1.60	goal0(0, 0) -> 3
3.36/1.60	S1(0) -> 4
3.36/1.60	addlist1(0, 0) -> 5
3.36/1.60	Cons1(4, 5) -> 1
3.36/1.60	Nil1() -> 1
3.36/1.60	True1() -> 2
3.36/1.60	False1() -> 2
3.36/1.60	addlist1(0, 0) -> 3
3.36/1.60	Cons1(4, 5) -> 3
3.36/1.60	Cons1(4, 5) -> 5
3.36/1.60	Nil1() -> 3
3.36/1.60	Nil1() -> 5
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(4)
3.36/1.60	BOUNDS(1, n^1)
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.36/1.60	Transformed a relative TRS into a decreasing-loop problem.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(6)
3.36/1.60	Obligation:
3.36/1.60	Analyzing the following TRS for decreasing loops:
3.36/1.60	
3.36/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.60	
3.36/1.60	
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   addlist(Cons(x, xs'), Cons(S(0), xs)) -> Cons(S(x), addlist(xs', xs))
3.36/1.60	   addlist(Cons(S(0), xs'), Cons(x, xs)) -> Cons(S(x), addlist(xs', xs))
3.36/1.60	   addlist(Nil, ys) -> Nil
3.36/1.60	   notEmpty(Cons(x, xs)) -> True
3.36/1.60	   notEmpty(Nil) -> False
3.36/1.60	   goal(xs, ys) -> addlist(xs, ys)
3.36/1.60	
3.36/1.60	S is empty.
3.36/1.60	Rewrite Strategy: INNERMOST
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.36/1.60	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.36/1.60	
3.36/1.60	The rewrite sequence
3.36/1.60	
3.36/1.60	addlist(Cons(x, xs'), Cons(S(0), xs)) ->^+ Cons(S(x), addlist(xs', xs))
3.36/1.60	
3.36/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.36/1.60	
3.36/1.60	The pumping substitution is [xs' / Cons(x, xs'), xs / Cons(S(0), xs)].
3.36/1.60	
3.36/1.60	The result substitution is [ ].
3.36/1.60	
3.36/1.60	
3.36/1.60	
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(8)
3.36/1.60	Complex Obligation (BEST)
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(9)
3.36/1.60	Obligation:
3.36/1.60	Proved the lower bound n^1 for the following obligation:
3.36/1.60	
3.36/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.60	
3.36/1.60	
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   addlist(Cons(x, xs'), Cons(S(0), xs)) -> Cons(S(x), addlist(xs', xs))
3.36/1.60	   addlist(Cons(S(0), xs'), Cons(x, xs)) -> Cons(S(x), addlist(xs', xs))
3.36/1.60	   addlist(Nil, ys) -> Nil
3.36/1.60	   notEmpty(Cons(x, xs)) -> True
3.36/1.60	   notEmpty(Nil) -> False
3.36/1.60	   goal(xs, ys) -> addlist(xs, ys)
3.36/1.60	
3.36/1.60	S is empty.
3.36/1.60	Rewrite Strategy: INNERMOST
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(10) LowerBoundPropagationProof (FINISHED)
3.36/1.60	Propagated lower bound.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(11)
3.36/1.60	BOUNDS(n^1, INF)
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(12)
3.36/1.60	Obligation:
3.36/1.60	Analyzing the following TRS for decreasing loops:
3.36/1.60	
3.36/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.36/1.60	
3.36/1.60	
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   addlist(Cons(x, xs'), Cons(S(0), xs)) -> Cons(S(x), addlist(xs', xs))
3.36/1.60	   addlist(Cons(S(0), xs'), Cons(x, xs)) -> Cons(S(x), addlist(xs', xs))
3.36/1.60	   addlist(Nil, ys) -> Nil
3.36/1.60	   notEmpty(Cons(x, xs)) -> True
3.36/1.60	   notEmpty(Nil) -> False
3.36/1.60	   goal(xs, ys) -> addlist(xs, ys)
3.36/1.60	
3.36/1.60	S is empty.
3.36/1.60	Rewrite Strategy: INNERMOST
3.36/1.61	EOF