3.27/1.71	WORST_CASE(Omega(n^1), O(n^1))
3.62/1.72	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.62/1.72	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.62/1.72	
3.62/1.72	
3.62/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.62/1.72	
3.62/1.72	(0) CpxTRS
3.62/1.72	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.62/1.72	(2) CpxTRS
3.62/1.72	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 35 ms]
3.62/1.72	    (4) BOUNDS(1, n^1)
3.62/1.72	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.62/1.72	(6) TRS for Loop Detection
3.62/1.72	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.62/1.72	    (8) BEST
3.62/1.72	        (9) proven lower bound
3.62/1.72	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.62/1.72	            (11) BOUNDS(n^1, INF)
3.62/1.72	        (12) TRS for Loop Detection
3.62/1.72	
3.62/1.72	
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(0)
3.62/1.72	Obligation:
3.62/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.62/1.72	
3.62/1.72	
3.62/1.72	The TRS R consists of the following rules:
3.62/1.72	
3.62/1.72	   concat(leaf, Y) -> Y
3.62/1.72	   concat(cons(U, V), Y) -> cons(U, concat(V, Y))
3.62/1.72	   lessleaves(X, leaf) -> false
3.62/1.72	   lessleaves(leaf, cons(W, Z)) -> true
3.62/1.72	   lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))
3.62/1.72	
3.62/1.72	S is empty.
3.62/1.72	Rewrite Strategy: INNERMOST
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.62/1.72	transformed relative TRS to TRS
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(2)
3.62/1.72	Obligation:
3.62/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.62/1.72	
3.62/1.72	
3.62/1.72	The TRS R consists of the following rules:
3.62/1.72	
3.62/1.72	   concat(leaf, Y) -> Y
3.62/1.72	   concat(cons(U, V), Y) -> cons(U, concat(V, Y))
3.62/1.72	   lessleaves(X, leaf) -> false
3.62/1.72	   lessleaves(leaf, cons(W, Z)) -> true
3.62/1.72	   lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))
3.62/1.72	
3.62/1.72	S is empty.
3.62/1.72	Rewrite Strategy: INNERMOST
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.62/1.72	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 2. 
3.62/1.72	
3.62/1.72	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.62/1.72	final states : [1, 2]
3.62/1.72	transitions: 
3.62/1.72	leaf0() -> 0
3.62/1.72	cons0(0, 0) -> 0
3.62/1.72	false0() -> 0
3.62/1.72	true0() -> 0
3.62/1.72	concat0(0, 0) -> 1
3.62/1.72	lessleaves0(0, 0) -> 2
3.62/1.72	concat1(0, 0) -> 3
3.62/1.72	cons1(0, 3) -> 1
3.62/1.72	false1() -> 2
3.62/1.72	true1() -> 2
3.62/1.72	concat1(0, 0) -> 4
3.62/1.72	concat1(0, 0) -> 5
3.62/1.72	lessleaves1(4, 5) -> 2
3.62/1.72	cons1(0, 3) -> 3
3.62/1.72	cons1(0, 3) -> 4
3.62/1.72	cons1(0, 3) -> 5
3.62/1.72	concat1(0, 3) -> 5
3.62/1.72	concat1(0, 3) -> 4
3.62/1.72	concat2(0, 3) -> 6
3.62/1.72	concat2(0, 3) -> 7
3.62/1.72	lessleaves2(6, 7) -> 2
3.62/1.72	concat1(0, 3) -> 3
3.62/1.72	0 -> 1
3.62/1.72	0 -> 3
3.62/1.72	0 -> 4
3.62/1.72	0 -> 5
3.62/1.72	3 -> 4
3.62/1.72	3 -> 5
3.62/1.72	3 -> 6
3.62/1.72	3 -> 7
3.62/1.72	
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(4)
3.62/1.72	BOUNDS(1, n^1)
3.62/1.72	
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.62/1.72	Transformed a relative TRS into a decreasing-loop problem.
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(6)
3.62/1.72	Obligation:
3.62/1.72	Analyzing the following TRS for decreasing loops:
3.62/1.72	
3.62/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.62/1.72	
3.62/1.72	
3.62/1.72	The TRS R consists of the following rules:
3.62/1.72	
3.62/1.72	   concat(leaf, Y) -> Y
3.62/1.72	   concat(cons(U, V), Y) -> cons(U, concat(V, Y))
3.62/1.72	   lessleaves(X, leaf) -> false
3.62/1.72	   lessleaves(leaf, cons(W, Z)) -> true
3.62/1.72	   lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))
3.62/1.72	
3.62/1.72	S is empty.
3.62/1.72	Rewrite Strategy: INNERMOST
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.62/1.72	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.62/1.72	
3.62/1.72	The rewrite sequence
3.62/1.72	
3.62/1.72	concat(cons(U, V), Y) ->^+ cons(U, concat(V, Y))
3.62/1.72	
3.62/1.72	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.62/1.72	
3.62/1.72	The pumping substitution is [V / cons(U, V)].
3.62/1.72	
3.62/1.72	The result substitution is [ ].
3.62/1.72	
3.62/1.72	
3.62/1.72	
3.62/1.72	
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(8)
3.62/1.72	Complex Obligation (BEST)
3.62/1.72	
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(9)
3.62/1.72	Obligation:
3.62/1.72	Proved the lower bound n^1 for the following obligation:
3.62/1.72	
3.62/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.62/1.72	
3.62/1.72	
3.62/1.72	The TRS R consists of the following rules:
3.62/1.72	
3.62/1.72	   concat(leaf, Y) -> Y
3.62/1.72	   concat(cons(U, V), Y) -> cons(U, concat(V, Y))
3.62/1.72	   lessleaves(X, leaf) -> false
3.62/1.72	   lessleaves(leaf, cons(W, Z)) -> true
3.62/1.72	   lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))
3.62/1.72	
3.62/1.72	S is empty.
3.62/1.72	Rewrite Strategy: INNERMOST
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(10) LowerBoundPropagationProof (FINISHED)
3.62/1.72	Propagated lower bound.
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(11)
3.62/1.72	BOUNDS(n^1, INF)
3.62/1.72	
3.62/1.72	----------------------------------------
3.62/1.72	
3.62/1.72	(12)
3.62/1.72	Obligation:
3.62/1.72	Analyzing the following TRS for decreasing loops:
3.62/1.72	
3.62/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.62/1.72	
3.62/1.72	
3.62/1.72	The TRS R consists of the following rules:
3.62/1.72	
3.62/1.72	   concat(leaf, Y) -> Y
3.62/1.72	   concat(cons(U, V), Y) -> cons(U, concat(V, Y))
3.62/1.72	   lessleaves(X, leaf) -> false
3.62/1.72	   lessleaves(leaf, cons(W, Z)) -> true
3.62/1.72	   lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))
3.62/1.72	
3.62/1.72	S is empty.
3.62/1.72	Rewrite Strategy: INNERMOST
3.62/1.76	EOF