3.95/1.80	WORST_CASE(Omega(n^1), O(n^1))
3.95/1.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.95/1.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.95/1.81	
3.95/1.81	
3.95/1.81	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
3.95/1.81	
3.95/1.81	(0) CpxRelTRS
3.95/1.81	(1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 141 ms]
3.95/1.81	(2) CpxRelTRS
3.95/1.81	(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.95/1.81	(4) CpxTRS
3.95/1.81	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 34 ms]
3.95/1.81	    (6) BOUNDS(1, n^1)
3.95/1.81	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.95/1.81	(8) TRS for Loop Detection
3.95/1.81	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.95/1.81	    (10) BEST
3.95/1.81	        (11) proven lower bound
3.95/1.81	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
3.95/1.81	            (13) BOUNDS(n^1, INF)
3.95/1.81	        (14) TRS for Loop Detection
3.95/1.81	
3.95/1.81	
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(0)
3.95/1.81	Obligation:
3.95/1.81	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
3.95/1.81	
3.95/1.81	
3.95/1.81	The TRS R consists of the following rules:
3.95/1.81	
3.95/1.81	   f(C(x1, x2)) -> C(f(x1), f(x2))
3.95/1.81	   f(Z) -> Z
3.95/1.81	   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
3.95/1.81	   eqZList(C(x1, x2), Z) -> False
3.95/1.81	   eqZList(Z, C(y1, y2)) -> False
3.95/1.81	   eqZList(Z, Z) -> True
3.95/1.81	   second(C(x1, x2)) -> x2
3.95/1.81	   first(C(x1, x2)) -> x1
3.95/1.81	   g(x) -> x
3.95/1.81	
3.95/1.81	The (relative) TRS S consists of the following rules:
3.95/1.81	
3.95/1.81	   and(False, False) -> False
3.95/1.81	   and(True, False) -> False
3.95/1.81	   and(False, True) -> False
3.95/1.81	   and(True, True) -> True
3.95/1.81	
3.95/1.81	Rewrite Strategy: INNERMOST
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
3.95/1.81	proved termination of relative rules
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(2)
3.95/1.81	Obligation:
3.95/1.81	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
3.95/1.81	
3.95/1.81	
3.95/1.81	The TRS R consists of the following rules:
3.95/1.81	
3.95/1.81	   f(C(x1, x2)) -> C(f(x1), f(x2))
3.95/1.81	   f(Z) -> Z
3.95/1.81	   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
3.95/1.81	   eqZList(C(x1, x2), Z) -> False
3.95/1.81	   eqZList(Z, C(y1, y2)) -> False
3.95/1.81	   eqZList(Z, Z) -> True
3.95/1.81	   second(C(x1, x2)) -> x2
3.95/1.81	   first(C(x1, x2)) -> x1
3.95/1.81	   g(x) -> x
3.95/1.81	
3.95/1.81	The (relative) TRS S consists of the following rules:
3.95/1.81	
3.95/1.81	   and(False, False) -> False
3.95/1.81	   and(True, False) -> False
3.95/1.81	   and(False, True) -> False
3.95/1.81	   and(True, True) -> True
3.95/1.81	
3.95/1.81	Rewrite Strategy: INNERMOST
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(3) RelTrsToTrsProof (UPPER BOUND(ID))
3.95/1.81	transformed relative TRS to TRS
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(4)
3.95/1.81	Obligation:
3.95/1.81	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.95/1.81	
3.95/1.81	
3.95/1.81	The TRS R consists of the following rules:
3.95/1.81	
3.95/1.81	   f(C(x1, x2)) -> C(f(x1), f(x2))
3.95/1.81	   f(Z) -> Z
3.95/1.81	   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
3.95/1.81	   eqZList(C(x1, x2), Z) -> False
3.95/1.81	   eqZList(Z, C(y1, y2)) -> False
3.95/1.81	   eqZList(Z, Z) -> True
3.95/1.81	   second(C(x1, x2)) -> x2
3.95/1.81	   first(C(x1, x2)) -> x1
3.95/1.81	   g(x) -> x
3.95/1.81	   and(False, False) -> False
3.95/1.81	   and(True, False) -> False
3.95/1.81	   and(False, True) -> False
3.95/1.81	   and(True, True) -> True
3.95/1.81	
3.95/1.81	S is empty.
3.95/1.81	Rewrite Strategy: INNERMOST
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(5) CpxTrsMatchBoundsTAProof (FINISHED)
3.95/1.81	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. 
3.95/1.81	
3.95/1.81	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.95/1.81	final states : [1, 2, 3, 4, 5, 6]
3.95/1.81	transitions: 
3.95/1.81	C0(0, 0) -> 0
3.95/1.81	Z0() -> 0
3.95/1.81	False0() -> 0
3.95/1.81	True0() -> 0
3.95/1.81	f0(0) -> 1
3.95/1.81	eqZList0(0, 0) -> 2
3.95/1.81	second0(0) -> 3
3.95/1.81	first0(0) -> 4
3.95/1.81	g0(0) -> 5
3.95/1.81	and0(0, 0) -> 6
3.95/1.81	f1(0) -> 7
3.95/1.81	f1(0) -> 8
3.95/1.81	C1(7, 8) -> 1
3.95/1.81	Z1() -> 1
3.95/1.81	eqZList1(0, 0) -> 9
3.95/1.81	eqZList1(0, 0) -> 10
3.95/1.81	and1(9, 10) -> 2
3.95/1.81	False1() -> 2
3.95/1.81	True1() -> 2
3.95/1.81	False1() -> 6
3.95/1.81	True1() -> 6
3.95/1.81	C1(7, 8) -> 7
3.95/1.81	C1(7, 8) -> 8
3.95/1.81	Z1() -> 7
3.95/1.81	Z1() -> 8
3.95/1.81	and1(9, 10) -> 9
3.95/1.81	and1(9, 10) -> 10
3.95/1.81	False1() -> 9
3.95/1.81	False1() -> 10
3.95/1.81	True1() -> 9
3.95/1.81	True1() -> 10
3.95/1.81	False2() -> 2
3.95/1.81	False2() -> 9
3.95/1.81	False2() -> 10
3.95/1.81	True2() -> 2
3.95/1.81	True2() -> 9
3.95/1.81	True2() -> 10
3.95/1.81	0 -> 3
3.95/1.81	0 -> 4
3.95/1.81	0 -> 5
3.95/1.81	
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(6)
3.95/1.81	BOUNDS(1, n^1)
3.95/1.81	
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.95/1.81	Transformed a relative TRS into a decreasing-loop problem.
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(8)
3.95/1.81	Obligation:
3.95/1.81	Analyzing the following TRS for decreasing loops:
3.95/1.81	
3.95/1.81	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
3.95/1.81	
3.95/1.81	
3.95/1.81	The TRS R consists of the following rules:
3.95/1.81	
3.95/1.81	   f(C(x1, x2)) -> C(f(x1), f(x2))
3.95/1.81	   f(Z) -> Z
3.95/1.81	   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
3.95/1.81	   eqZList(C(x1, x2), Z) -> False
3.95/1.81	   eqZList(Z, C(y1, y2)) -> False
3.95/1.81	   eqZList(Z, Z) -> True
3.95/1.81	   second(C(x1, x2)) -> x2
3.95/1.81	   first(C(x1, x2)) -> x1
3.95/1.81	   g(x) -> x
3.95/1.81	
3.95/1.81	The (relative) TRS S consists of the following rules:
3.95/1.81	
3.95/1.81	   and(False, False) -> False
3.95/1.81	   and(True, False) -> False
3.95/1.81	   and(False, True) -> False
3.95/1.81	   and(True, True) -> True
3.95/1.81	
3.95/1.81	Rewrite Strategy: INNERMOST
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(9) DecreasingLoopProof (LOWER BOUND(ID))
3.95/1.81	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.95/1.81	
3.95/1.81	The rewrite sequence
3.95/1.81	
3.95/1.81	eqZList(C(x1, x2), C(y1, y2)) ->^+ and(eqZList(x1, y1), eqZList(x2, y2))
3.95/1.81	
3.95/1.81	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.95/1.81	
3.95/1.81	The pumping substitution is [x1 / C(x1, x2), y1 / C(y1, y2)].
3.95/1.81	
3.95/1.81	The result substitution is [ ].
3.95/1.81	
3.95/1.81	
3.95/1.81	
3.95/1.81	
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(10)
3.95/1.81	Complex Obligation (BEST)
3.95/1.81	
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(11)
3.95/1.81	Obligation:
3.95/1.81	Proved the lower bound n^1 for the following obligation:
3.95/1.81	
3.95/1.81	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
3.95/1.81	
3.95/1.81	
3.95/1.81	The TRS R consists of the following rules:
3.95/1.81	
3.95/1.81	   f(C(x1, x2)) -> C(f(x1), f(x2))
3.95/1.81	   f(Z) -> Z
3.95/1.81	   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
3.95/1.81	   eqZList(C(x1, x2), Z) -> False
3.95/1.81	   eqZList(Z, C(y1, y2)) -> False
3.95/1.81	   eqZList(Z, Z) -> True
3.95/1.81	   second(C(x1, x2)) -> x2
3.95/1.81	   first(C(x1, x2)) -> x1
3.95/1.81	   g(x) -> x
3.95/1.81	
3.95/1.81	The (relative) TRS S consists of the following rules:
3.95/1.81	
3.95/1.81	   and(False, False) -> False
3.95/1.81	   and(True, False) -> False
3.95/1.81	   and(False, True) -> False
3.95/1.81	   and(True, True) -> True
3.95/1.81	
3.95/1.81	Rewrite Strategy: INNERMOST
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(12) LowerBoundPropagationProof (FINISHED)
3.95/1.81	Propagated lower bound.
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(13)
3.95/1.81	BOUNDS(n^1, INF)
3.95/1.81	
3.95/1.81	----------------------------------------
3.95/1.81	
3.95/1.81	(14)
3.95/1.81	Obligation:
3.95/1.81	Analyzing the following TRS for decreasing loops:
3.95/1.81	
3.95/1.81	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
3.95/1.81	
3.95/1.81	
3.95/1.81	The TRS R consists of the following rules:
3.95/1.81	
3.95/1.81	   f(C(x1, x2)) -> C(f(x1), f(x2))
3.95/1.81	   f(Z) -> Z
3.95/1.81	   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
3.95/1.81	   eqZList(C(x1, x2), Z) -> False
3.95/1.81	   eqZList(Z, C(y1, y2)) -> False
3.95/1.81	   eqZList(Z, Z) -> True
3.95/1.81	   second(C(x1, x2)) -> x2
3.95/1.81	   first(C(x1, x2)) -> x1
3.95/1.81	   g(x) -> x
3.95/1.81	
3.95/1.81	The (relative) TRS S consists of the following rules:
3.95/1.81	
3.95/1.81	   and(False, False) -> False
3.95/1.81	   and(True, False) -> False
3.95/1.81	   and(False, True) -> False
3.95/1.81	   and(True, True) -> True
3.95/1.81	
3.95/1.81	Rewrite Strategy: INNERMOST
4.06/1.84	EOF