22.82/6.87	WORST_CASE(Omega(n^1), O(n^1))
22.82/6.88	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
22.82/6.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.82/6.88	
22.82/6.88	
22.82/6.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.82/6.88	
22.82/6.88	(0) CpxTRS
22.82/6.88	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
22.82/6.88	(2) CpxTRS
22.82/6.88	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
22.82/6.88	    (4) CpxTRS
22.82/6.88	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 149 ms]
22.82/6.88	    (6) BOUNDS(1, n^1)
22.82/6.88	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
22.82/6.88	(8) TRS for Loop Detection
22.82/6.88	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
22.82/6.88	    (10) BEST
22.82/6.88	        (11) proven lower bound
22.82/6.88	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
22.82/6.88	            (13) BOUNDS(n^1, INF)
22.82/6.88	        (14) TRS for Loop Detection
22.82/6.88	
22.82/6.88	
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(0)
22.82/6.88	Obligation:
22.82/6.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.82/6.88	
22.82/6.88	
22.82/6.88	The TRS R consists of the following rules:
22.82/6.88	
22.82/6.88	   active(U11(tt, M, N)) -> mark(U12(tt, M, N))
22.82/6.88	   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
22.82/6.88	   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
22.82/6.88	   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
22.82/6.88	   active(plus(N, 0)) -> mark(N)
22.82/6.88	   active(plus(N, s(M))) -> mark(U11(tt, M, N))
22.82/6.88	   active(x(N, 0)) -> mark(0)
22.82/6.88	   active(x(N, s(M))) -> mark(U21(tt, M, N))
22.82/6.88	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
22.82/6.88	   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
22.82/6.88	   active(s(X)) -> s(active(X))
22.82/6.88	   active(plus(X1, X2)) -> plus(active(X1), X2)
22.82/6.88	   active(plus(X1, X2)) -> plus(X1, active(X2))
22.82/6.88	   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
22.82/6.88	   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
22.82/6.88	   active(x(X1, X2)) -> x(active(X1), X2)
22.82/6.88	   active(x(X1, X2)) -> x(X1, active(X2))
22.82/6.88	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
22.82/6.88	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
22.82/6.88	   s(mark(X)) -> mark(s(X))
22.82/6.88	   plus(mark(X1), X2) -> mark(plus(X1, X2))
22.82/6.88	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
22.82/6.88	   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
22.82/6.88	   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
22.82/6.88	   x(mark(X1), X2) -> mark(x(X1, X2))
22.82/6.88	   x(X1, mark(X2)) -> mark(x(X1, X2))
22.82/6.88	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(tt) -> ok(tt)
22.82/6.88	   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(s(X)) -> s(proper(X))
22.82/6.88	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
22.82/6.88	   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
22.82/6.88	   proper(0) -> ok(0)
22.82/6.88	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
22.82/6.88	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
22.82/6.88	   s(ok(X)) -> ok(s(X))
22.82/6.88	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
22.82/6.88	   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
22.82/6.88	   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
22.82/6.88	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
22.82/6.88	   top(mark(X)) -> top(proper(X))
22.82/6.88	   top(ok(X)) -> top(active(X))
22.82/6.88	
22.82/6.88	S is empty.
22.82/6.88	Rewrite Strategy: FULL
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
22.82/6.88	The following defined symbols can occur below the 0th argument of top: proper, active
22.82/6.88	The following defined symbols can occur below the 0th argument of proper: proper, active
22.82/6.88	The following defined symbols can occur below the 0th argument of active: proper, active
22.82/6.88	
22.82/6.88	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
22.82/6.88	active(U11(tt, M, N)) -> mark(U12(tt, M, N))
22.82/6.88	active(U12(tt, M, N)) -> mark(s(plus(N, M)))
22.82/6.88	active(U21(tt, M, N)) -> mark(U22(tt, M, N))
22.82/6.88	active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
22.82/6.88	active(plus(N, 0)) -> mark(N)
22.82/6.88	active(plus(N, s(M))) -> mark(U11(tt, M, N))
22.82/6.88	active(x(N, 0)) -> mark(0)
22.82/6.88	active(x(N, s(M))) -> mark(U21(tt, M, N))
22.82/6.88	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
22.82/6.88	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
22.82/6.88	active(s(X)) -> s(active(X))
22.82/6.88	active(plus(X1, X2)) -> plus(active(X1), X2)
22.82/6.88	active(plus(X1, X2)) -> plus(X1, active(X2))
22.82/6.88	active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
22.82/6.88	active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
22.82/6.88	active(x(X1, X2)) -> x(active(X1), X2)
22.82/6.88	active(x(X1, X2)) -> x(X1, active(X2))
22.82/6.88	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
22.82/6.88	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
22.82/6.88	proper(s(X)) -> s(proper(X))
22.82/6.88	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
22.82/6.88	proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
22.82/6.88	proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
22.82/6.88	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
22.82/6.88	
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(2)
22.82/6.88	Obligation:
22.82/6.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
22.82/6.88	
22.82/6.88	
22.82/6.88	The TRS R consists of the following rules:
22.82/6.88	
22.82/6.88	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
22.82/6.88	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
22.82/6.88	   s(mark(X)) -> mark(s(X))
22.82/6.88	   plus(mark(X1), X2) -> mark(plus(X1, X2))
22.82/6.88	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
22.82/6.88	   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
22.82/6.88	   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
22.82/6.88	   x(mark(X1), X2) -> mark(x(X1, X2))
22.82/6.88	   x(X1, mark(X2)) -> mark(x(X1, X2))
22.82/6.88	   proper(tt) -> ok(tt)
22.82/6.88	   proper(0) -> ok(0)
22.82/6.88	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
22.82/6.88	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
22.82/6.88	   s(ok(X)) -> ok(s(X))
22.82/6.88	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
22.82/6.88	   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
22.82/6.88	   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
22.82/6.88	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
22.82/6.88	   top(mark(X)) -> top(proper(X))
22.82/6.88	   top(ok(X)) -> top(active(X))
22.82/6.88	
22.82/6.88	S is empty.
22.82/6.88	Rewrite Strategy: FULL
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(3) RelTrsToTrsProof (UPPER BOUND(ID))
22.82/6.88	transformed relative TRS to TRS
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(4)
22.82/6.88	Obligation:
22.82/6.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
22.82/6.88	
22.82/6.88	
22.82/6.88	The TRS R consists of the following rules:
22.82/6.88	
22.82/6.88	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
22.82/6.88	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
22.82/6.88	   s(mark(X)) -> mark(s(X))
22.82/6.88	   plus(mark(X1), X2) -> mark(plus(X1, X2))
22.82/6.88	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
22.82/6.88	   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
22.82/6.88	   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
22.82/6.88	   x(mark(X1), X2) -> mark(x(X1, X2))
22.82/6.88	   x(X1, mark(X2)) -> mark(x(X1, X2))
22.82/6.88	   proper(tt) -> ok(tt)
22.82/6.88	   proper(0) -> ok(0)
22.82/6.88	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
22.82/6.88	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
22.82/6.88	   s(ok(X)) -> ok(s(X))
22.82/6.88	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
22.82/6.88	   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
22.82/6.88	   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
22.82/6.88	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
22.82/6.88	   top(mark(X)) -> top(proper(X))
22.82/6.88	   top(ok(X)) -> top(active(X))
22.82/6.88	
22.82/6.88	S is empty.
22.82/6.88	Rewrite Strategy: FULL
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(5) CpxTrsMatchBoundsTAProof (FINISHED)
22.82/6.88	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. 
22.82/6.88	
22.82/6.88	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
22.82/6.88	final states : [1, 2, 3, 4, 5, 6, 7, 8, 9]
22.82/6.88	transitions: 
22.82/6.88	mark0(0) -> 0
22.82/6.88	tt0() -> 0
22.82/6.88	ok0(0) -> 0
22.82/6.88	00() -> 0
22.82/6.88	active0(0) -> 0
22.82/6.88	U110(0, 0, 0) -> 1
22.82/6.88	U120(0, 0, 0) -> 2
22.82/6.88	s0(0) -> 3
22.82/6.88	plus0(0, 0) -> 4
22.82/6.88	U210(0, 0, 0) -> 5
22.82/6.88	U220(0, 0, 0) -> 6
22.82/6.88	x0(0, 0) -> 7
22.82/6.88	proper0(0) -> 8
22.82/6.88	top0(0) -> 9
22.82/6.88	U111(0, 0, 0) -> 10
22.82/6.88	mark1(10) -> 1
22.82/6.88	U121(0, 0, 0) -> 11
22.82/6.88	mark1(11) -> 2
22.82/6.88	s1(0) -> 12
22.82/6.88	mark1(12) -> 3
22.82/6.88	plus1(0, 0) -> 13
22.82/6.88	mark1(13) -> 4
22.82/6.88	U211(0, 0, 0) -> 14
22.82/6.88	mark1(14) -> 5
22.82/6.88	U221(0, 0, 0) -> 15
22.82/6.88	mark1(15) -> 6
22.82/6.88	x1(0, 0) -> 16
22.82/6.88	mark1(16) -> 7
22.82/6.88	tt1() -> 17
22.82/6.88	ok1(17) -> 8
22.82/6.88	01() -> 18
22.82/6.88	ok1(18) -> 8
22.82/6.88	U111(0, 0, 0) -> 19
22.82/6.88	ok1(19) -> 1
22.82/6.88	U121(0, 0, 0) -> 20
22.82/6.88	ok1(20) -> 2
22.82/6.88	s1(0) -> 21
22.82/6.88	ok1(21) -> 3
22.82/6.88	plus1(0, 0) -> 22
22.82/6.88	ok1(22) -> 4
22.82/6.88	U211(0, 0, 0) -> 23
22.82/6.88	ok1(23) -> 5
22.82/6.88	U221(0, 0, 0) -> 24
22.82/6.88	ok1(24) -> 6
22.82/6.88	x1(0, 0) -> 25
22.82/6.88	ok1(25) -> 7
22.82/6.88	proper1(0) -> 26
22.82/6.88	top1(26) -> 9
22.82/6.88	active1(0) -> 27
22.82/6.88	top1(27) -> 9
22.82/6.88	mark1(10) -> 10
22.82/6.88	mark1(10) -> 19
22.82/6.88	mark1(11) -> 11
22.82/6.88	mark1(11) -> 20
22.82/6.88	mark1(12) -> 12
22.82/6.88	mark1(12) -> 21
22.82/6.88	mark1(13) -> 13
22.82/6.88	mark1(13) -> 22
22.82/6.88	mark1(14) -> 14
22.82/6.88	mark1(14) -> 23
22.82/6.88	mark1(15) -> 15
22.82/6.88	mark1(15) -> 24
22.82/6.88	mark1(16) -> 16
22.82/6.88	mark1(16) -> 25
22.82/6.88	ok1(17) -> 26
22.82/6.88	ok1(18) -> 26
22.82/6.88	ok1(19) -> 10
22.82/6.88	ok1(19) -> 19
22.82/6.88	ok1(20) -> 11
22.82/6.88	ok1(20) -> 20
22.82/6.88	ok1(21) -> 12
22.82/6.88	ok1(21) -> 21
22.82/6.88	ok1(22) -> 13
22.82/6.88	ok1(22) -> 22
22.82/6.88	ok1(23) -> 14
22.82/6.88	ok1(23) -> 23
22.82/6.88	ok1(24) -> 15
22.82/6.88	ok1(24) -> 24
22.82/6.88	ok1(25) -> 16
22.82/6.88	ok1(25) -> 25
22.82/6.88	active2(17) -> 28
22.82/6.88	top2(28) -> 9
22.82/6.88	active2(18) -> 28
22.82/6.88	
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(6)
22.82/6.88	BOUNDS(1, n^1)
22.82/6.88	
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
22.82/6.88	Transformed a relative TRS into a decreasing-loop problem.
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(8)
22.82/6.88	Obligation:
22.82/6.88	Analyzing the following TRS for decreasing loops:
22.82/6.88	
22.82/6.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.82/6.88	
22.82/6.88	
22.82/6.88	The TRS R consists of the following rules:
22.82/6.88	
22.82/6.88	   active(U11(tt, M, N)) -> mark(U12(tt, M, N))
22.82/6.88	   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
22.82/6.88	   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
22.82/6.88	   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
22.82/6.88	   active(plus(N, 0)) -> mark(N)
22.82/6.88	   active(plus(N, s(M))) -> mark(U11(tt, M, N))
22.82/6.88	   active(x(N, 0)) -> mark(0)
22.82/6.88	   active(x(N, s(M))) -> mark(U21(tt, M, N))
22.82/6.88	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
22.82/6.88	   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
22.82/6.88	   active(s(X)) -> s(active(X))
22.82/6.88	   active(plus(X1, X2)) -> plus(active(X1), X2)
22.82/6.88	   active(plus(X1, X2)) -> plus(X1, active(X2))
22.82/6.88	   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
22.82/6.88	   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
22.82/6.88	   active(x(X1, X2)) -> x(active(X1), X2)
22.82/6.88	   active(x(X1, X2)) -> x(X1, active(X2))
22.82/6.88	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
22.82/6.88	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
22.82/6.88	   s(mark(X)) -> mark(s(X))
22.82/6.88	   plus(mark(X1), X2) -> mark(plus(X1, X2))
22.82/6.88	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
22.82/6.88	   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
22.82/6.88	   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
22.82/6.88	   x(mark(X1), X2) -> mark(x(X1, X2))
22.82/6.88	   x(X1, mark(X2)) -> mark(x(X1, X2))
22.82/6.88	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(tt) -> ok(tt)
22.82/6.88	   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(s(X)) -> s(proper(X))
22.82/6.88	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
22.82/6.88	   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
22.82/6.88	   proper(0) -> ok(0)
22.82/6.88	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
22.82/6.88	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
22.82/6.88	   s(ok(X)) -> ok(s(X))
22.82/6.88	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
22.82/6.88	   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
22.82/6.88	   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
22.82/6.88	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
22.82/6.88	   top(mark(X)) -> top(proper(X))
22.82/6.88	   top(ok(X)) -> top(active(X))
22.82/6.88	
22.82/6.88	S is empty.
22.82/6.88	Rewrite Strategy: FULL
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(9) DecreasingLoopProof (LOWER BOUND(ID))
22.82/6.88	The following loop(s) give(s) rise to the lower bound Omega(n^1):
22.82/6.88	
22.82/6.88	The rewrite sequence
22.82/6.88	
22.82/6.88	plus(X1, mark(X2)) ->^+ mark(plus(X1, X2))
22.82/6.88	
22.82/6.88	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
22.82/6.88	
22.82/6.88	The pumping substitution is [X2 / mark(X2)].
22.82/6.88	
22.82/6.88	The result substitution is [ ].
22.82/6.88	
22.82/6.88	
22.82/6.88	
22.82/6.88	
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(10)
22.82/6.88	Complex Obligation (BEST)
22.82/6.88	
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(11)
22.82/6.88	Obligation:
22.82/6.88	Proved the lower bound n^1 for the following obligation:
22.82/6.88	
22.82/6.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.82/6.88	
22.82/6.88	
22.82/6.88	The TRS R consists of the following rules:
22.82/6.88	
22.82/6.88	   active(U11(tt, M, N)) -> mark(U12(tt, M, N))
22.82/6.88	   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
22.82/6.88	   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
22.82/6.88	   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
22.82/6.88	   active(plus(N, 0)) -> mark(N)
22.82/6.88	   active(plus(N, s(M))) -> mark(U11(tt, M, N))
22.82/6.88	   active(x(N, 0)) -> mark(0)
22.82/6.88	   active(x(N, s(M))) -> mark(U21(tt, M, N))
22.82/6.88	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
22.82/6.88	   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
22.82/6.88	   active(s(X)) -> s(active(X))
22.82/6.88	   active(plus(X1, X2)) -> plus(active(X1), X2)
22.82/6.88	   active(plus(X1, X2)) -> plus(X1, active(X2))
22.82/6.88	   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
22.82/6.88	   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
22.82/6.88	   active(x(X1, X2)) -> x(active(X1), X2)
22.82/6.88	   active(x(X1, X2)) -> x(X1, active(X2))
22.82/6.88	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
22.82/6.88	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
22.82/6.88	   s(mark(X)) -> mark(s(X))
22.82/6.88	   plus(mark(X1), X2) -> mark(plus(X1, X2))
22.82/6.88	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
22.82/6.88	   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
22.82/6.88	   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
22.82/6.88	   x(mark(X1), X2) -> mark(x(X1, X2))
22.82/6.88	   x(X1, mark(X2)) -> mark(x(X1, X2))
22.82/6.88	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(tt) -> ok(tt)
22.82/6.88	   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(s(X)) -> s(proper(X))
22.82/6.88	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
22.82/6.88	   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
22.82/6.88	   proper(0) -> ok(0)
22.82/6.88	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
22.82/6.88	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
22.82/6.88	   s(ok(X)) -> ok(s(X))
22.82/6.88	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
22.82/6.88	   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
22.82/6.88	   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
22.82/6.88	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
22.82/6.88	   top(mark(X)) -> top(proper(X))
22.82/6.88	   top(ok(X)) -> top(active(X))
22.82/6.88	
22.82/6.88	S is empty.
22.82/6.88	Rewrite Strategy: FULL
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(12) LowerBoundPropagationProof (FINISHED)
22.82/6.88	Propagated lower bound.
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(13)
22.82/6.88	BOUNDS(n^1, INF)
22.82/6.88	
22.82/6.88	----------------------------------------
22.82/6.88	
22.82/6.88	(14)
22.82/6.88	Obligation:
22.82/6.88	Analyzing the following TRS for decreasing loops:
22.82/6.88	
22.82/6.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.82/6.88	
22.82/6.88	
22.82/6.88	The TRS R consists of the following rules:
22.82/6.88	
22.82/6.88	   active(U11(tt, M, N)) -> mark(U12(tt, M, N))
22.82/6.88	   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
22.82/6.88	   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
22.82/6.88	   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
22.82/6.88	   active(plus(N, 0)) -> mark(N)
22.82/6.88	   active(plus(N, s(M))) -> mark(U11(tt, M, N))
22.82/6.88	   active(x(N, 0)) -> mark(0)
22.82/6.88	   active(x(N, s(M))) -> mark(U21(tt, M, N))
22.82/6.88	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
22.82/6.88	   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
22.82/6.88	   active(s(X)) -> s(active(X))
22.82/6.88	   active(plus(X1, X2)) -> plus(active(X1), X2)
22.82/6.88	   active(plus(X1, X2)) -> plus(X1, active(X2))
22.82/6.88	   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
22.82/6.88	   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
22.82/6.88	   active(x(X1, X2)) -> x(active(X1), X2)
22.82/6.88	   active(x(X1, X2)) -> x(X1, active(X2))
22.82/6.88	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
22.82/6.88	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
22.82/6.88	   s(mark(X)) -> mark(s(X))
22.82/6.88	   plus(mark(X1), X2) -> mark(plus(X1, X2))
22.82/6.88	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
22.82/6.88	   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
22.82/6.88	   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
22.82/6.88	   x(mark(X1), X2) -> mark(x(X1, X2))
22.82/6.88	   x(X1, mark(X2)) -> mark(x(X1, X2))
22.82/6.88	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(tt) -> ok(tt)
22.82/6.88	   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(s(X)) -> s(proper(X))
22.82/6.88	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
22.82/6.88	   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
22.82/6.88	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
22.82/6.88	   proper(0) -> ok(0)
22.82/6.88	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
22.82/6.88	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
22.82/6.88	   s(ok(X)) -> ok(s(X))
22.82/6.88	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
22.82/6.88	   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
22.82/6.88	   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
22.82/6.88	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
22.82/6.88	   top(mark(X)) -> top(proper(X))
22.82/6.88	   top(ok(X)) -> top(active(X))
22.82/6.88	
22.82/6.88	S is empty.
22.82/6.88	Rewrite Strategy: FULL
22.93/6.93	EOF