5.07/2.44	WORST_CASE(Omega(n^1), O(n^1))
5.07/2.44	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
5.07/2.44	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.07/2.44	
5.07/2.44	
5.07/2.44	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
5.07/2.44	
5.07/2.44	(0) CpxIntTrs
5.07/2.44	(1) Koat Proof [FINISHED, 118 ms]
5.07/2.44	(2) BOUNDS(1, n^1)
5.07/2.44	(3) Loat Proof [FINISHED, 718 ms]
5.07/2.44	(4) BOUNDS(n^1, INF)
5.07/2.44	
5.07/2.44	
5.07/2.44	----------------------------------------
5.07/2.44	
5.07/2.44	(0)
5.07/2.44	Obligation:
5.07/2.44	Complexity Int TRS consisting of the following rules:
5.07/2.44	evalDis2start(A, B, C) -> Com_1(evalDis2entryin(A, B, C)) :|: TRUE
5.07/2.44	evalDis2entryin(A, B, C) -> Com_1(evalDis2bb3in(B, C, A)) :|: TRUE
5.07/2.44	evalDis2bb3in(A, B, C) -> Com_1(evalDis2bbin(A, B, C)) :|: A >= C + 1
5.07/2.44	evalDis2bb3in(A, B, C) -> Com_1(evalDis2returnin(A, B, C)) :|: C >= A
5.07/2.44	evalDis2bbin(A, B, C) -> Com_1(evalDis2bb1in(A, B, C)) :|: B >= C + 1
5.07/2.44	evalDis2bbin(A, B, C) -> Com_1(evalDis2bb2in(A, B, C)) :|: C >= B
5.07/2.44	evalDis2bb1in(A, B, C) -> Com_1(evalDis2bb3in(A, B, C + 1)) :|: TRUE
5.07/2.44	evalDis2bb2in(A, B, C) -> Com_1(evalDis2bb3in(A, B + 1, C)) :|: TRUE
5.07/2.44	evalDis2returnin(A, B, C) -> Com_1(evalDis2stop(A, B, C)) :|: TRUE
5.07/2.44	
5.07/2.44	The start-symbols are:[evalDis2start_3]
5.07/2.44	
5.07/2.44	
5.07/2.44	----------------------------------------
5.07/2.44	
5.07/2.44	(1) Koat Proof (FINISHED)
5.07/2.44	YES(?, 12*ar_1 + 6*ar_2 + 6*ar_0 + 7)
5.07/2.44	
5.07/2.44	
5.07/2.44	
5.07/2.44	Initial complexity problem:
5.07/2.44	
5.07/2.44	1:	T:
5.07/2.44	
5.07/2.44			(Comp: ?, Cost: 1)    evalDis2start(ar_0, ar_1, ar_2) -> Com_1(evalDis2entryin(ar_0, ar_1, ar_2))
5.07/2.44	
5.07/2.44			(Comp: ?, Cost: 1)    evalDis2entryin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_1, ar_2, ar_0))
5.07/2.44	
5.07/2.44			(Comp: ?, Cost: 1)    evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bbin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
5.07/2.44	
5.07/2.44			(Comp: ?, Cost: 1)    evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2returnin(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
5.07/2.44	
5.07/2.44			(Comp: ?, Cost: 1)    evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb1in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
5.07/2.44	
5.07/2.44			(Comp: ?, Cost: 1)    evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
5.07/2.44	
5.07/2.44			(Comp: ?, Cost: 1)    evalDis2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1, ar_2 + 1))
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1 + 1, ar_2))
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2returnin(ar_0, ar_1, ar_2) -> Com_1(evalDis2stop(ar_0, ar_1, ar_2))
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalDis2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
5.07/2.45	
5.07/2.45		start location:	koat_start
5.07/2.45	
5.07/2.45		leaf cost:	0
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Repeatedly propagating knowledge in problem 1 produces the following problem:
5.07/2.45	
5.07/2.45	2:	T:
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 1)    evalDis2start(ar_0, ar_1, ar_2) -> Com_1(evalDis2entryin(ar_0, ar_1, ar_2))
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 1)    evalDis2entryin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_1, ar_2, ar_0))
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bbin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2returnin(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb1in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1, ar_2 + 1))
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1 + 1, ar_2))
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2returnin(ar_0, ar_1, ar_2) -> Com_1(evalDis2stop(ar_0, ar_1, ar_2))
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalDis2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
5.07/2.45	
5.07/2.45		start location:	koat_start
5.07/2.45	
5.07/2.45		leaf cost:	0
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	A polynomial rank function with
5.07/2.45	
5.07/2.45		Pol(evalDis2start) = 2
5.07/2.45	
5.07/2.45		Pol(evalDis2entryin) = 2
5.07/2.45	
5.07/2.45		Pol(evalDis2bb3in) = 2
5.07/2.45	
5.07/2.45		Pol(evalDis2bbin) = 2
5.07/2.45	
5.07/2.45		Pol(evalDis2returnin) = 1
5.07/2.45	
5.07/2.45		Pol(evalDis2bb1in) = 2
5.07/2.45	
5.07/2.45		Pol(evalDis2bb2in) = 2
5.07/2.45	
5.07/2.45		Pol(evalDis2stop) = 0
5.07/2.45	
5.07/2.45		Pol(koat_start) = 2
5.07/2.45	
5.07/2.45	orients all transitions weakly and the transitions
5.07/2.45	
5.07/2.45		evalDis2returnin(ar_0, ar_1, ar_2) -> Com_1(evalDis2stop(ar_0, ar_1, ar_2))
5.07/2.45	
5.07/2.45		evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2returnin(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
5.07/2.45	
5.07/2.45	strictly and produces the following problem:
5.07/2.45	
5.07/2.45	3:	T:
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 1)    evalDis2start(ar_0, ar_1, ar_2) -> Com_1(evalDis2entryin(ar_0, ar_1, ar_2))
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 1)    evalDis2entryin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_1, ar_2, ar_0))
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bbin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45			(Comp: 2, Cost: 1)    evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2returnin(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb1in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1, ar_2 + 1))
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1 + 1, ar_2))
5.07/2.45	
5.07/2.45			(Comp: 2, Cost: 1)    evalDis2returnin(ar_0, ar_1, ar_2) -> Com_1(evalDis2stop(ar_0, ar_1, ar_2))
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalDis2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
5.07/2.45	
5.07/2.45		start location:	koat_start
5.07/2.45	
5.07/2.45		leaf cost:	0
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Applied AI with 'oct' on problem 3 to obtain the following invariants:
5.07/2.45	
5.07/2.45	  For symbol evalDis2bb1in: X_2 - X_3 - 1 >= 0 /\ X_1 - X_3 - 1 >= 0
5.07/2.45	
5.07/2.45	  For symbol evalDis2bb2in: X_1 - X_3 - 1 >= 0 /\ -X_2 + X_3 >= 0 /\ X_1 - X_2 - 1 >= 0
5.07/2.45	
5.07/2.45	  For symbol evalDis2bbin: X_1 - X_3 - 1 >= 0
5.07/2.45	
5.07/2.45	  For symbol evalDis2returnin: -X_1 + X_3 >= 0
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	This yielded the following problem:
5.07/2.45	
5.07/2.45	4:	T:
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalDis2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
5.07/2.45	
5.07/2.45			(Comp: 2, Cost: 1)    evalDis2returnin(ar_0, ar_1, ar_2) -> Com_1(evalDis2stop(ar_0, ar_1, ar_2)) [ -ar_0 + ar_2 >= 0 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 - ar_1 - 1 >= 0 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_0 - ar_2 - 1 >= 0 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb2in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= ar_1 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb1in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ ar_1 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45			(Comp: 2, Cost: 1)    evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2returnin(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)    evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bbin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 1)    evalDis2entryin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_1, ar_2, ar_0))
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 1)    evalDis2start(ar_0, ar_1, ar_2) -> Com_1(evalDis2entryin(ar_0, ar_1, ar_2))
5.07/2.45	
5.07/2.45		start location:	koat_start
5.07/2.45	
5.07/2.45		leaf cost:	0
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	A polynomial rank function with
5.07/2.45	
5.07/2.45		Pol(koat_start) = 2*V_2 - 2*V_3
5.07/2.45	
5.07/2.45		Pol(evalDis2start) = 2*V_2 - 2*V_3
5.07/2.45	
5.07/2.45		Pol(evalDis2returnin) = 2*V_1 - 2*V_2
5.07/2.45	
5.07/2.45		Pol(evalDis2stop) = 2*V_1 - 2*V_2
5.07/2.45	
5.07/2.45		Pol(evalDis2bb2in) = 2*V_1 - 2*V_2 - 1
5.07/2.45	
5.07/2.45		Pol(evalDis2bb3in) = 2*V_1 - 2*V_2
5.07/2.45	
5.07/2.45		Pol(evalDis2bb1in) = 2*V_1 - 2*V_2
5.07/2.45	
5.07/2.45		Pol(evalDis2bbin) = 2*V_1 - 2*V_2
5.07/2.45	
5.07/2.45		Pol(evalDis2entryin) = 2*V_2 - 2*V_3
5.07/2.45	
5.07/2.45	orients all transitions weakly and the transitions
5.07/2.45	
5.07/2.45		evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb2in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= ar_1 ]
5.07/2.45	
5.07/2.45		evalDis2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 - ar_1 - 1 >= 0 ]
5.07/2.45	
5.07/2.45	strictly and produces the following problem:
5.07/2.45	
5.07/2.45	5:	T:
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2) -> Com_1(evalDis2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
5.07/2.45	
5.07/2.45			(Comp: 2, Cost: 1)                  evalDis2returnin(ar_0, ar_1, ar_2) -> Com_1(evalDis2stop(ar_0, ar_1, ar_2)) [ -ar_0 + ar_2 >= 0 ]
5.07/2.45	
5.07/2.45			(Comp: 2*ar_1 + 2*ar_2, Cost: 1)    evalDis2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 - ar_1 - 1 >= 0 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)                  evalDis2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_0 - ar_2 - 1 >= 0 ]
5.07/2.45	
5.07/2.45			(Comp: 2*ar_1 + 2*ar_2, Cost: 1)    evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb2in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= ar_1 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)                  evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb1in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ ar_1 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45			(Comp: 2, Cost: 1)                  evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2returnin(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)                  evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bbin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 1)                  evalDis2entryin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_1, ar_2, ar_0))
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 1)                  evalDis2start(ar_0, ar_1, ar_2) -> Com_1(evalDis2entryin(ar_0, ar_1, ar_2))
5.07/2.45	
5.07/2.45		start location:	koat_start
5.07/2.45	
5.07/2.45		leaf cost:	0
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	A polynomial rank function with
5.07/2.45	
5.07/2.45		Pol(koat_start) = -2*V_1 + 2*V_2
5.07/2.45	
5.07/2.45		Pol(evalDis2start) = -2*V_1 + 2*V_2
5.07/2.45	
5.07/2.45		Pol(evalDis2returnin) = 2*V_1 - 2*V_3
5.07/2.45	
5.07/2.45		Pol(evalDis2stop) = 2*V_1 - 2*V_3
5.07/2.45	
5.07/2.45		Pol(evalDis2bb2in) = 2*V_1 - 2*V_3
5.07/2.45	
5.07/2.45		Pol(evalDis2bb3in) = 2*V_1 - 2*V_3
5.07/2.45	
5.07/2.45		Pol(evalDis2bb1in) = 2*V_1 - 2*V_3 - 1
5.07/2.45	
5.07/2.45		Pol(evalDis2bbin) = 2*V_1 - 2*V_3
5.07/2.45	
5.07/2.45		Pol(evalDis2entryin) = -2*V_1 + 2*V_2
5.07/2.45	
5.07/2.45	orients all transitions weakly and the transitions
5.07/2.45	
5.07/2.45		evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb1in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ ar_1 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45		evalDis2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_0 - ar_2 - 1 >= 0 ]
5.07/2.45	
5.07/2.45	strictly and produces the following problem:
5.07/2.45	
5.07/2.45	6:	T:
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2) -> Com_1(evalDis2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
5.07/2.45	
5.07/2.45			(Comp: 2, Cost: 1)                  evalDis2returnin(ar_0, ar_1, ar_2) -> Com_1(evalDis2stop(ar_0, ar_1, ar_2)) [ -ar_0 + ar_2 >= 0 ]
5.07/2.45	
5.07/2.45			(Comp: 2*ar_1 + 2*ar_2, Cost: 1)    evalDis2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 - ar_1 - 1 >= 0 ]
5.07/2.45	
5.07/2.45			(Comp: 2*ar_0 + 2*ar_1, Cost: 1)    evalDis2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_0 - ar_2 - 1 >= 0 ]
5.07/2.45	
5.07/2.45			(Comp: 2*ar_1 + 2*ar_2, Cost: 1)    evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb2in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= ar_1 ]
5.07/2.45	
5.07/2.45			(Comp: 2*ar_0 + 2*ar_1, Cost: 1)    evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb1in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ ar_1 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45			(Comp: 2, Cost: 1)                  evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2returnin(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
5.07/2.45	
5.07/2.45			(Comp: ?, Cost: 1)                  evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bbin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 1)                  evalDis2entryin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_1, ar_2, ar_0))
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 1)                  evalDis2start(ar_0, ar_1, ar_2) -> Com_1(evalDis2entryin(ar_0, ar_1, ar_2))
5.07/2.45	
5.07/2.45		start location:	koat_start
5.07/2.45	
5.07/2.45		leaf cost:	0
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Repeatedly propagating knowledge in problem 6 produces the following problem:
5.07/2.45	
5.07/2.45	7:	T:
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 0)                               koat_start(ar_0, ar_1, ar_2) -> Com_1(evalDis2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
5.07/2.45	
5.07/2.45			(Comp: 2, Cost: 1)                               evalDis2returnin(ar_0, ar_1, ar_2) -> Com_1(evalDis2stop(ar_0, ar_1, ar_2)) [ -ar_0 + ar_2 >= 0 ]
5.07/2.45	
5.07/2.45			(Comp: 2*ar_1 + 2*ar_2, Cost: 1)                 evalDis2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 - ar_1 - 1 >= 0 ]
5.07/2.45	
5.07/2.45			(Comp: 2*ar_0 + 2*ar_1, Cost: 1)                 evalDis2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_0, ar_1, ar_2 + 1)) [ ar_1 - ar_2 - 1 >= 0 /\ ar_0 - ar_2 - 1 >= 0 ]
5.07/2.45	
5.07/2.45			(Comp: 2*ar_1 + 2*ar_2, Cost: 1)                 evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb2in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= ar_1 ]
5.07/2.45	
5.07/2.45			(Comp: 2*ar_0 + 2*ar_1, Cost: 1)                 evalDis2bbin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb1in(ar_0, ar_1, ar_2)) [ ar_0 - ar_2 - 1 >= 0 /\ ar_1 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45			(Comp: 2, Cost: 1)                               evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2returnin(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
5.07/2.45	
5.07/2.45			(Comp: 2*ar_0 + 4*ar_1 + 2*ar_2 + 1, Cost: 1)    evalDis2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalDis2bbin(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 1)                               evalDis2entryin(ar_0, ar_1, ar_2) -> Com_1(evalDis2bb3in(ar_1, ar_2, ar_0))
5.07/2.45	
5.07/2.45			(Comp: 1, Cost: 1)                               evalDis2start(ar_0, ar_1, ar_2) -> Com_1(evalDis2entryin(ar_0, ar_1, ar_2))
5.07/2.45	
5.07/2.45		start location:	koat_start
5.07/2.45	
5.07/2.45		leaf cost:	0
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Complexity upper bound 12*ar_1 + 6*ar_2 + 6*ar_0 + 7
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Time: 0.205 sec (SMT: 0.184 sec)
5.07/2.45	
5.07/2.45	
5.07/2.45	----------------------------------------
5.07/2.45	
5.07/2.45	(2)
5.07/2.45	BOUNDS(1, n^1)
5.07/2.45	
5.07/2.45	----------------------------------------
5.07/2.45	
5.07/2.45	(3) Loat Proof (FINISHED)
5.07/2.45	
5.07/2.45	
5.07/2.45	### Pre-processing the ITS problem ###
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Initial linear ITS problem
5.07/2.45	
5.07/2.45	   Start location: evalDis2start
5.07/2.45	
5.07/2.45	      0: evalDis2start -> evalDis2entryin : [], cost: 1
5.07/2.45	
5.07/2.45	      1: evalDis2entryin -> evalDis2bb3in : A'=B, B'=C, C'=A, [], cost: 1
5.07/2.45	
5.07/2.45	      2: evalDis2bb3in -> evalDis2bbin : [ A>=1+C ], cost: 1
5.07/2.45	
5.07/2.45	      3: evalDis2bb3in -> evalDis2returnin : [ C>=A ], cost: 1
5.07/2.45	
5.07/2.45	      4: evalDis2bbin -> evalDis2bb1in : [ B>=1+C ], cost: 1
5.07/2.45	
5.07/2.45	      5: evalDis2bbin -> evalDis2bb2in : [ C>=B ], cost: 1
5.07/2.45	
5.07/2.45	      6: evalDis2bb1in -> evalDis2bb3in : C'=1+C, [], cost: 1
5.07/2.45	
5.07/2.45	      7: evalDis2bb2in -> evalDis2bb3in : B'=1+B, [], cost: 1
5.07/2.45	
5.07/2.45	      8: evalDis2returnin -> evalDis2stop : [], cost: 1
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Removed unreachable and leaf rules:
5.07/2.45	
5.07/2.45	   Start location: evalDis2start
5.07/2.45	
5.07/2.45	      0: evalDis2start -> evalDis2entryin : [], cost: 1
5.07/2.45	
5.07/2.45	      1: evalDis2entryin -> evalDis2bb3in : A'=B, B'=C, C'=A, [], cost: 1
5.07/2.45	
5.07/2.45	      2: evalDis2bb3in -> evalDis2bbin : [ A>=1+C ], cost: 1
5.07/2.45	
5.07/2.45	      4: evalDis2bbin -> evalDis2bb1in : [ B>=1+C ], cost: 1
5.07/2.45	
5.07/2.45	      5: evalDis2bbin -> evalDis2bb2in : [ C>=B ], cost: 1
5.07/2.45	
5.07/2.45	      6: evalDis2bb1in -> evalDis2bb3in : C'=1+C, [], cost: 1
5.07/2.45	
5.07/2.45	      7: evalDis2bb2in -> evalDis2bb3in : B'=1+B, [], cost: 1
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	### Simplification by acceleration and chaining ###
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Eliminated locations (on linear paths):
5.07/2.45	
5.07/2.45	   Start location: evalDis2start
5.07/2.45	
5.07/2.45	      9: evalDis2start -> evalDis2bb3in : A'=B, B'=C, C'=A, [], cost: 2
5.07/2.45	
5.07/2.45	      2: evalDis2bb3in -> evalDis2bbin : [ A>=1+C ], cost: 1
5.07/2.45	
5.07/2.45	     10: evalDis2bbin -> evalDis2bb3in : C'=1+C, [ B>=1+C ], cost: 2
5.07/2.45	
5.07/2.45	     11: evalDis2bbin -> evalDis2bb3in : B'=1+B, [ C>=B ], cost: 2
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Eliminated locations (on tree-shaped paths):
5.07/2.45	
5.07/2.45	   Start location: evalDis2start
5.07/2.45	
5.07/2.45	      9: evalDis2start -> evalDis2bb3in : A'=B, B'=C, C'=A, [], cost: 2
5.07/2.45	
5.07/2.45	     12: evalDis2bb3in -> evalDis2bb3in : C'=1+C, [ A>=1+C && B>=1+C ], cost: 3
5.07/2.45	
5.07/2.45	     13: evalDis2bb3in -> evalDis2bb3in : B'=1+B, [ A>=1+C && C>=B ], cost: 3
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Accelerating simple loops of location 2.
5.07/2.45	
5.07/2.45	   Accelerating the following rules:
5.07/2.45	
5.07/2.45	     12: evalDis2bb3in -> evalDis2bb3in : C'=1+C, [ A>=1+C && B>=1+C ], cost: 3
5.07/2.45	
5.07/2.45	     13: evalDis2bb3in -> evalDis2bb3in : B'=1+B, [ A>=1+C && C>=B ], cost: 3
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	   Accelerated rule 12 with backward acceleration, yielding the new rule 14.
5.07/2.45	
5.07/2.45	   Accelerated rule 12 with backward acceleration, yielding the new rule 15.
5.07/2.45	
5.07/2.45	   Accelerated rule 13 with metering function 1+C-B, yielding the new rule 16.
5.07/2.45	
5.07/2.45	   Removing the simple loops: 12 13.
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Accelerated all simple loops using metering functions (where possible):
5.07/2.45	
5.07/2.45	   Start location: evalDis2start
5.07/2.45	
5.07/2.45	      9: evalDis2start -> evalDis2bb3in : A'=B, B'=C, C'=A, [], cost: 2
5.07/2.45	
5.07/2.45	     14: evalDis2bb3in -> evalDis2bb3in : C'=A, [ A>=1+C && B>=1+C && B>=A ], cost: -3*C+3*A
5.07/2.45	
5.07/2.45	     15: evalDis2bb3in -> evalDis2bb3in : C'=B, [ A>=1+C && B>=1+C && A>=B ], cost: -3*C+3*B
5.07/2.45	
5.07/2.45	     16: evalDis2bb3in -> evalDis2bb3in : B'=1+C, [ A>=1+C && C>=B ], cost: 3+3*C-3*B
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Chained accelerated rules (with incoming rules):
5.07/2.45	
5.07/2.45	   Start location: evalDis2start
5.07/2.45	
5.07/2.45	      9: evalDis2start -> evalDis2bb3in : A'=B, B'=C, C'=A, [], cost: 2
5.07/2.45	
5.07/2.45	     17: evalDis2start -> evalDis2bb3in : A'=B, B'=C, C'=B, [ B>=1+A && C>=1+A && C>=B ], cost: 2-3*A+3*B
5.07/2.45	
5.07/2.45	     18: evalDis2start -> evalDis2bb3in : A'=B, B'=C, [ B>=1+A && C>=1+A && B>=C ], cost: 2+3*C-3*A
5.07/2.45	
5.07/2.45	     19: evalDis2start -> evalDis2bb3in : A'=B, B'=1+A, C'=A, [ B>=1+A && A>=C ], cost: 5-3*C+3*A
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Removed unreachable locations (and leaf rules with constant cost):
5.07/2.45	
5.07/2.45	   Start location: evalDis2start
5.07/2.45	
5.07/2.45	     17: evalDis2start -> evalDis2bb3in : A'=B, B'=C, C'=B, [ B>=1+A && C>=1+A && C>=B ], cost: 2-3*A+3*B
5.07/2.45	
5.07/2.45	     18: evalDis2start -> evalDis2bb3in : A'=B, B'=C, [ B>=1+A && C>=1+A && B>=C ], cost: 2+3*C-3*A
5.07/2.45	
5.07/2.45	     19: evalDis2start -> evalDis2bb3in : A'=B, B'=1+A, C'=A, [ B>=1+A && A>=C ], cost: 5-3*C+3*A
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	### Computing asymptotic complexity ###
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Fully simplified ITS problem
5.07/2.45	
5.07/2.45	   Start location: evalDis2start
5.07/2.45	
5.07/2.45	     17: evalDis2start -> evalDis2bb3in : A'=B, B'=C, C'=B, [ B>=1+A && C>=1+A && C>=B ], cost: 2-3*A+3*B
5.07/2.45	
5.07/2.45	     18: evalDis2start -> evalDis2bb3in : A'=B, B'=C, [ B>=1+A && C>=1+A && B>=C ], cost: 2+3*C-3*A
5.07/2.45	
5.07/2.45	     19: evalDis2start -> evalDis2bb3in : A'=B, B'=1+A, C'=A, [ B>=1+A && A>=C ], cost: 5-3*C+3*A
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Computing asymptotic complexity for rule 17
5.07/2.45	
5.07/2.45	   Solved the limit problem by the following transformations:
5.07/2.45	
5.07/2.45	      Created initial limit problem:
5.07/2.45	
5.07/2.45	      2-3*A+3*B (+), 1+C-B (+/+!), -A+B (+/+!), C-A (+/+!) [not solved]
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	      removing all constraints (solved by SMT)
5.07/2.45	
5.07/2.45	      resulting limit problem: [solved]
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	      applying transformation rule (C) using substitution {C==2*n,A==0,B==n}
5.07/2.45	
5.07/2.45	      resulting limit problem:
5.07/2.45	
5.07/2.45	      [solved]
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	   Solution:
5.07/2.45	
5.07/2.45	      C / 2*n
5.07/2.45	
5.07/2.45	      A / 0
5.07/2.45	
5.07/2.45	      B / n
5.07/2.45	
5.07/2.45	   Resulting cost 2+3*n has complexity: Poly(n^1)
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Found new complexity Poly(n^1).
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	Obtained the following overall complexity (w.r.t. the length of the input n):
5.07/2.45	
5.07/2.45	   Complexity:  Poly(n^1)
5.07/2.45	
5.07/2.45	   Cpx degree:  1
5.07/2.45	
5.07/2.45	   Solved cost: 2+3*n
5.07/2.45	
5.07/2.45	   Rule cost:   2-3*A+3*B
5.07/2.45	
5.07/2.45	   Rule guard:  [ B>=1+A && C>=1+A && C>=B ]
5.07/2.45	
5.07/2.45	
5.07/2.45	
5.07/2.45	WORST_CASE(Omega(n^1),?)
5.07/2.45	
5.07/2.45	
5.07/2.45	----------------------------------------
5.07/2.45	
5.07/2.45	(4)
5.07/2.45	BOUNDS(n^1, INF)
5.07/2.47	EOF