4.82/2.37	WORST_CASE(Omega(n^1), O(n^1))
5.00/2.38	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
5.00/2.38	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.00/2.38	
5.00/2.38	
5.00/2.38	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
5.00/2.38	
5.00/2.38	(0) CpxIntTrs
5.00/2.38	(1) Koat Proof [FINISHED, 129 ms]
5.00/2.38	(2) BOUNDS(1, n^1)
5.00/2.38	(3) Loat Proof [FINISHED, 629 ms]
5.00/2.38	(4) BOUNDS(n^1, INF)
5.00/2.38	
5.00/2.38	
5.00/2.38	----------------------------------------
5.00/2.38	
5.00/2.38	(0)
5.00/2.38	Obligation:
5.00/2.38	Complexity Int TRS consisting of the following rules:
5.00/2.38	eval_start_start(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb0_in(v_1, v_n, v_x_0, v_x_1)) :|: TRUE
5.00/2.38	eval_start_bb0_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_0(v_1, v_n, v_x_0, v_x_1)) :|: TRUE
5.00/2.38	eval_start_0(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_1(v_1, v_n, v_x_0, v_x_1)) :|: TRUE
5.00/2.38	eval_start_1(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_2(v_1, v_n, v_x_0, v_x_1)) :|: TRUE
5.00/2.38	eval_start_2(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_3(v_1, v_n, v_x_0, v_x_1)) :|: TRUE
5.00/2.38	eval_start_3(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb1_in(v_1, v_n, 0, v_x_1)) :|: TRUE
5.00/2.38	eval_start_bb1_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb2_in(v_1, v_n, v_x_0, v_x_1)) :|: v_x_0 < v_n
5.00/2.38	eval_start_bb1_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb4_in(v_1, v_n, v_x_0, v_x_0)) :|: v_x_0 >= v_n
5.00/2.38	eval_start_bb2_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_4(v_1, v_n, v_x_0, v_x_1)) :|: TRUE
5.00/2.38	eval_start_4(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_5(nondef_0, v_n, v_x_0, v_x_1)) :|: TRUE
5.00/2.38	eval_start_5(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb4_in(v_1, v_n, v_x_0, v_x_0)) :|: v_1 > 0
5.00/2.38	eval_start_5(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb3_in(v_1, v_n, v_x_0, v_x_1)) :|: v_1 <= 0
5.00/2.38	eval_start_bb3_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb1_in(v_1, v_n, v_x_0 + 1, v_x_1)) :|: TRUE
5.00/2.38	eval_start_bb4_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb5_in(v_1, v_n, v_x_0, v_x_1)) :|: v_x_1 < v_n
5.00/2.38	eval_start_bb4_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb6_in(v_1, v_n, v_x_0, v_x_1)) :|: v_x_1 >= v_n
5.00/2.38	eval_start_bb5_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_bb4_in(v_1, v_n, v_x_0, v_x_1 + 1)) :|: TRUE
5.00/2.38	eval_start_bb6_in(v_1, v_n, v_x_0, v_x_1) -> Com_1(eval_start_stop(v_1, v_n, v_x_0, v_x_1)) :|: TRUE
5.00/2.38	
5.00/2.38	The start-symbols are:[eval_start_start_4]
5.00/2.38	
5.00/2.38	
5.00/2.38	----------------------------------------
5.00/2.38	
5.00/2.38	(1) Koat Proof (FINISHED)
5.00/2.38	YES(?, 7*ar_1 + 32)
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Initial complexity problem:
5.00/2.38	
5.00/2.38	1:	T:
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_0 + 1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_0, ar_3)) [ ar_0 >= ar_1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, e))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_0, ar_3)) [ ar_3 >= 1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_3 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 + 1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb6in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2 + 1, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb6in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
5.00/2.38	
5.00/2.38		start location:	koat_start
5.00/2.38	
5.00/2.38		leaf cost:	0
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Repeatedly propagating knowledge in problem 1 produces the following problem:
5.00/2.38	
5.00/2.38	2:	T:
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)    evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)    evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)    evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)    evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)    evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)    evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_0 + 1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_0, ar_3)) [ ar_0 >= ar_1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, e))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_0, ar_3)) [ ar_3 >= 1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_3 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 + 1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb6in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2 + 1, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb6in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
5.00/2.38	
5.00/2.38		start location:	koat_start
5.00/2.38	
5.00/2.38		leaf cost:	0
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	A polynomial rank function with
5.00/2.38	
5.00/2.38		Pol(evalstartstart) = 3
5.00/2.38	
5.00/2.38		Pol(evalstartbb0in) = 3
5.00/2.38	
5.00/2.38		Pol(evalstart0) = 3
5.00/2.38	
5.00/2.38		Pol(evalstart1) = 3
5.00/2.38	
5.00/2.38		Pol(evalstart2) = 3
5.00/2.38	
5.00/2.38		Pol(evalstart3) = 3
5.00/2.38	
5.00/2.38		Pol(evalstartbb1in) = 3
5.00/2.38	
5.00/2.38		Pol(evalstartbb2in) = 3
5.00/2.38	
5.00/2.38		Pol(evalstartbb4in) = 2
5.00/2.38	
5.00/2.38		Pol(evalstart4) = 3
5.00/2.38	
5.00/2.38		Pol(evalstart5) = 3
5.00/2.38	
5.00/2.38		Pol(evalstartbb3in) = 3
5.00/2.38	
5.00/2.38		Pol(evalstartbb5in) = 2
5.00/2.38	
5.00/2.38		Pol(evalstartbb6in) = 1
5.00/2.38	
5.00/2.38		Pol(evalstartstop) = 0
5.00/2.38	
5.00/2.38		Pol(koat_start) = 3
5.00/2.38	
5.00/2.38	orients all transitions weakly and the transitions
5.00/2.38	
5.00/2.38		evalstartbb6in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38		evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb6in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 ]
5.00/2.38	
5.00/2.38		evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_0, ar_3)) [ ar_0 >= ar_1 ]
5.00/2.38	
5.00/2.38		evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_0, ar_3)) [ ar_3 >= 1 ]
5.00/2.38	
5.00/2.38	strictly and produces the following problem:
5.00/2.38	
5.00/2.38	3:	T:
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)    evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)    evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)    evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)    evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)    evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)    evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_0 + 1 ]
5.00/2.38	
5.00/2.38			(Comp: 3, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_0, ar_3)) [ ar_0 >= ar_1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, e))
5.00/2.38	
5.00/2.38			(Comp: 3, Cost: 1)    evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_0, ar_3)) [ ar_3 >= 1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_3 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 + 1 ]
5.00/2.38	
5.00/2.38			(Comp: 3, Cost: 1)    evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb6in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)    evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2 + 1, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 3, Cost: 1)    evalstartbb6in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
5.00/2.38	
5.00/2.38		start location:	koat_start
5.00/2.38	
5.00/2.38		leaf cost:	0
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	A polynomial rank function with
5.00/2.38	
5.00/2.38		Pol(evalstartstart) = V_2 + 2
5.00/2.38	
5.00/2.38		Pol(evalstartbb0in) = V_2 + 2
5.00/2.38	
5.00/2.38		Pol(evalstart0) = V_2 + 2
5.00/2.38	
5.00/2.38		Pol(evalstart1) = V_2 + 2
5.00/2.38	
5.00/2.38		Pol(evalstart2) = V_2 + 2
5.00/2.38	
5.00/2.38		Pol(evalstart3) = V_2 + 2
5.00/2.38	
5.00/2.38		Pol(evalstartbb1in) = -V_1 + V_2 + 2
5.00/2.38	
5.00/2.38		Pol(evalstartbb2in) = -V_1 + V_2 + 1
5.00/2.38	
5.00/2.38		Pol(evalstartbb4in) = V_2 - V_3 + 1
5.00/2.38	
5.00/2.38		Pol(evalstart4) = -V_1 + V_2 + 1
5.00/2.38	
5.00/2.38		Pol(evalstart5) = -V_1 + V_2 + 1
5.00/2.38	
5.00/2.38		Pol(evalstartbb3in) = -V_1 + V_2 + 1
5.00/2.38	
5.00/2.38		Pol(evalstartbb5in) = V_2 - V_3
5.00/2.38	
5.00/2.38		Pol(evalstartbb6in) = V_2 - V_3
5.00/2.38	
5.00/2.38		Pol(evalstartstop) = V_2 - V_3
5.00/2.38	
5.00/2.38		Pol(koat_start) = V_2 + 2
5.00/2.38	
5.00/2.38	orients all transitions weakly and the transitions
5.00/2.38	
5.00/2.38		evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 + 1 ]
5.00/2.38	
5.00/2.38		evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_0 + 1 ]
5.00/2.38	
5.00/2.38	strictly and produces the following problem:
5.00/2.38	
5.00/2.38	4:	T:
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)           evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)           evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)           evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)           evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)           evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)           evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ar_1 + 2, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_0 + 1 ]
5.00/2.38	
5.00/2.38			(Comp: 3, Cost: 1)           evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_0, ar_3)) [ ar_0 >= ar_1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)           evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)           evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, e))
5.00/2.38	
5.00/2.38			(Comp: 3, Cost: 1)           evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_0, ar_3)) [ ar_3 >= 1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)           evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_3 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)           evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ar_1 + 2, Cost: 1)    evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 + 1 ]
5.00/2.38	
5.00/2.38			(Comp: 3, Cost: 1)           evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb6in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 ]
5.00/2.38	
5.00/2.38			(Comp: ?, Cost: 1)           evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2 + 1, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 3, Cost: 1)           evalstartbb6in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
5.00/2.38	
5.00/2.38		start location:	koat_start
5.00/2.38	
5.00/2.38		leaf cost:	0
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Repeatedly propagating knowledge in problem 4 produces the following problem:
5.00/2.38	
5.00/2.38	5:	T:
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)           evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)           evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)           evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)           evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)           evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 1)           evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ar_1 + 2, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_0 + 1 ]
5.00/2.38	
5.00/2.38			(Comp: 3, Cost: 1)           evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_0, ar_3)) [ ar_0 >= ar_1 ]
5.00/2.38	
5.00/2.38			(Comp: ar_1 + 2, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ar_1 + 2, Cost: 1)    evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, e))
5.00/2.38	
5.00/2.38			(Comp: 3, Cost: 1)           evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_0, ar_3)) [ ar_3 >= 1 ]
5.00/2.38	
5.00/2.38			(Comp: ar_1 + 2, Cost: 1)    evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_3 ]
5.00/2.38	
5.00/2.38			(Comp: ar_1 + 2, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: ar_1 + 2, Cost: 1)    evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_2 + 1 ]
5.00/2.38	
5.00/2.38			(Comp: 3, Cost: 1)           evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb6in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_1 ]
5.00/2.38	
5.00/2.38			(Comp: ar_1 + 2, Cost: 1)    evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2 + 1, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 3, Cost: 1)           evalstartbb6in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))
5.00/2.38	
5.00/2.38			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
5.00/2.38	
5.00/2.38		start location:	koat_start
5.00/2.38	
5.00/2.38		leaf cost:	0
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Complexity upper bound 7*ar_1 + 32
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Time: 0.144 sec (SMT: 0.121 sec)
5.00/2.38	
5.00/2.38	
5.00/2.38	----------------------------------------
5.00/2.38	
5.00/2.38	(2)
5.00/2.38	BOUNDS(1, n^1)
5.00/2.38	
5.00/2.38	----------------------------------------
5.00/2.38	
5.00/2.38	(3) Loat Proof (FINISHED)
5.00/2.38	
5.00/2.38	
5.00/2.38	### Pre-processing the ITS problem ###
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Initial linear ITS problem
5.00/2.38	
5.00/2.38	   Start location: evalstartstart
5.00/2.38	
5.00/2.38	      0: evalstartstart -> evalstartbb0in : [], cost: 1
5.00/2.38	
5.00/2.38	      1: evalstartbb0in -> evalstart0 : [], cost: 1
5.00/2.38	
5.00/2.38	      2: evalstart0 -> evalstart1 : [], cost: 1
5.00/2.38	
5.00/2.38	      3: evalstart1 -> evalstart2 : [], cost: 1
5.00/2.38	
5.00/2.38	      4: evalstart2 -> evalstart3 : [], cost: 1
5.00/2.38	
5.00/2.38	      5: evalstart3 -> evalstartbb1in : A'=0, [], cost: 1
5.00/2.38	
5.00/2.38	      6: evalstartbb1in -> evalstartbb2in : [ B>=1+A ], cost: 1
5.00/2.38	
5.00/2.38	      7: evalstartbb1in -> evalstartbb4in : C'=A, [ A>=B ], cost: 1
5.00/2.38	
5.00/2.38	      8: evalstartbb2in -> evalstart4 : [], cost: 1
5.00/2.38	
5.00/2.38	      9: evalstart4 -> evalstart5 : D'=free, [], cost: 1
5.00/2.38	
5.00/2.38	     10: evalstart5 -> evalstartbb4in : C'=A, [ D>=1 ], cost: 1
5.00/2.38	
5.00/2.38	     11: evalstart5 -> evalstartbb3in : [ 0>=D ], cost: 1
5.00/2.38	
5.00/2.38	     12: evalstartbb3in -> evalstartbb1in : A'=1+A, [], cost: 1
5.00/2.38	
5.00/2.38	     13: evalstartbb4in -> evalstartbb5in : [ B>=1+C ], cost: 1
5.00/2.38	
5.00/2.38	     14: evalstartbb4in -> evalstartbb6in : [ C>=B ], cost: 1
5.00/2.38	
5.00/2.38	     15: evalstartbb5in -> evalstartbb4in : C'=1+C, [], cost: 1
5.00/2.38	
5.00/2.38	     16: evalstartbb6in -> evalstartstop : [], cost: 1
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Removed unreachable and leaf rules:
5.00/2.38	
5.00/2.38	   Start location: evalstartstart
5.00/2.38	
5.00/2.38	      0: evalstartstart -> evalstartbb0in : [], cost: 1
5.00/2.38	
5.00/2.38	      1: evalstartbb0in -> evalstart0 : [], cost: 1
5.00/2.38	
5.00/2.38	      2: evalstart0 -> evalstart1 : [], cost: 1
5.00/2.38	
5.00/2.38	      3: evalstart1 -> evalstart2 : [], cost: 1
5.00/2.38	
5.00/2.38	      4: evalstart2 -> evalstart3 : [], cost: 1
5.00/2.38	
5.00/2.38	      5: evalstart3 -> evalstartbb1in : A'=0, [], cost: 1
5.00/2.38	
5.00/2.38	      6: evalstartbb1in -> evalstartbb2in : [ B>=1+A ], cost: 1
5.00/2.38	
5.00/2.38	      7: evalstartbb1in -> evalstartbb4in : C'=A, [ A>=B ], cost: 1
5.00/2.38	
5.00/2.38	      8: evalstartbb2in -> evalstart4 : [], cost: 1
5.00/2.38	
5.00/2.38	      9: evalstart4 -> evalstart5 : D'=free, [], cost: 1
5.00/2.38	
5.00/2.38	     10: evalstart5 -> evalstartbb4in : C'=A, [ D>=1 ], cost: 1
5.00/2.38	
5.00/2.38	     11: evalstart5 -> evalstartbb3in : [ 0>=D ], cost: 1
5.00/2.38	
5.00/2.38	     12: evalstartbb3in -> evalstartbb1in : A'=1+A, [], cost: 1
5.00/2.38	
5.00/2.38	     13: evalstartbb4in -> evalstartbb5in : [ B>=1+C ], cost: 1
5.00/2.38	
5.00/2.38	     15: evalstartbb5in -> evalstartbb4in : C'=1+C, [], cost: 1
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	### Simplification by acceleration and chaining ###
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Eliminated locations (on linear paths):
5.00/2.38	
5.00/2.38	   Start location: evalstartstart
5.00/2.38	
5.00/2.38	     21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6
5.00/2.38	
5.00/2.38	      7: evalstartbb1in -> evalstartbb4in : C'=A, [ A>=B ], cost: 1
5.00/2.38	
5.00/2.38	     23: evalstartbb1in -> evalstart5 : D'=free, [ B>=1+A ], cost: 3
5.00/2.38	
5.00/2.38	     10: evalstart5 -> evalstartbb4in : C'=A, [ D>=1 ], cost: 1
5.00/2.38	
5.00/2.38	     24: evalstart5 -> evalstartbb1in : A'=1+A, [ 0>=D ], cost: 2
5.00/2.38	
5.00/2.38	     25: evalstartbb4in -> evalstartbb4in : C'=1+C, [ B>=1+C ], cost: 2
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Accelerating simple loops of location 11.
5.00/2.38	
5.00/2.38	   Accelerating the following rules:
5.00/2.38	
5.00/2.38	     25: evalstartbb4in -> evalstartbb4in : C'=1+C, [ B>=1+C ], cost: 2
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	   Accelerated rule 25 with metering function -C+B, yielding the new rule 26.
5.00/2.38	
5.00/2.38	   Removing the simple loops: 25.
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Accelerated all simple loops using metering functions (where possible):
5.00/2.38	
5.00/2.38	   Start location: evalstartstart
5.00/2.38	
5.00/2.38	     21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6
5.00/2.38	
5.00/2.38	      7: evalstartbb1in -> evalstartbb4in : C'=A, [ A>=B ], cost: 1
5.00/2.38	
5.00/2.38	     23: evalstartbb1in -> evalstart5 : D'=free, [ B>=1+A ], cost: 3
5.00/2.38	
5.00/2.38	     10: evalstart5 -> evalstartbb4in : C'=A, [ D>=1 ], cost: 1
5.00/2.38	
5.00/2.38	     24: evalstart5 -> evalstartbb1in : A'=1+A, [ 0>=D ], cost: 2
5.00/2.38	
5.00/2.38	     26: evalstartbb4in -> evalstartbb4in : C'=B, [ B>=1+C ], cost: -2*C+2*B
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Chained accelerated rules (with incoming rules):
5.00/2.38	
5.00/2.38	   Start location: evalstartstart
5.00/2.38	
5.00/2.38	     21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6
5.00/2.38	
5.00/2.38	      7: evalstartbb1in -> evalstartbb4in : C'=A, [ A>=B ], cost: 1
5.00/2.38	
5.00/2.38	     23: evalstartbb1in -> evalstart5 : D'=free, [ B>=1+A ], cost: 3
5.00/2.38	
5.00/2.38	     10: evalstart5 -> evalstartbb4in : C'=A, [ D>=1 ], cost: 1
5.00/2.38	
5.00/2.38	     24: evalstart5 -> evalstartbb1in : A'=1+A, [ 0>=D ], cost: 2
5.00/2.38	
5.00/2.38	     27: evalstart5 -> evalstartbb4in : C'=B, [ D>=1 && B>=1+A ], cost: 1-2*A+2*B
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Removed unreachable locations (and leaf rules with constant cost):
5.00/2.38	
5.00/2.38	   Start location: evalstartstart
5.00/2.38	
5.00/2.38	     21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6
5.00/2.38	
5.00/2.38	     23: evalstartbb1in -> evalstart5 : D'=free, [ B>=1+A ], cost: 3
5.00/2.38	
5.00/2.38	     24: evalstart5 -> evalstartbb1in : A'=1+A, [ 0>=D ], cost: 2
5.00/2.38	
5.00/2.38	     27: evalstart5 -> evalstartbb4in : C'=B, [ D>=1 && B>=1+A ], cost: 1-2*A+2*B
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Eliminated locations (on tree-shaped paths):
5.00/2.38	
5.00/2.38	   Start location: evalstartstart
5.00/2.38	
5.00/2.38	     21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6
5.00/2.38	
5.00/2.38	     28: evalstartbb1in -> evalstartbb1in : A'=1+A, D'=free, [ B>=1+A && 0>=free ], cost: 5
5.00/2.38	
5.00/2.38	     29: evalstartbb1in -> evalstartbb4in : C'=B, D'=free, [ B>=1+A && free>=1 ], cost: 4-2*A+2*B
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Accelerating simple loops of location 6.
5.00/2.38	
5.00/2.38	   Accelerating the following rules:
5.00/2.38	
5.00/2.38	     28: evalstartbb1in -> evalstartbb1in : A'=1+A, D'=free, [ B>=1+A && 0>=free ], cost: 5
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	   Accelerated rule 28 with metering function -A+B, yielding the new rule 30.
5.00/2.38	
5.00/2.38	   Removing the simple loops: 28.
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Accelerated all simple loops using metering functions (where possible):
5.00/2.38	
5.00/2.38	   Start location: evalstartstart
5.00/2.38	
5.00/2.38	     21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6
5.00/2.38	
5.00/2.38	     29: evalstartbb1in -> evalstartbb4in : C'=B, D'=free, [ B>=1+A && free>=1 ], cost: 4-2*A+2*B
5.00/2.38	
5.00/2.38	     30: evalstartbb1in -> evalstartbb1in : A'=B, D'=free, [ B>=1+A && 0>=free ], cost: -5*A+5*B
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Chained accelerated rules (with incoming rules):
5.00/2.38	
5.00/2.38	   Start location: evalstartstart
5.00/2.38	
5.00/2.38	     21: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6
5.00/2.38	
5.00/2.38	     31: evalstartstart -> evalstartbb1in : A'=B, D'=free, [ B>=1 && 0>=free ], cost: 6+5*B
5.00/2.38	
5.00/2.38	     29: evalstartbb1in -> evalstartbb4in : C'=B, D'=free, [ B>=1+A && free>=1 ], cost: 4-2*A+2*B
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Eliminated locations (on tree-shaped paths):
5.00/2.38	
5.00/2.38	   Start location: evalstartstart
5.00/2.38	
5.00/2.38	     32: evalstartstart -> evalstartbb4in : A'=0, C'=B, D'=free, [ B>=1 && free>=1 ], cost: 10+2*B
5.00/2.38	
5.00/2.38	     33: evalstartstart -> [17] : [ B>=1 && 0>=free ], cost: 6+5*B
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	### Computing asymptotic complexity ###
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Fully simplified ITS problem
5.00/2.38	
5.00/2.38	   Start location: evalstartstart
5.00/2.38	
5.00/2.38	     32: evalstartstart -> evalstartbb4in : A'=0, C'=B, D'=free, [ B>=1 && free>=1 ], cost: 10+2*B
5.00/2.38	
5.00/2.38	     33: evalstartstart -> [17] : [ B>=1 && 0>=free ], cost: 6+5*B
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	Computing asymptotic complexity for rule 32
5.00/2.38	
5.00/2.38	   Solved the limit problem by the following transformations:
5.00/2.38	
5.00/2.38	      Created initial limit problem:
5.00/2.38	
5.00/2.38	      free (+/+!), 10+2*B (+), B (+/+!) [not solved]
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	      removing all constraints (solved by SMT)
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5.00/2.38	      resulting limit problem: [solved]
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	      applying transformation rule (C) using substitution {free==1,B==n}
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5.00/2.38	      resulting limit problem:
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5.00/2.38	      [solved]
5.00/2.38	
5.00/2.38	
5.00/2.38	
5.00/2.38	   Solution:
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5.00/2.38	      free / 1
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5.00/2.38	      B / n
5.00/2.38	
5.00/2.38	   Resulting cost 10+2*n has complexity: Poly(n^1)
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5.00/2.38	
5.00/2.38	
5.00/2.38	Found new complexity Poly(n^1).
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5.00/2.38	
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5.00/2.38	Obtained the following overall complexity (w.r.t. the length of the input n):
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5.00/2.38	   Complexity:  Poly(n^1)
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5.00/2.38	   Cpx degree:  1
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5.00/2.38	   Solved cost: 10+2*n
5.00/2.38	
5.00/2.38	   Rule cost:   10+2*B
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5.00/2.38	   Rule guard:  [ B>=1 && free>=1 ]
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5.00/2.38	WORST_CASE(Omega(n^1),?)
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5.00/2.38	----------------------------------------
5.00/2.38	
5.00/2.38	(4)
5.00/2.38	BOUNDS(n^1, INF)
5.02/2.40	EOF