2.22/1.51	WORST_CASE(?, O(n^1))
2.35/1.52	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.35/1.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.35/1.52	
2.35/1.52	
2.35/1.52	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.35/1.52	
2.35/1.52	(0) CpxIntTrs
2.35/1.52	(1) Koat Proof [FINISHED, 75 ms]
2.35/1.52	(2) BOUNDS(1, n^1)
2.35/1.52	
2.35/1.52	
2.35/1.52	----------------------------------------
2.35/1.52	
2.35/1.52	(0)
2.35/1.52	Obligation:
2.35/1.52	Complexity Int TRS consisting of the following rules:
2.35/1.52	eval_terminate_start(v_.0, v_.01, v_.02, v_i, v_j, v_k) -> Com_1(eval_terminate_bb0_in(v_.0, v_.01, v_.02, v_i, v_j, v_k)) :|: TRUE
2.35/1.52	eval_terminate_bb0_in(v_.0, v_.01, v_.02, v_i, v_j, v_k) -> Com_1(eval_terminate_bb1_in(v_i, v_j, v_k, v_i, v_j, v_k)) :|: TRUE
2.35/1.52	eval_terminate_bb1_in(v_.0, v_.01, v_.02, v_i, v_j, v_k) -> Com_1(eval_terminate_bb2_in(v_.0, v_.01, v_.02, v_i, v_j, v_k)) :|: v_.0 <= 100 && v_.01 <= v_.02
2.35/1.52	eval_terminate_bb1_in(v_.0, v_.01, v_.02, v_i, v_j, v_k) -> Com_1(eval_terminate_bb3_in(v_.0, v_.01, v_.02, v_i, v_j, v_k)) :|: v_.0 > 100
2.35/1.52	eval_terminate_bb1_in(v_.0, v_.01, v_.02, v_i, v_j, v_k) -> Com_1(eval_terminate_bb3_in(v_.0, v_.01, v_.02, v_i, v_j, v_k)) :|: v_.01 > v_.02
2.35/1.52	eval_terminate_bb2_in(v_.0, v_.01, v_.02, v_i, v_j, v_k) -> Com_1(eval_terminate_bb1_in(v_.01, v_.0 + 1, v_.02 - 1, v_i, v_j, v_k)) :|: TRUE
2.35/1.52	eval_terminate_bb3_in(v_.0, v_.01, v_.02, v_i, v_j, v_k) -> Com_1(eval_terminate_stop(v_.0, v_.01, v_.02, v_i, v_j, v_k)) :|: TRUE
2.35/1.52	
2.35/1.52	The start-symbols are:[eval_terminate_start_6]
2.35/1.52	
2.35/1.52	
2.35/1.52	----------------------------------------
2.35/1.52	
2.35/1.52	(1) Koat Proof (FINISHED)
2.35/1.52	YES(?, 2*ar_1 + 2*ar_3 + 2*ar_5 + 210)
2.35/1.52	
2.35/1.52	
2.35/1.52	
2.35/1.52	Initial complexity problem:
2.35/1.52	
2.35/1.52	1:	T:
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatestart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb1in(ar_1, ar_1, ar_3, ar_3, ar_5, ar_5))
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 100 >= ar_0 /\ ar_4 >= ar_2 ]
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 101 ]
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_4 + 1 ]
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb1in(ar_2, ar_1, ar_0 + 1, ar_3, ar_4 - 1, ar_5))
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatestop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatestart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.35/1.52	
2.35/1.52		start location:	koat_start
2.35/1.52	
2.35/1.52		leaf cost:	0
2.35/1.52	
2.35/1.52	
2.35/1.52	
2.35/1.52	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.35/1.52	
2.35/1.52	2:	T:
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 1)    evalterminatestart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 1)    evalterminatebb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb1in(ar_1, ar_1, ar_3, ar_3, ar_5, ar_5))
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 100 >= ar_0 /\ ar_4 >= ar_2 ]
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 101 ]
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_4 + 1 ]
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb1in(ar_2, ar_1, ar_0 + 1, ar_3, ar_4 - 1, ar_5))
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatestop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatestart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.35/1.52	
2.35/1.52		start location:	koat_start
2.35/1.52	
2.35/1.52		leaf cost:	0
2.35/1.52	
2.35/1.52	
2.35/1.52	
2.35/1.52	A polynomial rank function with
2.35/1.52	
2.35/1.52		Pol(evalterminatestart) = 2
2.35/1.52	
2.35/1.52		Pol(evalterminatebb0in) = 2
2.35/1.52	
2.35/1.52		Pol(evalterminatebb1in) = 2
2.35/1.52	
2.35/1.52		Pol(evalterminatebb2in) = 2
2.35/1.52	
2.35/1.52		Pol(evalterminatebb3in) = 1
2.35/1.52	
2.35/1.52		Pol(evalterminatestop) = 0
2.35/1.52	
2.35/1.52		Pol(koat_start) = 2
2.35/1.52	
2.35/1.52	orients all transitions weakly and the transitions
2.35/1.52	
2.35/1.52		evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatestop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.35/1.52	
2.35/1.52		evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_4 + 1 ]
2.35/1.52	
2.35/1.52		evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 101 ]
2.35/1.52	
2.35/1.52	strictly and produces the following problem:
2.35/1.52	
2.35/1.52	3:	T:
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 1)    evalterminatestart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 1)    evalterminatebb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb1in(ar_1, ar_1, ar_3, ar_3, ar_5, ar_5))
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 100 >= ar_0 /\ ar_4 >= ar_2 ]
2.35/1.52	
2.35/1.52			(Comp: 2, Cost: 1)    evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 101 ]
2.35/1.52	
2.35/1.52			(Comp: 2, Cost: 1)    evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_4 + 1 ]
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)    evalterminatebb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb1in(ar_2, ar_1, ar_0 + 1, ar_3, ar_4 - 1, ar_5))
2.35/1.52	
2.35/1.52			(Comp: 2, Cost: 1)    evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatestop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatestart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.35/1.52	
2.35/1.52		start location:	koat_start
2.35/1.52	
2.35/1.52		leaf cost:	0
2.35/1.52	
2.35/1.52	
2.35/1.52	
2.35/1.52	A polynomial rank function with
2.35/1.52	
2.35/1.52		Pol(evalterminatestart) = -V_2 - V_4 + V_6 + 101
2.35/1.52	
2.35/1.52		Pol(evalterminatebb0in) = -V_2 - V_4 + V_6 + 101
2.35/1.52	
2.35/1.52		Pol(evalterminatebb1in) = -V_1 - V_3 + V_5 + 101
2.35/1.52	
2.35/1.52		Pol(evalterminatebb2in) = -V_1 - V_3 + V_5 + 99
2.35/1.52	
2.35/1.52		Pol(evalterminatebb3in) = -V_1 - V_3 + V_5
2.35/1.52	
2.35/1.52		Pol(evalterminatestop) = -V_1 - V_3 + V_5
2.35/1.52	
2.35/1.52		Pol(koat_start) = -V_2 - V_4 + V_6 + 101
2.35/1.52	
2.35/1.52	orients all transitions weakly and the transition
2.35/1.52	
2.35/1.52		evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 100 >= ar_0 /\ ar_4 >= ar_2 ]
2.35/1.52	
2.35/1.52	strictly and produces the following problem:
2.35/1.52	
2.35/1.52	4:	T:
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 1)                           evalterminatestart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 1)                           evalterminatebb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb1in(ar_1, ar_1, ar_3, ar_3, ar_5, ar_5))
2.35/1.52	
2.35/1.52			(Comp: ar_1 + ar_3 + ar_5 + 101, Cost: 1)    evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 100 >= ar_0 /\ ar_4 >= ar_2 ]
2.35/1.52	
2.35/1.52			(Comp: 2, Cost: 1)                           evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 101 ]
2.35/1.52	
2.35/1.52			(Comp: 2, Cost: 1)                           evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_4 + 1 ]
2.35/1.52	
2.35/1.52			(Comp: ?, Cost: 1)                           evalterminatebb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb1in(ar_2, ar_1, ar_0 + 1, ar_3, ar_4 - 1, ar_5))
2.35/1.52	
2.35/1.52			(Comp: 2, Cost: 1)                           evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatestop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 0)                           koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatestart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.35/1.52	
2.35/1.52		start location:	koat_start
2.35/1.52	
2.35/1.52		leaf cost:	0
2.35/1.52	
2.35/1.52	
2.35/1.52	
2.35/1.52	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.35/1.52	
2.35/1.52	5:	T:
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 1)                           evalterminatestart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 1)                           evalterminatebb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb1in(ar_1, ar_1, ar_3, ar_3, ar_5, ar_5))
2.35/1.52	
2.35/1.52			(Comp: ar_1 + ar_3 + ar_5 + 101, Cost: 1)    evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 100 >= ar_0 /\ ar_4 >= ar_2 ]
2.35/1.52	
2.35/1.52			(Comp: 2, Cost: 1)                           evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 101 ]
2.35/1.52	
2.35/1.52			(Comp: 2, Cost: 1)                           evalterminatebb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_4 + 1 ]
2.35/1.52	
2.35/1.52			(Comp: ar_1 + ar_3 + ar_5 + 101, Cost: 1)    evalterminatebb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatebb1in(ar_2, ar_1, ar_0 + 1, ar_3, ar_4 - 1, ar_5))
2.35/1.52	
2.35/1.52			(Comp: 2, Cost: 1)                           evalterminatebb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatestop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.35/1.52	
2.35/1.52			(Comp: 1, Cost: 0)                           koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalterminatestart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.35/1.52	
2.35/1.52		start location:	koat_start
2.35/1.52	
2.35/1.52		leaf cost:	0
2.35/1.52	
2.35/1.52	
2.35/1.52	
2.35/1.52	Complexity upper bound 2*ar_1 + 2*ar_3 + 2*ar_5 + 210
2.35/1.52	
2.35/1.52	
2.35/1.52	
2.35/1.52	Time: 0.096 sec (SMT: 0.081 sec)
2.35/1.52	
2.35/1.52	
2.35/1.52	----------------------------------------
2.35/1.52	
2.35/1.52	(2)
2.35/1.52	BOUNDS(1, n^1)
2.35/1.54	EOF