4.64/2.41	WORST_CASE(Omega(n^1), O(n^1))
4.87/2.42	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.87/2.42	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.87/2.42	
4.87/2.42	
4.87/2.42	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
4.87/2.42	
4.87/2.42	(0) CpxIntTrs
4.87/2.42	(1) Koat Proof [FINISHED, 116 ms]
4.87/2.42	(2) BOUNDS(1, n^1)
4.87/2.42	(3) Loat Proof [FINISHED, 629 ms]
4.87/2.42	(4) BOUNDS(n^1, INF)
4.87/2.42	
4.87/2.42	
4.87/2.42	----------------------------------------
4.87/2.42	
4.87/2.42	(0)
4.87/2.42	Obligation:
4.87/2.42	Complexity Int TRS consisting of the following rules:
4.87/2.42	eval_start_start(v__0, v_flag_0, v_n) -> Com_1(eval_start_bb0_in(v__0, v_flag_0, v_n)) :|: TRUE
4.87/2.42	eval_start_bb0_in(v__0, v_flag_0, v_n) -> Com_1(eval_start_0(v__0, v_flag_0, v_n)) :|: TRUE
4.87/2.42	eval_start_0(v__0, v_flag_0, v_n) -> Com_1(eval_start_1(v__0, v_flag_0, v_n)) :|: TRUE
4.87/2.42	eval_start_1(v__0, v_flag_0, v_n) -> Com_1(eval_start_2(v__0, v_flag_0, v_n)) :|: TRUE
4.87/2.42	eval_start_2(v__0, v_flag_0, v_n) -> Com_1(eval_start_3(v__0, v_flag_0, v_n)) :|: TRUE
4.87/2.42	eval_start_3(v__0, v_flag_0, v_n) -> Com_1(eval_start_4(v__0, v_flag_0, v_n)) :|: TRUE
4.87/2.42	eval_start_4(v__0, v_flag_0, v_n) -> Com_1(eval_start_5(v__0, v_flag_0, v_n)) :|: TRUE
4.87/2.42	eval_start_5(v__0, v_flag_0, v_n) -> Com_1(eval_start_bb1_in(v_n, 1, v_n)) :|: TRUE
4.87/2.42	eval_start_bb1_in(v__0, v_flag_0, v_n) -> Com_1(eval_start_bb2_in(v__0, v_flag_0, v_n)) :|: v_flag_0 > 0
4.87/2.42	eval_start_bb1_in(v__0, v_flag_0, v_n) -> Com_1(eval_start_bb3_in(v__0, v_flag_0, v_n)) :|: v_flag_0 <= 0
4.87/2.42	eval_start_bb2_in(v__0, v_flag_0, v_n) -> Com_1(eval_start_bb1_in(v__0 - 1, 1, v_n)) :|: v__0 > 0
4.87/2.42	eval_start_bb2_in(v__0, v_flag_0, v_n) -> Com_1(eval_start_bb1_in(v__0, 1, v_n)) :|: v__0 > 0 && v__0 <= 0
4.87/2.42	eval_start_bb2_in(v__0, v_flag_0, v_n) -> Com_1(eval_start_bb1_in(v__0 - 1, 0, v_n)) :|: v__0 <= 0 && v__0 > 0
4.87/2.42	eval_start_bb2_in(v__0, v_flag_0, v_n) -> Com_1(eval_start_bb1_in(v__0, 0, v_n)) :|: v__0 <= 0
4.87/2.42	eval_start_bb3_in(v__0, v_flag_0, v_n) -> Com_1(eval_start_stop(v__0, v_flag_0, v_n)) :|: TRUE
4.87/2.42	
4.87/2.42	The start-symbols are:[eval_start_start_3]
4.87/2.42	
4.87/2.42	
4.87/2.42	----------------------------------------
4.87/2.42	
4.87/2.42	(1) Koat Proof (FINISHED)
4.87/2.42	YES(?, 2*ar_1 + 14)
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Initial complexity problem:
4.87/2.42	
4.87/2.42	1:	T:
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_1, ar_1, 1))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1, 1)) [ ar_0 >= 1 /\ 0 >= ar_0 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 - 1, ar_1, 0)) [ 0 >= ar_0 /\ ar_0 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.87/2.42	
4.87/2.42		start location:	koat_start
4.87/2.42	
4.87/2.42		leaf cost:	0
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Testing for reachability in the complexity graph removes the following transitions from problem 1:
4.87/2.42	
4.87/2.42		evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1, 1)) [ ar_0 >= 1 /\ 0 >= ar_0 ]
4.87/2.42	
4.87/2.42		evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 - 1, ar_1, 0)) [ 0 >= ar_0 /\ ar_0 >= 1 ]
4.87/2.42	
4.87/2.42	We thus obtain the following problem:
4.87/2.42	
4.87/2.42	2:	T:
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_1, ar_1, 1))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.87/2.42	
4.87/2.42		start location:	koat_start
4.87/2.42	
4.87/2.42		leaf cost:	0
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Repeatedly propagating knowledge in problem 2 produces the following problem:
4.87/2.42	
4.87/2.42	3:	T:
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_1, ar_1, 1))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.87/2.42	
4.87/2.42		start location:	koat_start
4.87/2.42	
4.87/2.42		leaf cost:	0
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	A polynomial rank function with
4.87/2.42	
4.87/2.42		Pol(evalstartbb3in) = 1
4.87/2.42	
4.87/2.42		Pol(evalstartstop) = 0
4.87/2.42	
4.87/2.42		Pol(evalstartbb1in) = 2
4.87/2.42	
4.87/2.42		Pol(evalstartbb2in) = 2
4.87/2.42	
4.87/2.42		Pol(evalstart5) = 2
4.87/2.42	
4.87/2.42		Pol(evalstart4) = 2
4.87/2.42	
4.87/2.42		Pol(evalstart3) = 2
4.87/2.42	
4.87/2.42		Pol(evalstart2) = 2
4.87/2.42	
4.87/2.42		Pol(evalstart1) = 2
4.87/2.42	
4.87/2.42		Pol(evalstart0) = 2
4.87/2.42	
4.87/2.42		Pol(evalstartbb0in) = 2
4.87/2.42	
4.87/2.42		Pol(evalstartstart) = 2
4.87/2.42	
4.87/2.42		Pol(koat_start) = 2
4.87/2.42	
4.87/2.42	orients all transitions weakly and the transitions
4.87/2.42	
4.87/2.42		evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42		evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.87/2.42	
4.87/2.42	strictly and produces the following problem:
4.87/2.42	
4.87/2.42	4:	T:
4.87/2.42	
4.87/2.42			(Comp: 2, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 2, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_1, ar_1, 1))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.87/2.42	
4.87/2.42		start location:	koat_start
4.87/2.42	
4.87/2.42		leaf cost:	0
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	A polynomial rank function with
4.87/2.42	
4.87/2.42		Pol(evalstartbb3in) = V_3
4.87/2.42	
4.87/2.42		Pol(evalstartstop) = V_3
4.87/2.42	
4.87/2.42		Pol(evalstartbb1in) = V_3
4.87/2.42	
4.87/2.42		Pol(evalstartbb2in) = 1
4.87/2.42	
4.87/2.42		Pol(evalstart5) = 1
4.87/2.42	
4.87/2.42		Pol(evalstart4) = 1
4.87/2.42	
4.87/2.42		Pol(evalstart3) = 1
4.87/2.42	
4.87/2.42		Pol(evalstart2) = 1
4.87/2.42	
4.87/2.42		Pol(evalstart1) = 1
4.87/2.42	
4.87/2.42		Pol(evalstart0) = 1
4.87/2.42	
4.87/2.42		Pol(evalstartbb0in) = 1
4.87/2.42	
4.87/2.42		Pol(evalstartstart) = 1
4.87/2.42	
4.87/2.42		Pol(koat_start) = 1
4.87/2.42	
4.87/2.42	orients all transitions weakly and the transition
4.87/2.42	
4.87/2.42		evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
4.87/2.42	
4.87/2.42	strictly and produces the following problem:
4.87/2.42	
4.87/2.42	5:	T:
4.87/2.42	
4.87/2.42			(Comp: 2, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 2, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_1, ar_1, 1))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)    evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.87/2.42	
4.87/2.42		start location:	koat_start
4.87/2.42	
4.87/2.42		leaf cost:	0
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	A polynomial rank function with
4.87/2.42	
4.87/2.42		Pol(evalstartbb3in) = V_1
4.87/2.42	
4.87/2.42		Pol(evalstartstop) = V_1
4.87/2.42	
4.87/2.42		Pol(evalstartbb1in) = V_1
4.87/2.42	
4.87/2.42		Pol(evalstartbb2in) = V_1
4.87/2.42	
4.87/2.42		Pol(evalstart5) = V_2
4.87/2.42	
4.87/2.42		Pol(evalstart4) = V_2
4.87/2.42	
4.87/2.42		Pol(evalstart3) = V_2
4.87/2.42	
4.87/2.42		Pol(evalstart2) = V_2
4.87/2.42	
4.87/2.42		Pol(evalstart1) = V_2
4.87/2.42	
4.87/2.42		Pol(evalstart0) = V_2
4.87/2.42	
4.87/2.42		Pol(evalstartbb0in) = V_2
4.87/2.42	
4.87/2.42		Pol(evalstartstart) = V_2
4.87/2.42	
4.87/2.42		Pol(koat_start) = V_2
4.87/2.42	
4.87/2.42	orients all transitions weakly and the transition
4.87/2.42	
4.87/2.42		evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
4.87/2.42	
4.87/2.42	strictly and produces the following problem:
4.87/2.42	
4.87/2.42	6:	T:
4.87/2.42	
4.87/2.42			(Comp: 2, Cost: 1)       evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 2, Cost: 1)       evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)       evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
4.87/2.42	
4.87/2.42			(Comp: ar_1, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: ?, Cost: 1)       evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)       evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_1, ar_1, 1))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)       evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)       evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)       evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)       evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)       evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)       evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)       evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.87/2.42	
4.87/2.42		start location:	koat_start
4.87/2.42	
4.87/2.42		leaf cost:	0
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Repeatedly propagating knowledge in problem 6 produces the following problem:
4.87/2.42	
4.87/2.42	7:	T:
4.87/2.42	
4.87/2.42			(Comp: 2, Cost: 1)           evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 2, Cost: 1)           evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)           evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1, 0)) [ 0 >= ar_0 ]
4.87/2.42	
4.87/2.42			(Comp: ar_1, Cost: 1)        evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0 - 1, ar_1, 1)) [ ar_0 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: ar_1 + 1, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)           evalstart5(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_1, ar_1, 1))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)           evalstart4(ar_0, ar_1, ar_2) -> Com_1(evalstart5(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)           evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstart4(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)           evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)           evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)           evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)           evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 1)           evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))
4.87/2.42	
4.87/2.42			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
4.87/2.42	
4.87/2.42		start location:	koat_start
4.87/2.42	
4.87/2.42		leaf cost:	0
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Complexity upper bound 2*ar_1 + 14
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Time: 0.112 sec (SMT: 0.096 sec)
4.87/2.42	
4.87/2.42	
4.87/2.42	----------------------------------------
4.87/2.42	
4.87/2.42	(2)
4.87/2.42	BOUNDS(1, n^1)
4.87/2.42	
4.87/2.42	----------------------------------------
4.87/2.42	
4.87/2.42	(3) Loat Proof (FINISHED)
4.87/2.42	
4.87/2.42	
4.87/2.42	### Pre-processing the ITS problem ###
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Initial linear ITS problem
4.87/2.42	
4.87/2.42	   Start location: evalstartstart
4.87/2.42	
4.87/2.42	      0: evalstartstart -> evalstartbb0in : [], cost: 1
4.87/2.42	
4.87/2.42	      1: evalstartbb0in -> evalstart0 : [], cost: 1
4.87/2.42	
4.87/2.42	      2: evalstart0 -> evalstart1 : [], cost: 1
4.87/2.42	
4.87/2.42	      3: evalstart1 -> evalstart2 : [], cost: 1
4.87/2.42	
4.87/2.42	      4: evalstart2 -> evalstart3 : [], cost: 1
4.87/2.42	
4.87/2.42	      5: evalstart3 -> evalstart4 : [], cost: 1
4.87/2.42	
4.87/2.42	      6: evalstart4 -> evalstart5 : [], cost: 1
4.87/2.42	
4.87/2.42	      7: evalstart5 -> evalstartbb1in : A'=B, C'=1, [], cost: 1
4.87/2.42	
4.87/2.42	      8: evalstartbb1in -> evalstartbb2in : [ C>=1 ], cost: 1
4.87/2.42	
4.87/2.42	      9: evalstartbb1in -> evalstartbb3in : [ 0>=C ], cost: 1
4.87/2.42	
4.87/2.42	     10: evalstartbb2in -> evalstartbb1in : A'=-1+A, C'=1, [ A>=1 ], cost: 1
4.87/2.42	
4.87/2.42	     11: evalstartbb2in -> evalstartbb1in : C'=1, [ A>=1 && 0>=A ], cost: 1
4.87/2.42	
4.87/2.42	     12: evalstartbb2in -> evalstartbb1in : A'=-1+A, C'=0, [ 0>=A && A>=1 ], cost: 1
4.87/2.42	
4.87/2.42	     13: evalstartbb2in -> evalstartbb1in : C'=0, [ 0>=A ], cost: 1
4.87/2.42	
4.87/2.42	     14: evalstartbb3in -> evalstartstop : [], cost: 1
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Removed unreachable and leaf rules:
4.87/2.42	
4.87/2.42	   Start location: evalstartstart
4.87/2.42	
4.87/2.42	      0: evalstartstart -> evalstartbb0in : [], cost: 1
4.87/2.42	
4.87/2.42	      1: evalstartbb0in -> evalstart0 : [], cost: 1
4.87/2.42	
4.87/2.42	      2: evalstart0 -> evalstart1 : [], cost: 1
4.87/2.42	
4.87/2.42	      3: evalstart1 -> evalstart2 : [], cost: 1
4.87/2.42	
4.87/2.42	      4: evalstart2 -> evalstart3 : [], cost: 1
4.87/2.42	
4.87/2.42	      5: evalstart3 -> evalstart4 : [], cost: 1
4.87/2.42	
4.87/2.42	      6: evalstart4 -> evalstart5 : [], cost: 1
4.87/2.42	
4.87/2.42	      7: evalstart5 -> evalstartbb1in : A'=B, C'=1, [], cost: 1
4.87/2.42	
4.87/2.42	      8: evalstartbb1in -> evalstartbb2in : [ C>=1 ], cost: 1
4.87/2.42	
4.87/2.42	     10: evalstartbb2in -> evalstartbb1in : A'=-1+A, C'=1, [ A>=1 ], cost: 1
4.87/2.42	
4.87/2.42	     11: evalstartbb2in -> evalstartbb1in : C'=1, [ A>=1 && 0>=A ], cost: 1
4.87/2.42	
4.87/2.42	     12: evalstartbb2in -> evalstartbb1in : A'=-1+A, C'=0, [ 0>=A && A>=1 ], cost: 1
4.87/2.42	
4.87/2.42	     13: evalstartbb2in -> evalstartbb1in : C'=0, [ 0>=A ], cost: 1
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Removed rules with unsatisfiable guard:
4.87/2.42	
4.87/2.42	   Start location: evalstartstart
4.87/2.42	
4.87/2.42	      0: evalstartstart -> evalstartbb0in : [], cost: 1
4.87/2.42	
4.87/2.42	      1: evalstartbb0in -> evalstart0 : [], cost: 1
4.87/2.42	
4.87/2.42	      2: evalstart0 -> evalstart1 : [], cost: 1
4.87/2.42	
4.87/2.42	      3: evalstart1 -> evalstart2 : [], cost: 1
4.87/2.42	
4.87/2.42	      4: evalstart2 -> evalstart3 : [], cost: 1
4.87/2.42	
4.87/2.42	      5: evalstart3 -> evalstart4 : [], cost: 1
4.87/2.42	
4.87/2.42	      6: evalstart4 -> evalstart5 : [], cost: 1
4.87/2.42	
4.87/2.42	      7: evalstart5 -> evalstartbb1in : A'=B, C'=1, [], cost: 1
4.87/2.42	
4.87/2.42	      8: evalstartbb1in -> evalstartbb2in : [ C>=1 ], cost: 1
4.87/2.42	
4.87/2.42	     10: evalstartbb2in -> evalstartbb1in : A'=-1+A, C'=1, [ A>=1 ], cost: 1
4.87/2.42	
4.87/2.42	     13: evalstartbb2in -> evalstartbb1in : C'=0, [ 0>=A ], cost: 1
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	### Simplification by acceleration and chaining ###
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Eliminated locations (on linear paths):
4.87/2.42	
4.87/2.42	   Start location: evalstartstart
4.87/2.42	
4.87/2.42	     21: evalstartstart -> evalstartbb1in : A'=B, C'=1, [], cost: 8
4.87/2.42	
4.87/2.42	      8: evalstartbb1in -> evalstartbb2in : [ C>=1 ], cost: 1
4.87/2.42	
4.87/2.42	     10: evalstartbb2in -> evalstartbb1in : A'=-1+A, C'=1, [ A>=1 ], cost: 1
4.87/2.42	
4.87/2.42	     13: evalstartbb2in -> evalstartbb1in : C'=0, [ 0>=A ], cost: 1
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Eliminated locations (on tree-shaped paths):
4.87/2.42	
4.87/2.42	   Start location: evalstartstart
4.87/2.42	
4.87/2.42	     21: evalstartstart -> evalstartbb1in : A'=B, C'=1, [], cost: 8
4.87/2.42	
4.87/2.42	     22: evalstartbb1in -> evalstartbb1in : A'=-1+A, C'=1, [ C>=1 && A>=1 ], cost: 2
4.87/2.42	
4.87/2.42	     23: evalstartbb1in -> evalstartbb1in : C'=0, [ C>=1 && 0>=A ], cost: 2
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Accelerating simple loops of location 8.
4.87/2.42	
4.87/2.42	   Accelerating the following rules:
4.87/2.42	
4.87/2.42	     22: evalstartbb1in -> evalstartbb1in : A'=-1+A, C'=1, [ C>=1 && A>=1 ], cost: 2
4.87/2.42	
4.87/2.42	     23: evalstartbb1in -> evalstartbb1in : C'=0, [ C>=1 && 0>=A ], cost: 2
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	   Accelerated rule 22 with metering function A, yielding the new rule 24.
4.87/2.42	
4.87/2.42	   Found no metering function for rule 23.
4.87/2.42	
4.87/2.42	   Removing the simple loops: 22.
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Accelerated all simple loops using metering functions (where possible):
4.87/2.42	
4.87/2.42	   Start location: evalstartstart
4.87/2.42	
4.87/2.42	     21: evalstartstart -> evalstartbb1in : A'=B, C'=1, [], cost: 8
4.87/2.42	
4.87/2.42	     23: evalstartbb1in -> evalstartbb1in : C'=0, [ C>=1 && 0>=A ], cost: 2
4.87/2.42	
4.87/2.42	     24: evalstartbb1in -> evalstartbb1in : A'=0, C'=1, [ C>=1 && A>=1 ], cost: 2*A
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Chained accelerated rules (with incoming rules):
4.87/2.42	
4.87/2.42	   Start location: evalstartstart
4.87/2.42	
4.87/2.42	     21: evalstartstart -> evalstartbb1in : A'=B, C'=1, [], cost: 8
4.87/2.42	
4.87/2.42	     25: evalstartstart -> evalstartbb1in : A'=B, C'=0, [ 0>=B ], cost: 10
4.87/2.42	
4.87/2.42	     26: evalstartstart -> evalstartbb1in : A'=0, C'=1, [ B>=1 ], cost: 8+2*B
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Removed unreachable locations (and leaf rules with constant cost):
4.87/2.42	
4.87/2.42	   Start location: evalstartstart
4.87/2.42	
4.87/2.42	     26: evalstartstart -> evalstartbb1in : A'=0, C'=1, [ B>=1 ], cost: 8+2*B
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	### Computing asymptotic complexity ###
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Fully simplified ITS problem
4.87/2.42	
4.87/2.42	   Start location: evalstartstart
4.87/2.42	
4.87/2.42	     26: evalstartstart -> evalstartbb1in : A'=0, C'=1, [ B>=1 ], cost: 8+2*B
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Computing asymptotic complexity for rule 26
4.87/2.42	
4.87/2.42	   Solved the limit problem by the following transformations:
4.87/2.42	
4.87/2.42	      Created initial limit problem:
4.87/2.42	
4.87/2.42	      8+2*B (+), B (+/+!) [not solved]
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	      removing all constraints (solved by SMT)
4.87/2.42	
4.87/2.42	      resulting limit problem: [solved]
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	      applying transformation rule (C) using substitution {B==n}
4.87/2.42	
4.87/2.42	      resulting limit problem:
4.87/2.42	
4.87/2.42	      [solved]
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	   Solution:
4.87/2.42	
4.87/2.42	      B / n
4.87/2.42	
4.87/2.42	   Resulting cost 8+2*n has complexity: Poly(n^1)
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Found new complexity Poly(n^1).
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	Obtained the following overall complexity (w.r.t. the length of the input n):
4.87/2.42	
4.87/2.42	   Complexity:  Poly(n^1)
4.87/2.42	
4.87/2.42	   Cpx degree:  1
4.87/2.42	
4.87/2.42	   Solved cost: 8+2*n
4.87/2.42	
4.87/2.42	   Rule cost:   8+2*B
4.87/2.42	
4.87/2.42	   Rule guard:  [ B>=1 ]
4.87/2.42	
4.87/2.42	
4.87/2.42	
4.87/2.42	WORST_CASE(Omega(n^1),?)
4.87/2.42	
4.87/2.42	
4.87/2.42	----------------------------------------
4.87/2.42	
4.87/2.42	(4)
4.87/2.42	BOUNDS(n^1, INF)
4.89/3.64	EOF