4.98/2.38	WORST_CASE(Omega(n^1), O(n^1))
4.98/2.39	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.98/2.39	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.98/2.39	
4.98/2.39	
4.98/2.39	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
4.98/2.39	
4.98/2.39	(0) CpxIntTrs
4.98/2.39	(1) Koat Proof [FINISHED, 313 ms]
4.98/2.39	(2) BOUNDS(1, n^1)
4.98/2.39	(3) Loat Proof [FINISHED, 695 ms]
4.98/2.39	(4) BOUNDS(n^1, INF)
4.98/2.39	
4.98/2.39	
4.98/2.39	----------------------------------------
4.98/2.39	
4.98/2.39	(0)
4.98/2.39	Obligation:
4.98/2.39	Complexity Int TRS consisting of the following rules:
4.98/2.39	eval_start_start(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb0_in(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
4.98/2.39	eval_start_bb0_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_0(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
4.98/2.39	eval_start_0(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_1(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
4.98/2.39	eval_start_1(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_2(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
4.98/2.39	eval_start_2(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_3(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
4.98/2.39	eval_start_3(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_4(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
4.98/2.39	eval_start_4(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_5(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
4.98/2.39	eval_start_5(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_6(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
4.98/2.39	eval_start_6(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, 0, 0)) :|: TRUE
4.98/2.39	eval_start_bb1_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb2_in(v_m, v_n, v_x_0, v_y_0)) :|: v_x_0 < v_n
4.98/2.39	eval_start_bb1_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb3_in(v_m, v_n, v_x_0, v_y_0)) :|: v_x_0 >= v_n
4.98/2.39	eval_start_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, v_x_0, v_y_0 + 1)) :|: v_y_0 < v_m
4.98/2.39	eval_start_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, v_x_0 + 1, v_y_0 + 1)) :|: v_y_0 < v_m && v_y_0 >= v_m
4.98/2.39	eval_start_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, v_x_0, v_y_0)) :|: v_y_0 >= v_m && v_y_0 < v_m
4.98/2.39	eval_start_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_bb1_in(v_m, v_n, v_x_0 + 1, v_y_0)) :|: v_y_0 >= v_m
4.98/2.39	eval_start_bb3_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_start_stop(v_m, v_n, v_x_0, v_y_0)) :|: TRUE
4.98/2.39	
4.98/2.39	The start-symbols are:[eval_start_start_4]
4.98/2.39	
4.98/2.39	
4.98/2.39	----------------------------------------
4.98/2.39	
4.98/2.39	(1) Koat Proof (FINISHED)
4.98/2.39	YES(?, 2*ar_2 + 2*ar_3 + 14)
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Initial complexity problem:
4.98/2.39	
4.98/2.39	1:	T:
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, 0, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 /\ ar_1 >= ar_3 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_3 /\ ar_3 >= ar_1 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_1 >= ar_3 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(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 ]
4.98/2.39	
4.98/2.39		start location:	koat_start
4.98/2.39	
4.98/2.39		leaf cost:	0
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Testing for reachability in the complexity graph removes the following transitions from problem 1:
4.98/2.39	
4.98/2.39		evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 /\ ar_1 >= ar_3 ]
4.98/2.39	
4.98/2.39		evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= ar_3 /\ ar_3 >= ar_1 + 1 ]
4.98/2.39	
4.98/2.39	We thus obtain the following problem:
4.98/2.39	
4.98/2.39	2:	T:
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_1 >= ar_3 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, 0, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(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 ]
4.98/2.39	
4.98/2.39		start location:	koat_start
4.98/2.39	
4.98/2.39		leaf cost:	0
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Repeatedly propagating knowledge in problem 2 produces the following problem:
4.98/2.39	
4.98/2.39	3:	T:
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_1 >= ar_3 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, 0, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(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 ]
4.98/2.39	
4.98/2.39		start location:	koat_start
4.98/2.39	
4.98/2.39		leaf cost:	0
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	A polynomial rank function with
4.98/2.39	
4.98/2.39		Pol(evalstartbb3in) = 1
4.98/2.39	
4.98/2.39		Pol(evalstartstop) = 0
4.98/2.39	
4.98/2.39		Pol(evalstartbb2in) = 2
4.98/2.39	
4.98/2.39		Pol(evalstartbb1in) = 2
4.98/2.39	
4.98/2.39		Pol(evalstart6) = 2
4.98/2.39	
4.98/2.39		Pol(evalstart5) = 2
4.98/2.39	
4.98/2.39		Pol(evalstart4) = 2
4.98/2.39	
4.98/2.39		Pol(evalstart3) = 2
4.98/2.39	
4.98/2.39		Pol(evalstart2) = 2
4.98/2.39	
4.98/2.39		Pol(evalstart1) = 2
4.98/2.39	
4.98/2.39		Pol(evalstart0) = 2
4.98/2.39	
4.98/2.39		Pol(evalstartbb0in) = 2
4.98/2.39	
4.98/2.39		Pol(evalstartstart) = 2
4.98/2.39	
4.98/2.39		Pol(koat_start) = 2
4.98/2.39	
4.98/2.39	orients all transitions weakly and the transitions
4.98/2.39	
4.98/2.39		evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39		evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]
4.98/2.39	
4.98/2.39	strictly and produces the following problem:
4.98/2.39	
4.98/2.39	4:	T:
4.98/2.39	
4.98/2.39			(Comp: 2, Cost: 1)    evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_1 >= ar_3 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: 2, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, 0, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)    evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(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 ]
4.98/2.39	
4.98/2.39		start location:	koat_start
4.98/2.39	
4.98/2.39		leaf cost:	0
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	A polynomial rank function with
4.98/2.39	
4.98/2.39		Pol(evalstartbb3in) = -V_2 + V_4
4.98/2.39	
4.98/2.39		Pol(evalstartstop) = -V_2 + V_4
4.98/2.39	
4.98/2.39		Pol(evalstartbb2in) = -V_2 + V_4
4.98/2.39	
4.98/2.39		Pol(evalstartbb1in) = -V_2 + V_4
4.98/2.39	
4.98/2.39		Pol(evalstart6) = V_4
4.98/2.39	
4.98/2.39		Pol(evalstart5) = V_4
4.98/2.39	
4.98/2.39		Pol(evalstart4) = V_4
4.98/2.39	
4.98/2.39		Pol(evalstart3) = V_4
4.98/2.39	
4.98/2.39		Pol(evalstart2) = V_4
4.98/2.39	
4.98/2.39		Pol(evalstart1) = V_4
4.98/2.39	
4.98/2.39		Pol(evalstart0) = V_4
4.98/2.39	
4.98/2.39		Pol(evalstartbb0in) = V_4
4.98/2.39	
4.98/2.39		Pol(evalstartstart) = V_4
4.98/2.39	
4.98/2.39		Pol(koat_start) = V_4
4.98/2.39	
4.98/2.39	orients all transitions weakly and the transition
4.98/2.39	
4.98/2.39		evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ]
4.98/2.39	
4.98/2.39	strictly and produces the following problem:
4.98/2.39	
4.98/2.39	5:	T:
4.98/2.39	
4.98/2.39			(Comp: 2, Cost: 1)       evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)       evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_1 >= ar_3 ]
4.98/2.39	
4.98/2.39			(Comp: ar_3, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: 2, Cost: 1)       evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)       evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, 0, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(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 ]
4.98/2.39	
4.98/2.39		start location:	koat_start
4.98/2.39	
4.98/2.39		leaf cost:	0
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Applied AI with 'oct' on problem 5 to obtain the following invariants:
4.98/2.39	
4.98/2.39	  For symbol evalstartbb1in: X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0
4.98/2.39	
4.98/2.39	  For symbol evalstartbb2in: X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ -X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0
4.98/2.39	
4.98/2.39	  For symbol evalstartbb3in: X_1 - X_3 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	This yielded the following problem:
4.98/2.39	
4.98/2.39	6:	T:
4.98/2.39	
4.98/2.39			(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 ]
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, 0, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(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 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: 2, Cost: 1)       evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ]
4.98/2.39	
4.98/2.39			(Comp: ar_3, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_1 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: ?, Cost: 1)       evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_3 ]
4.98/2.39	
4.98/2.39			(Comp: 2, Cost: 1)       evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 - ar_2 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
4.98/2.39	
4.98/2.39		start location:	koat_start
4.98/2.39	
4.98/2.39		leaf cost:	0
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	A polynomial rank function with
4.98/2.39	
4.98/2.39		Pol(koat_start) = V_3
4.98/2.39	
4.98/2.39		Pol(evalstartstart) = V_3
4.98/2.39	
4.98/2.39		Pol(evalstartbb0in) = V_3
4.98/2.39	
4.98/2.39		Pol(evalstart0) = V_3
4.98/2.39	
4.98/2.39		Pol(evalstart1) = V_3
4.98/2.39	
4.98/2.39		Pol(evalstart2) = V_3
4.98/2.39	
4.98/2.39		Pol(evalstart3) = V_3
4.98/2.39	
4.98/2.39		Pol(evalstart4) = V_3
4.98/2.39	
4.98/2.39		Pol(evalstart5) = V_3
4.98/2.39	
4.98/2.39		Pol(evalstart6) = V_3
4.98/2.39	
4.98/2.39		Pol(evalstartbb1in) = -V_1 + V_3
4.98/2.39	
4.98/2.39		Pol(evalstartbb2in) = -V_1 + V_3
4.98/2.39	
4.98/2.39		Pol(evalstartbb3in) = -V_1 + V_3
4.98/2.39	
4.98/2.39		Pol(evalstartstop) = -V_1 + V_3
4.98/2.39	
4.98/2.39	orients all transitions weakly and the transition
4.98/2.39	
4.98/2.39		evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_3 ]
4.98/2.39	
4.98/2.39	strictly and produces the following problem:
4.98/2.39	
4.98/2.39	7:	T:
4.98/2.39	
4.98/2.39			(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 ]
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)       evalstart6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, 0, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(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 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: 2, Cost: 1)       evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ]
4.98/2.39	
4.98/2.39			(Comp: ar_3, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_1 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: ar_2, Cost: 1)    evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_3 ]
4.98/2.39	
4.98/2.39			(Comp: 2, Cost: 1)       evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 - ar_2 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
4.98/2.39	
4.98/2.39		start location:	koat_start
4.98/2.39	
4.98/2.39		leaf cost:	0
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Repeatedly propagating knowledge in problem 7 produces the following problem:
4.98/2.39	
4.98/2.39	8:	T:
4.98/2.39	
4.98/2.39			(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 ]
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)                  evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)                  evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)                  evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)                  evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)                  evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)                  evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)                  evalstart4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)                  evalstart5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: 1, Cost: 1)                  evalstart6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(0, 0, ar_2, ar_3))
4.98/2.39	
4.98/2.39			(Comp: ar_2 + ar_3 + 1, Cost: 1)    evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: 2, Cost: 1)                  evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ]
4.98/2.39	
4.98/2.39			(Comp: ar_3, Cost: 1)               evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_1 + 1 ]
4.98/2.39	
4.98/2.39			(Comp: ar_2, Cost: 1)               evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_3 ]
4.98/2.39	
4.98/2.39			(Comp: 2, Cost: 1)                  evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 - ar_2 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
4.98/2.39	
4.98/2.39		start location:	koat_start
4.98/2.39	
4.98/2.39		leaf cost:	0
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Complexity upper bound 2*ar_2 + 2*ar_3 + 14
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Time: 0.301 sec (SMT: 0.246 sec)
4.98/2.39	
4.98/2.39	
4.98/2.39	----------------------------------------
4.98/2.39	
4.98/2.39	(2)
4.98/2.39	BOUNDS(1, n^1)
4.98/2.39	
4.98/2.39	----------------------------------------
4.98/2.39	
4.98/2.39	(3) Loat Proof (FINISHED)
4.98/2.39	
4.98/2.39	
4.98/2.39	### Pre-processing the ITS problem ###
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Initial linear ITS problem
4.98/2.39	
4.98/2.39	   Start location: evalstartstart
4.98/2.39	
4.98/2.39	      0: evalstartstart -> evalstartbb0in : [], cost: 1
4.98/2.39	
4.98/2.39	      1: evalstartbb0in -> evalstart0 : [], cost: 1
4.98/2.39	
4.98/2.39	      2: evalstart0 -> evalstart1 : [], cost: 1
4.98/2.39	
4.98/2.39	      3: evalstart1 -> evalstart2 : [], cost: 1
4.98/2.39	
4.98/2.39	      4: evalstart2 -> evalstart3 : [], cost: 1
4.98/2.39	
4.98/2.39	      5: evalstart3 -> evalstart4 : [], cost: 1
4.98/2.39	
4.98/2.39	      6: evalstart4 -> evalstart5 : [], cost: 1
4.98/2.39	
4.98/2.39	      7: evalstart5 -> evalstart6 : [], cost: 1
4.98/2.39	
4.98/2.39	      8: evalstart6 -> evalstartbb1in : A'=0, B'=0, [], cost: 1
4.98/2.39	
4.98/2.39	      9: evalstartbb1in -> evalstartbb2in : [ C>=1+A ], cost: 1
4.98/2.39	
4.98/2.39	     10: evalstartbb1in -> evalstartbb3in : [ A>=C ], cost: 1
4.98/2.39	
4.98/2.39	     11: evalstartbb2in -> evalstartbb1in : B'=1+B, [ D>=1+B ], cost: 1
4.98/2.39	
4.98/2.39	     12: evalstartbb2in -> evalstartbb1in : A'=1+A, B'=1+B, [ D>=1+B && B>=D ], cost: 1
4.98/2.39	
4.98/2.39	     13: evalstartbb2in -> evalstartbb1in : [ B>=D && D>=1+B ], cost: 1
4.98/2.39	
4.98/2.39	     14: evalstartbb2in -> evalstartbb1in : A'=1+A, [ B>=D ], cost: 1
4.98/2.39	
4.98/2.39	     15: evalstartbb3in -> evalstartstop : [], cost: 1
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Removed unreachable and leaf rules:
4.98/2.39	
4.98/2.39	   Start location: evalstartstart
4.98/2.39	
4.98/2.39	      0: evalstartstart -> evalstartbb0in : [], cost: 1
4.98/2.39	
4.98/2.39	      1: evalstartbb0in -> evalstart0 : [], cost: 1
4.98/2.39	
4.98/2.39	      2: evalstart0 -> evalstart1 : [], cost: 1
4.98/2.39	
4.98/2.39	      3: evalstart1 -> evalstart2 : [], cost: 1
4.98/2.39	
4.98/2.39	      4: evalstart2 -> evalstart3 : [], cost: 1
4.98/2.39	
4.98/2.39	      5: evalstart3 -> evalstart4 : [], cost: 1
4.98/2.39	
4.98/2.39	      6: evalstart4 -> evalstart5 : [], cost: 1
4.98/2.39	
4.98/2.39	      7: evalstart5 -> evalstart6 : [], cost: 1
4.98/2.39	
4.98/2.39	      8: evalstart6 -> evalstartbb1in : A'=0, B'=0, [], cost: 1
4.98/2.39	
4.98/2.39	      9: evalstartbb1in -> evalstartbb2in : [ C>=1+A ], cost: 1
4.98/2.39	
4.98/2.39	     11: evalstartbb2in -> evalstartbb1in : B'=1+B, [ D>=1+B ], cost: 1
4.98/2.39	
4.98/2.39	     12: evalstartbb2in -> evalstartbb1in : A'=1+A, B'=1+B, [ D>=1+B && B>=D ], cost: 1
4.98/2.39	
4.98/2.39	     13: evalstartbb2in -> evalstartbb1in : [ B>=D && D>=1+B ], cost: 1
4.98/2.39	
4.98/2.39	     14: evalstartbb2in -> evalstartbb1in : A'=1+A, [ B>=D ], cost: 1
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Removed rules with unsatisfiable guard:
4.98/2.39	
4.98/2.39	   Start location: evalstartstart
4.98/2.39	
4.98/2.39	      0: evalstartstart -> evalstartbb0in : [], cost: 1
4.98/2.39	
4.98/2.39	      1: evalstartbb0in -> evalstart0 : [], cost: 1
4.98/2.39	
4.98/2.39	      2: evalstart0 -> evalstart1 : [], cost: 1
4.98/2.39	
4.98/2.39	      3: evalstart1 -> evalstart2 : [], cost: 1
4.98/2.39	
4.98/2.39	      4: evalstart2 -> evalstart3 : [], cost: 1
4.98/2.39	
4.98/2.39	      5: evalstart3 -> evalstart4 : [], cost: 1
4.98/2.39	
4.98/2.39	      6: evalstart4 -> evalstart5 : [], cost: 1
4.98/2.39	
4.98/2.39	      7: evalstart5 -> evalstart6 : [], cost: 1
4.98/2.39	
4.98/2.39	      8: evalstart6 -> evalstartbb1in : A'=0, B'=0, [], cost: 1
4.98/2.39	
4.98/2.39	      9: evalstartbb1in -> evalstartbb2in : [ C>=1+A ], cost: 1
4.98/2.39	
4.98/2.39	     11: evalstartbb2in -> evalstartbb1in : B'=1+B, [ D>=1+B ], cost: 1
4.98/2.39	
4.98/2.39	     14: evalstartbb2in -> evalstartbb1in : A'=1+A, [ B>=D ], cost: 1
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	### Simplification by acceleration and chaining ###
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Eliminated locations (on linear paths):
4.98/2.39	
4.98/2.39	   Start location: evalstartstart
4.98/2.39	
4.98/2.39	     23: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 9
4.98/2.39	
4.98/2.39	      9: evalstartbb1in -> evalstartbb2in : [ C>=1+A ], cost: 1
4.98/2.39	
4.98/2.39	     11: evalstartbb2in -> evalstartbb1in : B'=1+B, [ D>=1+B ], cost: 1
4.98/2.39	
4.98/2.39	     14: evalstartbb2in -> evalstartbb1in : A'=1+A, [ B>=D ], cost: 1
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Eliminated locations (on tree-shaped paths):
4.98/2.39	
4.98/2.39	   Start location: evalstartstart
4.98/2.39	
4.98/2.39	     23: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 9
4.98/2.39	
4.98/2.39	     24: evalstartbb1in -> evalstartbb1in : B'=1+B, [ C>=1+A && D>=1+B ], cost: 2
4.98/2.39	
4.98/2.39	     25: evalstartbb1in -> evalstartbb1in : A'=1+A, [ C>=1+A && B>=D ], cost: 2
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Accelerating simple loops of location 9.
4.98/2.39	
4.98/2.39	   Accelerating the following rules:
4.98/2.39	
4.98/2.39	     24: evalstartbb1in -> evalstartbb1in : B'=1+B, [ C>=1+A && D>=1+B ], cost: 2
4.98/2.39	
4.98/2.39	     25: evalstartbb1in -> evalstartbb1in : A'=1+A, [ C>=1+A && B>=D ], cost: 2
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	   Accelerated rule 24 with metering function D-B, yielding the new rule 26.
4.98/2.39	
4.98/2.39	   Accelerated rule 25 with metering function C-A, yielding the new rule 27.
4.98/2.39	
4.98/2.39	   Removing the simple loops: 24 25.
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Accelerated all simple loops using metering functions (where possible):
4.98/2.39	
4.98/2.39	   Start location: evalstartstart
4.98/2.39	
4.98/2.39	     23: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 9
4.98/2.39	
4.98/2.39	     26: evalstartbb1in -> evalstartbb1in : B'=D, [ C>=1+A && D>=1+B ], cost: 2*D-2*B
4.98/2.39	
4.98/2.39	     27: evalstartbb1in -> evalstartbb1in : A'=C, [ C>=1+A && B>=D ], cost: 2*C-2*A
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Chained accelerated rules (with incoming rules):
4.98/2.39	
4.98/2.39	   Start location: evalstartstart
4.98/2.39	
4.98/2.39	     23: evalstartstart -> evalstartbb1in : A'=0, B'=0, [], cost: 9
4.98/2.39	
4.98/2.39	     28: evalstartstart -> evalstartbb1in : A'=0, B'=D, [ C>=1 && D>=1 ], cost: 9+2*D
4.98/2.39	
4.98/2.39	     29: evalstartstart -> evalstartbb1in : A'=C, B'=0, [ C>=1 && 0>=D ], cost: 9+2*C
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Removed unreachable locations (and leaf rules with constant cost):
4.98/2.39	
4.98/2.39	   Start location: evalstartstart
4.98/2.39	
4.98/2.39	     28: evalstartstart -> evalstartbb1in : A'=0, B'=D, [ C>=1 && D>=1 ], cost: 9+2*D
4.98/2.39	
4.98/2.39	     29: evalstartstart -> evalstartbb1in : A'=C, B'=0, [ C>=1 && 0>=D ], cost: 9+2*C
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	### Computing asymptotic complexity ###
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Fully simplified ITS problem
4.98/2.39	
4.98/2.39	   Start location: evalstartstart
4.98/2.39	
4.98/2.39	     28: evalstartstart -> evalstartbb1in : A'=0, B'=D, [ C>=1 && D>=1 ], cost: 9+2*D
4.98/2.39	
4.98/2.39	     29: evalstartstart -> evalstartbb1in : A'=C, B'=0, [ C>=1 && 0>=D ], cost: 9+2*C
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Computing asymptotic complexity for rule 28
4.98/2.39	
4.98/2.39	   Solved the limit problem by the following transformations:
4.98/2.39	
4.98/2.39	      Created initial limit problem:
4.98/2.39	
4.98/2.39	      C (+/+!), 9+2*D (+), D (+/+!) [not solved]
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	      removing all constraints (solved by SMT)
4.98/2.39	
4.98/2.39	      resulting limit problem: [solved]
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	      applying transformation rule (C) using substitution {C==1,D==n}
4.98/2.39	
4.98/2.39	      resulting limit problem:
4.98/2.39	
4.98/2.39	      [solved]
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	   Solution:
4.98/2.39	
4.98/2.39	      C / 1
4.98/2.39	
4.98/2.39	      D / n
4.98/2.39	
4.98/2.39	   Resulting cost 9+2*n has complexity: Poly(n^1)
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Found new complexity Poly(n^1).
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	Obtained the following overall complexity (w.r.t. the length of the input n):
4.98/2.39	
4.98/2.39	   Complexity:  Poly(n^1)
4.98/2.39	
4.98/2.39	   Cpx degree:  1
4.98/2.39	
4.98/2.39	   Solved cost: 9+2*n
4.98/2.39	
4.98/2.39	   Rule cost:   9+2*D
4.98/2.39	
4.98/2.39	   Rule guard:  [ C>=1 && D>=1 ]
4.98/2.39	
4.98/2.39	
4.98/2.39	
4.98/2.39	WORST_CASE(Omega(n^1),?)
4.98/2.39	
4.98/2.39	
4.98/2.39	----------------------------------------
4.98/2.39	
4.98/2.39	(4)
4.98/2.39	BOUNDS(n^1, INF)
5.20/2.41	EOF