2.06/1.35	WORST_CASE(?, O(n^1))
2.06/1.36	proof of /export/starexec/sandbox2/output/output_files/bench.koat
2.06/1.36	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.06/1.36	
2.06/1.36	
2.06/1.36	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.06/1.36	
2.06/1.36	(0) CpxIntTrs
2.06/1.36	(1) Koat Proof [FINISHED, 183 ms]
2.06/1.36	(2) BOUNDS(1, n^1)
2.06/1.36	
2.06/1.36	
2.06/1.36	----------------------------------------
2.06/1.36	
2.06/1.36	(0)
2.06/1.36	Obligation:
2.06/1.36	Complexity Int TRS consisting of the following rules:
2.06/1.36	eval_speed_popl10_simple_single_start(v_n, v_x.0) -> Com_1(eval_speed_popl10_simple_single_bb0_in(v_n, v_x.0)) :|: TRUE
2.06/1.36	eval_speed_popl10_simple_single_bb0_in(v_n, v_x.0) -> Com_1(eval_speed_popl10_simple_single_bb1_in(v_n, 0)) :|: TRUE
2.06/1.36	eval_speed_popl10_simple_single_bb1_in(v_n, v_x.0) -> Com_1(eval_speed_popl10_simple_single_bb2_in(v_n, v_x.0)) :|: v_x.0 < v_n
2.06/1.36	eval_speed_popl10_simple_single_bb1_in(v_n, v_x.0) -> Com_1(eval_speed_popl10_simple_single_bb3_in(v_n, v_x.0)) :|: v_x.0 >= v_n
2.06/1.36	eval_speed_popl10_simple_single_bb2_in(v_n, v_x.0) -> Com_1(eval_speed_popl10_simple_single_0(v_n, v_x.0)) :|: TRUE
2.06/1.37	eval_speed_popl10_simple_single_0(v_n, v_x.0) -> Com_2(eval_nondet_start(v_n, v_x.0), eval_speed_popl10_simple_single_1(v_n, v_x.0)) :|: TRUE
2.06/1.37	eval_speed_popl10_simple_single_1(v_n, v_x.0) -> Com_1(eval_speed_popl10_simple_single_bb1_in(v_n, v_x.0 + 1)) :|: TRUE
2.06/1.37	eval_speed_popl10_simple_single_bb3_in(v_n, v_x.0) -> Com_1(eval_speed_popl10_simple_single_stop(v_n, v_x.0)) :|: TRUE
2.06/1.37	
2.06/1.37	The start-symbols are:[eval_speed_popl10_simple_single_start_2]
2.06/1.37	
2.06/1.37	
2.06/1.37	----------------------------------------
2.06/1.37	
2.06/1.37	(1) Koat Proof (FINISHED)
2.06/1.37	YES(?, 16*ar_1 + 6)
2.06/1.37	
2.06/1.37	
2.06/1.37	
2.06/1.37	Initial complexity problem:
2.06/1.37	
2.06/1.37	1:	T:
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglestart(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb0in(ar_0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb0in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb1in(0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb1in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb2in(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb1in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb3in(ar_0, ar_1)) [ ar_0 >= ar_1 ]
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb2in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesingle0(ar_0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesingle0(ar_0, ar_1) -> Com_2(evalnondetstart(ar_0, ar_1), evalspeedpopl10simplesingle1(ar_0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesingle1(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb1in(ar_0 + 1, ar_1))
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb3in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglestop(ar_0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglestart(ar_0, ar_1)) [ 0 <= 0 ]
2.06/1.37	
2.06/1.37		start location:	koat_start
2.06/1.37	
2.06/1.37		leaf cost:	0
2.06/1.37	
2.06/1.37	
2.06/1.37	
2.06/1.37	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.06/1.37	
2.06/1.37	2:	T:
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 1)    evalspeedpopl10simplesinglestart(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb0in(ar_0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 1)    evalspeedpopl10simplesinglebb0in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb1in(0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb1in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb2in(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb1in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb3in(ar_0, ar_1)) [ ar_0 >= ar_1 ]
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb2in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesingle0(ar_0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesingle0(ar_0, ar_1) -> Com_2(evalnondetstart(ar_0, ar_1), evalspeedpopl10simplesingle1(ar_0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesingle1(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb1in(ar_0 + 1, ar_1))
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb3in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglestop(ar_0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglestart(ar_0, ar_1)) [ 0 <= 0 ]
2.06/1.37	
2.06/1.37		start location:	koat_start
2.06/1.37	
2.06/1.37		leaf cost:	0
2.06/1.37	
2.06/1.37	
2.06/1.37	
2.06/1.37	A polynomial rank function with
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesinglestart) = 2
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesinglebb0in) = 2
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesinglebb1in) = 2
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesinglebb2in) = 2
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesinglebb3in) = 1
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesingle0) = 2
2.06/1.37	
2.06/1.37		Pol(evalnondetstart) = 0
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesingle1) = 2
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesinglestop) = 0
2.06/1.37	
2.06/1.37		Pol(koat_start) = 2
2.06/1.37	
2.06/1.37	orients all transitions weakly and the transitions
2.06/1.37	
2.06/1.37		evalspeedpopl10simplesinglebb3in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglestop(ar_0, ar_1))
2.06/1.37	
2.06/1.37		evalspeedpopl10simplesinglebb1in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb3in(ar_0, ar_1)) [ ar_0 >= ar_1 ]
2.06/1.37	
2.06/1.37	strictly and produces the following problem:
2.06/1.37	
2.06/1.37	3:	T:
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 1)    evalspeedpopl10simplesinglestart(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb0in(ar_0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 1)    evalspeedpopl10simplesinglebb0in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb1in(0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb1in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb2in(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
2.06/1.37	
2.06/1.37			(Comp: 2, Cost: 1)    evalspeedpopl10simplesinglebb1in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb3in(ar_0, ar_1)) [ ar_0 >= ar_1 ]
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb2in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesingle0(ar_0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesingle0(ar_0, ar_1) -> Com_2(evalnondetstart(ar_0, ar_1), evalspeedpopl10simplesingle1(ar_0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesingle1(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb1in(ar_0 + 1, ar_1))
2.06/1.37	
2.06/1.37			(Comp: 2, Cost: 1)    evalspeedpopl10simplesinglebb3in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglestop(ar_0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglestart(ar_0, ar_1)) [ 0 <= 0 ]
2.06/1.37	
2.06/1.37		start location:	koat_start
2.06/1.37	
2.06/1.37		leaf cost:	0
2.06/1.37	
2.06/1.37	
2.06/1.37	
2.06/1.37	Applied AI with 'oct' on problem 3 to obtain the following invariants:
2.06/1.37	
2.06/1.37	  For symbol evalspeedpopl10simplesingle0: X_2 - 1 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 >= 0
2.06/1.37	
2.06/1.37	  For symbol evalspeedpopl10simplesingle1: X_2 - 1 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 >= 0
2.06/1.37	
2.06/1.37	  For symbol evalspeedpopl10simplesinglebb1in: X_1 >= 0
2.06/1.37	
2.06/1.37	  For symbol evalspeedpopl10simplesinglebb2in: X_2 - 1 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 >= 0
2.06/1.37	
2.06/1.37	  For symbol evalspeedpopl10simplesinglebb3in: X_1 - X_2 >= 0 /\ X_1 >= 0
2.06/1.37	
2.06/1.37	
2.06/1.37	
2.06/1.37	
2.06/1.37	
2.06/1.37	This yielded the following problem:
2.06/1.37	
2.06/1.37	4:	T:
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglestart(ar_0, ar_1)) [ 0 <= 0 ]
2.06/1.37	
2.06/1.37			(Comp: 2, Cost: 1)    evalspeedpopl10simplesinglebb3in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglestop(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 >= 0 ]
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesingle1(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb1in(ar_0 + 1, ar_1)) [ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 >= 0 ]
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesingle0(ar_0, ar_1) -> Com_2(evalnondetstart(ar_0, ar_1), evalspeedpopl10simplesingle1(ar_0, ar_1)) [ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 >= 0 ]
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb2in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesingle0(ar_0, ar_1)) [ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 >= 0 ]
2.06/1.37	
2.06/1.37			(Comp: 2, Cost: 1)    evalspeedpopl10simplesinglebb1in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb3in(ar_0, ar_1)) [ ar_0 >= 0 /\ ar_0 >= ar_1 ]
2.06/1.37	
2.06/1.37			(Comp: ?, Cost: 1)    evalspeedpopl10simplesinglebb1in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb2in(ar_0, ar_1)) [ ar_0 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 1)    evalspeedpopl10simplesinglebb0in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb1in(0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 1)    evalspeedpopl10simplesinglestart(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb0in(ar_0, ar_1))
2.06/1.37	
2.06/1.37		start location:	koat_start
2.06/1.37	
2.06/1.37		leaf cost:	0
2.06/1.37	
2.06/1.37	
2.06/1.37	
2.06/1.37	A polynomial rank function with
2.06/1.37	
2.06/1.37		Pol(koat_start) = 4*V_2
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesinglestart) = 4*V_2
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesinglebb3in) = -4*V_1 + 4*V_2
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesinglestop) = -4*V_1 + 4*V_2
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesingle1) = -4*V_1 + 4*V_2 - 3
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesinglebb1in) = -4*V_1 + 4*V_2
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesingle0) = -4*V_1 + 4*V_2 - 2
2.06/1.37	
2.06/1.37		Pol(evalnondetstart) = -4*V_1
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesinglebb2in) = -4*V_1 + 4*V_2 - 1
2.06/1.37	
2.06/1.37		Pol(evalspeedpopl10simplesinglebb0in) = 4*V_2
2.06/1.37	
2.06/1.37	orients all transitions weakly and the transitions
2.06/1.37	
2.06/1.37		evalspeedpopl10simplesinglebb2in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesingle0(ar_0, ar_1)) [ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 >= 0 ]
2.06/1.37	
2.06/1.37		evalspeedpopl10simplesinglebb1in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb2in(ar_0, ar_1)) [ ar_0 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.06/1.37	
2.06/1.37		evalspeedpopl10simplesingle1(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb1in(ar_0 + 1, ar_1)) [ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 >= 0 ]
2.06/1.37	
2.06/1.37		evalspeedpopl10simplesingle0(ar_0, ar_1) -> Com_2(evalnondetstart(ar_0, ar_1), evalspeedpopl10simplesingle1(ar_0, ar_1)) [ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 >= 0 ]
2.06/1.37	
2.06/1.37	strictly and produces the following problem:
2.06/1.37	
2.06/1.37	5:	T:
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 0)         koat_start(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglestart(ar_0, ar_1)) [ 0 <= 0 ]
2.06/1.37	
2.06/1.37			(Comp: 2, Cost: 1)         evalspeedpopl10simplesinglebb3in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglestop(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 >= 0 ]
2.06/1.37	
2.06/1.37			(Comp: 4*ar_1, Cost: 1)    evalspeedpopl10simplesingle1(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb1in(ar_0 + 1, ar_1)) [ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 >= 0 ]
2.06/1.37	
2.06/1.37			(Comp: 4*ar_1, Cost: 1)    evalspeedpopl10simplesingle0(ar_0, ar_1) -> Com_2(evalnondetstart(ar_0, ar_1), evalspeedpopl10simplesingle1(ar_0, ar_1)) [ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 >= 0 ]
2.06/1.37	
2.06/1.37			(Comp: 4*ar_1, Cost: 1)    evalspeedpopl10simplesinglebb2in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesingle0(ar_0, ar_1)) [ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ -ar_0 + ar_1 - 1 >= 0 /\ ar_0 >= 0 ]
2.06/1.37	
2.06/1.37			(Comp: 2, Cost: 1)         evalspeedpopl10simplesinglebb1in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb3in(ar_0, ar_1)) [ ar_0 >= 0 /\ ar_0 >= ar_1 ]
2.06/1.37	
2.06/1.37			(Comp: 4*ar_1, Cost: 1)    evalspeedpopl10simplesinglebb1in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb2in(ar_0, ar_1)) [ ar_0 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 1)         evalspeedpopl10simplesinglebb0in(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb1in(0, ar_1))
2.06/1.37	
2.06/1.37			(Comp: 1, Cost: 1)         evalspeedpopl10simplesinglestart(ar_0, ar_1) -> Com_1(evalspeedpopl10simplesinglebb0in(ar_0, ar_1))
2.06/1.37	
2.06/1.37		start location:	koat_start
2.06/1.37	
2.06/1.37		leaf cost:	0
2.06/1.37	
2.06/1.37	
2.06/1.37	
2.06/1.37	Complexity upper bound 16*ar_1 + 6
2.06/1.37	
2.06/1.37	
2.06/1.37	
2.06/1.37	Time: 0.140 sec (SMT: 0.124 sec)
2.06/1.37	
2.06/1.37	
2.06/1.37	----------------------------------------
2.06/1.37	
2.06/1.37	(2)
2.06/1.37	BOUNDS(1, n^1)
2.26/1.39	EOF