2.29/1.34	WORST_CASE(?, O(n^1))
2.29/1.35	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.29/1.35	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.29/1.35	
2.29/1.35	
2.29/1.35	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.29/1.35	
2.29/1.35	(0) CpxIntTrs
2.29/1.35	(1) Koat Proof [FINISHED, 84 ms]
2.29/1.35	(2) BOUNDS(1, n^1)
2.29/1.35	
2.29/1.35	
2.29/1.35	----------------------------------------
2.29/1.35	
2.29/1.35	(0)
2.29/1.35	Obligation:
2.29/1.35	Complexity Int TRS consisting of the following rules:
2.29/1.35	eval_foo_start(v_.0, v_x, v_y) -> Com_1(eval_foo_bb0_in(v_.0, v_x, v_y)) :|: TRUE
2.29/1.35	eval_foo_bb0_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_y, v_x, v_y)) :|: TRUE
2.29/1.35	eval_foo_bb1_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.0, v_x, v_y)) :|: v_x > 0 && v_x > v_.0
2.29/1.35	eval_foo_bb1_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_x, v_y)) :|: v_x <= 0
2.29/1.35	eval_foo_bb1_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_x, v_y)) :|: v_x <= v_.0
2.29/1.35	eval_foo_bb2_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.0 + v_x, v_x, v_y)) :|: TRUE
2.29/1.35	eval_foo_bb3_in(v_.0, v_x, v_y) -> Com_1(eval_foo_stop(v_.0, v_x, v_y)) :|: TRUE
2.29/1.35	
2.29/1.35	The start-symbols are:[eval_foo_start_3]
2.29/1.35	
2.29/1.35	
2.29/1.35	----------------------------------------
2.29/1.35	
2.29/1.35	(1) Koat Proof (FINISHED)
2.29/1.35	YES(?, 2*ar_1 + 2*ar_2 + 8)
2.29/1.35	
2.29/1.35	
2.29/1.35	
2.29/1.35	Initial complexity problem:
2.29/1.35	
2.29/1.35	1:	T:
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 /\ ar_2 >= ar_0 + 1 ]
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 + ar_2, ar_1, ar_2))
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.29/1.35	
2.29/1.35			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.29/1.35	
2.29/1.35		start location:	koat_start
2.29/1.35	
2.29/1.35		leaf cost:	0
2.29/1.35	
2.29/1.35	
2.29/1.35	
2.29/1.35	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.29/1.35	
2.29/1.35	2:	T:
2.29/1.35	
2.29/1.35			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.29/1.35	
2.29/1.35			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 /\ ar_2 >= ar_0 + 1 ]
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 + ar_2, ar_1, ar_2))
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.29/1.35	
2.29/1.35			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.29/1.35	
2.29/1.35		start location:	koat_start
2.29/1.35	
2.29/1.35		leaf cost:	0
2.29/1.35	
2.29/1.35	
2.29/1.35	
2.29/1.35	A polynomial rank function with
2.29/1.35	
2.29/1.35		Pol(evalfoostart) = 2
2.29/1.35	
2.29/1.35		Pol(evalfoobb0in) = 2
2.29/1.35	
2.29/1.35		Pol(evalfoobb1in) = 2
2.29/1.35	
2.29/1.35		Pol(evalfoobb2in) = 2
2.29/1.35	
2.29/1.35		Pol(evalfoobb3in) = 1
2.29/1.35	
2.29/1.35		Pol(evalfoostop) = 0
2.29/1.35	
2.29/1.35		Pol(koat_start) = 2
2.29/1.35	
2.29/1.35	orients all transitions weakly and the transitions
2.29/1.35	
2.29/1.35		evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.29/1.35	
2.29/1.35		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.29/1.35	
2.29/1.35		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.29/1.35	
2.29/1.35	strictly and produces the following problem:
2.29/1.35	
2.29/1.35	3:	T:
2.29/1.35	
2.29/1.35			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.29/1.35	
2.29/1.35			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.29/1.35	
2.29/1.35			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 /\ ar_2 >= ar_0 + 1 ]
2.29/1.35	
2.29/1.35			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.29/1.36	
2.29/1.36			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.29/1.36	
2.29/1.36			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 + ar_2, ar_1, ar_2))
2.29/1.36	
2.29/1.36			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.29/1.36	
2.29/1.36			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.29/1.36	
2.29/1.36		start location:	koat_start
2.29/1.36	
2.29/1.36		leaf cost:	0
2.29/1.36	
2.29/1.36	
2.29/1.36	
2.29/1.36	A polynomial rank function with
2.29/1.36	
2.29/1.36		Pol(evalfoostart) = -V_2 + V_3
2.29/1.36	
2.29/1.36		Pol(evalfoobb0in) = -V_2 + V_3
2.29/1.36	
2.29/1.36		Pol(evalfoobb1in) = -V_1 + V_3
2.29/1.36	
2.29/1.36		Pol(evalfoobb2in) = -V_1
2.29/1.36	
2.29/1.36		Pol(evalfoobb3in) = -V_1 + V_3
2.29/1.36	
2.29/1.36		Pol(evalfoostop) = -V_1 + V_3
2.29/1.36	
2.29/1.36		Pol(koat_start) = -V_2 + V_3
2.29/1.36	
2.29/1.36	orients all transitions weakly and the transition
2.29/1.36	
2.29/1.36		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 /\ ar_2 >= ar_0 + 1 ]
2.29/1.36	
2.29/1.36	strictly and produces the following problem:
2.29/1.36	
2.29/1.36	4:	T:
2.29/1.36	
2.29/1.36			(Comp: 1, Cost: 1)              evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.29/1.36	
2.29/1.36			(Comp: 1, Cost: 1)              evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.29/1.36	
2.29/1.36			(Comp: ar_1 + ar_2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 /\ ar_2 >= ar_0 + 1 ]
2.29/1.36	
2.29/1.36			(Comp: 2, Cost: 1)              evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.29/1.36	
2.29/1.36			(Comp: 2, Cost: 1)              evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.29/1.36	
2.29/1.36			(Comp: ?, Cost: 1)              evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 + ar_2, ar_1, ar_2))
2.29/1.36	
2.29/1.36			(Comp: 2, Cost: 1)              evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.29/1.36	
2.29/1.36			(Comp: 1, Cost: 0)              koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.29/1.36	
2.29/1.36		start location:	koat_start
2.29/1.36	
2.29/1.36		leaf cost:	0
2.29/1.36	
2.29/1.36	
2.29/1.36	
2.29/1.36	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.29/1.36	
2.29/1.36	5:	T:
2.29/1.36	
2.29/1.36			(Comp: 1, Cost: 1)              evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.29/1.36	
2.29/1.36			(Comp: 1, Cost: 1)              evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.29/1.36	
2.29/1.36			(Comp: ar_1 + ar_2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 /\ ar_2 >= ar_0 + 1 ]
2.29/1.36	
2.29/1.36			(Comp: 2, Cost: 1)              evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.29/1.36	
2.29/1.36			(Comp: 2, Cost: 1)              evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.29/1.36	
2.29/1.36			(Comp: ar_1 + ar_2, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 + ar_2, ar_1, ar_2))
2.29/1.36	
2.29/1.36			(Comp: 2, Cost: 1)              evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.29/1.36	
2.29/1.36			(Comp: 1, Cost: 0)              koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.29/1.36	
2.29/1.36		start location:	koat_start
2.29/1.36	
2.29/1.36		leaf cost:	0
2.29/1.36	
2.29/1.36	
2.29/1.36	
2.29/1.36	Complexity upper bound 2*ar_1 + 2*ar_2 + 8
2.29/1.36	
2.29/1.36	
2.29/1.36	
2.29/1.36	Time: 0.081 sec (SMT: 0.074 sec)
2.29/1.36	
2.29/1.36	
2.29/1.36	----------------------------------------
2.29/1.36	
2.29/1.36	(2)
2.29/1.36	BOUNDS(1, n^1)
2.32/1.38	EOF