2.15/1.24	WORST_CASE(?, O(n^1))
2.22/1.25	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.22/1.25	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.22/1.25	
2.22/1.25	
2.22/1.25	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.22/1.25	
2.22/1.25	(0) CpxIntTrs
2.22/1.25	(1) Koat Proof [FINISHED, 72 ms]
2.22/1.25	(2) BOUNDS(1, n^1)
2.22/1.25	
2.22/1.25	
2.22/1.25	----------------------------------------
2.22/1.25	
2.22/1.25	(0)
2.22/1.25	Obligation:
2.22/1.25	Complexity Int TRS consisting of the following rules:
2.22/1.25	eval_foo_start(v_.0, v_bound, v_i, v_sum) -> Com_1(eval_foo_bb0_in(v_.0, v_bound, v_i, v_sum)) :|: TRUE
2.22/1.25	eval_foo_bb0_in(v_.0, v_bound, v_i, v_sum) -> Com_1(eval_foo_bb1_in(0, v_bound, v_i, v_sum)) :|: TRUE
2.22/1.25	eval_foo_bb1_in(v_.0, v_bound, v_i, v_sum) -> Com_1(eval_foo_bb2_in(v_.0, v_bound, v_i, v_sum)) :|: v_.0 < v_bound
2.22/1.25	eval_foo_bb1_in(v_.0, v_bound, v_i, v_sum) -> Com_1(eval_foo_bb3_in(v_.0, v_bound, v_i, v_sum)) :|: v_.0 >= v_bound
2.22/1.25	eval_foo_bb2_in(v_.0, v_bound, v_i, v_sum) -> Com_1(eval_foo_bb1_in(v_.0 + 1, v_bound, v_i, v_sum)) :|: TRUE
2.22/1.25	eval_foo_bb3_in(v_.0, v_bound, v_i, v_sum) -> Com_1(eval_foo_stop(v_.0, v_bound, v_i, v_sum)) :|: TRUE
2.22/1.25	
2.22/1.25	The start-symbols are:[eval_foo_start_4]
2.22/1.25	
2.22/1.25	
2.22/1.25	----------------------------------------
2.22/1.25	
2.22/1.25	(1) Koat Proof (FINISHED)
2.22/1.25	YES(?, 2*ar_1 + 6)
2.22/1.25	
2.22/1.25	
2.22/1.25	
2.22/1.25	Initial complexity problem:
2.22/1.25	
2.22/1.25	1:	T:
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_0 >= ar_1 ]
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1))
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.22/1.25	
2.22/1.25		start location:	koat_start
2.22/1.25	
2.22/1.25		leaf cost:	0
2.22/1.25	
2.22/1.25	
2.22/1.25	
2.22/1.25	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.22/1.25	
2.22/1.25	2:	T:
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_0 >= ar_1 ]
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1))
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.22/1.25	
2.22/1.25		start location:	koat_start
2.22/1.25	
2.22/1.25		leaf cost:	0
2.22/1.25	
2.22/1.25	
2.22/1.25	
2.22/1.25	A polynomial rank function with
2.22/1.25	
2.22/1.25		Pol(evalfoostart) = 2
2.22/1.25	
2.22/1.25		Pol(evalfoobb0in) = 2
2.22/1.25	
2.22/1.25		Pol(evalfoobb1in) = 2
2.22/1.25	
2.22/1.25		Pol(evalfoobb2in) = 2
2.22/1.25	
2.22/1.25		Pol(evalfoobb3in) = 1
2.22/1.25	
2.22/1.25		Pol(evalfoostop) = 0
2.22/1.25	
2.22/1.25		Pol(koat_start) = 2
2.22/1.25	
2.22/1.25	orients all transitions weakly and the transitions
2.22/1.25	
2.22/1.25		evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.22/1.25	
2.22/1.25		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_0 >= ar_1 ]
2.22/1.25	
2.22/1.25	strictly and produces the following problem:
2.22/1.25	
2.22/1.25	3:	T:
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
2.22/1.25	
2.22/1.25			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_0 >= ar_1 ]
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1))
2.22/1.25	
2.22/1.25			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.22/1.25	
2.22/1.25		start location:	koat_start
2.22/1.25	
2.22/1.25		leaf cost:	0
2.22/1.25	
2.22/1.25	
2.22/1.25	
2.22/1.25	A polynomial rank function with
2.22/1.25	
2.22/1.25		Pol(evalfoostart) = V_2
2.22/1.25	
2.22/1.25		Pol(evalfoobb0in) = V_2
2.22/1.25	
2.22/1.25		Pol(evalfoobb1in) = -V_1 + V_2
2.22/1.25	
2.22/1.25		Pol(evalfoobb2in) = -V_1 + V_2 - 1
2.22/1.25	
2.22/1.25		Pol(evalfoobb3in) = -V_1 + V_2
2.22/1.25	
2.22/1.25		Pol(evalfoostop) = -V_1 + V_2
2.22/1.25	
2.22/1.25		Pol(koat_start) = V_2
2.22/1.25	
2.22/1.25	orients all transitions weakly and the transition
2.22/1.25	
2.22/1.25		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
2.22/1.25	
2.22/1.25	strictly and produces the following problem:
2.22/1.25	
2.22/1.25	4:	T:
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 1)       evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: ar_1, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
2.22/1.25	
2.22/1.25			(Comp: 2, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_0 >= ar_1 ]
2.22/1.25	
2.22/1.25			(Comp: ?, Cost: 1)       evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1))
2.22/1.25	
2.22/1.25			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.22/1.25	
2.22/1.25		start location:	koat_start
2.22/1.25	
2.22/1.25		leaf cost:	0
2.22/1.25	
2.22/1.25	
2.22/1.25	
2.22/1.25	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.22/1.25	
2.22/1.25	5:	T:
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 1)       evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: ar_1, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= ar_0 + 1 ]
2.22/1.25	
2.22/1.25			(Comp: 2, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_0 >= ar_1 ]
2.22/1.25	
2.22/1.25			(Comp: ar_1, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1))
2.22/1.25	
2.22/1.25			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.22/1.25	
2.22/1.25			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.22/1.25	
2.22/1.25		start location:	koat_start
2.22/1.25	
2.22/1.25		leaf cost:	0
2.22/1.25	
2.22/1.25	
2.22/1.25	
2.22/1.25	Complexity upper bound 2*ar_1 + 6
2.22/1.25	
2.22/1.25	
2.22/1.25	
2.22/1.25	Time: 0.040 sec (SMT: 0.036 sec)
2.22/1.25	
2.22/1.25	
2.22/1.25	----------------------------------------
2.22/1.25	
2.22/1.25	(2)
2.22/1.25	BOUNDS(1, n^1)
2.24/1.26	EOF