2.15/1.38	WORST_CASE(?, O(n^1))
2.15/1.39	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.15/1.39	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.15/1.39	
2.15/1.39	
2.15/1.39	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.15/1.39	
2.15/1.39	(0) CpxIntTrs
2.15/1.39	(1) Koat Proof [FINISHED, 77 ms]
2.15/1.39	(2) BOUNDS(1, n^1)
2.15/1.39	
2.15/1.39	
2.15/1.39	----------------------------------------
2.15/1.39	
2.15/1.39	(0)
2.15/1.39	Obligation:
2.15/1.39	Complexity Int TRS consisting of the following rules:
2.15/1.39	eval_foo_start(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp) -> Com_1(eval_foo_bb0_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp)) :|: TRUE
2.15/1.39	eval_foo_bb0_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp) -> Com_1(eval_foo_bb1_in(v_x, v_.01, v_res, v_restmp, v_x, v_xtmp)) :|: TRUE
2.15/1.39	eval_foo_bb1_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp) -> Com_1(eval_foo_bb2_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp)) :|: v_.0 > 1
2.15/1.39	eval_foo_bb1_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp) -> Com_1(eval_foo_bb6_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp)) :|: v_.0 <= 1
2.15/1.39	eval_foo_bb2_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp) -> Com_1(eval_foo_bb3_in(v_.0, v_.0 - 2, v_res, v_restmp, v_x, v_xtmp)) :|: TRUE
2.15/1.39	eval_foo_bb3_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp) -> Com_1(eval_foo_bb4_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp)) :|: v_.01 > 1
2.15/1.39	eval_foo_bb3_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp) -> Com_1(eval_foo_bb5_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp)) :|: v_.01 <= 1
2.15/1.39	eval_foo_bb4_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp) -> Com_1(eval_foo_bb3_in(v_.0, v_.01 - 2, v_res, v_restmp, v_x, v_xtmp)) :|: TRUE
2.15/1.39	eval_foo_bb5_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp) -> Com_1(eval_foo_bb1_in(v_.01 + 1, v_.01, v_res, v_restmp, v_x, v_xtmp)) :|: TRUE
2.15/1.39	eval_foo_bb6_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp) -> Com_1(eval_foo_stop(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp)) :|: TRUE
2.15/1.39	
2.15/1.39	The start-symbols are:[eval_foo_start_6]
2.15/1.39	
2.15/1.39	
2.15/1.39	----------------------------------------
2.15/1.39	
2.15/1.39	(1) Koat Proof (FINISHED)
2.15/1.39	YES(?, 8*ar_1 + 6)
2.15/1.39	
2.15/1.39	
2.15/1.39	
2.15/1.39	Initial complexity problem:
2.15/1.39	
2.15/1.39	1:	T:
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 >= 2 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb6in(ar_0, ar_1, ar_2)) [ 1 >= ar_0 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_0 - 2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= 2 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 1 >= ar_2 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2 - 2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_2 + 1, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb6in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.15/1.39	
2.15/1.39		start location:	koat_start
2.15/1.39	
2.15/1.39		leaf cost:	0
2.15/1.39	
2.15/1.39	
2.15/1.39	
2.15/1.39	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.15/1.39	
2.15/1.39	2:	T:
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 >= 2 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb6in(ar_0, ar_1, ar_2)) [ 1 >= ar_0 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_0 - 2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= 2 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 1 >= ar_2 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2 - 2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_2 + 1, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb6in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.15/1.39	
2.15/1.39		start location:	koat_start
2.15/1.39	
2.15/1.39		leaf cost:	0
2.15/1.39	
2.15/1.39	
2.15/1.39	
2.15/1.39	A polynomial rank function with
2.15/1.39	
2.15/1.39		Pol(evalfoostart) = 2
2.15/1.39	
2.15/1.39		Pol(evalfoobb0in) = 2
2.15/1.39	
2.15/1.39		Pol(evalfoobb1in) = 2
2.15/1.39	
2.15/1.39		Pol(evalfoobb2in) = 2
2.15/1.39	
2.15/1.39		Pol(evalfoobb6in) = 1
2.15/1.39	
2.15/1.39		Pol(evalfoobb3in) = 2
2.15/1.39	
2.15/1.39		Pol(evalfoobb4in) = 2
2.15/1.39	
2.15/1.39		Pol(evalfoobb5in) = 2
2.15/1.39	
2.15/1.39		Pol(evalfoostop) = 0
2.15/1.39	
2.15/1.39		Pol(koat_start) = 2
2.15/1.39	
2.15/1.39	orients all transitions weakly and the transitions
2.15/1.39	
2.15/1.39		evalfoobb6in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.15/1.39	
2.15/1.39		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb6in(ar_0, ar_1, ar_2)) [ 1 >= ar_0 ]
2.15/1.39	
2.15/1.39	strictly and produces the following problem:
2.15/1.39	
2.15/1.39	3:	T:
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 >= 2 ]
2.15/1.39	
2.15/1.39			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb6in(ar_0, ar_1, ar_2)) [ 1 >= ar_0 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_0 - 2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= 2 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 1 >= ar_2 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2 - 2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)    evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_2 + 1, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: 2, Cost: 1)    evalfoobb6in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.15/1.39	
2.15/1.39		start location:	koat_start
2.15/1.39	
2.15/1.39		leaf cost:	0
2.15/1.39	
2.15/1.39	
2.15/1.39	
2.15/1.39	A polynomial rank function with
2.15/1.39	
2.15/1.39		Pol(evalfoostart) = V_2
2.15/1.39	
2.15/1.39		Pol(evalfoobb0in) = V_2
2.15/1.39	
2.15/1.39		Pol(evalfoobb1in) = V_1 - 1
2.15/1.39	
2.15/1.39		Pol(evalfoobb2in) = V_1 - 2
2.15/1.39	
2.15/1.39		Pol(evalfoobb6in) = V_1 - 1
2.15/1.39	
2.15/1.39		Pol(evalfoobb3in) = V_3
2.15/1.39	
2.15/1.39		Pol(evalfoobb4in) = V_3 - 1
2.15/1.39	
2.15/1.39		Pol(evalfoobb5in) = V_3
2.15/1.39	
2.15/1.39		Pol(evalfoostop) = V_1 - 1
2.15/1.39	
2.15/1.39		Pol(koat_start) = V_2
2.15/1.39	
2.15/1.39	orients all transitions weakly and the transitions
2.15/1.39	
2.15/1.39		evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= 2 ]
2.15/1.39	
2.15/1.39		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 >= 2 ]
2.15/1.39	
2.15/1.39	strictly and produces the following problem:
2.15/1.39	
2.15/1.39	4:	T:
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 1)       evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: ar_1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 >= 2 ]
2.15/1.39	
2.15/1.39			(Comp: 2, Cost: 1)       evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb6in(ar_0, ar_1, ar_2)) [ 1 >= ar_0 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)       evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_0 - 2))
2.15/1.39	
2.15/1.39			(Comp: ar_1, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= 2 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)       evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 1 >= ar_2 ]
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)       evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2 - 2))
2.15/1.39	
2.15/1.39			(Comp: ?, Cost: 1)       evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_2 + 1, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: 2, Cost: 1)       evalfoobb6in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.15/1.39	
2.15/1.39		start location:	koat_start
2.15/1.39	
2.15/1.39		leaf cost:	0
2.15/1.39	
2.15/1.39	
2.15/1.39	
2.15/1.39	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.15/1.39	
2.15/1.39	5:	T:
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 1)         evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 1)         evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: ar_1, Cost: 1)      evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 >= 2 ]
2.15/1.39	
2.15/1.39			(Comp: 2, Cost: 1)         evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb6in(ar_0, ar_1, ar_2)) [ 1 >= ar_0 ]
2.15/1.39	
2.15/1.39			(Comp: ar_1, Cost: 1)      evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_0 - 2))
2.15/1.39	
2.15/1.39			(Comp: ar_1, Cost: 1)      evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= 2 ]
2.15/1.39	
2.15/1.39			(Comp: 2*ar_1, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 1 >= ar_2 ]
2.15/1.39	
2.15/1.39			(Comp: ar_1, Cost: 1)      evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2 - 2))
2.15/1.39	
2.15/1.39			(Comp: 2*ar_1, Cost: 1)    evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_2 + 1, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: 2, Cost: 1)         evalfoobb6in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.15/1.39	
2.15/1.39			(Comp: 1, Cost: 0)         koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.15/1.39	
2.15/1.39		start location:	koat_start
2.15/1.39	
2.15/1.39		leaf cost:	0
2.15/1.39	
2.15/1.39	
2.15/1.39	
2.15/1.39	Complexity upper bound 8*ar_1 + 6
2.15/1.39	
2.15/1.39	
2.15/1.39	
2.15/1.39	Time: 0.083 sec (SMT: 0.076 sec)
2.15/1.39	
2.15/1.39	
2.15/1.39	----------------------------------------
2.15/1.39	
2.15/1.39	(2)
2.15/1.39	BOUNDS(1, n^1)
2.15/1.41	EOF