2.09/1.28	WORST_CASE(?, O(n^1))
2.23/1.29	proof of /export/starexec/sandbox2/output/output_files/bench.koat
2.23/1.29	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.23/1.29	
2.23/1.29	
2.23/1.29	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.23/1.29	
2.23/1.29	(0) CpxIntTrs
2.23/1.29	(1) Koat Proof [FINISHED, 75 ms]
2.23/1.29	(2) BOUNDS(1, n^1)
2.23/1.29	
2.23/1.29	
2.23/1.29	----------------------------------------
2.23/1.29	
2.23/1.29	(0)
2.23/1.29	Obligation:
2.23/1.29	Complexity Int TRS consisting of the following rules:
2.23/1.29	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
2.23/1.29	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
2.23/1.29	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_.0, v_res, v_restmp, v_x, v_xtmp)) :|: v_.0 > 1
2.23/1.29	eval_foo_bb1_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_.0 <= 1
2.23/1.29	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_.01, v_res, v_restmp, v_x, v_xtmp)) :|: v_.01 > 1
2.23/1.29	eval_foo_bb2_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
2.23/1.29	eval_foo_bb3_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp) -> Com_1(eval_foo_bb2_in(v_.0, v_.01 - 2, v_res, v_restmp, v_x, v_xtmp)) :|: TRUE
2.23/1.29	eval_foo_bb4_in(v_.0, v_.01, v_res, v_restmp, v_x, v_xtmp) -> Com_1(eval_foo_bb1_in(v_.01, v_.01, v_res, v_restmp, v_x, v_xtmp)) :|: TRUE
2.23/1.29	eval_foo_bb5_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
2.23/1.29	
2.23/1.29	The start-symbols are:[eval_foo_start_6]
2.23/1.29	
2.23/1.29	
2.23/1.29	----------------------------------------
2.23/1.29	
2.23/1.29	(1) Koat Proof (FINISHED)
2.23/1.29	YES(?, 15*ar_1 + 6)
2.23/1.29	
2.23/1.29	
2.23/1.29	
2.23/1.29	Initial complexity problem:
2.23/1.29	
2.23/1.29	1:	T:
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_0)) [ ar_0 >= 2 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 1 >= ar_0 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 2 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ 1 >= ar_2 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 - 2))
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_2, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.23/1.29	
2.23/1.29		start location:	koat_start
2.23/1.29	
2.23/1.29		leaf cost:	0
2.23/1.29	
2.23/1.29	
2.23/1.29	
2.23/1.29	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.23/1.29	
2.23/1.29	2:	T:
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_0)) [ ar_0 >= 2 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 1 >= ar_0 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 2 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ 1 >= ar_2 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 - 2))
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_2, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.23/1.29	
2.23/1.29		start location:	koat_start
2.23/1.29	
2.23/1.29		leaf cost:	0
2.23/1.29	
2.23/1.29	
2.23/1.29	
2.23/1.29	A polynomial rank function with
2.23/1.29	
2.23/1.29		Pol(evalfoostart) = 2
2.23/1.29	
2.23/1.29		Pol(evalfoobb0in) = 2
2.23/1.29	
2.23/1.29		Pol(evalfoobb1in) = 2
2.23/1.29	
2.23/1.29		Pol(evalfoobb2in) = 2
2.23/1.29	
2.23/1.29		Pol(evalfoobb5in) = 1
2.23/1.29	
2.23/1.29		Pol(evalfoobb3in) = 2
2.23/1.29	
2.23/1.29		Pol(evalfoobb4in) = 2
2.23/1.29	
2.23/1.29		Pol(evalfoostop) = 0
2.23/1.29	
2.23/1.29		Pol(koat_start) = 2
2.23/1.29	
2.23/1.29	orients all transitions weakly and the transitions
2.23/1.29	
2.23/1.29		evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.23/1.29	
2.23/1.29		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 1 >= ar_0 ]
2.23/1.29	
2.23/1.29	strictly and produces the following problem:
2.23/1.29	
2.23/1.29	3:	T:
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_0)) [ ar_0 >= 2 ]
2.23/1.29	
2.23/1.29			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 1 >= ar_0 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 2 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ 1 >= ar_2 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 - 2))
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)    evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_2, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 2, Cost: 1)    evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.23/1.29	
2.23/1.29		start location:	koat_start
2.23/1.29	
2.23/1.29		leaf cost:	0
2.23/1.29	
2.23/1.29	
2.23/1.29	
2.23/1.29	A polynomial rank function with
2.23/1.29	
2.23/1.29		Pol(evalfoostart) = 3*V_2
2.23/1.29	
2.23/1.29		Pol(evalfoobb0in) = 3*V_2
2.23/1.29	
2.23/1.29		Pol(evalfoobb1in) = 3*V_1
2.23/1.29	
2.23/1.29		Pol(evalfoobb2in) = V_3 + 3
2.23/1.29	
2.23/1.29		Pol(evalfoobb5in) = 3*V_1
2.23/1.29	
2.23/1.29		Pol(evalfoobb3in) = V_3 + 1
2.23/1.29	
2.23/1.29		Pol(evalfoobb4in) = 3*V_3
2.23/1.29	
2.23/1.29		Pol(evalfoostop) = 3*V_1
2.23/1.29	
2.23/1.29		Pol(koat_start) = 3*V_2
2.23/1.29	
2.23/1.29	orients all transitions weakly and the transitions
2.23/1.29	
2.23/1.29		evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 2 ]
2.23/1.29	
2.23/1.29		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_0)) [ ar_0 >= 2 ]
2.23/1.29	
2.23/1.29	strictly and produces the following problem:
2.23/1.29	
2.23/1.29	4:	T:
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 1)         evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 1)         evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 3*ar_1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_0)) [ ar_0 >= 2 ]
2.23/1.29	
2.23/1.29			(Comp: 2, Cost: 1)         evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 1 >= ar_0 ]
2.23/1.29	
2.23/1.29			(Comp: 3*ar_1, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 2 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)         evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ 1 >= ar_2 ]
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)         evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 - 2))
2.23/1.29	
2.23/1.29			(Comp: ?, Cost: 1)         evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_2, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 2, Cost: 1)         evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 0)         koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.23/1.29	
2.23/1.29		start location:	koat_start
2.23/1.29	
2.23/1.29		leaf cost:	0
2.23/1.29	
2.23/1.29	
2.23/1.29	
2.23/1.29	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.23/1.29	
2.23/1.29	5:	T:
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 1)         evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 1)         evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 3*ar_1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_0)) [ ar_0 >= 2 ]
2.23/1.29	
2.23/1.29			(Comp: 2, Cost: 1)         evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 1 >= ar_0 ]
2.23/1.29	
2.23/1.29			(Comp: 3*ar_1, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 2 ]
2.23/1.29	
2.23/1.29			(Comp: 3*ar_1, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ 1 >= ar_2 ]
2.23/1.29	
2.23/1.29			(Comp: 3*ar_1, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 - 2))
2.23/1.29	
2.23/1.29			(Comp: 3*ar_1, Cost: 1)    evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_2, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 2, Cost: 1)         evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.23/1.29	
2.23/1.29			(Comp: 1, Cost: 0)         koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.23/1.29	
2.23/1.29		start location:	koat_start
2.23/1.29	
2.23/1.29		leaf cost:	0
2.23/1.29	
2.23/1.29	
2.23/1.29	
2.23/1.29	Complexity upper bound 15*ar_1 + 6
2.23/1.29	
2.23/1.29	
2.23/1.29	
2.23/1.29	Time: 0.067 sec (SMT: 0.060 sec)
2.23/1.29	
2.23/1.29	
2.23/1.29	----------------------------------------
2.23/1.29	
2.23/1.29	(2)
2.23/1.29	BOUNDS(1, n^1)
2.23/1.30	EOF