2.20/1.25	WORST_CASE(?, O(n^1))
2.20/1.26	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.20/1.26	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.20/1.26	
2.20/1.26	
2.20/1.26	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.20/1.26	
2.20/1.26	(0) CpxIntTrs
2.20/1.26	(1) Koat Proof [FINISHED, 35 ms]
2.20/1.26	(2) BOUNDS(1, n^1)
2.20/1.26	
2.20/1.26	
2.20/1.26	----------------------------------------
2.20/1.26	
2.20/1.26	(0)
2.20/1.26	Obligation:
2.20/1.26	Complexity Int TRS consisting of the following rules:
2.20/1.26	eval_foo_start(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb0_in(v_.0, v_.01, v_x, v_y)) :|: TRUE
2.20/1.26	eval_foo_bb0_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_x, v_y, v_x, v_y)) :|: TRUE
2.20/1.26	eval_foo_bb1_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.0, v_.01, v_x, v_y)) :|: v_.0 > 0 && v_.01 > 0
2.20/1.26	eval_foo_bb1_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_.01, v_x, v_y)) :|: v_.0 <= 0
2.20/1.26	eval_foo_bb1_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_.01, v_x, v_y)) :|: v_.01 <= 0
2.20/1.26	eval_foo_bb2_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.0 - 1, v_.01 - 1, v_x, v_y)) :|: TRUE
2.20/1.26	eval_foo_bb3_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_stop(v_.0, v_.01, v_x, v_y)) :|: TRUE
2.20/1.26	
2.20/1.26	The start-symbols are:[eval_foo_start_4]
2.20/1.26	
2.20/1.26	
2.20/1.26	----------------------------------------
2.20/1.26	
2.20/1.26	(1) Koat Proof (FINISHED)
2.20/1.26	YES(?, 2*ar_3 + 10)
2.20/1.26	
2.20/1.26	
2.20/1.26	
2.20/1.26	Initial complexity problem:
2.20/1.26	
2.20/1.26	1:	T:
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3))
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 1 /\ ar_2 >= 1 ]
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 ]
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 ]
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2 - 1, ar_3))
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.20/1.26	
2.20/1.26		start location:	koat_start
2.20/1.26	
2.20/1.26		leaf cost:	0
2.20/1.26	
2.20/1.26	
2.20/1.26	
2.20/1.26	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.20/1.26	
2.20/1.26	2:	T:
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3))
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 1 /\ ar_2 >= 1 ]
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 ]
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 ]
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2 - 1, ar_3))
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.20/1.26	
2.20/1.26		start location:	koat_start
2.20/1.26	
2.20/1.26		leaf cost:	0
2.20/1.26	
2.20/1.26	
2.20/1.26	
2.20/1.26	A polynomial rank function with
2.20/1.26	
2.20/1.26		Pol(evalfoostart) = 2
2.20/1.26	
2.20/1.26		Pol(evalfoobb0in) = 2
2.20/1.26	
2.20/1.26		Pol(evalfoobb1in) = 2
2.20/1.26	
2.20/1.26		Pol(evalfoobb2in) = 2
2.20/1.26	
2.20/1.26		Pol(evalfoobb3in) = 1
2.20/1.26	
2.20/1.26		Pol(evalfoostop) = 0
2.20/1.26	
2.20/1.26		Pol(koat_start) = 2
2.20/1.26	
2.20/1.26	orients all transitions weakly and the transitions
2.20/1.26	
2.20/1.26		evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.20/1.26	
2.20/1.26		evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 ]
2.20/1.26	
2.20/1.26		evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 ]
2.20/1.26	
2.20/1.26	strictly and produces the following problem:
2.20/1.26	
2.20/1.26	3:	T:
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3))
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 1 /\ ar_2 >= 1 ]
2.20/1.26	
2.20/1.26			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 ]
2.20/1.26	
2.20/1.26			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 ]
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2 - 1, ar_3))
2.20/1.26	
2.20/1.26			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.20/1.26	
2.20/1.26		start location:	koat_start
2.20/1.26	
2.20/1.26		leaf cost:	0
2.20/1.26	
2.20/1.26	
2.20/1.26	
2.20/1.26	A polynomial rank function with
2.20/1.26	
2.20/1.26		Pol(evalfoostart) = V_4 + 1
2.20/1.26	
2.20/1.26		Pol(evalfoobb0in) = V_4 + 1
2.20/1.26	
2.20/1.26		Pol(evalfoobb1in) = V_3 + 1
2.20/1.26	
2.20/1.26		Pol(evalfoobb2in) = V_3
2.20/1.26	
2.20/1.26		Pol(evalfoobb3in) = V_3
2.20/1.26	
2.20/1.26		Pol(evalfoostop) = V_3
2.20/1.26	
2.20/1.26		Pol(koat_start) = V_4 + 1
2.20/1.26	
2.20/1.26	orients all transitions weakly and the transition
2.20/1.26	
2.20/1.26		evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 1 /\ ar_2 >= 1 ]
2.20/1.26	
2.20/1.26	strictly and produces the following problem:
2.20/1.26	
2.20/1.26	4:	T:
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 1)           evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 1)           evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3))
2.20/1.26	
2.20/1.26			(Comp: ar_3 + 1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 1 /\ ar_2 >= 1 ]
2.20/1.26	
2.20/1.26			(Comp: 2, Cost: 1)           evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 ]
2.20/1.26	
2.20/1.26			(Comp: 2, Cost: 1)           evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 ]
2.20/1.26	
2.20/1.26			(Comp: ?, Cost: 1)           evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2 - 1, ar_3))
2.20/1.26	
2.20/1.26			(Comp: 2, Cost: 1)           evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.20/1.26	
2.20/1.26		start location:	koat_start
2.20/1.26	
2.20/1.26		leaf cost:	0
2.20/1.26	
2.20/1.26	
2.20/1.26	
2.20/1.26	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.20/1.26	
2.20/1.26	5:	T:
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 1)           evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 1)           evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3))
2.20/1.26	
2.20/1.26			(Comp: ar_3 + 1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 1 /\ ar_2 >= 1 ]
2.20/1.26	
2.20/1.26			(Comp: 2, Cost: 1)           evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 ]
2.20/1.26	
2.20/1.26			(Comp: 2, Cost: 1)           evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 ]
2.20/1.26	
2.20/1.26			(Comp: ar_3 + 1, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2 - 1, ar_3))
2.20/1.26	
2.20/1.26			(Comp: 2, Cost: 1)           evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.20/1.26	
2.20/1.26			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.20/1.26	
2.20/1.26		start location:	koat_start
2.20/1.26	
2.20/1.26		leaf cost:	0
2.20/1.26	
2.20/1.26	
2.20/1.26	
2.20/1.26	Complexity upper bound 2*ar_3 + 10
2.20/1.26	
2.20/1.26	
2.20/1.26	
2.20/1.26	Time: 0.071 sec (SMT: 0.063 sec)
2.20/1.26	
2.20/1.26	
2.20/1.26	----------------------------------------
2.20/1.26	
2.20/1.26	(2)
2.20/1.26	BOUNDS(1, n^1)
2.20/1.28	EOF