2.34/1.38	WORST_CASE(?, O(n^1))
2.34/1.39	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.34/1.39	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.34/1.39	
2.34/1.39	
2.34/1.39	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.34/1.39	
2.34/1.39	(0) CpxIntTrs
2.34/1.39	(1) Koat Proof [FINISHED, 184 ms]
2.34/1.39	(2) BOUNDS(1, n^1)
2.34/1.39	
2.34/1.39	
2.34/1.39	----------------------------------------
2.34/1.39	
2.34/1.39	(0)
2.34/1.39	Obligation:
2.34/1.39	Complexity Int TRS consisting of the following rules:
2.34/1.39	eval_foo_start(v_.0, v_.01, v_x, v_y, v_z) -> Com_1(eval_foo_bb0_in(v_.0, v_.01, v_x, v_y, v_z)) :|: TRUE
2.34/1.39	eval_foo_bb0_in(v_.0, v_.01, v_x, v_y, v_z) -> Com_1(eval_foo_bb1_in(v_x, v_y, v_x, v_y, v_z)) :|: TRUE
2.34/1.39	eval_foo_bb1_in(v_.0, v_.01, v_x, v_y, v_z) -> Com_1(eval_foo_bb2_in(v_.0, v_.01, v_x, v_y, v_z)) :|: v_.0 >= 0
2.34/1.39	eval_foo_bb1_in(v_.0, v_.01, v_x, v_y, v_z) -> Com_1(eval_foo_.critedge_in(v_.0, v_.01, v_x, v_y, v_z)) :|: v_.0 < 0
2.34/1.39	eval_foo_bb2_in(v_.0, v_.01, v_x, v_y, v_z) -> Com_1(eval_foo_bb3_in(v_.0, v_.01, v_x, v_y, v_z)) :|: v_.0 + v_.01 >= 0
2.34/1.39	eval_foo_bb2_in(v_.0, v_.01, v_x, v_y, v_z) -> Com_1(eval_foo_.critedge_in(v_.0, v_.01, v_x, v_y, v_z)) :|: v_.0 + v_.01 < 0
2.34/1.39	eval_foo_bb3_in(v_.0, v_.01, v_x, v_y, v_z) -> Com_1(eval_foo_bb1_in(v_.0 + v_.01 + v_z, -(v_z) - 1, v_x, v_y, v_z)) :|: TRUE
2.34/1.39	eval_foo_.critedge_in(v_.0, v_.01, v_x, v_y, v_z) -> Com_1(eval_foo_stop(v_.0, v_.01, v_x, v_y, v_z)) :|: TRUE
2.34/1.39	
2.34/1.39	The start-symbols are:[eval_foo_start_5]
2.34/1.39	
2.34/1.39	
2.34/1.39	----------------------------------------
2.34/1.39	
2.34/1.39	(1) Koat Proof (FINISHED)
2.34/1.39	YES(?, 3*ar_1 + 3*ar_3 + 12)
2.34/1.39	
2.34/1.39	
2.34/1.39	
2.34/1.39	Initial complexity problem:
2.34/1.39	
2.34/1.39	1:	T:
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= 0 ]
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 >= ar_0 + 1 ]
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 + ar_2 >= 0 ]
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 >= ar_0 + ar_2 + 1 ]
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_0 + ar_2 + ar_4, ar_1, -ar_4 - 1, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ]
2.34/1.39	
2.34/1.39		start location:	koat_start
2.34/1.39	
2.34/1.39		leaf cost:	0
2.34/1.39	
2.34/1.39	
2.34/1.39	
2.34/1.39	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.34/1.39	
2.34/1.39	2:	T:
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= 0 ]
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 >= ar_0 + 1 ]
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 + ar_2 >= 0 ]
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 >= ar_0 + ar_2 + 1 ]
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_0 + ar_2 + ar_4, ar_1, -ar_4 - 1, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ]
2.34/1.39	
2.34/1.39		start location:	koat_start
2.34/1.39	
2.34/1.39		leaf cost:	0
2.34/1.39	
2.34/1.39	
2.34/1.39	
2.34/1.39	A polynomial rank function with
2.34/1.39	
2.34/1.39		Pol(evalfoostart) = 2
2.34/1.39	
2.34/1.39		Pol(evalfoobb0in) = 2
2.34/1.39	
2.34/1.39		Pol(evalfoobb1in) = 2
2.34/1.39	
2.34/1.39		Pol(evalfoobb2in) = 2
2.34/1.39	
2.34/1.39		Pol(evalfoocritedgein) = 1
2.34/1.39	
2.34/1.39		Pol(evalfoobb3in) = 2
2.34/1.39	
2.34/1.39		Pol(evalfoostop) = 0
2.34/1.39	
2.34/1.39		Pol(koat_start) = 2
2.34/1.39	
2.34/1.39	orients all transitions weakly and the transitions
2.34/1.39	
2.34/1.39		evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4))
2.34/1.39	
2.34/1.39		evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 >= ar_0 + ar_2 + 1 ]
2.34/1.39	
2.34/1.39		evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 >= ar_0 + 1 ]
2.34/1.39	
2.34/1.39	strictly and produces the following problem:
2.34/1.39	
2.34/1.39	3:	T:
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= 0 ]
2.34/1.39	
2.34/1.39			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 >= ar_0 + 1 ]
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 + ar_2 >= 0 ]
2.34/1.39	
2.34/1.39			(Comp: 2, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 >= ar_0 + ar_2 + 1 ]
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_0 + ar_2 + ar_4, ar_1, -ar_4 - 1, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: 2, Cost: 1)    evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ]
2.34/1.39	
2.34/1.39		start location:	koat_start
2.34/1.39	
2.34/1.39		leaf cost:	0
2.34/1.39	
2.34/1.39	
2.34/1.39	
2.34/1.39	A polynomial rank function with
2.34/1.39	
2.34/1.39		Pol(evalfoostart) = V_2 + V_4 + 1
2.34/1.39	
2.34/1.39		Pol(evalfoobb0in) = V_2 + V_4 + 1
2.34/1.39	
2.34/1.39		Pol(evalfoobb1in) = V_1 + V_3 + 1
2.34/1.39	
2.34/1.39		Pol(evalfoobb2in) = V_1 + V_3 + 1
2.34/1.39	
2.34/1.39		Pol(evalfoocritedgein) = V_1 + V_3
2.34/1.39	
2.34/1.39		Pol(evalfoobb3in) = V_1 + V_3
2.34/1.39	
2.34/1.39		Pol(evalfoostop) = V_1 + V_3
2.34/1.39	
2.34/1.39		Pol(koat_start) = V_2 + V_4 + 1
2.34/1.39	
2.34/1.39	orients all transitions weakly and the transition
2.34/1.39	
2.34/1.39		evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 + ar_2 >= 0 ]
2.34/1.39	
2.34/1.39	strictly and produces the following problem:
2.34/1.39	
2.34/1.39	4:	T:
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 1)                  evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 1)                  evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= 0 ]
2.34/1.39	
2.34/1.39			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 >= ar_0 + 1 ]
2.34/1.39	
2.34/1.39			(Comp: ar_1 + ar_3 + 1, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 + ar_2 >= 0 ]
2.34/1.39	
2.34/1.39			(Comp: 2, Cost: 1)                  evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 >= ar_0 + ar_2 + 1 ]
2.34/1.39	
2.34/1.39			(Comp: ?, Cost: 1)                  evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_0 + ar_2 + ar_4, ar_1, -ar_4 - 1, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: 2, Cost: 1)                  evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ]
2.34/1.39	
2.34/1.39		start location:	koat_start
2.34/1.39	
2.34/1.39		leaf cost:	0
2.34/1.39	
2.34/1.39	
2.34/1.39	
2.34/1.39	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.34/1.39	
2.34/1.39	5:	T:
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 1)                  evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 1)                  evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: ar_1 + ar_3 + 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= 0 ]
2.34/1.39	
2.34/1.39			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 >= ar_0 + 1 ]
2.34/1.39	
2.34/1.39			(Comp: ar_1 + ar_3 + 1, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 + ar_2 >= 0 ]
2.34/1.39	
2.34/1.39			(Comp: 2, Cost: 1)                  evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 >= ar_0 + ar_2 + 1 ]
2.34/1.39	
2.34/1.39			(Comp: ar_1 + ar_3 + 1, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_0 + ar_2 + ar_4, ar_1, -ar_4 - 1, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: 2, Cost: 1)                  evalfoocritedgein(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4))
2.34/1.39	
2.34/1.39			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ]
2.34/1.39	
2.34/1.39		start location:	koat_start
2.34/1.39	
2.34/1.39		leaf cost:	0
2.34/1.39	
2.34/1.39	
2.34/1.39	
2.34/1.39	Complexity upper bound 3*ar_1 + 3*ar_3 + 12
2.34/1.39	
2.34/1.39	
2.34/1.39	
2.34/1.39	Time: 0.142 sec (SMT: 0.129 sec)
2.34/1.39	
2.34/1.39	
2.34/1.39	----------------------------------------
2.34/1.39	
2.34/1.39	(2)
2.34/1.39	BOUNDS(1, n^1)
2.34/1.42	EOF