2.10/1.32	WORST_CASE(?, O(n^1))
2.23/1.33	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.23/1.33	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.23/1.33	
2.23/1.33	
2.23/1.33	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.23/1.33	
2.23/1.33	(0) CpxIntTrs
2.23/1.33	(1) Koat Proof [FINISHED, 76 ms]
2.23/1.33	(2) BOUNDS(1, n^1)
2.23/1.33	
2.23/1.33	
2.23/1.33	----------------------------------------
2.23/1.33	
2.23/1.33	(0)
2.23/1.33	Obligation:
2.23/1.33	Complexity Int TRS consisting of the following rules:
2.23/1.33	eval_foo_start(v_.0, v_.01, v_.02, v_ell, v_i, v_j, v_k) -> Com_1(eval_foo_bb0_in(v_.0, v_.01, v_.02, v_ell, v_i, v_j, v_k)) :|: TRUE
2.23/1.33	eval_foo_bb0_in(v_.0, v_.01, v_.02, v_ell, v_i, v_j, v_k) -> Com_1(eval_foo_bb1_in(v_i, v_j, v_k, v_ell, v_i, v_j, v_k)) :|: TRUE
2.23/1.33	eval_foo_bb1_in(v_.0, v_.01, v_.02, v_ell, v_i, v_j, v_k) -> Com_1(eval_foo_bb2_in(v_.0, v_.01, v_.02, v_ell, v_i, v_j, v_k)) :|: v_.0 <= 100 && v_.01 <= v_.02
2.23/1.33	eval_foo_bb1_in(v_.0, v_.01, v_.02, v_ell, v_i, v_j, v_k) -> Com_1(eval_foo_bb3_in(v_.0, v_.01, v_.02, v_ell, v_i, v_j, v_k)) :|: v_.0 > 100
2.23/1.33	eval_foo_bb1_in(v_.0, v_.01, v_.02, v_ell, v_i, v_j, v_k) -> Com_1(eval_foo_bb3_in(v_.0, v_.01, v_.02, v_ell, v_i, v_j, v_k)) :|: v_.01 > v_.02
2.23/1.33	eval_foo_bb2_in(v_.0, v_.01, v_.02, v_ell, v_i, v_j, v_k) -> Com_1(eval_foo_bb1_in(v_.01, v_.0 + 1, v_.02 - 1, v_ell, v_i, v_j, v_k)) :|: TRUE
2.23/1.33	eval_foo_bb3_in(v_.0, v_.01, v_.02, v_ell, v_i, v_j, v_k) -> Com_1(eval_foo_stop(v_.0, v_.01, v_.02, v_ell, v_i, v_j, v_k)) :|: TRUE
2.23/1.33	
2.23/1.33	The start-symbols are:[eval_foo_start_7]
2.23/1.33	
2.23/1.33	
2.23/1.33	----------------------------------------
2.23/1.33	
2.23/1.33	(1) Koat Proof (FINISHED)
2.23/1.33	YES(?, 2*ar_1 + 2*ar_3 + 2*ar_5 + 210)
2.23/1.33	
2.23/1.33	
2.23/1.33	
2.23/1.33	Initial complexity problem:
2.23/1.33	
2.23/1.33	1:	T:
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_5, ar_5))
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 100 >= ar_0 /\ ar_4 >= ar_2 ]
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 101 ]
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_4 + 1 ]
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_2, ar_1, ar_0 + 1, ar_3, ar_4 - 1, ar_5))
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.23/1.33	
2.23/1.33		start location:	koat_start
2.23/1.33	
2.23/1.33		leaf cost:	0
2.23/1.33	
2.23/1.33	
2.23/1.33	
2.23/1.33	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.23/1.33	
2.23/1.33	2:	T:
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_5, ar_5))
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 100 >= ar_0 /\ ar_4 >= ar_2 ]
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 101 ]
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_4 + 1 ]
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_2, ar_1, ar_0 + 1, ar_3, ar_4 - 1, ar_5))
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.23/1.33	
2.23/1.33		start location:	koat_start
2.23/1.33	
2.23/1.33		leaf cost:	0
2.23/1.33	
2.23/1.33	
2.23/1.33	
2.23/1.33	A polynomial rank function with
2.23/1.33	
2.23/1.33		Pol(evalfoostart) = 2
2.23/1.33	
2.23/1.33		Pol(evalfoobb0in) = 2
2.23/1.33	
2.23/1.33		Pol(evalfoobb1in) = 2
2.23/1.33	
2.23/1.33		Pol(evalfoobb2in) = 2
2.23/1.33	
2.23/1.33		Pol(evalfoobb3in) = 1
2.23/1.33	
2.23/1.33		Pol(evalfoostop) = 0
2.23/1.33	
2.23/1.33		Pol(koat_start) = 2
2.23/1.33	
2.23/1.33	orients all transitions weakly and the transitions
2.23/1.33	
2.23/1.33		evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.23/1.33	
2.23/1.33		evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_4 + 1 ]
2.23/1.33	
2.23/1.33		evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 101 ]
2.23/1.33	
2.23/1.33	strictly and produces the following problem:
2.23/1.33	
2.23/1.33	3:	T:
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_5, ar_5))
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 100 >= ar_0 /\ ar_4 >= ar_2 ]
2.23/1.33	
2.23/1.33			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 101 ]
2.23/1.33	
2.23/1.33			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_4 + 1 ]
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_2, ar_1, ar_0 + 1, ar_3, ar_4 - 1, ar_5))
2.23/1.33	
2.23/1.33			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.23/1.33	
2.23/1.33		start location:	koat_start
2.23/1.33	
2.23/1.33		leaf cost:	0
2.23/1.33	
2.23/1.33	
2.23/1.33	
2.23/1.33	A polynomial rank function with
2.23/1.33	
2.23/1.33		Pol(evalfoostart) = -V_2 - V_4 + V_6 + 101
2.23/1.33	
2.23/1.33		Pol(evalfoobb0in) = -V_2 - V_4 + V_6 + 101
2.23/1.33	
2.23/1.33		Pol(evalfoobb1in) = -V_1 - V_3 + V_5 + 101
2.23/1.33	
2.23/1.33		Pol(evalfoobb2in) = -V_1 - V_3 + V_5 + 99
2.23/1.33	
2.23/1.33		Pol(evalfoobb3in) = -V_1 - V_3 + V_5
2.23/1.33	
2.23/1.33		Pol(evalfoostop) = -V_1 - V_3 + V_5
2.23/1.33	
2.23/1.33		Pol(koat_start) = -V_2 - V_4 + V_6 + 101
2.23/1.33	
2.23/1.33	orients all transitions weakly and the transition
2.23/1.33	
2.23/1.33		evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 100 >= ar_0 /\ ar_4 >= ar_2 ]
2.23/1.33	
2.23/1.33	strictly and produces the following problem:
2.23/1.33	
2.23/1.33	4:	T:
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 1)                           evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 1)                           evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_5, ar_5))
2.23/1.33	
2.23/1.33			(Comp: ar_1 + ar_3 + ar_5 + 101, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 100 >= ar_0 /\ ar_4 >= ar_2 ]
2.23/1.33	
2.23/1.33			(Comp: 2, Cost: 1)                           evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 101 ]
2.23/1.33	
2.23/1.33			(Comp: 2, Cost: 1)                           evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_4 + 1 ]
2.23/1.33	
2.23/1.33			(Comp: ?, Cost: 1)                           evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_2, ar_1, ar_0 + 1, ar_3, ar_4 - 1, ar_5))
2.23/1.33	
2.23/1.33			(Comp: 2, Cost: 1)                           evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 0)                           koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.23/1.33	
2.23/1.33		start location:	koat_start
2.23/1.33	
2.23/1.33		leaf cost:	0
2.23/1.33	
2.23/1.33	
2.23/1.33	
2.23/1.33	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.23/1.33	
2.23/1.33	5:	T:
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 1)                           evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 1)                           evalfoobb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_3, ar_3, ar_5, ar_5))
2.23/1.33	
2.23/1.33			(Comp: ar_1 + ar_3 + ar_5 + 101, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 100 >= ar_0 /\ ar_4 >= ar_2 ]
2.23/1.33	
2.23/1.33			(Comp: 2, Cost: 1)                           evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= 101 ]
2.23/1.33	
2.23/1.33			(Comp: 2, Cost: 1)                           evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_4 + 1 ]
2.23/1.33	
2.23/1.33			(Comp: ar_1 + ar_3 + ar_5 + 101, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoobb1in(ar_2, ar_1, ar_0 + 1, ar_3, ar_4 - 1, ar_5))
2.23/1.33	
2.23/1.33			(Comp: 2, Cost: 1)                           evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5))
2.23/1.33	
2.23/1.33			(Comp: 1, Cost: 0)                           koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
2.23/1.33	
2.23/1.33		start location:	koat_start
2.23/1.33	
2.23/1.33		leaf cost:	0
2.23/1.33	
2.23/1.33	
2.23/1.33	
2.23/1.33	Complexity upper bound 2*ar_1 + 2*ar_3 + 2*ar_5 + 210
2.23/1.33	
2.23/1.33	
2.23/1.33	
2.23/1.33	Time: 0.094 sec (SMT: 0.080 sec)
2.23/1.33	
2.23/1.33	
2.23/1.33	----------------------------------------
2.23/1.33	
2.23/1.33	(2)
2.23/1.33	BOUNDS(1, n^1)
2.23/1.34	EOF