2.23/1.36	WORST_CASE(?, O(n^1))
2.23/1.37	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.23/1.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.23/1.37	
2.23/1.37	
2.23/1.37	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.23/1.37	
2.23/1.37	(0) CpxIntTrs
2.23/1.37	(1) Koat Proof [FINISHED, 84 ms]
2.23/1.37	(2) BOUNDS(1, n^1)
2.23/1.37	
2.23/1.37	
2.23/1.37	----------------------------------------
2.23/1.37	
2.23/1.37	(0)
2.23/1.37	Obligation:
2.23/1.37	Complexity Int TRS consisting of the following rules:
2.23/1.37	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
2.23/1.37	eval_foo_bb0_in(v_.0, v_.01, v_x, v_y, v_z) -> Com_1(eval_foo_bb1_in(v_y, v_z, v_x, v_y, v_z)) :|: TRUE
2.23/1.37	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_x > v_.0 && v_x > v_.01
2.23/1.37	eval_foo_bb1_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_x <= v_.0
2.23/1.37	eval_foo_bb1_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_x <= v_.01
2.23/1.37	eval_foo_bb2_in(v_.0, v_.01, v_x, v_y, v_z) -> Com_1(eval_foo_bb1_in(v_.0 + 1, v_.01 + 1, v_x, v_y, v_z)) :|: TRUE
2.23/1.37	eval_foo_bb3_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
2.23/1.37	
2.23/1.37	The start-symbols are:[eval_foo_start_5]
2.23/1.37	
2.23/1.37	
2.23/1.37	----------------------------------------
2.23/1.37	
2.23/1.37	(1) Koat Proof (FINISHED)
2.23/1.37	YES(?, 2*ar_3 + 2*ar_4 + 10)
2.23/1.37	
2.23/1.37	
2.23/1.37	
2.23/1.37	Initial complexity problem:
2.23/1.37	
2.23/1.37	1:	T:
2.23/1.37	
2.23/1.37			(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))
2.23/1.37	
2.23/1.37			(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))
2.23/1.37	
2.23/1.37			(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_4 >= ar_0 + 1 /\ ar_4 >= ar_2 + 1 ]
2.23/1.37	
2.23/1.37			(Comp: ?, Cost: 1)    evalfoobb1in(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_4 ]
2.23/1.37	
2.23/1.37			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_4 ]
2.23/1.37	
2.23/1.37			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4))
2.23/1.37	
2.23/1.37			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4))
2.23/1.37	
2.23/1.37			(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 ]
2.23/1.37	
2.23/1.37		start location:	koat_start
2.23/1.37	
2.23/1.37		leaf cost:	0
2.23/1.37	
2.23/1.37	
2.23/1.37	
2.23/1.37	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.23/1.37	
2.23/1.37	2:	T:
2.23/1.37	
2.23/1.37			(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))
2.23/1.37	
2.23/1.37			(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))
2.23/1.37	
2.23/1.37			(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_4 >= ar_0 + 1 /\ ar_4 >= ar_2 + 1 ]
2.23/1.37	
2.23/1.37			(Comp: ?, Cost: 1)    evalfoobb1in(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_4 ]
2.23/1.37	
2.23/1.37			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_4 ]
2.23/1.37	
2.23/1.37			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4))
2.23/1.37	
2.23/1.37			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4))
2.23/1.37	
2.23/1.37			(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 ]
2.23/1.37	
2.23/1.37		start location:	koat_start
2.23/1.37	
2.23/1.37		leaf cost:	0
2.23/1.37	
2.23/1.37	
2.23/1.37	
2.23/1.37	A polynomial rank function with
2.23/1.37	
2.23/1.37		Pol(evalfoostart) = 2
2.23/1.37	
2.23/1.37		Pol(evalfoobb0in) = 2
2.23/1.37	
2.23/1.37		Pol(evalfoobb1in) = 2
2.23/1.37	
2.23/1.37		Pol(evalfoobb2in) = 2
2.23/1.37	
2.23/1.37		Pol(evalfoobb3in) = 1
2.23/1.37	
2.23/1.37		Pol(evalfoostop) = 0
2.23/1.37	
2.23/1.37		Pol(koat_start) = 2
2.23/1.37	
2.23/1.37	orients all transitions weakly and the transitions
2.23/1.37	
2.23/1.37		evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4))
2.23/1.37	
2.23/1.37		evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_4 ]
2.23/1.38	
2.23/1.38		evalfoobb1in(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_4 ]
2.23/1.38	
2.23/1.38	strictly and produces the following problem:
2.23/1.38	
2.23/1.38	3:	T:
2.23/1.38	
2.23/1.38			(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))
2.23/1.38	
2.23/1.38			(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))
2.23/1.38	
2.23/1.38			(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_4 >= ar_0 + 1 /\ ar_4 >= ar_2 + 1 ]
2.23/1.38	
2.23/1.38			(Comp: 2, Cost: 1)    evalfoobb1in(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_4 ]
2.23/1.38	
2.23/1.38			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_4 ]
2.23/1.38	
2.23/1.38			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4))
2.23/1.38	
2.23/1.38			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4))
2.23/1.38	
2.23/1.38			(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 ]
2.23/1.38	
2.23/1.38		start location:	koat_start
2.23/1.38	
2.23/1.38		leaf cost:	0
2.23/1.38	
2.23/1.38	
2.23/1.38	
2.23/1.38	A polynomial rank function with
2.23/1.38	
2.23/1.38		Pol(evalfoostart) = -V_4 + V_5 + 1
2.23/1.38	
2.23/1.38		Pol(evalfoobb0in) = -V_4 + V_5 + 1
2.23/1.38	
2.23/1.38		Pol(evalfoobb1in) = -V_3 + V_5 + 1
2.23/1.38	
2.23/1.38		Pol(evalfoobb2in) = -V_3 + V_5
2.23/1.38	
2.23/1.38		Pol(evalfoobb3in) = -V_3 + V_5
2.23/1.38	
2.23/1.38		Pol(evalfoostop) = -V_3 + V_5
2.23/1.38	
2.23/1.38		Pol(koat_start) = -V_4 + V_5 + 1
2.23/1.38	
2.23/1.38	orients all transitions weakly and the transition
2.23/1.38	
2.23/1.38		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_4 >= ar_0 + 1 /\ ar_4 >= ar_2 + 1 ]
2.23/1.38	
2.23/1.38	strictly and produces the following problem:
2.23/1.38	
2.23/1.38	4:	T:
2.23/1.38	
2.23/1.38			(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))
2.23/1.38	
2.23/1.38			(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))
2.23/1.38	
2.23/1.38			(Comp: ar_3 + ar_4 + 1, 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_4 >= ar_0 + 1 /\ ar_4 >= ar_2 + 1 ]
2.23/1.38	
2.23/1.38			(Comp: 2, Cost: 1)                  evalfoobb1in(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_4 ]
2.23/1.38	
2.23/1.38			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_4 ]
2.23/1.38	
2.23/1.38			(Comp: ?, Cost: 1)                  evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4))
2.23/1.38	
2.23/1.38			(Comp: 2, Cost: 1)                  evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4))
2.23/1.38	
2.23/1.38			(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 ]
2.23/1.38	
2.23/1.38		start location:	koat_start
2.23/1.38	
2.23/1.38		leaf cost:	0
2.23/1.38	
2.23/1.38	
2.23/1.38	
2.23/1.38	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.23/1.38	
2.23/1.38	5:	T:
2.23/1.38	
2.23/1.38			(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))
2.23/1.38	
2.23/1.38			(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))
2.23/1.38	
2.23/1.38			(Comp: ar_3 + ar_4 + 1, 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_4 >= ar_0 + 1 /\ ar_4 >= ar_2 + 1 ]
2.23/1.38	
2.23/1.38			(Comp: 2, Cost: 1)                  evalfoobb1in(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_4 ]
2.23/1.38	
2.23/1.38			(Comp: 2, Cost: 1)                  evalfoobb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_4 ]
2.23/1.38	
2.23/1.38			(Comp: ar_3 + ar_4 + 1, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4))
2.23/1.38	
2.23/1.38			(Comp: 2, Cost: 1)                  evalfoobb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3, ar_4))
2.23/1.38	
2.23/1.38			(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 ]
2.23/1.38	
2.23/1.38		start location:	koat_start
2.23/1.38	
2.23/1.38		leaf cost:	0
2.23/1.38	
2.23/1.38	
2.23/1.38	
2.23/1.38	Complexity upper bound 2*ar_3 + 2*ar_4 + 10
2.23/1.38	
2.23/1.38	
2.23/1.38	
2.23/1.38	Time: 0.084 sec (SMT: 0.073 sec)
2.23/1.38	
2.23/1.38	
2.23/1.38	----------------------------------------
2.23/1.38	
2.23/1.38	(2)
2.23/1.38	BOUNDS(1, n^1)
2.38/1.39	EOF