2.14/2.49	WORST_CASE(?, O(1))
2.14/2.50	proof of /export/starexec/sandbox2/output/output_files/bench.koat
2.14/2.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.14/2.50	
2.14/2.50	
2.14/2.50	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
2.14/2.50	
2.14/2.50	(0) CpxIntTrs
2.14/2.50	(1) Koat Proof [FINISHED, 45 ms]
2.14/2.50	(2) BOUNDS(1, 1)
2.14/2.50	
2.14/2.50	
2.14/2.50	----------------------------------------
2.14/2.50	
2.14/2.50	(0)
2.14/2.50	Obligation:
2.14/2.50	Complexity Int TRS consisting of the following rules:
2.14/2.50	eval_foo_start(v_.0, v_.01, v_i, v_j) -> Com_1(eval_foo_bb0_in(v_.0, v_.01, v_i, v_j)) :|: TRUE
2.14/2.50	eval_foo_bb0_in(v_.0, v_.01, v_i, v_j) -> Com_1(eval_foo_bb1_in(0, v_.01, v_i, v_j)) :|: TRUE
2.14/2.50	eval_foo_bb1_in(v_.0, v_.01, v_i, v_j) -> Com_1(eval_foo_bb2_in(v_.0, v_.01, v_i, v_j)) :|: v_.0 < 100
2.14/2.50	eval_foo_bb1_in(v_.0, v_.01, v_i, v_j) -> Com_1(eval_foo_bb3_in(v_.0, 5, v_i, v_j)) :|: v_.0 >= 100
2.14/2.50	eval_foo_bb2_in(v_.0, v_.01, v_i, v_j) -> Com_1(eval_foo_bb1_in(v_.0 + 1, v_.01, v_i, v_j)) :|: TRUE
2.14/2.50	eval_foo_bb3_in(v_.0, v_.01, v_i, v_j) -> Com_1(eval_foo_bb4_in(v_.0, v_.01, v_i, v_j)) :|: v_.01 < 21
2.14/2.50	eval_foo_bb3_in(v_.0, v_.01, v_i, v_j) -> Com_1(eval_foo_bb5_in(v_.0, v_.01, v_i, v_j)) :|: v_.01 >= 21
2.14/2.50	eval_foo_bb4_in(v_.0, v_.01, v_i, v_j) -> Com_1(eval_foo_bb3_in(v_.0, v_.01 + 3, v_i, v_j)) :|: TRUE
2.14/2.50	eval_foo_bb5_in(v_.0, v_.01, v_i, v_j) -> Com_1(eval_foo_stop(v_.0, v_.01, v_i, v_j)) :|: TRUE
2.14/2.50	
2.14/2.50	The start-symbols are:[eval_foo_start_4]
2.14/2.50	
2.14/2.50	
2.14/2.50	----------------------------------------
2.14/2.50	
2.14/2.50	(1) Koat Proof (FINISHED)
2.14/2.50	YES(?, 243)
2.14/2.50	
2.14/2.50	
2.14/2.50	
2.14/2.50	Initial complexity problem:
2.14/2.50	
2.14/2.50	1:	T:
2.14/2.50	
2.14/2.50			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.14/2.50	
2.14/2.50			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(0, ar_1))
2.14/2.50	
2.14/2.50			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 99 >= ar_0 ]
2.14/2.50	
2.14/2.50			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, 5)) [ ar_0 >= 100 ]
2.14/2.50	
2.14/2.50			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1))
2.14/2.50	
2.14/2.50			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb4in(ar_0, ar_1)) [ 20 >= ar_1 ]
2.14/2.50	
2.14/2.50			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb5in(ar_0, ar_1)) [ ar_1 >= 21 ]
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb4in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1 + 3))
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb5in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.25/2.50	
2.25/2.50		start location:	koat_start
2.25/2.50	
2.25/2.50		leaf cost:	0
2.25/2.50	
2.25/2.50	
2.25/2.50	
2.25/2.50	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.25/2.50	
2.25/2.50	2:	T:
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 99 >= ar_0 ]
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, 5)) [ ar_0 >= 100 ]
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1))
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb4in(ar_0, ar_1)) [ 20 >= ar_1 ]
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb5in(ar_0, ar_1)) [ ar_1 >= 21 ]
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb4in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1 + 3))
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb5in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.25/2.50	
2.25/2.50		start location:	koat_start
2.25/2.50	
2.25/2.50		leaf cost:	0
2.25/2.50	
2.25/2.50	
2.25/2.50	
2.25/2.50	A polynomial rank function with
2.25/2.50	
2.25/2.50		Pol(evalfoostart) = 3
2.25/2.50	
2.25/2.50		Pol(evalfoobb0in) = 3
2.25/2.50	
2.25/2.50		Pol(evalfoobb1in) = 3
2.25/2.50	
2.25/2.50		Pol(evalfoobb2in) = 3
2.25/2.50	
2.25/2.50		Pol(evalfoobb3in) = 2
2.25/2.50	
2.25/2.50		Pol(evalfoobb4in) = 2
2.25/2.50	
2.25/2.50		Pol(evalfoobb5in) = 1
2.25/2.50	
2.25/2.50		Pol(evalfoostop) = 0
2.25/2.50	
2.25/2.50		Pol(koat_start) = 3
2.25/2.50	
2.25/2.50	orients all transitions weakly and the transitions
2.25/2.50	
2.25/2.50		evalfoobb5in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.25/2.50	
2.25/2.50		evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb5in(ar_0, ar_1)) [ ar_1 >= 21 ]
2.25/2.50	
2.25/2.50		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, 5)) [ ar_0 >= 100 ]
2.25/2.50	
2.25/2.50	strictly and produces the following problem:
2.25/2.50	
2.25/2.50	3:	T:
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 99 >= ar_0 ]
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, 5)) [ ar_0 >= 100 ]
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1))
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb4in(ar_0, ar_1)) [ 20 >= ar_1 ]
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb5in(ar_0, ar_1)) [ ar_1 >= 21 ]
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)    evalfoobb4in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1 + 3))
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)    evalfoobb5in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.25/2.50	
2.25/2.50		start location:	koat_start
2.25/2.50	
2.25/2.50		leaf cost:	0
2.25/2.50	
2.25/2.50	
2.25/2.50	
2.25/2.50	A polynomial rank function with
2.25/2.50	
2.25/2.50		Pol(evalfoostart) = 100
2.25/2.50	
2.25/2.50		Pol(evalfoobb0in) = 100
2.25/2.50	
2.25/2.50		Pol(evalfoobb1in) = -V_1 + 100
2.25/2.50	
2.25/2.50		Pol(evalfoobb2in) = -V_1 + 99
2.25/2.50	
2.25/2.50		Pol(evalfoobb3in) = -V_1 - 6*V_2 + 121
2.25/2.50	
2.25/2.50		Pol(evalfoobb4in) = -V_1 - 6*V_2 + 103
2.25/2.50	
2.25/2.50		Pol(evalfoobb5in) = -V_1 - 6*V_2
2.25/2.50	
2.25/2.50		Pol(evalfoostop) = -V_1 - 6*V_2
2.25/2.50	
2.25/2.50		Pol(koat_start) = 100
2.25/2.50	
2.25/2.50	orients all transitions weakly and the transition
2.25/2.50	
2.25/2.50		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 99 >= ar_0 ]
2.25/2.50	
2.25/2.50	strictly and produces the following problem:
2.25/2.50	
2.25/2.50	4:	T:
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 1)      evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 1)      evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 100, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 99 >= ar_0 ]
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)      evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, 5)) [ ar_0 >= 100 ]
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)      evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1))
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)      evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb4in(ar_0, ar_1)) [ 20 >= ar_1 ]
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)      evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb5in(ar_0, ar_1)) [ ar_1 >= 21 ]
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)      evalfoobb4in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1 + 3))
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)      evalfoobb5in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 0)      koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.25/2.50	
2.25/2.50		start location:	koat_start
2.25/2.50	
2.25/2.50		leaf cost:	0
2.25/2.50	
2.25/2.50	
2.25/2.50	
2.25/2.50	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.25/2.50	
2.25/2.50	5:	T:
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 1)      evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 1)      evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 100, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 99 >= ar_0 ]
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)      evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, 5)) [ ar_0 >= 100 ]
2.25/2.50	
2.25/2.50			(Comp: 100, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1))
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)      evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb4in(ar_0, ar_1)) [ 20 >= ar_1 ]
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)      evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb5in(ar_0, ar_1)) [ ar_1 >= 21 ]
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)      evalfoobb4in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1 + 3))
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)      evalfoobb5in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 0)      koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.25/2.50	
2.25/2.50		start location:	koat_start
2.25/2.50	
2.25/2.50		leaf cost:	0
2.25/2.50	
2.25/2.50	
2.25/2.50	
2.25/2.50	A polynomial rank function with
2.25/2.50	
2.25/2.50		Pol(evalfoostart) = 16
2.25/2.50	
2.25/2.50		Pol(evalfoobb0in) = 16
2.25/2.50	
2.25/2.50		Pol(evalfoobb1in) = 16
2.25/2.50	
2.25/2.50		Pol(evalfoobb2in) = 16
2.25/2.50	
2.25/2.50		Pol(evalfoobb3in) = -V_2 + 21
2.25/2.50	
2.25/2.50		Pol(evalfoobb4in) = -V_2 + 18
2.25/2.50	
2.25/2.50		Pol(evalfoobb5in) = -V_2
2.25/2.50	
2.25/2.50		Pol(evalfoostop) = -V_2
2.25/2.50	
2.25/2.50		Pol(koat_start) = 16
2.25/2.50	
2.25/2.50	orients all transitions weakly and the transition
2.25/2.50	
2.25/2.50		evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb4in(ar_0, ar_1)) [ 20 >= ar_1 ]
2.25/2.50	
2.25/2.50	strictly and produces the following problem:
2.25/2.50	
2.25/2.50	6:	T:
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 1)      evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 1)      evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 100, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 99 >= ar_0 ]
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)      evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, 5)) [ ar_0 >= 100 ]
2.25/2.50	
2.25/2.50			(Comp: 100, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 16, Cost: 1)     evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb4in(ar_0, ar_1)) [ 20 >= ar_1 ]
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)      evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb5in(ar_0, ar_1)) [ ar_1 >= 21 ]
2.25/2.50	
2.25/2.50			(Comp: ?, Cost: 1)      evalfoobb4in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1 + 3))
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)      evalfoobb5in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 0)      koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.25/2.50	
2.25/2.50		start location:	koat_start
2.25/2.50	
2.25/2.50		leaf cost:	0
2.25/2.50	
2.25/2.50	
2.25/2.50	
2.25/2.50	Repeatedly propagating knowledge in problem 6 produces the following problem:
2.25/2.50	
2.25/2.50	7:	T:
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 1)      evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 1)      evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 100, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 99 >= ar_0 ]
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)      evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, 5)) [ ar_0 >= 100 ]
2.25/2.50	
2.25/2.50			(Comp: 100, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 + 1, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 16, Cost: 1)     evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb4in(ar_0, ar_1)) [ 20 >= ar_1 ]
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)      evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoobb5in(ar_0, ar_1)) [ ar_1 >= 21 ]
2.25/2.50	
2.25/2.50			(Comp: 16, Cost: 1)     evalfoobb4in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1 + 3))
2.25/2.50	
2.25/2.50			(Comp: 3, Cost: 1)      evalfoobb5in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.25/2.50	
2.25/2.50			(Comp: 1, Cost: 0)      koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.25/2.50	
2.25/2.50		start location:	koat_start
2.25/2.50	
2.25/2.50		leaf cost:	0
2.25/2.50	
2.25/2.50	
2.25/2.50	
2.25/2.50	Complexity upper bound 243
2.25/2.50	
2.25/2.50	
2.25/2.50	
2.25/2.50	Time: 0.071 sec (SMT: 0.063 sec)
2.25/2.50	
2.25/2.50	
2.25/2.50	----------------------------------------
2.25/2.50	
2.25/2.50	(2)
2.25/2.50	BOUNDS(1, 1)
2.25/2.52	EOF