2.12/1.16	WORST_CASE(?, O(n^1))
2.12/1.17	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.12/1.17	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.12/1.17	
2.12/1.17	
2.12/1.17	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.12/1.17	
2.12/1.17	(0) CpxIntTrs
2.12/1.17	(1) Koat Proof [FINISHED, 82 ms]
2.12/1.17	(2) BOUNDS(1, n^1)
2.12/1.17	
2.12/1.17	
2.12/1.17	----------------------------------------
2.12/1.17	
2.12/1.17	(0)
2.12/1.17	Obligation:
2.12/1.17	Complexity Int TRS consisting of the following rules:
2.12/1.17	eval_foo_start(v_.0, v_i, v_n) -> Com_1(eval_foo_bb0_in(v_.0, v_i, v_n)) :|: TRUE
2.12/1.17	eval_foo_bb0_in(v_.0, v_i, v_n) -> Com_1(eval_foo_bb1_in(v_n - 1, v_i, v_n)) :|: TRUE
2.12/1.17	eval_foo_bb1_in(v_.0, v_i, v_n) -> Com_1(eval_foo_bb2_in(v_.0, v_i, v_n)) :|: v_.0 > 1
2.12/1.17	eval_foo_bb1_in(v_.0, v_i, v_n) -> Com_1(eval_foo_bb3_in(v_.0, v_i, v_n)) :|: v_.0 <= 1
2.12/1.17	eval_foo_bb2_in(v_.0, v_i, v_n) -> Com_1(eval_foo_bb1_in(v_.0 - 1, v_i, v_n)) :|: TRUE
2.12/1.17	eval_foo_bb3_in(v_.0, v_i, v_n) -> Com_1(eval_foo_stop(v_.0, v_i, v_n)) :|: TRUE
2.12/1.17	
2.12/1.17	The start-symbols are:[eval_foo_start_3]
2.12/1.17	
2.12/1.17	
2.12/1.17	----------------------------------------
2.12/1.17	
2.12/1.17	(1) Koat Proof (FINISHED)
2.12/1.17	YES(?, 2*ar_1 + 6)
2.12/1.17	
2.12/1.17	
2.12/1.17	
2.12/1.17	Initial complexity problem:
2.12/1.17	
2.12/1.17	1:	T:
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_1 - 1, ar_1))
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 >= 2 ]
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 1 >= ar_0 ]
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1))
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.12/1.17	
2.12/1.17		start location:	koat_start
2.12/1.17	
2.12/1.17		leaf cost:	0
2.12/1.17	
2.12/1.17	
2.12/1.17	
2.12/1.17	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.12/1.17	
2.12/1.17	2:	T:
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_1 - 1, ar_1))
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 >= 2 ]
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 1 >= ar_0 ]
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1))
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.12/1.17	
2.12/1.17		start location:	koat_start
2.12/1.17	
2.12/1.17		leaf cost:	0
2.12/1.17	
2.12/1.17	
2.12/1.17	
2.12/1.17	A polynomial rank function with
2.12/1.17	
2.12/1.17		Pol(evalfoostart) = 2
2.12/1.17	
2.12/1.17		Pol(evalfoobb0in) = 2
2.12/1.17	
2.12/1.17		Pol(evalfoobb1in) = 2
2.12/1.17	
2.12/1.17		Pol(evalfoobb2in) = 2
2.12/1.17	
2.12/1.17		Pol(evalfoobb3in) = 1
2.12/1.17	
2.12/1.17		Pol(evalfoostop) = 0
2.12/1.17	
2.12/1.17		Pol(koat_start) = 2
2.12/1.17	
2.12/1.17	orients all transitions weakly and the transitions
2.12/1.17	
2.12/1.17		evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.12/1.17	
2.12/1.17		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 1 >= ar_0 ]
2.12/1.17	
2.12/1.17	strictly and produces the following problem:
2.12/1.17	
2.12/1.17	3:	T:
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_1 - 1, ar_1))
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 >= 2 ]
2.12/1.17	
2.12/1.17			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 1 >= ar_0 ]
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1))
2.12/1.17	
2.12/1.17			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.12/1.17	
2.12/1.17		start location:	koat_start
2.12/1.17	
2.12/1.17		leaf cost:	0
2.12/1.17	
2.12/1.17	
2.12/1.17	
2.12/1.17	A polynomial rank function with
2.12/1.17	
2.12/1.17		Pol(evalfoostart) = V_2
2.12/1.17	
2.12/1.17		Pol(evalfoobb0in) = V_2
2.12/1.17	
2.12/1.17		Pol(evalfoobb1in) = V_1 + 1
2.12/1.17	
2.12/1.17		Pol(evalfoobb2in) = V_1
2.12/1.17	
2.12/1.17		Pol(evalfoobb3in) = V_1
2.12/1.17	
2.12/1.17		Pol(evalfoostop) = V_1
2.12/1.17	
2.12/1.17		Pol(koat_start) = V_2
2.12/1.17	
2.12/1.17	orients all transitions weakly and the transition
2.12/1.17	
2.12/1.17		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 >= 2 ]
2.12/1.17	
2.12/1.17	strictly and produces the following problem:
2.12/1.17	
2.12/1.17	4:	T:
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 1)       evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_1 - 1, ar_1))
2.12/1.17	
2.12/1.17			(Comp: ar_1, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 >= 2 ]
2.12/1.17	
2.12/1.17			(Comp: 2, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 1 >= ar_0 ]
2.12/1.17	
2.12/1.17			(Comp: ?, Cost: 1)       evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1))
2.12/1.17	
2.12/1.17			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.12/1.17	
2.12/1.17		start location:	koat_start
2.12/1.17	
2.12/1.17		leaf cost:	0
2.12/1.17	
2.12/1.17	
2.12/1.17	
2.12/1.17	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.12/1.17	
2.12/1.17	5:	T:
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 1)       evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_1 - 1, ar_1))
2.12/1.17	
2.12/1.17			(Comp: ar_1, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 >= 2 ]
2.12/1.17	
2.12/1.17			(Comp: 2, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 1 >= ar_0 ]
2.12/1.17	
2.12/1.17			(Comp: ar_1, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1))
2.12/1.17	
2.12/1.17			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.12/1.17	
2.12/1.17			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.12/1.17	
2.12/1.17		start location:	koat_start
2.12/1.17	
2.12/1.17		leaf cost:	0
2.12/1.17	
2.12/1.17	
2.12/1.17	
2.12/1.17	Complexity upper bound 2*ar_1 + 6
2.12/1.17	
2.12/1.17	
2.12/1.17	
2.12/1.17	Time: 0.038 sec (SMT: 0.035 sec)
2.12/1.17	
2.12/1.17	
2.12/1.17	----------------------------------------
2.12/1.17	
2.12/1.17	(2)
2.12/1.17	BOUNDS(1, n^1)
2.12/1.18	EOF