2.13/1.22	WORST_CASE(?, O(1))
2.13/1.23	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.13/1.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.13/1.23	
2.13/1.23	
2.13/1.23	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
2.13/1.23	
2.13/1.23	(0) CpxIntTrs
2.13/1.23	(1) Koat Proof [FINISHED, 74 ms]
2.13/1.23	(2) BOUNDS(1, 1)
2.13/1.23	
2.13/1.23	
2.13/1.23	----------------------------------------
2.13/1.23	
2.13/1.23	(0)
2.13/1.23	Obligation:
2.13/1.23	Complexity Int TRS consisting of the following rules:
2.13/1.23	eval_foo_start(v_.0, v_c, v_i) -> Com_1(eval_foo_bb0_in(v_.0, v_c, v_i)) :|: TRUE
2.13/1.23	eval_foo_bb0_in(v_.0, v_c, v_i) -> Com_1(eval_foo_bb1_in(0, v_c, v_i)) :|: TRUE
2.13/1.23	eval_foo_bb1_in(v_.0, v_c, v_i) -> Com_1(eval_foo_bb2_in(v_.0, v_c, v_i)) :|: v_.0 < 20
2.13/1.23	eval_foo_bb1_in(v_.0, v_c, v_i) -> Com_1(eval_foo_bb3_in(v_.0, v_c, v_i)) :|: v_.0 >= 20
2.13/1.23	eval_foo_bb2_in(v_.0, v_c, v_i) -> Com_1(eval_foo_bb1_in(v_.0 + 1, v_c, v_i)) :|: v_.0 + 1 <= 10
2.13/1.23	eval_foo_bb2_in(v_.0, v_c, v_i) -> Com_1(eval_foo_bb1_in(v_.0 + 1, v_c, v_i)) :|: v_.0 + 1 > 10
2.13/1.23	eval_foo_bb3_in(v_.0, v_c, v_i) -> Com_1(eval_foo_stop(v_.0, v_c, v_i)) :|: TRUE
2.13/1.23	
2.13/1.23	The start-symbols are:[eval_foo_start_3]
2.13/1.23	
2.13/1.23	
2.13/1.23	----------------------------------------
2.13/1.23	
2.13/1.23	(1) Koat Proof (FINISHED)
2.13/1.23	YES(?, 123)
2.13/1.23	
2.13/1.23	
2.13/1.23	
2.13/1.23	Initial complexity problem:
2.13/1.23	
2.13/1.23	1:	T:
2.13/1.23	
2.13/1.23			(Comp: ?, Cost: 1)    evalfoostart(ar_0) -> Com_1(evalfoobb0in(ar_0))
2.13/1.23	
2.13/1.23			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0) -> Com_1(evalfoobb1in(0))
2.13/1.23	
2.13/1.23			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0) -> Com_1(evalfoobb2in(ar_0)) [ 19 >= ar_0 ]
2.13/1.23	
2.13/1.23			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0) -> Com_1(evalfoobb3in(ar_0)) [ ar_0 >= 20 ]
2.13/1.23	
2.13/1.23			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0) -> Com_1(evalfoobb1in(ar_0 + 1)) [ 9 >= ar_0 ]
2.13/1.23	
2.13/1.23			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0) -> Com_1(evalfoobb1in(ar_0 + 1)) [ ar_0 >= 10 ]
2.13/1.23	
2.13/1.23			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0) -> Com_1(evalfoostop(ar_0))
2.13/1.23	
2.13/1.23			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(evalfoostart(ar_0)) [ 0 <= 0 ]
2.13/1.23	
2.13/1.23		start location:	koat_start
2.13/1.23	
2.13/1.23		leaf cost:	0
2.13/1.23	
2.13/1.23	
2.13/1.23	
2.13/1.23	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.13/1.23	
2.13/1.23	2:	T:
2.13/1.23	
2.13/1.23			(Comp: 1, Cost: 1)    evalfoostart(ar_0) -> Com_1(evalfoobb0in(ar_0))
2.13/1.23	
2.13/1.23			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0) -> Com_1(evalfoobb1in(0))
2.13/1.23	
2.13/1.23			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0) -> Com_1(evalfoobb2in(ar_0)) [ 19 >= ar_0 ]
2.13/1.23	
2.13/1.23			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0) -> Com_1(evalfoobb3in(ar_0)) [ ar_0 >= 20 ]
2.13/1.23	
2.13/1.23			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0) -> Com_1(evalfoobb1in(ar_0 + 1)) [ 9 >= ar_0 ]
2.13/1.23	
2.13/1.23			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0) -> Com_1(evalfoobb1in(ar_0 + 1)) [ ar_0 >= 10 ]
2.13/1.23	
2.13/1.23			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0) -> Com_1(evalfoostop(ar_0))
2.13/1.23	
2.13/1.23			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(evalfoostart(ar_0)) [ 0 <= 0 ]
2.13/1.23	
2.13/1.23		start location:	koat_start
2.13/1.23	
2.13/1.23		leaf cost:	0
2.13/1.23	
2.13/1.23	
2.13/1.23	
2.13/1.23	A polynomial rank function with
2.13/1.23	
2.13/1.23		Pol(evalfoostart) = 2
2.13/1.23	
2.13/1.23		Pol(evalfoobb0in) = 2
2.13/1.23	
2.13/1.23		Pol(evalfoobb1in) = 2
2.13/1.23	
2.13/1.23		Pol(evalfoobb2in) = 2
2.13/1.23	
2.13/1.23		Pol(evalfoobb3in) = 1
2.13/1.23	
2.13/1.23		Pol(evalfoostop) = 0
2.13/1.23	
2.13/1.23		Pol(koat_start) = 2
2.13/1.23	
2.13/1.23	orients all transitions weakly and the transitions
2.13/1.23	
2.13/1.23		evalfoobb3in(ar_0) -> Com_1(evalfoostop(ar_0))
2.13/1.23	
2.13/1.23		evalfoobb1in(ar_0) -> Com_1(evalfoobb3in(ar_0)) [ ar_0 >= 20 ]
2.13/1.23	
2.13/1.23	strictly and produces the following problem:
2.19/1.23	
2.19/1.23	3:	T:
2.19/1.23	
2.19/1.23			(Comp: 1, Cost: 1)    evalfoostart(ar_0) -> Com_1(evalfoobb0in(ar_0))
2.19/1.23	
2.19/1.23			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0) -> Com_1(evalfoobb1in(0))
2.19/1.23	
2.19/1.23			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0) -> Com_1(evalfoobb2in(ar_0)) [ 19 >= ar_0 ]
2.19/1.23	
2.19/1.23			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0) -> Com_1(evalfoobb3in(ar_0)) [ ar_0 >= 20 ]
2.19/1.23	
2.19/1.23			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0) -> Com_1(evalfoobb1in(ar_0 + 1)) [ 9 >= ar_0 ]
2.19/1.23	
2.19/1.23			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0) -> Com_1(evalfoobb1in(ar_0 + 1)) [ ar_0 >= 10 ]
2.19/1.23	
2.19/1.23			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0) -> Com_1(evalfoostop(ar_0))
2.19/1.23	
2.19/1.23			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(evalfoostart(ar_0)) [ 0 <= 0 ]
2.19/1.23	
2.19/1.23		start location:	koat_start
2.19/1.23	
2.19/1.23		leaf cost:	0
2.19/1.23	
2.19/1.23	
2.19/1.23	
2.19/1.23	A polynomial rank function with
2.19/1.23	
2.19/1.23		Pol(evalfoostart) = 39
2.19/1.23	
2.19/1.23		Pol(evalfoobb0in) = 39
2.19/1.23	
2.19/1.23		Pol(evalfoobb1in) = -2*V_1 + 39
2.19/1.23	
2.19/1.23		Pol(evalfoobb2in) = -2*V_1 + 38
2.19/1.23	
2.19/1.23		Pol(evalfoobb3in) = -2*V_1
2.19/1.23	
2.19/1.23		Pol(evalfoostop) = -2*V_1
2.19/1.23	
2.19/1.23		Pol(koat_start) = 39
2.19/1.23	
2.19/1.23	orients all transitions weakly and the transitions
2.19/1.23	
2.19/1.23		evalfoobb2in(ar_0) -> Com_1(evalfoobb1in(ar_0 + 1)) [ 9 >= ar_0 ]
2.19/1.23	
2.19/1.23		evalfoobb1in(ar_0) -> Com_1(evalfoobb2in(ar_0)) [ 19 >= ar_0 ]
2.19/1.23	
2.19/1.23	strictly and produces the following problem:
2.19/1.23	
2.19/1.23	4:	T:
2.19/1.23	
2.19/1.23			(Comp: 1, Cost: 1)     evalfoostart(ar_0) -> Com_1(evalfoobb0in(ar_0))
2.19/1.23	
2.19/1.23			(Comp: 1, Cost: 1)     evalfoobb0in(ar_0) -> Com_1(evalfoobb1in(0))
2.19/1.23	
2.19/1.23			(Comp: 39, Cost: 1)    evalfoobb1in(ar_0) -> Com_1(evalfoobb2in(ar_0)) [ 19 >= ar_0 ]
2.19/1.23	
2.19/1.23			(Comp: 2, Cost: 1)     evalfoobb1in(ar_0) -> Com_1(evalfoobb3in(ar_0)) [ ar_0 >= 20 ]
2.19/1.23	
2.19/1.23			(Comp: 39, Cost: 1)    evalfoobb2in(ar_0) -> Com_1(evalfoobb1in(ar_0 + 1)) [ 9 >= ar_0 ]
2.19/1.23	
2.19/1.23			(Comp: ?, Cost: 1)     evalfoobb2in(ar_0) -> Com_1(evalfoobb1in(ar_0 + 1)) [ ar_0 >= 10 ]
2.19/1.23	
2.19/1.23			(Comp: 2, Cost: 1)     evalfoobb3in(ar_0) -> Com_1(evalfoostop(ar_0))
2.19/1.23	
2.19/1.23			(Comp: 1, Cost: 0)     koat_start(ar_0) -> Com_1(evalfoostart(ar_0)) [ 0 <= 0 ]
2.19/1.23	
2.19/1.23		start location:	koat_start
2.19/1.23	
2.19/1.23		leaf cost:	0
2.19/1.23	
2.19/1.23	
2.19/1.23	
2.19/1.23	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.19/1.23	
2.19/1.23	5:	T:
2.19/1.23	
2.19/1.23			(Comp: 1, Cost: 1)     evalfoostart(ar_0) -> Com_1(evalfoobb0in(ar_0))
2.19/1.23	
2.19/1.23			(Comp: 1, Cost: 1)     evalfoobb0in(ar_0) -> Com_1(evalfoobb1in(0))
2.19/1.23	
2.19/1.23			(Comp: 39, Cost: 1)    evalfoobb1in(ar_0) -> Com_1(evalfoobb2in(ar_0)) [ 19 >= ar_0 ]
2.19/1.23	
2.19/1.23			(Comp: 2, Cost: 1)     evalfoobb1in(ar_0) -> Com_1(evalfoobb3in(ar_0)) [ ar_0 >= 20 ]
2.19/1.23	
2.19/1.23			(Comp: 39, Cost: 1)    evalfoobb2in(ar_0) -> Com_1(evalfoobb1in(ar_0 + 1)) [ 9 >= ar_0 ]
2.19/1.23	
2.19/1.23			(Comp: 39, Cost: 1)    evalfoobb2in(ar_0) -> Com_1(evalfoobb1in(ar_0 + 1)) [ ar_0 >= 10 ]
2.19/1.23	
2.19/1.23			(Comp: 2, Cost: 1)     evalfoobb3in(ar_0) -> Com_1(evalfoostop(ar_0))
2.19/1.23	
2.19/1.23			(Comp: 1, Cost: 0)     koat_start(ar_0) -> Com_1(evalfoostart(ar_0)) [ 0 <= 0 ]
2.19/1.23	
2.19/1.23		start location:	koat_start
2.19/1.23	
2.19/1.23		leaf cost:	0
2.19/1.23	
2.19/1.23	
2.19/1.23	
2.19/1.23	Complexity upper bound 123
2.19/1.23	
2.19/1.23	
2.19/1.23	
2.19/1.23	Time: 0.054 sec (SMT: 0.051 sec)
2.19/1.23	
2.19/1.23	
2.19/1.23	----------------------------------------
2.19/1.23	
2.19/1.23	(2)
2.19/1.23	BOUNDS(1, 1)
2.19/1.25	EOF