2.34/1.44	WORST_CASE(?, O(n^1))
2.34/1.45	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.34/1.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.34/1.45	
2.34/1.45	
2.34/1.45	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.34/1.45	
2.34/1.45	(0) CpxIntTrs
2.34/1.45	(1) Koat Proof [FINISHED, 189 ms]
2.34/1.45	(2) BOUNDS(1, n^1)
2.34/1.45	
2.34/1.45	
2.34/1.45	----------------------------------------
2.34/1.45	
2.34/1.45	(0)
2.34/1.45	Obligation:
2.34/1.45	Complexity Int TRS consisting of the following rules:
2.34/1.45	eval_foo_start(v_.0, v_.01, v_.02, v_d1, v_d1old, v_d2, v_x) -> Com_1(eval_foo_bb0_in(v_.0, v_.01, v_.02, v_d1, v_d1old, v_d2, v_x)) :|: TRUE
2.34/1.45	eval_foo_bb0_in(v_.0, v_.01, v_.02, v_d1, v_d1old, v_d2, v_x) -> Com_1(eval_foo_bb1_in(v_x, 73, 74, v_d1, v_d1old, v_d2, v_x)) :|: TRUE
2.34/1.45	eval_foo_bb1_in(v_.0, v_.01, v_.02, v_d1, v_d1old, v_d2, v_x) -> Com_1(eval_foo_bb2_in(v_.0, v_.01, v_.02, v_d1, v_d1old, v_d2, v_x)) :|: v_.0 >= 0
2.34/1.45	eval_foo_bb1_in(v_.0, v_.01, v_.02, v_d1, v_d1old, v_d2, v_x) -> Com_1(eval_foo_bb3_in(v_.0, v_.01, v_.02, v_d1, v_d1old, v_d2, v_x)) :|: v_.0 < 0
2.34/1.45	eval_foo_bb2_in(v_.0, v_.01, v_.02, v_d1, v_d1old, v_d2, v_x) -> Com_1(eval_foo_bb1_in(v_.0 - v_.01, v_.02 + 1, v_.01 + 1, v_d1, v_d1old, v_d2, v_x)) :|: TRUE
2.34/1.45	eval_foo_bb3_in(v_.0, v_.01, v_.02, v_d1, v_d1old, v_d2, v_x) -> Com_1(eval_foo_stop(v_.0, v_.01, v_.02, v_d1, v_d1old, v_d2, v_x)) :|: TRUE
2.34/1.45	
2.34/1.45	The start-symbols are:[eval_foo_start_7]
2.34/1.45	
2.34/1.45	
2.34/1.45	----------------------------------------
2.34/1.45	
2.34/1.45	(1) Koat Proof (FINISHED)
2.34/1.45	YES(?, 4*ar_1 + 154)
2.34/1.45	
2.34/1.45	
2.34/1.45	
2.34/1.45	Initial complexity problem:
2.34/1.45	
2.34/1.45	1:	T:
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, 73, 74))
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 ]
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 - ar_2, ar_1, ar_3 + 1, ar_2 + 1))
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.34/1.45	
2.34/1.45		start location:	koat_start
2.34/1.45	
2.34/1.45		leaf cost:	0
2.34/1.45	
2.34/1.45	
2.34/1.45	
2.34/1.45	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.34/1.45	
2.34/1.45	2:	T:
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, 73, 74))
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 ]
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 - ar_2, ar_1, ar_3 + 1, ar_2 + 1))
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.34/1.45	
2.34/1.45		start location:	koat_start
2.34/1.45	
2.34/1.45		leaf cost:	0
2.34/1.45	
2.34/1.45	
2.34/1.45	
2.34/1.45	A polynomial rank function with
2.34/1.45	
2.34/1.45		Pol(evalfoostart) = 2
2.34/1.45	
2.34/1.45		Pol(evalfoobb0in) = 2
2.34/1.45	
2.34/1.45		Pol(evalfoobb1in) = 2
2.34/1.45	
2.34/1.45		Pol(evalfoobb2in) = 2
2.34/1.45	
2.34/1.45		Pol(evalfoobb3in) = 1
2.34/1.45	
2.34/1.45		Pol(evalfoostop) = 0
2.34/1.45	
2.34/1.45		Pol(koat_start) = 2
2.34/1.45	
2.34/1.45	orients all transitions weakly and the transitions
2.34/1.45	
2.34/1.45		evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.34/1.45	
2.34/1.45		evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.34/1.45	
2.34/1.45	strictly and produces the following problem:
2.34/1.45	
2.34/1.45	3:	T:
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, 73, 74))
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= 0 ]
2.34/1.45	
2.34/1.45			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 - ar_2, ar_1, ar_3 + 1, ar_2 + 1))
2.34/1.45	
2.34/1.45			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3))
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.34/1.45	
2.34/1.45		start location:	koat_start
2.34/1.45	
2.34/1.45		leaf cost:	0
2.34/1.45	
2.34/1.45	
2.34/1.45	
2.34/1.45	Applied AI with 'oct' on problem 3 to obtain the following invariants:
2.34/1.45	
2.34/1.45	  For symbol evalfoobb1in: X_4 - 74 >= 0 /\ X_3 + X_4 - 147 >= 0 /\ X_3 - 73 >= 0 /\ -X_1 + X_2 >= 0
2.34/1.45	
2.34/1.45	  For symbol evalfoobb2in: X_4 - 74 >= 0 /\ X_3 + X_4 - 147 >= 0 /\ X_2 + X_4 - 74 >= 0 /\ X_1 + X_4 - 74 >= 0 /\ X_3 - 73 >= 0 /\ X_2 + X_3 - 73 >= 0 /\ X_1 + X_3 - 73 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ -X_1 + X_2 >= 0 /\ X_1 >= 0
2.34/1.45	
2.34/1.45	  For symbol evalfoobb3in: X_4 - 74 >= 0 /\ X_3 + X_4 - 147 >= 0 /\ -X_1 + X_4 - 75 >= 0 /\ X_3 - 73 >= 0 /\ -X_1 + X_3 - 74 >= 0 /\ -X_1 + X_2 >= 0 /\ -X_1 - 1 >= 0
2.34/1.45	
2.34/1.45	
2.34/1.45	
2.34/1.45	
2.34/1.45	
2.34/1.45	This yielded the following problem:
2.34/1.45	
2.34/1.45	4:	T:
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.34/1.45	
2.34/1.45			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3)) [ ar_3 - 74 >= 0 /\ ar_2 + ar_3 - 147 >= 0 /\ -ar_0 + ar_3 - 75 >= 0 /\ ar_2 - 73 >= 0 /\ -ar_0 + ar_2 - 74 >= 0 /\ -ar_0 + ar_1 >= 0 /\ -ar_0 - 1 >= 0 ]
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 - ar_2, ar_1, ar_3 + 1, ar_2 + 1)) [ ar_3 - 74 >= 0 /\ ar_2 + ar_3 - 147 >= 0 /\ ar_1 + ar_3 - 74 >= 0 /\ ar_0 + ar_3 - 74 >= 0 /\ ar_2 - 73 >= 0 /\ ar_1 + ar_2 - 73 >= 0 /\ ar_0 + ar_2 - 73 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ -ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
2.34/1.45	
2.34/1.45			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 - 74 >= 0 /\ ar_2 + ar_3 - 147 >= 0 /\ ar_2 - 73 >= 0 /\ -ar_0 + ar_1 >= 0 /\ 0 >= ar_0 + 1 ]
2.34/1.45	
2.34/1.45			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 - 74 >= 0 /\ ar_2 + ar_3 - 147 >= 0 /\ ar_2 - 73 >= 0 /\ -ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, 73, 74))
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.34/1.45	
2.34/1.45		start location:	koat_start
2.34/1.45	
2.34/1.45		leaf cost:	0
2.34/1.45	
2.34/1.45	
2.34/1.45	
2.34/1.45	A polynomial rank function with
2.34/1.45	
2.34/1.45		Pol(koat_start) = 2*V_2 + 74
2.34/1.45	
2.34/1.45		Pol(evalfoostart) = 2*V_2 + 74
2.34/1.45	
2.34/1.45		Pol(evalfoobb3in) = 2*V_1
2.34/1.45	
2.34/1.45		Pol(evalfoostop) = 2*V_1
2.34/1.45	
2.34/1.45		Pol(evalfoobb2in) = 2*V_1 + 1
2.34/1.45	
2.34/1.45		Pol(evalfoobb1in) = 2*V_1 + 74
2.34/1.45	
2.34/1.45		Pol(evalfoobb0in) = 2*V_2 + 74
2.34/1.45	
2.34/1.45	orients all transitions weakly and the transitions
2.34/1.45	
2.34/1.45		evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 - ar_2, ar_1, ar_3 + 1, ar_2 + 1)) [ ar_3 - 74 >= 0 /\ ar_2 + ar_3 - 147 >= 0 /\ ar_1 + ar_3 - 74 >= 0 /\ ar_0 + ar_3 - 74 >= 0 /\ ar_2 - 73 >= 0 /\ ar_1 + ar_2 - 73 >= 0 /\ ar_0 + ar_2 - 73 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ -ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
2.34/1.45	
2.34/1.45		evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 - 74 >= 0 /\ ar_2 + ar_3 - 147 >= 0 /\ ar_2 - 73 >= 0 /\ -ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
2.34/1.45	
2.34/1.45	strictly and produces the following problem:
2.34/1.45	
2.34/1.45	5:	T:
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 0)              koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
2.34/1.45	
2.34/1.45			(Comp: 2, Cost: 1)              evalfoobb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoostop(ar_0, ar_1, ar_2, ar_3)) [ ar_3 - 74 >= 0 /\ ar_2 + ar_3 - 147 >= 0 /\ -ar_0 + ar_3 - 75 >= 0 /\ ar_2 - 73 >= 0 /\ -ar_0 + ar_2 - 74 >= 0 /\ -ar_0 + ar_1 >= 0 /\ -ar_0 - 1 >= 0 ]
2.34/1.45	
2.34/1.45			(Comp: 2*ar_1 + 74, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_0 - ar_2, ar_1, ar_3 + 1, ar_2 + 1)) [ ar_3 - 74 >= 0 /\ ar_2 + ar_3 - 147 >= 0 /\ ar_1 + ar_3 - 74 >= 0 /\ ar_0 + ar_3 - 74 >= 0 /\ ar_2 - 73 >= 0 /\ ar_1 + ar_2 - 73 >= 0 /\ ar_0 + ar_2 - 73 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ -ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
2.34/1.45	
2.34/1.45			(Comp: 2, Cost: 1)              evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 - 74 >= 0 /\ ar_2 + ar_3 - 147 >= 0 /\ ar_2 - 73 >= 0 /\ -ar_0 + ar_1 >= 0 /\ 0 >= ar_0 + 1 ]
2.34/1.45	
2.34/1.45			(Comp: 2*ar_1 + 74, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 - 74 >= 0 /\ ar_2 + ar_3 - 147 >= 0 /\ ar_2 - 73 >= 0 /\ -ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ]
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 1)              evalfoobb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb1in(ar_1, ar_1, 73, 74))
2.34/1.45	
2.34/1.45			(Comp: 1, Cost: 1)              evalfoostart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2, ar_3))
2.34/1.45	
2.34/1.45		start location:	koat_start
2.34/1.45	
2.34/1.45		leaf cost:	0
2.34/1.45	
2.34/1.45	
2.34/1.45	
2.34/1.45	Complexity upper bound 4*ar_1 + 154
2.34/1.45	
2.34/1.45	
2.34/1.45	
2.34/1.45	Time: 0.225 sec (SMT: 0.209 sec)
2.34/1.45	
2.34/1.45	
2.34/1.45	----------------------------------------
2.34/1.45	
2.34/1.45	(2)
2.34/1.45	BOUNDS(1, n^1)
2.37/1.47	EOF