2.11/1.35	WORST_CASE(?, O(1))
2.11/1.36	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.11/1.36	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.11/1.36	
2.11/1.36	
2.11/1.36	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
2.11/1.36	
2.11/1.36	(0) CpxIntTrs
2.11/1.36	(1) Koat Proof [FINISHED, 174 ms]
2.11/1.36	(2) BOUNDS(1, 1)
2.11/1.36	
2.11/1.36	
2.11/1.36	----------------------------------------
2.11/1.36	
2.11/1.36	(0)
2.11/1.36	Obligation:
2.11/1.36	Complexity Int TRS consisting of the following rules:
2.11/1.36	eval_foo_start(v_.0, v_x, v_y) -> Com_1(eval_foo_bb0_in(v_.0, v_x, v_y)) :|: TRUE
2.11/1.36	eval_foo_bb0_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_x, v_x, v_y)) :|: v_y > v_x
2.11/1.36	eval_foo_bb0_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_x, v_y)) :|: v_y <= v_x
2.11/1.36	eval_foo_bb1_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.0, v_x, v_y)) :|: v_.0 >= 0
2.11/1.36	eval_foo_bb1_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_x, v_y)) :|: v_.0 < 0
2.11/1.36	eval_foo_bb2_in(v_.0, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.0 - v_y, v_x, v_y)) :|: TRUE
2.11/1.36	eval_foo_bb3_in(v_.0, v_x, v_y) -> Com_1(eval_foo_stop(v_.0, v_x, v_y)) :|: TRUE
2.11/1.36	
2.11/1.36	The start-symbols are:[eval_foo_start_3]
2.11/1.36	
2.11/1.36	
2.11/1.36	----------------------------------------
2.11/1.36	
2.11/1.36	(1) Koat Proof (FINISHED)
2.11/1.36	YES(?, 9)
2.11/1.36	
2.11/1.36	
2.11/1.36	
2.11/1.36	Initial complexity problem:
2.11/1.36	
2.11/1.36	1:	T:
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_1)) [ ar_0 >= ar_1 + 1 ]
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 0 ]
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 + 1 ]
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 - ar_0))
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.11/1.36	
2.11/1.36		start location:	koat_start
2.11/1.36	
2.11/1.36		leaf cost:	0
2.11/1.36	
2.11/1.36	
2.11/1.36	
2.11/1.36	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.11/1.36	
2.11/1.36	2:	T:
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_1)) [ ar_0 >= ar_1 + 1 ]
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 0 ]
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 + 1 ]
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 - ar_0))
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.11/1.36	
2.11/1.36		start location:	koat_start
2.11/1.36	
2.11/1.36		leaf cost:	0
2.11/1.36	
2.11/1.36	
2.11/1.36	
2.11/1.36	A polynomial rank function with
2.11/1.36	
2.11/1.36		Pol(evalfoostart) = 2
2.11/1.36	
2.11/1.36		Pol(evalfoobb0in) = 2
2.11/1.36	
2.11/1.36		Pol(evalfoobb1in) = 2
2.11/1.36	
2.11/1.36		Pol(evalfoobb3in) = 1
2.11/1.36	
2.11/1.36		Pol(evalfoobb2in) = 2
2.11/1.36	
2.11/1.36		Pol(evalfoostop) = 0
2.11/1.36	
2.11/1.36		Pol(koat_start) = 2
2.11/1.36	
2.11/1.36	orients all transitions weakly and the transitions
2.11/1.36	
2.11/1.36		evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.11/1.36	
2.11/1.36		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 + 1 ]
2.11/1.36	
2.11/1.36	strictly and produces the following problem:
2.11/1.36	
2.11/1.36	3:	T:
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_1)) [ ar_0 >= ar_1 + 1 ]
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_2 >= 0 ]
2.11/1.36	
2.11/1.36			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 + 1 ]
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 - ar_0))
2.11/1.36	
2.11/1.36			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.11/1.36	
2.11/1.36		start location:	koat_start
2.11/1.36	
2.11/1.36		leaf cost:	0
2.11/1.36	
2.11/1.36	
2.11/1.36	
2.11/1.36	Applied AI with 'oct' on problem 3 to obtain the following invariants:
2.11/1.36	
2.11/1.36	  For symbol evalfoobb1in: X_2 - X_3 >= 0 /\ X_1 - X_3 - 1 >= 0 /\ X_1 - X_2 - 1 >= 0
2.11/1.36	
2.11/1.36	  For symbol evalfoobb2in: X_2 - X_3 >= 0 /\ X_1 - X_3 - 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_1 - X_2 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0
2.11/1.36	
2.11/1.36	
2.11/1.36	
2.11/1.36	
2.11/1.36	
2.11/1.36	This yielded the following problem:
2.11/1.36	
2.11/1.36	4:	T:
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.11/1.36	
2.11/1.36			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 - ar_0)) [ ar_1 - ar_2 >= 0 /\ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_0 - ar_1 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ]
2.11/1.36	
2.11/1.36			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ ar_0 - ar_2 - 1 >= 0 /\ ar_0 - ar_1 - 1 >= 0 /\ 0 >= ar_2 + 1 ]
2.11/1.36	
2.11/1.36			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ ar_0 - ar_2 - 1 >= 0 /\ ar_0 - ar_1 - 1 >= 0 /\ ar_2 >= 0 ]
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_1)) [ ar_0 >= ar_1 + 1 ]
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.11/1.36	
2.11/1.36		start location:	koat_start
2.11/1.36	
2.11/1.36		leaf cost:	0
2.11/1.36	
2.11/1.36	
2.11/1.36	
2.11/1.36	Repeatedly propagating knowledge in problem 4 produces the following problem:
2.11/1.36	
2.11/1.36	5:	T:
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.11/1.36	
2.11/1.36			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_2 - ar_0)) [ ar_1 - ar_2 >= 0 /\ ar_0 - ar_2 - 1 >= 0 /\ ar_2 >= 0 /\ ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ ar_0 - ar_1 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ]
2.11/1.36	
2.11/1.36			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ ar_0 - ar_2 - 1 >= 0 /\ ar_0 - ar_1 - 1 >= 0 /\ 0 >= ar_2 + 1 ]
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_1 - ar_2 >= 0 /\ ar_0 - ar_2 - 1 >= 0 /\ ar_0 - ar_1 - 1 >= 0 /\ ar_2 >= 0 ]
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1, ar_1)) [ ar_0 >= ar_1 + 1 ]
2.11/1.36	
2.11/1.36			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.11/1.36	
2.11/1.36		start location:	koat_start
2.11/1.36	
2.11/1.36		leaf cost:	0
2.11/1.36	
2.11/1.36	
2.11/1.36	
2.11/1.36	Complexity upper bound 9
2.11/1.36	
2.11/1.36	
2.11/1.36	
2.11/1.36	Time: 0.150 sec (SMT: 0.138 sec)
2.11/1.36	
2.11/1.36	
2.11/1.36	----------------------------------------
2.11/1.36	
2.11/1.36	(2)
2.11/1.36	BOUNDS(1, 1)
2.11/1.37	EOF