2.49/1.54	WORST_CASE(?, O(n^1))
2.49/1.55	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.49/1.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.49/1.55	
2.49/1.55	
2.49/1.55	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.49/1.55	
2.49/1.55	(0) CpxIntTrs
2.49/1.55	(1) Koat Proof [FINISHED, 277 ms]
2.49/1.55	(2) BOUNDS(1, n^1)
2.49/1.55	
2.49/1.55	
2.49/1.55	----------------------------------------
2.49/1.55	
2.49/1.55	(0)
2.49/1.55	Obligation:
2.49/1.55	Complexity Int TRS consisting of the following rules:
2.49/1.55	eval_foo_start(v_.0, v_M, v_x) -> Com_1(eval_foo_bb0_in(v_.0, v_M, v_x)) :|: TRUE
2.49/1.55	eval_foo_bb0_in(v_.0, v_M, v_x) -> Com_1(eval_foo_bb1_in(v_x, v_M, v_x)) :|: v_M > 0
2.49/1.55	eval_foo_bb0_in(v_.0, v_M, v_x) -> Com_1(eval_foo_bb3_in(v_.0, v_M, v_x)) :|: v_M <= 0
2.49/1.55	eval_foo_bb1_in(v_.0, v_M, v_x) -> Com_1(eval_foo_bb2_in(v_.0, v_M, v_x)) :|: v_.0 < v_M
2.49/1.55	eval_foo_bb1_in(v_.0, v_M, v_x) -> Com_1(eval_foo_bb2_in(v_.0, v_M, v_x)) :|: v_.0 > v_M
2.49/1.55	eval_foo_bb1_in(v_.0, v_M, v_x) -> Com_1(eval_foo_bb3_in(v_.0, v_M, v_x)) :|: v_.0 >= v_M && v_.0 <= v_M
2.49/1.55	eval_foo_bb2_in(v_.0, v_M, v_x) -> Com_1(eval_foo_bb1_in(0, v_M, v_x)) :|: v_.0 > v_M
2.49/1.55	eval_foo_bb2_in(v_.0, v_M, v_x) -> Com_1(eval_foo_bb1_in(v_.0 + 1, v_M, v_x)) :|: v_.0 <= v_M
2.49/1.55	eval_foo_bb3_in(v_.0, v_M, v_x) -> Com_1(eval_foo_stop(v_.0, v_M, v_x)) :|: TRUE
2.49/1.55	
2.49/1.55	The start-symbols are:[eval_foo_start_3]
2.49/1.55	
2.49/1.55	
2.49/1.55	----------------------------------------
2.49/1.55	
2.49/1.55	(1) Koat Proof (FINISHED)
2.49/1.55	YES(?, 4*ar_0 + 2*ar_2 + 17)
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	Initial complexity problem:
2.49/1.55	
2.49/1.55	1:	T:
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 0, ar_2)) [ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55		start location:	koat_start
2.49/1.55	
2.49/1.55		leaf cost:	0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.49/1.55	
2.49/1.55	2:	T:
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 0, ar_2)) [ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55		start location:	koat_start
2.49/1.55	
2.49/1.55		leaf cost:	0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	A polynomial rank function with
2.49/1.55	
2.49/1.55		Pol(evalfoostart) = 2
2.49/1.55	
2.49/1.55		Pol(evalfoobb0in) = 2
2.49/1.55	
2.49/1.55		Pol(evalfoobb1in) = 2
2.49/1.55	
2.49/1.55		Pol(evalfoobb3in) = 1
2.49/1.55	
2.49/1.55		Pol(evalfoobb2in) = 2
2.49/1.55	
2.49/1.55		Pol(evalfoostop) = 0
2.49/1.55	
2.49/1.55		Pol(koat_start) = 2
2.49/1.55	
2.49/1.55	orients all transitions weakly and the transitions
2.49/1.55	
2.49/1.55		evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55	strictly and produces the following problem:
2.49/1.55	
2.49/1.55	3:	T:
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 0, ar_2)) [ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55		start location:	koat_start
2.49/1.55	
2.49/1.55		leaf cost:	0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	Applied AI with 'oct' on problem 3 to obtain the following invariants:
2.49/1.55	
2.49/1.55	  For symbol evalfoobb1in: X_1 - 1 >= 0
2.49/1.55	
2.49/1.55	  For symbol evalfoobb2in: X_1 - 1 >= 0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	This yielded the following problem:
2.49/1.55	
2.49/1.55	4:	T:
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 0, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55		start location:	koat_start
2.49/1.55	
2.49/1.55		leaf cost:	0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	By chaining the transition koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] with all transitions in problem 4, the following new transition is obtained:
2.49/1.55	
2.49/1.55		koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55	We thus obtain the following problem:
2.49/1.55	
2.49/1.55	5:	T:
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 0, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55		start location:	koat_start
2.49/1.55	
2.49/1.55		leaf cost:	0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	Testing for reachability in the complexity graph removes the following transition from problem 5:
2.49/1.55	
2.49/1.55		evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55	We thus obtain the following problem:
2.49/1.55	
2.49/1.55	6:	T:
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 0, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55		start location:	koat_start
2.49/1.55	
2.49/1.55		leaf cost:	0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	By chaining the transition evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 0, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ] with all transitions in problem 6, the following new transition is obtained:
2.49/1.55	
2.49/1.55		evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, 0, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55	We thus obtain the following problem:
2.49/1.55	
2.49/1.55	7:	T:
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 2)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, 0, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55		start location:	koat_start
2.49/1.55	
2.49/1.55		leaf cost:	0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	By chaining the transition evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, 0, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 1 ] with all transitions in problem 7, the following new transition is obtained:
2.49/1.55	
2.49/1.55		evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 1 /\ ar_0 >= 0 ]
2.49/1.55	
2.49/1.55	We thus obtain the following problem:
2.49/1.55	
2.49/1.55	8:	T:
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 3)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 1 /\ ar_0 >= 0 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55		start location:	koat_start
2.49/1.55	
2.49/1.55		leaf cost:	0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	By chaining the transition evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 ] with all transitions in problem 8, the following new transition is obtained:
2.49/1.55	
2.49/1.55		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55	We thus obtain the following problem:
2.49/1.55	
2.49/1.55	9:	T:
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 2)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 3)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 1 /\ ar_0 >= 0 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55		start location:	koat_start
2.49/1.55	
2.49/1.55		leaf cost:	0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	Testing for reachability in the complexity graph removes the following transition from problem 9:
2.49/1.55	
2.49/1.55		evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55	We thus obtain the following problem:
2.49/1.55	
2.49/1.55	10:	T:
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 3)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 1 /\ ar_0 >= 0 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 2)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55		start location:	koat_start
2.49/1.55	
2.49/1.55		leaf cost:	0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	Repeatedly propagating knowledge in problem 10 produces the following problem:
2.49/1.55	
2.49/1.55	11:	T:
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 3)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 1 /\ ar_0 >= 0 ]
2.49/1.55	
2.49/1.55			(Comp: ?, Cost: 2)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55		start location:	koat_start
2.49/1.55	
2.49/1.55		leaf cost:	0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	A polynomial rank function with
2.49/1.55	
2.49/1.55		Pol(evalfoobb1in) = V_1 - V_2 + 1
2.49/1.55	
2.49/1.55	and size complexities
2.49/1.55	
2.49/1.55		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
2.49/1.55	
2.49/1.55		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
2.49/1.55	
2.49/1.55		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
2.49/1.55	
2.49/1.55		S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-0) = ar_0
2.49/1.55	
2.49/1.55		S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-1) = ar_1
2.49/1.55	
2.49/1.55		S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-2) = ar_2
2.49/1.55	
2.49/1.55		S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]", 0-0) = ar_0
2.49/1.55	
2.49/1.55		S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]", 0-1) = ar_2
2.49/1.55	
2.49/1.55		S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]", 0-2) = ar_2
2.49/1.55	
2.49/1.55		S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-0) = ar_0
2.49/1.55	
2.49/1.55		S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-1) = ?
2.49/1.55	
2.49/1.55		S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.49/1.55	
2.49/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\\ ar_1 = ar_0 ]", 0-0) = ar_0
2.49/1.55	
2.49/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\\ ar_1 = ar_0 ]", 0-1) = ?
2.49/1.55	
2.49/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\\ ar_1 = ar_0 ]", 0-2) = ar_2
2.49/1.55	
2.49/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\\ ar_1 >= ar_0 + 1 ]", 0-0) = ar_0
2.49/1.55	
2.49/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\\ ar_1 >= ar_0 + 1 ]", 0-1) = ar_2
2.49/1.55	
2.49/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\\ ar_1 >= ar_0 + 1 ]", 0-2) = ar_2
2.49/1.55	
2.49/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= ar_1 ]", 0-0) = ar_0
2.49/1.55	
2.49/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= ar_1 ]", 0-1) = ?
2.49/1.55	
2.49/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\\ ar_0 >= ar_1 + 1 /\\ ar_0 >= ar_1 ]", 0-2) = ar_2
2.49/1.55	
2.49/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 1, ar_2)) [ ar_0 - 1 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 1 /\\ ar_0 >= 0 ]", 0-0) = ar_0
2.49/1.55	
2.49/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 1, ar_2)) [ ar_0 - 1 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 1 /\\ ar_0 >= 0 ]", 0-1) = 1
2.49/1.55	
2.49/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 1, ar_2)) [ ar_0 - 1 >= 0 /\\ ar_1 >= ar_0 + 1 /\\ ar_0 >= 1 /\\ ar_0 >= 0 ]", 0-2) = ar_2
2.49/1.55	
2.49/1.55	orients the transitions
2.49/1.55	
2.49/1.55		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55	weakly and the transition
2.49/1.55	
2.49/1.55		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55	strictly and produces the following problem:
2.49/1.55	
2.49/1.55	12:	T:
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 3)                    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 /\ ar_0 >= 1 /\ ar_0 >= 0 ]
2.49/1.55	
2.49/1.55			(Comp: 2*ar_0 + ar_2 + 3, Cost: 2)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_1 + 1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 /\ ar_0 >= ar_1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)                    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 + 1 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)                    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 - 1 >= 0 /\ ar_1 = ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 2, Cost: 1)                    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)                    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)                    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.49/1.55	
2.49/1.55			(Comp: 1, Cost: 1)                    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.49/1.55	
2.49/1.55		start location:	koat_start
2.49/1.55	
2.49/1.55		leaf cost:	0
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	Complexity upper bound 4*ar_0 + 2*ar_2 + 17
2.49/1.55	
2.49/1.55	
2.49/1.55	
2.49/1.55	Time: 0.298 sec (SMT: 0.254 sec)
2.49/1.55	
2.49/1.55	
2.49/1.55	----------------------------------------
2.49/1.55	
2.49/1.55	(2)
2.49/1.55	BOUNDS(1, n^1)
2.51/1.57	EOF