2.14/1.39	WORST_CASE(?, O(n^1))
2.44/1.40	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.44/1.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.44/1.40	
2.44/1.40	
2.44/1.40	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
2.44/1.40	
2.44/1.40	(0) CpxIntTrs
2.44/1.40	(1) Koat Proof [FINISHED, 275 ms]
2.44/1.40	(2) BOUNDS(1, n^1)
2.44/1.40	
2.44/1.40	
2.44/1.40	----------------------------------------
2.44/1.40	
2.44/1.40	(0)
2.44/1.40	Obligation:
2.44/1.40	Complexity Int TRS consisting of the following rules:
2.44/1.40	eval_foo_start(v_.0, v_x) -> Com_1(eval_foo_bb0_in(v_.0, v_x)) :|: TRUE
2.44/1.40	eval_foo_bb0_in(v_.0, v_x) -> Com_1(eval_foo_bb1_in(v_x, v_x)) :|: v_x > 0
2.44/1.40	eval_foo_bb0_in(v_.0, v_x) -> Com_1(eval_foo_bb3_in(v_.0, v_x)) :|: v_x <= 0
2.44/1.40	eval_foo_bb1_in(v_.0, v_x) -> Com_1(eval_foo_bb2_in(v_.0, v_x)) :|: v_.0 < 0
2.44/1.40	eval_foo_bb1_in(v_.0, v_x) -> Com_1(eval_foo_bb2_in(v_.0, v_x)) :|: v_.0 > 0
2.44/1.40	eval_foo_bb1_in(v_.0, v_x) -> Com_1(eval_foo_bb3_in(v_.0, v_x)) :|: v_.0 >= 0 && v_.0 <= 0
2.44/1.40	eval_foo_bb2_in(v_.0, v_x) -> Com_1(eval_foo_bb1_in(v_.0 - 1, v_x)) :|: TRUE
2.44/1.40	eval_foo_bb3_in(v_.0, v_x) -> Com_1(eval_foo_stop(v_.0, v_x)) :|: TRUE
2.44/1.40	
2.44/1.40	The start-symbols are:[eval_foo_start_2]
2.44/1.40	
2.44/1.40	
2.44/1.40	----------------------------------------
2.44/1.40	
2.44/1.40	(1) Koat Proof (FINISHED)
2.44/1.40	YES(?, 2*ar_0 + 9)
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	Initial complexity problem:
2.44/1.40	
2.44/1.40	1:	T:
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_1 = 0 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1))
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40		start location:	koat_start
2.44/1.40	
2.44/1.40		leaf cost:	0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.44/1.40	
2.44/1.40	2:	T:
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_1 = 0 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1))
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40		start location:	koat_start
2.44/1.40	
2.44/1.40		leaf cost:	0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	A polynomial rank function with
2.44/1.40	
2.44/1.40		Pol(evalfoostart) = 2
2.44/1.40	
2.44/1.40		Pol(evalfoobb0in) = 2
2.44/1.40	
2.44/1.40		Pol(evalfoobb1in) = 2
2.44/1.40	
2.44/1.40		Pol(evalfoobb3in) = 1
2.44/1.40	
2.44/1.40		Pol(evalfoobb2in) = 2
2.44/1.40	
2.44/1.40		Pol(evalfoostop) = 0
2.44/1.40	
2.44/1.40		Pol(koat_start) = 2
2.44/1.40	
2.44/1.40	orients all transitions weakly and the transitions
2.44/1.40	
2.44/1.40		evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_1 = 0 ]
2.44/1.40	
2.44/1.40	strictly and produces the following problem:
2.44/1.40	
2.44/1.40	3:	T:
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_1 = 0 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1))
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)    evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40		start location:	koat_start
2.44/1.40	
2.44/1.40		leaf cost:	0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	A polynomial rank function with
2.44/1.40	
2.44/1.40		Pol(evalfoobb2in) = V_2 - 1
2.44/1.40	
2.44/1.40		Pol(evalfoobb1in) = V_2
2.44/1.40	
2.44/1.40	and size complexities
2.44/1.40	
2.44/1.40		S("koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
2.44/1.40	
2.44/1.40		S("koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
2.44/1.40	
2.44/1.40		S("evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))", 0-0) = ar_0
2.44/1.40	
2.44/1.40		S("evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))", 0-1) = ar_1
2.44/1.40	
2.44/1.40		S("evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1))", 0-0) = ar_0
2.44/1.40	
2.44/1.40		S("evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1))", 0-1) = ?
2.44/1.40	
2.44/1.40		S("evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_1 = 0 ]", 0-0) = ar_0
2.44/1.40	
2.44/1.40		S("evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_1 = 0 ]", 0-1) = 0
2.44/1.40	
2.44/1.40		S("evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= 1 ]", 0-0) = ar_0
2.44/1.40	
2.44/1.40		S("evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= 1 ]", 0-1) = ?
2.44/1.40	
2.44/1.40		S("evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 0 >= ar_1 + 1 ]", 0-0) = ar_0
2.44/1.40	
2.44/1.40		S("evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 0 >= ar_1 + 1 ]", 0-1) = ?
2.44/1.40	
2.44/1.40		S("evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]", 0-0) = ar_0
2.44/1.40	
2.44/1.40		S("evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]", 0-1) = ar_1
2.44/1.40	
2.44/1.40		S("evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]", 0-0) = ar_0
2.44/1.40	
2.44/1.40		S("evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]", 0-1) = ar_0
2.44/1.40	
2.44/1.40		S("evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))", 0-0) = ar_0
2.44/1.40	
2.44/1.40		S("evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))", 0-1) = ar_1
2.44/1.40	
2.44/1.40	orients the transitions
2.44/1.40	
2.44/1.40		evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1))
2.44/1.40	
2.44/1.40		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40	weakly and the transition
2.44/1.40	
2.44/1.40		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40	strictly and produces the following problem:
2.44/1.40	
2.44/1.40	4:	T:
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40			(Comp: ar_0, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_1 = 0 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)       evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1))
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40		start location:	koat_start
2.44/1.40	
2.44/1.40		leaf cost:	0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	Applied AI with 'oct' on problem 4 to obtain the following invariants:
2.44/1.40	
2.44/1.40	  For symbol evalfoobb1in: X_1 - X_2 >= 0 /\ X_1 - 1 >= 0
2.44/1.40	
2.44/1.40	  For symbol evalfoobb2in: X_1 - X_2 >= 0 /\ X_1 - 1 >= 0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	This yielded the following problem:
2.44/1.40	
2.44/1.40	5:	T:
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)       evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 = 0 ]
2.44/1.40	
2.44/1.40			(Comp: ar_0, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.44/1.40	
2.44/1.40		start location:	koat_start
2.44/1.40	
2.44/1.40		leaf cost:	0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	By chaining the transition koat_start(ar_0, ar_1) -> Com_1(evalfoostart(ar_0, ar_1)) [ 0 <= 0 ] with all transitions in problem 5, the following new transition is obtained:
2.44/1.40	
2.44/1.40		koat_start(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40	We thus obtain the following problem:
2.44/1.40	
2.44/1.40	6:	T:
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       koat_start(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)       evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 = 0 ]
2.44/1.40	
2.44/1.40			(Comp: ar_0, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.44/1.40	
2.44/1.40		start location:	koat_start
2.44/1.40	
2.44/1.40		leaf cost:	0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	Testing for reachability in the complexity graph removes the following transition from problem 6:
2.44/1.40	
2.44/1.40		evalfoostart(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1))
2.44/1.40	
2.44/1.40	We thus obtain the following problem:
2.44/1.40	
2.44/1.40	7:	T:
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 = 0 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)       evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 ]
2.44/1.40	
2.44/1.40			(Comp: ar_0, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       koat_start(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40		start location:	koat_start
2.44/1.40	
2.44/1.40		leaf cost:	0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	By chaining the transition evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 + 1 ] with all transitions in problem 7, the following new transition is obtained:
2.44/1.40	
2.44/1.40		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40	We thus obtain the following problem:
2.44/1.40	
2.44/1.40	8:	T:
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 2)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 = 0 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 1)       evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 ]
2.44/1.40	
2.44/1.40			(Comp: ar_0, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       koat_start(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40		start location:	koat_start
2.44/1.40	
2.44/1.40		leaf cost:	0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	Repeatedly propagating knowledge in problem 8 produces the following problem:
2.44/1.40	
2.44/1.40	9:	T:
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 2)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 = 0 ]
2.44/1.40	
2.44/1.40			(Comp: ar_0, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 ]
2.44/1.40	
2.44/1.40			(Comp: ar_0, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       koat_start(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40		start location:	koat_start
2.44/1.40	
2.44/1.40		leaf cost:	0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	By chaining the transition evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 = 0 ] with all transitions in problem 9, the following new transition is obtained:
2.44/1.40	
2.44/1.40		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 = 0 ]
2.44/1.40	
2.44/1.40	We thus obtain the following problem:
2.44/1.40	
2.44/1.40	10:	T:
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 2)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 = 0 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 2)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40			(Comp: ar_0, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 ]
2.44/1.40	
2.44/1.40			(Comp: ar_0, Cost: 1)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       koat_start(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40		start location:	koat_start
2.44/1.40	
2.44/1.40		leaf cost:	0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	By chaining the transition evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb2in(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ] with all transitions in problem 10, the following new transition is obtained:
2.44/1.40	
2.44/1.40		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40	We thus obtain the following problem:
2.44/1.40	
2.44/1.40	11:	T:
2.44/1.40	
2.44/1.40			(Comp: ar_0, Cost: 2)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 2)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 = 0 ]
2.44/1.40	
2.44/1.40			(Comp: ?, Cost: 2)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40			(Comp: ar_0, Cost: 1)    evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       koat_start(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40		start location:	koat_start
2.44/1.40	
2.44/1.40		leaf cost:	0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	Testing for reachability in the complexity graph removes the following transitions from problem 11:
2.44/1.40	
2.44/1.40		evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 + 1 ]
2.44/1.40	
2.44/1.40		evalfoobb2in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 ]
2.44/1.40	
2.44/1.40	We thus obtain the following problem:
2.44/1.40	
2.44/1.40	12:	T:
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 2)       evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 = 0 ]
2.44/1.40	
2.44/1.40			(Comp: ar_0, Cost: 2)    evalfoobb1in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_1 - 1)) [ ar_0 - ar_1 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 2, Cost: 1)       evalfoobb3in(ar_0, ar_1) -> Com_1(evalfoostop(ar_0, ar_1))
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb1in(ar_0, ar_0)) [ ar_0 >= 1 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       evalfoobb0in(ar_0, ar_1) -> Com_1(evalfoobb3in(ar_0, ar_1)) [ 0 >= ar_0 ]
2.44/1.40	
2.44/1.40			(Comp: 1, Cost: 1)       koat_start(ar_0, ar_1) -> Com_1(evalfoobb0in(ar_0, ar_1)) [ 0 <= 0 ]
2.44/1.40	
2.44/1.40		start location:	koat_start
2.44/1.40	
2.44/1.40		leaf cost:	0
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	Complexity upper bound 2*ar_0 + 9
2.44/1.40	
2.44/1.40	
2.44/1.40	
2.44/1.40	Time: 0.252 sec (SMT: 0.227 sec)
2.44/1.40	
2.44/1.40	
2.44/1.40	----------------------------------------
2.44/1.40	
2.44/1.40	(2)
2.44/1.40	BOUNDS(1, n^1)
2.44/1.42	EOF