2.26/1.54	WORST_CASE(?, O(n^2))
2.26/1.55	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.26/1.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.26/1.55	
2.26/1.55	
2.26/1.55	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2).
2.26/1.55	
2.26/1.55	(0) CpxIntTrs
2.26/1.55	(1) Koat Proof [FINISHED, 75 ms]
2.26/1.55	(2) BOUNDS(1, n^2)
2.26/1.55	
2.26/1.55	
2.26/1.55	----------------------------------------
2.26/1.55	
2.26/1.55	(0)
2.26/1.55	Obligation:
2.26/1.55	Complexity Int TRS consisting of the following rules:
2.26/1.55	eval_foo_start(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb0_in(v_.0, v_.01, v_x, v_y)) :|: TRUE
2.26/1.55	eval_foo_bb0_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_x, v_.01, v_x, v_y)) :|: TRUE
2.26/1.55	eval_foo_bb1_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.0, 1, v_x, v_y)) :|: v_.0 >= 0
2.26/1.55	eval_foo_bb1_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb5_in(v_.0, v_.01, v_x, v_y)) :|: v_.0 < 0
2.26/1.55	eval_foo_bb2_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_.01, v_x, v_y)) :|: v_.01 < v_.0
2.26/1.55	eval_foo_bb2_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb4_in(v_.0, v_.01, v_x, v_y)) :|: v_.01 >= v_.0
2.26/1.55	eval_foo_bb3_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.0, v_.01 + 1, v_x, v_y)) :|: TRUE
2.26/1.55	eval_foo_bb4_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.0 - 1, v_.01, v_x, v_y)) :|: TRUE
2.26/1.55	eval_foo_bb5_in(v_.0, v_.01, v_x, v_y) -> Com_1(eval_foo_stop(v_.0, v_.01, v_x, v_y)) :|: TRUE
2.26/1.55	
2.26/1.55	The start-symbols are:[eval_foo_start_4]
2.26/1.55	
2.26/1.55	
2.26/1.55	----------------------------------------
2.26/1.55	
2.26/1.55	(1) Koat Proof (FINISHED)
2.26/1.55	YES(?, 51*ar_1 + 6*ar_1^2 + 51)
2.26/1.55	
2.26/1.55	
2.26/1.55	
2.26/1.55	Initial complexity problem:
2.26/1.55	
2.26/1.55	1:	T:
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.26/1.55	
2.26/1.55		start location:	koat_start
2.26/1.55	
2.26/1.55		leaf cost:	0
2.26/1.55	
2.26/1.55	
2.26/1.55	
2.26/1.55	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.26/1.55	
2.26/1.55	2:	T:
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.26/1.55	
2.26/1.55		start location:	koat_start
2.26/1.55	
2.26/1.55		leaf cost:	0
2.26/1.55	
2.26/1.55	
2.26/1.55	
2.26/1.55	A polynomial rank function with
2.26/1.55	
2.26/1.55		Pol(evalfoostart) = 2
2.26/1.55	
2.26/1.55		Pol(evalfoobb0in) = 2
2.26/1.55	
2.26/1.55		Pol(evalfoobb1in) = 2
2.26/1.55	
2.26/1.55		Pol(evalfoobb2in) = 2
2.26/1.55	
2.26/1.55		Pol(evalfoobb5in) = 1
2.26/1.55	
2.26/1.55		Pol(evalfoobb3in) = 2
2.26/1.55	
2.26/1.55		Pol(evalfoobb4in) = 2
2.26/1.55	
2.26/1.55		Pol(evalfoostop) = 0
2.26/1.55	
2.26/1.55		Pol(koat_start) = 2
2.26/1.55	
2.26/1.55	orients all transitions weakly and the transitions
2.26/1.55	
2.26/1.55		evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]
2.26/1.55	
2.26/1.55	strictly and produces the following problem:
2.26/1.55	
2.26/1.55	3:	T:
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 1)    evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 1)    evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]
2.26/1.55	
2.26/1.55			(Comp: 2, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)    evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 2, Cost: 1)    evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.26/1.55	
2.26/1.55		start location:	koat_start
2.26/1.55	
2.26/1.55		leaf cost:	0
2.26/1.55	
2.26/1.55	
2.26/1.55	
2.26/1.55	A polynomial rank function with
2.26/1.55	
2.26/1.55		Pol(evalfoostart) = V_2 + 1
2.26/1.55	
2.26/1.55		Pol(evalfoobb0in) = V_2 + 1
2.26/1.55	
2.26/1.55		Pol(evalfoobb1in) = V_1 + 1
2.26/1.55	
2.26/1.55		Pol(evalfoobb2in) = V_1
2.26/1.55	
2.26/1.55		Pol(evalfoobb5in) = V_1
2.26/1.55	
2.26/1.55		Pol(evalfoobb3in) = V_1
2.26/1.55	
2.26/1.55		Pol(evalfoobb4in) = V_1
2.26/1.55	
2.26/1.55		Pol(evalfoostop) = V_1
2.26/1.55	
2.26/1.55		Pol(koat_start) = V_2 + 1
2.26/1.55	
2.26/1.55	orients all transitions weakly and the transition
2.26/1.55	
2.26/1.55		evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]
2.26/1.55	
2.26/1.55	strictly and produces the following problem:
2.26/1.55	
2.26/1.55	4:	T:
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 1)           evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 1)           evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: ar_1 + 1, Cost: 1)    evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]
2.26/1.55	
2.26/1.55			(Comp: 2, Cost: 1)           evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)           evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)           evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)           evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)           evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 2, Cost: 1)           evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.26/1.55	
2.26/1.55		start location:	koat_start
2.26/1.55	
2.26/1.55		leaf cost:	0
2.26/1.55	
2.26/1.55	
2.26/1.55	
2.26/1.55	A polynomial rank function with
2.26/1.55	
2.26/1.55		Pol(evalfoobb4in) = 1
2.26/1.55	
2.26/1.55		Pol(evalfoobb1in) = 0
2.26/1.55	
2.26/1.55		Pol(evalfoobb3in) = 2
2.26/1.55	
2.26/1.55		Pol(evalfoobb2in) = 2
2.26/1.55	
2.26/1.55	and size complexities
2.26/1.55	
2.26/1.55		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
2.26/1.55	
2.26/1.55		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
2.26/1.55	
2.26/1.55		S("evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-0) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-2) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))", 0-0) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))", 0-2) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))", 0-0) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))", 0-2) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]", 0-0) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]", 0-2) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]", 0-0) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]", 0-2) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]", 0-0) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]", 0-2) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]", 0-0) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]", 0-2) = 1
2.26/1.55	
2.26/1.55		S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))", 0-0) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))", 0-2) = ar_2
2.26/1.55	
2.26/1.55		S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0
2.26/1.55	
2.26/1.55		S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.26/1.55	
2.26/1.55	orients the transitions
2.26/1.55	
2.26/1.55		evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))
2.26/1.55	
2.26/1.55		evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))
2.26/1.55	
2.26/1.55		evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
2.26/1.55	
2.26/1.55		evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
2.26/1.55	
2.26/1.55	weakly and the transitions
2.26/1.55	
2.26/1.55		evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))
2.26/1.55	
2.26/1.55		evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
2.26/1.55	
2.26/1.55	strictly and produces the following problem:
2.26/1.55	
2.26/1.55	5:	T:
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 1)             evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 1)             evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: ar_1 + 1, Cost: 1)      evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]
2.26/1.55	
2.26/1.55			(Comp: 2, Cost: 1)             evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)             evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: 2*ar_1 + 2, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)             evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))
2.26/1.55	
2.26/1.55			(Comp: 2*ar_1 + 2, Cost: 1)    evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 2, Cost: 1)             evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 0)             koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.26/1.55	
2.26/1.55		start location:	koat_start
2.26/1.55	
2.26/1.55		leaf cost:	0
2.26/1.55	
2.26/1.55	
2.26/1.55	
2.26/1.55	A polynomial rank function with
2.26/1.55	
2.26/1.55		Pol(evalfoobb3in) = V_1 - V_3
2.26/1.55	
2.26/1.55		Pol(evalfoobb2in) = V_1 - V_3 + 1
2.26/1.55	
2.26/1.55	and size complexities
2.26/1.55	
2.26/1.55		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
2.26/1.55	
2.26/1.55		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
2.26/1.55	
2.26/1.55		S("evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-0) = 3*ar_1 + 162
2.26/1.55	
2.26/1.55		S("evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-2) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))", 0-0) = 3*ar_1 + 18
2.26/1.55	
2.26/1.55		S("evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))", 0-2) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))", 0-0) = 3*ar_1 + 18
2.26/1.55	
2.26/1.55		S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))", 0-2) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]", 0-0) = 3*ar_1 + 18
2.26/1.55	
2.26/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]", 0-2) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]", 0-0) = 3*ar_1 + 18
2.26/1.55	
2.26/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]", 0-2) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]", 0-0) = 3*ar_1 + 54
2.26/1.55	
2.26/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]", 0-2) = ?
2.26/1.55	
2.26/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]", 0-0) = 3*ar_1 + 18
2.26/1.55	
2.26/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]", 0-2) = 1
2.26/1.55	
2.26/1.55		S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))", 0-0) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))", 0-2) = ar_2
2.26/1.55	
2.26/1.55		S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0
2.26/1.55	
2.26/1.55		S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.26/1.55	
2.26/1.55		S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.26/1.55	
2.26/1.55	orients the transitions
2.26/1.55	
2.26/1.55		evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))
2.26/1.55	
2.26/1.55		evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
2.26/1.55	
2.26/1.55	weakly and the transition
2.26/1.55	
2.26/1.55		evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
2.26/1.55	
2.26/1.55	strictly and produces the following problem:
2.26/1.55	
2.26/1.55	6:	T:
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 1)                          evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 1)                          evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: ar_1 + 1, Cost: 1)                   evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]
2.26/1.55	
2.26/1.55			(Comp: 2, Cost: 1)                          evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: 3*ar_1^2 + 23*ar_1 + 20, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: 2*ar_1 + 2, Cost: 1)                 evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
2.26/1.55	
2.26/1.55			(Comp: ?, Cost: 1)                          evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))
2.26/1.55	
2.26/1.55			(Comp: 2*ar_1 + 2, Cost: 1)                 evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 2, Cost: 1)                          evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 0)                          koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.26/1.55	
2.26/1.55		start location:	koat_start
2.26/1.55	
2.26/1.55		leaf cost:	0
2.26/1.55	
2.26/1.55	
2.26/1.55	
2.26/1.55	Repeatedly propagating knowledge in problem 6 produces the following problem:
2.26/1.55	
2.26/1.55	7:	T:
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 1)                          evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 1)                          evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: ar_1 + 1, Cost: 1)                   evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 1)) [ ar_0 >= 0 ]
2.26/1.55	
2.26/1.55			(Comp: 2, Cost: 1)                          evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: 3*ar_1^2 + 23*ar_1 + 20, Cost: 1)    evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
2.26/1.55	
2.26/1.55			(Comp: 2*ar_1 + 2, Cost: 1)                 evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
2.26/1.55	
2.26/1.55			(Comp: 3*ar_1^2 + 23*ar_1 + 20, Cost: 1)    evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))
2.26/1.55	
2.26/1.55			(Comp: 2*ar_1 + 2, Cost: 1)                 evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 2, Cost: 1)                          evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))
2.26/1.55	
2.26/1.55			(Comp: 1, Cost: 0)                          koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.26/1.55	
2.26/1.55		start location:	koat_start
2.26/1.55	
2.26/1.55		leaf cost:	0
2.26/1.55	
2.26/1.55	
2.26/1.55	
2.26/1.55	Complexity upper bound 51*ar_1 + 6*ar_1^2 + 51
2.26/1.55	
2.26/1.55	
2.26/1.55	
2.26/1.55	Time: 0.089 sec (SMT: 0.078 sec)
2.26/1.55	
2.26/1.55	
2.26/1.55	----------------------------------------
2.26/1.55	
2.26/1.55	(2)
2.26/1.55	BOUNDS(1, n^2)
2.26/1.58	EOF