2.25/1.33	WORST_CASE(?, O(n^2))
2.25/1.34	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.25/1.34	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.25/1.34	
2.25/1.34	
2.25/1.34	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2).
2.25/1.34	
2.25/1.34	(0) CpxIntTrs
2.25/1.34	(1) Koat Proof [FINISHED, 74 ms]
2.25/1.34	(2) BOUNDS(1, n^2)
2.25/1.34	
2.25/1.34	
2.25/1.34	----------------------------------------
2.25/1.34	
2.25/1.34	(0)
2.25/1.34	Obligation:
2.25/1.34	Complexity Int TRS consisting of the following rules:
2.25/1.34	eval_while2_start(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb0_in(v_N, v_i.0, v_j.0)) :|: TRUE
2.25/1.34	eval_while2_bb0_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb1_in(v_N, v_N, v_j.0)) :|: TRUE
2.25/1.34	eval_while2_bb1_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb2_in(v_N, v_i.0, v_N)) :|: v_i.0 > 0
2.25/1.34	eval_while2_bb1_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb5_in(v_N, v_i.0, v_j.0)) :|: v_i.0 <= 0
2.25/1.34	eval_while2_bb2_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb3_in(v_N, v_i.0, v_j.0)) :|: v_j.0 > 0
2.25/1.34	eval_while2_bb2_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb4_in(v_N, v_i.0, v_j.0)) :|: v_j.0 <= 0
2.25/1.34	eval_while2_bb3_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb2_in(v_N, v_i.0, v_j.0 - 1)) :|: TRUE
2.25/1.34	eval_while2_bb4_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_bb1_in(v_N, v_i.0 - 1, v_j.0)) :|: TRUE
2.25/1.34	eval_while2_bb5_in(v_N, v_i.0, v_j.0) -> Com_1(eval_while2_stop(v_N, v_i.0, v_j.0)) :|: TRUE
2.25/1.34	
2.25/1.34	The start-symbols are:[eval_while2_start_3]
2.25/1.34	
2.25/1.34	
2.25/1.34	----------------------------------------
2.25/1.34	
2.25/1.34	(1) Koat Proof (FINISHED)
2.25/1.34	YES(?, 7*ar_1 + 2*ar_1^2 + 11)
2.25/1.34	
2.25/1.34	
2.25/1.34	
2.25/1.34	Initial complexity problem:
2.25/1.34	
2.25/1.34	1:	T:
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.25/1.34	
2.25/1.34		start location:	koat_start
2.25/1.34	
2.25/1.34		leaf cost:	0
2.25/1.34	
2.25/1.34	
2.25/1.34	
2.25/1.34	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.25/1.34	
2.25/1.34	2:	T:
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 1)    evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 1)    evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.25/1.34	
2.25/1.34		start location:	koat_start
2.25/1.34	
2.25/1.34		leaf cost:	0
2.25/1.34	
2.25/1.34	
2.25/1.34	
2.25/1.34	A polynomial rank function with
2.25/1.34	
2.25/1.34		Pol(evalwhile2start) = 2
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb0in) = 2
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb1in) = 2
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb2in) = 2
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb5in) = 1
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb3in) = 2
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb4in) = 2
2.25/1.34	
2.25/1.34		Pol(evalwhile2stop) = 0
2.25/1.34	
2.25/1.34		Pol(koat_start) = 2
2.25/1.34	
2.25/1.34	orients all transitions weakly and the transitions
2.25/1.34	
2.25/1.34		evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
2.25/1.34	
2.25/1.34		evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.25/1.34	
2.25/1.34	strictly and produces the following problem:
2.25/1.34	
2.25/1.34	3:	T:
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 1)    evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 1)    evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
2.25/1.34	
2.25/1.34			(Comp: 2, Cost: 1)    evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)    evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: 2, Cost: 1)    evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.25/1.34	
2.25/1.34		start location:	koat_start
2.25/1.34	
2.25/1.34		leaf cost:	0
2.25/1.34	
2.25/1.34	
2.25/1.34	
2.25/1.34	A polynomial rank function with
2.25/1.34	
2.25/1.34		Pol(evalwhile2start) = V_2 + 1
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb0in) = V_2 + 1
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb1in) = V_1 + 1
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb2in) = V_1
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb5in) = V_1
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb3in) = V_1
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb4in) = V_1
2.25/1.34	
2.25/1.34		Pol(evalwhile2stop) = V_1
2.25/1.34	
2.25/1.34		Pol(koat_start) = V_2 + 1
2.25/1.34	
2.25/1.34	orients all transitions weakly and the transition
2.25/1.34	
2.25/1.34		evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
2.25/1.34	
2.25/1.34	strictly and produces the following problem:
2.25/1.34	
2.25/1.34	4:	T:
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 1)           evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 1)           evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: ar_1 + 1, Cost: 1)    evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
2.25/1.34	
2.25/1.34			(Comp: 2, Cost: 1)           evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)           evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)           evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)           evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)           evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: 2, Cost: 1)           evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.25/1.34	
2.25/1.34		start location:	koat_start
2.25/1.34	
2.25/1.34		leaf cost:	0
2.25/1.34	
2.25/1.34	
2.25/1.34	
2.25/1.34	A polynomial rank function with
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb4in) = 1
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb1in) = 0
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb3in) = 2
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb2in) = 2
2.25/1.34	
2.25/1.34	and size complexities
2.25/1.34	
2.25/1.34		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
2.25/1.34	
2.25/1.34		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
2.25/1.34	
2.25/1.34		S("evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-0) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-2) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))", 0-0) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))", 0-2) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-0) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-2) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-0) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-2) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-0) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-2) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-0) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-2) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-0) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-2) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))", 0-0) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))", 0-2) = ar_2
2.25/1.34	
2.25/1.34		S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0
2.25/1.34	
2.25/1.34		S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.25/1.34	
2.25/1.34	orients the transitions
2.25/1.34	
2.25/1.34		evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))
2.25/1.34	
2.25/1.34		evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
2.25/1.34	
2.25/1.34		evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.25/1.34	
2.25/1.34		evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.25/1.34	
2.25/1.34	weakly and the transitions
2.25/1.34	
2.25/1.34		evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))
2.25/1.34	
2.25/1.34		evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.25/1.34	
2.25/1.34	strictly and produces the following problem:
2.25/1.34	
2.25/1.34	5:	T:
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 1)             evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 1)             evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: ar_1 + 1, Cost: 1)      evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
2.25/1.34	
2.25/1.34			(Comp: 2, Cost: 1)             evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)             evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.25/1.34	
2.25/1.34			(Comp: 2*ar_1 + 2, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.25/1.34	
2.25/1.34			(Comp: ?, Cost: 1)             evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
2.25/1.34	
2.25/1.34			(Comp: 2*ar_1 + 2, Cost: 1)    evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: 2, Cost: 1)             evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
2.25/1.34	
2.25/1.34			(Comp: 1, Cost: 0)             koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.25/1.34	
2.25/1.34		start location:	koat_start
2.25/1.34	
2.25/1.34		leaf cost:	0
2.25/1.34	
2.25/1.34	
2.25/1.34	
2.25/1.34	A polynomial rank function with
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb3in) = V_3 - 1
2.25/1.34	
2.25/1.34		Pol(evalwhile2bb2in) = V_3
2.25/1.34	
2.25/1.34	and size complexities
2.25/1.34	
2.25/1.34		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
2.25/1.34	
2.25/1.34		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
2.25/1.34	
2.25/1.34		S("evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-0) = 3*ar_1 + 162
2.25/1.34	
2.25/1.34		S("evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))", 0-2) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))", 0-0) = 3*ar_1 + 18
2.25/1.34	
2.25/1.34		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))", 0-2) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-0) = 3*ar_1 + 18
2.25/1.34	
2.25/1.34		S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))", 0-2) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-0) = 3*ar_1 + 18
2.25/1.34	
2.25/1.34		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]", 0-2) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-0) = 3*ar_1 + 18
2.25/1.34	
2.25/1.34		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]", 0-2) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-0) = 3*ar_1 + 54
2.25/1.34	
2.25/1.34		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-2) = ?
2.25/1.34	
2.25/1.34		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-0) = 3*ar_1 + 18
2.25/1.34	
2.25/1.34		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]", 0-2) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))", 0-0) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))", 0-1) = ar_1
2.25/1.34	
2.25/1.34		S("evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))", 0-2) = ar_2
2.25/1.35	
2.25/1.35		S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0
2.25/1.35	
2.25/1.35		S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.25/1.35	
2.25/1.35		S("evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.25/1.35	
2.25/1.35	orients the transitions
2.25/1.35	
2.25/1.35		evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
2.25/1.35	
2.25/1.35		evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.25/1.35	
2.25/1.35	weakly and the transition
2.25/1.35	
2.25/1.35		evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.25/1.35	
2.25/1.35	strictly and produces the following problem:
2.25/1.35	
2.25/1.35	6:	T:
2.25/1.35	
2.25/1.35			(Comp: 1, Cost: 1)                evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))
2.25/1.35	
2.25/1.35			(Comp: 1, Cost: 1)                evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))
2.25/1.35	
2.25/1.35			(Comp: ar_1 + 1, Cost: 1)         evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
2.25/1.35	
2.25/1.35			(Comp: 2, Cost: 1)                evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.25/1.35	
2.25/1.35			(Comp: ar_1^2 + ar_1, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.25/1.35	
2.25/1.35			(Comp: 2*ar_1 + 2, Cost: 1)       evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.25/1.35	
2.25/1.35			(Comp: ?, Cost: 1)                evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
2.25/1.35	
2.25/1.35			(Comp: 2*ar_1 + 2, Cost: 1)       evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))
2.25/1.35	
2.25/1.35			(Comp: 2, Cost: 1)                evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
2.25/1.35	
2.25/1.35			(Comp: 1, Cost: 0)                koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.25/1.35	
2.25/1.35		start location:	koat_start
2.25/1.35	
2.25/1.35		leaf cost:	0
2.25/1.35	
2.25/1.35	
2.25/1.35	
2.25/1.35	Repeatedly propagating knowledge in problem 6 produces the following problem:
2.25/1.35	
2.25/1.35	7:	T:
2.25/1.35	
2.25/1.35			(Comp: 1, Cost: 1)                evalwhile2start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb0in(ar_0, ar_1, ar_2))
2.25/1.35	
2.25/1.35			(Comp: 1, Cost: 1)                evalwhile2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_1, ar_1, ar_2))
2.25/1.35	
2.25/1.35			(Comp: ar_1 + 1, Cost: 1)         evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_1)) [ ar_0 >= 1 ]
2.25/1.35	
2.25/1.35			(Comp: 2, Cost: 1)                evalwhile2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]
2.25/1.35	
2.25/1.35			(Comp: ar_1^2 + ar_1, Cost: 1)    evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb3in(ar_0, ar_1, ar_2)) [ ar_2 >= 1 ]
2.25/1.35	
2.25/1.35			(Comp: 2*ar_1 + 2, Cost: 1)       evalwhile2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_2 ]
2.25/1.35	
2.25/1.35			(Comp: ar_1^2 + ar_1, Cost: 1)    evalwhile2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb2in(ar_0, ar_1, ar_2 - 1))
2.25/1.35	
2.25/1.35			(Comp: 2*ar_1 + 2, Cost: 1)       evalwhile2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2bb1in(ar_0 - 1, ar_1, ar_2))
2.25/1.35	
2.25/1.35			(Comp: 2, Cost: 1)                evalwhile2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalwhile2stop(ar_0, ar_1, ar_2))
2.25/1.35	
2.25/1.35			(Comp: 1, Cost: 0)                koat_start(ar_0, ar_1, ar_2) -> Com_1(evalwhile2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.25/1.35	
2.25/1.35		start location:	koat_start
2.25/1.35	
2.25/1.35		leaf cost:	0
2.25/1.35	
2.25/1.35	
2.25/1.35	
2.25/1.35	Complexity upper bound 7*ar_1 + 2*ar_1^2 + 11
2.25/1.35	
2.25/1.35	
2.25/1.35	
2.25/1.35	Time: 0.089 sec (SMT: 0.078 sec)
2.25/1.35	
2.25/1.35	
2.25/1.35	----------------------------------------
2.25/1.35	
2.25/1.35	(2)
2.25/1.35	BOUNDS(1, n^2)
2.25/1.36	EOF