2.19/1.30	WORST_CASE(?, O(n^2))
2.19/1.31	proof of /export/starexec/sandbox2/output/output_files/bench.koat
2.19/1.31	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.19/1.31	
2.19/1.31	
2.19/1.31	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2).
2.19/1.31	
2.19/1.31	(0) CpxIntTrs
2.19/1.31	(1) Koat Proof [FINISHED, 75 ms]
2.19/1.31	(2) BOUNDS(1, n^2)
2.19/1.31	
2.19/1.31	
2.19/1.31	----------------------------------------
2.19/1.31	
2.19/1.31	(0)
2.19/1.31	Obligation:
2.19/1.31	Complexity Int TRS consisting of the following rules:
2.19/1.31	eval_ex_paper2_start(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb0_in(v_n, v_x.0, v_y.0)) :|: TRUE
2.19/1.31	eval_ex_paper2_bb0_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb1_in(v_n, 1, v_y.0)) :|: TRUE
2.19/1.31	eval_ex_paper2_bb1_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb2_in(v_n, v_x.0, v_x.0)) :|: v_x.0 <= v_n
2.19/1.31	eval_ex_paper2_bb1_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb5_in(v_n, v_x.0, v_y.0)) :|: v_x.0 > v_n
2.19/1.31	eval_ex_paper2_bb2_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb3_in(v_n, v_x.0, v_y.0)) :|: v_y.0 <= v_n
2.19/1.31	eval_ex_paper2_bb2_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb4_in(v_n, v_x.0, v_y.0)) :|: v_y.0 > v_n
2.19/1.31	eval_ex_paper2_bb3_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb2_in(v_n, v_x.0, v_y.0 + 1)) :|: TRUE
2.19/1.31	eval_ex_paper2_bb4_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_bb1_in(v_n, v_x.0 + 1, v_y.0)) :|: TRUE
2.19/1.31	eval_ex_paper2_bb5_in(v_n, v_x.0, v_y.0) -> Com_1(eval_ex_paper2_stop(v_n, v_x.0, v_y.0)) :|: TRUE
2.19/1.31	
2.19/1.31	The start-symbols are:[eval_ex_paper2_start_3]
2.19/1.31	
2.19/1.31	
2.19/1.31	----------------------------------------
2.19/1.31	
2.19/1.31	(1) Koat Proof (FINISHED)
2.19/1.31	YES(?, 27*ar_1 + 6*ar_1^2 + 6)
2.19/1.31	
2.19/1.31	
2.19/1.31	
2.19/1.31	Initial complexity problem:
2.19/1.31	
2.19/1.31	1:	T:
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.19/1.31	
2.19/1.31		start location:	koat_start
2.19/1.31	
2.19/1.31		leaf cost:	0
2.19/1.31	
2.19/1.31	
2.19/1.31	
2.19/1.31	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.19/1.31	
2.19/1.31	2:	T:
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 1)    evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 1)    evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.19/1.31	
2.19/1.31		start location:	koat_start
2.19/1.31	
2.19/1.31		leaf cost:	0
2.19/1.31	
2.19/1.31	
2.19/1.31	
2.19/1.31	A polynomial rank function with
2.19/1.31	
2.19/1.31		Pol(evalexpaper2start) = 2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb0in) = 2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb1in) = 2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb2in) = 2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb5in) = 1
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb3in) = 2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb4in) = 2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2stop) = 0
2.19/1.31	
2.19/1.31		Pol(koat_start) = 2
2.19/1.31	
2.19/1.31	orients all transitions weakly and the transitions
2.19/1.31	
2.19/1.31		evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31		evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31	strictly and produces the following problem:
2.19/1.31	
2.19/1.31	3:	T:
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 1)    evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 1)    evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]
2.19/1.31	
2.19/1.31			(Comp: 2, Cost: 1)    evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)    evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 2, Cost: 1)    evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.19/1.31	
2.19/1.31		start location:	koat_start
2.19/1.31	
2.19/1.31		leaf cost:	0
2.19/1.31	
2.19/1.31	
2.19/1.31	
2.19/1.31	A polynomial rank function with
2.19/1.31	
2.19/1.31		Pol(evalexpaper2start) = V_2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb0in) = V_2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb1in) = -V_1 + V_2 + 1
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb2in) = -V_1 + V_2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb5in) = -V_1 + V_2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb3in) = -V_1 + V_2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb4in) = -V_1 + V_2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2stop) = -V_1 + V_2
2.19/1.31	
2.19/1.31		Pol(koat_start) = V_2
2.19/1.31	
2.19/1.31	orients all transitions weakly and the transition
2.19/1.31	
2.19/1.31		evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]
2.19/1.31	
2.19/1.31	strictly and produces the following problem:
2.19/1.31	
2.19/1.31	4:	T:
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 1)       evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 1)       evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: ar_1, Cost: 1)    evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]
2.19/1.31	
2.19/1.31			(Comp: 2, Cost: 1)       evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)       evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)       evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)       evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)       evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 2, Cost: 1)       evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.19/1.31	
2.19/1.31		start location:	koat_start
2.19/1.31	
2.19/1.31		leaf cost:	0
2.19/1.31	
2.19/1.31	
2.19/1.31	
2.19/1.31	A polynomial rank function with
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb4in) = 1
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb1in) = 0
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb3in) = 2
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb2in) = 2
2.19/1.31	
2.19/1.31	and size complexities
2.19/1.31	
2.19/1.31		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
2.19/1.31	
2.19/1.31		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
2.19/1.31	
2.19/1.31		S("evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))", 0-0) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))", 0-0) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))", 0-0) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]", 0-0) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]", 0-0) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]", 0-0) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]", 0-0) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))", 0-0) = 1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))", 0-2) = ar_2
2.19/1.31	
2.19/1.31		S("evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0
2.19/1.31	
2.19/1.31		S("evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.19/1.31	
2.19/1.31	orients the transitions
2.19/1.31	
2.19/1.31		evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))
2.19/1.31	
2.19/1.31		evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))
2.19/1.31	
2.19/1.31		evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31		evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]
2.19/1.31	
2.19/1.31	weakly and the transitions
2.19/1.31	
2.19/1.31		evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))
2.19/1.31	
2.19/1.31		evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31	strictly and produces the following problem:
2.19/1.31	
2.19/1.31	5:	T:
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 1)         evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 1)         evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: ar_1, Cost: 1)      evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]
2.19/1.31	
2.19/1.31			(Comp: 2, Cost: 1)         evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)         evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]
2.19/1.31	
2.19/1.31			(Comp: 2*ar_1, Cost: 1)    evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)         evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))
2.19/1.31	
2.19/1.31			(Comp: 2*ar_1, Cost: 1)    evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 2, Cost: 1)         evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 0)         koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.19/1.31	
2.19/1.31		start location:	koat_start
2.19/1.31	
2.19/1.31		leaf cost:	0
2.19/1.31	
2.19/1.31	
2.19/1.31	
2.19/1.31	A polynomial rank function with
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb3in) = V_2 - V_3
2.19/1.31	
2.19/1.31		Pol(evalexpaper2bb2in) = V_2 - V_3 + 1
2.19/1.31	
2.19/1.31	and size complexities
2.19/1.31	
2.19/1.31		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
2.19/1.31	
2.19/1.31		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
2.19/1.31	
2.19/1.31		S("evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))", 0-0) = 2*ar_1 + 20
2.19/1.31	
2.19/1.31		S("evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))", 0-0) = 2*ar_1 + 4
2.19/1.31	
2.19/1.31		S("evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))", 0-0) = 2*ar_1 + 4
2.19/1.31	
2.19/1.31		S("evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]", 0-0) = 2*ar_1 + 4
2.19/1.31	
2.19/1.31		S("evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]", 0-0) = 2*ar_1 + 4
2.19/1.31	
2.19/1.31		S("evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]", 0-0) = 2*ar_1 + 10
2.19/1.31	
2.19/1.31		S("evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]", 0-2) = ?
2.19/1.31	
2.19/1.31		S("evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]", 0-0) = 2*ar_1 + 4
2.19/1.31	
2.19/1.31		S("evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]", 0-2) = 2*ar_1 + 10
2.19/1.31	
2.19/1.31		S("evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))", 0-0) = 1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))", 0-2) = ar_2
2.19/1.31	
2.19/1.31		S("evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0
2.19/1.31	
2.19/1.31		S("evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.19/1.31	
2.19/1.31		S("evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.19/1.31	
2.19/1.31	orients the transitions
2.19/1.31	
2.19/1.31		evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))
2.19/1.31	
2.19/1.31		evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]
2.19/1.31	
2.19/1.31	weakly and the transition
2.19/1.31	
2.19/1.31		evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]
2.19/1.31	
2.19/1.31	strictly and produces the following problem:
2.19/1.31	
2.19/1.31	6:	T:
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 1)                     evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 1)                     evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: ar_1, Cost: 1)                  evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]
2.19/1.31	
2.19/1.31			(Comp: 2, Cost: 1)                     evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: 3*ar_1^2 + 11*ar_1, Cost: 1)    evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]
2.19/1.31	
2.19/1.31			(Comp: 2*ar_1, Cost: 1)                evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: ?, Cost: 1)                     evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))
2.19/1.31	
2.19/1.31			(Comp: 2*ar_1, Cost: 1)                evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 2, Cost: 1)                     evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 0)                     koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.19/1.31	
2.19/1.31		start location:	koat_start
2.19/1.31	
2.19/1.31		leaf cost:	0
2.19/1.31	
2.19/1.31	
2.19/1.31	
2.19/1.31	Repeatedly propagating knowledge in problem 6 produces the following problem:
2.19/1.31	
2.19/1.31	7:	T:
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 1)                     evalexpaper2start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb0in(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 1)                     evalexpaper2bb0in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: ar_1, Cost: 1)                  evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_0)) [ ar_1 >= ar_0 ]
2.19/1.31	
2.19/1.31			(Comp: 2, Cost: 1)                     evalexpaper2bb1in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: 3*ar_1^2 + 11*ar_1, Cost: 1)    evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb3in(ar_0, ar_1, ar_2)) [ ar_1 >= ar_2 ]
2.19/1.31	
2.19/1.31			(Comp: 2*ar_1, Cost: 1)                evalexpaper2bb2in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_1 + 1 ]
2.19/1.31	
2.19/1.31			(Comp: 3*ar_1^2 + 11*ar_1, Cost: 1)    evalexpaper2bb3in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb2in(ar_0, ar_1, ar_2 + 1))
2.19/1.31	
2.19/1.31			(Comp: 2*ar_1, Cost: 1)                evalexpaper2bb4in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2bb1in(ar_0 + 1, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 2, Cost: 1)                     evalexpaper2bb5in(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2stop(ar_0, ar_1, ar_2))
2.19/1.31	
2.19/1.31			(Comp: 1, Cost: 0)                     koat_start(ar_0, ar_1, ar_2) -> Com_1(evalexpaper2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.19/1.31	
2.19/1.31		start location:	koat_start
2.19/1.31	
2.19/1.31		leaf cost:	0
2.19/1.31	
2.19/1.31	
2.19/1.31	
2.19/1.31	Complexity upper bound 27*ar_1 + 6*ar_1^2 + 6
2.19/1.31	
2.19/1.31	
2.19/1.31	
2.19/1.31	Time: 0.088 sec (SMT: 0.077 sec)
2.19/1.31	
2.19/1.31	
2.19/1.31	----------------------------------------
2.19/1.31	
2.19/1.31	(2)
2.19/1.31	BOUNDS(1, n^2)
2.19/1.32	EOF