2.00/1.39	WORST_CASE(?, O(n^2))
2.00/1.40	proof of /export/starexec/sandbox/output/output_files/bench.koat
2.00/1.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.00/1.40	
2.00/1.40	
2.00/1.40	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2).
2.00/1.40	
2.00/1.40	(0) CpxIntTrs
2.00/1.40	(1) Koat Proof [FINISHED, 76 ms]
2.00/1.40	(2) BOUNDS(1, n^2)
2.00/1.40	
2.00/1.40	
2.00/1.40	----------------------------------------
2.00/1.40	
2.00/1.40	(0)
2.00/1.40	Obligation:
2.00/1.40	Complexity Int TRS consisting of the following rules:
2.00/1.40	eval_textbook_ex2_start(v_i.0, v_j.0, v_n) -> Com_1(eval_textbook_ex2_bb0_in(v_i.0, v_j.0, v_n)) :|: TRUE
2.00/1.40	eval_textbook_ex2_bb0_in(v_i.0, v_j.0, v_n) -> Com_1(eval_textbook_ex2_bb1_in(0, v_j.0, v_n)) :|: TRUE
2.00/1.40	eval_textbook_ex2_bb1_in(v_i.0, v_j.0, v_n) -> Com_1(eval_textbook_ex2_bb2_in(v_i.0, 0, v_n)) :|: v_i.0 <= v_n - 1
2.00/1.40	eval_textbook_ex2_bb1_in(v_i.0, v_j.0, v_n) -> Com_1(eval_textbook_ex2_bb5_in(v_i.0, v_j.0, v_n)) :|: v_i.0 > v_n - 1
2.00/1.40	eval_textbook_ex2_bb2_in(v_i.0, v_j.0, v_n) -> Com_1(eval_textbook_ex2_bb3_in(v_i.0, v_j.0, v_n)) :|: v_j.0 <= v_i.0
2.00/1.40	eval_textbook_ex2_bb2_in(v_i.0, v_j.0, v_n) -> Com_1(eval_textbook_ex2_bb4_in(v_i.0, v_j.0, v_n)) :|: v_j.0 > v_i.0
2.00/1.40	eval_textbook_ex2_bb3_in(v_i.0, v_j.0, v_n) -> Com_1(eval_textbook_ex2_bb2_in(v_i.0, v_j.0 + 1, v_n)) :|: TRUE
2.00/1.40	eval_textbook_ex2_bb4_in(v_i.0, v_j.0, v_n) -> Com_1(eval_textbook_ex2_bb1_in(v_i.0 + 1, v_j.0, v_n)) :|: TRUE
2.00/1.40	eval_textbook_ex2_bb5_in(v_i.0, v_j.0, v_n) -> Com_1(eval_textbook_ex2_stop(v_i.0, v_j.0, v_n)) :|: TRUE
2.00/1.40	
2.00/1.40	The start-symbols are:[eval_textbook_ex2_start_3]
2.00/1.40	
2.00/1.40	
2.00/1.40	----------------------------------------
2.00/1.40	
2.00/1.40	(1) Koat Proof (FINISHED)
2.00/1.40	YES(?, 27*ar_1 + 4*ar_1^2 + 29)
2.00/1.40	
2.00/1.40	
2.00/1.40	
2.00/1.40	Initial complexity problem:
2.00/1.40	
2.00/1.40	1:	T:
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.00/1.40	
2.00/1.40		start location:	koat_start
2.00/1.40	
2.00/1.40		leaf cost:	0
2.00/1.40	
2.00/1.40	
2.00/1.40	
2.00/1.40	Repeatedly propagating knowledge in problem 1 produces the following problem:
2.00/1.40	
2.00/1.40	2:	T:
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 1)    evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 1)    evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.00/1.40	
2.00/1.40		start location:	koat_start
2.00/1.40	
2.00/1.40		leaf cost:	0
2.00/1.40	
2.00/1.40	
2.00/1.40	
2.00/1.40	A polynomial rank function with
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2start) = 2
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb0in) = 2
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb1in) = 2
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb2in) = 2
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb5in) = 1
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb3in) = 2
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb4in) = 2
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2stop) = 0
2.00/1.40	
2.00/1.40		Pol(koat_start) = 2
2.00/1.40	
2.00/1.40	orients all transitions weakly and the transitions
2.00/1.40	
2.00/1.40		evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40		evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
2.00/1.40	
2.00/1.40	strictly and produces the following problem:
2.00/1.40	
2.00/1.40	3:	T:
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 1)    evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 1)    evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: 2, Cost: 1)    evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)    evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 2, Cost: 1)    evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.00/1.40	
2.00/1.40		start location:	koat_start
2.00/1.40	
2.00/1.40		leaf cost:	0
2.00/1.40	
2.00/1.40	
2.00/1.40	
2.00/1.40	A polynomial rank function with
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2start) = V_2 + 1
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb0in) = V_2 + 1
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb1in) = -V_1 + V_2 + 1
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb2in) = -V_1 + V_2
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb5in) = -V_1 + V_2
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb3in) = -V_1 + V_2
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb4in) = -V_1 + V_2
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2stop) = -V_1 + V_2
2.00/1.40	
2.00/1.40		Pol(koat_start) = V_2 + 1
2.00/1.40	
2.00/1.40	orients all transitions weakly and the transition
2.00/1.40	
2.00/1.40		evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40	strictly and produces the following problem:
2.00/1.40	
2.00/1.40	4:	T:
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 1)           evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 1)           evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: ar_1 + 1, Cost: 1)    evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: 2, Cost: 1)           evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)           evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)           evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)           evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)           evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 2, Cost: 1)           evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 0)           koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.00/1.40	
2.00/1.40		start location:	koat_start
2.00/1.40	
2.00/1.40		leaf cost:	0
2.00/1.40	
2.00/1.40	
2.00/1.40	
2.00/1.40	A polynomial rank function with
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb4in) = 1
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb1in) = 0
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb3in) = 2
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb2in) = 2
2.00/1.40	
2.00/1.40	and size complexities
2.00/1.40	
2.00/1.40		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
2.00/1.40	
2.00/1.40		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))", 0-0) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))", 0-2) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))", 0-0) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))", 0-2) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))", 0-0) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))", 0-2) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]", 0-0) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]", 0-2) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]", 0-0) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]", 0-2) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]", 0-0) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]", 0-2) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]", 0-0) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]", 0-2) = 0
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))", 0-0) = 0
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))", 0-2) = ar_2
2.00/1.40	
2.00/1.40		S("evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0
2.00/1.40	
2.00/1.40		S("evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.00/1.40	
2.00/1.40	orients the transitions
2.00/1.40	
2.00/1.40		evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))
2.00/1.40	
2.00/1.40		evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))
2.00/1.40	
2.00/1.40		evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40		evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.00/1.40	
2.00/1.40	weakly and the transitions
2.00/1.40	
2.00/1.40		evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))
2.00/1.40	
2.00/1.40		evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40	strictly and produces the following problem:
2.00/1.40	
2.00/1.40	5:	T:
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 1)             evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 1)             evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: ar_1 + 1, Cost: 1)      evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: 2, Cost: 1)             evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)             evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.00/1.40	
2.00/1.40			(Comp: 2*ar_1 + 2, Cost: 1)    evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)             evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))
2.00/1.40	
2.00/1.40			(Comp: 2*ar_1 + 2, Cost: 1)    evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 2, Cost: 1)             evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 0)             koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.00/1.40	
2.00/1.40		start location:	koat_start
2.00/1.40	
2.00/1.40		leaf cost:	0
2.00/1.40	
2.00/1.40	
2.00/1.40	
2.00/1.40	A polynomial rank function with
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb3in) = V_1 - V_3
2.00/1.40	
2.00/1.40		Pol(evaltextbookex2bb2in) = V_1 - V_3 + 1
2.00/1.40	
2.00/1.40	and size complexities
2.00/1.40	
2.00/1.40		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
2.00/1.40	
2.00/1.40		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))", 0-0) = 2*ar_1 + 32
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))", 0-2) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))", 0-0) = 2*ar_1 + 8
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))", 0-2) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))", 0-0) = 2*ar_1 + 8
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))", 0-2) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]", 0-0) = 2*ar_1 + 8
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]", 0-2) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]", 0-0) = 2*ar_1 + 8
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]", 0-2) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]", 0-0) = 2*ar_1 + 16
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]", 0-2) = ?
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]", 0-0) = 2*ar_1 + 8
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]", 0-2) = 0
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))", 0-0) = 0
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))", 0-2) = ar_2
2.00/1.40	
2.00/1.40		S("evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0
2.00/1.40	
2.00/1.40		S("evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1
2.00/1.40	
2.00/1.40		S("evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2
2.00/1.40	
2.00/1.40	orients the transitions
2.00/1.40	
2.00/1.40		evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))
2.00/1.40	
2.00/1.40		evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.00/1.40	
2.00/1.40	weakly and the transition
2.00/1.40	
2.00/1.40		evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.00/1.40	
2.00/1.40	strictly and produces the following problem:
2.00/1.40	
2.00/1.40	6:	T:
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 1)                         evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 1)                         evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: ar_1 + 1, Cost: 1)                  evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: 2, Cost: 1)                         evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
2.00/1.40	
2.00/1.40			(Comp: 2*ar_1^2 + 11*ar_1 + 9, Cost: 1)    evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.00/1.40	
2.00/1.40			(Comp: 2*ar_1 + 2, Cost: 1)                evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: ?, Cost: 1)                         evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))
2.00/1.40	
2.00/1.40			(Comp: 2*ar_1 + 2, Cost: 1)                evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 2, Cost: 1)                         evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 0)                         koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.00/1.40	
2.00/1.40		start location:	koat_start
2.00/1.40	
2.00/1.40		leaf cost:	0
2.00/1.40	
2.00/1.40	
2.00/1.40	
2.00/1.40	Repeatedly propagating knowledge in problem 6 produces the following problem:
2.00/1.40	
2.00/1.40	7:	T:
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 1)                         evaltextbookex2start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb0in(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 1)                         evaltextbookex2bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: ar_1 + 1, Cost: 1)                  evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, 0)) [ ar_1 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: 2, Cost: 1)                         evaltextbookex2bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb5in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 ]
2.00/1.40	
2.00/1.40			(Comp: 2*ar_1^2 + 11*ar_1 + 9, Cost: 1)    evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 ]
2.00/1.40	
2.00/1.40			(Comp: 2*ar_1 + 2, Cost: 1)                evaltextbookex2bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 + 1 ]
2.00/1.40	
2.00/1.40			(Comp: 2*ar_1^2 + 11*ar_1 + 9, Cost: 1)    evaltextbookex2bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb2in(ar_0, ar_1, ar_2 + 1))
2.00/1.40	
2.00/1.40			(Comp: 2*ar_1 + 2, Cost: 1)                evaltextbookex2bb4in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2bb1in(ar_0 + 1, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 2, Cost: 1)                         evaltextbookex2bb5in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2stop(ar_0, ar_1, ar_2))
2.00/1.40	
2.00/1.40			(Comp: 1, Cost: 0)                         koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex2start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
2.00/1.40	
2.00/1.40		start location:	koat_start
2.00/1.40	
2.00/1.40		leaf cost:	0
2.00/1.40	
2.00/1.40	
2.00/1.40	
2.00/1.40	Complexity upper bound 27*ar_1 + 4*ar_1^2 + 29
2.00/1.40	
2.00/1.40	
2.00/1.40	
2.00/1.40	Time: 0.091 sec (SMT: 0.081 sec)
2.00/1.40	
2.00/1.40	
2.00/1.40	----------------------------------------
2.00/1.40	
2.00/1.40	(2)
2.00/1.40	BOUNDS(1, n^2)
2.00/1.43	EOF