1.97/1.25	WORST_CASE(?, O(n^1))
1.97/1.26	proof of /export/starexec/sandbox/output/output_files/bench.koat
1.97/1.26	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1.97/1.26	
1.97/1.26	
1.97/1.26	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1).
1.97/1.26	
1.97/1.26	(0) CpxIntTrs
1.97/1.26	(1) Koat Proof [FINISHED, 72 ms]
1.97/1.26	(2) BOUNDS(1, n^1)
1.97/1.26	
1.97/1.26	
1.97/1.26	----------------------------------------
1.97/1.26	
1.97/1.26	(0)
1.97/1.26	Obligation:
1.97/1.26	Complexity Int TRS consisting of the following rules:
1.97/1.26	eval_textbook_ex1_start(v_a, v_b, v_i.0) -> Com_1(eval_textbook_ex1_bb0_in(v_a, v_b, v_i.0)) :|: TRUE
1.97/1.26	eval_textbook_ex1_bb0_in(v_a, v_b, v_i.0) -> Com_1(eval_textbook_ex1_bb1_in(v_a, v_b, v_a)) :|: TRUE
1.97/1.26	eval_textbook_ex1_bb1_in(v_a, v_b, v_i.0) -> Com_1(eval_textbook_ex1_bb2_in(v_a, v_b, v_i.0)) :|: v_i.0 <= v_b
1.97/1.26	eval_textbook_ex1_bb1_in(v_a, v_b, v_i.0) -> Com_1(eval_textbook_ex1_bb3_in(v_a, v_b, v_i.0)) :|: v_i.0 > v_b
1.97/1.26	eval_textbook_ex1_bb2_in(v_a, v_b, v_i.0) -> Com_1(eval_textbook_ex1_bb1_in(v_a, v_b, v_i.0 + 1)) :|: TRUE
1.97/1.26	eval_textbook_ex1_bb3_in(v_a, v_b, v_i.0) -> Com_1(eval_textbook_ex1_stop(v_a, v_b, v_i.0)) :|: TRUE
1.97/1.26	
1.97/1.26	The start-symbols are:[eval_textbook_ex1_start_3]
1.97/1.26	
1.97/1.26	
1.97/1.26	----------------------------------------
1.97/1.26	
1.97/1.26	(1) Koat Proof (FINISHED)
1.97/1.26	YES(?, 2*ar_1 + 2*ar_2 + 8)
1.97/1.26	
1.97/1.26	
1.97/1.26	
1.97/1.26	Initial complexity problem:
1.97/1.26	
1.97/1.26	1:	T:
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)    evaltextbookex1start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb0in(ar_0, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)    evaltextbookex1bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb1in(ar_1, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)    evaltextbookex1bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)    evaltextbookex1bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)    evaltextbookex1bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb1in(ar_0 + 1, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)    evaltextbookex1bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1stop(ar_0, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
1.97/1.26	
1.97/1.26		start location:	koat_start
1.97/1.26	
1.97/1.26		leaf cost:	0
1.97/1.26	
1.97/1.26	
1.97/1.26	
1.97/1.26	Repeatedly propagating knowledge in problem 1 produces the following problem:
1.97/1.26	
1.97/1.26	2:	T:
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 1)    evaltextbookex1start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb0in(ar_0, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 1)    evaltextbookex1bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb1in(ar_1, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)    evaltextbookex1bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)    evaltextbookex1bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)    evaltextbookex1bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb1in(ar_0 + 1, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)    evaltextbookex1bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1stop(ar_0, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
1.97/1.26	
1.97/1.26		start location:	koat_start
1.97/1.26	
1.97/1.26		leaf cost:	0
1.97/1.26	
1.97/1.26	
1.97/1.26	
1.97/1.26	A polynomial rank function with
1.97/1.26	
1.97/1.26		Pol(evaltextbookex1start) = 2
1.97/1.26	
1.97/1.26		Pol(evaltextbookex1bb0in) = 2
1.97/1.26	
1.97/1.26		Pol(evaltextbookex1bb1in) = 2
1.97/1.26	
1.97/1.26		Pol(evaltextbookex1bb2in) = 2
1.97/1.26	
1.97/1.26		Pol(evaltextbookex1bb3in) = 1
1.97/1.26	
1.97/1.26		Pol(evaltextbookex1stop) = 0
1.97/1.26	
1.97/1.26		Pol(koat_start) = 2
1.97/1.26	
1.97/1.26	orients all transitions weakly and the transitions
1.97/1.26	
1.97/1.26		evaltextbookex1bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1stop(ar_0, ar_1, ar_2))
1.97/1.26	
1.97/1.26		evaltextbookex1bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
1.97/1.26	
1.97/1.26	strictly and produces the following problem:
1.97/1.26	
1.97/1.26	3:	T:
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 1)    evaltextbookex1start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb0in(ar_0, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 1)    evaltextbookex1bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb1in(ar_1, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)    evaltextbookex1bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
1.97/1.26	
1.97/1.26			(Comp: 2, Cost: 1)    evaltextbookex1bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)    evaltextbookex1bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb1in(ar_0 + 1, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: 2, Cost: 1)    evaltextbookex1bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1stop(ar_0, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
1.97/1.26	
1.97/1.26		start location:	koat_start
1.97/1.26	
1.97/1.26		leaf cost:	0
1.97/1.26	
1.97/1.26	
1.97/1.26	
1.97/1.26	A polynomial rank function with
1.97/1.26	
1.97/1.26		Pol(evaltextbookex1start) = -V_2 + V_3 + 1
1.97/1.26	
1.97/1.26		Pol(evaltextbookex1bb0in) = -V_2 + V_3 + 1
1.97/1.26	
1.97/1.26		Pol(evaltextbookex1bb1in) = -V_1 + V_3 + 1
1.97/1.26	
1.97/1.26		Pol(evaltextbookex1bb2in) = -V_1 + V_3
1.97/1.26	
1.97/1.26		Pol(evaltextbookex1bb3in) = -V_1 + V_3
1.97/1.26	
1.97/1.26		Pol(evaltextbookex1stop) = -V_1 + V_3
1.97/1.26	
1.97/1.26		Pol(koat_start) = -V_2 + V_3 + 1
1.97/1.26	
1.97/1.26	orients all transitions weakly and the transition
1.97/1.26	
1.97/1.26		evaltextbookex1bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
1.97/1.26	
1.97/1.26	strictly and produces the following problem:
1.97/1.26	
1.97/1.26	4:	T:
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 1)                  evaltextbookex1start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb0in(ar_0, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 1)                  evaltextbookex1bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb1in(ar_1, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: ar_1 + ar_2 + 1, Cost: 1)    evaltextbookex1bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
1.97/1.26	
1.97/1.26			(Comp: 2, Cost: 1)                  evaltextbookex1bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
1.97/1.26	
1.97/1.26			(Comp: ?, Cost: 1)                  evaltextbookex1bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb1in(ar_0 + 1, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: 2, Cost: 1)                  evaltextbookex1bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1stop(ar_0, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
1.97/1.26	
1.97/1.26		start location:	koat_start
1.97/1.26	
1.97/1.26		leaf cost:	0
1.97/1.26	
1.97/1.26	
1.97/1.26	
1.97/1.26	Repeatedly propagating knowledge in problem 4 produces the following problem:
1.97/1.26	
1.97/1.26	5:	T:
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 1)                  evaltextbookex1start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb0in(ar_0, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 1)                  evaltextbookex1bb0in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb1in(ar_1, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: ar_1 + ar_2 + 1, Cost: 1)    evaltextbookex1bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb2in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]
1.97/1.26	
1.97/1.26			(Comp: 2, Cost: 1)                  evaltextbookex1bb1in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]
1.97/1.26	
1.97/1.26			(Comp: ar_1 + ar_2 + 1, Cost: 1)    evaltextbookex1bb2in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1bb1in(ar_0 + 1, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: 2, Cost: 1)                  evaltextbookex1bb3in(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1stop(ar_0, ar_1, ar_2))
1.97/1.26	
1.97/1.26			(Comp: 1, Cost: 0)                  koat_start(ar_0, ar_1, ar_2) -> Com_1(evaltextbookex1start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
1.97/1.26	
1.97/1.26		start location:	koat_start
1.97/1.26	
1.97/1.26		leaf cost:	0
1.97/1.26	
1.97/1.26	
1.97/1.26	
1.97/1.26	Complexity upper bound 2*ar_1 + 2*ar_2 + 8
1.97/1.26	
1.97/1.26	
1.97/1.26	
1.97/1.26	Time: 0.049 sec (SMT: 0.043 sec)
1.97/1.26	
1.97/1.26	
1.97/1.26	----------------------------------------
1.97/1.26	
1.97/1.26	(2)
1.97/1.26	BOUNDS(1, n^1)
2.05/1.30	EOF