3.23/1.65	WORST_CASE(?, O(1))
3.23/1.66	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.23/1.66	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.23/1.66	
3.23/1.66	
3.23/1.66	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.23/1.66	
3.23/1.66	(0) CpxIntTrs
3.23/1.66	(1) Koat Proof [FINISHED, 19 ms]
3.23/1.66	(2) BOUNDS(1, 1)
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(0)
3.23/1.66	Obligation:
3.23/1.66	Complexity Int TRS consisting of the following rules:
3.23/1.66	f0(A, B, C) -> Com_1(f1(A, B, 2)) :|: A >= 0 && 3 >= A && 3 >= B && B >= 0
3.23/1.66	f1(A, B, C) -> Com_1(f1(A, B + 1, C)) :|: C + A >= 2 * B + 1
3.23/1.66	f1(A, B, C) -> Com_1(f1(A, B - 1, C)) :|: 2 * B >= 2 + C + A
3.23/1.66	
3.23/1.66	The start-symbols are:[f0_3]
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(1) Koat Proof (FINISHED)
3.23/1.66	YES(?, 23)
3.23/1.66	
3.23/1.66	
3.23/1.66	
3.23/1.66	Initial complexity problem:
3.23/1.66	
3.23/1.66	1:	T:
3.23/1.66	
3.23/1.66			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ]
3.23/1.66	
3.23/1.66			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.23/1.66	
3.23/1.66			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.23/1.66	
3.23/1.66			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.23/1.66	
3.23/1.66		start location:	koat_start
3.23/1.66	
3.23/1.66		leaf cost:	0
3.23/1.66	
3.23/1.66	
3.23/1.66	
3.23/1.66	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.23/1.66	
3.23/1.66	2:	T:
3.23/1.66	
3.23/1.66			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ]
3.23/1.66	
3.23/1.66			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.23/1.66	
3.23/1.66			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.23/1.66	
3.23/1.66			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.23/1.66	
3.23/1.66		start location:	koat_start
3.23/1.66	
3.23/1.66		leaf cost:	0
3.23/1.66	
3.23/1.66	
3.23/1.66	
3.23/1.66	A polynomial rank function with
3.23/1.66	
3.23/1.66		Pol(f1) = -V_1 + 2*V_2 - V_3
3.23/1.66	
3.23/1.66	and size complexities
3.23/1.66	
3.23/1.66		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
3.23/1.66	
3.23/1.66		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
3.23/1.66	
3.23/1.66		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
3.23/1.66	
3.23/1.66		S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-0) = 3
3.23/1.66	
3.23/1.66		S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-1) = ?
3.23/1.66	
3.23/1.66		S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-2) = 2
3.23/1.66	
3.23/1.66		S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-0) = 3
3.23/1.66	
3.23/1.66		S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-1) = ?
3.23/1.66	
3.23/1.66		S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-2) = 2
3.23/1.66	
3.23/1.66		S("f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-0) = 3
3.23/1.66	
3.23/1.66		S("f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-1) = 3
3.23/1.66	
3.23/1.66		S("f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-2) = 2
3.23/1.66	
3.23/1.66	orients the transitions
3.23/1.66	
3.23/1.66		f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.23/1.66	
3.23/1.66	weakly and the transition
3.23/1.66	
3.23/1.66		f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.23/1.66	
3.23/1.66	strictly and produces the following problem:
3.23/1.66	
3.23/1.66	3:	T:
3.23/1.66	
3.23/1.66			(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ]
3.23/1.66	
3.23/1.66			(Comp: ?, Cost: 1)     f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.23/1.66	
3.23/1.66			(Comp: 11, Cost: 1)    f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.23/1.66	
3.23/1.66			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.23/1.66	
3.23/1.66		start location:	koat_start
3.23/1.66	
3.23/1.66		leaf cost:	0
3.23/1.66	
3.23/1.66	
3.23/1.66	
3.23/1.66	A polynomial rank function with
3.23/1.66	
3.23/1.66		Pol(f1) = V_1 - 2*V_2 + V_3
3.23/1.66	
3.23/1.66	and size complexities
3.23/1.66	
3.23/1.66		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
3.23/1.66	
3.23/1.66		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
3.23/1.66	
3.23/1.66		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
3.23/1.66	
3.23/1.66		S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-0) = 3
3.23/1.66	
3.23/1.66		S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-1) = 14
3.23/1.66	
3.23/1.66		S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-2) = 2
3.23/1.66	
3.23/1.66		S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-0) = 3
3.23/1.66	
3.23/1.66		S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-1) = ?
3.23/1.66	
3.23/1.66		S("f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-2) = 2
3.23/1.66	
3.23/1.66		S("f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-0) = 3
3.23/1.66	
3.23/1.66		S("f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-1) = 3
3.23/1.66	
3.23/1.66		S("f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-2) = 2
3.23/1.66	
3.23/1.66	orients the transitions
3.23/1.66	
3.23/1.66		f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.23/1.66	
3.23/1.66	weakly and the transition
3.23/1.66	
3.23/1.66		f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.23/1.66	
3.23/1.66	strictly and produces the following problem:
3.23/1.66	
3.23/1.66	4:	T:
3.23/1.66	
3.23/1.66			(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1, 2)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ]
3.23/1.66	
3.23/1.66			(Comp: 11, Cost: 1)    f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 + 1, ar_2)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.23/1.66	
3.23/1.66			(Comp: 11, Cost: 1)    f1(ar_0, ar_1, ar_2) -> Com_1(f1(ar_0, ar_1 - 1, ar_2)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.23/1.66	
3.23/1.66			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1, ar_2) -> Com_1(f0(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.23/1.66	
3.23/1.66		start location:	koat_start
3.23/1.66	
3.23/1.66		leaf cost:	0
3.23/1.66	
3.23/1.66	
3.23/1.66	
3.23/1.66	Complexity upper bound 23
3.23/1.66	
3.23/1.66	
3.23/1.66	
3.23/1.66	Time: 0.077 sec (SMT: 0.071 sec)
3.23/1.66	
3.23/1.66	
3.23/1.66	----------------------------------------
3.23/1.66	
3.23/1.66	(2)
3.23/1.66	BOUNDS(1, 1)
3.23/1.68	EOF