3.41/1.81	WORST_CASE(?, O(1))
3.41/1.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.41/1.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.41/1.81	
3.41/1.81	
3.41/1.81	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.41/1.81	
3.41/1.81	(0) CpxIntTrs
3.41/1.81	(1) Koat Proof [FINISHED, 133 ms]
3.41/1.81	(2) BOUNDS(1, 1)
3.41/1.81	
3.41/1.81	
3.41/1.81	----------------------------------------
3.41/1.81	
3.41/1.81	(0)
3.41/1.81	Obligation:
3.41/1.81	Complexity Int TRS consisting of the following rules:
3.41/1.81	f0(A, B, C, D) -> Com_1(f1(A, B, 2, D)) :|: A >= 0 && 3 >= A && 3 >= B && B >= 0
3.41/1.81	f1(A, B, C, D) -> Com_1(f1(A, B + 1, C, B + 1)) :|: C + A >= 2 * B + 1 && 0 >= 2
3.41/1.81	f1(A, B, C, D) -> Com_1(f1(A, B + 1, C, B + 1)) :|: C + A >= 2 * B + 1
3.41/1.81	f1(A, B, C, D) -> Com_1(f1(A, B - 1, C, B - 1)) :|: 2 * B >= 2 + C + A
3.41/1.81	f1(A, B, C, D) -> Com_1(f1(A, B - 1, C, B - 1)) :|: 2 * B >= 2 + C + A && 0 >= 2
3.41/1.81	f1(A, B, C, D) -> Com_1(f1(A, B, C, B)) :|: 0 >= 1 && 2 * B >= C + A && C + A + 1 >= 2 * B
3.41/1.81	
3.41/1.81	The start-symbols are:[f0_4]
3.41/1.81	
3.41/1.81	
3.41/1.81	----------------------------------------
3.41/1.81	
3.41/1.81	(1) Koat Proof (FINISHED)
3.41/1.81	YES(?, 23)
3.41/1.81	
3.41/1.81	
3.41/1.81	
3.41/1.81	Initial complexity problem:
3.41/1.81	
3.41/1.81	1:	T:
3.41/1.81	
3.41/1.81			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ]
3.41/1.81	
3.41/1.81			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 /\ 0 >= 2 ]
3.41/1.81	
3.41/1.81			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.41/1.81	
3.41/1.81			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.41/1.81	
3.41/1.81			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 /\ 0 >= 2 ]
3.41/1.81	
3.41/1.81			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, ar_2, ar_1)) [ 0 >= 1 /\ 2*ar_1 >= ar_2 + ar_0 /\ ar_2 + ar_0 + 1 >= 2*ar_1 ]
3.41/1.81	
3.41/1.81			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.41/1.81	
3.41/1.81		start location:	koat_start
3.41/1.81	
3.41/1.81		leaf cost:	0
3.41/1.81	
3.41/1.81	
3.41/1.81	
3.41/1.81	Testing for reachability in the complexity graph removes the following transitions from problem 1:
3.41/1.81	
3.41/1.81		f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 /\ 0 >= 2 ]
3.41/1.81	
3.41/1.81		f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 /\ 0 >= 2 ]
3.41/1.81	
3.41/1.81		f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, ar_2, ar_1)) [ 0 >= 1 /\ 2*ar_1 >= ar_2 + ar_0 /\ ar_2 + ar_0 + 1 >= 2*ar_1 ]
3.41/1.81	
3.41/1.81	We thus obtain the following problem:
3.41/1.81	
3.41/1.81	2:	T:
3.41/1.81	
3.41/1.81			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.41/1.81	
3.41/1.81			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.41/1.81	
3.41/1.81			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ]
3.41/1.81	
3.41/1.81			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.41/1.81	
3.41/1.81		start location:	koat_start
3.41/1.81	
3.41/1.81		leaf cost:	0
3.41/1.81	
3.41/1.82	
3.41/1.82	
3.41/1.82	Repeatedly propagating knowledge in problem 2 produces the following problem:
3.41/1.82	
3.41/1.82	3:	T:
3.41/1.82	
3.41/1.82			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.41/1.82	
3.41/1.82			(Comp: ?, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.41/1.82	
3.41/1.82			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ]
3.41/1.82	
3.41/1.82			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.41/1.82	
3.41/1.82		start location:	koat_start
3.41/1.82	
3.41/1.82		leaf cost:	0
3.41/1.82	
3.41/1.82	
3.41/1.82	
3.41/1.82	A polynomial rank function with
3.41/1.82	
3.41/1.82		Pol(f1) = -V_1 + 2*V_2 - V_3
3.41/1.82	
3.41/1.82	and size complexities
3.41/1.82	
3.41/1.82		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-0) = ar_0
3.41/1.82	
3.41/1.82		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-1) = ar_1
3.41/1.82	
3.41/1.82		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-2) = ar_2
3.41/1.82	
3.41/1.82		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3
3.41/1.82	
3.41/1.82		S("f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-0) = 3
3.41/1.82	
3.41/1.82		S("f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-1) = 3
3.41/1.82	
3.41/1.82		S("f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-2) = 2
3.41/1.82	
3.41/1.82		S("f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-3) = ar_3
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-0) = 3
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-1) = ?
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-2) = 2
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-3) = ?
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-0) = 3
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-1) = ?
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-2) = 2
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-3) = ?
3.41/1.82	
3.41/1.82	orients the transitions
3.41/1.82	
3.41/1.82		f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.41/1.82	
3.41/1.82	weakly and the transition
3.41/1.82	
3.41/1.82		f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.41/1.82	
3.41/1.82	strictly and produces the following problem:
3.41/1.82	
3.41/1.82	4:	T:
3.41/1.82	
3.41/1.82			(Comp: 11, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.41/1.82	
3.41/1.82			(Comp: ?, Cost: 1)     f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.41/1.82	
3.41/1.82			(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ]
3.41/1.82	
3.41/1.82			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.41/1.82	
3.41/1.82		start location:	koat_start
3.41/1.82	
3.41/1.82		leaf cost:	0
3.41/1.82	
3.41/1.82	
3.41/1.82	
3.41/1.82	A polynomial rank function with
3.41/1.82	
3.41/1.82		Pol(f1) = V_1 - 2*V_2 + V_3
3.41/1.82	
3.41/1.82	and size complexities
3.41/1.82	
3.41/1.82		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-0) = ar_0
3.41/1.82	
3.41/1.82		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-1) = ar_1
3.41/1.82	
3.41/1.82		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-2) = ar_2
3.41/1.82	
3.41/1.82		S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3
3.41/1.82	
3.41/1.82		S("f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-0) = 3
3.41/1.82	
3.41/1.82		S("f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-1) = 3
3.41/1.82	
3.41/1.82		S("f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-2) = 2
3.41/1.82	
3.41/1.82		S("f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\\ 3 >= ar_0 /\\ 3 >= ar_1 /\\ ar_1 >= 0 ]", 0-3) = ar_3
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-0) = 3
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-1) = ?
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-2) = 2
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]", 0-3) = ?
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-0) = 3
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-1) = 14
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-2) = 2
3.41/1.82	
3.41/1.82		S("f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]", 0-3) = 15
3.41/1.82	
3.41/1.82	orients the transitions
3.41/1.82	
3.41/1.82		f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.41/1.82	
3.41/1.82	weakly and the transition
3.41/1.82	
3.41/1.82		f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.41/1.82	
3.41/1.82	strictly and produces the following problem:
3.41/1.82	
3.41/1.82	5:	T:
3.41/1.82	
3.41/1.82			(Comp: 11, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 - 1, ar_2, ar_1 - 1)) [ 2*ar_1 >= ar_2 + ar_0 + 2 ]
3.41/1.82	
3.41/1.82			(Comp: 11, Cost: 1)    f1(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1 + 1, ar_2, ar_1 + 1)) [ ar_2 + ar_0 >= 2*ar_1 + 1 ]
3.41/1.82	
3.41/1.82			(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f1(ar_0, ar_1, 2, ar_3)) [ ar_0 >= 0 /\ 3 >= ar_0 /\ 3 >= ar_1 /\ ar_1 >= 0 ]
3.41/1.82	
3.41/1.82			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]
3.41/1.82	
3.41/1.82		start location:	koat_start
3.41/1.82	
3.41/1.82		leaf cost:	0
3.41/1.82	
3.41/1.82	
3.41/1.82	
3.41/1.82	Complexity upper bound 23
3.41/1.82	
3.41/1.82	
3.41/1.82	
3.41/1.82	Time: 0.115 sec (SMT: 0.108 sec)
3.41/1.82	
3.41/1.82	
3.41/1.82	----------------------------------------
3.41/1.82	
3.41/1.82	(2)
3.41/1.82	BOUNDS(1, 1)
3.88/1.83	EOF