3.45/1.71	WORST_CASE(?, O(1))
3.45/1.72	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.45/1.72	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.45/1.72	
3.45/1.72	
3.45/1.72	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.45/1.72	
3.45/1.72	(0) CpxIntTrs
3.45/1.72	(1) Koat Proof [FINISHED, 11 ms]
3.45/1.72	(2) BOUNDS(1, 1)
3.45/1.72	
3.45/1.72	
3.45/1.72	----------------------------------------
3.45/1.72	
3.45/1.72	(0)
3.45/1.72	Obligation:
3.45/1.72	Complexity Int TRS consisting of the following rules:
3.45/1.72	f18(A, B, C, D, E, F) -> Com_1(f18(A, B + 1, C, D, E, F)) :|: A >= B + 1
3.45/1.72	f24(A, B, C, D, E, F) -> Com_1(f24(A, B + 1, C, D, E, F)) :|: A >= B + 1
3.45/1.72	f31(A, B, C, D, E, F) -> Com_1(f31(A, B + 1, C, D, E, F)) :|: A >= B + 1
3.45/1.72	f31(A, B, C, D, E, F) -> Com_1(f39(A, B, C, D, E, F)) :|: B >= A
3.45/1.72	f24(A, B, C, D, E, F) -> Com_1(f31(A, 0, C, D, E, F)) :|: B >= A
3.45/1.72	f18(A, B, C, D, E, F) -> Com_1(f24(A, 0, C, D, E, F)) :|: B >= A
3.45/1.72	f0(A, B, C, D, E, F) -> Com_1(f18(10, 0, 10, G, 10, H)) :|: TRUE
3.45/1.72	
3.45/1.72	The start-symbols are:[f0_6]
3.45/1.72	
3.45/1.72	
3.45/1.72	----------------------------------------
3.45/1.72	
3.45/1.72	(1) Koat Proof (FINISHED)
3.45/1.72	YES(?, 100)
3.45/1.72	
3.45/1.72	
3.45/1.72	
3.45/1.72	Initial complexity problem:
3.45/1.72	
3.45/1.72	1:	T:
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f18(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(ar_0, ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f24(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f24(ar_0, ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f31(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f31(ar_0, ar_1 + 1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f31(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f39(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f24(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f31(ar_0, 0, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f18(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f24(ar_0, 0, ar_2, ar_3, ar_4, ar_5)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f18(10, 0, 10, g, 10, h))
3.45/1.72	
3.45/1.72			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ]
3.45/1.72	
3.45/1.72		start location:	koat_start
3.45/1.72	
3.45/1.72		leaf cost:	0
3.45/1.72	
3.45/1.72	
3.45/1.72	
3.45/1.72	Slicing away variables that do not contribute to conditions from problem 1 leaves variables [ar_0, ar_1].
3.45/1.72	
3.45/1.72	We thus obtain the following problem:
3.45/1.72	
3.45/1.72	2:	T:
3.45/1.72	
3.45/1.72			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f18(10, 0))
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f18(ar_0, ar_1) -> Com_1(f24(ar_0, 0)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f24(ar_0, ar_1) -> Com_1(f31(ar_0, 0)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f31(ar_0, ar_1) -> Com_1(f39(ar_0, ar_1)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f31(ar_0, ar_1) -> Com_1(f31(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f24(ar_0, ar_1) -> Com_1(f24(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f18(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72		start location:	koat_start
3.45/1.72	
3.45/1.72		leaf cost:	0
3.45/1.72	
3.45/1.72	
3.45/1.72	
3.45/1.72	Repeatedly propagating knowledge in problem 2 produces the following problem:
3.45/1.72	
3.45/1.72	3:	T:
3.45/1.72	
3.45/1.72			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.45/1.72	
3.45/1.72			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f18(10, 0))
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f18(ar_0, ar_1) -> Com_1(f24(ar_0, 0)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f24(ar_0, ar_1) -> Com_1(f31(ar_0, 0)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f31(ar_0, ar_1) -> Com_1(f39(ar_0, ar_1)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f31(ar_0, ar_1) -> Com_1(f31(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f24(ar_0, ar_1) -> Com_1(f24(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f18(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72		start location:	koat_start
3.45/1.72	
3.45/1.72		leaf cost:	0
3.45/1.72	
3.45/1.72	
3.45/1.72	
3.45/1.72	A polynomial rank function with
3.45/1.72	
3.45/1.72		Pol(koat_start) = 3
3.45/1.72	
3.45/1.72		Pol(f0) = 3
3.45/1.72	
3.45/1.72		Pol(f18) = 3
3.45/1.72	
3.45/1.72		Pol(f24) = 2
3.45/1.72	
3.45/1.72		Pol(f31) = 1
3.45/1.72	
3.45/1.72		Pol(f39) = 0
3.45/1.72	
3.45/1.72	orients all transitions weakly and the transitions
3.45/1.72	
3.45/1.72		f31(ar_0, ar_1) -> Com_1(f39(ar_0, ar_1)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72		f24(ar_0, ar_1) -> Com_1(f31(ar_0, 0)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72		f18(ar_0, ar_1) -> Com_1(f24(ar_0, 0)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72	strictly and produces the following problem:
3.45/1.72	
3.45/1.72	4:	T:
3.45/1.72	
3.45/1.72			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.45/1.72	
3.45/1.72			(Comp: 1, Cost: 1)    f0(ar_0, ar_1) -> Com_1(f18(10, 0))
3.45/1.72	
3.45/1.72			(Comp: 3, Cost: 1)    f18(ar_0, ar_1) -> Com_1(f24(ar_0, 0)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: 3, Cost: 1)    f24(ar_0, ar_1) -> Com_1(f31(ar_0, 0)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: 3, Cost: 1)    f31(ar_0, ar_1) -> Com_1(f39(ar_0, ar_1)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f31(ar_0, ar_1) -> Com_1(f31(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f24(ar_0, ar_1) -> Com_1(f24(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)    f18(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72		start location:	koat_start
3.45/1.72	
3.45/1.72		leaf cost:	0
3.45/1.72	
3.45/1.72	
3.45/1.72	
3.45/1.72	A polynomial rank function with
3.45/1.72	
3.45/1.72		Pol(koat_start) = 10
3.45/1.72	
3.45/1.72		Pol(f0) = 10
3.45/1.72	
3.45/1.72		Pol(f18) = V_1
3.45/1.72	
3.45/1.72		Pol(f24) = V_1
3.45/1.72	
3.45/1.72		Pol(f31) = V_1 - V_2
3.45/1.72	
3.45/1.72		Pol(f39) = V_1 - V_2
3.45/1.72	
3.45/1.72	orients all transitions weakly and the transition
3.45/1.72	
3.45/1.72		f31(ar_0, ar_1) -> Com_1(f31(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72	strictly and produces the following problem:
3.45/1.72	
3.45/1.72	5:	T:
3.45/1.72	
3.45/1.72			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.45/1.72	
3.45/1.72			(Comp: 1, Cost: 1)     f0(ar_0, ar_1) -> Com_1(f18(10, 0))
3.45/1.72	
3.45/1.72			(Comp: 3, Cost: 1)     f18(ar_0, ar_1) -> Com_1(f24(ar_0, 0)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: 3, Cost: 1)     f24(ar_0, ar_1) -> Com_1(f31(ar_0, 0)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: 3, Cost: 1)     f31(ar_0, ar_1) -> Com_1(f39(ar_0, ar_1)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: 10, Cost: 1)    f31(ar_0, ar_1) -> Com_1(f31(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)     f24(ar_0, ar_1) -> Com_1(f24(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: ?, Cost: 1)     f18(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72		start location:	koat_start
3.45/1.72	
3.45/1.72		leaf cost:	0
3.45/1.72	
3.45/1.72	
3.45/1.72	
3.45/1.72	A polynomial rank function with
3.45/1.72	
3.45/1.72		Pol(f24) = V_1 - V_2
3.45/1.72	
3.45/1.72		Pol(f18) = V_1 - V_2
3.45/1.72	
3.45/1.72	and size complexities
3.45/1.72	
3.45/1.72		S("f18(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]", 0-0) = 10
3.45/1.72	
3.45/1.72		S("f18(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ?
3.45/1.72	
3.45/1.72		S("f24(ar_0, ar_1) -> Com_1(f24(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]", 0-0) = 10
3.45/1.72	
3.45/1.72		S("f24(ar_0, ar_1) -> Com_1(f24(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ?
3.45/1.72	
3.45/1.72		S("f31(ar_0, ar_1) -> Com_1(f31(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]", 0-0) = 10
3.45/1.72	
3.45/1.72		S("f31(ar_0, ar_1) -> Com_1(f31(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]", 0-1) = 10
3.45/1.72	
3.45/1.72		S("f31(ar_0, ar_1) -> Com_1(f39(ar_0, ar_1)) [ ar_1 >= ar_0 ]", 0-0) = 10
3.45/1.72	
3.45/1.72		S("f31(ar_0, ar_1) -> Com_1(f39(ar_0, ar_1)) [ ar_1 >= ar_0 ]", 0-1) = 10
3.45/1.72	
3.45/1.72		S("f24(ar_0, ar_1) -> Com_1(f31(ar_0, 0)) [ ar_1 >= ar_0 ]", 0-0) = 10
3.45/1.72	
3.45/1.72		S("f24(ar_0, ar_1) -> Com_1(f31(ar_0, 0)) [ ar_1 >= ar_0 ]", 0-1) = 0
3.45/1.72	
3.45/1.72		S("f18(ar_0, ar_1) -> Com_1(f24(ar_0, 0)) [ ar_1 >= ar_0 ]", 0-0) = 10
3.45/1.72	
3.45/1.72		S("f18(ar_0, ar_1) -> Com_1(f24(ar_0, 0)) [ ar_1 >= ar_0 ]", 0-1) = 0
3.45/1.72	
3.45/1.72		S("f0(ar_0, ar_1) -> Com_1(f18(10, 0))", 0-0) = 10
3.45/1.72	
3.45/1.72		S("f0(ar_0, ar_1) -> Com_1(f18(10, 0))", 0-1) = 0
3.45/1.72	
3.45/1.72		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
3.45/1.72	
3.45/1.72		S("koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
3.45/1.72	
3.45/1.72	orients the transitions
3.45/1.72	
3.45/1.72		f24(ar_0, ar_1) -> Com_1(f24(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72		f18(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72	weakly and the transitions
3.45/1.72	
3.45/1.72		f24(ar_0, ar_1) -> Com_1(f24(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72		f18(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72	strictly and produces the following problem:
3.45/1.72	
3.45/1.72	6:	T:
3.45/1.72	
3.45/1.72			(Comp: 1, Cost: 0)     koat_start(ar_0, ar_1) -> Com_1(f0(ar_0, ar_1)) [ 0 <= 0 ]
3.45/1.72	
3.45/1.72			(Comp: 1, Cost: 1)     f0(ar_0, ar_1) -> Com_1(f18(10, 0))
3.45/1.72	
3.45/1.72			(Comp: 3, Cost: 1)     f18(ar_0, ar_1) -> Com_1(f24(ar_0, 0)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: 3, Cost: 1)     f24(ar_0, ar_1) -> Com_1(f31(ar_0, 0)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: 3, Cost: 1)     f31(ar_0, ar_1) -> Com_1(f39(ar_0, ar_1)) [ ar_1 >= ar_0 ]
3.45/1.72	
3.45/1.72			(Comp: 10, Cost: 1)    f31(ar_0, ar_1) -> Com_1(f31(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: 40, Cost: 1)    f24(ar_0, ar_1) -> Com_1(f24(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72			(Comp: 40, Cost: 1)    f18(ar_0, ar_1) -> Com_1(f18(ar_0, ar_1 + 1)) [ ar_0 >= ar_1 + 1 ]
3.45/1.72	
3.45/1.72		start location:	koat_start
3.45/1.72	
3.45/1.72		leaf cost:	0
3.45/1.72	
3.45/1.72	
3.45/1.72	
3.45/1.72	Complexity upper bound 100
3.45/1.72	
3.45/1.72	
3.45/1.72	
3.45/1.72	Time: 0.075 sec (SMT: 0.069 sec)
3.45/1.72	
3.45/1.72	
3.45/1.72	----------------------------------------
3.45/1.72	
3.45/1.72	(2)
3.45/1.72	BOUNDS(1, 1)
3.45/1.73	EOF