3.55/1.76	WORST_CASE(?, O(1))
3.55/1.77	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
3.55/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.55/1.77	
3.55/1.77	
3.55/1.77	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.55/1.77	
3.55/1.77	(0) CpxIntTrs
3.55/1.77	(1) Koat Proof [FINISHED, 99 ms]
3.55/1.77	(2) BOUNDS(1, 1)
3.55/1.77	
3.55/1.77	
3.55/1.77	----------------------------------------
3.55/1.77	
3.55/1.77	(0)
3.55/1.77	Obligation:
3.55/1.77	Complexity Int TRS consisting of the following rules:
3.55/1.77	f8(A, B, C, D) -> Com_1(f8(A - 1, B, C, D)) :|: A >= 0
3.55/1.77	f19(A, B, C, D) -> Com_1(f19(A, B - 1, C, D)) :|: B >= 0
3.55/1.77	f28(A, B, C, D) -> Com_1(f28(A, B, C - 1, D)) :|: C >= 0
3.55/1.77	f28(A, B, C, D) -> Com_1(f36(A, B, C, D)) :|: 0 >= C + 1
3.55/1.77	f19(A, B, C, D) -> Com_1(f28(A, B, 999, D)) :|: 0 >= B + 1
3.55/1.77	f0(A, B, C, D) -> Com_1(f19(A, 999, C, 1)) :|: TRUE
3.55/1.77	f8(A, B, C, D) -> Com_1(f19(A, 999, C, D)) :|: 0 >= A + 1
3.55/1.77	
3.55/1.77	The start-symbols are:[f0_4]
3.55/1.77	
3.55/1.77	
3.55/1.77	----------------------------------------
3.55/1.77	
3.55/1.77	(1) Koat Proof (FINISHED)
3.55/1.77	YES(?, 2005)
3.55/1.77	
3.55/1.77	
3.55/1.77	
3.55/1.77	Initial complexity problem:
3.55/1.77	
3.55/1.77	1:	T:
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 - 1, ar_1, ar_2, ar_3)) [ ar_0 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, 1))
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
3.55/1.77	
3.55/1.77			(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 ]
3.55/1.77	
3.55/1.77		start location:	koat_start
3.55/1.77	
3.55/1.77		leaf cost:	0
3.55/1.77	
3.55/1.77	
3.55/1.77	
3.55/1.77	Testing for reachability in the complexity graph removes the following transitions from problem 1:
3.55/1.77	
3.55/1.77		f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f8(ar_0 - 1, ar_1, ar_2, ar_3)) [ ar_0 >= 0 ]
3.55/1.77	
3.55/1.77		f8(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, ar_3)) [ 0 >= ar_0 + 1 ]
3.55/1.77	
3.55/1.77	We thus obtain the following problem:
3.55/1.77	
3.55/1.77	2:	T:
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, 1))
3.55/1.77	
3.55/1.77			(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 ]
3.55/1.77	
3.55/1.77		start location:	koat_start
3.55/1.77	
3.55/1.77		leaf cost:	0
3.55/1.77	
3.55/1.77	
3.55/1.77	
3.55/1.77	Repeatedly propagating knowledge in problem 2 produces the following problem:
3.55/1.77	
3.55/1.77	3:	T:
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, 1))
3.55/1.77	
3.55/1.77			(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 ]
3.55/1.77	
3.55/1.77		start location:	koat_start
3.55/1.77	
3.55/1.77		leaf cost:	0
3.55/1.77	
3.55/1.77	
3.55/1.77	
3.55/1.77	A polynomial rank function with
3.55/1.77	
3.55/1.77		Pol(f28) = 1
3.55/1.77	
3.55/1.77		Pol(f36) = 0
3.55/1.77	
3.55/1.77		Pol(f19) = 2
3.55/1.77	
3.55/1.77		Pol(f0) = 2
3.55/1.77	
3.55/1.77		Pol(koat_start) = 2
3.55/1.77	
3.55/1.77	orients all transitions weakly and the transitions
3.55/1.77	
3.55/1.77		f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
3.55/1.77	
3.55/1.77		f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ]
3.55/1.77	
3.55/1.77	strictly and produces the following problem:
3.55/1.77	
3.55/1.77	4:	T:
3.55/1.77	
3.55/1.77			(Comp: 2, Cost: 1)    f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: 2, Cost: 1)    f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)    f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, 1))
3.55/1.77	
3.55/1.77			(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 ]
3.55/1.77	
3.55/1.77		start location:	koat_start
3.55/1.77	
3.55/1.77		leaf cost:	0
3.55/1.77	
3.55/1.77	
3.55/1.77	
3.55/1.77	A polynomial rank function with
3.55/1.77	
3.55/1.77		Pol(f28) = V_3 + 1
3.55/1.77	
3.55/1.77		Pol(f36) = V_3
3.55/1.77	
3.55/1.77		Pol(f19) = 1000
3.55/1.77	
3.55/1.77		Pol(f0) = 1000
3.55/1.77	
3.55/1.77		Pol(koat_start) = 1000
3.55/1.77	
3.55/1.77	orients all transitions weakly and the transition
3.55/1.77	
3.55/1.77		f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ]
3.55/1.77	
3.55/1.77	strictly and produces the following problem:
3.55/1.77	
3.55/1.77	5:	T:
3.55/1.77	
3.55/1.77			(Comp: 2, Cost: 1)       f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
3.55/1.77	
3.55/1.77			(Comp: 1000, Cost: 1)    f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: 2, Cost: 1)       f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ]
3.55/1.77	
3.55/1.77			(Comp: ?, Cost: 1)       f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: 1, Cost: 1)       f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, 1))
3.55/1.77	
3.55/1.77			(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 ]
3.55/1.77	
3.55/1.77		start location:	koat_start
3.55/1.77	
3.55/1.77		leaf cost:	0
3.55/1.77	
3.55/1.77	
3.55/1.77	
3.55/1.77	A polynomial rank function with
3.55/1.77	
3.55/1.77		Pol(f28) = V_2
3.55/1.77	
3.55/1.77		Pol(f36) = V_2
3.55/1.77	
3.55/1.77		Pol(f19) = V_2 + 1
3.55/1.77	
3.55/1.77		Pol(f0) = 1000
3.55/1.77	
3.55/1.77		Pol(koat_start) = 1000
3.55/1.77	
3.55/1.77	orients all transitions weakly and the transition
3.55/1.77	
3.55/1.77		f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ]
3.55/1.77	
3.55/1.77	strictly and produces the following problem:
3.55/1.77	
3.55/1.77	6:	T:
3.55/1.77	
3.55/1.77			(Comp: 2, Cost: 1)       f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3)) [ 0 >= ar_2 + 1 ]
3.55/1.77	
3.55/1.77			(Comp: 1000, Cost: 1)    f28(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, ar_2 - 1, ar_3)) [ ar_2 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: 2, Cost: 1)       f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f28(ar_0, ar_1, 999, ar_3)) [ 0 >= ar_1 + 1 ]
3.55/1.77	
3.55/1.77			(Comp: 1000, Cost: 1)    f19(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, ar_1 - 1, ar_2, ar_3)) [ ar_1 >= 0 ]
3.55/1.77	
3.55/1.77			(Comp: 1, Cost: 1)       f0(ar_0, ar_1, ar_2, ar_3) -> Com_1(f19(ar_0, 999, ar_2, 1))
3.55/1.77	
3.55/1.77			(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 ]
3.55/1.77	
3.55/1.77		start location:	koat_start
3.55/1.77	
3.55/1.77		leaf cost:	0
3.55/1.77	
3.55/1.77	
3.55/1.77	
3.55/1.77	Complexity upper bound 2005
3.55/1.77	
3.55/1.77	
3.55/1.77	
3.55/1.77	Time: 0.093 sec (SMT: 0.085 sec)
3.55/1.77	
3.55/1.77	
3.55/1.77	----------------------------------------
3.55/1.77	
3.55/1.77	(2)
3.55/1.77	BOUNDS(1, 1)
3.55/1.79	EOF