3.44/1.85	WORST_CASE(?, O(1))
3.44/1.85	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.44/1.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.44/1.85	
3.44/1.85	
3.44/1.85	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.44/1.85	
3.44/1.85	(0) CpxIntTrs
3.44/1.85	(1) Koat Proof [FINISHED, 12 ms]
3.44/1.85	(2) BOUNDS(1, 1)
3.44/1.85	
3.44/1.85	
3.44/1.85	----------------------------------------
3.44/1.85	
3.44/1.85	(0)
3.44/1.85	Obligation:
3.44/1.85	Complexity Int TRS consisting of the following rules:
3.44/1.85	f0(A, B, C, D, E) -> Com_1(f7(F, 0, 0, D, E)) :|: 0 >= F + 1
3.44/1.85	f0(A, B, C, D, E) -> Com_1(f7(F, 0, 0, D, E)) :|: F >= 1
3.44/1.85	f0(A, B, C, D, E) -> Com_1(f7(0, 1023, 0, D, E)) :|: TRUE
3.44/1.85	f7(A, B, C, D, E) -> Com_1(f7(A, B, C + 1, D + 2, E)) :|: B >= C
3.44/1.85	f7(A, B, C, D, E) -> Com_1(f21(A, B, C, D, E)) :|: E >= 0 && C >= 1 + B && 1022 >= E
3.44/1.85	f7(A, B, C, D, E) -> Com_1(f21(A, B, C, D, E)) :|: C >= 1 + B && E >= 1023
3.44/1.85	f7(A, B, C, D, E) -> Com_1(f21(A, B, C, D, E)) :|: C >= 1 + B && 0 >= E + 1
3.44/1.85	
3.44/1.85	The start-symbols are:[f0_5]
3.44/1.85	
3.44/1.85	
3.44/1.85	----------------------------------------
3.44/1.85	
3.44/1.85	(1) Koat Proof (FINISHED)
3.44/1.85	YES(?, 1030)
3.44/1.85	
3.44/1.85	
3.44/1.85	
3.44/1.85	Initial complexity problem:
3.44/1.85	
3.44/1.85	1:	T:
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f7(f, 0, 0, ar_3, ar_4)) [ 0 >= f + 1 ]
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f7(f, 0, 0, ar_3, ar_4)) [ f >= 1 ]
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f7(0, 1023, 0, ar_3, ar_4))
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f7(ar_0, ar_1, ar_2 + 1, ar_3 + 2, ar_4)) [ ar_1 >= ar_2 ]
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f21(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= 0 /\ ar_2 >= ar_1 + 1 /\ 1022 >= ar_4 ]
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f21(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_1 + 1 /\ ar_4 >= 1023 ]
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f21(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_1 + 1 /\ 0 >= ar_4 + 1 ]
3.44/1.85	
3.44/1.85			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ]
3.44/1.85	
3.44/1.85		start location:	koat_start
3.44/1.85	
3.44/1.85		leaf cost:	0
3.44/1.85	
3.44/1.85	
3.44/1.85	
3.44/1.85	Slicing away variables that do not contribute to conditions from problem 1 leaves variables [ar_1, ar_2, ar_4].
3.44/1.85	
3.44/1.85	We thus obtain the following problem:
3.44/1.85	
3.44/1.85	2:	T:
3.44/1.85	
3.44/1.85			(Comp: 1, Cost: 0)    koat_start(ar_1, ar_2, ar_4) -> Com_1(f0(ar_1, ar_2, ar_4)) [ 0 <= 0 ]
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_2 >= ar_1 + 1 /\ 0 >= ar_4 + 1 ]
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_2 >= ar_1 + 1 /\ ar_4 >= 1023 ]
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_4 >= 0 /\ ar_2 >= ar_1 + 1 /\ 1022 >= ar_4 ]
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f7(ar_1, ar_2 + 1, ar_4)) [ ar_1 >= ar_2 ]
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f0(ar_1, ar_2, ar_4) -> Com_1(f7(1023, 0, ar_4))
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f0(ar_1, ar_2, ar_4) -> Com_1(f7(0, 0, ar_4)) [ f >= 1 ]
3.44/1.85	
3.44/1.85			(Comp: ?, Cost: 1)    f0(ar_1, ar_2, ar_4) -> Com_1(f7(0, 0, ar_4)) [ 0 >= f + 1 ]
3.44/1.85	
3.44/1.85		start location:	koat_start
3.44/1.85	
3.44/1.85		leaf cost:	0
3.44/1.86	
3.44/1.86	
3.44/1.86	
3.44/1.86	Repeatedly propagating knowledge in problem 2 produces the following problem:
3.44/1.86	
3.44/1.86	3:	T:
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 0)    koat_start(ar_1, ar_2, ar_4) -> Com_1(f0(ar_1, ar_2, ar_4)) [ 0 <= 0 ]
3.44/1.86	
3.44/1.86			(Comp: ?, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_2 >= ar_1 + 1 /\ 0 >= ar_4 + 1 ]
3.44/1.86	
3.44/1.86			(Comp: ?, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_2 >= ar_1 + 1 /\ ar_4 >= 1023 ]
3.44/1.86	
3.44/1.86			(Comp: ?, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_4 >= 0 /\ ar_2 >= ar_1 + 1 /\ 1022 >= ar_4 ]
3.44/1.86	
3.44/1.86			(Comp: ?, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f7(ar_1, ar_2 + 1, ar_4)) [ ar_1 >= ar_2 ]
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)    f0(ar_1, ar_2, ar_4) -> Com_1(f7(1023, 0, ar_4))
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)    f0(ar_1, ar_2, ar_4) -> Com_1(f7(0, 0, ar_4)) [ f >= 1 ]
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)    f0(ar_1, ar_2, ar_4) -> Com_1(f7(0, 0, ar_4)) [ 0 >= f + 1 ]
3.44/1.86	
3.44/1.86		start location:	koat_start
3.44/1.86	
3.44/1.86		leaf cost:	0
3.44/1.86	
3.44/1.86	
3.44/1.86	
3.44/1.86	A polynomial rank function with
3.44/1.86	
3.44/1.86		Pol(koat_start) = 1
3.44/1.86	
3.44/1.86		Pol(f0) = 1
3.44/1.86	
3.44/1.86		Pol(f7) = 1
3.44/1.86	
3.44/1.86		Pol(f21) = 0
3.44/1.86	
3.44/1.86	orients all transitions weakly and the transitions
3.44/1.86	
3.44/1.86		f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_4 >= 0 /\ ar_2 >= ar_1 + 1 /\ 1022 >= ar_4 ]
3.44/1.86	
3.44/1.86		f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_2 >= ar_1 + 1 /\ ar_4 >= 1023 ]
3.44/1.86	
3.44/1.86		f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_2 >= ar_1 + 1 /\ 0 >= ar_4 + 1 ]
3.44/1.86	
3.44/1.86	strictly and produces the following problem:
3.44/1.86	
3.44/1.86	4:	T:
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 0)    koat_start(ar_1, ar_2, ar_4) -> Com_1(f0(ar_1, ar_2, ar_4)) [ 0 <= 0 ]
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_2 >= ar_1 + 1 /\ 0 >= ar_4 + 1 ]
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_2 >= ar_1 + 1 /\ ar_4 >= 1023 ]
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_4 >= 0 /\ ar_2 >= ar_1 + 1 /\ 1022 >= ar_4 ]
3.44/1.86	
3.44/1.86			(Comp: ?, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f7(ar_1, ar_2 + 1, ar_4)) [ ar_1 >= ar_2 ]
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)    f0(ar_1, ar_2, ar_4) -> Com_1(f7(1023, 0, ar_4))
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)    f0(ar_1, ar_2, ar_4) -> Com_1(f7(0, 0, ar_4)) [ f >= 1 ]
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)    f0(ar_1, ar_2, ar_4) -> Com_1(f7(0, 0, ar_4)) [ 0 >= f + 1 ]
3.44/1.86	
3.44/1.86		start location:	koat_start
3.44/1.86	
3.44/1.86		leaf cost:	0
3.44/1.86	
3.44/1.86	
3.44/1.86	
3.44/1.86	A polynomial rank function with
3.44/1.86	
3.44/1.86		Pol(koat_start) = 1024
3.44/1.86	
3.44/1.86		Pol(f0) = 1024
3.44/1.86	
3.44/1.86		Pol(f7) = V_1 - V_2 + 1
3.44/1.86	
3.44/1.86		Pol(f21) = V_1 - V_2
3.44/1.86	
3.44/1.86	orients all transitions weakly and the transition
3.44/1.86	
3.44/1.86		f7(ar_1, ar_2, ar_4) -> Com_1(f7(ar_1, ar_2 + 1, ar_4)) [ ar_1 >= ar_2 ]
3.44/1.86	
3.44/1.86	strictly and produces the following problem:
3.44/1.86	
3.44/1.86	5:	T:
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 0)       koat_start(ar_1, ar_2, ar_4) -> Com_1(f0(ar_1, ar_2, ar_4)) [ 0 <= 0 ]
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)       f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_2 >= ar_1 + 1 /\ 0 >= ar_4 + 1 ]
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)       f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_2 >= ar_1 + 1 /\ ar_4 >= 1023 ]
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)       f7(ar_1, ar_2, ar_4) -> Com_1(f21(ar_1, ar_2, ar_4)) [ ar_4 >= 0 /\ ar_2 >= ar_1 + 1 /\ 1022 >= ar_4 ]
3.44/1.86	
3.44/1.86			(Comp: 1024, Cost: 1)    f7(ar_1, ar_2, ar_4) -> Com_1(f7(ar_1, ar_2 + 1, ar_4)) [ ar_1 >= ar_2 ]
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)       f0(ar_1, ar_2, ar_4) -> Com_1(f7(1023, 0, ar_4))
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)       f0(ar_1, ar_2, ar_4) -> Com_1(f7(0, 0, ar_4)) [ f >= 1 ]
3.44/1.86	
3.44/1.86			(Comp: 1, Cost: 1)       f0(ar_1, ar_2, ar_4) -> Com_1(f7(0, 0, ar_4)) [ 0 >= f + 1 ]
3.44/1.86	
3.44/1.86		start location:	koat_start
3.44/1.86	
3.44/1.86		leaf cost:	0
3.44/1.86	
3.44/1.86	
3.44/1.86	
3.44/1.86	Complexity upper bound 1030
3.44/1.86	
3.44/1.86	
3.44/1.86	
3.44/1.86	Time: 0.073 sec (SMT: 0.065 sec)
3.44/1.86	
3.44/1.86	
3.44/1.86	----------------------------------------
3.44/1.86	
3.44/1.86	(2)
3.44/1.86	BOUNDS(1, 1)
3.75/1.87	EOF