3.40/1.67	WORST_CASE(?, O(1))
3.40/1.68	proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
3.40/1.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.40/1.68	
3.40/1.68	
3.40/1.68	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.40/1.68	
3.40/1.68	(0) CpxIntTrs
3.40/1.68	(1) Koat Proof [FINISHED, 36 ms]
3.40/1.68	(2) BOUNDS(1, 1)
3.40/1.68	
3.40/1.68	
3.40/1.68	----------------------------------------
3.40/1.68	
3.40/1.68	(0)
3.40/1.68	Obligation:
3.40/1.68	Complexity Int TRS consisting of the following rules:
3.40/1.68	f4(A) -> Com_1(f5(A)) :|: 0 >= B + 1
3.40/1.68	f4(A) -> Com_1(f5(A)) :|: TRUE
3.40/1.68	f0(A) -> Com_1(f4(0)) :|: TRUE
3.40/1.68	f5(A) -> Com_1(f11(A)) :|: A >= 3
3.40/1.68	f4(A) -> Com_1(f11(A)) :|: TRUE
3.40/1.68	f5(A) -> Com_1(f4(A + 1)) :|: 2 >= A
3.40/1.68	f11(A) -> Com_1(f14(A)) :|: 1 >= A
3.40/1.68	f11(A) -> Com_1(f14(A)) :|: A >= 2
3.40/1.68	
3.40/1.68	The start-symbols are:[f0_1]
3.40/1.68	
3.40/1.68	
3.40/1.68	----------------------------------------
3.40/1.68	
3.40/1.68	(1) Koat Proof (FINISHED)
3.40/1.68	YES(?, 20)
3.40/1.68	
3.40/1.68	
3.40/1.68	
3.40/1.68	Initial complexity problem:
3.40/1.68	
3.40/1.68	1:	T:
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f4(ar_0) -> Com_1(f5(ar_0)) [ 0 >= b + 1 ]
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f4(ar_0) -> Com_1(f5(ar_0))
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f0(ar_0) -> Com_1(f4(0))
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f11(ar_0)) [ ar_0 >= 3 ]
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f4(ar_0) -> Com_1(f11(ar_0))
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f4(ar_0 + 1)) [ 2 >= ar_0 ]
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f11(ar_0) -> Com_1(f14(ar_0)) [ 1 >= ar_0 ]
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f11(ar_0) -> Com_1(f14(ar_0)) [ ar_0 >= 2 ]
3.40/1.68	
3.40/1.68			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.40/1.68	
3.40/1.68		start location:	koat_start
3.40/1.68	
3.40/1.68		leaf cost:	0
3.40/1.68	
3.40/1.68	
3.40/1.68	
3.40/1.68	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.40/1.68	
3.40/1.68	2:	T:
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f4(ar_0) -> Com_1(f5(ar_0)) [ 0 >= b + 1 ]
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f4(ar_0) -> Com_1(f5(ar_0))
3.40/1.68	
3.40/1.68			(Comp: 1, Cost: 1)    f0(ar_0) -> Com_1(f4(0))
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f11(ar_0)) [ ar_0 >= 3 ]
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f4(ar_0) -> Com_1(f11(ar_0))
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f4(ar_0 + 1)) [ 2 >= ar_0 ]
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f11(ar_0) -> Com_1(f14(ar_0)) [ 1 >= ar_0 ]
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f11(ar_0) -> Com_1(f14(ar_0)) [ ar_0 >= 2 ]
3.40/1.68	
3.40/1.68			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.40/1.68	
3.40/1.68		start location:	koat_start
3.40/1.68	
3.40/1.68		leaf cost:	0
3.40/1.68	
3.40/1.68	
3.40/1.68	
3.40/1.68	A polynomial rank function with
3.40/1.68	
3.40/1.68		Pol(f4) = 2
3.40/1.68	
3.40/1.68		Pol(f5) = 2
3.40/1.68	
3.40/1.68		Pol(f0) = 2
3.40/1.68	
3.40/1.68		Pol(f11) = 1
3.40/1.68	
3.40/1.68		Pol(f14) = 0
3.40/1.68	
3.40/1.68		Pol(koat_start) = 2
3.40/1.68	
3.40/1.68	orients all transitions weakly and the transitions
3.40/1.68	
3.40/1.68		f5(ar_0) -> Com_1(f11(ar_0)) [ ar_0 >= 3 ]
3.40/1.68	
3.40/1.68		f4(ar_0) -> Com_1(f11(ar_0))
3.40/1.68	
3.40/1.68		f11(ar_0) -> Com_1(f14(ar_0)) [ 1 >= ar_0 ]
3.40/1.68	
3.40/1.68		f11(ar_0) -> Com_1(f14(ar_0)) [ ar_0 >= 2 ]
3.40/1.68	
3.40/1.68	strictly and produces the following problem:
3.40/1.68	
3.40/1.68	3:	T:
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f4(ar_0) -> Com_1(f5(ar_0)) [ 0 >= b + 1 ]
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f4(ar_0) -> Com_1(f5(ar_0))
3.40/1.68	
3.40/1.68			(Comp: 1, Cost: 1)    f0(ar_0) -> Com_1(f4(0))
3.40/1.68	
3.40/1.68			(Comp: 2, Cost: 1)    f5(ar_0) -> Com_1(f11(ar_0)) [ ar_0 >= 3 ]
3.40/1.68	
3.40/1.68			(Comp: 2, Cost: 1)    f4(ar_0) -> Com_1(f11(ar_0))
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f4(ar_0 + 1)) [ 2 >= ar_0 ]
3.40/1.68	
3.40/1.68			(Comp: 2, Cost: 1)    f11(ar_0) -> Com_1(f14(ar_0)) [ 1 >= ar_0 ]
3.40/1.68	
3.40/1.68			(Comp: 2, Cost: 1)    f11(ar_0) -> Com_1(f14(ar_0)) [ ar_0 >= 2 ]
3.40/1.68	
3.40/1.68			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.40/1.68	
3.40/1.68		start location:	koat_start
3.40/1.68	
3.40/1.68		leaf cost:	0
3.40/1.68	
3.40/1.68	
3.40/1.68	
3.40/1.68	A polynomial rank function with
3.40/1.68	
3.40/1.68		Pol(f4) = -V_1 + 3
3.40/1.68	
3.40/1.68		Pol(f5) = -V_1 + 3
3.40/1.68	
3.40/1.68		Pol(f0) = 3
3.40/1.68	
3.40/1.68		Pol(f11) = -V_1
3.40/1.68	
3.40/1.68		Pol(f14) = -V_1
3.40/1.68	
3.40/1.68		Pol(koat_start) = 3
3.40/1.68	
3.40/1.68	orients all transitions weakly and the transition
3.40/1.68	
3.40/1.68		f5(ar_0) -> Com_1(f4(ar_0 + 1)) [ 2 >= ar_0 ]
3.40/1.68	
3.40/1.68	strictly and produces the following problem:
3.40/1.68	
3.40/1.68	4:	T:
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f4(ar_0) -> Com_1(f5(ar_0)) [ 0 >= b + 1 ]
3.40/1.68	
3.40/1.68			(Comp: ?, Cost: 1)    f4(ar_0) -> Com_1(f5(ar_0))
3.40/1.68	
3.40/1.68			(Comp: 1, Cost: 1)    f0(ar_0) -> Com_1(f4(0))
3.40/1.68	
3.40/1.68			(Comp: 2, Cost: 1)    f5(ar_0) -> Com_1(f11(ar_0)) [ ar_0 >= 3 ]
3.40/1.68	
3.40/1.68			(Comp: 2, Cost: 1)    f4(ar_0) -> Com_1(f11(ar_0))
3.40/1.68	
3.40/1.68			(Comp: 3, Cost: 1)    f5(ar_0) -> Com_1(f4(ar_0 + 1)) [ 2 >= ar_0 ]
3.40/1.68	
3.40/1.68			(Comp: 2, Cost: 1)    f11(ar_0) -> Com_1(f14(ar_0)) [ 1 >= ar_0 ]
3.40/1.68	
3.40/1.68			(Comp: 2, Cost: 1)    f11(ar_0) -> Com_1(f14(ar_0)) [ ar_0 >= 2 ]
3.40/1.68	
3.40/1.68			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.40/1.68	
3.40/1.68		start location:	koat_start
3.40/1.68	
3.40/1.68		leaf cost:	0
3.40/1.68	
3.40/1.68	
3.40/1.68	
3.40/1.68	Repeatedly propagating knowledge in problem 4 produces the following problem:
3.40/1.68	
3.40/1.68	5:	T:
3.40/1.68	
3.40/1.68			(Comp: 4, Cost: 1)    f4(ar_0) -> Com_1(f5(ar_0)) [ 0 >= b + 1 ]
3.40/1.68	
3.40/1.68			(Comp: 4, Cost: 1)    f4(ar_0) -> Com_1(f5(ar_0))
3.40/1.68	
3.40/1.68			(Comp: 1, Cost: 1)    f0(ar_0) -> Com_1(f4(0))
3.40/1.68	
3.40/1.68			(Comp: 2, Cost: 1)    f5(ar_0) -> Com_1(f11(ar_0)) [ ar_0 >= 3 ]
3.40/1.68	
3.40/1.68			(Comp: 2, Cost: 1)    f4(ar_0) -> Com_1(f11(ar_0))
3.40/1.68	
3.40/1.68			(Comp: 3, Cost: 1)    f5(ar_0) -> Com_1(f4(ar_0 + 1)) [ 2 >= ar_0 ]
3.40/1.68	
3.40/1.68			(Comp: 2, Cost: 1)    f11(ar_0) -> Com_1(f14(ar_0)) [ 1 >= ar_0 ]
3.40/1.68	
3.40/1.68			(Comp: 2, Cost: 1)    f11(ar_0) -> Com_1(f14(ar_0)) [ ar_0 >= 2 ]
3.40/1.68	
3.40/1.68			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.40/1.68	
3.40/1.68		start location:	koat_start
3.40/1.68	
3.40/1.68		leaf cost:	0
3.40/1.68	
3.40/1.68	
3.40/1.68	
3.40/1.68	Complexity upper bound 20
3.40/1.68	
3.40/1.68	
3.40/1.68	
3.40/1.68	Time: 0.059 sec (SMT: 0.056 sec)
3.40/1.68	
3.40/1.68	
3.40/1.68	----------------------------------------
3.40/1.68	
3.40/1.68	(2)
3.40/1.68	BOUNDS(1, 1)
3.55/1.70	EOF