3.46/1.70	WORST_CASE(?, O(1))
3.66/1.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.66/1.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.66/1.71	
3.66/1.71	
3.66/1.71	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.66/1.71	
3.66/1.71	(0) CpxIntTrs
3.66/1.71	(1) Koat Proof [FINISHED, 18 ms]
3.66/1.71	(2) BOUNDS(1, 1)
3.66/1.71	
3.66/1.71	
3.66/1.71	----------------------------------------
3.66/1.71	
3.66/1.71	(0)
3.66/1.71	Obligation:
3.66/1.71	Complexity Int TRS consisting of the following rules:
3.66/1.71	f0(A, B, C, D, E, F, G) -> Com_1(f5(0, B, C, D, E, F, G)) :|: TRUE
3.66/1.71	f5(A, B, C, D, E, F, G) -> Com_1(f5(A + 1, B, C, D, E, F, G)) :|: 99 >= A
3.66/1.71	f17(A, B, C, D, E, F, G) -> Com_1(f17(A, B, C, D, E, F, G)) :|: TRUE
3.66/1.71	f17(A, B, C, D, E, F, G) -> Com_1(f17(A, B + 1, C, D, E, F, G)) :|: 0 >= H + 1
3.66/1.71	f17(A, B, C, D, E, F, G) -> Com_1(f17(A, B + 1, C, D, E, F, G)) :|: TRUE
3.66/1.71	f32(A, B, C, D, E, F, G) -> Com_1(f32(A, B, C, D, E, F, G)) :|: TRUE
3.66/1.71	f32(A, B, C, D, E, F, G) -> Com_1(f32(A, B, C + 1, D, E, F, G)) :|: 0 >= H + 1
3.66/1.71	f32(A, B, C, D, E, F, G) -> Com_1(f32(A, B, C + 1, D, E, F, G)) :|: TRUE
3.66/1.71	f32(A, B, C, D, E, F, G) -> Com_1(f13(A, B, C, C, C, F, G)) :|: TRUE
3.66/1.71	f17(A, B, C, D, E, F, G) -> Com_1(f32(A, B, B, B, E, B, H)) :|: 0 >= I + 1
3.66/1.71	f17(A, B, C, D, E, F, G) -> Com_1(f32(A, B, B, B, E, B, H)) :|: TRUE
3.66/1.71	f17(A, B, C, D, E, F, G) -> Com_1(f13(A, B, C, B, E, B, H)) :|: TRUE
3.66/1.71	f5(A, B, C, D, E, F, G) -> Com_1(f13(A, B, C, A - 2, E, F, G)) :|: A >= 100
3.66/1.71	f5(A, B, C, D, E, F, G) -> Com_1(f17(A, A - 2, C, A - 2, E, F, G)) :|: 0 >= A + 1 && A >= 100
3.66/1.71	
3.66/1.71	The start-symbols are:[f0_7]
3.66/1.71	
3.66/1.71	
3.66/1.71	----------------------------------------
3.66/1.71	
3.66/1.71	(1) Koat Proof (FINISHED)
3.66/1.71	YES(?, 102)
3.66/1.71	
3.66/1.71	
3.66/1.71	
3.66/1.71	Initial complexity problem:
3.66/1.71	
3.66/1.71	1:	T:
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f5(0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f5(ar_0 + 1, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6)) [ 99 >= ar_0 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f17(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f17(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f17(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f17(ar_0, ar_1 + 1, ar_2, ar_3, ar_4, ar_5, ar_6)) [ 0 >= h + 1 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f17(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f17(ar_0, ar_1 + 1, ar_2, ar_3, ar_4, ar_5, ar_6))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f32(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f32(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f32(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f32(ar_0, ar_1, ar_2 + 1, ar_3, ar_4, ar_5, ar_6)) [ 0 >= h + 1 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f32(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f32(ar_0, ar_1, ar_2 + 1, ar_3, ar_4, ar_5, ar_6))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f32(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f13(ar_0, ar_1, ar_2, ar_2, ar_2, ar_5, ar_6))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f17(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f32(ar_0, ar_1, ar_1, ar_1, ar_4, ar_1, h)) [ 0 >= i + 1 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f17(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f32(ar_0, ar_1, ar_1, ar_1, ar_4, ar_1, h))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f17(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f13(ar_0, ar_1, ar_2, ar_1, ar_4, ar_1, h))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f13(ar_0, ar_1, ar_2, ar_0 - 2, ar_4, ar_5, ar_6)) [ ar_0 >= 100 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f17(ar_0, ar_0 - 2, ar_2, ar_0 - 2, ar_4, ar_5, ar_6)) [ 0 >= ar_0 + 1 /\ ar_0 >= 100 ]
3.66/1.71	
3.66/1.71			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6)) [ 0 <= 0 ]
3.66/1.71	
3.66/1.71		start location:	koat_start
3.66/1.71	
3.66/1.71		leaf cost:	0
3.66/1.71	
3.66/1.71	
3.66/1.71	
3.66/1.71	Slicing away variables that do not contribute to conditions from problem 1 leaves variables [ar_0].
3.66/1.71	
3.66/1.71	We thus obtain the following problem:
3.66/1.71	
3.66/1.71	2:	T:
3.66/1.71	
3.66/1.71			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f17(ar_0)) [ 0 >= ar_0 + 1 /\ ar_0 >= 100 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f13(ar_0)) [ ar_0 >= 100 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f17(ar_0) -> Com_1(f13(ar_0))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f17(ar_0) -> Com_1(f32(ar_0))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f17(ar_0) -> Com_1(f32(ar_0)) [ 0 >= i + 1 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f32(ar_0) -> Com_1(f13(ar_0))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f32(ar_0) -> Com_1(f32(ar_0))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f32(ar_0) -> Com_1(f32(ar_0)) [ 0 >= h + 1 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f32(ar_0) -> Com_1(f32(ar_0))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f17(ar_0) -> Com_1(f17(ar_0))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f17(ar_0) -> Com_1(f17(ar_0)) [ 0 >= h + 1 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f17(ar_0) -> Com_1(f17(ar_0))
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f5(ar_0 + 1)) [ 99 >= ar_0 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f0(ar_0) -> Com_1(f5(0))
3.66/1.71	
3.66/1.71		start location:	koat_start
3.66/1.71	
3.66/1.71		leaf cost:	0
3.66/1.71	
3.66/1.71	
3.66/1.71	
3.66/1.71	Testing for reachability in the complexity graph removes the following transitions from problem 2:
3.66/1.71	
3.66/1.71		f5(ar_0) -> Com_1(f17(ar_0)) [ 0 >= ar_0 + 1 /\ ar_0 >= 100 ]
3.66/1.71	
3.66/1.71		f17(ar_0) -> Com_1(f13(ar_0))
3.66/1.71	
3.66/1.71		f17(ar_0) -> Com_1(f32(ar_0))
3.66/1.71	
3.66/1.71		f17(ar_0) -> Com_1(f32(ar_0)) [ 0 >= i + 1 ]
3.66/1.71	
3.66/1.71		f32(ar_0) -> Com_1(f13(ar_0))
3.66/1.71	
3.66/1.71		f32(ar_0) -> Com_1(f32(ar_0))
3.66/1.71	
3.66/1.71		f32(ar_0) -> Com_1(f32(ar_0)) [ 0 >= h + 1 ]
3.66/1.71	
3.66/1.71		f32(ar_0) -> Com_1(f32(ar_0))
3.66/1.71	
3.66/1.71		f17(ar_0) -> Com_1(f17(ar_0))
3.66/1.71	
3.66/1.71		f17(ar_0) -> Com_1(f17(ar_0)) [ 0 >= h + 1 ]
3.66/1.71	
3.66/1.71		f17(ar_0) -> Com_1(f17(ar_0))
3.66/1.71	
3.66/1.71	We thus obtain the following problem:
3.66/1.71	
3.66/1.71	3:	T:
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f13(ar_0)) [ ar_0 >= 100 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f5(ar_0 + 1)) [ 99 >= ar_0 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f0(ar_0) -> Com_1(f5(0))
3.66/1.71	
3.66/1.71			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.66/1.71	
3.66/1.71		start location:	koat_start
3.66/1.71	
3.66/1.71		leaf cost:	0
3.66/1.71	
3.66/1.71	
3.66/1.71	
3.66/1.71	Repeatedly propagating knowledge in problem 3 produces the following problem:
3.66/1.71	
3.66/1.71	4:	T:
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f13(ar_0)) [ ar_0 >= 100 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f5(ar_0 + 1)) [ 99 >= ar_0 ]
3.66/1.71	
3.66/1.71			(Comp: 1, Cost: 1)    f0(ar_0) -> Com_1(f5(0))
3.66/1.71	
3.66/1.71			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.66/1.71	
3.66/1.71		start location:	koat_start
3.66/1.71	
3.66/1.71		leaf cost:	0
3.66/1.71	
3.66/1.71	
3.66/1.71	
3.66/1.71	A polynomial rank function with
3.66/1.71	
3.66/1.71		Pol(f5) = 1
3.66/1.71	
3.66/1.71		Pol(f13) = 0
3.66/1.71	
3.66/1.71		Pol(f0) = 1
3.66/1.71	
3.66/1.71		Pol(koat_start) = 1
3.66/1.71	
3.66/1.71	orients all transitions weakly and the transition
3.66/1.71	
3.66/1.71		f5(ar_0) -> Com_1(f13(ar_0)) [ ar_0 >= 100 ]
3.66/1.71	
3.66/1.71	strictly and produces the following problem:
3.66/1.71	
3.66/1.71	5:	T:
3.66/1.71	
3.66/1.71			(Comp: 1, Cost: 1)    f5(ar_0) -> Com_1(f13(ar_0)) [ ar_0 >= 100 ]
3.66/1.71	
3.66/1.71			(Comp: ?, Cost: 1)    f5(ar_0) -> Com_1(f5(ar_0 + 1)) [ 99 >= ar_0 ]
3.66/1.71	
3.66/1.71			(Comp: 1, Cost: 1)    f0(ar_0) -> Com_1(f5(0))
3.66/1.71	
3.66/1.71			(Comp: 1, Cost: 0)    koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.66/1.71	
3.66/1.71		start location:	koat_start
3.66/1.71	
3.66/1.71		leaf cost:	0
3.66/1.71	
3.66/1.71	
3.66/1.71	
3.66/1.71	A polynomial rank function with
3.66/1.71	
3.66/1.71		Pol(f5) = -V_1 + 100
3.66/1.71	
3.66/1.71		Pol(f13) = -V_1
3.66/1.71	
3.66/1.71		Pol(f0) = 100
3.66/1.71	
3.66/1.71		Pol(koat_start) = 100
3.66/1.71	
3.66/1.71	orients all transitions weakly and the transition
3.66/1.71	
3.66/1.71		f5(ar_0) -> Com_1(f5(ar_0 + 1)) [ 99 >= ar_0 ]
3.66/1.71	
3.66/1.71	strictly and produces the following problem:
3.66/1.71	
3.66/1.71	6:	T:
3.66/1.71	
3.66/1.71			(Comp: 1, Cost: 1)      f5(ar_0) -> Com_1(f13(ar_0)) [ ar_0 >= 100 ]
3.66/1.71	
3.66/1.71			(Comp: 100, Cost: 1)    f5(ar_0) -> Com_1(f5(ar_0 + 1)) [ 99 >= ar_0 ]
3.66/1.71	
3.66/1.71			(Comp: 1, Cost: 1)      f0(ar_0) -> Com_1(f5(0))
3.66/1.71	
3.66/1.71			(Comp: 1, Cost: 0)      koat_start(ar_0) -> Com_1(f0(ar_0)) [ 0 <= 0 ]
3.66/1.71	
3.66/1.71		start location:	koat_start
3.66/1.71	
3.66/1.71		leaf cost:	0
3.66/1.71	
3.66/1.71	
3.66/1.71	
3.66/1.71	Complexity upper bound 102
3.66/1.71	
3.66/1.71	
3.66/1.71	
3.66/1.71	Time: 0.074 sec (SMT: 0.069 sec)
3.66/1.71	
3.66/1.71	
3.66/1.71	----------------------------------------
3.66/1.71	
3.66/1.71	(2)
3.66/1.71	BOUNDS(1, 1)
3.66/1.72	EOF