3.26/1.71	WORST_CASE(?, O(1))
3.26/1.72	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.26/1.72	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.26/1.72	
3.26/1.72	
3.26/1.72	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.26/1.72	
3.26/1.72	(0) CpxIntTrs
3.26/1.72	(1) Koat Proof [FINISHED, 29 ms]
3.26/1.72	(2) BOUNDS(1, 1)
3.26/1.72	
3.26/1.72	
3.26/1.72	----------------------------------------
3.26/1.72	
3.26/1.72	(0)
3.26/1.72	Obligation:
3.26/1.72	Complexity Int TRS consisting of the following rules:
3.26/1.72	f12(A, B, C, D, E) -> Com_1(f12(A + 1, B, C, D, E)) :|: 9 >= A
3.26/1.72	f25(A, B, C, D, E) -> Com_1(f25(A, B + 1, C, D, E)) :|: 9 >= B
3.26/1.72	f25(A, B, C, D, E) -> Com_1(f36(A, B, C, D, E)) :|: B >= 10
3.26/1.72	f12(A, B, C, D, E) -> Com_1(f25(A, 0, F, D, E)) :|: A >= 10
3.26/1.72	f0(A, B, C, D, E) -> Com_1(f12(0, B, C, F, G)) :|: TRUE
3.26/1.72	
3.26/1.72	The start-symbols are:[f0_5]
3.26/1.72	
3.26/1.72	
3.26/1.72	----------------------------------------
3.26/1.72	
3.26/1.72	(1) Koat Proof (FINISHED)
3.26/1.72	YES(?, 25)
3.26/1.72	
3.26/1.72	
3.26/1.72	
3.26/1.72	Initial complexity problem:
3.26/1.72	
3.26/1.72	1:	T:
3.26/1.72	
3.26/1.72			(Comp: ?, Cost: 1)    f12(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f12(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ 9 >= ar_0 ]
3.26/1.72	
3.26/1.72			(Comp: ?, Cost: 1)    f25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f25(ar_0, ar_1 + 1, ar_2, ar_3, ar_4)) [ 9 >= ar_1 ]
3.26/1.72	
3.26/1.72			(Comp: ?, Cost: 1)    f25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_1 >= 10 ]
3.26/1.72	
3.26/1.72			(Comp: ?, Cost: 1)    f12(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f25(ar_0, 0, f, ar_3, ar_4)) [ ar_0 >= 10 ]
3.26/1.72	
3.26/1.72			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f12(0, ar_1, ar_2, f, g))
3.26/1.72	
3.26/1.72			(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 ]
3.26/1.72	
3.26/1.72		start location:	koat_start
3.26/1.72	
3.26/1.72		leaf cost:	0
3.26/1.72	
3.26/1.72	
3.26/1.72	
3.26/1.72	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.26/1.72	
3.26/1.72	2:	T:
3.26/1.72	
3.26/1.72			(Comp: ?, Cost: 1)    f12(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f12(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ 9 >= ar_0 ]
3.26/1.72	
3.26/1.72			(Comp: ?, Cost: 1)    f25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f25(ar_0, ar_1 + 1, ar_2, ar_3, ar_4)) [ 9 >= ar_1 ]
3.26/1.72	
3.26/1.72			(Comp: ?, Cost: 1)    f25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_1 >= 10 ]
3.26/1.72	
3.26/1.72			(Comp: ?, Cost: 1)    f12(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f25(ar_0, 0, f, ar_3, ar_4)) [ ar_0 >= 10 ]
3.26/1.72	
3.26/1.72			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f12(0, ar_1, ar_2, f, g))
3.26/1.72	
3.26/1.72			(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 ]
3.26/1.72	
3.26/1.72		start location:	koat_start
3.26/1.72	
3.26/1.72		leaf cost:	0
3.26/1.72	
3.26/1.72	
3.26/1.72	
3.26/1.72	A polynomial rank function with
3.26/1.72	
3.26/1.72		Pol(f12) = 2
3.26/1.72	
3.26/1.72		Pol(f25) = 1
3.26/1.72	
3.26/1.72		Pol(f36) = 0
3.26/1.72	
3.26/1.72		Pol(f0) = 2
3.26/1.72	
3.26/1.72		Pol(koat_start) = 2
3.26/1.72	
3.26/1.72	orients all transitions weakly and the transitions
3.26/1.72	
3.26/1.72		f25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_1 >= 10 ]
3.26/1.72	
3.26/1.72		f12(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f25(ar_0, 0, f, ar_3, ar_4)) [ ar_0 >= 10 ]
3.26/1.72	
3.26/1.72	strictly and produces the following problem:
3.26/1.72	
3.26/1.72	3:	T:
3.26/1.72	
3.26/1.72			(Comp: ?, Cost: 1)    f12(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f12(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ 9 >= ar_0 ]
3.26/1.72	
3.26/1.72			(Comp: ?, Cost: 1)    f25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f25(ar_0, ar_1 + 1, ar_2, ar_3, ar_4)) [ 9 >= ar_1 ]
3.26/1.72	
3.26/1.72			(Comp: 2, Cost: 1)    f25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_1 >= 10 ]
3.26/1.72	
3.26/1.72			(Comp: 2, Cost: 1)    f12(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f25(ar_0, 0, f, ar_3, ar_4)) [ ar_0 >= 10 ]
3.26/1.72	
3.26/1.72			(Comp: 1, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f12(0, ar_1, ar_2, f, g))
3.26/1.72	
3.26/1.72			(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 ]
3.26/1.72	
3.26/1.72		start location:	koat_start
3.26/1.72	
3.26/1.72		leaf cost:	0
3.26/1.72	
3.26/1.72	
3.26/1.72	
3.26/1.72	A polynomial rank function with
3.26/1.72	
3.26/1.72		Pol(f12) = -V_1 + 10
3.26/1.72	
3.26/1.72		Pol(f25) = -V_1
3.26/1.72	
3.26/1.72		Pol(f36) = -V_1
3.26/1.72	
3.26/1.72		Pol(f0) = 10
3.26/1.72	
3.26/1.72		Pol(koat_start) = 10
3.26/1.72	
3.26/1.72	orients all transitions weakly and the transition
3.26/1.72	
3.26/1.72		f12(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f12(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ 9 >= ar_0 ]
3.26/1.72	
3.26/1.72	strictly and produces the following problem:
3.26/1.72	
3.26/1.72	4:	T:
3.26/1.72	
3.26/1.72			(Comp: 10, Cost: 1)    f12(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f12(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ 9 >= ar_0 ]
3.26/1.72	
3.26/1.72			(Comp: ?, Cost: 1)     f25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f25(ar_0, ar_1 + 1, ar_2, ar_3, ar_4)) [ 9 >= ar_1 ]
3.26/1.72	
3.26/1.72			(Comp: 2, Cost: 1)     f25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_1 >= 10 ]
3.26/1.72	
3.26/1.72			(Comp: 2, Cost: 1)     f12(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f25(ar_0, 0, f, ar_3, ar_4)) [ ar_0 >= 10 ]
3.26/1.72	
3.26/1.72			(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f12(0, ar_1, ar_2, f, g))
3.26/1.72	
3.26/1.72			(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 ]
3.26/1.72	
3.26/1.72		start location:	koat_start
3.26/1.72	
3.26/1.72		leaf cost:	0
3.26/1.72	
3.26/1.72	
3.26/1.72	
3.26/1.72	A polynomial rank function with
3.26/1.72	
3.26/1.72		Pol(f12) = 10
3.26/1.72	
3.26/1.72		Pol(f25) = -V_2 + 10
3.26/1.72	
3.26/1.72		Pol(f36) = -V_2
3.26/1.72	
3.26/1.72		Pol(f0) = 10
3.26/1.72	
3.26/1.72		Pol(koat_start) = 10
3.26/1.72	
3.26/1.72	orients all transitions weakly and the transition
3.26/1.72	
3.26/1.72		f25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f25(ar_0, ar_1 + 1, ar_2, ar_3, ar_4)) [ 9 >= ar_1 ]
3.26/1.72	
3.26/1.72	strictly and produces the following problem:
3.26/1.72	
3.26/1.72	5:	T:
3.26/1.72	
3.26/1.72			(Comp: 10, Cost: 1)    f12(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f12(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ 9 >= ar_0 ]
3.26/1.72	
3.26/1.72			(Comp: 10, Cost: 1)    f25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f25(ar_0, ar_1 + 1, ar_2, ar_3, ar_4)) [ 9 >= ar_1 ]
3.26/1.72	
3.26/1.72			(Comp: 2, Cost: 1)     f25(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f36(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_1 >= 10 ]
3.26/1.72	
3.26/1.72			(Comp: 2, Cost: 1)     f12(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f25(ar_0, 0, f, ar_3, ar_4)) [ ar_0 >= 10 ]
3.26/1.72	
3.26/1.72			(Comp: 1, Cost: 1)     f0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(f12(0, ar_1, ar_2, f, g))
3.26/1.72	
3.26/1.72			(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 ]
3.26/1.72	
3.26/1.72		start location:	koat_start
3.26/1.72	
3.26/1.72		leaf cost:	0
3.26/1.72	
3.26/1.72	
3.26/1.72	
3.26/1.72	Complexity upper bound 25
3.26/1.72	
3.26/1.72	
3.26/1.72	
3.26/1.72	Time: 0.099 sec (SMT: 0.089 sec)
3.26/1.72	
3.26/1.72	
3.26/1.72	----------------------------------------
3.26/1.72	
3.26/1.72	(2)
3.26/1.72	BOUNDS(1, 1)
3.65/1.73	EOF