3.78/1.73	WORST_CASE(?, O(1))
3.78/1.74	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.78/1.74	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.78/1.74	
3.78/1.74	
3.78/1.74	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1).
3.78/1.74	
3.78/1.74	(0) CpxIntTrs
3.78/1.74	(1) Koat Proof [FINISHED, 98 ms]
3.78/1.74	(2) BOUNDS(1, 1)
3.78/1.74	
3.78/1.74	
3.78/1.74	----------------------------------------
3.78/1.74	
3.78/1.74	(0)
3.78/1.74	Obligation:
3.78/1.74	Complexity Int TRS consisting of the following rules:
3.78/1.74	f0(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f7(8, 0, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: TRUE
3.78/1.74	f7(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f7(A, B + 1, U + V, W, X + Y, Z, A1 + B1, C1, D1 + E1, F1, U + V + D1 + E1, U + V - D1 - E1, X + Y + A1 + B1, X + Y - A1 - B1, -(3196), G1, H1, I1 + J1, K1 + J1, J1)) :|: 7 >= B
3.78/1.74	f62(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f62(A, B + 1, U + V, W, X + Y, Z, A1 + B1, C1, D1 + E1, F1, U + V + D1 + E1, U + V - D1 - E1, X + Y + A1 + B1, X + Y - A1 - B1, -(3196), G1, H1, I1 + J1, K1 + J1, J1)) :|: 7 >= B
3.78/1.74	f62(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f118(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: B >= 8
3.78/1.74	f7(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) -> Com_1(f62(A, 0, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)) :|: B >= 8
3.78/1.74	
3.78/1.74	The start-symbols are:[f0_20]
3.78/1.74	
3.78/1.74	
3.78/1.74	----------------------------------------
3.78/1.74	
3.78/1.74	(1) Koat Proof (FINISHED)
3.78/1.74	YES(?, 21)
3.78/1.74	
3.78/1.74	
3.78/1.74	
3.78/1.74	Initial complexity problem:
3.78/1.74	
3.78/1.74	1:	T:
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18, ar_19) -> Com_1(f7(8, 0, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18, ar_19))
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18, ar_19) -> Com_1(f7(ar_0, ar_1 + 1, u + v, w, x + y, z, a1 + b1, c1, d1 + e1, f1, u + v + d1 + e1, u + v - d1 - e1, x + y + a1 + b1, x + y - a1 - b1, -3196, g1, h1, i1 + j1, k1 + j1, j1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18, ar_19) -> Com_1(f62(ar_0, ar_1 + 1, u + v, w, x + y, z, a1 + b1, c1, d1 + e1, f1, u + v + d1 + e1, u + v - d1 - e1, x + y + a1 + b1, x + y - a1 - b1, -3196, g1, h1, i1 + j1, k1 + j1, j1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f62(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18, ar_19) -> Com_1(f118(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18, ar_19)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f7(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18, ar_19) -> Com_1(f62(ar_0, 0, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18, ar_19)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18, ar_19) -> Com_1(f0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11, ar_12, ar_13, ar_14, ar_15, ar_16, ar_17, ar_18, ar_19)) [ 0 <= 0 ]
3.78/1.74	
3.78/1.74		start location:	koat_start
3.78/1.74	
3.78/1.74		leaf cost:	0
3.78/1.74	
3.78/1.74	
3.78/1.74	
3.78/1.74	Slicing away variables that do not contribute to conditions from problem 1 leaves variables [ar_1].
3.78/1.74	
3.78/1.74	We thus obtain the following problem:
3.78/1.74	
3.78/1.74	2:	T:
3.78/1.74	
3.78/1.74			(Comp: 1, Cost: 0)    koat_start(ar_1) -> Com_1(f0(ar_1)) [ 0 <= 0 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f7(ar_1) -> Com_1(f62(0)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f62(ar_1) -> Com_1(f118(ar_1)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f62(ar_1) -> Com_1(f62(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f7(ar_1) -> Com_1(f7(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f0(ar_1) -> Com_1(f7(0))
3.78/1.74	
3.78/1.74		start location:	koat_start
3.78/1.74	
3.78/1.74		leaf cost:	0
3.78/1.74	
3.78/1.74	
3.78/1.74	
3.78/1.74	Repeatedly propagating knowledge in problem 2 produces the following problem:
3.78/1.74	
3.78/1.74	3:	T:
3.78/1.74	
3.78/1.74			(Comp: 1, Cost: 0)    koat_start(ar_1) -> Com_1(f0(ar_1)) [ 0 <= 0 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f7(ar_1) -> Com_1(f62(0)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f62(ar_1) -> Com_1(f118(ar_1)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f62(ar_1) -> Com_1(f62(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f7(ar_1) -> Com_1(f7(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74			(Comp: 1, Cost: 1)    f0(ar_1) -> Com_1(f7(0))
3.78/1.74	
3.78/1.74		start location:	koat_start
3.78/1.74	
3.78/1.74		leaf cost:	0
3.78/1.74	
3.78/1.74	
3.78/1.74	
3.78/1.74	A polynomial rank function with
3.78/1.74	
3.78/1.74		Pol(koat_start) = 2
3.78/1.74	
3.78/1.74		Pol(f0) = 2
3.78/1.74	
3.78/1.74		Pol(f7) = 2
3.78/1.74	
3.78/1.74		Pol(f62) = 1
3.78/1.74	
3.78/1.74		Pol(f118) = 0
3.78/1.74	
3.78/1.74	orients all transitions weakly and the transitions
3.78/1.74	
3.78/1.74		f7(ar_1) -> Com_1(f62(0)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74		f62(ar_1) -> Com_1(f118(ar_1)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74	strictly and produces the following problem:
3.78/1.74	
3.78/1.74	4:	T:
3.78/1.74	
3.78/1.74			(Comp: 1, Cost: 0)    koat_start(ar_1) -> Com_1(f0(ar_1)) [ 0 <= 0 ]
3.78/1.74	
3.78/1.74			(Comp: 2, Cost: 1)    f7(ar_1) -> Com_1(f62(0)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74			(Comp: 2, Cost: 1)    f62(ar_1) -> Com_1(f118(ar_1)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f62(ar_1) -> Com_1(f62(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f7(ar_1) -> Com_1(f7(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74			(Comp: 1, Cost: 1)    f0(ar_1) -> Com_1(f7(0))
3.78/1.74	
3.78/1.74		start location:	koat_start
3.78/1.74	
3.78/1.74		leaf cost:	0
3.78/1.74	
3.78/1.74	
3.78/1.74	
3.78/1.74	A polynomial rank function with
3.78/1.74	
3.78/1.74		Pol(koat_start) = 8
3.78/1.74	
3.78/1.74		Pol(f0) = 8
3.78/1.74	
3.78/1.74		Pol(f7) = 8
3.78/1.74	
3.78/1.74		Pol(f62) = -V_1 + 8
3.78/1.74	
3.78/1.74		Pol(f118) = -V_1
3.78/1.74	
3.78/1.74	orients all transitions weakly and the transition
3.78/1.74	
3.78/1.74		f62(ar_1) -> Com_1(f62(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74	strictly and produces the following problem:
3.78/1.74	
3.78/1.74	5:	T:
3.78/1.74	
3.78/1.74			(Comp: 1, Cost: 0)    koat_start(ar_1) -> Com_1(f0(ar_1)) [ 0 <= 0 ]
3.78/1.74	
3.78/1.74			(Comp: 2, Cost: 1)    f7(ar_1) -> Com_1(f62(0)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74			(Comp: 2, Cost: 1)    f62(ar_1) -> Com_1(f118(ar_1)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74			(Comp: 8, Cost: 1)    f62(ar_1) -> Com_1(f62(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74			(Comp: ?, Cost: 1)    f7(ar_1) -> Com_1(f7(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74			(Comp: 1, Cost: 1)    f0(ar_1) -> Com_1(f7(0))
3.78/1.74	
3.78/1.74		start location:	koat_start
3.78/1.74	
3.78/1.74		leaf cost:	0
3.78/1.74	
3.78/1.74	
3.78/1.74	
3.78/1.74	A polynomial rank function with
3.78/1.74	
3.78/1.74		Pol(f7) = -V_1 + 8
3.78/1.74	
3.78/1.74	and size complexities
3.78/1.74	
3.78/1.74		S("f0(ar_1) -> Com_1(f7(0))", 0-0) = 0
3.78/1.74	
3.78/1.74		S("f7(ar_1) -> Com_1(f7(ar_1 + 1)) [ 7 >= ar_1 ]", 0-0) = 8
3.78/1.74	
3.78/1.74		S("f62(ar_1) -> Com_1(f62(ar_1 + 1)) [ 7 >= ar_1 ]", 0-0) = 8
3.78/1.74	
3.78/1.74		S("f62(ar_1) -> Com_1(f118(ar_1)) [ ar_1 >= 8 ]", 0-0) = 8
3.78/1.74	
3.78/1.74		S("f7(ar_1) -> Com_1(f62(0)) [ ar_1 >= 8 ]", 0-0) = 0
3.78/1.74	
3.78/1.74		S("koat_start(ar_1) -> Com_1(f0(ar_1)) [ 0 <= 0 ]", 0-0) = ar_1
3.78/1.74	
3.78/1.74	orients the transitions
3.78/1.74	
3.78/1.74		f7(ar_1) -> Com_1(f7(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74	weakly and the transition
3.78/1.74	
3.78/1.74		f7(ar_1) -> Com_1(f7(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74	strictly and produces the following problem:
3.78/1.74	
3.78/1.74	6:	T:
3.78/1.74	
3.78/1.74			(Comp: 1, Cost: 0)    koat_start(ar_1) -> Com_1(f0(ar_1)) [ 0 <= 0 ]
3.78/1.74	
3.78/1.74			(Comp: 2, Cost: 1)    f7(ar_1) -> Com_1(f62(0)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74			(Comp: 2, Cost: 1)    f62(ar_1) -> Com_1(f118(ar_1)) [ ar_1 >= 8 ]
3.78/1.74	
3.78/1.74			(Comp: 8, Cost: 1)    f62(ar_1) -> Com_1(f62(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74			(Comp: 8, Cost: 1)    f7(ar_1) -> Com_1(f7(ar_1 + 1)) [ 7 >= ar_1 ]
3.78/1.74	
3.78/1.74			(Comp: 1, Cost: 1)    f0(ar_1) -> Com_1(f7(0))
3.78/1.74	
3.78/1.74		start location:	koat_start
3.78/1.74	
3.78/1.74		leaf cost:	0
3.78/1.74	
3.78/1.74	
3.78/1.74	
3.78/1.74	Complexity upper bound 21
3.78/1.74	
3.78/1.74	
3.78/1.74	
3.78/1.74	Time: 0.082 sec (SMT: 0.079 sec)
3.78/1.74	
3.78/1.74	
3.78/1.74	----------------------------------------
3.78/1.74	
3.78/1.74	(2)
3.78/1.74	BOUNDS(1, 1)
3.78/1.75	EOF