8.21/4.97 WORST_CASE(Omega(n^1), O(n^2)) 8.21/4.98 proof of /export/starexec/sandbox/benchmark/theBenchmark.koat 8.21/4.98 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 8.21/4.98 8.21/4.98 8.21/4.98 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^2). 8.21/4.98 8.21/4.98 (0) CpxIntTrs 8.21/4.98 (1) Koat Proof [FINISHED, 138 ms] 8.21/4.98 (2) BOUNDS(1, n^2) 8.21/4.98 (3) Loat Proof [FINISHED, 1208 ms] 8.21/4.98 (4) BOUNDS(n^1, INF) 8.21/4.98 8.21/4.98 8.21/4.98 ---------------------------------------- 8.21/4.98 8.21/4.98 (0) 8.21/4.98 Obligation: 8.21/4.98 Complexity Int TRS consisting of the following rules: 8.21/4.98 f0(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f3(A, B, C, D, E, F, G, H, I, J, K, L)) :|: TRUE 8.21/4.98 f0(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f4(A, B, C, D, E, F, G, H, I, J, K, L)) :|: TRUE 8.21/4.98 f0(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f8(A, B, C, D, E, F, G, H, I, J, K, L)) :|: TRUE 8.21/4.98 f3(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f4(A, B, D, D, F, F, A, B, I, J, K, L)) :|: A >= B + 1 8.21/4.98 f3(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f4(A, B, D, D, F, F, A, B, I, J, K, L)) :|: B >= A + 1 8.21/4.98 f0(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f3(F, D, D, D, F, F, A, B, I, J, K, L)) :|: TRUE 8.21/4.98 f0(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f3(F, D, D, D, F, F, M, N, M, N, K, L)) :|: TRUE 8.21/4.98 f0(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f8(O, P, D, M, F, N, A, B, M, N, O, P)) :|: TRUE 8.21/4.98 f3(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f8(O, P, D, M, F, N, A, A, M, N, O, P)) :|: A >= B && A <= B 8.21/4.98 f0(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f3(A + 1, B, D, D, F, F, A, B, I, J, K, L)) :|: TRUE 8.21/4.98 f4(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f3(A + 1, B, D, D, F, F, A, B, I, J, K, L)) :|: B >= A + 1 8.21/4.98 f0(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f3(A, B + 1, D, D, F, F, A, B, I, J, K, L)) :|: TRUE 8.21/4.98 f4(A, B, C, D, E, F, G, H, I, J, K, L) -> Com_1(f3(A, B + 1, D, D, F, F, A, B, I, J, K, L)) :|: A >= B 8.21/4.98 8.21/4.98 The start-symbols are:[f0_12] 8.21/4.98 8.21/4.98 8.21/4.98 ---------------------------------------- 8.21/4.98 8.21/4.98 (1) Koat Proof (FINISHED) 8.21/4.98 YES(?, 42136*ar_0 + 42076*ar_1 + 21368*ar_5 + 21308*ar_3 + 320*ar_0^2 + 608*ar_0*ar_1 + 480*ar_0*ar_5 + 448*ar_0*ar_3 + 464*ar_1*ar_5 + 160*ar_5^2 + 304*ar_3*ar_5 + 288*ar_1^2 + 432*ar_1*ar_3 + 144*ar_3^2 + 77869) 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Initial complexity problem: 8.21/4.98 8.21/4.98 1: T: 8.21/4.98 8.21/4.98 (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) -> Com_1(f3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11)) 8.21/4.98 8.21/4.98 (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) -> Com_1(f4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11)) 8.21/4.98 8.21/4.98 (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) -> Com_1(f8(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11) -> Com_1(f4(ar_0, ar_1, ar_3, ar_3, ar_5, ar_5, ar_0, ar_1, ar_8, ar_9, ar_10, ar_11)) [ ar_0 >= ar_1 + 1 ] 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11) -> Com_1(f4(ar_0, ar_1, ar_3, ar_3, ar_5, ar_5, ar_0, ar_1, ar_8, ar_9, ar_10, ar_11)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 (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) -> Com_1(f3(ar_5, ar_3, ar_3, ar_3, ar_5, ar_5, ar_0, ar_1, ar_8, ar_9, ar_10, ar_11)) 8.21/4.98 8.21/4.98 (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) -> Com_1(f3(ar_5, ar_3, ar_3, ar_3, ar_5, ar_5, m, n, m, n, ar_10, ar_11)) 8.21/4.98 8.21/4.98 (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) -> Com_1(f8(m, n, ar_3, o, ar_5, p, ar_0, ar_1, o, p, m, n)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11) -> Com_1(f8(m, n, ar_3, o, ar_5, p, ar_0, ar_0, o, p, m, n)) [ ar_0 = ar_1 ] 8.21/4.98 8.21/4.98 (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) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_3, ar_5, ar_5, ar_0, ar_1, ar_8, ar_9, ar_10, ar_11)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_3, ar_5, ar_5, ar_0, ar_1, ar_8, ar_9, ar_10, ar_11)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 (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) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_3, ar_5, ar_5, ar_0, ar_1, ar_8, ar_9, ar_10, ar_11)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5, ar_6, ar_7, ar_8, ar_9, ar_10, ar_11) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_3, ar_5, ar_5, ar_0, ar_1, ar_8, ar_9, ar_10, ar_11)) [ ar_0 >= ar_1 ] 8.21/4.98 8.21/4.98 (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) -> 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)) [ 0 <= 0 ] 8.21/4.98 8.21/4.98 start location: koat_start 8.21/4.98 8.21/4.98 leaf cost: 0 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [ar_0, ar_1, ar_3, ar_5]. 8.21/4.98 8.21/4.98 We thus obtain the following problem: 8.21/4.98 8.21/4.98 2: T: 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ] 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ] 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ] 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ] 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 start location: koat_start 8.21/4.98 8.21/4.98 leaf cost: 0 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Repeatedly propagating knowledge in problem 2 produces the following problem: 8.21/4.98 8.21/4.98 3: T: 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ] 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 start location: koat_start 8.21/4.98 8.21/4.98 leaf cost: 0 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 A polynomial rank function with 8.21/4.98 8.21/4.98 Pol(koat_start) = 1 8.21/4.98 8.21/4.98 Pol(f0) = 1 8.21/4.98 8.21/4.98 Pol(f4) = 1 8.21/4.98 8.21/4.98 Pol(f3) = 1 8.21/4.98 8.21/4.98 Pol(f8) = 0 8.21/4.98 8.21/4.98 orients all transitions weakly and the transition 8.21/4.98 8.21/4.98 f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ] 8.21/4.98 8.21/4.98 strictly and produces the following problem: 8.21/4.98 8.21/4.98 4: T: 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ] 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 start location: koat_start 8.21/4.98 8.21/4.98 leaf cost: 0 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 A polynomial rank function with 8.21/4.98 8.21/4.98 Pol(f4) = 2*V_1 - 2*V_2 + 1 8.21/4.98 8.21/4.98 Pol(f3) = 2*V_1 - 2*V_2 + 2 8.21/4.98 8.21/4.98 and size complexities 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5))", 0-0) = ar_0 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5))", 0-1) = ar_1 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5))", 0-0) = ar_0 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5))", 0-1) = ar_1 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5))", 0-0) = ar_0 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5))", 0-1) = ar_1 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ]", 0-0) = ar_0 + ar_5 + 1 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ? 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ]", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ]", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-0) = ? 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-1) = ? 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-0) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-1) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-0) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-1) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-0) = ? 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-1) = ? 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-2) = ? 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-3) = ? 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-0) = ? 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-1) = ? 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-2) = ? 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-3) = ? 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5))", 0-0) = ar_0 + 1 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5))", 0-1) = ar_1 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-0) = ? 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-1) = ? 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5))", 0-0) = ar_0 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5))", 0-1) = ar_1 + 1 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ]", 0-0) = ar_0 + ar_5 + 1 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ]", 0-1) = ? 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ]", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ]", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ]", 0-0) = ar_0 8.21/4.98 8.21/4.98 S("koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ]", 0-1) = ar_1 8.21/4.98 8.21/4.98 S("koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ]", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ]", 0-3) = ar_5 8.21/4.98 8.21/4.98 orients the transitions 8.21/4.98 8.21/4.98 f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ] 8.21/4.98 8.21/4.98 f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ] 8.21/4.98 8.21/4.98 weakly and the transitions 8.21/4.98 8.21/4.98 f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ] 8.21/4.98 8.21/4.98 f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ] 8.21/4.98 8.21/4.98 strictly and produces the following problem: 8.21/4.98 8.21/4.98 5: T: 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ] 8.21/4.98 8.21/4.98 (Comp: 8*ar_0 + 8*ar_1 + 4*ar_5 + 4*ar_3 + 15, Cost: 1) f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 (Comp: 8*ar_0 + 8*ar_1 + 4*ar_5 + 4*ar_3 + 15, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 start location: koat_start 8.21/4.98 8.21/4.98 leaf cost: 0 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 A polynomial rank function with 8.21/4.98 8.21/4.98 Pol(f4) = -2*V_1 + 2*V_2 - 1 8.21/4.98 8.21/4.98 Pol(f3) = -2*V_1 + 2*V_2 8.21/4.98 8.21/4.98 and size complexities 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5))", 0-0) = ar_0 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5))", 0-1) = ar_1 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5))", 0-0) = ar_0 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5))", 0-1) = ar_1 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5))", 0-0) = ar_0 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5))", 0-1) = ar_1 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ]", 0-0) = ar_0 + ar_5 + 1 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ]", 0-1) = 9*ar_0 + 9*ar_1 + 9*ar_3 + 9*ar_5 + 1296 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ]", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ]", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-0) = ? 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-1) = 9*ar_0 + 9*ar_1 + 9*ar_3 + 9*ar_5 + 11664 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-0) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-1) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-0) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-1) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-0) = ? 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-1) = ? 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-2) = ? 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-3) = ? 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-0) = ? 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-1) = ? 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-2) = ? 8.21/4.98 8.21/4.98 S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-3) = ? 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5))", 0-0) = ar_0 + 1 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5))", 0-1) = ar_1 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-0) = ? 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-1) = 9*ar_0 + 9*ar_1 + 9*ar_3 + 9*ar_5 + 11664 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ]", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5))", 0-0) = ar_0 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5))", 0-1) = ar_1 + 1 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5))", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5))", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ]", 0-0) = ar_0 + ar_5 + 1 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ]", 0-1) = 9*ar_0 + 9*ar_1 + 9*ar_3 + 9*ar_5 + 1296 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ]", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ]", 0-3) = ar_5 8.21/4.98 8.21/4.98 S("koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ]", 0-0) = ar_0 8.21/4.98 8.21/4.98 S("koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ]", 0-1) = ar_1 8.21/4.98 8.21/4.98 S("koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ]", 0-2) = ar_3 8.21/4.98 8.21/4.98 S("koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ]", 0-3) = ar_5 8.21/4.98 8.21/4.98 orients the transitions 8.21/4.98 8.21/4.98 f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 weakly and the transitions 8.21/4.98 8.21/4.98 f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 strictly and produces the following problem: 8.21/4.98 8.21/4.98 6: T: 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_3, ar_5) -> Com_1(f0(ar_0, ar_1, ar_3, ar_5)) [ 0 <= 0 ] 8.21/4.98 8.21/4.98 (Comp: 8*ar_0 + 8*ar_1 + 4*ar_5 + 4*ar_3 + 15, Cost: 1) f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) [ ar_0 >= ar_1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 21060*ar_0 + 21030*ar_1 + 10680*ar_5 + 10650*ar_3 + 160*ar_0^2 + 304*ar_0*ar_1 + 240*ar_0*ar_5 + 224*ar_0*ar_3 + 232*ar_1*ar_5 + 80*ar_5^2 + 152*ar_3*ar_5 + 144*ar_1^2 + 216*ar_1*ar_3 + 72*ar_3^2 + 38915, Cost: 1) f4(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 21060*ar_0 + 21030*ar_1 + 10680*ar_5 + 10650*ar_3 + 160*ar_0^2 + 304*ar_0*ar_1 + 240*ar_0*ar_5 + 224*ar_0*ar_3 + 232*ar_1*ar_5 + 80*ar_5^2 + 152*ar_3*ar_5 + 144*ar_1^2 + 216*ar_1*ar_3 + 72*ar_3^2 + 38915, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_1 >= ar_0 + 1 ] 8.21/4.98 8.21/4.98 (Comp: 8*ar_0 + 8*ar_1 + 4*ar_5 + 4*ar_3 + 15, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) [ ar_0 >= ar_1 + 1 ] 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5)) 8.21/4.98 8.21/4.98 start location: koat_start 8.21/4.98 8.21/4.98 leaf cost: 0 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Complexity upper bound 42136*ar_0 + 42076*ar_1 + 21368*ar_5 + 21308*ar_3 + 320*ar_0^2 + 608*ar_0*ar_1 + 480*ar_0*ar_5 + 448*ar_0*ar_3 + 464*ar_1*ar_5 + 160*ar_5^2 + 304*ar_3*ar_5 + 288*ar_1^2 + 432*ar_1*ar_3 + 144*ar_3^2 + 77869 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Time: 0.179 sec (SMT: 0.145 sec) 8.21/4.98 8.21/4.98 8.21/4.98 ---------------------------------------- 8.21/4.98 8.21/4.98 (2) 8.21/4.98 BOUNDS(1, n^2) 8.21/4.98 8.21/4.98 ---------------------------------------- 8.21/4.98 8.21/4.98 (3) Loat Proof (FINISHED) 8.21/4.98 8.21/4.98 8.21/4.98 ### Pre-processing the ITS problem ### 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Initial linear ITS problem 8.21/4.98 8.21/4.98 Start location: f0 8.21/4.98 8.21/4.98 0: f0 -> f3 : [], cost: 1 8.21/4.98 8.21/4.98 1: f0 -> f4 : [], cost: 1 8.21/4.98 8.21/4.98 2: f0 -> f8 : [], cost: 1 8.21/4.98 8.21/4.98 5: f0 -> f3 : A'=F, B'=D, C'=D, E'=F, G'=A, H'=B, [], cost: 1 8.21/4.98 8.21/4.98 6: f0 -> f3 : A'=F, B'=D, C'=D, E'=F, G'=free, H'=free_1, Q'=free, J'=free_1, [], cost: 1 8.21/4.98 8.21/4.98 7: f0 -> f8 : A'=free_3, B'=free_4, C'=D, D'=free_5, E'=F, F'=free_2, G'=A, H'=B, Q'=free_5, J'=free_2, K'=free_3, L'=free_4, [], cost: 1 8.21/4.98 8.21/4.98 9: f0 -> f3 : A'=1+A, C'=D, E'=F, G'=A, H'=B, [], cost: 1 8.21/4.98 8.21/4.98 11: f0 -> f3 : B'=1+B, C'=D, E'=F, G'=A, H'=B, [], cost: 1 8.21/4.98 8.21/4.98 3: f3 -> f4 : C'=D, E'=F, G'=A, H'=B, [ A>=1+B ], cost: 1 8.21/4.98 8.21/4.98 4: f3 -> f4 : C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 1 8.21/4.98 8.21/4.98 8: f3 -> f8 : A'=free_7, B'=free_8, C'=D, D'=free_9, E'=F, F'=free_6, G'=A, H'=A, Q'=free_9, J'=free_6, K'=free_7, L'=free_8, [ A==B ], cost: 1 8.21/4.98 8.21/4.98 10: f4 -> f3 : A'=1+A, C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 1 8.21/4.98 8.21/4.98 12: f4 -> f3 : B'=1+B, C'=D, E'=F, G'=A, H'=B, [ A>=B ], cost: 1 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Removed unreachable and leaf rules: 8.21/4.98 8.21/4.98 Start location: f0 8.21/4.98 8.21/4.98 0: f0 -> f3 : [], cost: 1 8.21/4.98 8.21/4.98 1: f0 -> f4 : [], cost: 1 8.21/4.98 8.21/4.98 5: f0 -> f3 : A'=F, B'=D, C'=D, E'=F, G'=A, H'=B, [], cost: 1 8.21/4.98 8.21/4.98 6: f0 -> f3 : A'=F, B'=D, C'=D, E'=F, G'=free, H'=free_1, Q'=free, J'=free_1, [], cost: 1 8.21/4.98 8.21/4.98 9: f0 -> f3 : A'=1+A, C'=D, E'=F, G'=A, H'=B, [], cost: 1 8.21/4.98 8.21/4.98 11: f0 -> f3 : B'=1+B, C'=D, E'=F, G'=A, H'=B, [], cost: 1 8.21/4.98 8.21/4.98 3: f3 -> f4 : C'=D, E'=F, G'=A, H'=B, [ A>=1+B ], cost: 1 8.21/4.98 8.21/4.98 4: f3 -> f4 : C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 1 8.21/4.98 8.21/4.98 10: f4 -> f3 : A'=1+A, C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 1 8.21/4.98 8.21/4.98 12: f4 -> f3 : B'=1+B, C'=D, E'=F, G'=A, H'=B, [ A>=B ], cost: 1 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 ### Simplification by acceleration and chaining ### 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Eliminated location f3 (as a last resort): 8.21/4.98 8.21/4.98 Start location: f0 8.21/4.98 8.21/4.98 1: f0 -> f4 : [], cost: 1 8.21/4.98 8.21/4.98 13: f0 -> f4 : C'=D, E'=F, G'=A, H'=B, [ A>=1+B ], cost: 2 8.21/4.98 8.21/4.98 14: f0 -> f4 : C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 2 8.21/4.98 8.21/4.98 15: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, [ F>=1+D ], cost: 2 8.21/4.98 8.21/4.98 16: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, [ D>=1+F ], cost: 2 8.21/4.98 8.21/4.98 17: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, Q'=free, J'=free_1, [ F>=1+D ], cost: 2 8.21/4.98 8.21/4.98 18: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, Q'=free, J'=free_1, [ D>=1+F ], cost: 2 8.21/4.98 8.21/4.98 19: f0 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ 1+A>=1+B ], cost: 2 8.21/4.98 8.21/4.98 20: f0 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ B>=2+A ], cost: 2 8.21/4.98 8.21/4.98 22: f0 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ A>=2+B ], cost: 2 8.21/4.98 8.21/4.98 23: f0 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ 1+B>=1+A ], cost: 2 8.21/4.98 8.21/4.98 21: f4 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ B>=2+A ], cost: 2 8.21/4.98 8.21/4.98 24: f4 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ A>=2+B ], cost: 2 8.21/4.98 8.21/4.98 25: f4 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ A>=B && 1+B>=1+A ], cost: 2 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Accelerating simple loops of location 2. 8.21/4.98 8.21/4.98 Simplified some of the simple loops (and removed duplicate rules). 8.21/4.98 8.21/4.98 Accelerating the following rules: 8.21/4.98 8.21/4.98 21: f4 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ B>=2+A ], cost: 2 8.21/4.98 8.21/4.98 24: f4 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ A>=2+B ], cost: 2 8.21/4.98 8.21/4.98 25: f4 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ -A+B==0 ], cost: 2 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Accelerated rule 21 with metering function -1-A+B, yielding the new rule 26. 8.21/4.98 8.21/4.98 Accelerated rule 24 with metering function -1+A-B, yielding the new rule 27. 8.21/4.98 8.21/4.98 Accelerated rule 25 with metering function 1+A-B, yielding the new rule 28. 8.21/4.98 8.21/4.98 Removing the simple loops: 21 24 25. 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Accelerated all simple loops using metering functions (where possible): 8.21/4.98 8.21/4.98 Start location: f0 8.21/4.98 8.21/4.98 1: f0 -> f4 : [], cost: 1 8.21/4.98 8.21/4.98 13: f0 -> f4 : C'=D, E'=F, G'=A, H'=B, [ A>=1+B ], cost: 2 8.21/4.98 8.21/4.98 14: f0 -> f4 : C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 2 8.21/4.98 8.21/4.98 15: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, [ F>=1+D ], cost: 2 8.21/4.98 8.21/4.98 16: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, [ D>=1+F ], cost: 2 8.21/4.98 8.21/4.98 17: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, Q'=free, J'=free_1, [ F>=1+D ], cost: 2 8.21/4.98 8.21/4.98 18: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, Q'=free, J'=free_1, [ D>=1+F ], cost: 2 8.21/4.98 8.21/4.98 19: f0 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ 1+A>=1+B ], cost: 2 8.21/4.98 8.21/4.98 20: f0 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ B>=2+A ], cost: 2 8.21/4.98 8.21/4.98 22: f0 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ A>=2+B ], cost: 2 8.21/4.98 8.21/4.98 23: f0 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ 1+B>=1+A ], cost: 2 8.21/4.98 8.21/4.98 26: f4 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=2+A ], cost: -2-2*A+2*B 8.21/4.98 8.21/4.98 27: f4 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=2+B ], cost: -2+2*A-2*B 8.21/4.98 8.21/4.98 28: f4 -> f4 : B'=1+A, C'=D, E'=F, G'=A, H'=1+A, [ -A+B==0 ], cost: 2+2*A-2*B 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Chained accelerated rules (with incoming rules): 8.21/4.98 8.21/4.98 Start location: f0 8.21/4.98 8.21/4.98 1: f0 -> f4 : [], cost: 1 8.21/4.98 8.21/4.98 13: f0 -> f4 : C'=D, E'=F, G'=A, H'=B, [ A>=1+B ], cost: 2 8.21/4.98 8.21/4.98 14: f0 -> f4 : C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 2 8.21/4.98 8.21/4.98 15: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, [ F>=1+D ], cost: 2 8.21/4.98 8.21/4.98 16: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, [ D>=1+F ], cost: 2 8.21/4.98 8.21/4.98 17: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, Q'=free, J'=free_1, [ F>=1+D ], cost: 2 8.21/4.98 8.21/4.98 18: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, Q'=free, J'=free_1, [ D>=1+F ], cost: 2 8.21/4.98 8.21/4.98 19: f0 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ 1+A>=1+B ], cost: 2 8.21/4.98 8.21/4.98 20: f0 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ B>=2+A ], cost: 2 8.21/4.98 8.21/4.98 22: f0 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ A>=2+B ], cost: 2 8.21/4.98 8.21/4.98 23: f0 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ 1+B>=1+A ], cost: 2 8.21/4.98 8.21/4.98 29: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=2+A ], cost: -1-2*A+2*B 8.21/4.98 8.21/4.98 30: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=2+A ], cost: -2*A+2*B 8.21/4.98 8.21/4.98 31: f0 -> f4 : A'=-1+D, B'=D, C'=D, E'=F, G'=-1+D, H'=D, [ D>=2+F ], cost: -2*F+2*D 8.21/4.98 8.21/4.98 32: f0 -> f4 : A'=-1+D, B'=D, C'=D, E'=F, G'=-1+D, H'=D, Q'=free, J'=free_1, [ D>=2+F ], cost: -2*F+2*D 8.21/4.98 8.21/4.98 33: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=3+A ], cost: -2-2*A+2*B 8.21/4.98 8.21/4.98 34: f0 -> f4 : A'=B, B'=1+B, C'=D, E'=F, G'=B, H'=1+B, [ 1+B>=2+A ], cost: 2-2*A+2*B 8.21/4.98 8.21/4.98 35: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=2+B ], cost: -1+2*A-2*B 8.21/4.98 8.21/4.98 36: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=2+B ], cost: 2*A-2*B 8.21/4.98 8.21/4.98 37: f0 -> f4 : A'=F, B'=-1+F, C'=D, E'=F, G'=F, H'=-1+F, [ F>=2+D ], cost: 2*F-2*D 8.21/4.98 8.21/4.98 38: f0 -> f4 : A'=F, B'=-1+F, C'=D, E'=F, G'=F, H'=-1+F, Q'=free, J'=free_1, [ F>=2+D ], cost: 2*F-2*D 8.21/4.98 8.21/4.98 39: f0 -> f4 : A'=1+A, B'=A, C'=D, E'=F, G'=1+A, H'=A, [ 1+A>=2+B ], cost: 2+2*A-2*B 8.21/4.98 8.21/4.98 40: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=3+B ], cost: -2+2*A-2*B 8.21/4.98 8.21/4.98 41: f0 -> f4 : B'=1+A, C'=D, E'=F, G'=A, H'=1+A, [ -A+B==0 ], cost: 3+2*A-2*B 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Removed unreachable locations (and leaf rules with constant cost): 8.21/4.98 8.21/4.98 Start location: f0 8.21/4.98 8.21/4.98 29: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=2+A ], cost: -1-2*A+2*B 8.21/4.98 8.21/4.98 30: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=2+A ], cost: -2*A+2*B 8.21/4.98 8.21/4.98 31: f0 -> f4 : A'=-1+D, B'=D, C'=D, E'=F, G'=-1+D, H'=D, [ D>=2+F ], cost: -2*F+2*D 8.21/4.98 8.21/4.98 32: f0 -> f4 : A'=-1+D, B'=D, C'=D, E'=F, G'=-1+D, H'=D, Q'=free, J'=free_1, [ D>=2+F ], cost: -2*F+2*D 8.21/4.98 8.21/4.98 33: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=3+A ], cost: -2-2*A+2*B 8.21/4.98 8.21/4.98 34: f0 -> f4 : A'=B, B'=1+B, C'=D, E'=F, G'=B, H'=1+B, [ 1+B>=2+A ], cost: 2-2*A+2*B 8.21/4.98 8.21/4.98 35: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=2+B ], cost: -1+2*A-2*B 8.21/4.98 8.21/4.98 36: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=2+B ], cost: 2*A-2*B 8.21/4.98 8.21/4.98 37: f0 -> f4 : A'=F, B'=-1+F, C'=D, E'=F, G'=F, H'=-1+F, [ F>=2+D ], cost: 2*F-2*D 8.21/4.98 8.21/4.98 38: f0 -> f4 : A'=F, B'=-1+F, C'=D, E'=F, G'=F, H'=-1+F, Q'=free, J'=free_1, [ F>=2+D ], cost: 2*F-2*D 8.21/4.98 8.21/4.98 39: f0 -> f4 : A'=1+A, B'=A, C'=D, E'=F, G'=1+A, H'=A, [ 1+A>=2+B ], cost: 2+2*A-2*B 8.21/4.98 8.21/4.98 40: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=3+B ], cost: -2+2*A-2*B 8.21/4.98 8.21/4.98 41: f0 -> f4 : B'=1+A, C'=D, E'=F, G'=A, H'=1+A, [ -A+B==0 ], cost: 3+2*A-2*B 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 ### Computing asymptotic complexity ### 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Fully simplified ITS problem 8.21/4.98 8.21/4.98 Start location: f0 8.21/4.98 8.21/4.98 30: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=2+A ], cost: -2*A+2*B 8.21/4.98 8.21/4.98 32: f0 -> f4 : A'=-1+D, B'=D, C'=D, E'=F, G'=-1+D, H'=D, Q'=free, J'=free_1, [ D>=2+F ], cost: -2*F+2*D 8.21/4.98 8.21/4.98 33: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=3+A ], cost: -2-2*A+2*B 8.21/4.98 8.21/4.98 34: f0 -> f4 : A'=B, B'=1+B, C'=D, E'=F, G'=B, H'=1+B, [ 1+B>=2+A ], cost: 2-2*A+2*B 8.21/4.98 8.21/4.98 36: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=2+B ], cost: 2*A-2*B 8.21/4.98 8.21/4.98 38: f0 -> f4 : A'=F, B'=-1+F, C'=D, E'=F, G'=F, H'=-1+F, Q'=free, J'=free_1, [ F>=2+D ], cost: 2*F-2*D 8.21/4.98 8.21/4.98 39: f0 -> f4 : A'=1+A, B'=A, C'=D, E'=F, G'=1+A, H'=A, [ 1+A>=2+B ], cost: 2+2*A-2*B 8.21/4.98 8.21/4.98 40: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=3+B ], cost: -2+2*A-2*B 8.21/4.98 8.21/4.98 41: f0 -> f4 : B'=1+A, C'=D, E'=F, G'=A, H'=1+A, [ -A+B==0 ], cost: 3+2*A-2*B 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Computing asymptotic complexity for rule 30 8.21/4.98 8.21/4.98 Solved the limit problem by the following transformations: 8.21/4.98 8.21/4.98 Created initial limit problem: 8.21/4.98 8.21/4.98 -2*A+2*B (+), -1-A+B (+/+!) [not solved] 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 removing all constraints (solved by SMT) 8.21/4.98 8.21/4.98 resulting limit problem: [solved] 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 applying transformation rule (C) using substitution {A==0,B==n} 8.21/4.98 8.21/4.98 resulting limit problem: 8.21/4.98 8.21/4.98 [solved] 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Solution: 8.21/4.98 8.21/4.98 A / 0 8.21/4.98 8.21/4.98 B / n 8.21/4.98 8.21/4.98 Resulting cost 2*n has complexity: Poly(n^1) 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Found new complexity Poly(n^1). 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 Obtained the following overall complexity (w.r.t. the length of the input n): 8.21/4.98 8.21/4.98 Complexity: Poly(n^1) 8.21/4.98 8.21/4.98 Cpx degree: 1 8.21/4.98 8.21/4.98 Solved cost: 2*n 8.21/4.98 8.21/4.98 Rule cost: -2*A+2*B 8.21/4.98 8.21/4.98 Rule guard: [ B>=2+A ] 8.21/4.98 8.21/4.98 8.21/4.98 8.21/4.98 WORST_CASE(Omega(n^1),?) 8.21/4.98 8.21/4.98 8.21/4.98 ---------------------------------------- 8.21/4.98 8.21/4.98 (4) 8.21/4.98 BOUNDS(n^1, INF) 8.32/5.01 EOF