/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 116 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 1231 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: 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 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 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 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 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 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 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 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 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 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 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 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 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 The start-symbols are:[f0_12] ---------------------------------------- (1) Koat Proof (FINISHED) 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) Initial complexity problem: 1: T: (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)) (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)) (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)) (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 ] (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 ] (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)) (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)) (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)) (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 ] (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)) (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 ] (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)) (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 ] (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 ] start location: koat_start leaf cost: 0 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [ar_0, ar_1, ar_3, ar_5]. We thus obtain the following problem: 2: T: (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 ] (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 ] (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1 + 1, ar_3, ar_5)) (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 ] (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0 + 1, ar_1, ar_3, ar_5)) (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ] (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) (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 ] (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 ] (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5)) (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) (Comp: ?, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5)) start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (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 ] (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 ] (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)) (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 ] (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)) (Comp: ?, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ] (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) (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 ] (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 ] (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 1 Pol(f0) = 1 Pol(f4) = 1 Pol(f3) = 1 Pol(f8) = 0 orients all transitions weakly and the transition f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ] strictly and produces the following problem: 4: T: (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 ] (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 ] (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)) (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 ] (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)) (Comp: 1, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ] (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) (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 ] (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 ] (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f4) = 2*V_1 - 2*V_2 + 1 Pol(f3) = 2*V_1 - 2*V_2 + 2 and size complexities 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 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 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 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 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 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 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 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 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 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 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 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 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 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) = ? 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 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 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) = ? 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) = ? 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 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 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 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 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 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 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 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 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 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 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-0) = ? S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-1) = ? S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-2) = ? S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-3) = ? S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-0) = ? S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-1) = ? S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-2) = ? S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-3) = ? 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 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 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 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 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) = ? 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) = ? 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 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 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 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 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 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 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 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) = ? 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 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 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 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 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 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 orients the transitions 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 ] 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 ] weakly and the transitions 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 ] 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 ] strictly and produces the following problem: 5: T: (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 ] (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 ] (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)) (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 ] (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)) (Comp: 1, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ] (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) (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 ] (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 ] (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f4) = -2*V_1 + 2*V_2 - 1 Pol(f3) = -2*V_1 + 2*V_2 and size complexities 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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) = ? 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 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 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 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 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 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 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 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 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 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 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 S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-0) = ? S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-1) = ? S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-2) = ? S("f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p))", 0-3) = ? S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-0) = ? S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-1) = ? S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-2) = ? S("f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ]", 0-3) = ? 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 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 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 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 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) = ? 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 orients the transitions 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 ] 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 ] weakly and the transitions 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 ] 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 ] strictly and produces the following problem: 6: T: (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 ] (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 ] (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)) (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 ] (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)) (Comp: 1, Cost: 1) f3(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) [ ar_0 = ar_1 ] (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(m, n, o, p)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_5, ar_3, ar_3, ar_5)) (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 ] (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 ] (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f8(ar_0, ar_1, ar_3, ar_5)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f4(ar_0, ar_1, ar_3, ar_5)) (Comp: 1, Cost: 1) f0(ar_0, ar_1, ar_3, ar_5) -> Com_1(f3(ar_0, ar_1, ar_3, ar_5)) start location: koat_start leaf cost: 0 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 Time: 0.166 sec (SMT: 0.133 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: f0 0: f0 -> f3 : [], cost: 1 1: f0 -> f4 : [], cost: 1 2: f0 -> f8 : [], cost: 1 5: f0 -> f3 : A'=F, B'=D, C'=D, E'=F, G'=A, H'=B, [], cost: 1 6: f0 -> f3 : A'=F, B'=D, C'=D, E'=F, G'=free, H'=free_1, Q'=free, J'=free_1, [], cost: 1 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 9: f0 -> f3 : A'=1+A, C'=D, E'=F, G'=A, H'=B, [], cost: 1 11: f0 -> f3 : B'=1+B, C'=D, E'=F, G'=A, H'=B, [], cost: 1 3: f3 -> f4 : C'=D, E'=F, G'=A, H'=B, [ A>=1+B ], cost: 1 4: f3 -> f4 : C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 1 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 10: f4 -> f3 : A'=1+A, C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 1 12: f4 -> f3 : B'=1+B, C'=D, E'=F, G'=A, H'=B, [ A>=B ], cost: 1 Removed unreachable and leaf rules: Start location: f0 0: f0 -> f3 : [], cost: 1 1: f0 -> f4 : [], cost: 1 5: f0 -> f3 : A'=F, B'=D, C'=D, E'=F, G'=A, H'=B, [], cost: 1 6: f0 -> f3 : A'=F, B'=D, C'=D, E'=F, G'=free, H'=free_1, Q'=free, J'=free_1, [], cost: 1 9: f0 -> f3 : A'=1+A, C'=D, E'=F, G'=A, H'=B, [], cost: 1 11: f0 -> f3 : B'=1+B, C'=D, E'=F, G'=A, H'=B, [], cost: 1 3: f3 -> f4 : C'=D, E'=F, G'=A, H'=B, [ A>=1+B ], cost: 1 4: f3 -> f4 : C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 1 10: f4 -> f3 : A'=1+A, C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 1 12: f4 -> f3 : B'=1+B, C'=D, E'=F, G'=A, H'=B, [ A>=B ], cost: 1 ### Simplification by acceleration and chaining ### Eliminated location f3 (as a last resort): Start location: f0 1: f0 -> f4 : [], cost: 1 13: f0 -> f4 : C'=D, E'=F, G'=A, H'=B, [ A>=1+B ], cost: 2 14: f0 -> f4 : C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 2 15: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, [ F>=1+D ], cost: 2 16: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, [ D>=1+F ], cost: 2 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 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 19: f0 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ 1+A>=1+B ], cost: 2 20: f0 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ B>=2+A ], cost: 2 22: f0 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ A>=2+B ], cost: 2 23: f0 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ 1+B>=1+A ], cost: 2 21: f4 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ B>=2+A ], cost: 2 24: f4 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ A>=2+B ], cost: 2 25: f4 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ A>=B && 1+B>=1+A ], cost: 2 Accelerating simple loops of location 2. Simplified some of the simple loops (and removed duplicate rules). Accelerating the following rules: 21: f4 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ B>=2+A ], cost: 2 24: f4 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ A>=2+B ], cost: 2 25: f4 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ -A+B==0 ], cost: 2 Accelerated rule 21 with metering function -1-A+B, yielding the new rule 26. Accelerated rule 24 with metering function -1+A-B, yielding the new rule 27. Accelerated rule 25 with metering function 1+A-B, yielding the new rule 28. Removing the simple loops: 21 24 25. Accelerated all simple loops using metering functions (where possible): Start location: f0 1: f0 -> f4 : [], cost: 1 13: f0 -> f4 : C'=D, E'=F, G'=A, H'=B, [ A>=1+B ], cost: 2 14: f0 -> f4 : C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 2 15: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, [ F>=1+D ], cost: 2 16: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, [ D>=1+F ], cost: 2 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 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 19: f0 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ 1+A>=1+B ], cost: 2 20: f0 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ B>=2+A ], cost: 2 22: f0 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ A>=2+B ], cost: 2 23: f0 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ 1+B>=1+A ], cost: 2 26: f4 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=2+A ], cost: -2-2*A+2*B 27: f4 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=2+B ], cost: -2+2*A-2*B 28: f4 -> f4 : B'=1+A, C'=D, E'=F, G'=A, H'=1+A, [ -A+B==0 ], cost: 2+2*A-2*B Chained accelerated rules (with incoming rules): Start location: f0 1: f0 -> f4 : [], cost: 1 13: f0 -> f4 : C'=D, E'=F, G'=A, H'=B, [ A>=1+B ], cost: 2 14: f0 -> f4 : C'=D, E'=F, G'=A, H'=B, [ B>=1+A ], cost: 2 15: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, [ F>=1+D ], cost: 2 16: f0 -> f4 : A'=F, B'=D, C'=D, E'=F, G'=F, H'=D, [ D>=1+F ], cost: 2 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 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 19: f0 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ 1+A>=1+B ], cost: 2 20: f0 -> f4 : A'=1+A, C'=D, E'=F, G'=1+A, H'=B, [ B>=2+A ], cost: 2 22: f0 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ A>=2+B ], cost: 2 23: f0 -> f4 : B'=1+B, C'=D, E'=F, G'=A, H'=1+B, [ 1+B>=1+A ], cost: 2 29: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=2+A ], cost: -1-2*A+2*B 30: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=2+A ], cost: -2*A+2*B 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 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 33: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=3+A ], cost: -2-2*A+2*B 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 35: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=2+B ], cost: -1+2*A-2*B 36: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=2+B ], cost: 2*A-2*B 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 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 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 40: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=3+B ], cost: -2+2*A-2*B 41: f0 -> f4 : B'=1+A, C'=D, E'=F, G'=A, H'=1+A, [ -A+B==0 ], cost: 3+2*A-2*B Removed unreachable locations (and leaf rules with constant cost): Start location: f0 29: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=2+A ], cost: -1-2*A+2*B 30: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=2+A ], cost: -2*A+2*B 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 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 33: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=3+A ], cost: -2-2*A+2*B 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 35: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=2+B ], cost: -1+2*A-2*B 36: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=2+B ], cost: 2*A-2*B 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 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 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 40: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=3+B ], cost: -2+2*A-2*B 41: f0 -> f4 : B'=1+A, C'=D, E'=F, G'=A, H'=1+A, [ -A+B==0 ], cost: 3+2*A-2*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: f0 30: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=2+A ], cost: -2*A+2*B 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 33: f0 -> f4 : A'=-1+B, C'=D, E'=F, G'=-1+B, H'=B, [ B>=3+A ], cost: -2-2*A+2*B 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 36: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=2+B ], cost: 2*A-2*B 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 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 40: f0 -> f4 : B'=-1+A, C'=D, E'=F, G'=A, H'=-1+A, [ A>=3+B ], cost: -2+2*A-2*B 41: f0 -> f4 : B'=1+A, C'=D, E'=F, G'=A, H'=1+A, [ -A+B==0 ], cost: 3+2*A-2*B Computing asymptotic complexity for rule 30 Solved the limit problem by the following transformations: Created initial limit problem: -2*A+2*B (+), -1-A+B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {A==0,B==n} resulting limit problem: [solved] Solution: A / 0 B / n Resulting cost 2*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^1) Cpx degree: 1 Solved cost: 2*n Rule cost: -2*A+2*B Rule guard: [ B>=2+A ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)