WORST_CASE(Omega(1),?) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: __init 0: f161_0_createList_Return -> f227_0_isCyclic_NONNULL : arg1'=arg1P_1, arg2'=arg2P_1, [ arg1P_1<=arg1 && arg1>-1 && arg1P_1>-1 ], cost: 1 2: f227_0_isCyclic_NONNULL -> f331_0_isCyclic_NULL : arg1'=arg1P_3, arg2'=arg2P_3, [ arg1P_3<=arg1 && 1+arg2P_3<=arg1 && arg1>0 && arg1P_3>0 && arg2P_3>-1 ], cost: 1 1: f1_0_main_Load -> f227_0_isCyclic_NONNULL : arg1'=arg1P_2, arg2'=arg2P_2, [ arg1>0 && arg1P_2>-1 ], cost: 1 4: f1_0_main_Load -> f201_0_createList_LE : arg1'=arg1P_5, arg2'=arg2P_5, [ arg2>-1 && arg1P_5>-1 && arg1>0 ], cost: 1 3: f331_0_isCyclic_NULL -> f331_0_isCyclic_NULL : arg1'=arg1P_4, arg2'=arg2P_4, [ arg1>=1+arg1P_4 && arg2>=3+arg2P_4 && arg1>0 && arg2>2 && arg1P_4>-1 && arg2P_4>-1 ], cost: 1 5: f201_0_createList_LE -> f201_0_createList_LE : arg1'=arg1P_6, arg2'=arg2P_6, [ arg1>0 && -1+arg1==arg1P_6 ], cost: 1 6: __init -> f1_0_main_Load : arg1'=arg1P_7, arg2'=arg2P_7, [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 6: __init -> f1_0_main_Load : arg1'=arg1P_7, arg2'=arg2P_7, [], cost: 1 Removed unreachable and leaf rules: Start location: __init 2: f227_0_isCyclic_NONNULL -> f331_0_isCyclic_NULL : arg1'=arg1P_3, arg2'=arg2P_3, [ arg1P_3<=arg1 && 1+arg2P_3<=arg1 && arg1>0 && arg1P_3>0 && arg2P_3>-1 ], cost: 1 1: f1_0_main_Load -> f227_0_isCyclic_NONNULL : arg1'=arg1P_2, arg2'=arg2P_2, [ arg1>0 && arg1P_2>-1 ], cost: 1 4: f1_0_main_Load -> f201_0_createList_LE : arg1'=arg1P_5, arg2'=arg2P_5, [ arg2>-1 && arg1P_5>-1 && arg1>0 ], cost: 1 3: f331_0_isCyclic_NULL -> f331_0_isCyclic_NULL : arg1'=arg1P_4, arg2'=arg2P_4, [ arg1>=1+arg1P_4 && arg2>=3+arg2P_4 && arg1>0 && arg2>2 && arg1P_4>-1 && arg2P_4>-1 ], cost: 1 5: f201_0_createList_LE -> f201_0_createList_LE : arg1'=arg1P_6, arg2'=arg2P_6, [ arg1>0 && -1+arg1==arg1P_6 ], cost: 1 6: __init -> f1_0_main_Load : arg1'=arg1P_7, arg2'=arg2P_7, [], cost: 1 Simplified all rules, resulting in: Start location: __init 2: f227_0_isCyclic_NONNULL -> f331_0_isCyclic_NULL : arg1'=arg1P_3, arg2'=arg2P_3, [ arg1P_3<=arg1 && 1+arg2P_3<=arg1 && arg1>0 && arg1P_3>0 && arg2P_3>-1 ], cost: 1 1: f1_0_main_Load -> f227_0_isCyclic_NONNULL : arg1'=arg1P_2, arg2'=arg2P_2, [ arg1>0 && arg1P_2>-1 ], cost: 1 4: f1_0_main_Load -> f201_0_createList_LE : arg1'=arg1P_5, arg2'=arg2P_5, [ arg2>-1 && arg1P_5>-1 && arg1>0 ], cost: 1 3: f331_0_isCyclic_NULL -> f331_0_isCyclic_NULL : arg1'=arg1P_4, arg2'=arg2P_4, [ arg1>=1+arg1P_4 && arg2>=3+arg2P_4 && arg1>0 && arg2>2 && arg1P_4>-1 && arg2P_4>-1 ], cost: 1 5: f201_0_createList_LE -> f201_0_createList_LE : arg1'=-1+arg1, arg2'=arg2P_6, [ arg1>0 ], cost: 1 6: __init -> f1_0_main_Load : arg1'=arg1P_7, arg2'=arg2P_7, [], cost: 1 ### Simplification by acceleration and chaining ### Accelerating simple loops of location 3. Accelerating the following rules: 3: f331_0_isCyclic_NULL -> f331_0_isCyclic_NULL : arg1'=arg1P_4, arg2'=arg2P_4, [ arg1>=1+arg1P_4 && arg2>=3+arg2P_4 && arg1>0 && arg2>2 && arg1P_4>-1 && arg2P_4>-1 ], cost: 1 Failed to prove monotonicity of the guard of rule 3. [accelerate] Nesting with 1 inner and 1 outer candidates Accelerating simple loops of location 4. Accelerating the following rules: 5: f201_0_createList_LE -> f201_0_createList_LE : arg1'=-1+arg1, arg2'=arg2P_6, [ arg1>0 ], cost: 1 Accelerated rule 5 with backward acceleration, yielding the new rule 7. [accelerate] Nesting with 1 inner and 1 outer candidates Removing the simple loops: 5. Accelerated all simple loops using metering functions (where possible): Start location: __init 2: f227_0_isCyclic_NONNULL -> f331_0_isCyclic_NULL : arg1'=arg1P_3, arg2'=arg2P_3, [ arg1P_3<=arg1 && 1+arg2P_3<=arg1 && arg1>0 && arg1P_3>0 && arg2P_3>-1 ], cost: 1 1: f1_0_main_Load -> f227_0_isCyclic_NONNULL : arg1'=arg1P_2, arg2'=arg2P_2, [ arg1>0 && arg1P_2>-1 ], cost: 1 4: f1_0_main_Load -> f201_0_createList_LE : arg1'=arg1P_5, arg2'=arg2P_5, [ arg2>-1 && arg1P_5>-1 && arg1>0 ], cost: 1 3: f331_0_isCyclic_NULL -> f331_0_isCyclic_NULL : arg1'=arg1P_4, arg2'=arg2P_4, [ arg1>=1+arg1P_4 && arg2>=3+arg2P_4 && arg1>0 && arg2>2 && arg1P_4>-1 && arg2P_4>-1 ], cost: 1 7: f201_0_createList_LE -> f201_0_createList_LE : arg1'=0, arg2'=arg2P_6, [ arg1>=1 ], cost: arg1 6: __init -> f1_0_main_Load : arg1'=arg1P_7, arg2'=arg2P_7, [], cost: 1 Chained accelerated rules (with incoming rules): Start location: __init 2: f227_0_isCyclic_NONNULL -> f331_0_isCyclic_NULL : arg1'=arg1P_3, arg2'=arg2P_3, [ arg1P_3<=arg1 && 1+arg2P_3<=arg1 && arg1>0 && arg1P_3>0 && arg2P_3>-1 ], cost: 1 8: f227_0_isCyclic_NONNULL -> f331_0_isCyclic_NULL : arg1'=arg1P_4, arg2'=arg2P_4, [ arg1P_4>-1 && arg2P_4>-1 && 1+arg1P_4<=arg1 && 3+arg2P_4<=-1+arg1 && 3<=-1+arg1 ], cost: 2 1: f1_0_main_Load -> f227_0_isCyclic_NONNULL : arg1'=arg1P_2, arg2'=arg2P_2, [ arg1>0 && arg1P_2>-1 ], cost: 1 4: f1_0_main_Load -> f201_0_createList_LE : arg1'=arg1P_5, arg2'=arg2P_5, [ arg2>-1 && arg1P_5>-1 && arg1>0 ], cost: 1 9: f1_0_main_Load -> f201_0_createList_LE : arg1'=0, arg2'=arg2P_6, [ arg2>-1 && arg1>0 && arg1P_5>=1 ], cost: 1+arg1P_5 6: __init -> f1_0_main_Load : arg1'=arg1P_7, arg2'=arg2P_7, [], cost: 1 Removed unreachable locations (and leaf rules with constant cost): Start location: __init 9: f1_0_main_Load -> f201_0_createList_LE : arg1'=0, arg2'=arg2P_6, [ arg2>-1 && arg1>0 && arg1P_5>=1 ], cost: 1+arg1P_5 6: __init -> f1_0_main_Load : arg1'=arg1P_7, arg2'=arg2P_7, [], cost: 1 Eliminated locations (on linear paths): Start location: __init 10: __init -> f201_0_createList_LE : arg1'=0, arg2'=arg2P_6, [ arg2P_7>-1 && arg1P_7>0 && arg1P_5>=1 ], cost: 2+arg1P_5 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: __init 10: __init -> f201_0_createList_LE : arg1'=0, arg2'=arg2P_6, [ arg2P_7>-1 && arg1P_7>0 && arg1P_5>=1 ], cost: 2+arg1P_5 Computing asymptotic complexity for rule 10 Resulting cost 0 has complexity: Unknown Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Constant Cpx degree: 0 Solved cost: 1 Rule cost: 1 Rule guard: [] WORST_CASE(Omega(1),?)