WORST_CASE(INF,?) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: __init 0: f167_0_createList_Return -> f276_0_main_NULL : arg1'=arg1P_1, arg2'=arg2P_1, [ arg2P_1<=arg1 && arg1>0 && arg2P_1>0 && 2+arg2<=arg1 && 0==arg1P_1 ], cost: 1 2: f276_0_main_NULL -> f276_0_main_NULL : arg1'=arg1P_3, arg2'=arg2P_3, [ 1+arg2P_3<=arg2 && arg2>0 && arg2P_3>-1 && 0==arg1 && 1==arg1P_3 ], cost: 1 3: f276_0_main_NULL -> f276_0_main_NULL : arg1'=arg1P_4, arg2'=arg2P_4, [ arg2>=1+arg2P_4 && arg2>0 && arg2P_4>-1 && 2==arg1 && 0==arg1P_4 ], cost: 1 4: f276_0_main_NULL -> f276_0_main_NULL : arg1'=arg1P_5, arg2'=arg2P_5, [ -2+arg2P_5<=arg2 && arg2>0 && arg2P_5>2 && 1==arg1 && 2==arg1P_5 ], cost: 1 1: f1_0_main_Load -> f276_0_main_NULL : arg1'=arg1P_2, arg2'=arg2P_2, [ x6_1>-1 && arg2>0 && arg1>0 && arg2P_2>0 && 0==arg1P_2 ], cost: 1 5: f1_0_main_Load -> f190_0_createList_LE : arg1'=arg1P_6, arg2'=arg2P_6, [ arg1P_6>-1 && arg2>0 && arg1>0 ], cost: 1 6: f190_0_createList_LE -> f190_0_createList_LE : arg1'=arg1P_7, arg2'=arg2P_7, [ arg1>0 && -1+arg1==arg1P_7 ], cost: 1 7: __init -> f1_0_main_Load : arg1'=arg1P_8, arg2'=arg2P_8, [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 7: __init -> f1_0_main_Load : arg1'=arg1P_8, arg2'=arg2P_8, [], cost: 1 Removed unreachable and leaf rules: Start location: __init 2: f276_0_main_NULL -> f276_0_main_NULL : arg1'=arg1P_3, arg2'=arg2P_3, [ 1+arg2P_3<=arg2 && arg2>0 && arg2P_3>-1 && 0==arg1 && 1==arg1P_3 ], cost: 1 3: f276_0_main_NULL -> f276_0_main_NULL : arg1'=arg1P_4, arg2'=arg2P_4, [ arg2>=1+arg2P_4 && arg2>0 && arg2P_4>-1 && 2==arg1 && 0==arg1P_4 ], cost: 1 4: f276_0_main_NULL -> f276_0_main_NULL : arg1'=arg1P_5, arg2'=arg2P_5, [ -2+arg2P_5<=arg2 && arg2>0 && arg2P_5>2 && 1==arg1 && 2==arg1P_5 ], cost: 1 1: f1_0_main_Load -> f276_0_main_NULL : arg1'=arg1P_2, arg2'=arg2P_2, [ x6_1>-1 && arg2>0 && arg1>0 && arg2P_2>0 && 0==arg1P_2 ], cost: 1 5: f1_0_main_Load -> f190_0_createList_LE : arg1'=arg1P_6, arg2'=arg2P_6, [ arg1P_6>-1 && arg2>0 && arg1>0 ], cost: 1 6: f190_0_createList_LE -> f190_0_createList_LE : arg1'=arg1P_7, arg2'=arg2P_7, [ arg1>0 && -1+arg1==arg1P_7 ], cost: 1 7: __init -> f1_0_main_Load : arg1'=arg1P_8, arg2'=arg2P_8, [], cost: 1 Simplified all rules, resulting in: Start location: __init 2: f276_0_main_NULL -> f276_0_main_NULL : arg1'=1, arg2'=arg2P_3, [ 1+arg2P_3<=arg2 && arg2>0 && arg2P_3>-1 && 0==arg1 ], cost: 1 3: f276_0_main_NULL -> f276_0_main_NULL : arg1'=0, arg2'=arg2P_4, [ arg2>=1+arg2P_4 && arg2>0 && arg2P_4>-1 && 2==arg1 ], cost: 1 4: f276_0_main_NULL -> f276_0_main_NULL : arg1'=2, arg2'=arg2P_5, [ -2+arg2P_5<=arg2 && arg2>0 && arg2P_5>2 && 1==arg1 ], cost: 1 1: f1_0_main_Load -> f276_0_main_NULL : arg1'=0, arg2'=arg2P_2, [ arg2>0 && arg1>0 && arg2P_2>0 ], cost: 1 5: f1_0_main_Load -> f190_0_createList_LE : arg1'=arg1P_6, arg2'=arg2P_6, [ arg1P_6>-1 && arg2>0 && arg1>0 ], cost: 1 6: f190_0_createList_LE -> f190_0_createList_LE : arg1'=-1+arg1, arg2'=arg2P_7, [ arg1>0 ], cost: 1 7: __init -> f1_0_main_Load : arg1'=arg1P_8, arg2'=arg2P_8, [], cost: 1 ### Simplification by acceleration and chaining ### Accelerating simple loops of location 1. Accelerating the following rules: 2: f276_0_main_NULL -> f276_0_main_NULL : arg1'=1, arg2'=arg2P_3, [ 1+arg2P_3<=arg2 && arg2>0 && arg2P_3>-1 && 0==arg1 ], cost: 1 3: f276_0_main_NULL -> f276_0_main_NULL : arg1'=0, arg2'=arg2P_4, [ arg2>=1+arg2P_4 && arg2>0 && arg2P_4>-1 && 2==arg1 ], cost: 1 4: f276_0_main_NULL -> f276_0_main_NULL : arg1'=2, arg2'=arg2P_5, [ -2+arg2P_5<=arg2 && arg2>0 && arg2P_5>2 && 1==arg1 ], cost: 1 Accelerated rule 2 with metering function -arg1, yielding the new rule 8. Accelerated rule 3 with metering function meter (where 2*meter==-2+arg1), yielding the new rule 9. Accelerated rule 4 with metering function 2-arg1, yielding the new rule 10. Removing the simple loops: 2 3 4. Accelerating simple loops of location 3. Accelerating the following rules: 6: f190_0_createList_LE -> f190_0_createList_LE : arg1'=-1+arg1, arg2'=arg2P_7, [ arg1>0 ], cost: 1 Accelerated rule 6 with metering function arg1, yielding the new rule 11. Removing the simple loops: 6. Accelerated all simple loops using metering functions (where possible): Start location: __init 8: f276_0_main_NULL -> f276_0_main_NULL : arg1'=1, arg2'=arg2P_3, [ 1+arg2P_3<=arg2 && arg2>0 && arg2P_3>-1 && 0==arg1 && -arg1>=1 ], cost: -arg1 9: f276_0_main_NULL -> f276_0_main_NULL : arg1'=0, arg2'=arg2P_4, [ arg2>=1+arg2P_4 && arg2>0 && arg2P_4>-1 && 2==arg1 && 2*meter==-2+arg1 && meter>=1 ], cost: meter 10: f276_0_main_NULL -> f276_0_main_NULL : arg1'=2, arg2'=arg2P_5, [ -2+arg2P_5<=arg2 && arg2>0 && arg2P_5>2 && 1==arg1 ], cost: 2-arg1 1: f1_0_main_Load -> f276_0_main_NULL : arg1'=0, arg2'=arg2P_2, [ arg2>0 && arg1>0 && arg2P_2>0 ], cost: 1 5: f1_0_main_Load -> f190_0_createList_LE : arg1'=arg1P_6, arg2'=arg2P_6, [ arg1P_6>-1 && arg2>0 && arg1>0 ], cost: 1 11: f190_0_createList_LE -> f190_0_createList_LE : arg1'=0, arg2'=arg2P_7, [ arg1>0 ], cost: arg1 7: __init -> f1_0_main_Load : arg1'=arg1P_8, arg2'=arg2P_8, [], cost: 1 Chained accelerated rules (with incoming rules): Start location: __init 1: f1_0_main_Load -> f276_0_main_NULL : arg1'=0, arg2'=arg2P_2, [ arg2>0 && arg1>0 && arg2P_2>0 ], cost: 1 5: f1_0_main_Load -> f190_0_createList_LE : arg1'=arg1P_6, arg2'=arg2P_6, [ arg1P_6>-1 && arg2>0 && arg1>0 ], cost: 1 12: f1_0_main_Load -> f190_0_createList_LE : arg1'=0, arg2'=arg2P_7, [ arg2>0 && arg1>0 && arg1P_6>0 ], cost: 1+arg1P_6 7: __init -> f1_0_main_Load : arg1'=arg1P_8, arg2'=arg2P_8, [], cost: 1 Removed unreachable locations (and leaf rules with constant cost): Start location: __init 12: f1_0_main_Load -> f190_0_createList_LE : arg1'=0, arg2'=arg2P_7, [ arg2>0 && arg1>0 && arg1P_6>0 ], cost: 1+arg1P_6 7: __init -> f1_0_main_Load : arg1'=arg1P_8, arg2'=arg2P_8, [], cost: 1 Eliminated locations (on linear paths): Start location: __init 13: __init -> f190_0_createList_LE : arg1'=0, arg2'=arg2P_7, [ arg2P_8>0 && arg1P_8>0 && arg1P_6>0 ], cost: 2+arg1P_6 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: __init 13: __init -> f190_0_createList_LE : arg1'=0, arg2'=arg2P_7, [ arg2P_8>0 && arg1P_8>0 && arg1P_6>0 ], cost: 2+arg1P_6 Computing asymptotic complexity for rule 13 Solved the limit problem by the following transformations: Created initial limit problem: arg2P_8 (+/+!), arg1P_6 (+/+!), arg1P_8 (+/+!), 2+arg1P_6 (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {arg2P_8==n,arg1P_6==n,arg1P_8==n} resulting limit problem: [solved] Solution: arg2P_8 / n arg1P_6 / n arg1P_8 / n Resulting cost 2+n has complexity: Unbounded Found new complexity Unbounded. Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Unbounded Cpx degree: Unbounded Solved cost: 2+n Rule cost: 2+arg1P_6 Rule guard: [ arg2P_8>0 && arg1P_8>0 && arg1P_6>0 ] WORST_CASE(INF,?)