WORST_CASE(INF,?) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: __init 0: f1_0_main_Load -> f135_0_f_LE : arg1'=arg1P_1, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1>0 && arg2>-1 && arg2==arg1P_1 && arg2==arg2P_1 ], cost: 1 1: f135_0_f_LE -> f182_0_f_LE : arg1'=arg1P_2, arg2'=arg2P_2, arg3'=arg3P_2, [ arg1>0 && arg1==arg1P_2 && 2==arg2P_2 && arg2==arg3P_2 ], cost: 1 2: f182_0_f_LE -> f135_0_f_LE : arg1'=arg1P_3, arg2'=arg2P_3, arg3'=arg3P_3, [ arg2>0 && arg1>0 && arg1-arg2==arg1P_3 && -1+arg3==arg2P_3 ], cost: 1 3: f182_0_f_LE -> f182_0_f_LE : arg1'=arg1P_4, arg2'=arg2P_4, arg3'=arg3P_4, [ arg2>0 && arg1>0 && arg1==arg1P_4 && -1+arg2==arg2P_4 && -1+arg3==arg3P_4 ], cost: 1 4: f182_0_f_LE -> f182_0_f_LE : arg1'=arg1P_5, arg2'=arg2P_5, arg3'=arg3P_5, [ arg2>0 && arg1>0 && arg1==arg1P_5 && -1+arg2==arg2P_5 ], cost: 1 5: __init -> f1_0_main_Load : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 5: __init -> f1_0_main_Load : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, [], cost: 1 Simplified all rules, resulting in: Start location: __init 0: f1_0_main_Load -> f135_0_f_LE : arg1'=arg2, arg3'=arg3P_1, [ arg1>0 && arg2>-1 ], cost: 1 1: f135_0_f_LE -> f182_0_f_LE : arg2'=2, arg3'=arg2, [ arg1>0 ], cost: 1 2: f182_0_f_LE -> f135_0_f_LE : arg1'=arg1-arg2, arg2'=-1+arg3, arg3'=arg3P_3, [ arg2>0 && arg1>0 ], cost: 1 3: f182_0_f_LE -> f182_0_f_LE : arg2'=-1+arg2, arg3'=-1+arg3, [ arg2>0 && arg1>0 ], cost: 1 4: f182_0_f_LE -> f182_0_f_LE : arg2'=-1+arg2, arg3'=arg3P_5, [ arg2>0 && arg1>0 ], cost: 1 5: __init -> f1_0_main_Load : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, [], cost: 1 ### Simplification by acceleration and chaining ### Accelerating simple loops of location 2. Accelerating the following rules: 3: f182_0_f_LE -> f182_0_f_LE : arg2'=-1+arg2, arg3'=-1+arg3, [ arg2>0 && arg1>0 ], cost: 1 4: f182_0_f_LE -> f182_0_f_LE : arg2'=-1+arg2, arg3'=arg3P_5, [ arg2>0 && arg1>0 ], cost: 1 Accelerated rule 3 with metering function arg2, yielding the new rule 6. Accelerated rule 4 with metering function arg2, yielding the new rule 7. Removing the simple loops: 3 4. Accelerated all simple loops using metering functions (where possible): Start location: __init 0: f1_0_main_Load -> f135_0_f_LE : arg1'=arg2, arg3'=arg3P_1, [ arg1>0 && arg2>-1 ], cost: 1 1: f135_0_f_LE -> f182_0_f_LE : arg2'=2, arg3'=arg2, [ arg1>0 ], cost: 1 2: f182_0_f_LE -> f135_0_f_LE : arg1'=arg1-arg2, arg2'=-1+arg3, arg3'=arg3P_3, [ arg2>0 && arg1>0 ], cost: 1 6: f182_0_f_LE -> f182_0_f_LE : arg2'=0, arg3'=arg3-arg2, [ arg2>0 && arg1>0 ], cost: arg2 7: f182_0_f_LE -> f182_0_f_LE : arg2'=0, arg3'=arg3P_5, [ arg2>0 && arg1>0 ], cost: arg2 5: __init -> f1_0_main_Load : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, [], cost: 1 Chained accelerated rules (with incoming rules): Start location: __init 0: f1_0_main_Load -> f135_0_f_LE : arg1'=arg2, arg3'=arg3P_1, [ arg1>0 && arg2>-1 ], cost: 1 1: f135_0_f_LE -> f182_0_f_LE : arg2'=2, arg3'=arg2, [ arg1>0 ], cost: 1 8: f135_0_f_LE -> f182_0_f_LE : arg2'=0, arg3'=-2+arg2, [ arg1>0 ], cost: 3 9: f135_0_f_LE -> f182_0_f_LE : arg2'=0, arg3'=arg3P_5, [ arg1>0 ], cost: 3 2: f182_0_f_LE -> f135_0_f_LE : arg1'=arg1-arg2, arg2'=-1+arg3, arg3'=arg3P_3, [ arg2>0 && arg1>0 ], cost: 1 5: __init -> f1_0_main_Load : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, [], cost: 1 Eliminated locations (on linear paths): Start location: __init 1: f135_0_f_LE -> f182_0_f_LE : arg2'=2, arg3'=arg2, [ arg1>0 ], cost: 1 8: f135_0_f_LE -> f182_0_f_LE : arg2'=0, arg3'=-2+arg2, [ arg1>0 ], cost: 3 9: f135_0_f_LE -> f182_0_f_LE : arg2'=0, arg3'=arg3P_5, [ arg1>0 ], cost: 3 2: f182_0_f_LE -> f135_0_f_LE : arg1'=arg1-arg2, arg2'=-1+arg3, arg3'=arg3P_3, [ arg2>0 && arg1>0 ], cost: 1 10: __init -> f135_0_f_LE : arg1'=arg2P_6, arg2'=arg2P_6, arg3'=arg3P_1, [ arg1P_6>0 && arg2P_6>-1 ], cost: 2 Eliminated locations (on tree-shaped paths): Start location: __init 11: f135_0_f_LE -> f135_0_f_LE : arg1'=-2+arg1, arg2'=-1+arg2, arg3'=arg3P_3, [ arg1>0 ], cost: 2 10: __init -> f135_0_f_LE : arg1'=arg2P_6, arg2'=arg2P_6, arg3'=arg3P_1, [ arg1P_6>0 && arg2P_6>-1 ], cost: 2 Accelerating simple loops of location 1. Accelerating the following rules: 11: f135_0_f_LE -> f135_0_f_LE : arg1'=-2+arg1, arg2'=-1+arg2, arg3'=arg3P_3, [ arg1>0 ], cost: 2 Accelerated rule 11 with metering function meter (where 2*meter==arg1), yielding the new rule 12. Removing the simple loops: 11. Accelerated all simple loops using metering functions (where possible): Start location: __init 12: f135_0_f_LE -> f135_0_f_LE : arg1'=-2*meter+arg1, arg2'=-meter+arg2, arg3'=arg3P_3, [ arg1>0 && 2*meter==arg1 && meter>=1 ], cost: 2*meter 10: __init -> f135_0_f_LE : arg1'=arg2P_6, arg2'=arg2P_6, arg3'=arg3P_1, [ arg1P_6>0 && arg2P_6>-1 ], cost: 2 Chained accelerated rules (with incoming rules): Start location: __init 10: __init -> f135_0_f_LE : arg1'=arg2P_6, arg2'=arg2P_6, arg3'=arg3P_1, [ arg1P_6>0 && arg2P_6>-1 ], cost: 2 13: __init -> f135_0_f_LE : arg1'=0, arg2'=meter, arg3'=arg3P_3, [ 2*meter>0 && meter>=1 ], cost: 2+2*meter Removed unreachable locations (and leaf rules with constant cost): Start location: __init 13: __init -> f135_0_f_LE : arg1'=0, arg2'=meter, arg3'=arg3P_3, [ 2*meter>0 && meter>=1 ], cost: 2+2*meter ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: __init 13: __init -> f135_0_f_LE : arg1'=0, arg2'=meter, arg3'=arg3P_3, [ 2*meter>0 && meter>=1 ], cost: 2+2*meter Computing asymptotic complexity for rule 13 Solved the limit problem by the following transformations: Created initial limit problem: 2+2*meter (+), 2*meter (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {meter==n} resulting limit problem: [solved] Solution: meter / n Resulting cost 2+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+2*n Rule cost: 2+2*meter Rule guard: [ 2*meter>0 ] WORST_CASE(INF,?)