WORST_CASE(Omega(1),?) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: __init 0: f1_0_main_Load -> f213_0_increase_LE : arg1'=arg1P_1, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1P_1>-1 && arg2>1 && arg2P_1>-1 && arg1>0 && arg1P_1+arg2P_1==arg3P_1 ], cost: 1 1: f213_0_increase_LE -> f213_0_increase_LE\' : arg1'=arg1P_2, arg2'=arg2P_2, arg3'=arg3P_2, [ arg1>-2 && arg2-2*x13_1<0 && arg3>0 && arg1==arg1P_2 && arg2==arg2P_2 && arg3==arg3P_2 ], cost: 1 2: f213_0_increase_LE -> f213_0_increase_LE\' : arg1'=arg1P_3, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1>-2 && -2*x17_1+arg2>0 && arg3>0 && arg1==arg1P_3 && arg2==arg2P_3 && arg3==arg3P_3 ], cost: 1 4: f213_0_increase_LE -> f213_0_increase_LE\' : arg1'=arg1P_5, arg2'=arg2P_5, arg3'=arg3P_5, [ arg1>-2 && -2*x25_1+arg2==0 && arg3>0 && arg1==arg1P_5 && arg2==arg2P_5 && arg3==arg3P_5 ], cost: 1 3: f213_0_increase_LE\' -> f213_0_increase_LE : arg1'=arg1P_4, arg2'=arg2P_4, arg3'=arg3P_4, [ -2*x21_1+arg2>0 && arg1>-2 && -2*x21_1+arg2<2 && arg3>0 && 1+arg1==arg1P_4 && arg2==arg2P_4 && 1+arg1+arg2==arg3P_4 ], cost: 1 5: f213_0_increase_LE\' -> f213_0_increase_LE : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, [ arg1>-2 && arg3>0 && -2*x29_1+arg2==0 && -2*x29_1+arg2<2 && -2*x29_1+arg2>=0 && 1+arg1==arg1P_6 && -2+arg2==arg2P_6 && -1+arg1+arg2==arg3P_6 ], cost: 1 6: __init -> f1_0_main_Load : arg1'=arg1P_7, arg2'=arg2P_7, arg3'=arg3P_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, arg3'=arg3P_7, [], cost: 1 Simplified all rules, resulting in: Start location: __init 0: f1_0_main_Load -> f213_0_increase_LE : arg1'=arg1P_1, arg2'=arg2P_1, arg3'=arg1P_1+arg2P_1, [ arg1P_1>-1 && arg2>1 && arg2P_1>-1 && arg1>0 ], cost: 1 1: f213_0_increase_LE -> f213_0_increase_LE\' : [ arg1>-2 && arg2-2*x13_1<0 && arg3>0 ], cost: 1 2: f213_0_increase_LE -> f213_0_increase_LE\' : [ arg1>-2 && -2*x17_1+arg2>0 && arg3>0 ], cost: 1 4: f213_0_increase_LE -> f213_0_increase_LE\' : [ arg1>-2 && -2*x25_1+arg2==0 && arg3>0 ], cost: 1 3: f213_0_increase_LE\' -> f213_0_increase_LE : arg1'=1+arg1, arg3'=1+arg1+arg2, [ 1+2*x21_1-arg2==0 && arg1>-2 && arg3>0 ], cost: 1 5: f213_0_increase_LE\' -> f213_0_increase_LE : arg1'=1+arg1, arg2'=-2+arg2, arg3'=-1+arg1+arg2, [ arg1>-2 && arg3>0 && -2*x29_1+arg2==0 ], cost: 1 6: __init -> f1_0_main_Load : arg1'=arg1P_7, arg2'=arg2P_7, arg3'=arg3P_7, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: __init 1: f213_0_increase_LE -> f213_0_increase_LE\' : [ arg1>-2 && arg2-2*x13_1<0 && arg3>0 ], cost: 1 2: f213_0_increase_LE -> f213_0_increase_LE\' : [ arg1>-2 && -2*x17_1+arg2>0 && arg3>0 ], cost: 1 4: f213_0_increase_LE -> f213_0_increase_LE\' : [ arg1>-2 && -2*x25_1+arg2==0 && arg3>0 ], cost: 1 3: f213_0_increase_LE\' -> f213_0_increase_LE : arg1'=1+arg1, arg3'=1+arg1+arg2, [ 1+2*x21_1-arg2==0 && arg1>-2 && arg3>0 ], cost: 1 5: f213_0_increase_LE\' -> f213_0_increase_LE : arg1'=1+arg1, arg2'=-2+arg2, arg3'=-1+arg1+arg2, [ arg1>-2 && arg3>0 && -2*x29_1+arg2==0 ], cost: 1 7: __init -> f213_0_increase_LE : arg1'=arg1P_1, arg2'=arg2P_1, arg3'=arg1P_1+arg2P_1, [ arg1P_1>-1 && arg2P_7>1 && arg2P_1>-1 && arg1P_7>0 ], cost: 2 Eliminated locations (on tree-shaped paths): Start location: __init 8: f213_0_increase_LE -> f213_0_increase_LE : arg1'=1+arg1, arg3'=1+arg1+arg2, [ arg1>-2 && arg2-2*x13_1<0 && arg3>0 && 1+2*x21_1-arg2==0 ], cost: 2 9: f213_0_increase_LE -> f213_0_increase_LE : arg1'=1+arg1, arg2'=-2+arg2, arg3'=-1+arg1+arg2, [ arg1>-2 && arg2-2*x13_1<0 && arg3>0 && -2*x29_1+arg2==0 ], cost: 2 10: f213_0_increase_LE -> f213_0_increase_LE : arg1'=1+arg1, arg3'=1+arg1+arg2, [ arg1>-2 && -2*x17_1+arg2>0 && arg3>0 && 1+2*x21_1-arg2==0 ], cost: 2 11: f213_0_increase_LE -> f213_0_increase_LE : arg1'=1+arg1, arg2'=-2+arg2, arg3'=-1+arg1+arg2, [ arg1>-2 && -2*x17_1+arg2>0 && arg3>0 && -2*x29_1+arg2==0 ], cost: 2 12: f213_0_increase_LE -> f213_0_increase_LE : arg1'=1+arg1, arg2'=-2+arg2, arg3'=-1+arg1+arg2, [ arg1>-2 && -2*x25_1+arg2==0 && arg3>0 && -2*x29_1+arg2==0 ], cost: 2 7: __init -> f213_0_increase_LE : arg1'=arg1P_1, arg2'=arg2P_1, arg3'=arg1P_1+arg2P_1, [ arg1P_1>-1 && arg2P_7>1 && arg2P_1>-1 && arg1P_7>0 ], cost: 2 Accelerating simple loops of location 1. Accelerating the following rules: 8: f213_0_increase_LE -> f213_0_increase_LE : arg1'=1+arg1, arg3'=1+arg1+arg2, [ arg1>-2 && arg2-2*x13_1<0 && arg3>0 && 1+2*x21_1-arg2==0 ], cost: 2 9: f213_0_increase_LE -> f213_0_increase_LE : arg1'=1+arg1, arg2'=-2+arg2, arg3'=-1+arg1+arg2, [ arg1>-2 && arg2-2*x13_1<0 && arg3>0 && -2*x29_1+arg2==0 ], cost: 2 10: f213_0_increase_LE -> f213_0_increase_LE : arg1'=1+arg1, arg3'=1+arg1+arg2, [ arg1>-2 && -2*x17_1+arg2>0 && arg3>0 && 1+2*x21_1-arg2==0 ], cost: 2 11: f213_0_increase_LE -> f213_0_increase_LE : arg1'=1+arg1, arg2'=-2+arg2, arg3'=-1+arg1+arg2, [ arg1>-2 && -2*x17_1+arg2>0 && arg3>0 && -2*x29_1+arg2==0 ], cost: 2 12: f213_0_increase_LE -> f213_0_increase_LE : arg1'=1+arg1, arg2'=-2+arg2, arg3'=-1+arg1+arg2, [ arg1>-2 && -2*x25_1+arg2==0 && arg3>0 && -2*x29_1+arg2==0 ], cost: 2 Found no metering function for rule 8. Accelerated rule 9 with metering function meter (where 2*meter==-2*x29_1+arg2), yielding the new rule 13. Found no metering function for rule 10. Accelerated rule 11 with metering function meter_1 (where 2*meter_1==-2*x29_1+arg2), yielding the new rule 14. Accelerated rule 12 with metering function meter_2 (where 2*meter_2==-x25_1-x29_1+arg2), yielding the new rule 15. Removing the simple loops: 9 11 12. Accelerated all simple loops using metering functions (where possible): Start location: __init 8: f213_0_increase_LE -> f213_0_increase_LE : arg1'=1+arg1, arg3'=1+arg1+arg2, [ arg1>-2 && arg2-2*x13_1<0 && arg3>0 && 1+2*x21_1-arg2==0 ], cost: 2 10: f213_0_increase_LE -> f213_0_increase_LE : arg1'=1+arg1, arg3'=1+arg1+arg2, [ arg1>-2 && -2*x17_1+arg2>0 && arg3>0 && 1+2*x21_1-arg2==0 ], cost: 2 13: f213_0_increase_LE -> f213_0_increase_LE : arg1'=meter+arg1, arg2'=-2*meter+arg2, arg3'=-meter+arg1+arg2, [ arg1>-2 && arg2-2*x13_1<0 && arg3>0 && -2*x29_1+arg2==0 && 2*meter==-2*x29_1+arg2 && meter>=1 ], cost: 2*meter 14: f213_0_increase_LE -> f213_0_increase_LE : arg1'=meter_1+arg1, arg2'=-2*meter_1+arg2, arg3'=-meter_1+arg1+arg2, [ arg1>-2 && -2*x17_1+arg2>0 && arg3>0 && -2*x29_1+arg2==0 && 2*meter_1==-2*x29_1+arg2 && meter_1>=1 ], cost: 2*meter_1 15: f213_0_increase_LE -> f213_0_increase_LE : arg1'=meter_2+arg1, arg2'=-2*meter_2+arg2, arg3'=-meter_2+arg1+arg2, [ arg1>-2 && -2*x25_1+arg2==0 && arg3>0 && -2*x29_1+arg2==0 && 2*meter_2==-x25_1-x29_1+arg2 && meter_2>=1 ], cost: 2*meter_2 7: __init -> f213_0_increase_LE : arg1'=arg1P_1, arg2'=arg2P_1, arg3'=arg1P_1+arg2P_1, [ arg1P_1>-1 && arg2P_7>1 && arg2P_1>-1 && arg1P_7>0 ], cost: 2 Chained accelerated rules (with incoming rules): Start location: __init 7: __init -> f213_0_increase_LE : arg1'=arg1P_1, arg2'=arg2P_1, arg3'=arg1P_1+arg2P_1, [ arg1P_1>-1 && arg2P_7>1 && arg2P_1>-1 && arg1P_7>0 ], cost: 2 16: __init -> f213_0_increase_LE : arg1'=1+arg1P_1, arg2'=1+2*x21_1, arg3'=2+2*x21_1+arg1P_1, [ arg1P_1>-1 && 1+2*x21_1>-1 && 1+2*x21_1-2*x13_1<0 && 1+2*x21_1+arg1P_1>0 ], cost: 4 17: __init -> f213_0_increase_LE : arg1'=1+arg1P_1, arg2'=1+2*x21_1, arg3'=2+2*x21_1+arg1P_1, [ arg1P_1>-1 && 1+2*x21_1>-1 && 1+2*x21_1-2*x17_1>0 && 1+2*x21_1+arg1P_1>0 ], cost: 4 Removed unreachable locations (and leaf rules with constant cost): Start location: __init ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: __init 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),?)