WORST_CASE(INF,?) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: __init 0: f1_0_main_Load -> f33_0_rec_Cmp : arg1'=arg1P_1, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1>0 && arg2>-1 && arg2==arg1P_1 ], cost: 1 1: f33_0_rec_Cmp -> f48_0_rec_GE : arg1'=arg1P_2, arg2'=arg2P_2, arg3'=arg3P_2, [ arg1==arg1P_2 && arg1==arg2P_2 && arg1==arg3P_2 ], cost: 1 2: f48_0_rec_GE -> f33_0_rec_Cmp : arg1'=arg1P_3, arg2'=arg2P_3, arg3'=arg3P_3, [ -1+arg10 && arg2>99 && arg2==arg3 && -1+arg1==arg1P_3 ], cost: 1 3: f48_0_rec_GE -> f77_0_descend_LE : arg1'=arg1P_4, arg2'=arg2P_4, arg3'=arg3P_4, [ arg2>0 && arg2<100 && arg2==arg3 && arg2==arg1P_4 ], cost: 1 4: f48_0_rec_GE -> f77_0_descend_LE : arg1'=arg1P_5, arg2'=arg2P_5, arg3'=arg3P_5, [ arg2<100 && arg2>0 && arg2==arg3 && arg2==arg1P_5 ], cost: 1 5: f48_0_rec_GE -> f48_0_rec_GE : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, [ arg2<100 && arg2>0 && arg2==arg3 && arg1==arg1P_6 && 1+arg2==arg2P_6 && 1+arg2==arg3P_6 ], cost: 1 6: f77_0_descend_LE -> f77_0_descend_LE : arg1'=arg1P_7, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1>0 && -1+arg1 f1_0_main_Load : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_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, arg3'=arg3P_8, [], cost: 1 Simplified all rules, resulting in: Start location: __init 0: f1_0_main_Load -> f33_0_rec_Cmp : arg1'=arg2, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1>0 && arg2>-1 ], cost: 1 1: f33_0_rec_Cmp -> f48_0_rec_GE : arg2'=arg1, arg3'=arg1, [], cost: 1 2: f48_0_rec_GE -> f33_0_rec_Cmp : arg1'=-1+arg1, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1>0 && arg2>99 && arg2==arg3 ], cost: 1 3: f48_0_rec_GE -> f77_0_descend_LE : arg1'=arg2, arg2'=arg2P_4, arg3'=arg3P_4, [ arg2>0 && arg2<100 && arg2==arg3 ], cost: 1 4: f48_0_rec_GE -> f77_0_descend_LE : arg1'=arg2, arg2'=arg2P_5, arg3'=arg3P_5, [ arg2<100 && arg2>0 && arg2==arg3 ], cost: 1 5: f48_0_rec_GE -> f48_0_rec_GE : arg2'=1+arg2, arg3'=1+arg2, [ arg2<100 && arg2>0 && arg2==arg3 ], cost: 1 6: f77_0_descend_LE -> f77_0_descend_LE : arg1'=-1+arg1, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1>0 ], cost: 1 7: __init -> f1_0_main_Load : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, [], cost: 1 ### Simplification by acceleration and chaining ### Accelerating simple loops of location 2. Accelerating the following rules: 5: f48_0_rec_GE -> f48_0_rec_GE : arg2'=1+arg2, arg3'=1+arg2, [ arg2<100 && arg2>0 && arg2==arg3 ], cost: 1 Accelerated rule 5 with metering function 100-arg2, yielding the new rule 8. Removing the simple loops: 5. Accelerating simple loops of location 3. Accelerating the following rules: 6: f77_0_descend_LE -> f77_0_descend_LE : arg1'=-1+arg1, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1>0 ], cost: 1 Accelerated rule 6 with metering function arg1, yielding the new rule 9. Removing the simple loops: 6. Accelerated all simple loops using metering functions (where possible): Start location: __init 0: f1_0_main_Load -> f33_0_rec_Cmp : arg1'=arg2, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1>0 && arg2>-1 ], cost: 1 1: f33_0_rec_Cmp -> f48_0_rec_GE : arg2'=arg1, arg3'=arg1, [], cost: 1 2: f48_0_rec_GE -> f33_0_rec_Cmp : arg1'=-1+arg1, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1>0 && arg2>99 && arg2==arg3 ], cost: 1 3: f48_0_rec_GE -> f77_0_descend_LE : arg1'=arg2, arg2'=arg2P_4, arg3'=arg3P_4, [ arg2>0 && arg2<100 && arg2==arg3 ], cost: 1 4: f48_0_rec_GE -> f77_0_descend_LE : arg1'=arg2, arg2'=arg2P_5, arg3'=arg3P_5, [ arg2<100 && arg2>0 && arg2==arg3 ], cost: 1 8: f48_0_rec_GE -> f48_0_rec_GE : arg2'=100, arg3'=100, [ arg2<100 && arg2>0 && arg2==arg3 ], cost: 100-arg2 9: f77_0_descend_LE -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1>0 ], cost: arg1 7: __init -> f1_0_main_Load : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, [], cost: 1 Chained accelerated rules (with incoming rules): Start location: __init 0: f1_0_main_Load -> f33_0_rec_Cmp : arg1'=arg2, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1>0 && arg2>-1 ], cost: 1 1: f33_0_rec_Cmp -> f48_0_rec_GE : arg2'=arg1, arg3'=arg1, [], cost: 1 10: f33_0_rec_Cmp -> f48_0_rec_GE : arg2'=100, arg3'=100, [ arg1<100 && arg1>0 ], cost: 101-arg1 2: f48_0_rec_GE -> f33_0_rec_Cmp : arg1'=-1+arg1, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1>0 && arg2>99 && arg2==arg3 ], cost: 1 3: f48_0_rec_GE -> f77_0_descend_LE : arg1'=arg2, arg2'=arg2P_4, arg3'=arg3P_4, [ arg2>0 && arg2<100 && arg2==arg3 ], cost: 1 4: f48_0_rec_GE -> f77_0_descend_LE : arg1'=arg2, arg2'=arg2P_5, arg3'=arg3P_5, [ arg2<100 && arg2>0 && arg2==arg3 ], cost: 1 11: f48_0_rec_GE -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg2>0 && arg2<100 && arg2==arg3 ], cost: 1+arg2 12: f48_0_rec_GE -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg2<100 && arg2>0 && arg2==arg3 ], cost: 1+arg2 7: __init -> f1_0_main_Load : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, [], cost: 1 Removed unreachable locations (and leaf rules with constant cost): Start location: __init 0: f1_0_main_Load -> f33_0_rec_Cmp : arg1'=arg2, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1>0 && arg2>-1 ], cost: 1 1: f33_0_rec_Cmp -> f48_0_rec_GE : arg2'=arg1, arg3'=arg1, [], cost: 1 10: f33_0_rec_Cmp -> f48_0_rec_GE : arg2'=100, arg3'=100, [ arg1<100 && arg1>0 ], cost: 101-arg1 2: f48_0_rec_GE -> f33_0_rec_Cmp : arg1'=-1+arg1, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1>0 && arg2>99 && arg2==arg3 ], cost: 1 11: f48_0_rec_GE -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg2>0 && arg2<100 && arg2==arg3 ], cost: 1+arg2 12: f48_0_rec_GE -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg2<100 && arg2>0 && arg2==arg3 ], cost: 1+arg2 7: __init -> f1_0_main_Load : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, [], cost: 1 Eliminated locations (on linear paths): Start location: __init 1: f33_0_rec_Cmp -> f48_0_rec_GE : arg2'=arg1, arg3'=arg1, [], cost: 1 10: f33_0_rec_Cmp -> f48_0_rec_GE : arg2'=100, arg3'=100, [ arg1<100 && arg1>0 ], cost: 101-arg1 2: f48_0_rec_GE -> f33_0_rec_Cmp : arg1'=-1+arg1, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1>0 && arg2>99 && arg2==arg3 ], cost: 1 11: f48_0_rec_GE -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg2>0 && arg2<100 && arg2==arg3 ], cost: 1+arg2 12: f48_0_rec_GE -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg2<100 && arg2>0 && arg2==arg3 ], cost: 1+arg2 13: __init -> f33_0_rec_Cmp : arg1'=arg2P_8, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1P_8>0 && arg2P_8>-1 ], cost: 2 Eliminated locations (on tree-shaped paths): Start location: __init 14: f33_0_rec_Cmp -> f33_0_rec_Cmp : arg1'=-1+arg1, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1>99 ], cost: 2 15: f33_0_rec_Cmp -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1>0 && arg1<100 ], cost: 2+arg1 16: f33_0_rec_Cmp -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1<100 && arg1>0 ], cost: 2+arg1 17: f33_0_rec_Cmp -> f33_0_rec_Cmp : arg1'=-1+arg1, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1<100 && arg1>0 ], cost: 102-arg1 18: f33_0_rec_Cmp -> [7] : [ arg1<100 && arg1>0 ], cost: 101-arg1 13: __init -> f33_0_rec_Cmp : arg1'=arg2P_8, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1P_8>0 && arg2P_8>-1 ], cost: 2 Accelerating simple loops of location 1. Accelerating the following rules: 14: f33_0_rec_Cmp -> f33_0_rec_Cmp : arg1'=-1+arg1, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1>99 ], cost: 2 17: f33_0_rec_Cmp -> f33_0_rec_Cmp : arg1'=-1+arg1, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1<100 && arg1>0 ], cost: 102-arg1 Accelerated rule 14 with metering function -99+arg1, yielding the new rule 19. Accelerated rule 17 with metering function arg1, yielding the new rule 20. Removing the simple loops: 14 17. Accelerated all simple loops using metering functions (where possible): Start location: __init 15: f33_0_rec_Cmp -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1>0 && arg1<100 ], cost: 2+arg1 16: f33_0_rec_Cmp -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1<100 && arg1>0 ], cost: 2+arg1 18: f33_0_rec_Cmp -> [7] : [ arg1<100 && arg1>0 ], cost: 101-arg1 19: f33_0_rec_Cmp -> f33_0_rec_Cmp : arg1'=99, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1>99 ], cost: -198+2*arg1 20: f33_0_rec_Cmp -> f33_0_rec_Cmp : arg1'=0, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1<100 && arg1>0 ], cost: 203/2*arg1-1/2*arg1^2 13: __init -> f33_0_rec_Cmp : arg1'=arg2P_8, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1P_8>0 && arg2P_8>-1 ], cost: 2 Chained accelerated rules (with incoming rules): Start location: __init 15: f33_0_rec_Cmp -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1>0 && arg1<100 ], cost: 2+arg1 16: f33_0_rec_Cmp -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1<100 && arg1>0 ], cost: 2+arg1 18: f33_0_rec_Cmp -> [7] : [ arg1<100 && arg1>0 ], cost: 101-arg1 13: __init -> f33_0_rec_Cmp : arg1'=arg2P_8, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1P_8>0 && arg2P_8>-1 ], cost: 2 21: __init -> f33_0_rec_Cmp : arg1'=99, arg2'=arg2P_3, arg3'=arg3P_3, [ arg2P_8>99 ], cost: -196+2*arg2P_8 22: __init -> f33_0_rec_Cmp : arg1'=0, arg2'=arg2P_3, arg3'=arg3P_3, [ arg2P_8<100 && arg2P_8>0 ], cost: 2-1/2*arg2P_8^2+203/2*arg2P_8 Eliminated locations (on tree-shaped paths): Start location: __init 23: __init -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1P_8>0 && arg2P_8>0 && arg2P_8<100 ], cost: 4+arg2P_8 24: __init -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1P_8>0 && arg2P_8<100 && arg2P_8>0 ], cost: 4+arg2P_8 25: __init -> [7] : arg1'=arg2P_8, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1P_8>0 && arg2P_8<100 && arg2P_8>0 ], cost: 103-arg2P_8 26: __init -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg2P_8>99 ], cost: -95+2*arg2P_8 27: __init -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg2P_8>99 ], cost: -95+2*arg2P_8 28: __init -> [7] : arg1'=99, arg2'=arg2P_3, arg3'=arg3P_3, [ arg2P_8>99 ], cost: -194+2*arg2P_8 29: __init -> [9] : [ arg2P_8<100 && arg2P_8>0 ], cost: 2-1/2*arg2P_8^2+203/2*arg2P_8 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: __init 23: __init -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1P_8>0 && arg2P_8>0 && arg2P_8<100 ], cost: 4+arg2P_8 24: __init -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1P_8>0 && arg2P_8<100 && arg2P_8>0 ], cost: 4+arg2P_8 25: __init -> [7] : arg1'=arg2P_8, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1P_8>0 && arg2P_8<100 && arg2P_8>0 ], cost: 103-arg2P_8 27: __init -> f77_0_descend_LE : arg1'=0, arg2'=arg2P_7, arg3'=arg3P_7, [ arg2P_8>99 ], cost: -95+2*arg2P_8 29: __init -> [9] : [ arg2P_8<100 && arg2P_8>0 ], cost: 2-1/2*arg2P_8^2+203/2*arg2P_8 Computing asymptotic complexity for rule 23 Could not solve the limit problem. Resulting cost 0 has complexity: Unknown Computing asymptotic complexity for rule 24 Could not solve the limit problem. Resulting cost 0 has complexity: Unknown Computing asymptotic complexity for rule 25 Could not solve the limit problem. Resulting cost 0 has complexity: Unknown Computing asymptotic complexity for rule 27 Solved the limit problem by the following transformations: Created initial limit problem: -99+arg2P_8 (+/+!), -95+2*arg2P_8 (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {arg2P_8==n} resulting limit problem: [solved] Solution: arg2P_8 / n Resulting cost -95+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: -95+2*n Rule cost: -95+2*arg2P_8 Rule guard: [ arg2P_8>99 ] WORST_CASE(INF,?)