WORST_CASE(INF,?) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: __init 0: f1_0_main_ConstantStackPush -> f99_0_loop_aux_LE : arg1'=arg1P_1, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1>0 && arg2>-1 && 0==arg1P_1 && 0==arg2P_1 && 10*arg2==arg3P_1 ], cost: 1 1: f99_0_loop_aux_LE -> f128_0_loop_aux_EQ : arg1'=arg1P_2, arg2'=arg2P_2, arg3'=arg3P_2, [ arg3<10 && arg3>1 && arg1==arg1P_2 && arg3==arg2P_2 && arg2==arg3P_2 ], cost: 1 2: f99_0_loop_aux_LE -> f128_0_loop_aux_EQ : arg1'=arg1P_3, arg2'=arg2P_3, arg3'=arg3P_3, [ arg3>10 && arg1==arg1P_3 && arg3==arg2P_3 && arg2==arg3P_3 ], cost: 1 4: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=arg1P_5, arg2'=arg2P_5, arg3'=arg3P_5, [ arg1<2 && 1==arg3 && 1==arg1P_5 && 1==arg2P_5 && 2==arg3P_5 ], cost: 1 5: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, [ arg1<2 && 10==arg3 && 0==arg1P_6 && 0==arg2P_6 && 9==arg3P_6 ], cost: 1 3: f128_0_loop_aux_EQ -> f99_0_loop_aux_LE : arg1'=arg1P_4, arg2'=arg2P_4, arg3'=arg3P_4, [ arg1<2 && arg2>1 && 0==arg3 && 0==arg1P_4 && 0==arg2P_4 && -1+arg2==arg3P_4 ], cost: 1 6: f128_0_loop_aux_EQ -> f99_0_loop_aux_LE : arg1'=arg1P_7, arg2'=arg2P_7, arg3'=arg3P_7, [ arg1<2 && arg2>1 && 1==arg3 && 1==arg1P_7 && 1==arg2P_7 && 1+arg2==arg3P_7 ], cost: 1 7: __init -> f1_0_main_ConstantStackPush : 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_ConstantStackPush : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, [], cost: 1 Simplified all rules, resulting in: Start location: __init 0: f1_0_main_ConstantStackPush -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=10*arg2, [ arg1>0 && arg2>-1 ], cost: 1 1: f99_0_loop_aux_LE -> f128_0_loop_aux_EQ : arg2'=arg3, arg3'=arg2, [ arg3<10 && arg3>1 ], cost: 1 2: f99_0_loop_aux_LE -> f128_0_loop_aux_EQ : arg2'=arg3, arg3'=arg2, [ arg3>10 ], cost: 1 4: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=2, [ arg1<2 && 1==arg3 ], cost: 1 5: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1<2 && 10==arg3 ], cost: 1 3: f128_0_loop_aux_EQ -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=-1+arg2, [ arg1<2 && arg2>1 && 0==arg3 ], cost: 1 6: f128_0_loop_aux_EQ -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=1+arg2, [ arg1<2 && arg2>1 && 1==arg3 ], cost: 1 7: __init -> f1_0_main_ConstantStackPush : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, [], cost: 1 ### Simplification by acceleration and chaining ### Accelerating simple loops of location 1. Accelerating the following rules: 4: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=2, [ arg1<2 && 1==arg3 ], cost: 1 5: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1<2 && 10==arg3 ], cost: 1 Accelerated rule 4 with metering function 2-arg3, yielding the new rule 8. Accelerated rule 5 with metering function -9+arg3, yielding the new rule 9. Removing the simple loops: 4 5. Accelerated all simple loops using metering functions (where possible): Start location: __init 0: f1_0_main_ConstantStackPush -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=10*arg2, [ arg1>0 && arg2>-1 ], cost: 1 1: f99_0_loop_aux_LE -> f128_0_loop_aux_EQ : arg2'=arg3, arg3'=arg2, [ arg3<10 && arg3>1 ], cost: 1 2: f99_0_loop_aux_LE -> f128_0_loop_aux_EQ : arg2'=arg3, arg3'=arg2, [ arg3>10 ], cost: 1 8: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=2, [ arg1<2 && 1==arg3 ], cost: 2-arg3 9: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1<2 && 10==arg3 ], cost: -9+arg3 3: f128_0_loop_aux_EQ -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=-1+arg2, [ arg1<2 && arg2>1 && 0==arg3 ], cost: 1 6: f128_0_loop_aux_EQ -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=1+arg2, [ arg1<2 && arg2>1 && 1==arg3 ], cost: 1 7: __init -> f1_0_main_ConstantStackPush : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, [], cost: 1 Chained accelerated rules (with incoming rules): Start location: __init 0: f1_0_main_ConstantStackPush -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=10*arg2, [ arg1>0 && arg2>-1 ], cost: 1 11: f1_0_main_ConstantStackPush -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1>0 && arg2>-1 && 10==10*arg2 ], cost: -8+10*arg2 1: f99_0_loop_aux_LE -> f128_0_loop_aux_EQ : arg2'=arg3, arg3'=arg2, [ arg3<10 && arg3>1 ], cost: 1 2: f99_0_loop_aux_LE -> f128_0_loop_aux_EQ : arg2'=arg3, arg3'=arg2, [ arg3>10 ], cost: 1 3: f128_0_loop_aux_EQ -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=-1+arg2, [ arg1<2 && arg2>1 && 0==arg3 ], cost: 1 6: f128_0_loop_aux_EQ -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=1+arg2, [ arg1<2 && arg2>1 && 1==arg3 ], cost: 1 10: f128_0_loop_aux_EQ -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=2, [ arg1<2 && 0==arg3 && 1==-1+arg2 ], cost: 4-arg2 12: f128_0_loop_aux_EQ -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1<2 && 0==arg3 && 10==-1+arg2 ], cost: -9+arg2 13: f128_0_loop_aux_EQ -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1<2 && 1==arg3 && 10==1+arg2 ], cost: -7+arg2 7: __init -> f1_0_main_ConstantStackPush : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, [], cost: 1 Eliminated locations (on tree-shaped paths): Start location: __init 16: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=-1+arg3, [ arg3<10 && arg3>1 && arg1<2 && 0==arg2 ], cost: 2 17: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=1+arg3, [ arg3<10 && arg3>1 && arg1<2 && 1==arg2 ], cost: 2 18: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=2, [ arg1<2 && 0==arg2 && 1==-1+arg3 ], cost: 5-arg3 19: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1<2 && 1==arg2 && 10==1+arg3 ], cost: -6+arg3 20: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=-1+arg3, [ arg3>10 && arg1<2 && 0==arg2 ], cost: 2 21: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=1+arg3, [ arg3>10 && arg1<2 && 1==arg2 ], cost: 2 22: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1<2 && 0==arg2 && 10==-1+arg3 ], cost: -8+arg3 14: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=10*arg2P_8, [ arg1P_8>0 && arg2P_8>-1 ], cost: 2 15: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1P_8>0 && arg2P_8>-1 && 10==10*arg2P_8 ], cost: -7+10*arg2P_8 Applied pruning (of leafs and parallel rules): Start location: __init 16: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=-1+arg3, [ arg3<10 && arg3>1 && arg1<2 && 0==arg2 ], cost: 2 17: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=1+arg3, [ arg3<10 && arg3>1 && arg1<2 && 1==arg2 ], cost: 2 18: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=2, [ arg1<2 && 0==arg2 && 1==-1+arg3 ], cost: 5-arg3 19: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1<2 && 1==arg2 && 10==1+arg3 ], cost: -6+arg3 20: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=-1+arg3, [ arg3>10 && arg1<2 && 0==arg2 ], cost: 2 14: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=10*arg2P_8, [ arg1P_8>0 && arg2P_8>-1 ], cost: 2 15: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1P_8>0 && arg2P_8>-1 && 10==10*arg2P_8 ], cost: -7+10*arg2P_8 Accelerating simple loops of location 1. Accelerating the following rules: 16: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=-1+arg3, [ arg3<10 && arg3>1 && arg1<2 && 0==arg2 ], cost: 2 17: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=1+arg3, [ arg3<10 && arg3>1 && arg1<2 && 1==arg2 ], cost: 2 18: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=2, [ arg1<2 && 0==arg2 && 1==-1+arg3 ], cost: 5-arg3 19: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1<2 && 1==arg2 && 10==1+arg3 ], cost: -6+arg3 20: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=-1+arg3, [ arg3>10 && arg1<2 && 0==arg2 ], cost: 2 Accelerated rule 16 with metering function -1+arg3, yielding the new rule 23. Accelerated rule 17 with metering function 10-arg3, yielding the new rule 24. Accelerated rule 18 with metering function 1-arg2, yielding the new rule 25. Accelerated rule 19 with metering function arg2, yielding the new rule 26. Accelerated rule 20 with metering function -10+arg3, yielding the new rule 27. Removing the simple loops: 16 17 18 19 20. Accelerated all simple loops using metering functions (where possible): Start location: __init 23: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=1, [ arg3<10 && arg3>1 && arg1<2 && 0==arg2 ], cost: -2+2*arg3 24: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=10, [ arg3<10 && arg3>1 && arg1<2 && 1==arg2 ], cost: 20-2*arg3 25: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=1, arg2'=1, arg3'=2, [ arg1<2 && 0==arg2 && 1==-1+arg3 ], cost: 3-3*arg2 26: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1<2 && 1==arg2 && 10==1+arg3 ], cost: 3*arg2 27: f99_0_loop_aux_LE -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=10, [ arg3>10 && arg1<2 && 0==arg2 ], cost: -20+2*arg3 14: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=10*arg2P_8, [ arg1P_8>0 && arg2P_8>-1 ], cost: 2 15: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1P_8>0 && arg2P_8>-1 && 10==10*arg2P_8 ], cost: -7+10*arg2P_8 Chained accelerated rules (with incoming rules): Start location: __init 14: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=10*arg2P_8, [ arg1P_8>0 && arg2P_8>-1 ], cost: 2 15: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1P_8>0 && arg2P_8>-1 && 10==10*arg2P_8 ], cost: -7+10*arg2P_8 28: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=1, [], cost: 19 29: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=10, [ arg2P_8>-1 && 10*arg2P_8>10 ], cost: -18+20*arg2P_8 Removed unreachable locations (and leaf rules with constant cost): Start location: __init 15: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1P_8>0 && arg2P_8>-1 && 10==10*arg2P_8 ], cost: -7+10*arg2P_8 29: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=10, [ arg2P_8>-1 && 10*arg2P_8>10 ], cost: -18+20*arg2P_8 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: __init 15: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1P_8>0 && arg2P_8>-1 && 10==10*arg2P_8 ], cost: -7+10*arg2P_8 29: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=10, [ arg2P_8>-1 && 10*arg2P_8>10 ], cost: -18+20*arg2P_8 Computing asymptotic complexity for rule 15 Simplified the guard: 15: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=9, [ arg1P_8>0 && 10==10*arg2P_8 ], cost: -7+10*arg2P_8 Could not solve the limit problem. Resulting cost 0 has complexity: Unknown Computing asymptotic complexity for rule 29 Simplified the guard: 29: __init -> f99_0_loop_aux_LE : arg1'=0, arg2'=0, arg3'=10, [ 10*arg2P_8>10 ], cost: -18+20*arg2P_8 Solved the limit problem by the following transformations: Created initial limit problem: -18+20*arg2P_8 (+), -10+10*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 -18+20*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: -18+20*n Rule cost: -18+20*arg2P_8 Rule guard: [ 10*arg2P_8>10 ] WORST_CASE(INF,?)