WORST_CASE(Omega(1),?) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: __init 0: f1 -> f2 : arg1'=arg1P_1, arg2'=arg2P_1, [ arg2==arg2P_1 ], cost: 1 1: f2 -> f3 : arg1'=arg1P_2, arg2'=arg2P_2, [ arg1==arg1P_2 ], cost: 1 8: f3 -> f4 : arg1'=arg1P_9, arg2'=arg2P_9, [ arg1>=0 && arg1==arg1P_9 && arg2==arg2P_9 ], cost: 1 10: f3 -> f10 : arg1'=arg1P_11, arg2'=arg2P_11, [ arg1<0 && arg1==arg1P_11 && arg2==arg2P_11 ], cost: 1 2: f4 -> f5 : arg1'=arg1P_3, arg2'=arg2P_3, [ arg1P_3==arg2+arg1 && arg2==arg2P_3 ], cost: 1 4: f5 -> f6 : arg1'=arg1P_5, arg2'=arg2P_5, [ arg2>=0 && arg1==arg1P_5 && arg2==arg2P_5 ], cost: 1 5: f5 -> f7 : arg1'=arg1P_6, arg2'=arg2P_6, [ arg2<0 && arg1==arg1P_6 && arg2==arg2P_6 ], cost: 1 3: f6 -> f9 : arg1'=arg1P_4, arg2'=arg2P_4, [ arg2P_4==-1+arg2 && arg1==arg1P_4 ], cost: 1 6: f9 -> f8 : arg1'=arg1P_7, arg2'=arg2P_7, [ arg1==arg1P_7 && arg2==arg2P_7 ], cost: 1 7: f7 -> f8 : arg1'=arg1P_8, arg2'=arg2P_8, [ arg1==arg1P_8 && arg2==arg2P_8 ], cost: 1 9: f8 -> f3 : arg1'=arg1P_10, arg2'=arg2P_10, [ arg1==arg1P_10 && arg2==arg2P_10 ], cost: 1 11: __init -> f1 : arg1'=arg1P_12, arg2'=arg2P_12, [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 11: __init -> f1 : arg1'=arg1P_12, arg2'=arg2P_12, [], cost: 1 Removed unreachable and leaf rules: Start location: __init 0: f1 -> f2 : arg1'=arg1P_1, arg2'=arg2P_1, [ arg2==arg2P_1 ], cost: 1 1: f2 -> f3 : arg1'=arg1P_2, arg2'=arg2P_2, [ arg1==arg1P_2 ], cost: 1 8: f3 -> f4 : arg1'=arg1P_9, arg2'=arg2P_9, [ arg1>=0 && arg1==arg1P_9 && arg2==arg2P_9 ], cost: 1 2: f4 -> f5 : arg1'=arg1P_3, arg2'=arg2P_3, [ arg1P_3==arg2+arg1 && arg2==arg2P_3 ], cost: 1 4: f5 -> f6 : arg1'=arg1P_5, arg2'=arg2P_5, [ arg2>=0 && arg1==arg1P_5 && arg2==arg2P_5 ], cost: 1 5: f5 -> f7 : arg1'=arg1P_6, arg2'=arg2P_6, [ arg2<0 && arg1==arg1P_6 && arg2==arg2P_6 ], cost: 1 3: f6 -> f9 : arg1'=arg1P_4, arg2'=arg2P_4, [ arg2P_4==-1+arg2 && arg1==arg1P_4 ], cost: 1 6: f9 -> f8 : arg1'=arg1P_7, arg2'=arg2P_7, [ arg1==arg1P_7 && arg2==arg2P_7 ], cost: 1 7: f7 -> f8 : arg1'=arg1P_8, arg2'=arg2P_8, [ arg1==arg1P_8 && arg2==arg2P_8 ], cost: 1 9: f8 -> f3 : arg1'=arg1P_10, arg2'=arg2P_10, [ arg1==arg1P_10 && arg2==arg2P_10 ], cost: 1 11: __init -> f1 : arg1'=arg1P_12, arg2'=arg2P_12, [], cost: 1 Simplified all rules, resulting in: Start location: __init 0: f1 -> f2 : arg1'=arg1P_1, [], cost: 1 1: f2 -> f3 : arg2'=arg2P_2, [], cost: 1 8: f3 -> f4 : [ arg1>=0 ], cost: 1 2: f4 -> f5 : arg1'=arg2+arg1, [], cost: 1 4: f5 -> f6 : [ arg2>=0 ], cost: 1 5: f5 -> f7 : [ arg2<0 ], cost: 1 3: f6 -> f9 : arg2'=-1+arg2, [], cost: 1 6: f9 -> f8 : [], cost: 1 7: f7 -> f8 : [], cost: 1 9: f8 -> f3 : [], cost: 1 11: __init -> f1 : arg1'=arg1P_12, arg2'=arg2P_12, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: __init 14: f3 -> f5 : arg1'=arg2+arg1, [ arg1>=0 ], cost: 2 16: f5 -> f8 : [ arg2<0 ], cost: 2 17: f5 -> f8 : arg2'=-1+arg2, [ arg2>=0 ], cost: 3 9: f8 -> f3 : [], cost: 1 13: __init -> f3 : arg1'=arg1P_1, arg2'=arg2P_2, [], cost: 3 Eliminated locations (on tree-shaped paths): Start location: __init 18: f3 -> f8 : arg1'=arg2+arg1, [ arg1>=0 && arg2<0 ], cost: 4 19: f3 -> f8 : arg1'=arg2+arg1, arg2'=-1+arg2, [ arg1>=0 && arg2>=0 ], cost: 5 9: f8 -> f3 : [], cost: 1 13: __init -> f3 : arg1'=arg1P_1, arg2'=arg2P_2, [], cost: 3 Eliminated locations (on tree-shaped paths): Start location: __init 20: f3 -> f3 : arg1'=arg2+arg1, [ arg1>=0 && arg2<0 ], cost: 5 21: f3 -> f3 : arg1'=arg2+arg1, arg2'=-1+arg2, [ arg1>=0 && arg2>=0 ], cost: 6 13: __init -> f3 : arg1'=arg1P_1, arg2'=arg2P_2, [], cost: 3 Accelerating simple loops of location 2. Accelerating the following rules: 20: f3 -> f3 : arg1'=arg2+arg1, [ arg1>=0 && arg2<0 ], cost: 5 21: f3 -> f3 : arg1'=arg2+arg1, arg2'=-1+arg2, [ arg1>=0 && arg2>=0 ], cost: 6 Accelerated rule 20 with backward acceleration, yielding the new rule 22. Accelerated rule 21 with backward acceleration, yielding the new rule 23. [accelerate] Nesting with 2 inner and 2 outer candidates Removing the simple loops: 20 21. Accelerated all simple loops using metering functions (where possible): Start location: __init 22: f3 -> f3 : arg1'=arg2*k+arg1, [ arg2<0 && k>=0 && arg1+arg2*(-1+k)>=0 ], cost: 5*k 23: f3 -> f3 : arg1'=1/2+1/2*arg2+arg2*(1+arg2)-1/2*(1+arg2)^2+arg1, arg2'=-1, [ arg1>=0 && 1+arg2>=0 ], cost: 6+6*arg2 13: __init -> f3 : arg1'=arg1P_1, arg2'=arg2P_2, [], cost: 3 Chained accelerated rules (with incoming rules): Start location: __init 13: __init -> f3 : arg1'=arg1P_1, arg2'=arg2P_2, [], cost: 3 24: __init -> f3 : arg1'=arg1P_1+arg2P_2*k, arg2'=arg2P_2, [ arg2P_2<0 && k>=0 && arg1P_1+arg2P_2*(-1+k)>=0 ], cost: 3+5*k 25: __init -> f3 : arg1'=1/2+1/2*arg2P_2+arg1P_1-1/2*(1+arg2P_2)^2+arg2P_2*(1+arg2P_2), arg2'=-1, [ arg1P_1>=0 && 1+arg2P_2>=0 ], cost: 9+6*arg2P_2 Removed unreachable locations (and leaf rules with constant cost): Start location: __init 24: __init -> f3 : arg1'=arg1P_1+arg2P_2*k, arg2'=arg2P_2, [ arg2P_2<0 && k>=0 && arg1P_1+arg2P_2*(-1+k)>=0 ], cost: 3+5*k 25: __init -> f3 : arg1'=1/2+1/2*arg2P_2+arg1P_1-1/2*(1+arg2P_2)^2+arg2P_2*(1+arg2P_2), arg2'=-1, [ arg1P_1>=0 && 1+arg2P_2>=0 ], cost: 9+6*arg2P_2 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: __init 24: __init -> f3 : arg1'=arg1P_1+arg2P_2*k, arg2'=arg2P_2, [ arg2P_2<0 && k>=0 && arg1P_1+arg2P_2*(-1+k)>=0 ], cost: 3+5*k 25: __init -> f3 : arg1'=1/2+1/2*arg2P_2+arg1P_1-1/2*(1+arg2P_2)^2+arg2P_2*(1+arg2P_2), arg2'=-1, [ arg1P_1>=0 && 1+arg2P_2>=0 ], cost: 9+6*arg2P_2 Computing asymptotic complexity for rule 25 Resulting cost 0 has complexity: Unknown Computing asymptotic complexity for rule 24 Resulting cost 0 has complexity: Unknown 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),?)