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 12: f3 -> f4 : arg1'=arg1P_13, arg2'=arg2P_13, [ arg1>=0 && arg2>=0 && arg1==arg1P_13 && arg2==arg2P_13 ], cost: 1 13: f3 -> f5 : arg1'=arg1P_14, arg2'=arg2P_14, [ arg1<0 && arg1==arg1P_14 && arg2==arg2P_14 ], cost: 1 14: f3 -> f5 : arg1'=arg1P_15, arg2'=arg2P_15, [ arg2<0 && arg1==arg1P_15 && arg2==arg2P_15 ], cost: 1 2: f8 -> f11 : arg1'=arg1P_3, arg2'=arg2P_3, [ arg1P_3==1+arg1 && arg2==arg2P_3 ], cost: 1 6: f11 -> f10 : arg1'=arg1P_7, arg2'=arg2P_7, [ arg1==arg1P_7 && arg2==arg2P_7 ], cost: 1 3: f9 -> f12 : arg1'=arg1P_4, arg2'=arg2P_4, [ arg2P_4==1+arg2 && arg1==arg1P_4 ], cost: 1 7: f12 -> f10 : arg1'=arg1P_8, arg2'=arg2P_8, [ arg1==arg1P_8 && arg2==arg2P_8 ], cost: 1 4: f7 -> f8 : arg1'=arg1P_5, arg2'=arg2P_5, [ arg1 f9 : arg1'=arg1P_6, arg2'=arg2P_6, [ arg1>=arg2 && arg1==arg1P_6 && arg2==arg2P_6 ], cost: 1 10: f10 -> f4 : arg1'=arg1P_11, arg2'=arg2P_11, [ arg1==arg1P_11 && arg2==arg2P_11 ], cost: 1 8: f4 -> f7 : arg1'=arg1P_9, arg2'=arg2P_9, [ -arg2+arg1>2 && arg1==arg1P_9 && arg2==arg2P_9 ], cost: 1 9: f4 -> f7 : arg1'=arg1P_10, arg2'=arg2P_10, [ arg2-arg1>2 && arg1==arg1P_10 && arg2==arg2P_10 ], cost: 1 11: f4 -> f13 : arg1'=arg1P_12, arg2'=arg2P_12, [ -arg2+arg1<=2 && arg2-arg1<=2 && arg1==arg1P_12 && arg2==arg2P_12 ], cost: 1 15: f13 -> f6 : arg1'=arg1P_16, arg2'=arg2P_16, [ arg1==arg1P_16 && arg2==arg2P_16 ], cost: 1 16: f5 -> f6 : arg1'=arg1P_17, arg2'=arg2P_17, [ arg1==arg1P_17 && arg2==arg2P_17 ], cost: 1 17: __init -> f1 : arg1'=arg1P_18, arg2'=arg2P_18, [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 17: __init -> f1 : arg1'=arg1P_18, arg2'=arg2P_18, [], 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 12: f3 -> f4 : arg1'=arg1P_13, arg2'=arg2P_13, [ arg1>=0 && arg2>=0 && arg1==arg1P_13 && arg2==arg2P_13 ], cost: 1 2: f8 -> f11 : arg1'=arg1P_3, arg2'=arg2P_3, [ arg1P_3==1+arg1 && arg2==arg2P_3 ], cost: 1 6: f11 -> f10 : arg1'=arg1P_7, arg2'=arg2P_7, [ arg1==arg1P_7 && arg2==arg2P_7 ], cost: 1 3: f9 -> f12 : arg1'=arg1P_4, arg2'=arg2P_4, [ arg2P_4==1+arg2 && arg1==arg1P_4 ], cost: 1 7: f12 -> f10 : arg1'=arg1P_8, arg2'=arg2P_8, [ arg1==arg1P_8 && arg2==arg2P_8 ], cost: 1 4: f7 -> f8 : arg1'=arg1P_5, arg2'=arg2P_5, [ arg1 f9 : arg1'=arg1P_6, arg2'=arg2P_6, [ arg1>=arg2 && arg1==arg1P_6 && arg2==arg2P_6 ], cost: 1 10: f10 -> f4 : arg1'=arg1P_11, arg2'=arg2P_11, [ arg1==arg1P_11 && arg2==arg2P_11 ], cost: 1 8: f4 -> f7 : arg1'=arg1P_9, arg2'=arg2P_9, [ -arg2+arg1>2 && arg1==arg1P_9 && arg2==arg2P_9 ], cost: 1 9: f4 -> f7 : arg1'=arg1P_10, arg2'=arg2P_10, [ arg2-arg1>2 && arg1==arg1P_10 && arg2==arg2P_10 ], cost: 1 17: __init -> f1 : arg1'=arg1P_18, arg2'=arg2P_18, [], 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 12: f3 -> f4 : [ arg1>=0 && arg2>=0 ], cost: 1 2: f8 -> f11 : arg1'=1+arg1, [], cost: 1 6: f11 -> f10 : [], cost: 1 3: f9 -> f12 : arg2'=1+arg2, [], cost: 1 7: f12 -> f10 : [], cost: 1 4: f7 -> f8 : [ arg1 f9 : [ arg1>=arg2 ], cost: 1 10: f10 -> f4 : [], cost: 1 8: f4 -> f7 : [ -arg2+arg1>2 ], cost: 1 9: f4 -> f7 : [ arg2-arg1>2 ], cost: 1 17: __init -> f1 : arg1'=arg1P_18, arg2'=arg2P_18, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: __init 23: f7 -> f10 : arg1'=1+arg1, [ arg1 f10 : arg2'=1+arg2, [ arg1>=arg2 ], cost: 3 10: f10 -> f4 : [], cost: 1 8: f4 -> f7 : [ -arg2+arg1>2 ], cost: 1 9: f4 -> f7 : [ arg2-arg1>2 ], cost: 1 20: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, [ arg1P_1>=0 && arg2P_2>=0 ], cost: 4 Eliminated locations (on tree-shaped paths): Start location: __init 10: f10 -> f4 : [], cost: 1 25: f4 -> f10 : arg2'=1+arg2, [ -arg2+arg1>2 ], cost: 4 26: f4 -> f10 : arg1'=1+arg1, [ arg2-arg1>2 ], cost: 4 20: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, [ arg1P_1>=0 && arg2P_2>=0 ], cost: 4 Eliminated locations (on tree-shaped paths): Start location: __init 27: f4 -> f4 : arg2'=1+arg2, [ -arg2+arg1>2 ], cost: 5 28: f4 -> f4 : arg1'=1+arg1, [ arg2-arg1>2 ], cost: 5 20: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, [ arg1P_1>=0 && arg2P_2>=0 ], cost: 4 Accelerating simple loops of location 9. Accelerating the following rules: 27: f4 -> f4 : arg2'=1+arg2, [ -arg2+arg1>2 ], cost: 5 28: f4 -> f4 : arg1'=1+arg1, [ arg2-arg1>2 ], cost: 5 Accelerated rule 27 with backward acceleration, yielding the new rule 29. Accelerated rule 28 with backward acceleration, yielding the new rule 30. [accelerate] Nesting with 2 inner and 2 outer candidates Removing the simple loops: 27 28. Accelerated all simple loops using metering functions (where possible): Start location: __init 29: f4 -> f4 : arg2'=-2+arg1, [ -2-arg2+arg1>=0 ], cost: -10-5*arg2+5*arg1 30: f4 -> f4 : arg1'=-2+arg2, [ -2+arg2-arg1>=0 ], cost: -10+5*arg2-5*arg1 20: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, [ arg1P_1>=0 && arg2P_2>=0 ], cost: 4 Chained accelerated rules (with incoming rules): Start location: __init 20: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, [ arg1P_1>=0 && arg2P_2>=0 ], cost: 4 31: __init -> f4 : arg1'=arg1P_1, arg2'=-2+arg1P_1, [ arg1P_1>=0 && arg2P_2>=0 && -2-arg2P_2+arg1P_1>=0 ], cost: -6-5*arg2P_2+5*arg1P_1 32: __init -> f4 : arg1'=-2+arg2P_2, arg2'=arg2P_2, [ arg1P_1>=0 && arg2P_2>=0 && -2+arg2P_2-arg1P_1>=0 ], cost: -6+5*arg2P_2-5*arg1P_1 Removed unreachable locations (and leaf rules with constant cost): Start location: __init 31: __init -> f4 : arg1'=arg1P_1, arg2'=-2+arg1P_1, [ arg1P_1>=0 && arg2P_2>=0 && -2-arg2P_2+arg1P_1>=0 ], cost: -6-5*arg2P_2+5*arg1P_1 32: __init -> f4 : arg1'=-2+arg2P_2, arg2'=arg2P_2, [ arg1P_1>=0 && arg2P_2>=0 && -2+arg2P_2-arg1P_1>=0 ], cost: -6+5*arg2P_2-5*arg1P_1 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: __init 31: __init -> f4 : arg1'=arg1P_1, arg2'=-2+arg1P_1, [ arg1P_1>=0 && arg2P_2>=0 && -2-arg2P_2+arg1P_1>=0 ], cost: -6-5*arg2P_2+5*arg1P_1 32: __init -> f4 : arg1'=-2+arg2P_2, arg2'=arg2P_2, [ arg1P_1>=0 && arg2P_2>=0 && -2+arg2P_2-arg1P_1>=0 ], cost: -6+5*arg2P_2-5*arg1P_1 Computing asymptotic complexity for rule 31 Simplified the guard: 31: __init -> f4 : arg1'=arg1P_1, arg2'=-2+arg1P_1, [ arg2P_2>=0 && -2-arg2P_2+arg1P_1>=0 ], cost: -6-5*arg2P_2+5*arg1P_1 Resulting cost 0 has complexity: Unknown Computing asymptotic complexity for rule 32 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),?)