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, arg3'=arg3P_1, [ arg1==arg1P_1 && arg3==arg3P_1 ], cost: 1 1: f2 -> f3 : arg1'=arg1P_2, arg2'=arg2P_2, arg3'=arg3P_2, [ arg1==arg1P_2 && arg2==arg2P_2 ], cost: 1 2: f3 -> f4 : arg1'=arg1P_3, arg2'=arg2P_3, arg3'=arg3P_3, [ 0==arg1P_3 && arg2==arg2P_3 && arg3==arg3P_3 ], cost: 1 10: f4 -> f5 : arg1'=arg1P_11, arg2'=arg2P_11, arg3'=arg3P_11, [ arg2>0 && arg1==arg1P_11 && arg2==arg2P_11 && arg3==arg3P_11 ], cost: 1 12: f4 -> f12 : arg1'=arg1P_13, arg2'=arg2P_13, arg3'=arg3P_13, [ arg2<=0 && arg1==arg1P_13 && arg2==arg2P_13 && arg3==arg3P_13 ], cost: 1 3: f5 -> f6 : arg1'=arg1P_4, arg2'=arg2P_4, arg3'=arg3P_4, [ arg1==arg1P_4 && arg2==arg2P_4 && 0==arg3P_4 ], cost: 1 6: f6 -> f7 : arg1'=arg1P_7, arg2'=arg2P_7, arg3'=arg3P_7, [ arg3 f10 : arg1'=arg1P_9, arg2'=arg2P_9, arg3'=arg3P_9, [ arg3>=arg2 && arg1==arg1P_9 && arg2==arg2P_9 && arg3==arg3P_9 ], cost: 1 4: f7 -> f8 : arg1'=arg1P_5, arg2'=arg2P_5, arg3'=arg3P_5, [ arg3P_5==1+arg3 && arg1==arg1P_5 && arg2==arg2P_5 ], cost: 1 5: f8 -> f9 : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, [ arg1P_6==1+arg1 && arg2==arg2P_6 && arg3==arg3P_6 ], cost: 1 7: f9 -> f6 : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, [ arg1==arg1P_8 && arg2==arg2P_8 && arg3==arg3P_8 ], cost: 1 9: f10 -> f11 : arg1'=arg1P_10, arg2'=arg2P_10, arg3'=arg3P_10, [ arg2P_10==-1+arg2 && arg1==arg1P_10 && arg3==arg3P_10 ], cost: 1 11: f11 -> f4 : arg1'=arg1P_12, arg2'=arg2P_12, arg3'=arg3P_12, [ arg1==arg1P_12 && arg2==arg2P_12 && arg3==arg3P_12 ], cost: 1 13: __init -> f1 : arg1'=arg1P_14, arg2'=arg2P_14, arg3'=arg3P_14, [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 13: __init -> f1 : arg1'=arg1P_14, arg2'=arg2P_14, arg3'=arg3P_14, [], cost: 1 Removed unreachable and leaf rules: Start location: __init 0: f1 -> f2 : arg1'=arg1P_1, arg2'=arg2P_1, arg3'=arg3P_1, [ arg1==arg1P_1 && arg3==arg3P_1 ], cost: 1 1: f2 -> f3 : arg1'=arg1P_2, arg2'=arg2P_2, arg3'=arg3P_2, [ arg1==arg1P_2 && arg2==arg2P_2 ], cost: 1 2: f3 -> f4 : arg1'=arg1P_3, arg2'=arg2P_3, arg3'=arg3P_3, [ 0==arg1P_3 && arg2==arg2P_3 && arg3==arg3P_3 ], cost: 1 10: f4 -> f5 : arg1'=arg1P_11, arg2'=arg2P_11, arg3'=arg3P_11, [ arg2>0 && arg1==arg1P_11 && arg2==arg2P_11 && arg3==arg3P_11 ], cost: 1 3: f5 -> f6 : arg1'=arg1P_4, arg2'=arg2P_4, arg3'=arg3P_4, [ arg1==arg1P_4 && arg2==arg2P_4 && 0==arg3P_4 ], cost: 1 6: f6 -> f7 : arg1'=arg1P_7, arg2'=arg2P_7, arg3'=arg3P_7, [ arg3 f10 : arg1'=arg1P_9, arg2'=arg2P_9, arg3'=arg3P_9, [ arg3>=arg2 && arg1==arg1P_9 && arg2==arg2P_9 && arg3==arg3P_9 ], cost: 1 4: f7 -> f8 : arg1'=arg1P_5, arg2'=arg2P_5, arg3'=arg3P_5, [ arg3P_5==1+arg3 && arg1==arg1P_5 && arg2==arg2P_5 ], cost: 1 5: f8 -> f9 : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, [ arg1P_6==1+arg1 && arg2==arg2P_6 && arg3==arg3P_6 ], cost: 1 7: f9 -> f6 : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, [ arg1==arg1P_8 && arg2==arg2P_8 && arg3==arg3P_8 ], cost: 1 9: f10 -> f11 : arg1'=arg1P_10, arg2'=arg2P_10, arg3'=arg3P_10, [ arg2P_10==-1+arg2 && arg1==arg1P_10 && arg3==arg3P_10 ], cost: 1 11: f11 -> f4 : arg1'=arg1P_12, arg2'=arg2P_12, arg3'=arg3P_12, [ arg1==arg1P_12 && arg2==arg2P_12 && arg3==arg3P_12 ], cost: 1 13: __init -> f1 : arg1'=arg1P_14, arg2'=arg2P_14, arg3'=arg3P_14, [], cost: 1 Simplified all rules, resulting in: Start location: __init 0: f1 -> f2 : arg2'=arg2P_1, [], cost: 1 1: f2 -> f3 : arg3'=arg3P_2, [], cost: 1 2: f3 -> f4 : arg1'=0, [], cost: 1 10: f4 -> f5 : [ arg2>0 ], cost: 1 3: f5 -> f6 : arg3'=0, [], cost: 1 6: f6 -> f7 : [ arg3 f10 : [ arg3>=arg2 ], cost: 1 4: f7 -> f8 : arg3'=1+arg3, [], cost: 1 5: f8 -> f9 : arg1'=1+arg1, [], cost: 1 7: f9 -> f6 : [], cost: 1 9: f10 -> f11 : arg2'=-1+arg2, [], cost: 1 11: f11 -> f4 : [], cost: 1 13: __init -> f1 : arg1'=arg1P_14, arg2'=arg2P_14, arg3'=arg3P_14, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: __init 17: f4 -> f6 : arg3'=0, [ arg2>0 ], cost: 2 21: f6 -> f4 : arg2'=-1+arg2, [ arg3>=arg2 ], cost: 3 22: f6 -> f6 : arg1'=1+arg1, arg3'=1+arg3, [ arg3 f4 : arg1'=0, arg2'=arg2P_1, arg3'=arg3P_2, [], cost: 4 Accelerating simple loops of location 5. Accelerating the following rules: 22: f6 -> f6 : arg1'=1+arg1, arg3'=1+arg3, [ arg3 f6 : arg3'=0, [ arg2>0 ], cost: 2 21: f6 -> f4 : arg2'=-1+arg2, [ arg3>=arg2 ], cost: 3 23: f6 -> f6 : arg1'=arg2-arg3+arg1, arg3'=arg2, [ arg2-arg3>=0 ], cost: 4*arg2-4*arg3 16: __init -> f4 : arg1'=0, arg2'=arg2P_1, arg3'=arg3P_2, [], cost: 4 Chained accelerated rules (with incoming rules): Start location: __init 17: f4 -> f6 : arg3'=0, [ arg2>0 ], cost: 2 24: f4 -> f6 : arg1'=arg2+arg1, arg3'=arg2, [ arg2>0 ], cost: 2+4*arg2 21: f6 -> f4 : arg2'=-1+arg2, [ arg3>=arg2 ], cost: 3 16: __init -> f4 : arg1'=0, arg2'=arg2P_1, arg3'=arg3P_2, [], cost: 4 Eliminated locations (on tree-shaped paths): Start location: __init 25: f4 -> f4 : arg1'=arg2+arg1, arg2'=-1+arg2, arg3'=arg2, [ arg2>0 ], cost: 5+4*arg2 16: __init -> f4 : arg1'=0, arg2'=arg2P_1, arg3'=arg3P_2, [], cost: 4 Accelerating simple loops of location 3. Accelerating the following rules: 25: f4 -> f4 : arg1'=arg2+arg1, arg2'=-1+arg2, arg3'=arg2, [ arg2>0 ], cost: 5+4*arg2 Accelerated rule 25 with backward acceleration, yielding the new rule 26. [accelerate] Nesting with 1 inner and 1 outer candidates Removing the simple loops: 25. Accelerated all simple loops using metering functions (where possible): Start location: __init 26: f4 -> f4 : arg1'=1/2*arg2+1/2*arg2^2+arg1, arg2'=0, arg3'=1, [ arg2>=1 ], cost: 7*arg2+2*arg2^2 16: __init -> f4 : arg1'=0, arg2'=arg2P_1, arg3'=arg3P_2, [], cost: 4 Chained accelerated rules (with incoming rules): Start location: __init 16: __init -> f4 : arg1'=0, arg2'=arg2P_1, arg3'=arg3P_2, [], cost: 4 27: __init -> f4 : arg1'=1/2*arg2P_1^2+1/2*arg2P_1, arg2'=0, arg3'=1, [ arg2P_1>=1 ], cost: 4+2*arg2P_1^2+7*arg2P_1 Removed unreachable locations (and leaf rules with constant cost): Start location: __init 27: __init -> f4 : arg1'=1/2*arg2P_1^2+1/2*arg2P_1, arg2'=0, arg3'=1, [ arg2P_1>=1 ], cost: 4+2*arg2P_1^2+7*arg2P_1 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: __init 27: __init -> f4 : arg1'=1/2*arg2P_1^2+1/2*arg2P_1, arg2'=0, arg3'=1, [ arg2P_1>=1 ], cost: 4+2*arg2P_1^2+7*arg2P_1 Computing asymptotic complexity for rule 27 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),?)