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 && arg2==arg2P_1 ], cost: 1 1: f2 -> f3 : arg1'=arg1P_2, arg2'=arg2P_2, arg3'=arg3P_2, [ arg2==arg2P_2 && arg3==arg3P_2 ], cost: 1 2: f3 -> f4 : arg1'=arg1P_3, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1==arg1P_3 && arg3==arg3P_3 ], cost: 1 10: f4 -> f5 : arg1'=arg1P_11, arg2'=arg2P_11, arg3'=arg3P_11, [ arg1>=arg2 && arg3>=0 && arg1==arg1P_11 && arg2==arg2P_11 && arg3==arg3P_11 ], cost: 1 12: f4 -> f11 : arg1'=arg1P_13, arg2'=arg2P_13, arg3'=arg3P_13, [ arg1 f11 : arg1'=arg1P_14, arg2'=arg2P_14, arg3'=arg3P_14, [ arg3<0 && arg1==arg1P_14 && arg2==arg2P_14 && arg3==arg3P_14 ], cost: 1 3: f6 -> f9 : arg1'=arg1P_4, arg2'=arg2P_4, arg3'=arg3P_4, [ arg1P_4==-1-arg3+arg1 && arg2==arg2P_4 && arg3==arg3P_4 ], cost: 1 8: f9 -> f8 : arg1'=arg1P_9, arg2'=arg2P_9, arg3'=arg3P_9, [ arg1==arg1P_9 && arg2==arg2P_9 && arg3==arg3P_9 ], cost: 1 4: f7 -> f10 : arg1'=arg1P_5, arg2'=arg2P_5, arg3'=arg3P_5, [ arg2P_5==1+arg2+arg3 && arg1==arg1P_5 && arg3==arg3P_5 ], cost: 1 9: f10 -> f8 : arg1'=arg1P_10, arg2'=arg2P_10, arg3'=arg3P_10, [ arg1==arg1P_10 && arg2==arg2P_10 && arg3==arg3P_10 ], cost: 1 5: f5 -> f6 : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, [ x18_1<0 && arg1==arg1P_6 && arg2==arg2P_6 && arg3==arg3P_6 ], cost: 1 6: f5 -> f6 : arg1'=arg1P_7, arg2'=arg2P_7, arg3'=arg3P_7, [ x45_1>0 && arg1==arg1P_7 && arg2==arg2P_7 && arg3==arg3P_7 ], cost: 1 7: f5 -> f7 : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, [ x22_1==0 && arg1==arg1P_8 && arg2==arg2P_8 && arg3==arg3P_8 ], cost: 1 11: f8 -> f4 : arg1'=arg1P_12, arg2'=arg2P_12, arg3'=arg3P_12, [ arg1==arg1P_12 && arg2==arg2P_12 && arg3==arg3P_12 ], cost: 1 14: __init -> f1 : arg1'=arg1P_15, arg2'=arg2P_15, arg3'=arg3P_15, [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 14: __init -> f1 : arg1'=arg1P_15, arg2'=arg2P_15, arg3'=arg3P_15, [], 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 && arg2==arg2P_1 ], cost: 1 1: f2 -> f3 : arg1'=arg1P_2, arg2'=arg2P_2, arg3'=arg3P_2, [ arg2==arg2P_2 && arg3==arg3P_2 ], cost: 1 2: f3 -> f4 : arg1'=arg1P_3, arg2'=arg2P_3, arg3'=arg3P_3, [ arg1==arg1P_3 && arg3==arg3P_3 ], cost: 1 10: f4 -> f5 : arg1'=arg1P_11, arg2'=arg2P_11, arg3'=arg3P_11, [ arg1>=arg2 && arg3>=0 && arg1==arg1P_11 && arg2==arg2P_11 && arg3==arg3P_11 ], cost: 1 3: f6 -> f9 : arg1'=arg1P_4, arg2'=arg2P_4, arg3'=arg3P_4, [ arg1P_4==-1-arg3+arg1 && arg2==arg2P_4 && arg3==arg3P_4 ], cost: 1 8: f9 -> f8 : arg1'=arg1P_9, arg2'=arg2P_9, arg3'=arg3P_9, [ arg1==arg1P_9 && arg2==arg2P_9 && arg3==arg3P_9 ], cost: 1 4: f7 -> f10 : arg1'=arg1P_5, arg2'=arg2P_5, arg3'=arg3P_5, [ arg2P_5==1+arg2+arg3 && arg1==arg1P_5 && arg3==arg3P_5 ], cost: 1 9: f10 -> f8 : arg1'=arg1P_10, arg2'=arg2P_10, arg3'=arg3P_10, [ arg1==arg1P_10 && arg2==arg2P_10 && arg3==arg3P_10 ], cost: 1 5: f5 -> f6 : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, [ x18_1<0 && arg1==arg1P_6 && arg2==arg2P_6 && arg3==arg3P_6 ], cost: 1 6: f5 -> f6 : arg1'=arg1P_7, arg2'=arg2P_7, arg3'=arg3P_7, [ x45_1>0 && arg1==arg1P_7 && arg2==arg2P_7 && arg3==arg3P_7 ], cost: 1 7: f5 -> f7 : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, [ x22_1==0 && arg1==arg1P_8 && arg2==arg2P_8 && arg3==arg3P_8 ], cost: 1 11: f8 -> f4 : arg1'=arg1P_12, arg2'=arg2P_12, arg3'=arg3P_12, [ arg1==arg1P_12 && arg2==arg2P_12 && arg3==arg3P_12 ], cost: 1 14: __init -> f1 : arg1'=arg1P_15, arg2'=arg2P_15, arg3'=arg3P_15, [], cost: 1 Simplified all rules, resulting in: Start location: __init 0: f1 -> f2 : arg3'=arg3P_1, [], cost: 1 1: f2 -> f3 : arg1'=arg1P_2, [], cost: 1 2: f3 -> f4 : arg2'=arg2P_3, [], cost: 1 10: f4 -> f5 : [ arg1>=arg2 && arg3>=0 ], cost: 1 3: f6 -> f9 : arg1'=-1-arg3+arg1, [], cost: 1 8: f9 -> f8 : [], cost: 1 4: f7 -> f10 : arg2'=1+arg2+arg3, [], cost: 1 9: f10 -> f8 : [], cost: 1 6: f5 -> f6 : [], cost: 1 7: f5 -> f7 : [], cost: 1 11: f8 -> f4 : [], cost: 1 14: __init -> f1 : arg1'=arg1P_15, arg2'=arg2P_15, arg3'=arg3P_15, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: __init 10: f4 -> f5 : [ arg1>=arg2 && arg3>=0 ], cost: 1 20: f5 -> f8 : arg1'=-1-arg3+arg1, [], cost: 3 21: f5 -> f8 : arg2'=1+arg2+arg3, [], cost: 3 11: f8 -> f4 : [], cost: 1 17: __init -> f4 : arg1'=arg1P_2, arg2'=arg2P_3, arg3'=arg3P_1, [], cost: 4 Eliminated locations (on tree-shaped paths): Start location: __init 22: f4 -> f8 : arg1'=-1-arg3+arg1, [ arg1>=arg2 && arg3>=0 ], cost: 4 23: f4 -> f8 : arg2'=1+arg2+arg3, [ arg1>=arg2 && arg3>=0 ], cost: 4 11: f8 -> f4 : [], cost: 1 17: __init -> f4 : arg1'=arg1P_2, arg2'=arg2P_3, arg3'=arg3P_1, [], cost: 4 Eliminated locations (on tree-shaped paths): Start location: __init 24: f4 -> f4 : arg1'=-1-arg3+arg1, [ arg1>=arg2 && arg3>=0 ], cost: 5 25: f4 -> f4 : arg2'=1+arg2+arg3, [ arg1>=arg2 && arg3>=0 ], cost: 5 17: __init -> f4 : arg1'=arg1P_2, arg2'=arg2P_3, arg3'=arg3P_1, [], cost: 4 Accelerating simple loops of location 3. Accelerating the following rules: 24: f4 -> f4 : arg1'=-1-arg3+arg1, [ arg1>=arg2 && arg3>=0 ], cost: 5 25: f4 -> f4 : arg2'=1+arg2+arg3, [ arg1>=arg2 && arg3>=0 ], cost: 5 Accelerated rule 24 with backward acceleration, yielding the new rule 26. Accelerated rule 25 with backward acceleration, yielding the new rule 27. [accelerate] Nesting with 2 inner and 2 outer candidates Removing the simple loops: 24 25. Accelerated all simple loops using metering functions (where possible): Start location: __init 26: f4 -> f4 : arg1'=-arg3*k-k+arg1, [ arg3>=0 && k>=0 && 1-k-(-1+k)*arg3+arg1>=arg2 ], cost: 5*k 27: f4 -> f4 : arg2'=k_1*arg3+arg2+k_1, [ arg3>=0 && k_1>=0 && arg1>=-1+arg2+k_1+(-1+k_1)*arg3 ], cost: 5*k_1 17: __init -> f4 : arg1'=arg1P_2, arg2'=arg2P_3, arg3'=arg3P_1, [], cost: 4 Chained accelerated rules (with incoming rules): Start location: __init 17: __init -> f4 : arg1'=arg1P_2, arg2'=arg2P_3, arg3'=arg3P_1, [], cost: 4 28: __init -> f4 : arg1'=arg1P_2-k-arg3P_1*k, arg2'=arg2P_3, arg3'=arg3P_1, [ arg3P_1>=0 && k>=0 && 1+arg1P_2-k-(-1+k)*arg3P_1>=arg2P_3 ], cost: 4+5*k 29: __init -> f4 : arg1'=arg1P_2, arg2'=k_1+arg2P_3+arg3P_1*k_1, arg3'=arg3P_1, [ arg3P_1>=0 && k_1>=0 && arg1P_2>=-1+k_1+arg3P_1*(-1+k_1)+arg2P_3 ], cost: 4+5*k_1 Removed unreachable locations (and leaf rules with constant cost): Start location: __init 28: __init -> f4 : arg1'=arg1P_2-k-arg3P_1*k, arg2'=arg2P_3, arg3'=arg3P_1, [ arg3P_1>=0 && k>=0 && 1+arg1P_2-k-(-1+k)*arg3P_1>=arg2P_3 ], cost: 4+5*k 29: __init -> f4 : arg1'=arg1P_2, arg2'=k_1+arg2P_3+arg3P_1*k_1, arg3'=arg3P_1, [ arg3P_1>=0 && k_1>=0 && arg1P_2>=-1+k_1+arg3P_1*(-1+k_1)+arg2P_3 ], cost: 4+5*k_1 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: __init 28: __init -> f4 : arg1'=arg1P_2-k-arg3P_1*k, arg2'=arg2P_3, arg3'=arg3P_1, [ arg3P_1>=0 && k>=0 && 1+arg1P_2-k-(-1+k)*arg3P_1>=arg2P_3 ], cost: 4+5*k 29: __init -> f4 : arg1'=arg1P_2, arg2'=k_1+arg2P_3+arg3P_1*k_1, arg3'=arg3P_1, [ arg3P_1>=0 && k_1>=0 && arg1P_2>=-1+k_1+arg3P_1*(-1+k_1)+arg2P_3 ], cost: 4+5*k_1 Computing asymptotic complexity for rule 28 Resulting cost 0 has complexity: Unknown Computing asymptotic complexity for rule 29 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),?)