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, arg4'=arg4P_1, [ arg2==arg2P_1 && arg3==arg3P_1 && arg4==arg4P_1 ], cost: 1 1: f2 -> f3 : arg1'=arg1P_2, arg2'=arg2P_2, arg3'=arg3P_2, arg4'=arg4P_2, [ arg1==arg1P_2 && arg3==arg3P_2 && arg4==arg4P_2 ], cost: 1 2: f3 -> f4 : arg1'=arg1P_3, arg2'=arg2P_3, arg3'=arg3P_3, arg4'=arg4P_3, [ arg1==arg1P_3 && arg2==arg2P_3 && 0==arg3P_3 && arg4==arg4P_3 ], cost: 1 5: f4 -> f5 : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, arg4'=arg4P_6, [ arg1 f6 : arg1'=arg1P_7, arg2'=arg2P_7, arg3'=arg3P_7, arg4'=arg4P_7, [ arg1>=arg2 && arg1==arg1P_7 && arg2==arg2P_7 && arg3==arg3P_7 && arg4==arg4P_7 ], cost: 1 3: f5 -> f8 : arg1'=arg1P_4, arg2'=arg2P_4, arg3'=arg3P_4, arg4'=arg4P_4, [ arg1==arg1P_4 && arg2==arg2P_4 && arg3==arg3P_4 && arg1==arg4P_4 ], cost: 1 7: f8 -> f7 : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, arg4'=arg4P_8, [ arg1==arg1P_8 && arg2==arg2P_8 && arg3==arg3P_8 && arg4==arg4P_8 ], cost: 1 4: f6 -> f9 : arg1'=arg1P_5, arg2'=arg2P_5, arg3'=arg3P_5, arg4'=arg4P_5, [ arg1==arg1P_5 && arg2==arg2P_5 && arg3==arg3P_5 && arg2==arg4P_5 ], cost: 1 8: f9 -> f7 : arg1'=arg1P_9, arg2'=arg2P_9, arg3'=arg3P_9, arg4'=arg4P_9, [ arg1==arg1P_9 && arg2==arg2P_9 && arg3==arg3P_9 && arg4==arg4P_9 ], cost: 1 17: f7 -> f10 : arg1'=arg1P_18, arg2'=arg2P_18, arg3'=arg3P_18, arg4'=arg4P_18, [ arg4==arg2 && arg1==arg1P_18 && arg2==arg2P_18 && arg3==arg3P_18 && arg4==arg4P_18 ], cost: 1 19: f7 -> f18 : arg1'=arg1P_20, arg2'=arg2P_20, arg3'=arg3P_20, arg4'=arg4P_20, [ arg4 f18 : arg1'=arg1P_21, arg2'=arg2P_21, arg3'=arg3P_21, arg4'=arg4P_21, [ arg4>arg2 && arg1==arg1P_21 && arg2==arg2P_21 && arg3==arg3P_21 && arg4==arg4P_21 ], cost: 1 9: f10 -> f11 : arg1'=arg1P_10, arg2'=arg2P_10, arg3'=arg3P_10, arg4'=arg4P_10, [ arg2P_10==1+arg2 && arg1==arg1P_10 && arg3==arg3P_10 && arg4==arg4P_10 ], cost: 1 10: f11 -> f12 : arg1'=arg1P_11, arg2'=arg2P_11, arg3'=arg3P_11, arg4'=arg4P_11, [ arg3P_11==1+arg3 && arg1==arg1P_11 && arg2==arg2P_11 && arg4==arg4P_11 ], cost: 1 13: f12 -> f13 : arg1'=arg1P_14, arg2'=arg2P_14, arg3'=arg3P_14, arg4'=arg4P_14, [ arg1 f14 : arg1'=arg1P_15, arg2'=arg2P_15, arg3'=arg3P_15, arg4'=arg4P_15, [ arg1>=arg2 && arg1==arg1P_15 && arg2==arg2P_15 && arg3==arg3P_15 && arg4==arg4P_15 ], cost: 1 11: f13 -> f16 : arg1'=arg1P_12, arg2'=arg2P_12, arg3'=arg3P_12, arg4'=arg4P_12, [ arg1==arg1P_12 && arg2==arg2P_12 && arg3==arg3P_12 && arg1==arg4P_12 ], cost: 1 15: f16 -> f15 : arg1'=arg1P_16, arg2'=arg2P_16, arg3'=arg3P_16, arg4'=arg4P_16, [ arg1==arg1P_16 && arg2==arg2P_16 && arg3==arg3P_16 && arg4==arg4P_16 ], cost: 1 12: f14 -> f17 : arg1'=arg1P_13, arg2'=arg2P_13, arg3'=arg3P_13, arg4'=arg4P_13, [ arg1==arg1P_13 && arg2==arg2P_13 && arg3==arg3P_13 && arg2==arg4P_13 ], cost: 1 16: f17 -> f15 : arg1'=arg1P_17, arg2'=arg2P_17, arg3'=arg3P_17, arg4'=arg4P_17, [ arg1==arg1P_17 && arg2==arg2P_17 && arg3==arg3P_17 && arg4==arg4P_17 ], cost: 1 18: f15 -> f7 : arg1'=arg1P_19, arg2'=arg2P_19, arg3'=arg3P_19, arg4'=arg4P_19, [ arg1==arg1P_19 && arg2==arg2P_19 && arg3==arg3P_19 && arg4==arg4P_19 ], cost: 1 21: __init -> f1 : arg1'=arg1P_22, arg2'=arg2P_22, arg3'=arg3P_22, arg4'=arg4P_22, [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 21: __init -> f1 : arg1'=arg1P_22, arg2'=arg2P_22, arg3'=arg3P_22, arg4'=arg4P_22, [], cost: 1 Removed unreachable and leaf rules: Start location: __init 0: f1 -> f2 : arg1'=arg1P_1, arg2'=arg2P_1, arg3'=arg3P_1, arg4'=arg4P_1, [ arg2==arg2P_1 && arg3==arg3P_1 && arg4==arg4P_1 ], cost: 1 1: f2 -> f3 : arg1'=arg1P_2, arg2'=arg2P_2, arg3'=arg3P_2, arg4'=arg4P_2, [ arg1==arg1P_2 && arg3==arg3P_2 && arg4==arg4P_2 ], cost: 1 2: f3 -> f4 : arg1'=arg1P_3, arg2'=arg2P_3, arg3'=arg3P_3, arg4'=arg4P_3, [ arg1==arg1P_3 && arg2==arg2P_3 && 0==arg3P_3 && arg4==arg4P_3 ], cost: 1 5: f4 -> f5 : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, arg4'=arg4P_6, [ arg1 f6 : arg1'=arg1P_7, arg2'=arg2P_7, arg3'=arg3P_7, arg4'=arg4P_7, [ arg1>=arg2 && arg1==arg1P_7 && arg2==arg2P_7 && arg3==arg3P_7 && arg4==arg4P_7 ], cost: 1 3: f5 -> f8 : arg1'=arg1P_4, arg2'=arg2P_4, arg3'=arg3P_4, arg4'=arg4P_4, [ arg1==arg1P_4 && arg2==arg2P_4 && arg3==arg3P_4 && arg1==arg4P_4 ], cost: 1 7: f8 -> f7 : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, arg4'=arg4P_8, [ arg1==arg1P_8 && arg2==arg2P_8 && arg3==arg3P_8 && arg4==arg4P_8 ], cost: 1 4: f6 -> f9 : arg1'=arg1P_5, arg2'=arg2P_5, arg3'=arg3P_5, arg4'=arg4P_5, [ arg1==arg1P_5 && arg2==arg2P_5 && arg3==arg3P_5 && arg2==arg4P_5 ], cost: 1 8: f9 -> f7 : arg1'=arg1P_9, arg2'=arg2P_9, arg3'=arg3P_9, arg4'=arg4P_9, [ arg1==arg1P_9 && arg2==arg2P_9 && arg3==arg3P_9 && arg4==arg4P_9 ], cost: 1 17: f7 -> f10 : arg1'=arg1P_18, arg2'=arg2P_18, arg3'=arg3P_18, arg4'=arg4P_18, [ arg4==arg2 && arg1==arg1P_18 && arg2==arg2P_18 && arg3==arg3P_18 && arg4==arg4P_18 ], cost: 1 9: f10 -> f11 : arg1'=arg1P_10, arg2'=arg2P_10, arg3'=arg3P_10, arg4'=arg4P_10, [ arg2P_10==1+arg2 && arg1==arg1P_10 && arg3==arg3P_10 && arg4==arg4P_10 ], cost: 1 10: f11 -> f12 : arg1'=arg1P_11, arg2'=arg2P_11, arg3'=arg3P_11, arg4'=arg4P_11, [ arg3P_11==1+arg3 && arg1==arg1P_11 && arg2==arg2P_11 && arg4==arg4P_11 ], cost: 1 13: f12 -> f13 : arg1'=arg1P_14, arg2'=arg2P_14, arg3'=arg3P_14, arg4'=arg4P_14, [ arg1 f14 : arg1'=arg1P_15, arg2'=arg2P_15, arg3'=arg3P_15, arg4'=arg4P_15, [ arg1>=arg2 && arg1==arg1P_15 && arg2==arg2P_15 && arg3==arg3P_15 && arg4==arg4P_15 ], cost: 1 11: f13 -> f16 : arg1'=arg1P_12, arg2'=arg2P_12, arg3'=arg3P_12, arg4'=arg4P_12, [ arg1==arg1P_12 && arg2==arg2P_12 && arg3==arg3P_12 && arg1==arg4P_12 ], cost: 1 15: f16 -> f15 : arg1'=arg1P_16, arg2'=arg2P_16, arg3'=arg3P_16, arg4'=arg4P_16, [ arg1==arg1P_16 && arg2==arg2P_16 && arg3==arg3P_16 && arg4==arg4P_16 ], cost: 1 12: f14 -> f17 : arg1'=arg1P_13, arg2'=arg2P_13, arg3'=arg3P_13, arg4'=arg4P_13, [ arg1==arg1P_13 && arg2==arg2P_13 && arg3==arg3P_13 && arg2==arg4P_13 ], cost: 1 16: f17 -> f15 : arg1'=arg1P_17, arg2'=arg2P_17, arg3'=arg3P_17, arg4'=arg4P_17, [ arg1==arg1P_17 && arg2==arg2P_17 && arg3==arg3P_17 && arg4==arg4P_17 ], cost: 1 18: f15 -> f7 : arg1'=arg1P_19, arg2'=arg2P_19, arg3'=arg3P_19, arg4'=arg4P_19, [ arg1==arg1P_19 && arg2==arg2P_19 && arg3==arg3P_19 && arg4==arg4P_19 ], cost: 1 21: __init -> f1 : arg1'=arg1P_22, arg2'=arg2P_22, arg3'=arg3P_22, arg4'=arg4P_22, [], 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 2: f3 -> f4 : arg3'=0, [], cost: 1 5: f4 -> f5 : [ arg1 f6 : [ arg1>=arg2 ], cost: 1 3: f5 -> f8 : arg4'=arg1, [], cost: 1 7: f8 -> f7 : [], cost: 1 4: f6 -> f9 : arg4'=arg2, [], cost: 1 8: f9 -> f7 : [], cost: 1 17: f7 -> f10 : [ arg4==arg2 ], cost: 1 9: f10 -> f11 : arg2'=1+arg2, [], cost: 1 10: f11 -> f12 : arg3'=1+arg3, [], cost: 1 13: f12 -> f13 : [ arg1 f14 : [ arg1>=arg2 ], cost: 1 11: f13 -> f16 : arg4'=arg1, [], cost: 1 15: f16 -> f15 : [], cost: 1 12: f14 -> f17 : arg4'=arg2, [], cost: 1 16: f17 -> f15 : [], cost: 1 18: f15 -> f7 : [], cost: 1 21: __init -> f1 : arg1'=arg1P_22, arg2'=arg2P_22, arg3'=arg3P_22, arg4'=arg4P_22, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: __init 27: f4 -> f7 : arg4'=arg1, [ arg1 f7 : arg4'=arg2, [ arg1>=arg2 ], cost: 3 30: f7 -> f12 : arg2'=1+arg2, arg3'=1+arg3, [ arg4==arg2 ], cost: 3 33: f12 -> f15 : arg4'=arg1, [ arg1 f15 : arg4'=arg2, [ arg1>=arg2 ], cost: 3 18: f15 -> f7 : [], cost: 1 24: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=0, arg4'=arg4P_22, [], cost: 4 Eliminated locations (on tree-shaped paths): Start location: __init 37: f7 -> f15 : arg2'=1+arg2, arg3'=1+arg3, arg4'=arg1, [ arg4==arg2 && arg1<1+arg2 ], cost: 6 38: f7 -> f15 : arg2'=1+arg2, arg3'=1+arg3, arg4'=1+arg2, [ arg4==arg2 && arg1>=1+arg2 ], cost: 6 18: f15 -> f7 : [], cost: 1 35: __init -> f7 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=0, arg4'=arg1P_1, [ arg1P_1 f7 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=0, arg4'=arg2P_2, [ arg1P_1>=arg2P_2 ], cost: 7 Eliminated locations (on tree-shaped paths): Start location: __init 39: f7 -> f7 : arg2'=1+arg2, arg3'=1+arg3, arg4'=arg1, [ arg4==arg2 && arg1<1+arg2 ], cost: 7 40: f7 -> f7 : arg2'=1+arg2, arg3'=1+arg3, arg4'=1+arg2, [ arg4==arg2 && arg1>=1+arg2 ], cost: 7 35: __init -> f7 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=0, arg4'=arg1P_1, [ arg1P_1 f7 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=0, arg4'=arg2P_2, [ arg1P_1>=arg2P_2 ], cost: 7 Accelerating simple loops of location 8. Accelerating the following rules: 39: f7 -> f7 : arg2'=1+arg2, arg3'=1+arg3, arg4'=arg1, [ arg4==arg2 && arg1<1+arg2 ], cost: 7 40: f7 -> f7 : arg2'=1+arg2, arg3'=1+arg3, arg4'=1+arg2, [ arg4==arg2 && arg1>=1+arg2 ], cost: 7 Failed to prove monotonicity of the guard of rule 39. Accelerated rule 40 with backward acceleration, yielding the new rule 41. [accelerate] Nesting with 2 inner and 2 outer candidates Removing the simple loops: 40. Accelerated all simple loops using metering functions (where possible): Start location: __init 39: f7 -> f7 : arg2'=1+arg2, arg3'=1+arg3, arg4'=arg1, [ arg4==arg2 && arg1<1+arg2 ], cost: 7 41: f7 -> f7 : arg2'=arg1, arg3'=-arg2+arg3+arg1, arg4'=arg1, [ arg4==arg2 && -arg2+arg1>=1 ], cost: -7*arg2+7*arg1 35: __init -> f7 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=0, arg4'=arg1P_1, [ arg1P_1 f7 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=0, arg4'=arg2P_2, [ arg1P_1>=arg2P_2 ], cost: 7 Chained accelerated rules (with incoming rules): Start location: __init 35: __init -> f7 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=0, arg4'=arg1P_1, [ arg1P_1 f7 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=0, arg4'=arg2P_2, [ arg1P_1>=arg2P_2 ], cost: 7 42: __init -> f7 : arg1'=arg1P_1, arg2'=1+arg1P_1, arg3'=1, arg4'=arg1P_1, [], cost: 14 43: __init -> f7 : arg1'=arg1P_1, arg2'=arg1P_1, arg3'=-arg2P_2+arg1P_1, arg4'=arg1P_1, [ -arg2P_2+arg1P_1>=1 ], cost: 7-7*arg2P_2+7*arg1P_1 Removed unreachable locations (and leaf rules with constant cost): Start location: __init 43: __init -> f7 : arg1'=arg1P_1, arg2'=arg1P_1, arg3'=-arg2P_2+arg1P_1, arg4'=arg1P_1, [ -arg2P_2+arg1P_1>=1 ], cost: 7-7*arg2P_2+7*arg1P_1 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: __init 43: __init -> f7 : arg1'=arg1P_1, arg2'=arg1P_1, arg3'=-arg2P_2+arg1P_1, arg4'=arg1P_1, [ -arg2P_2+arg1P_1>=1 ], cost: 7-7*arg2P_2+7*arg1P_1 Computing asymptotic complexity for rule 43 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),?)