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 && arg3==arg3P_3 && 0==arg4P_3 ], cost: 1
     17: f4 -> f5 : arg1'=arg1P_18, arg2'=arg2P_18, arg3'=arg3P_18, arg4'=arg4P_18, [ arg1>=arg2 && arg2>0 && arg1==arg1P_18 && arg2==arg2P_18 && arg3==arg3P_18 && arg4==arg4P_18 ], cost: 1
     19: f4 -> f17 : arg1'=arg1P_20, arg2'=arg2P_20, arg3'=arg3P_20, arg4'=arg4P_20, [ arg1<arg2 && arg1==arg1P_20 && arg2==arg2P_20 && arg3==arg3P_20 && arg4==arg4P_20 ], cost: 1
     20: f4 -> f17 : arg1'=arg1P_21, arg2'=arg2P_21, arg3'=arg3P_21, arg4'=arg4P_21, [ arg2<=0 && arg1==arg1P_21 && arg2==arg2P_21 && arg3==arg3P_21 && arg4==arg4P_21 ], cost: 1
      3: f5 -> f6 : arg1'=arg1P_4, arg2'=arg2P_4, arg3'=arg3P_4, arg4'=arg4P_4, [ arg1==arg1P_4 && arg2==arg2P_4 && arg2==arg3P_4 && arg4==arg4P_4 ], cost: 1
     12: f6 -> f7 : arg1'=arg1P_13, arg2'=arg2P_13, arg3'=arg3P_13, arg4'=arg4P_13, [ arg3<0 && arg1==arg1P_13 && arg2==arg2P_13 && arg3==arg3P_13 && arg4==arg4P_13 ], cost: 1
     13: f6 -> f7 : arg1'=arg1P_14, arg2'=arg2P_14, arg3'=arg3P_14, arg4'=arg4P_14, [ arg3>0 && arg1==arg1P_14 && arg2==arg2P_14 && arg3==arg3P_14 && arg4==arg4P_14 ], cost: 1
     15: f6 -> f15 : arg1'=arg1P_16, arg2'=arg2P_16, arg3'=arg3P_16, arg4'=arg4P_16, [ arg3==0 && arg1==arg1P_16 && arg2==arg2P_16 && arg3==arg3P_16 && arg4==arg4P_16 ], cost: 1
      4: f8 -> f11 : arg1'=arg1P_5, arg2'=arg2P_5, arg3'=arg3P_5, arg4'=arg4P_5, [ arg3P_5==-1+arg3 && arg1==arg1P_5 && arg2==arg2P_5 && arg4==arg4P_5 ], cost: 1
      5: f11 -> f12 : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, arg4'=arg4P_6, [ arg1P_6==-1+arg1 && arg2==arg2P_6 && arg3==arg3P_6 && arg4==arg4P_6 ], cost: 1
     10: f12 -> f10 : arg1'=arg1P_11, arg2'=arg2P_11, arg3'=arg3P_11, arg4'=arg4P_11, [ arg1==arg1P_11 && arg2==arg2P_11 && arg3==arg3P_11 && arg4==arg4P_11 ], cost: 1
      6: f9 -> f13 : arg1'=arg1P_7, arg2'=arg2P_7, arg3'=arg3P_7, arg4'=arg4P_7, [ arg3P_7==1+arg3 && arg1==arg1P_7 && arg2==arg2P_7 && arg4==arg4P_7 ], cost: 1
      7: f13 -> f14 : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, arg4'=arg4P_8, [ arg1P_8==1+arg1 && arg2==arg2P_8 && arg3==arg3P_8 && arg4==arg4P_8 ], cost: 1
     11: f14 -> f10 : arg1'=arg1P_12, arg2'=arg2P_12, arg3'=arg3P_12, arg4'=arg4P_12, [ arg1==arg1P_12 && arg2==arg2P_12 && arg3==arg3P_12 && arg4==arg4P_12 ], cost: 1
      8: f7 -> f8 : arg1'=arg1P_9, arg2'=arg2P_9, arg3'=arg3P_9, arg4'=arg4P_9, [ arg3>0 && arg1==arg1P_9 && arg2==arg2P_9 && arg3==arg3P_9 && arg4==arg4P_9 ], cost: 1
      9: f7 -> f9 : arg1'=arg1P_10, arg2'=arg2P_10, arg3'=arg3P_10, arg4'=arg4P_10, [ arg3<=0 && arg1==arg1P_10 && arg2==arg2P_10 && arg3==arg3P_10 && arg4==arg4P_10 ], cost: 1
     14: f10 -> f6 : arg1'=arg1P_15, arg2'=arg2P_15, arg3'=arg3P_15, arg4'=arg4P_15, [ arg1==arg1P_15 && arg2==arg2P_15 && arg3==arg3P_15 && arg4==arg4P_15 ], cost: 1
     16: f15 -> f16 : arg1'=arg1P_17, arg2'=arg2P_17, arg3'=arg3P_17, arg4'=arg4P_17, [ arg4P_17==1+arg4 && arg1==arg1P_17 && arg2==arg2P_17 && arg3==arg3P_17 ], cost: 1
     18: f16 -> f4 : 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 && arg3==arg3P_3 && 0==arg4P_3 ], cost: 1
     17: f4 -> f5 : arg1'=arg1P_18, arg2'=arg2P_18, arg3'=arg3P_18, arg4'=arg4P_18, [ arg1>=arg2 && arg2>0 && arg1==arg1P_18 && arg2==arg2P_18 && arg3==arg3P_18 && arg4==arg4P_18 ], cost: 1
      3: f5 -> f6 : arg1'=arg1P_4, arg2'=arg2P_4, arg3'=arg3P_4, arg4'=arg4P_4, [ arg1==arg1P_4 && arg2==arg2P_4 && arg2==arg3P_4 && arg4==arg4P_4 ], cost: 1
     12: f6 -> f7 : arg1'=arg1P_13, arg2'=arg2P_13, arg3'=arg3P_13, arg4'=arg4P_13, [ arg3<0 && arg1==arg1P_13 && arg2==arg2P_13 && arg3==arg3P_13 && arg4==arg4P_13 ], cost: 1
     13: f6 -> f7 : arg1'=arg1P_14, arg2'=arg2P_14, arg3'=arg3P_14, arg4'=arg4P_14, [ arg3>0 && arg1==arg1P_14 && arg2==arg2P_14 && arg3==arg3P_14 && arg4==arg4P_14 ], cost: 1
     15: f6 -> f15 : arg1'=arg1P_16, arg2'=arg2P_16, arg3'=arg3P_16, arg4'=arg4P_16, [ arg3==0 && arg1==arg1P_16 && arg2==arg2P_16 && arg3==arg3P_16 && arg4==arg4P_16 ], cost: 1
      4: f8 -> f11 : arg1'=arg1P_5, arg2'=arg2P_5, arg3'=arg3P_5, arg4'=arg4P_5, [ arg3P_5==-1+arg3 && arg1==arg1P_5 && arg2==arg2P_5 && arg4==arg4P_5 ], cost: 1
      5: f11 -> f12 : arg1'=arg1P_6, arg2'=arg2P_6, arg3'=arg3P_6, arg4'=arg4P_6, [ arg1P_6==-1+arg1 && arg2==arg2P_6 && arg3==arg3P_6 && arg4==arg4P_6 ], cost: 1
     10: f12 -> f10 : arg1'=arg1P_11, arg2'=arg2P_11, arg3'=arg3P_11, arg4'=arg4P_11, [ arg1==arg1P_11 && arg2==arg2P_11 && arg3==arg3P_11 && arg4==arg4P_11 ], cost: 1
      6: f9 -> f13 : arg1'=arg1P_7, arg2'=arg2P_7, arg3'=arg3P_7, arg4'=arg4P_7, [ arg3P_7==1+arg3 && arg1==arg1P_7 && arg2==arg2P_7 && arg4==arg4P_7 ], cost: 1
      7: f13 -> f14 : arg1'=arg1P_8, arg2'=arg2P_8, arg3'=arg3P_8, arg4'=arg4P_8, [ arg1P_8==1+arg1 && arg2==arg2P_8 && arg3==arg3P_8 && arg4==arg4P_8 ], cost: 1
     11: f14 -> f10 : arg1'=arg1P_12, arg2'=arg2P_12, arg3'=arg3P_12, arg4'=arg4P_12, [ arg1==arg1P_12 && arg2==arg2P_12 && arg3==arg3P_12 && arg4==arg4P_12 ], cost: 1
      8: f7 -> f8 : arg1'=arg1P_9, arg2'=arg2P_9, arg3'=arg3P_9, arg4'=arg4P_9, [ arg3>0 && arg1==arg1P_9 && arg2==arg2P_9 && arg3==arg3P_9 && arg4==arg4P_9 ], cost: 1
      9: f7 -> f9 : arg1'=arg1P_10, arg2'=arg2P_10, arg3'=arg3P_10, arg4'=arg4P_10, [ arg3<=0 && arg1==arg1P_10 && arg2==arg2P_10 && arg3==arg3P_10 && arg4==arg4P_10 ], cost: 1
     14: f10 -> f6 : arg1'=arg1P_15, arg2'=arg2P_15, arg3'=arg3P_15, arg4'=arg4P_15, [ arg1==arg1P_15 && arg2==arg2P_15 && arg3==arg3P_15 && arg4==arg4P_15 ], cost: 1
     16: f15 -> f16 : arg1'=arg1P_17, arg2'=arg2P_17, arg3'=arg3P_17, arg4'=arg4P_17, [ arg4P_17==1+arg4 && arg1==arg1P_17 && arg2==arg2P_17 && arg3==arg3P_17 ], cost: 1
     18: f16 -> f4 : 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 : arg4'=0, [], cost: 1
     17: f4 -> f5 : [ arg1>=arg2 && arg2>0 ], cost: 1
      3: f5 -> f6 : arg3'=arg2, [], cost: 1
     12: f6 -> f7 : [ arg3<0 ], cost: 1
     13: f6 -> f7 : [ arg3>0 ], cost: 1
     15: f6 -> f15 : [ arg3==0 ], cost: 1
      4: f8 -> f11 : arg3'=-1+arg3, [], cost: 1
      5: f11 -> f12 : arg1'=-1+arg1, [], cost: 1
     10: f12 -> f10 : [], cost: 1
      6: f9 -> f13 : arg3'=1+arg3, [], cost: 1
      7: f13 -> f14 : arg1'=1+arg1, [], cost: 1
     11: f14 -> f10 : [], cost: 1
      8: f7 -> f8 : [ arg3>0 ], cost: 1
      9: f7 -> f9 : [ arg3<=0 ], cost: 1
     14: f10 -> f6 : [], cost: 1
     16: f15 -> f16 : arg4'=1+arg4, [], cost: 1
     18: f16 -> f4 : [], 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
     25: f4 -> f6 : arg3'=arg2, [ arg1>=arg2 && arg2>0 ], cost: 2
     12: f6 -> f7 : [ arg3<0 ], cost: 1
     13: f6 -> f7 : [ arg3>0 ], cost: 1
     27: f6 -> f4 : arg4'=1+arg4, [ arg3==0 ], cost: 3
     32: f7 -> f10 : arg1'=-1+arg1, arg3'=-1+arg3, [ arg3>0 ], cost: 4
     33: f7 -> f10 : arg1'=1+arg1, arg3'=1+arg3, [ arg3<=0 ], cost: 4
     14: f10 -> f6 : [], cost: 1
     24: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=arg3P_22, arg4'=0, [], cost: 4

Eliminated locations (on tree-shaped paths):
   Start location: __init
     25: f4 -> f6 : arg3'=arg2, [ arg1>=arg2 && arg2>0 ], cost: 2
     27: f6 -> f4 : arg4'=1+arg4, [ arg3==0 ], cost: 3
     34: f6 -> f10 : arg1'=1+arg1, arg3'=1+arg3, [ arg3<0 ], cost: 5
     35: f6 -> f10 : arg1'=-1+arg1, arg3'=-1+arg3, [ arg3>0 ], cost: 5
     14: f10 -> f6 : [], cost: 1
     24: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=arg3P_22, arg4'=0, [], cost: 4

Eliminated locations (on tree-shaped paths):
   Start location: __init
     25: f4 -> f6 : arg3'=arg2, [ arg1>=arg2 && arg2>0 ], cost: 2
     27: f6 -> f4 : arg4'=1+arg4, [ arg3==0 ], cost: 3
     36: f6 -> f6 : arg1'=1+arg1, arg3'=1+arg3, [ arg3<0 ], cost: 6
     37: f6 -> f6 : arg1'=-1+arg1, arg3'=-1+arg3, [ arg3>0 ], cost: 6
     24: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=arg3P_22, arg4'=0, [], cost: 4

Accelerating simple loops of location 5.
   Accelerating the following rules:
     36: f6 -> f6 : arg1'=1+arg1, arg3'=1+arg3, [ arg3<0 ], cost: 6
     37: f6 -> f6 : arg1'=-1+arg1, arg3'=-1+arg3, [ arg3>0 ], cost: 6

   Accelerated rule 36 with backward acceleration, yielding the new rule 38.
   Accelerated rule 37 with backward acceleration, yielding the new rule 39.
[accelerate] Nesting with 2 inner and 2 outer candidates
   Removing the simple loops: 36 37.

Accelerated all simple loops using metering functions (where possible):
   Start location: __init
     25: f4 -> f6 : arg3'=arg2, [ arg1>=arg2 && arg2>0 ], cost: 2
     27: f6 -> f4 : arg4'=1+arg4, [ arg3==0 ], cost: 3
     38: f6 -> f6 : arg1'=-arg3+arg1, arg3'=0, [ -arg3>=0 ], cost: -6*arg3
     39: f6 -> f6 : arg1'=-arg3+arg1, arg3'=0, [ arg3>=0 ], cost: 6*arg3
     24: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=arg3P_22, arg4'=0, [], cost: 4

Chained accelerated rules (with incoming rules):
   Start location: __init
     25: f4 -> f6 : arg3'=arg2, [ arg1>=arg2 && arg2>0 ], cost: 2
     40: f4 -> f6 : arg1'=-arg2+arg1, arg3'=0, [ arg1>=arg2 && arg2>0 ], cost: 2+6*arg2
     27: f6 -> f4 : arg4'=1+arg4, [ arg3==0 ], cost: 3
     24: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=arg3P_22, arg4'=0, [], cost: 4

Eliminated locations (on tree-shaped paths):
   Start location: __init
     41: f4 -> f4 : arg1'=-arg2+arg1, arg3'=0, arg4'=1+arg4, [ arg1>=arg2 && arg2>0 ], cost: 5+6*arg2
     24: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=arg3P_22, arg4'=0, [], cost: 4

Accelerating simple loops of location 3.
   Accelerating the following rules:
     41: f4 -> f4 : arg1'=-arg2+arg1, arg3'=0, arg4'=1+arg4, [ arg1>=arg2 && arg2>0 ], cost: 5+6*arg2

   Accelerated rule 41 with backward acceleration, yielding the new rule 42.
[accelerate] Nesting with 1 inner and 1 outer candidates
   Removing the simple loops: 41.

Accelerated all simple loops using metering functions (where possible):
   Start location: __init
     42: f4 -> f4 : arg1'=-k_2*arg2+arg1, arg3'=0, arg4'=k_2+arg4, [ arg2>0 && k_2>=1 && arg1-arg2*(-1+k_2)>=arg2 ], cost: 5*k_2+6*k_2*arg2
     24: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=arg3P_22, arg4'=0, [], cost: 4

Chained accelerated rules (with incoming rules):
   Start location: __init
     24: __init -> f4 : arg1'=arg1P_1, arg2'=arg2P_2, arg3'=arg3P_22, arg4'=0, [], cost: 4
     43: __init -> f4 : arg1'=-k_2*arg2P_2+arg1P_1, arg2'=arg2P_2, arg3'=0, arg4'=k_2, [ arg2P_2>0 && k_2>=1 && arg1P_1-arg2P_2*(-1+k_2)>=arg2P_2 ], cost: 4+5*k_2+6*k_2*arg2P_2

Removed unreachable locations (and leaf rules with constant cost):
   Start location: __init
     43: __init -> f4 : arg1'=-k_2*arg2P_2+arg1P_1, arg2'=arg2P_2, arg3'=0, arg4'=k_2, [ arg2P_2>0 && k_2>=1 && arg1P_1-arg2P_2*(-1+k_2)>=arg2P_2 ], cost: 4+5*k_2+6*k_2*arg2P_2

### Computing asymptotic complexity ###

Fully simplified ITS problem
   Start location: __init
     43: __init -> f4 : arg1'=-k_2*arg2P_2+arg1P_1, arg2'=arg2P_2, arg3'=0, arg4'=k_2, [ arg2P_2>0 && k_2>=1 && arg1P_1-arg2P_2*(-1+k_2)>=arg2P_2 ], cost: 4+5*k_2+6*k_2*arg2P_2

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),?)