WORST_CASE(Omega(1),?)

### Pre-processing the ITS problem ###

Initial linear ITS problem
   Start location: l7
      0: l0 -> l1 : a^0'=a^post_1, b^0'=b^post_1, c^0'=c^post_1, d^0'=d^post_1, e^0'=e^post_1, sum^0'=sum^post_1, [ a^post_1==a^post_1 && b^post_1==b^post_1 && c^post_1==c^post_1 && d^post_1==d^post_1 && e^post_1==e^post_1 && 1<=b^post_1+d^post_1+a^post_1+c^post_1+e^post_1 && sum^post_1==b^post_1+d^post_1+a^post_1+c^post_1+e^post_1 ], cost: 1
      1: l1 -> l2 : a^0'=a^post_2, b^0'=b^post_2, c^0'=c^post_2, d^0'=d^post_2, e^0'=e^post_2, sum^0'=sum^post_2, [ 1+e^0<=0 && e^post_2==-e^0 && d^post_2==d^0-e^post_2 && a^post_2==a^0-e^post_2 && b^0==b^post_2 && c^0==c^post_2 && sum^0==sum^post_2 ], cost: 1
      3: l1 -> l3 : a^0'=a^post_4, b^0'=b^post_4, c^0'=c^post_4, d^0'=d^post_4, e^0'=e^post_4, sum^0'=sum^post_4, [ 1+d^0<=0 && d^post_4==-d^0 && c^post_4==c^0-d^post_4 && e^post_4==e^0-d^post_4 && a^0==a^post_4 && b^0==b^post_4 && sum^0==sum^post_4 ], cost: 1
      5: l1 -> l4 : a^0'=a^post_6, b^0'=b^post_6, c^0'=c^post_6, d^0'=d^post_6, e^0'=e^post_6, sum^0'=sum^post_6, [ 1+c^0<=0 && c^post_6==-c^0 && b^post_6==-c^post_6+b^0 && d^post_6==-c^post_6+d^0 && a^0==a^post_6 && e^0==e^post_6 && sum^0==sum^post_6 ], cost: 1
      7: l1 -> l5 : a^0'=a^post_8, b^0'=b^post_8, c^0'=c^post_8, d^0'=d^post_8, e^0'=e^post_8, sum^0'=sum^post_8, [ 1+b^0<=0 && b^post_8==-b^0 && a^post_8==-b^post_8+a^0 && c^post_8==-b^post_8+c^0 && d^0==d^post_8 && e^0==e^post_8 && sum^0==sum^post_8 ], cost: 1
      9: l1 -> l6 : a^0'=a^post_10, b^0'=b^post_10, c^0'=c^post_10, d^0'=d^post_10, e^0'=e^post_10, sum^0'=sum^post_10, [ 1+a^0<=0 && a^post_10==-a^0 && b^post_10==-a^post_10+b^0 && e^post_10==-a^post_10+e^0 && c^0==c^post_10 && d^0==d^post_10 && sum^0==sum^post_10 ], cost: 1
      2: l2 -> l1 : a^0'=a^post_3, b^0'=b^post_3, c^0'=c^post_3, d^0'=d^post_3, e^0'=e^post_3, sum^0'=sum^post_3, [ a^0==a^post_3 && b^0==b^post_3 && c^0==c^post_3 && d^0==d^post_3 && e^0==e^post_3 && sum^0==sum^post_3 ], cost: 1
      4: l3 -> l1 : a^0'=a^post_5, b^0'=b^post_5, c^0'=c^post_5, d^0'=d^post_5, e^0'=e^post_5, sum^0'=sum^post_5, [ a^0==a^post_5 && b^0==b^post_5 && c^0==c^post_5 && d^0==d^post_5 && e^0==e^post_5 && sum^0==sum^post_5 ], cost: 1
      6: l4 -> l1 : a^0'=a^post_7, b^0'=b^post_7, c^0'=c^post_7, d^0'=d^post_7, e^0'=e^post_7, sum^0'=sum^post_7, [ a^0==a^post_7 && b^0==b^post_7 && c^0==c^post_7 && d^0==d^post_7 && e^0==e^post_7 && sum^0==sum^post_7 ], cost: 1
      8: l5 -> l1 : a^0'=a^post_9, b^0'=b^post_9, c^0'=c^post_9, d^0'=d^post_9, e^0'=e^post_9, sum^0'=sum^post_9, [ a^0==a^post_9 && b^0==b^post_9 && c^0==c^post_9 && d^0==d^post_9 && e^0==e^post_9 && sum^0==sum^post_9 ], cost: 1
     10: l6 -> l1 : a^0'=a^post_11, b^0'=b^post_11, c^0'=c^post_11, d^0'=d^post_11, e^0'=e^post_11, sum^0'=sum^post_11, [ a^0==a^post_11 && b^0==b^post_11 && c^0==c^post_11 && d^0==d^post_11 && e^0==e^post_11 && sum^0==sum^post_11 ], cost: 1
     11: l7 -> l0 : a^0'=a^post_12, b^0'=b^post_12, c^0'=c^post_12, d^0'=d^post_12, e^0'=e^post_12, sum^0'=sum^post_12, [ a^0==a^post_12 && b^0==b^post_12 && c^0==c^post_12 && d^0==d^post_12 && e^0==e^post_12 && sum^0==sum^post_12 ], cost: 1

Checking for constant complexity:
   The following rule is satisfiable with cost >= 1, yielding constant complexity:
     11: l7 -> l0 : a^0'=a^post_12, b^0'=b^post_12, c^0'=c^post_12, d^0'=d^post_12, e^0'=e^post_12, sum^0'=sum^post_12, [ a^0==a^post_12 && b^0==b^post_12 && c^0==c^post_12 && d^0==d^post_12 && e^0==e^post_12 && sum^0==sum^post_12 ], cost: 1

Simplified all rules, resulting in:
   Start location: l7
      0: l0 -> l1 : a^0'=a^post_1, b^0'=-d^post_1+sum^post_1-a^post_1-c^post_1-e^post_1, c^0'=c^post_1, d^0'=d^post_1, e^0'=e^post_1, sum^0'=sum^post_1, [ 1<=sum^post_1 ], cost: 1
      1: l1 -> l2 : a^0'=a^0+e^0, d^0'=d^0+e^0, e^0'=-e^0, [ 1+e^0<=0 ], cost: 1
      3: l1 -> l3 : c^0'=c^0+d^0, d^0'=-d^0, e^0'=d^0+e^0, [ 1+d^0<=0 ], cost: 1
      5: l1 -> l4 : b^0'=c^0+b^0, c^0'=-c^0, d^0'=c^0+d^0, [ 1+c^0<=0 ], cost: 1
      7: l1 -> l5 : a^0'=a^0+b^0, b^0'=-b^0, c^0'=c^0+b^0, [ 1+b^0<=0 ], cost: 1
      9: l1 -> l6 : a^0'=-a^0, b^0'=a^0+b^0, e^0'=a^0+e^0, [ 1+a^0<=0 ], cost: 1
      2: l2 -> l1 : [], cost: 1
      4: l3 -> l1 : [], cost: 1
      6: l4 -> l1 : [], cost: 1
      8: l5 -> l1 : [], cost: 1
     10: l6 -> l1 : [], cost: 1
     11: l7 -> l0 : [], cost: 1

### Simplification by acceleration and chaining ###

Eliminated locations (on linear paths):
   Start location: l7
     13: l1 -> l1 : a^0'=a^0+e^0, d^0'=d^0+e^0, e^0'=-e^0, [ 1+e^0<=0 ], cost: 2
     14: l1 -> l1 : c^0'=c^0+d^0, d^0'=-d^0, e^0'=d^0+e^0, [ 1+d^0<=0 ], cost: 2
     15: l1 -> l1 : b^0'=c^0+b^0, c^0'=-c^0, d^0'=c^0+d^0, [ 1+c^0<=0 ], cost: 2
     16: l1 -> l1 : a^0'=a^0+b^0, b^0'=-b^0, c^0'=c^0+b^0, [ 1+b^0<=0 ], cost: 2
     17: l1 -> l1 : a^0'=-a^0, b^0'=a^0+b^0, e^0'=a^0+e^0, [ 1+a^0<=0 ], cost: 2
     12: l7 -> l1 : a^0'=a^post_1, b^0'=-d^post_1+sum^post_1-a^post_1-c^post_1-e^post_1, c^0'=c^post_1, d^0'=d^post_1, e^0'=e^post_1, sum^0'=sum^post_1, [ 1<=sum^post_1 ], cost: 2

Accelerating simple loops of location 1.
   Accelerating the following rules:
     13: l1 -> l1 : a^0'=a^0+e^0, d^0'=d^0+e^0, e^0'=-e^0, [ 1+e^0<=0 ], cost: 2
     14: l1 -> l1 : c^0'=c^0+d^0, d^0'=-d^0, e^0'=d^0+e^0, [ 1+d^0<=0 ], cost: 2
     15: l1 -> l1 : b^0'=c^0+b^0, c^0'=-c^0, d^0'=c^0+d^0, [ 1+c^0<=0 ], cost: 2
     16: l1 -> l1 : a^0'=a^0+b^0, b^0'=-b^0, c^0'=c^0+b^0, [ 1+b^0<=0 ], cost: 2
     17: l1 -> l1 : a^0'=-a^0, b^0'=a^0+b^0, e^0'=a^0+e^0, [ 1+a^0<=0 ], cost: 2

   Found no metering function for rule 13.
   Found no metering function for rule 14.
   Found no metering function for rule 15.
   Found no metering function for rule 16.
   Found no metering function for rule 17.
   Removing the simple loops:.

Accelerated all simple loops using metering functions (where possible):
   Start location: l7
     13: l1 -> l1 : a^0'=a^0+e^0, d^0'=d^0+e^0, e^0'=-e^0, [ 1+e^0<=0 ], cost: 2
     14: l1 -> l1 : c^0'=c^0+d^0, d^0'=-d^0, e^0'=d^0+e^0, [ 1+d^0<=0 ], cost: 2
     15: l1 -> l1 : b^0'=c^0+b^0, c^0'=-c^0, d^0'=c^0+d^0, [ 1+c^0<=0 ], cost: 2
     16: l1 -> l1 : a^0'=a^0+b^0, b^0'=-b^0, c^0'=c^0+b^0, [ 1+b^0<=0 ], cost: 2
     17: l1 -> l1 : a^0'=-a^0, b^0'=a^0+b^0, e^0'=a^0+e^0, [ 1+a^0<=0 ], cost: 2
     12: l7 -> l1 : a^0'=a^post_1, b^0'=-d^post_1+sum^post_1-a^post_1-c^post_1-e^post_1, c^0'=c^post_1, d^0'=d^post_1, e^0'=e^post_1, sum^0'=sum^post_1, [ 1<=sum^post_1 ], cost: 2

Chained accelerated rules (with incoming rules):
   Start location: l7
     12: l7 -> l1 : a^0'=a^post_1, b^0'=-d^post_1+sum^post_1-a^post_1-c^post_1-e^post_1, c^0'=c^post_1, d^0'=d^post_1, e^0'=e^post_1, sum^0'=sum^post_1, [ 1<=sum^post_1 ], cost: 2
     18: l7 -> l1 : a^0'=a^post_1+e^post_1, b^0'=-d^post_1+sum^post_1-a^post_1-c^post_1-e^post_1, c^0'=c^post_1, d^0'=d^post_1+e^post_1, e^0'=-e^post_1, sum^0'=sum^post_1, [ 1<=sum^post_1 && 1+e^post_1<=0 ], cost: 4
     19: l7 -> l1 : a^0'=a^post_1, b^0'=-d^post_1+sum^post_1-a^post_1-c^post_1-e^post_1, c^0'=d^post_1+c^post_1, d^0'=-d^post_1, e^0'=d^post_1+e^post_1, sum^0'=sum^post_1, [ 1<=sum^post_1 && 1+d^post_1<=0 ], cost: 4
     20: l7 -> l1 : a^0'=a^post_1, b^0'=-d^post_1+sum^post_1-a^post_1-e^post_1, c^0'=-c^post_1, d^0'=d^post_1+c^post_1, e^0'=e^post_1, sum^0'=sum^post_1, [ 1<=sum^post_1 && 1+c^post_1<=0 ], cost: 4
     21: l7 -> l1 : a^0'=-d^post_1+sum^post_1-c^post_1-e^post_1, b^0'=d^post_1-sum^post_1+a^post_1+c^post_1+e^post_1, c^0'=-d^post_1+sum^post_1-a^post_1-e^post_1, d^0'=d^post_1, e^0'=e^post_1, sum^0'=sum^post_1, [ 1<=sum^post_1 && 1-d^post_1+sum^post_1-a^post_1-c^post_1-e^post_1<=0 ], cost: 4
     22: l7 -> l1 : a^0'=-a^post_1, b^0'=-d^post_1+sum^post_1-c^post_1-e^post_1, c^0'=c^post_1, d^0'=d^post_1, e^0'=a^post_1+e^post_1, sum^0'=sum^post_1, [ 1<=sum^post_1 && 1+a^post_1<=0 ], cost: 4

Removed unreachable locations (and leaf rules with constant cost):
   Start location: l7
     <empty>

### Computing asymptotic complexity ###

Fully simplified ITS problem
   Start location: l7
     <empty>

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:  [ a^0==a^post_12 && b^0==b^post_12 && c^0==c^post_12 && d^0==d^post_12 && e^0==e^post_12 && sum^0==sum^post_12 ]

WORST_CASE(Omega(1),?)