WORST_CASE(Omega(n^1),?)

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

Initial linear ITS problem
   Start location: l3
      0: l0 -> l1 : id^0'=id^post_1, m^0'=m^post_1, x^0'=x^post_1, [ x^0<=m^0 && x^post_1==1+x^0 && x^post_1<=1+m^0 && 0<=x^post_1 && id^0==id^post_1 && m^0==m^post_1 ], cost: 1
      1: l0 -> l1 : id^0'=id^post_2, m^0'=m^post_2, x^0'=x^post_2, [ 1+m^0<=x^0 && 1<=x^0 && x^post_2==0 && x^post_2<=1+m^0 && 0<=x^post_2 && id^0==id^post_2 && m^0==m^post_2 ], cost: 1
      2: l1 -> l0 : id^0'=id^post_3, m^0'=m^post_3, x^0'=x^post_3, [ 1+id^0<=x^0 && id^0==id^post_3 && m^0==m^post_3 && x^0==x^post_3 ], cost: 1
      3: l1 -> l0 : id^0'=id^post_4, m^0'=m^post_4, x^0'=x^post_4, [ 1+x^0<=id^0 && id^0==id^post_4 && m^0==m^post_4 && x^0==x^post_4 ], cost: 1
      4: l2 -> l1 : id^0'=id^post_5, m^0'=m^post_5, x^0'=x^post_5, [ id^0<=m^0 && 1<=id^0 && 1<=m^0 && x^post_5==1+id^0 && x^post_5<=1+m^0 && 0<=x^post_5 && id^0==id^post_5 && m^0==m^post_5 ], cost: 1
      5: l3 -> l2 : id^0'=id^post_6, m^0'=m^post_6, x^0'=x^post_6, [ id^0==id^post_6 && m^0==m^post_6 && x^0==x^post_6 ], cost: 1

Checking for constant complexity:
   The following rule is satisfiable with cost >= 1, yielding constant complexity:
      5: l3 -> l2 : id^0'=id^post_6, m^0'=m^post_6, x^0'=x^post_6, [ id^0==id^post_6 && m^0==m^post_6 && x^0==x^post_6 ], cost: 1

Simplified all rules, resulting in:
   Start location: l3
      0: l0 -> l1 : x^0'=1+x^0, [ x^0<=m^0 && 0<=1+x^0 ], cost: 1
      1: l0 -> l1 : x^0'=0, [ 1+m^0<=x^0 && 1<=x^0 && 0<=1+m^0 ], cost: 1
      2: l1 -> l0 : [ 1+id^0<=x^0 ], cost: 1
      3: l1 -> l0 : [ 1+x^0<=id^0 ], cost: 1
      4: l2 -> l1 : x^0'=1+id^0, [ id^0<=m^0 && 1<=id^0 ], cost: 1
      5: l3 -> l2 : [], cost: 1

### Simplification by acceleration and chaining ###

Eliminated locations (on linear paths):
   Start location: l3
      0: l0 -> l1 : x^0'=1+x^0, [ x^0<=m^0 && 0<=1+x^0 ], cost: 1
      1: l0 -> l1 : x^0'=0, [ 1+m^0<=x^0 && 1<=x^0 && 0<=1+m^0 ], cost: 1
      2: l1 -> l0 : [ 1+id^0<=x^0 ], cost: 1
      3: l1 -> l0 : [ 1+x^0<=id^0 ], cost: 1
      6: l3 -> l1 : x^0'=1+id^0, [ id^0<=m^0 && 1<=id^0 ], cost: 2

Eliminated locations (on tree-shaped paths):
   Start location: l3
      7: l1 -> l1 : x^0'=1+x^0, [ 1+id^0<=x^0 && x^0<=m^0 && 0<=1+x^0 ], cost: 2
      8: l1 -> l1 : x^0'=0, [ 1+id^0<=x^0 && 1+m^0<=x^0 && 1<=x^0 && 0<=1+m^0 ], cost: 2
      9: l1 -> l1 : x^0'=1+x^0, [ 1+x^0<=id^0 && x^0<=m^0 && 0<=1+x^0 ], cost: 2
     10: l1 -> l1 : x^0'=0, [ 1+x^0<=id^0 && 1+m^0<=x^0 && 1<=x^0 && 0<=1+m^0 ], cost: 2
      6: l3 -> l1 : x^0'=1+id^0, [ id^0<=m^0 && 1<=id^0 ], cost: 2

Accelerating simple loops of location 1.
   Accelerating the following rules:
      7: l1 -> l1 : x^0'=1+x^0, [ 1+id^0<=x^0 && x^0<=m^0 && 0<=1+x^0 ], cost: 2
      8: l1 -> l1 : x^0'=0, [ 1+id^0<=x^0 && 1+m^0<=x^0 && 1<=x^0 && 0<=1+m^0 ], cost: 2
      9: l1 -> l1 : x^0'=1+x^0, [ 1+x^0<=id^0 && x^0<=m^0 && 0<=1+x^0 ], cost: 2
     10: l1 -> l1 : x^0'=0, [ 1+x^0<=id^0 && 1+m^0<=x^0 && 1<=x^0 && 0<=1+m^0 ], cost: 2

   Accelerated rule 7 with metering function 1-x^0+m^0, yielding the new rule 11.
   Found no metering function for rule 8.
   Found no metering function for rule 9.
   Found no metering function for rule 10.
   Nested simple loops 8 (outer loop) and 11 (inner loop) with NONTERM, resulting in the new rules: 12, 13.
   Removing the simple loops: 7 8.

Accelerated all simple loops using metering functions (where possible):
   Start location: l3
      9: l1 -> l1 : x^0'=1+x^0, [ 1+x^0<=id^0 && x^0<=m^0 && 0<=1+x^0 ], cost: 2
     10: l1 -> l1 : x^0'=0, [ 1+x^0<=id^0 && 1+m^0<=x^0 && 1<=x^0 && 0<=1+m^0 ], cost: 2
     11: l1 -> l1 : x^0'=1+m^0, [ 1+id^0<=x^0 && x^0<=m^0 && 0<=1+x^0 ], cost: 2-2*x^0+2*m^0
     12: l1 -> [4] : [ 1+id^0<=x^0 && 1+m^0<=x^0 && 1<=x^0 && 1+id^0<=0 && 0<=m^0 && 4+2*m^0>=1 ], cost: NONTERM
     13: l1 -> [4] : x^0'=1+m^0, [ 1+id^0<=x^0 && x^0<=m^0 && 0<=1+x^0 && 1+id^0<=1+m^0 && 1<=1+m^0 && 1+id^0<=0 && 4+2*m^0>=1 ], cost: NONTERM
      6: l3 -> l1 : x^0'=1+id^0, [ id^0<=m^0 && 1<=id^0 ], cost: 2

Chained accelerated rules (with incoming rules):
   Start location: l3
      6: l3 -> l1 : x^0'=1+id^0, [ id^0<=m^0 && 1<=id^0 ], cost: 2
     14: l3 -> l1 : x^0'=1+m^0, [ 1<=id^0 && 1+id^0<=m^0 ], cost: 2-2*id^0+2*m^0

Removed unreachable locations (and leaf rules with constant cost):
   Start location: l3
     14: l3 -> l1 : x^0'=1+m^0, [ 1<=id^0 && 1+id^0<=m^0 ], cost: 2-2*id^0+2*m^0

### Computing asymptotic complexity ###

Fully simplified ITS problem
   Start location: l3
     14: l3 -> l1 : x^0'=1+m^0, [ 1<=id^0 && 1+id^0<=m^0 ], cost: 2-2*id^0+2*m^0

Computing asymptotic complexity for rule 14
   Solved the limit problem by the following transformations:
      Created initial limit problem:
      2-2*id^0+2*m^0 (+), id^0 (+/+!), -id^0+m^0 (+/+!) [not solved]

      removing all constraints (solved by SMT)
      resulting limit problem: [solved]

      applying transformation rule (C) using substitution {id^0==1,m^0==n}
      resulting limit problem:
      [solved]

   Solution:
      id^0 / 1
      m^0 / n
   Resulting cost 2*n has complexity: Poly(n^1)

Found new complexity Poly(n^1).

Obtained the following overall complexity (w.r.t. the length of the input n):
   Complexity:  Poly(n^1)
   Cpx degree:  1
   Solved cost: 2*n
   Rule cost:   2-2*id^0+2*m^0
   Rule guard:  [ 1<=id^0 && 1+id^0<=m^0 ]

WORST_CASE(Omega(n^1),?)