/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, INF).

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 889 ms]
(2) BOUNDS(n^1, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
evalcyclicstart(A, B, C) -> Com_1(evalcyclicentryin(A, B, C)) :|: TRUE
evalcyclicentryin(A, B, C) -> Com_1(evalcyclicbb3in(A, B, A + 1)) :|: A >= 0 && B >= A + 1
evalcyclicbb3in(A, B, C) -> Com_1(evalcyclicreturnin(A, B, C)) :|: C >= A && C <= A
evalcyclicbb3in(A, B, C) -> Com_1(evalcyclicbb4in(A, B, C)) :|: A >= C + 1
evalcyclicbb3in(A, B, C) -> Com_1(evalcyclicbb4in(A, B, C)) :|: C >= A + 1
evalcyclicbb4in(A, B, C) -> Com_1(evalcyclicbbin(A, B, C)) :|: 0 >= D + 1
evalcyclicbb4in(A, B, C) -> Com_1(evalcyclicbbin(A, B, C)) :|: D >= 1
evalcyclicbb4in(A, B, C) -> Com_1(evalcyclicreturnin(A, B, C)) :|: TRUE
evalcyclicbbin(A, B, C) -> Com_1(evalcyclicbb3in(A, B, C + 1)) :|: B >= C
evalcyclicbbin(A, B, C) -> Com_1(evalcyclicbb3in(A, B, 0)) :|: C >= B + 1
evalcyclicreturnin(A, B, C) -> Com_1(evalcyclicstop(A, B, C)) :|: TRUE

The start-symbols are:[evalcyclicstart_3]


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(1) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: evalcyclicstart

      0: evalcyclicstart -> evalcyclicentryin : [], cost: 1

      1: evalcyclicentryin -> evalcyclicbb3in : C'=1+A, [ A>=0 && B>=1+A ], cost: 1

      2: evalcyclicbb3in -> evalcyclicreturnin : [ C==A ], cost: 1

      3: evalcyclicbb3in -> evalcyclicbb4in : [ A>=1+C ], cost: 1

      4: evalcyclicbb3in -> evalcyclicbb4in : [ C>=1+A ], cost: 1

      5: evalcyclicbb4in -> evalcyclicbbin : [ 0>=1+free ], cost: 1

      6: evalcyclicbb4in -> evalcyclicbbin : [ free_1>=1 ], cost: 1

      7: evalcyclicbb4in -> evalcyclicreturnin : [], cost: 1

      8: evalcyclicbbin -> evalcyclicbb3in : C'=1+C, [ B>=C ], cost: 1

      9: evalcyclicbbin -> evalcyclicbb3in : C'=0, [ C>=1+B ], cost: 1

     10: evalcyclicreturnin -> evalcyclicstop : [], cost: 1



Checking for constant complexity:

   The following rule is satisfiable with cost >= 1, yielding constant complexity:

      0: evalcyclicstart -> evalcyclicentryin : [], cost: 1



Removed unreachable and leaf rules:

   Start location: evalcyclicstart

      0: evalcyclicstart -> evalcyclicentryin : [], cost: 1

      1: evalcyclicentryin -> evalcyclicbb3in : C'=1+A, [ A>=0 && B>=1+A ], cost: 1

      3: evalcyclicbb3in -> evalcyclicbb4in : [ A>=1+C ], cost: 1

      4: evalcyclicbb3in -> evalcyclicbb4in : [ C>=1+A ], cost: 1

      5: evalcyclicbb4in -> evalcyclicbbin : [ 0>=1+free ], cost: 1

      6: evalcyclicbb4in -> evalcyclicbbin : [ free_1>=1 ], cost: 1

      8: evalcyclicbbin -> evalcyclicbb3in : C'=1+C, [ B>=C ], cost: 1

      9: evalcyclicbbin -> evalcyclicbb3in : C'=0, [ C>=1+B ], cost: 1



Simplified all rules, resulting in:

   Start location: evalcyclicstart

      0: evalcyclicstart -> evalcyclicentryin : [], cost: 1

      1: evalcyclicentryin -> evalcyclicbb3in : C'=1+A, [ A>=0 && B>=1+A ], cost: 1

      3: evalcyclicbb3in -> evalcyclicbb4in : [ A>=1+C ], cost: 1

      4: evalcyclicbb3in -> evalcyclicbb4in : [ C>=1+A ], cost: 1

      6: evalcyclicbb4in -> evalcyclicbbin : [], cost: 1

      8: evalcyclicbbin -> evalcyclicbb3in : C'=1+C, [ B>=C ], cost: 1

      9: evalcyclicbbin -> evalcyclicbb3in : C'=0, [ C>=1+B ], cost: 1



### Simplification by acceleration and chaining ###



Eliminated locations (on linear paths):

   Start location: evalcyclicstart

     11: evalcyclicstart -> evalcyclicbb3in : C'=1+A, [ A>=0 && B>=1+A ], cost: 2

      3: evalcyclicbb3in -> evalcyclicbb4in : [ A>=1+C ], cost: 1

      4: evalcyclicbb3in -> evalcyclicbb4in : [ C>=1+A ], cost: 1

      6: evalcyclicbb4in -> evalcyclicbbin : [], cost: 1

      8: evalcyclicbbin -> evalcyclicbb3in : C'=1+C, [ B>=C ], cost: 1

      9: evalcyclicbbin -> evalcyclicbb3in : C'=0, [ C>=1+B ], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: evalcyclicstart

     11: evalcyclicstart -> evalcyclicbb3in : C'=1+A, [ A>=0 && B>=1+A ], cost: 2

     12: evalcyclicbb3in -> evalcyclicbbin : [ A>=1+C ], cost: 2

     13: evalcyclicbb3in -> evalcyclicbbin : [ C>=1+A ], cost: 2

      8: evalcyclicbbin -> evalcyclicbb3in : C'=1+C, [ B>=C ], cost: 1

      9: evalcyclicbbin -> evalcyclicbb3in : C'=0, [ C>=1+B ], cost: 1



Eliminated locations (on tree-shaped paths):

   Start location: evalcyclicstart

     11: evalcyclicstart -> evalcyclicbb3in : C'=1+A, [ A>=0 && B>=1+A ], cost: 2

     14: evalcyclicbb3in -> evalcyclicbb3in : C'=1+C, [ A>=1+C && B>=C ], cost: 3

     15: evalcyclicbb3in -> evalcyclicbb3in : C'=0, [ A>=1+C && C>=1+B ], cost: 3

     16: evalcyclicbb3in -> evalcyclicbb3in : C'=1+C, [ C>=1+A && B>=C ], cost: 3

     17: evalcyclicbb3in -> evalcyclicbb3in : C'=0, [ C>=1+A && C>=1+B ], cost: 3



Accelerating simple loops of location 2.

   Accelerating the following rules:

     14: evalcyclicbb3in -> evalcyclicbb3in : C'=1+C, [ A>=1+C && B>=C ], cost: 3

     15: evalcyclicbb3in -> evalcyclicbb3in : C'=0, [ A>=1+C && C>=1+B ], cost: 3

     16: evalcyclicbb3in -> evalcyclicbb3in : C'=1+C, [ C>=1+A && B>=C ], cost: 3

     17: evalcyclicbb3in -> evalcyclicbb3in : C'=0, [ C>=1+A && C>=1+B ], cost: 3



   Found no metering function for rule 14.

   Accelerated rule 15 with NONTERM (after strengthening guard), yielding the new rule 18.

   Accelerated rule 16 with metering function 1-C+B, yielding the new rule 19.

   Accelerated rule 17 with NONTERM (after strengthening guard), yielding the new rule 20.

   Nested simple loops 17 (outer loop) and 19 (inner loop) with NONTERM, resulting in the new rules: 21, 22.

   Removing the simple loops: 16 17.



Accelerated all simple loops using metering functions (where possible):

   Start location: evalcyclicstart

     11: evalcyclicstart -> evalcyclicbb3in : C'=1+A, [ A>=0 && B>=1+A ], cost: 2

     14: evalcyclicbb3in -> evalcyclicbb3in : C'=1+C, [ A>=1+C && B>=C ], cost: 3

     15: evalcyclicbb3in -> evalcyclicbb3in : C'=0, [ A>=1+C && C>=1+B ], cost: 3

     18: evalcyclicbb3in -> [7] : [ A>=1+C && C>=1+B && A>=1 && 0>=1+B ], cost: NONTERM

     19: evalcyclicbb3in -> evalcyclicbb3in : C'=1+B, [ C>=1+A && B>=C ], cost: 3-3*C+3*B

     20: evalcyclicbb3in -> [7] : [ C>=1+A && C>=1+B && 0>=1+A && 0>=1+B ], cost: NONTERM

     21: evalcyclicbb3in -> [7] : [ C>=1+A && C>=1+B && 0>=1+A && B>=0 && 6+3*B>=1 ], cost: NONTERM

     22: evalcyclicbb3in -> [7] : C'=1+B, [ C>=1+A && B>=C && 1+B>=1+A && 0>=1+A && B>=0 && 6+3*B>=1 ], cost: NONTERM



Chained accelerated rules (with incoming rules):

   Start location: evalcyclicstart

     11: evalcyclicstart -> evalcyclicbb3in : C'=1+A, [ A>=0 && B>=1+A ], cost: 2

     23: evalcyclicstart -> evalcyclicbb3in : C'=1+B, [ A>=0 && B>=1+A ], cost: 2-3*A+3*B



Removed unreachable locations (and leaf rules with constant cost):

   Start location: evalcyclicstart

     23: evalcyclicstart -> evalcyclicbb3in : C'=1+B, [ A>=0 && B>=1+A ], cost: 2-3*A+3*B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: evalcyclicstart

     23: evalcyclicstart -> evalcyclicbb3in : C'=1+B, [ A>=0 && B>=1+A ], cost: 2-3*A+3*B



Computing asymptotic complexity for rule 23

   Solved the limit problem by the following transformations:

      Created initial limit problem:

      2-3*A+3*B (+), -A+B (+/+!), 1+A (+/+!) [not solved]



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



      applying transformation rule (C) using substitution {A==0,B==n}

      resulting limit problem:

      [solved]



   Solution:

      A / 0

      B / n

   Resulting cost 2+3*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+3*n

   Rule cost:   2-3*A+3*B

   Rule guard:  [ A>=0 && B>=1+A ]



WORST_CASE(Omega(n^1),?)


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(2)
BOUNDS(n^1, INF)