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


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, max(6 + 2 * Arg_0, 6) + max(1, 1 + 2 * Arg_0) + nat(Arg_0)).

(0) CpxIntTrs
(1) Koat2 Proof [FINISHED, 822 ms]
(2) BOUNDS(1, max(6 + 2 * Arg_0, 6) + max(1, 1 + 2 * Arg_0) + nat(Arg_0))
(3) Loat Proof [FINISHED, 710 ms]
(4) BOUNDS(n^1, INF)


----------------------------------------

(0)
Obligation:
Complexity Int TRS consisting of the following rules:
evalspeedpldi2start(A, B, C) -> Com_1(evalspeedpldi2entryin(A, B, C)) :|: TRUE
evalspeedpldi2entryin(A, B, C) -> Com_1(evalspeedpldi2bb5in(B, 0, A)) :|: A >= 0 && B >= 1
evalspeedpldi2entryin(A, B, C) -> Com_1(evalspeedpldi2returnin(A, B, C)) :|: 0 >= A + 1
evalspeedpldi2entryin(A, B, C) -> Com_1(evalspeedpldi2returnin(A, B, C)) :|: 0 >= B
evalspeedpldi2bb5in(A, B, C) -> Com_1(evalspeedpldi2bb2in(A, B, C)) :|: C >= 1
evalspeedpldi2bb5in(A, B, C) -> Com_1(evalspeedpldi2returnin(A, B, C)) :|: 0 >= C
evalspeedpldi2bb2in(A, B, C) -> Com_1(evalspeedpldi2bb3in(A, B, C)) :|: A >= B + 1
evalspeedpldi2bb2in(A, B, C) -> Com_1(evalspeedpldi2bb5in(A, 0, C)) :|: B >= A
evalspeedpldi2bb3in(A, B, C) -> Com_1(evalspeedpldi2bb5in(A, B + 1, C - 1)) :|: TRUE
evalspeedpldi2returnin(A, B, C) -> Com_1(evalspeedpldi2stop(A, B, C)) :|: TRUE

The start-symbols are:[evalspeedpldi2start_3]


----------------------------------------

(1) Koat2 Proof (FINISHED)
YES( ?, 6+2*max([0, Arg_0])+max([1, 1+2*Arg_0])+max([0, Arg_0]) {O(n)})



Initial Complexity Problem:

  Start:  evalspeedpldi2start  

  Program_Vars:  Arg_0, Arg_1, Arg_2  

  Temp_Vars:    

  Locations:  evalspeedpldi2bb2in, evalspeedpldi2bb3in, evalspeedpldi2bb5in, evalspeedpldi2entryin, evalspeedpldi2returnin, evalspeedpldi2start, evalspeedpldi2stop  

  Transitions:

  evalspeedpldi2bb2in(Arg_0,Arg_1,Arg_2) -> evalspeedpldi2bb3in(Arg_0,Arg_1,Arg_2):|:1 <= Arg_2 && 1 <= Arg_1+Arg_2 && 2 <= Arg_0+Arg_2 && 0 <= Arg_1 && 1 <= Arg_0+Arg_1 && 1 <= Arg_0 && Arg_1+1 <= Arg_0

  evalspeedpldi2bb2in(Arg_0,Arg_1,Arg_2) -> evalspeedpldi2bb5in(Arg_0,0,Arg_2):|:1 <= Arg_2 && 1 <= Arg_1+Arg_2 && 2 <= Arg_0+Arg_2 && 0 <= Arg_1 && 1 <= Arg_0+Arg_1 && 1 <= Arg_0 && Arg_0 <= Arg_1

  evalspeedpldi2bb3in(Arg_0,Arg_1,Arg_2) -> evalspeedpldi2bb5in(Arg_0,Arg_1+1,Arg_2-1):|:1 <= Arg_2 && 1 <= Arg_1+Arg_2 && 2 <= Arg_0+Arg_2 && 1+Arg_1 <= Arg_0 && 0 <= Arg_1 && 1 <= Arg_0+Arg_1 && 1 <= Arg_0

  evalspeedpldi2bb5in(Arg_0,Arg_1,Arg_2) -> evalspeedpldi2bb2in(Arg_0,Arg_1,Arg_2):|:0 <= Arg_2 && 0 <= Arg_1+Arg_2 && 1 <= Arg_0+Arg_2 && 0 <= Arg_1 && 1 <= Arg_0+Arg_1 && 1 <= Arg_0 && 1 <= Arg_2

  evalspeedpldi2bb5in(Arg_0,Arg_1,Arg_2) -> evalspeedpldi2returnin(Arg_0,Arg_1,Arg_2):|:0 <= Arg_2 && 0 <= Arg_1+Arg_2 && 1 <= Arg_0+Arg_2 && 0 <= Arg_1 && 1 <= Arg_0+Arg_1 && 1 <= Arg_0 && Arg_2 <= 0

  evalspeedpldi2entryin(Arg_0,Arg_1,Arg_2) -> evalspeedpldi2bb5in(Arg_1,0,Arg_0):|:0 <= Arg_0 && 1 <= Arg_1

  evalspeedpldi2entryin(Arg_0,Arg_1,Arg_2) -> evalspeedpldi2returnin(Arg_0,Arg_1,Arg_2):|:Arg_0+1 <= 0

  evalspeedpldi2entryin(Arg_0,Arg_1,Arg_2) -> evalspeedpldi2returnin(Arg_0,Arg_1,Arg_2):|:Arg_1 <= 0

  evalspeedpldi2returnin(Arg_0,Arg_1,Arg_2) -> evalspeedpldi2stop(Arg_0,Arg_1,Arg_2):|:

  evalspeedpldi2start(Arg_0,Arg_1,Arg_2) -> evalspeedpldi2entryin(Arg_0,Arg_1,Arg_2):|:  



Timebounds: 

  Overall timebound: 6+2*max([0, Arg_0])+max([1, 1+2*Arg_0])+max([0, Arg_0]) {O(n)}

  6: evalspeedpldi2bb2in->evalspeedpldi2bb3in: max([0, Arg_0]) {O(n)}

  7: evalspeedpldi2bb2in->evalspeedpldi2bb5in: max([0, Arg_0]) {O(n)}

  8: evalspeedpldi2bb3in->evalspeedpldi2bb5in: max([0, Arg_0]) {O(n)}

  4: evalspeedpldi2bb5in->evalspeedpldi2bb2in: max([1, 1+2*Arg_0]) {O(n)}

  5: evalspeedpldi2bb5in->evalspeedpldi2returnin: 1 {O(1)}

  1: evalspeedpldi2entryin->evalspeedpldi2bb5in: 1 {O(1)}

  2: evalspeedpldi2entryin->evalspeedpldi2returnin: 1 {O(1)}

  3: evalspeedpldi2entryin->evalspeedpldi2returnin: 1 {O(1)}

  9: evalspeedpldi2returnin->evalspeedpldi2stop: 1 {O(1)}

  0: evalspeedpldi2start->evalspeedpldi2entryin: 1 {O(1)}



Costbounds:

  Overall costbound: 6+2*max([0, Arg_0])+max([1, 1+2*Arg_0])+max([0, Arg_0]) {O(n)}

  6: evalspeedpldi2bb2in->evalspeedpldi2bb3in: max([0, Arg_0]) {O(n)}

  7: evalspeedpldi2bb2in->evalspeedpldi2bb5in: max([0, Arg_0]) {O(n)}

  8: evalspeedpldi2bb3in->evalspeedpldi2bb5in: max([0, Arg_0]) {O(n)}

  4: evalspeedpldi2bb5in->evalspeedpldi2bb2in: max([1, 1+2*Arg_0]) {O(n)}

  5: evalspeedpldi2bb5in->evalspeedpldi2returnin: 1 {O(1)}

  1: evalspeedpldi2entryin->evalspeedpldi2bb5in: 1 {O(1)}

  2: evalspeedpldi2entryin->evalspeedpldi2returnin: 1 {O(1)}

  3: evalspeedpldi2entryin->evalspeedpldi2returnin: 1 {O(1)}

  9: evalspeedpldi2returnin->evalspeedpldi2stop: 1 {O(1)}

  0: evalspeedpldi2start->evalspeedpldi2entryin: 1 {O(1)}



Sizebounds:

`Lower:

  6: evalspeedpldi2bb2in->evalspeedpldi2bb3in, Arg_0: 1 {O(1)}

  6: evalspeedpldi2bb2in->evalspeedpldi2bb3in, Arg_1: 0 {O(1)}

  6: evalspeedpldi2bb2in->evalspeedpldi2bb3in, Arg_2: 1 {O(1)}

  7: evalspeedpldi2bb2in->evalspeedpldi2bb5in, Arg_0: 1 {O(1)}

  7: evalspeedpldi2bb2in->evalspeedpldi2bb5in, Arg_1: 0 {O(1)}

  7: evalspeedpldi2bb2in->evalspeedpldi2bb5in, Arg_2: 1 {O(1)}

  8: evalspeedpldi2bb3in->evalspeedpldi2bb5in, Arg_0: 1 {O(1)}

  8: evalspeedpldi2bb3in->evalspeedpldi2bb5in, Arg_1: 1 {O(1)}

  8: evalspeedpldi2bb3in->evalspeedpldi2bb5in, Arg_2: 0 {O(1)}

  4: evalspeedpldi2bb5in->evalspeedpldi2bb2in, Arg_0: 1 {O(1)}

  4: evalspeedpldi2bb5in->evalspeedpldi2bb2in, Arg_1: 0 {O(1)}

  4: evalspeedpldi2bb5in->evalspeedpldi2bb2in, Arg_2: 1 {O(1)}

  5: evalspeedpldi2bb5in->evalspeedpldi2returnin, Arg_0: 1 {O(1)}

  5: evalspeedpldi2bb5in->evalspeedpldi2returnin, Arg_1: 0 {O(1)}

  5: evalspeedpldi2bb5in->evalspeedpldi2returnin, Arg_2: 0 {O(1)}

  1: evalspeedpldi2entryin->evalspeedpldi2bb5in, Arg_0: 1 {O(1)}

  1: evalspeedpldi2entryin->evalspeedpldi2bb5in, Arg_1: 0 {O(1)}

  1: evalspeedpldi2entryin->evalspeedpldi2bb5in, Arg_2: 0 {O(1)}

  2: evalspeedpldi2entryin->evalspeedpldi2returnin, Arg_0: Arg_0 {O(n)}

  2: evalspeedpldi2entryin->evalspeedpldi2returnin, Arg_1: Arg_1 {O(n)}

  2: evalspeedpldi2entryin->evalspeedpldi2returnin, Arg_2: Arg_2 {O(n)}

  3: evalspeedpldi2entryin->evalspeedpldi2returnin, Arg_0: Arg_0 {O(n)}

  3: evalspeedpldi2entryin->evalspeedpldi2returnin, Arg_1: Arg_1 {O(n)}

  3: evalspeedpldi2entryin->evalspeedpldi2returnin, Arg_2: Arg_2 {O(n)}

  9: evalspeedpldi2returnin->evalspeedpldi2stop, Arg_0: min([1, Arg_0]) {O(n)}

  9: evalspeedpldi2returnin->evalspeedpldi2stop, Arg_1: min([0, Arg_1]) {O(n)}

  9: evalspeedpldi2returnin->evalspeedpldi2stop, Arg_2: min([0, Arg_2]) {O(n)}

  0: evalspeedpldi2start->evalspeedpldi2entryin, Arg_0: Arg_0 {O(n)}

  0: evalspeedpldi2start->evalspeedpldi2entryin, Arg_1: Arg_1 {O(n)}

  0: evalspeedpldi2start->evalspeedpldi2entryin, Arg_2: Arg_2 {O(n)}

`Upper:

  6: evalspeedpldi2bb2in->evalspeedpldi2bb3in, Arg_0: Arg_1 {O(n)}

  6: evalspeedpldi2bb2in->evalspeedpldi2bb3in, Arg_1: max([0, Arg_0]) {O(n)}

  6: evalspeedpldi2bb2in->evalspeedpldi2bb3in, Arg_2: Arg_0 {O(n)}

  7: evalspeedpldi2bb2in->evalspeedpldi2bb5in, Arg_0: Arg_1 {O(n)}

  7: evalspeedpldi2bb2in->evalspeedpldi2bb5in, Arg_1: 0 {O(1)}

  7: evalspeedpldi2bb2in->evalspeedpldi2bb5in, Arg_2: Arg_0 {O(n)}

  8: evalspeedpldi2bb3in->evalspeedpldi2bb5in, Arg_0: Arg_1 {O(n)}

  8: evalspeedpldi2bb3in->evalspeedpldi2bb5in, Arg_1: max([0, Arg_0]) {O(n)}

  8: evalspeedpldi2bb3in->evalspeedpldi2bb5in, Arg_2: Arg_0 {O(n)}

  4: evalspeedpldi2bb5in->evalspeedpldi2bb2in, Arg_0: Arg_1 {O(n)}

  4: evalspeedpldi2bb5in->evalspeedpldi2bb2in, Arg_1: max([0, Arg_0]) {O(n)}

  4: evalspeedpldi2bb5in->evalspeedpldi2bb2in, Arg_2: Arg_0 {O(n)}

  5: evalspeedpldi2bb5in->evalspeedpldi2returnin, Arg_0: Arg_1 {O(n)}

  5: evalspeedpldi2bb5in->evalspeedpldi2returnin, Arg_1: max([0, Arg_0]) {O(n)}

  5: evalspeedpldi2bb5in->evalspeedpldi2returnin, Arg_2: 0 {O(1)}

  1: evalspeedpldi2entryin->evalspeedpldi2bb5in, Arg_0: Arg_1 {O(n)}

  1: evalspeedpldi2entryin->evalspeedpldi2bb5in, Arg_1: 0 {O(1)}

  1: evalspeedpldi2entryin->evalspeedpldi2bb5in, Arg_2: Arg_0 {O(n)}

  2: evalspeedpldi2entryin->evalspeedpldi2returnin, Arg_0: -1 {O(1)}

  2: evalspeedpldi2entryin->evalspeedpldi2returnin, Arg_1: Arg_1 {O(n)}

  2: evalspeedpldi2entryin->evalspeedpldi2returnin, Arg_2: Arg_2 {O(n)}

  3: evalspeedpldi2entryin->evalspeedpldi2returnin, Arg_0: Arg_0 {O(n)}

  3: evalspeedpldi2entryin->evalspeedpldi2returnin, Arg_1: 0 {O(1)}

  3: evalspeedpldi2entryin->evalspeedpldi2returnin, Arg_2: Arg_2 {O(n)}

  9: evalspeedpldi2returnin->evalspeedpldi2stop, Arg_0: max([-1, max([Arg_1, Arg_0])]) {O(n)}

  9: evalspeedpldi2returnin->evalspeedpldi2stop, Arg_1: max([0, max([Arg_0, Arg_1])]) {O(n)}

  9: evalspeedpldi2returnin->evalspeedpldi2stop, Arg_2: max([0, Arg_2]) {O(n)}

  0: evalspeedpldi2start->evalspeedpldi2entryin, Arg_0: Arg_0 {O(n)}

  0: evalspeedpldi2start->evalspeedpldi2entryin, Arg_1: Arg_1 {O(n)}

  0: evalspeedpldi2start->evalspeedpldi2entryin, Arg_2: Arg_2 {O(n)}


----------------------------------------

(2)
BOUNDS(1, max(6 + 2 * Arg_0, 6) + max(1, 1 + 2 * Arg_0) + nat(Arg_0))

----------------------------------------

(3) Loat Proof (FINISHED)


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



Initial linear ITS problem

   Start location: evalspeedpldi2start

      0: evalspeedpldi2start -> evalspeedpldi2entryin : [], cost: 1

      1: evalspeedpldi2entryin -> evalspeedpldi2bb5in : A'=B, B'=0, C'=A, [ A>=0 && B>=1 ], cost: 1

      2: evalspeedpldi2entryin -> evalspeedpldi2returnin : [ 0>=1+A ], cost: 1

      3: evalspeedpldi2entryin -> evalspeedpldi2returnin : [ 0>=B ], cost: 1

      4: evalspeedpldi2bb5in -> evalspeedpldi2bb2in : [ C>=1 ], cost: 1

      5: evalspeedpldi2bb5in -> evalspeedpldi2returnin : [ 0>=C ], cost: 1

      6: evalspeedpldi2bb2in -> evalspeedpldi2bb3in : [ A>=1+B ], cost: 1

      7: evalspeedpldi2bb2in -> evalspeedpldi2bb5in : B'=0, [ B>=A ], cost: 1

      8: evalspeedpldi2bb3in -> evalspeedpldi2bb5in : B'=1+B, C'=-1+C, [], cost: 1

      9: evalspeedpldi2returnin -> evalspeedpldi2stop : [], cost: 1



Removed unreachable and leaf rules:

   Start location: evalspeedpldi2start

      0: evalspeedpldi2start -> evalspeedpldi2entryin : [], cost: 1

      1: evalspeedpldi2entryin -> evalspeedpldi2bb5in : A'=B, B'=0, C'=A, [ A>=0 && B>=1 ], cost: 1

      4: evalspeedpldi2bb5in -> evalspeedpldi2bb2in : [ C>=1 ], cost: 1

      6: evalspeedpldi2bb2in -> evalspeedpldi2bb3in : [ A>=1+B ], cost: 1

      7: evalspeedpldi2bb2in -> evalspeedpldi2bb5in : B'=0, [ B>=A ], cost: 1

      8: evalspeedpldi2bb3in -> evalspeedpldi2bb5in : B'=1+B, C'=-1+C, [], cost: 1



### Simplification by acceleration and chaining ###



Eliminated locations (on linear paths):

   Start location: evalspeedpldi2start

     10: evalspeedpldi2start -> evalspeedpldi2bb5in : A'=B, B'=0, C'=A, [ A>=0 && B>=1 ], cost: 2

      4: evalspeedpldi2bb5in -> evalspeedpldi2bb2in : [ C>=1 ], cost: 1

      7: evalspeedpldi2bb2in -> evalspeedpldi2bb5in : B'=0, [ B>=A ], cost: 1

     11: evalspeedpldi2bb2in -> evalspeedpldi2bb5in : B'=1+B, C'=-1+C, [ A>=1+B ], cost: 2



Eliminated locations (on tree-shaped paths):

   Start location: evalspeedpldi2start

     10: evalspeedpldi2start -> evalspeedpldi2bb5in : A'=B, B'=0, C'=A, [ A>=0 && B>=1 ], cost: 2

     12: evalspeedpldi2bb5in -> evalspeedpldi2bb5in : B'=0, [ C>=1 && B>=A ], cost: 2

     13: evalspeedpldi2bb5in -> evalspeedpldi2bb5in : B'=1+B, C'=-1+C, [ C>=1 && A>=1+B ], cost: 3



Accelerating simple loops of location 2.

   Accelerating the following rules:

     12: evalspeedpldi2bb5in -> evalspeedpldi2bb5in : B'=0, [ C>=1 && B>=A ], cost: 2

     13: evalspeedpldi2bb5in -> evalspeedpldi2bb5in : B'=1+B, C'=-1+C, [ C>=1 && A>=1+B ], cost: 3



   Accelerated rule 12 with NONTERM (after strengthening guard), yielding the new rule 14.

   Accelerated rule 13 with backward acceleration, yielding the new rule 15.

   Accelerated rule 13 with backward acceleration, yielding the new rule 16.

   Removing the simple loops: 13.



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

   Start location: evalspeedpldi2start

     10: evalspeedpldi2start -> evalspeedpldi2bb5in : A'=B, B'=0, C'=A, [ A>=0 && B>=1 ], cost: 2

     12: evalspeedpldi2bb5in -> evalspeedpldi2bb5in : B'=0, [ C>=1 && B>=A ], cost: 2

     14: evalspeedpldi2bb5in -> [7] : [ C>=1 && B>=A && 0>=A ], cost: INF

     15: evalspeedpldi2bb5in -> evalspeedpldi2bb5in : B'=C+B, C'=0, [ C>=1 && A>=1+B && A>=C+B ], cost: 3*C

     16: evalspeedpldi2bb5in -> evalspeedpldi2bb5in : B'=A, C'=C-A+B, [ C>=1 && A>=1+B && 1+C-A+B>=1 ], cost: 3*A-3*B



Chained accelerated rules (with incoming rules):

   Start location: evalspeedpldi2start

     10: evalspeedpldi2start -> evalspeedpldi2bb5in : A'=B, B'=0, C'=A, [ A>=0 && B>=1 ], cost: 2

     17: evalspeedpldi2start -> evalspeedpldi2bb5in : A'=B, B'=A, C'=0, [ B>=1 && A>=1 && B>=A ], cost: 2+3*A

     18: evalspeedpldi2start -> evalspeedpldi2bb5in : A'=B, C'=A-B, [ B>=1 && A>=1 && 1+A-B>=1 ], cost: 2+3*B



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

   Start location: evalspeedpldi2start

     17: evalspeedpldi2start -> evalspeedpldi2bb5in : A'=B, B'=A, C'=0, [ B>=1 && A>=1 && B>=A ], cost: 2+3*A

     18: evalspeedpldi2start -> evalspeedpldi2bb5in : A'=B, C'=A-B, [ B>=1 && A>=1 && 1+A-B>=1 ], cost: 2+3*B



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: evalspeedpldi2start

     17: evalspeedpldi2start -> evalspeedpldi2bb5in : A'=B, B'=A, C'=0, [ B>=1 && A>=1 && B>=A ], cost: 2+3*A

     18: evalspeedpldi2start -> evalspeedpldi2bb5in : A'=B, C'=A-B, [ B>=1 && A>=1 && 1+A-B>=1 ], cost: 2+3*B



Computing asymptotic complexity for rule 17

   Simplified the guard:

     17: evalspeedpldi2start -> evalspeedpldi2bb5in : A'=B, B'=A, C'=0, [ A>=1 && B>=A ], cost: 2+3*A

   Solved the limit problem by the following transformations:

      Created initial limit problem:

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



      removing all constraints (solved by SMT)

      resulting limit problem: [solved]



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

      resulting limit problem:

      [solved]



   Solution:

      A / n

      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

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



WORST_CASE(Omega(n^1),?)


----------------------------------------

(4)
BOUNDS(n^1, INF)