Problem SK90 4.29

Tool CaT

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	SK90 4.29

stdout:

YES(?,O(n^1))

Problem:
merge(x,nil()) -> x
merge(nil(),y) -> y
merge(++(x,y),++(u(),v())) -> ++(x,merge(y,++(u(),v())))
merge(++(x,y),++(u(),v())) -> ++(u(),merge(++(x,y),v()))

Proof:
Bounds Processor:
  bound: 1
  enrichment: match
  automaton:
   final states: {5}
   transitions:
    ++1(3,1) -> 18*
    ++1(3,3) -> 18*
    ++1(3,13) -> 15,5
    ++1(3,15) -> 15,5
    ++1(4,2) -> 18*
    ++1(4,4) -> 18*
    ++1(1,2) -> 18*
    ++1(1,4) -> 18*
    ++1(11,10) -> 12*
    ++1(2,1) -> 18*
    ++1(2,3) -> 18*
    ++1(2,13) -> 15,5
    ++1(2,15) -> 15,5
    ++1(3,2) -> 18*
    ++1(3,4) -> 18*
    ++1(4,1) -> 18*
    ++1(4,3) -> 18*
    ++1(4,13) -> 15,5
    ++1(4,15) -> 15,5
    ++1(1,1) -> 18*
    ++1(1,3) -> 18*
    ++1(1,13) -> 15,5
    ++1(1,15) -> 15,5
    ++1(11,19) -> 15,5
    ++1(2,2) -> 18*
    ++1(2,4) -> 18*
    u1() -> 11*
    merge1(2,12) -> 15*
    merge1(4,12) -> 15*
    merge1(1,12) -> 13*
    merge1(18,10) -> 19*
    merge1(3,12) -> 13*
    v1() -> 10*
    merge0(3,1) -> 5*
    merge0(3,3) -> 5*
    merge0(4,2) -> 5*
    merge0(4,4) -> 5*
    merge0(1,2) -> 5*
    merge0(1,4) -> 5*
    merge0(2,1) -> 5*
    merge0(2,3) -> 5*
    merge0(3,2) -> 5*
    merge0(3,4) -> 5*
    merge0(4,1) -> 5*
    merge0(4,3) -> 5*
    merge0(1,1) -> 5*
    merge0(1,3) -> 5*
    merge0(2,2) -> 5*
    merge0(2,4) -> 5*
    nil0() -> 1*
    ++0(3,1) -> 2*
    ++0(3,3) -> 2*
    ++0(4,2) -> 2*
    ++0(4,4) -> 2*
    ++0(1,2) -> 2*
    ++0(1,4) -> 2*
    ++0(2,1) -> 2*
    ++0(2,3) -> 2*
    ++0(3,2) -> 2*
    ++0(3,4) -> 2*
    ++0(4,1) -> 2*
    ++0(4,3) -> 2*
    ++0(1,1) -> 2*
    ++0(1,3) -> 2*
    ++0(2,2) -> 2*
    ++0(2,4) -> 2*
    u0() -> 3*
    v0() -> 4*
    1 -> 5*
    2 -> 5*
    3 -> 5*
    4 -> 5*
    12 -> 13*
  problem:

  Qed

Tool IRC1

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	SK90 4.29

stdout:

YES(?,O(n^1))

Tool IRC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	SK90 4.29

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  merge(x, nil()) -> x
     , merge(nil(), y) -> y
     , merge(++(x, y), ++(u(), v())) -> ++(x, merge(y, ++(u(), v())))
     , merge(++(x, y), ++(u(), v())) -> ++(u(), merge(++(x, y), v()))}

Proof Output:
  'Bounds with minimal-enrichment and initial automaton 'match'' proved the best result:

  Details:
  --------
    'Bounds with minimal-enrichment and initial automaton 'match'' succeeded with the following output:
     'Bounds with minimal-enrichment and initial automaton 'match''
     --------------------------------------------------------------
     Answer:           YES(?,O(n^1))
     Input Problem:    innermost runtime-complexity with respect to
       Rules:
         {  merge(x, nil()) -> x
          , merge(nil(), y) -> y
          , merge(++(x, y), ++(u(), v())) -> ++(x, merge(y, ++(u(), v())))
          , merge(++(x, y), ++(u(), v())) -> ++(u(), merge(++(x, y), v()))}

     Proof Output:
       The problem is match-bounded by 1.
       The enriched problem is compatible with the following automaton:
       {  merge_0(2, 2) -> 1
        , merge_1(2, 4) -> 3
        , merge_1(8, 6) -> 7
        , nil_0() -> 1
        , nil_0() -> 2
        , ++_0(2, 2) -> 1
        , ++_0(2, 2) -> 2
        , ++_1(2, 2) -> 8
        , ++_1(2, 3) -> 1
        , ++_1(2, 3) -> 3
        , ++_1(5, 6) -> 3
        , ++_1(5, 6) -> 4
        , ++_1(5, 7) -> 1
        , ++_1(5, 7) -> 3
        , u_0() -> 1
        , u_0() -> 2
        , u_1() -> 5
        , v_0() -> 1
        , v_0() -> 2
        , v_1() -> 6}

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	SK90 4.29

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	SK90 4.29

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    runtime-complexity with respect to
  Rules:
    {  merge(x, nil()) -> x
     , merge(nil(), y) -> y
     , merge(++(x, y), ++(u(), v())) -> ++(x, merge(y, ++(u(), v())))
     , merge(++(x, y), ++(u(), v())) -> ++(u(), merge(++(x, y), v()))}

Proof Output:
  'Bounds with minimal-enrichment and initial automaton 'match'' proved the best result:

  Details:
  --------
    'Bounds with minimal-enrichment and initial automaton 'match'' succeeded with the following output:
     'Bounds with minimal-enrichment and initial automaton 'match''
     --------------------------------------------------------------
     Answer:           YES(?,O(n^1))
     Input Problem:    runtime-complexity with respect to
       Rules:
         {  merge(x, nil()) -> x
          , merge(nil(), y) -> y
          , merge(++(x, y), ++(u(), v())) -> ++(x, merge(y, ++(u(), v())))
          , merge(++(x, y), ++(u(), v())) -> ++(u(), merge(++(x, y), v()))}

     Proof Output:
       The problem is match-bounded by 1.
       The enriched problem is compatible with the following automaton:
       {  merge_0(2, 2) -> 1
        , merge_1(2, 4) -> 3
        , merge_1(8, 6) -> 7
        , nil_0() -> 1
        , nil_0() -> 2
        , ++_0(2, 2) -> 1
        , ++_0(2, 2) -> 2
        , ++_1(2, 2) -> 8
        , ++_1(2, 3) -> 1
        , ++_1(2, 3) -> 3
        , ++_1(5, 6) -> 3
        , ++_1(5, 6) -> 4
        , ++_1(5, 7) -> 1
        , ++_1(5, 7) -> 3
        , u_0() -> 1
        , u_0() -> 2
        , u_1() -> 5
        , v_0() -> 1
        , v_0() -> 2
        , v_1() -> 6}

Tool pair1rc

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	SK90 4.29

stdout:

Tool pair2rc

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	SK90 4.29

stdout:

Tool pair3irc

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	SK90 4.29

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  merge(x, nil()) -> x
     , merge(nil(), y) -> y
     , merge(++(x, y), ++(u(), v())) -> ++(x, merge(y, ++(u(), v())))
     , merge(++(x, y), ++(u(), v())) -> ++(u(), merge(++(x, y), v()))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,O(n^1))

Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
  The input problem contains no overlaps that give rise to inapplicable rules.

  We abort the transformation and continue with the subprocessor on the problem

  Strict Trs:
    {  merge(x, nil()) -> x
     , merge(nil(), y) -> y
     , merge(++(x, y), ++(u(), v())) -> ++(x, merge(y, ++(u(), v())))
     , merge(++(x, y), ++(u(), v())) -> ++(u(), merge(++(x, y), v()))}
  StartTerms: basic terms
  Strategy: innermost

  1) 'Fastest' proved the goal fastest:

     'Fastest' proved the goal fastest:

     'Bounds with perSymbol-enrichment and initial automaton 'match'' proved the goal fastest:

     The problem is match-bounded by 1.
     The enriched problem is compatible with the following automaton:
     {  merge_0(2, 2) -> 1
      , merge_0(2, 3) -> 1
      , merge_0(2, 4) -> 1
      , merge_0(2, 5) -> 1
      , merge_0(3, 2) -> 1
      , merge_0(3, 3) -> 1
      , merge_0(3, 4) -> 1
      , merge_0(3, 5) -> 1
      , merge_0(4, 2) -> 1
      , merge_0(4, 3) -> 1
      , merge_0(4, 4) -> 1
      , merge_0(4, 5) -> 1
      , merge_0(5, 2) -> 1
      , merge_0(5, 3) -> 1
      , merge_0(5, 4) -> 1
      , merge_0(5, 5) -> 1
      , merge_1(2, 7) -> 6
      , merge_1(3, 7) -> 6
      , merge_1(4, 7) -> 6
      , merge_1(5, 7) -> 6
      , merge_1(11, 9) -> 10
      , nil_0() -> 1
      , nil_0() -> 2
      , ++_0(2, 2) -> 1
      , ++_0(2, 2) -> 3
      , ++_0(2, 3) -> 1
      , ++_0(2, 3) -> 3
      , ++_0(2, 4) -> 1
      , ++_0(2, 4) -> 3
      , ++_0(2, 5) -> 1
      , ++_0(2, 5) -> 3
      , ++_0(3, 2) -> 1
      , ++_0(3, 2) -> 3
      , ++_0(3, 3) -> 1
      , ++_0(3, 3) -> 3
      , ++_0(3, 4) -> 1
      , ++_0(3, 4) -> 3
      , ++_0(3, 5) -> 1
      , ++_0(3, 5) -> 3
      , ++_0(4, 2) -> 1
      , ++_0(4, 2) -> 3
      , ++_0(4, 3) -> 1
      , ++_0(4, 3) -> 3
      , ++_0(4, 4) -> 1
      , ++_0(4, 4) -> 3
      , ++_0(4, 5) -> 1
      , ++_0(4, 5) -> 3
      , ++_0(5, 2) -> 1
      , ++_0(5, 2) -> 3
      , ++_0(5, 3) -> 1
      , ++_0(5, 3) -> 3
      , ++_0(5, 4) -> 1
      , ++_0(5, 4) -> 3
      , ++_0(5, 5) -> 1
      , ++_0(5, 5) -> 3
      , ++_1(2, 2) -> 11
      , ++_1(2, 3) -> 11
      , ++_1(2, 4) -> 11
      , ++_1(2, 5) -> 11
      , ++_1(2, 6) -> 1
      , ++_1(2, 6) -> 6
      , ++_1(3, 2) -> 11
      , ++_1(3, 3) -> 11
      , ++_1(3, 4) -> 11
      , ++_1(3, 5) -> 11
      , ++_1(3, 6) -> 1
      , ++_1(3, 6) -> 6
      , ++_1(4, 2) -> 11
      , ++_1(4, 3) -> 11
      , ++_1(4, 4) -> 11
      , ++_1(4, 5) -> 11
      , ++_1(4, 6) -> 1
      , ++_1(4, 6) -> 6
      , ++_1(5, 2) -> 11
      , ++_1(5, 3) -> 11
      , ++_1(5, 4) -> 11
      , ++_1(5, 5) -> 11
      , ++_1(5, 6) -> 1
      , ++_1(5, 6) -> 6
      , ++_1(8, 9) -> 6
      , ++_1(8, 9) -> 7
      , ++_1(8, 10) -> 1
      , ++_1(8, 10) -> 6
      , u_0() -> 1
      , u_0() -> 4
      , u_1() -> 8
      , v_0() -> 1
      , v_0() -> 5
      , v_1() -> 9}


Hurray, we answered YES(?,O(n^1))

Tool pair3rc

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	SK90 4.29

stdout:

Tool rc

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	SK90 4.29

stdout:

Tool tup3irc

Execution Time	0.84229684ms
Answer	YES(?,O(n^1))
Input	SK90 4.29