Problem Transformed CSR 04 Ex26 Luc03b L

Tool CaT

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex26 Luc03b L

stdout:

YES(?,O(n^1))

Problem:
terms(N) -> cons(recip(sqr(N)))
sqr(0()) -> 0()
sqr(s()) -> s()
dbl(0()) -> 0()
dbl(s()) -> s()
add(0(),X) -> X
add(s(),Y) -> s()
first(0(),X) -> nil()
first(s(),cons(Y)) -> cons(Y)

Proof:
Bounds Processor:
  bound: 1
  enrichment: match
  automaton:
   final states: {10,9,8,7,6}
   transitions:
    cons1(5) -> 10*
    cons1(12) -> 6*
    cons1(2) -> 10*
    cons1(4) -> 10*
    cons1(1) -> 10*
    cons1(3) -> 10*
    nil1() -> 10*
    s1() -> 11,9,8,7
    01() -> 11,8,7
    recip1(11) -> 12*
    sqr1(5) -> 11*
    sqr1(2) -> 11*
    sqr1(4) -> 11*
    sqr1(1) -> 11*
    sqr1(3) -> 11*
    terms0(5) -> 6*
    terms0(2) -> 6*
    terms0(4) -> 6*
    terms0(1) -> 6*
    terms0(3) -> 6*
    cons0(5) -> 1*
    cons0(2) -> 1*
    cons0(4) -> 1*
    cons0(1) -> 1*
    cons0(3) -> 1*
    recip0(5) -> 2*
    recip0(2) -> 2*
    recip0(4) -> 2*
    recip0(1) -> 2*
    recip0(3) -> 2*
    sqr0(5) -> 7*
    sqr0(2) -> 7*
    sqr0(4) -> 7*
    sqr0(1) -> 7*
    sqr0(3) -> 7*
    00() -> 3*
    s0() -> 4*
    dbl0(5) -> 8*
    dbl0(2) -> 8*
    dbl0(4) -> 8*
    dbl0(1) -> 8*
    dbl0(3) -> 8*
    add0(3,1) -> 9*
    add0(3,3) -> 9*
    add0(3,5) -> 9*
    add0(4,2) -> 9*
    add0(4,4) -> 9*
    add0(5,1) -> 9*
    add0(5,3) -> 9*
    add0(5,5) -> 9*
    add0(1,2) -> 9*
    add0(1,4) -> 9*
    add0(2,1) -> 9*
    add0(2,3) -> 9*
    add0(2,5) -> 9*
    add0(3,2) -> 9*
    add0(3,4) -> 9*
    add0(4,1) -> 9*
    add0(4,3) -> 9*
    add0(4,5) -> 9*
    add0(5,2) -> 9*
    add0(5,4) -> 9*
    add0(1,1) -> 9*
    add0(1,3) -> 9*
    add0(1,5) -> 9*
    add0(2,2) -> 9*
    add0(2,4) -> 9*
    first0(3,1) -> 10*
    first0(3,3) -> 10*
    first0(3,5) -> 10*
    first0(4,2) -> 10*
    first0(4,4) -> 10*
    first0(5,1) -> 10*
    first0(5,3) -> 10*
    first0(5,5) -> 10*
    first0(1,2) -> 10*
    first0(1,4) -> 10*
    first0(2,1) -> 10*
    first0(2,3) -> 10*
    first0(2,5) -> 10*
    first0(3,2) -> 10*
    first0(3,4) -> 10*
    first0(4,1) -> 10*
    first0(4,3) -> 10*
    first0(4,5) -> 10*
    first0(5,2) -> 10*
    first0(5,4) -> 10*
    first0(1,1) -> 10*
    first0(1,3) -> 10*
    first0(1,5) -> 10*
    first0(2,2) -> 10*
    first0(2,4) -> 10*
    nil0() -> 5*
    1 -> 9*
    2 -> 9*
    3 -> 9*
    4 -> 9*
    5 -> 9*
  problem:

  Qed

Tool IRC1

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex26 Luc03b L

stdout:

YES(?,O(n^1))

Tool IRC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex26 Luc03b L

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:
    {  terms(N) -> cons(recip(sqr(N)))
     , sqr(0()) -> 0()
     , sqr(s()) -> s()
     , dbl(0()) -> 0()
     , dbl(s()) -> s()
     , add(0(), X) -> X
     , add(s(), Y) -> s()
     , first(0(), X) -> nil()
     , first(s(), cons(Y)) -> cons(Y)}

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:
         {  terms(N) -> cons(recip(sqr(N)))
          , sqr(0()) -> 0()
          , sqr(s()) -> s()
          , dbl(0()) -> 0()
          , dbl(s()) -> s()
          , add(0(), X) -> X
          , add(s(), Y) -> s()
          , first(0(), X) -> nil()
          , first(s(), cons(Y)) -> cons(Y)}

     Proof Output:
       The problem is match-bounded by 1.
       The enriched problem is compatible with the following automaton:
       {  terms_0(2) -> 1
        , cons_0(2) -> 1
        , cons_0(2) -> 2
        , cons_1(2) -> 1
        , cons_1(3) -> 1
        , recip_0(2) -> 1
        , recip_0(2) -> 2
        , recip_1(4) -> 3
        , sqr_0(2) -> 1
        , sqr_1(2) -> 4
        , 0_0() -> 1
        , 0_0() -> 2
        , 0_1() -> 1
        , 0_1() -> 4
        , s_0() -> 1
        , s_0() -> 2
        , s_1() -> 1
        , s_1() -> 4
        , dbl_0(2) -> 1
        , add_0(2, 2) -> 1
        , first_0(2, 2) -> 1
        , nil_0() -> 1
        , nil_0() -> 2
        , nil_1() -> 1}

Tool RC1

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex26 Luc03b L

stdout:

YES(?,O(n^1))

Tool RC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex26 Luc03b L

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    runtime-complexity with respect to
  Rules:
    {  terms(N) -> cons(recip(sqr(N)))
     , sqr(0()) -> 0()
     , sqr(s()) -> s()
     , dbl(0()) -> 0()
     , dbl(s()) -> s()
     , add(0(), X) -> X
     , add(s(), Y) -> s()
     , first(0(), X) -> nil()
     , first(s(), cons(Y)) -> cons(Y)}

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:
         {  terms(N) -> cons(recip(sqr(N)))
          , sqr(0()) -> 0()
          , sqr(s()) -> s()
          , dbl(0()) -> 0()
          , dbl(s()) -> s()
          , add(0(), X) -> X
          , add(s(), Y) -> s()
          , first(0(), X) -> nil()
          , first(s(), cons(Y)) -> cons(Y)}

     Proof Output:
       The problem is match-bounded by 1.
       The enriched problem is compatible with the following automaton:
       {  terms_0(2) -> 1
        , cons_0(2) -> 1
        , cons_0(2) -> 2
        , cons_1(2) -> 1
        , cons_1(3) -> 1
        , recip_0(2) -> 1
        , recip_0(2) -> 2
        , recip_1(4) -> 3
        , sqr_0(2) -> 1
        , sqr_1(2) -> 4
        , 0_0() -> 1
        , 0_0() -> 2
        , 0_1() -> 1
        , 0_1() -> 4
        , s_0() -> 1
        , s_0() -> 2
        , s_1() -> 1
        , s_1() -> 4
        , dbl_0(2) -> 1
        , add_0(2, 2) -> 1
        , first_0(2, 2) -> 1
        , nil_0() -> 1
        , nil_0() -> 2
        , nil_1() -> 1}