LMPO

Execution Time (secs)	0.035
Answer	YES(?,ELEMENTARY)
Input	SK90 2.20

YES(?,ELEMENTARY)

We consider the following Problem:

  Strict Trs:
    {  sum(0()) -> 0()
     , sum(s(x)) -> +(sqr(s(x)), sum(x))
     , sqr(x) -> *(x, x)
     , sum(s(x)) -> +(*(s(x), s(x)), sum(x))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,ELEMENTARY)

Proof:
  The input was oriented with the instance of
  Lightweight Multiset Path Order () as induced by the safe mapping

   safe(sum) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {1, 2},
   safe(sqr) = {}, safe(*) = {1, 2}

  and precedence

   sum > sqr .

  Following symbols are considered recursive:

   {sum, sqr}

  The recursion depth is 2 .

  For your convenience, here are the oriented rules in predicative
  notation (possibly applying argument filtering):

   Strict DPs: {}
   Weak DPs  : {}
   Strict Trs:
     {  sum(0();) -> 0()
      , sum(s(; x);) -> +(; sqr(s(; x);), sum(x;))
      , sqr(x;) -> *(; x, x)
      , sum(s(; x);) -> +(; *(; s(; x), s(; x)), sum(x;))}
   Weak Trs  : {}

Hurray, we answered YES(?,ELEMENTARY)

MPO

Execution Time (secs)	0.046
Answer	YES(?,PRIMREC)
Input	SK90 2.20

YES(?,PRIMREC)

We consider the following Problem:

  Strict Trs:
    {  sum(0()) -> 0()
     , sum(s(x)) -> +(sqr(s(x)), sum(x))
     , sqr(x) -> *(x, x)
     , sum(s(x)) -> +(*(s(x), s(x)), sum(x))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,PRIMREC)

Proof:
  The input was oriented with the instance of
  'multiset path orders' as induced by the precedence

   sum > +, sum > sqr, sum > *, sqr > * .

Hurray, we answered YES(?,PRIMREC)

POP*

Execution Time (secs)	0.034
Answer	YES(?,POLY)
Input	SK90 2.20

YES(?,POLY)

We consider the following Problem:

  Strict Trs:
    {  sum(0()) -> 0()
     , sum(s(x)) -> +(sqr(s(x)), sum(x))
     , sqr(x) -> *(x, x)
     , sum(s(x)) -> +(*(s(x), s(x)), sum(x))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,POLY)

Proof:
  The input was oriented with the instance of
  Polynomial Path Order () as induced by the safe mapping

   safe(sum) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {1, 2},
   safe(sqr) = {}, safe(*) = {1, 2}

  and precedence

   sum > sqr .

  Following symbols are considered recursive:

   {sum}

  The recursion depth is 1 .

  For your convenience, here are the oriented rules in predicative
  notation (possibly applying argument filtering):

   Strict DPs: {}
   Weak DPs  : {}
   Strict Trs:
     {  sum(0();) -> 0()
      , sum(s(; x);) -> +(; sqr(s(; x);), sum(x;))
      , sqr(x;) -> *(; x, x)
      , sum(s(; x);) -> +(; *(; s(; x), s(; x)), sum(x;))}
   Weak Trs  : {}

Hurray, we answered YES(?,POLY)

POP* (PS)

Execution Time (secs)	0.040
Answer	YES(?,POLY)
Input	SK90 2.20

YES(?,POLY)

We consider the following Problem:

  Strict Trs:
    {  sum(0()) -> 0()
     , sum(s(x)) -> +(sqr(s(x)), sum(x))
     , sqr(x) -> *(x, x)
     , sum(s(x)) -> +(*(s(x), s(x)), sum(x))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,POLY)

Proof:
  The input was oriented with the instance of
  Polynomial Path Order (PS) as induced by the safe mapping

   safe(sum) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {1, 2},
   safe(sqr) = {}, safe(*) = {1, 2}

  and precedence

   sum > sqr .

  Following symbols are considered recursive:

   {sum}

  The recursion depth is 1 .

  For your convenience, here are the oriented rules in predicative
  notation (possibly applying argument filtering):

   Strict DPs: {}
   Weak DPs  : {}
   Strict Trs:
     {  sum(0();) -> 0()
      , sum(s(; x);) -> +(; sqr(s(; x);), sum(x;))
      , sqr(x;) -> *(; x, x)
      , sum(s(; x);) -> +(; *(; s(; x), s(; x)), sum(x;))}
   Weak Trs  : {}

Hurray, we answered YES(?,POLY)

Small POP*

Execution Time (secs)	0.042
Answer	YES(?,O(n^1))
Input	SK90 2.20

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  sum(0()) -> 0()
     , sum(s(x)) -> +(sqr(s(x)), sum(x))
     , sqr(x) -> *(x, x)
     , sum(s(x)) -> +(*(s(x), s(x)), sum(x))}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  The input was oriented with the instance of
  Small Polynomial Path Order (WSC) as induced by the safe mapping

   safe(sum) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {1, 2},
   safe(sqr) = {1}, safe(*) = {1, 2}

  and precedence

   sum > sqr .

  Following symbols are considered recursive:

   {sum}

  The recursion depth is 1 .

  For your convenience, here are the oriented rules in predicative
  notation (possibly applying argument filtering):

   Strict DPs: {}
   Weak DPs  : {}
   Strict Trs:
     {  sum(0();) -> 0()
      , sum(s(; x);) -> +(; sqr(; s(; x)), sum(x;))
      , sqr(; x) -> *(; x, x)
      , sum(s(; x);) -> +(; *(; s(; x), s(; x)), sum(x;))}
   Weak Trs  : {}

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

Small POP* (PS)

Execution Time (secs)	0.068
Answer	YES(?,O(n^1))
Input	SK90 2.20

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  sum(0()) -> 0()
     , sum(s(x)) -> +(sqr(s(x)), sum(x))
     , sqr(x) -> *(x, x)
     , sum(s(x)) -> +(*(s(x), s(x)), sum(x))}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  The input was oriented with the instance of
  Small Polynomial Path Order (WSC,
                             PS,
                             Nat 1-bounded) as induced by the safe mapping

   safe(sum) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {1, 2},
   safe(sqr) = {1}, safe(*) = {1, 2}

  and precedence

   sum > sqr .

  Following symbols are considered recursive:

   {sum}

  The recursion depth is 1 .

  For your convenience, here are the oriented rules in predicative
  notation (possibly applying argument filtering):

   Strict DPs: {}
   Weak DPs  : {}
   Strict Trs:
     {  sum(0();) -> 0()
      , sum(s(; x);) -> +(; sqr(; s(; x)), sum(x;))
      , sqr(; x) -> *(; x, x)
      , sum(s(; x);) -> +(; *(; s(; x), s(; x)), sum(x;))}
   Weak Trs  : {}

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

Problem SK90 2.20

LMPO

MPO

POP*

POP* (PS)

Small POP*

Small POP* (PS)