LMPO

Execution Time (secs)	0.021
Answer	YES(?,ELEMENTARY)
Input	AProVE 10 ex5

YES(?,ELEMENTARY)

We consider the following Problem:

  Strict Trs:
    {  g(0()) -> 0()
     , g(s(x)) -> f(g(x))
     , f(0()) -> 0()}
  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(g) = {}, safe(0) = {}, safe(s) = {1}, safe(f) = {1}

  and precedence

   g > f .

  Following symbols are considered recursive:

   {g, f}

  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:
     {  g(0();) -> 0()
      , g(s(; x);) -> f(; g(x;))
      , f(; 0()) -> 0()}
   Weak Trs  : {}

Hurray, we answered YES(?,ELEMENTARY)

MPO

Execution Time (secs)	0.039
Answer	YES(?,PRIMREC)
Input	AProVE 10 ex5

YES(?,PRIMREC)

We consider the following Problem:

  Strict Trs:
    {  g(0()) -> 0()
     , g(s(x)) -> f(g(x))
     , f(0()) -> 0()}
  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

   s > g, s > f .

Hurray, we answered YES(?,PRIMREC)

POP*

Execution Time (secs)	0.024
Answer	YES(?,POLY)
Input	AProVE 10 ex5

YES(?,POLY)

We consider the following Problem:

  Strict Trs:
    {  g(0()) -> 0()
     , g(s(x)) -> f(g(x))
     , f(0()) -> 0()}
  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(g) = {}, safe(0) = {}, safe(s) = {1}, safe(f) = {1}

  and precedence

   g > f .

  Following symbols are considered recursive:

   {g, f}

  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:
     {  g(0();) -> 0()
      , g(s(; x);) -> f(; g(x;))
      , f(; 0()) -> 0()}
   Weak Trs  : {}

Hurray, we answered YES(?,POLY)

POP* (PS)

Execution Time (secs)	0.025
Answer	YES(?,POLY)
Input	AProVE 10 ex5

YES(?,POLY)

We consider the following Problem:

  Strict Trs:
    {  g(0()) -> 0()
     , g(s(x)) -> f(g(x))
     , f(0()) -> 0()}
  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(g) = {}, safe(0) = {}, safe(s) = {1}, safe(f) = {1}

  and precedence

   g > f .

  Following symbols are considered recursive:

   {g, f}

  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:
     {  g(0();) -> 0()
      , g(s(; x);) -> f(; g(x;))
      , f(; 0()) -> 0()}
   Weak Trs  : {}

Hurray, we answered YES(?,POLY)

Small POP*

Execution Time (secs)	0.049
Answer	YES(?,O(n^1))
Input	AProVE 10 ex5

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  g(0()) -> 0()
     , g(s(x)) -> f(g(x))
     , f(0()) -> 0()}
  StartTerms: basic terms
  Strategy: innermost

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

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

   safe(g) = {}, safe(0) = {}, safe(s) = {1}, safe(f) = {1}

  and precedence

   g > f .

  Following symbols are considered recursive:

   {g}

  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:
     {  g(0();) -> 0()
      , g(s(; x);) -> f(; g(x;))
      , f(; 0()) -> 0()}
   Weak Trs  : {}

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

Small POP* (PS)

Execution Time (secs)	0.041
Answer	YES(?,O(n^1))
Input	AProVE 10 ex5

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  g(0()) -> 0()
     , g(s(x)) -> f(g(x))
     , f(0()) -> 0()}
  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(g) = {}, safe(0) = {}, safe(s) = {1}, safe(f) = {1}

  and precedence

   g > f .

  Following symbols are considered recursive:

   {g}

  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:
     {  g(0();) -> 0()
      , g(s(; x);) -> f(; g(x;))
      , f(; 0()) -> 0()}
   Weak Trs  : {}

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

Problem AProVE 10 ex5

LMPO

MPO

POP*

POP* (PS)

Small POP*

Small POP* (PS)