Term Rewriting System R:
[x, y, w, z]
app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y)
APP(app(lt, app(s, x)), app(s, y)) -> APP(lt, x)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(if, app(app(lt, w), y))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(lt, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(lt, w)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(if, app(app(eq, w), y)), true)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(if, app(app(eq, w), y))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(eq, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(eq, w)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(eq, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(lt, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))
APP(app(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y)


Rules:


app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(eq, w), y)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Polynomial Ordering


Dependency Pairs:

APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(lt, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z)


Rules:


app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(lt, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))


Additionally, the following usable rules for innermost can be oriented:

app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(fork)=  0  
  POL(if)=  0  
  POL(0)=  0  
  POL(eq)=  0  
  POL(false)=  0  
  POL(null)=  0  
  POL(member)=  1  
  POL(lt)=  0  
  POL(true)=  0  
  POL(s)=  0  
  POL(app(x1, x2))=  x1  
  POL(APP(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pairs:

APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x)
APP(app(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z)


Rules:


app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))


Strategy:

innermost




Using the Dependency Graph the DP problem was split into 2 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:

Innermost Termination of R could not be shown.
Duration:
0:04 minutes