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)))

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


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.


SCC1:

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)

On this Scc, 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.

none
The transformation is resulting in one subcycle:


SCC1.Nar1:

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)

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   app(app(eq, app(s, x)), 0) -> false
   app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
   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)))
   app(app(lt, y), 0) -> false
   app(app(eq, x), x) -> true
   app(app(eq, 0), app(s, x)) -> false
   app(app(lt, 0), app(s, y)) -> true


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(null) = 0
POL(s) = 0
POL(APP(x_1, x_2)) = x_1
POL(eq) = 0
POL(true) = 0
POL(lt) = 0
POL(member) = 1
POL(app(x_1, x_2)) = x_1
POL(0) = 0
POL(if) = 0
POL(false) = 0
POL(fork) = 0

 resulting in two subcycles.


SCC1.Nar1.Polo1:

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(member, w), x)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(APP(x_1, x_2)) = 1 + x_1 + x_2
POL(member) = 1
POL(app(x_1, x_2)) = 1 + x_1 + x_2
POL(fork) = 1

The following Dependency Pairs can be deleted:

   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(member, w), x)



 This transformation is resulting in no new subcycles.


SCC1.Nar1.Polo2:

APP(app(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y)

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(null) = 0
POL(s) = 0
POL(APP(x_1, x_2)) = x_2
POL(eq) = 0
POL(true) = 0
POL(lt) = 0
POL(member) = 0
POL(app(x_1, x_2)) = 1 + x_2
POL(0) = 0
POL(if) = 0
POL(false) = 0
POL(fork) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 9.595 seconds.