Term Rewriting System R:
[x, y, z]
a(lambda(x), y) -> lambda(a(x, p(1, a(y, t))))
a(p(x, y), z) -> p(a(x, z), a(y, z))
a(a(x, y), z) -> a(x, a(y, z))
a(x, y) -> x
a(x, y) -> y
lambda(x) -> x
p(x, y) -> x
p(x, y) -> y

Innermost Termination of R to be shown.



   R
Removing Redundant Rules for Innermost Termination



Removing the following rules from R which left hand sides contain non normal subterms

a(lambda(x), y) -> lambda(a(x, p(1, a(y, t))))
a(p(x, y), z) -> p(a(x, z), a(y, z))
a(a(x, y), z) -> a(x, a(y, z))


   R
RRRI
       →TRS2
Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

a(x, y) -> x
a(x, y) -> y

where the Polynomial interpretation:
  POL(lambda(x1))=  x1  
  POL(a(x1, x2))=  1 + x1 + x2  
  POL(p(x1, x2))=  x1 + x2  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.


   R
RRRI
       →TRS2
RRRPolo
           →TRS3
Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

p(x, y) -> y
p(x, y) -> x

where the Polynomial interpretation:
  POL(lambda(x1))=  x1  
  POL(p(x1, x2))=  1 + x1 + x2  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.


   R
RRRI
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS4
Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

lambda(x) -> x

where the Polynomial interpretation:
  POL(lambda(x1))=  1 + x1  
was used.

All Rules of R can be deleted.


   R
RRRI
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS5
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes