Term Rewriting System R:
[Y, X]
minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X

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

div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)


   R
RRRI
       →TRS2
Removing Redundant Rules



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

minus(n0, Y) -> 0

where the Polynomial interpretation:
  POL(activate(x1))=  x1  
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(0)=  0  
  POL(geq(x1, x2))=  x1 + x2  
  POL(false)=  0  
  POL(minus(x1, x2))=  1 + x1 + x2  
  POL(n__s(x1))=  x1  
  POL(true)=  0  
  POL(n__0)=  0  
  POL(s(x1))=  x1  
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:

0 -> n0
geq(X, n0) -> true
geq(n0, ns(Y)) -> false

where the Polynomial interpretation:
  POL(activate(x1))=  2·x1  
  POL(if(x1, x2, x3))=  x1 + 2·x2 + 2·x3  
  POL(0)=  2  
  POL(geq(x1, x2))=  x1 + x2  
  POL(false)=  0  
  POL(minus(x1, x2))=  x1 + x2  
  POL(true)=  0  
  POL(n__s(x1))=  2·x1  
  POL(n__0)=  1  
  POL(s(x1))=  2·x1  
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:

if(false, X, Y) -> activate(Y)

where the Polynomial interpretation:
  POL(activate(x1))=  x1  
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(0)=  0  
  POL(geq(x1, x2))=  x1 + x2  
  POL(false)=  1  
  POL(minus(x1, x2))=  x1 + x2  
  POL(true)=  0  
  POL(n__s(x1))=  x1  
  POL(n__0)=  0  
  POL(s(x1))=  x1  
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
             ...
               →TRS5
Removing Redundant Rules



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

activate(n0) -> 0

where the Polynomial interpretation:
  POL(activate(x1))=  x1  
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(geq(x1, x2))=  x1 + x2  
  POL(0)=  0  
  POL(minus(x1, x2))=  x1 + x2  
  POL(n__s(x1))=  x1  
  POL(true)=  0  
  POL(s(x1))=  x1  
  POL(n__0)=  1  
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
             ...
               →TRS6
Removing Redundant Rules



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

s(X) -> ns(X)
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))

where the Polynomial interpretation:
  POL(activate(x1))=  2·x1  
  POL(if(x1, x2, x3))=  x1 + 2·x2 + x3  
  POL(geq(x1, x2))=  x1 + x2  
  POL(minus(x1, x2))=  x1 + x2  
  POL(true)=  0  
  POL(n__s(x1))=  1 + 2·x1  
  POL(s(x1))=  2 + 2·x1  
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
             ...
               →TRS7
Removing Redundant Rules



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

if(true, X, Y) -> activate(X)

where the Polynomial interpretation:
  POL(activate(x1))=  x1  
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(n__s(x1))=  x1  
  POL(true)=  1  
  POL(s(x1))=  x1  
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
             ...
               →TRS8
Removing Redundant Rules



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

activate(ns(X)) -> s(X)
activate(X) -> X

where the Polynomial interpretation:
  POL(activate(x1))=  1 + x1  
  POL(n__s(x1))=  x1  
  POL(s(x1))=  x1  
was used.

All Rules of R can be deleted.


   R
RRRI
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS9
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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