Term Rewriting System R:
[X, XS, N, Y, YS, X1, X2]
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nzWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
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

sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))


   R
     ↳RRRI
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(nfrom(X)) -> FROM(activate(X))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nzWquot(X1, X2)) -> ZWQUOT(activate(X1), activate(X2))
ACTIVATE(nzWquot(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nzWquot(X1, X2)) -> ACTIVATE(X2)

Furthermore, R contains one SCC. 


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Negative Polynomial Order


Dependency Pairs:

ACTIVATE(nzWquot(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nzWquot(X1, X2)) -> ACTIVATE(X1)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(nzWquot(X1, X2)) -> ZWQUOT(activate(X1), activate(X2))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
minus(X, 0) -> 0
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nzWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X
s(X) -> ns(X)





The following Dependency Pairs can be strictly oriented using the given order.

ACTIVATE(nzWquot(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nzWquot(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nzWquot(X1, X2)) -> ZWQUOT(activate(X1), activate(X2))


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nzWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)


Used ordering:
Polynomial Order with Interpretation:

POL( ACTIVATE(x1) ) = x1

POL( nzWquot(x1, x2) ) = x1 + x2 + 1

POL( ZWQUOT(x1, x2) ) = x1 + x2

POL( cons(x1, x2) ) = x2

POL( ns(x1) ) = x1

POL( nfrom(x1) ) = x1

POL( activate(x1) ) = x1

POL( from(x1) ) = x1

POL( s(x1) ) = x1

POL( zWquot(x1, x2) ) = x1 + x2 + 1

POL( nil ) = 0


This results in one new DP problem. 


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Neg POLO
             ...
               →DP Problem 2
                 ↳Dependency Graph


Dependency Pairs:

ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
minus(X, 0) -> 0
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nzWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X
s(X) -> ns(X)





Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Neg POLO
             ...
               →DP Problem 3
                 ↳Size-Change Principle


Dependency Pairs:

ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
minus(X, 0) -> 0
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nzWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X
s(X) -> ns(X)





We number the DPs as follows:
  1. ACTIVATE(ns(X)) -> ACTIVATE(X)
  2. ACTIVATE(nfrom(X)) -> ACTIVATE(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
nfrom(x1) -> nfrom(x1)
ns(x1) -> ns(x1)

We obtain no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes