Term Rewriting System R:
[X, L]
incr(nil) -> nil
incr(cons(X, L)) -> cons(s(X), incr(L))
adx(nil) -> nil
adx(cons(X, L)) -> incr(cons(X, adx(L)))
nats -> adx(zeros)
zeros -> cons(0, zeros)
head(cons(X, L)) -> X
tail(cons(X, L)) -> L

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

INCR(cons(X, L)) -> INCR(L)
ADX(cons(X, L)) -> INCR(cons(X, adx(L)))
ADX(cons(X, L)) -> ADX(L)
NATS -> ADX(zeros)
NATS -> ZEROS
ZEROS -> ZEROS

Furthermore, R contains three SCCs.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering
       →DP Problem 2
Remaining
       →DP Problem 3
Remaining


Dependency Pair:

INCR(cons(X, L)) -> INCR(L)


Rules:


incr(nil) -> nil
incr(cons(X, L)) -> cons(s(X), incr(L))
adx(nil) -> nil
adx(cons(X, L)) -> incr(cons(X, adx(L)))
nats -> adx(zeros)
zeros -> cons(0, zeros)
head(cons(X, L)) -> X
tail(cons(X, L)) -> L


Strategy:

innermost




The following dependency pair can be strictly oriented:

INCR(cons(X, L)) -> INCR(L)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(cons(x1, x2))=  1 + x1 + x2  
  POL(INCR(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
INCR(x1) -> INCR(x1)
cons(x1, x2) -> cons(x1, x2)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 4
Dependency Graph
       →DP Problem 2
Remaining
       →DP Problem 3
Remaining


Dependency Pair:


Rules:


incr(nil) -> nil
incr(cons(X, L)) -> cons(s(X), incr(L))
adx(nil) -> nil
adx(cons(X, L)) -> incr(cons(X, adx(L)))
nats -> adx(zeros)
zeros -> cons(0, zeros)
head(cons(X, L)) -> X
tail(cons(X, L)) -> L


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)




The following remains to be proven:


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)




The following remains to be proven:

Innermost Termination of R could not be shown.
Duration:
0:00 minutes