(VAR X XS N Y YS)
(STRATEGY CONTEXTSENSITIVE 
  (from 1)
  (cons 1)
  (s 1)
  (sel 1 2)
  (0)
  (minus 1 2)
  (quot 1 2)
  (zWquot 1 2)
  (nil)
)
(RULES 
from(X) -> cons(X,from(s(X)))
sel(0,cons(X,XS)) -> X
sel(s(N),cons(X,XS)) -> sel(N,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),zWquot(XS,YS))
)

Proving termination of context-sensitive rewriting for Ex4_7_37_Bor03:

-> Dependency pairs:
nF_sel(s(N),cons(X,XS)) -> nF_sel(N,XS)
nF_sel(s(N),cons(X,XS)) -> XS
nF_minus(s(X),s(Y)) -> nF_minus(X,Y)
nF_quot(s(X),s(Y)) -> nF_quot(minus(X,Y),s(Y))
nF_quot(s(X),s(Y)) -> nF_minus(X,Y)
nF_zWquot(cons(X,XS),cons(Y,YS)) -> nF_quot(X,Y)

-> Proof of termination for Ex4_7_37_Bor03_1_1:
-> -> Dependency pairs in cycle:
nF_sel(s(N),cons(X,XS)) -> nF_sel(N,XS)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for Ex4_7_37_Bor03_1_2:
-> -> Dependency pairs in cycle:
nF_quot(s(X),s(Y)) -> nF_quot(minus(X,Y),s(Y))

Usable Rules:
minus(X,0) -> 0
minus(s(X),s(Y)) -> minus(X,Y)

Polynomial Interpretation:

[from](X) = 0
[cons](X1,X2) = 0
[s](X) = 1
[sel](X1,X2) = 0
[0] = 0
[minus](X1,X2) = 0
[quot](X1,X2) = 0
[zWquot](X1,X2) = 0
[nil] = 0
[nF_quot](X1,X2) = X1

TIME: 0.107523


-> Proof of termination for Ex4_7_37_Bor03_1_3:
-> -> Dependency pairs in cycle:
nF_minus(s(X),s(Y)) -> nF_minus(X,Y)

Termination proved: Cycles verify subterm criterion.

SETTINGS:
Base ordering: Polynomial ordering
Proof mode: SCCs in CSDG + base ordering
Upper bound for coeffs: 1
Rationals below 1 for all non-replacing args: No
Polynomial interpretation: Linear
Coeffs in polynomials: No rationals
Delta: automatic



Termination was proved succesfully.