(VAR X Y Z V V1 V2 I P)
(STRATEGY CONTEXTSENSITIVE 
  (__ 1 2)
  (nil)
  (and 1)
  (tt)
  (isList)
  (isNeList)
  (isQid)
  (isNePal)
  (isPal)
  (a)
  (e)
  (i)
  (o)
  (u)
)
(RULES 
__(__(X,Y),Z) -> __(X,__(Y,Z))
__(X,nil) -> X
__(nil,X) -> X
and(tt,X) -> X
isList(V) -> isNeList(V)
isList(nil) -> tt
isList(__(V1,V2)) -> and(isList(V1),isList(V2))
isNeList(V) -> isQid(V)
isNeList(__(V1,V2)) -> and(isList(V1),isNeList(V2))
isNeList(__(V1,V2)) -> and(isNeList(V1),isList(V2))
isNePal(V) -> isQid(V)
isNePal(__(I,__(P,I))) -> and(isQid(I),isPal(P))
isPal(V) -> isNePal(V)
isPal(nil) -> tt
isQid(a) -> tt
isQid(e) -> tt
isQid(i) -> tt
isQid(o) -> tt
isQid(u) -> tt
)

Proving termination of context-sensitive rewriting for PALINDROME_nokinds:

-> Dependency pairs:
nF___(__(X,Y),Z) -> nF___(X,__(Y,Z))
nF___(__(X,Y),Z) -> nF___(Y,Z)
nF_and(tt,X) -> X
nF_isList(V) -> nF_isNeList(V)
nF_isList(__(V1,V2)) -> nF_and(isList(V1),isList(V2))
nF_isList(__(V1,V2)) -> nF_isList(V1)
nF_isNeList(V) -> nF_isQid(V)
nF_isNeList(__(V1,V2)) -> nF_and(isList(V1),isNeList(V2))
nF_isNeList(__(V1,V2)) -> nF_isList(V1)
nF_isNeList(__(V1,V2)) -> nF_and(isNeList(V1),isList(V2))
nF_isNeList(__(V1,V2)) -> nF_isNeList(V1)
nF_isNePal(V) -> nF_isQid(V)
nF_isNePal(__(I,__(P,I))) -> nF_and(isQid(I),isPal(P))
nF_isNePal(__(I,__(P,I))) -> nF_isQid(I)
nF_isPal(V) -> nF_isNePal(V)

-> Proof of termination for PALINDROME_nokinds_1_1:
-> -> Dependency pairs in cycle:
nF_and(tt,X) -> X
nF_isNePal(__(I,__(P,I))) -> nF_and(isQid(I),isPal(P))
nF_isPal(V) -> nF_isNePal(V)
nF_isNeList(__(V1,V2)) -> nF_and(isNeList(V1),isList(V2))
nF_isNeList(__(V1,V2)) -> nF_isNeList(V1)
nF_isList(V) -> nF_isNeList(V)
nF_isNeList(__(V1,V2)) -> nF_isList(V1)
nF_isList(__(V1,V2)) -> nF_isList(V1)
nF_isNeList(__(V1,V2)) -> nF_and(isList(V1),isNeList(V2))
nF_isList(__(V1,V2)) -> nF_and(isList(V1),isList(V2))


Polynomial Interpretation:

[__](X1,X2) = X1 + X2 + 1
[__](X1,X2) = X1 + X2 + 1
[isPal](X) = X + 1
[__](X1,X2) = X1 + X2 + 1
[isList](X) = X + 1
[__](X1,X2) = X1 + X2 + 1
[__](X1,X2) = X1 + X2 + 1
[__](X1,X2) = X1 + X2 + 1
[__](X1,X2) = X1 + X2 + 1
[isNeList](X) = X + 1
[and](X1,X2) = X1 + X2 + 1
[isNePal](X) = X + 1

TIME: 0.15382799999999996

Termination proved: There exists a projection such that there are no minimal mu-rewrite sequences in cycle.
-> -> Dependency pairs in cycle:
nF_isNeList(__(V1,V2)) -> nF_isNeList(V1)
nF_isList(V) -> nF_isNeList(V)
nF_isList(__(V1,V2)) -> nF_isList(V1)
nF_isNeList(__(V1,V2)) -> nF_isList(V1)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for PALINDROME_nokinds_1_2:
-> -> Dependency pairs in cycle:
nF___(__(X,Y),Z) -> nF___(X,__(Y,Z))
nF___(__(X,Y),Z) -> nF___(Y,Z)

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.