(VAR X Z N Y)
(STRATEGY CONTEXTSENSITIVE 
  (from 1)
  (cons 1)
  (s 1)
  (2ndspos 1 2)
  (0)
  (rnil)
  (rcons 1 2)
  (posrecip 1)
  (2ndsneg 1 2)
  (negrecip 1)
  (pi 1)
  (plus 1 2)
  (times 1 2)
  (square 1)
)
(RULES 
from(X) -> cons(X,from(s(X)))
2ndspos(0,Z) -> rnil
2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
2ndsneg(0,Z) -> rnil
2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
pi(X) -> 2ndspos(X,from(0))
plus(0,Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0,Y) -> 0
times(s(X),Y) -> plus(Y,times(X,Y))
square(X) -> times(X,X)
)

The TRS is orthogonal, thus termination of innermost context-sensitive rewriting is equivalent to termination of context-sensitive rewriting.

Proving termination of context-sensitive rewriting for Ex1_2_AEL03:

-> Dependency pairs:
nF_2ndspos(s(N),cons(X,cons(Y,Z))) -> nF_2ndsneg(N,Z)
nF_2ndspos(s(N),cons(X,cons(Y,Z))) -> Y
nF_2ndspos(s(N),cons(X,cons(Y,Z))) -> Z
nF_2ndsneg(s(N),cons(X,cons(Y,Z))) -> nF_2ndspos(N,Z)
nF_2ndsneg(s(N),cons(X,cons(Y,Z))) -> Y
nF_2ndsneg(s(N),cons(X,cons(Y,Z))) -> Z
nF_pi(X) -> nF_2ndspos(X,from(0))
nF_pi(X) -> nF_from(0)
nF_plus(s(X),Y) -> nF_plus(X,Y)
nF_times(s(X),Y) -> nF_plus(Y,times(X,Y))
nF_times(s(X),Y) -> nF_times(X,Y)
nF_square(X) -> nF_times(X,X)

-> Proof of termination for Ex1_2_AEL03_1_1:
-> -> Dependency pairs in cycle:
nF_times(s(X),Y) -> nF_times(X,Y)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for Ex1_2_AEL03_1_2:
-> -> Dependency pairs in cycle:
nF_plus(s(X),Y) -> nF_plus(X,Y)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for Ex1_2_AEL03_1_3:
-> -> Dependency pairs in cycle:
nF_2ndspos(s(N),cons(X,cons(Y,Z))) -> nF_2ndsneg(N,Z)
nF_2ndsneg(s(N),cons(X,cons(Y,Z))) -> nF_2ndspos(N,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.