(VAR X Y Z)
(STRATEGY CONTEXTSENSITIVE 
  (first 1 2)
  (s 1)
  (cons 1)
  (0)
  (nil)
)
(RULES 
first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
first(0,X) -> nil
)

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 Ex6_Luc98_1:

-> Dependency pairs:
No dependency pairs found.

-> Proof of termination for Ex6_Luc98_1:
Termination proved: No cycles in dependency graph.

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

(VAR X)
(STRATEGY CONTEXTSENSITIVE 
  (from 1)
  (cons 1)
  (s 1)
)
(RULES 
from(X) -> cons(X,from(s(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 Ex6_Luc98_2:

-> Dependency pairs:
No dependency pairs found.

-> Proof of termination for Ex6_Luc98_2:
Termination proved: No cycles in dependency graph.

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.