(VAR X Y Z)
(STRATEGY CONTEXTSENSITIVE 
  (fst 1 2)
  (s)
  (cons 1)
  (0)
  (nil)
)
(RULES 
fst(s(X),cons(Y,Z)) -> cons(Y,fst(X,Z))
fst(0,Z) -> 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 Ex2_Luc03b_1:

-> Dependency pairs:
No dependency pairs found.

-> Proof of termination for Ex2_Luc03b_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)
)
(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 Ex2_Luc03b_2:

-> Dependency pairs:
No dependency pairs found.

-> Proof of termination for Ex2_Luc03b_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

(VAR X Y)
(STRATEGY CONTEXTSENSITIVE 
  (add 1 2)
  (s)
  (0)
)
(RULES 
add(s(X),Y) -> s(add(X,Y))
add(0,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 Ex2_Luc03b_3:

-> Dependency pairs:
No dependency pairs found.

-> Proof of termination for Ex2_Luc03b_3:
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 Z)
(STRATEGY CONTEXTSENSITIVE 
  (len 1)
  (cons 1)
  (s)
  (nil)
  (0)
)
(RULES 
len(cons(X,Z)) -> s(len(Z))
len(nil) -> 0
)

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 Ex2_Luc03b_4:

-> Dependency pairs:
No dependency pairs found.

-> Proof of termination for Ex2_Luc03b_4:
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.