(VAR X Y)
(STRATEGY CONTEXTSENSITIVE 
  (eq)
  (false)
  (s)
  (0)
  (true)
)
(RULES 
eq(X,Y) -> false
eq(s(X),s(Y)) -> eq(X,Y)
eq(0,0) -> true
)

Proving termination of context-sensitive rewriting for Ex1_GL02a_1:

-> Dependency pairs:
nF_eq(s(X),s(Y)) -> nF_eq(X,Y)

-> Proof of termination for Ex1_GL02a_1:
-> -> Dependency pairs in cycle:
nF_eq(s(X),s(Y)) -> nF_eq(X,Y)


Polynomial Interpretation:

[s](X) = X + 1

TIME: 1.488299999999998e-2

Termination proved: There exists a projection such that there are no minimal mu-rewrite sequences in cycle.

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 
  (inf 1)
  (cons)
  (s)
)
(RULES 
inf(X) -> cons(X,inf(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 Ex1_GL02a_2:

-> Dependency pairs:
No dependency pairs found.

-> Proof of termination for Ex1_GL02a_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 L)
(STRATEGY CONTEXTSENSITIVE 
  (take 1 2)
  (s)
  (cons)
  (0)
  (nil)
)
(RULES 
take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
take(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 Ex1_GL02a_3:

-> Dependency pairs:
No dependency pairs found.

-> Proof of termination for Ex1_GL02a_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 L)
(STRATEGY CONTEXTSENSITIVE 
  (length 1)
  (cons)
  (s)
  (nil)
  (0)
)
(RULES 
length(cons(X,L)) -> s(length(L))
length(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 Ex1_GL02a_4:

-> Dependency pairs:
No dependency pairs found.

-> Proof of termination for Ex1_GL02a_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.