Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
  ↳7 RisEmptyProof (⇔)
    ↳8 TRUE
  ↳9 RisEmptyProof (⇔)
    ↳10 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

rev(nil) → nil
rev(.(x, y)) → ++(rev(y), .(x, nil))
car(.(x, y)) → x
cdr(.(x, y)) → y
null(nil) → true
null(.(x, y)) → false
++(nil, y) → y
++(.(x, y), z) → .(x, ++(y, z))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(++(x1, x2)) = x1 + x2   
POL(.(x1, x2)) = x1 + x2   
POL(car(x1)) = 1 + x1   
POL(cdr(x1)) = 1 + x1   
POL(false) = 0   
POL(nil) = 0   
POL(null(x1)) = 1 + x1   
POL(rev(x1)) = 1 + x1   
POL(true) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

rev(nil) → nil
car(.(x, y)) → x
cdr(.(x, y)) → y
null(nil) → true
null(.(x, y)) → false


(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

rev(.(x, y)) → ++(rev(y), .(x, nil))
++(nil, y) → y
++(.(x, y), z) → .(x, ++(y, z))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(++(x1, x2)) = x1 + x2   
POL(.(x1, x2)) = 2 + 2·x1 + x2   
POL(nil) = 1   
POL(rev(x1)) = 2·x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

rev(.(x, y)) → ++(rev(y), .(x, nil))
++(nil, y) → y


(4) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

++(.(x, y), z) → .(x, ++(y, z))

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(++(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(.(x1, x2)) = 1 + 2·x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

++(.(x, y), z) → .(x, ++(y, z))


(6) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(7) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(8) TRUE

(9) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(10) TRUE