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 QTRSRRRProof (⇔)
↳8 QTRS
↳9 QTRSRRRProof (⇔)
↳10 QTRS
↳11 QTRSRRRProof (⇔)
↳12 QTRS
↳13 QTRSRRRProof (⇔)
↳14 QTRS
  ↳15 RisEmptyProof (⇔)
    ↳16 TRUE
  ↳17 RisEmptyProof (⇔)
    ↳18 TRUE
  ↳19 RisEmptyProof (⇔)
    ↳20 TRUE

(0) Obligation:

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

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

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

make(x) → .(x, nil)


(2) Obligation:

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

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

++(nil, y) → y
++(x, nil) → x


(4) Obligation:

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

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

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(++(x1, x2)) = x1 + x2   
POL(.(x1, x2)) = 1 + 2·x1 + x2   
POL(nil) = 2   
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(nil) → nil


(6) Obligation:

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

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

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(++(x1, x2)) = 2 + x1 + x2   
POL(.(x1, x2)) = 2·x1 + x2   
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), rev(x))


(8) Obligation:

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

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

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

rev(rev(x)) → x


(10) Obligation:

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

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

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(++(x1, x2)) = 2 + x1 + 2·x2   
POL(.(x1, x2)) = 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)


(12) Obligation:

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

++(.(x, y), z) → .(x, ++(y, z))

Q is empty.

(13) 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))


(14) Obligation:

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

(15) RisEmptyProof (EQUIVALENT transformation)

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

(16) TRUE

(17) RisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) RisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE