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
  ↳11 RisEmptyProof (⇔)
    ↳12 TRUE

(0) Obligation:

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

double(0) → 0
double(s(x)) → s(s(double(x)))
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
if(0, y, z) → y
if(s(x), y, z) → z
half(double(x)) → x

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(-(x1, x2)) = 1 + x1 + x2   
POL(0) = 0   
POL(double(x1)) = 1 + x1   
POL(half(x1)) = 1 + x1   
POL(if(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(s(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

double(0) → 0
half(0) → 0
half(s(0)) → 0
-(x, 0) → x
if(0, y, z) → y
if(s(x), y, z) → z
half(double(x)) → x


(2) Obligation:

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

double(s(x)) → s(s(double(x)))
half(s(s(x))) → s(half(x))
-(s(x), s(y)) → -(x, y)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

half(s(s(x))) → s(half(x))
-(s(x), s(y)) → -(x, y)


(4) Obligation:

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

double(s(x)) → s(s(double(x)))

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
double1 > s1

Status:
s1: [1]
double1: [1]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

double(s(x)) → s(s(double(x)))


(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

(11) RisEmptyProof (EQUIVALENT transformation)

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

(12) TRUE