(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)) = 2 + x1 + 2·x2
POL(0) = 0
POL(double(x1)) = 2·x1
POL(half(x1)) = 2 + 2·x1
POL(if(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(s(x1)) = 2 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
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
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
double(0) → 0
double(s(x)) → s(s(double(x)))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(double(x1)) = 1 + 2·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:
double(0) → 0
(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:
double1: multiset
s1: multiset
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) YES