YES
Termination w.r.t. Q proof of ../tpdb/TRS/AG01/#3.7.trs
Termination w.r.t. Q of the following Term Rewriting System could be proven:
Q restricted rewrite system:
The TRS R consists of the following rules:
half(0) → 0
half(s(s(x))) → s(half(x))
log(s(0)) → 0
log(s(s(x))) → s(log(s(half(x))))
Q is empty.
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
half(0) → 0
half(s(s(x))) → s(half(x))
log(s(0)) → 0
log(s(s(x))) → s(log(s(half(x))))
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
half(0) → 0
half(s(s(x))) → s(half(x))
log(s(0)) → 0
log(s(s(x))) → s(log(s(half(x))))
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
log(s(0)) → 0
Used ordering:
Polynomial interpretation [25]:
POL(0) = 2
POL(half(x1)) = x1
POL(log(x1)) = 2·x1
POL(s(x1)) = 2·x1
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
half(0) → 0
half(s(s(x))) → s(half(x))
log(s(s(x))) → s(log(s(half(x))))
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
half(0) → 0
half(s(s(x))) → s(half(x))
log(s(s(x))) → s(log(s(half(x))))
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
half(s(s(x))) → s(half(x))
log(s(s(x))) → s(log(s(half(x))))
Used ordering:
Polynomial interpretation [25]:
POL(0) = 0
POL(half(x1)) = x1
POL(log(x1)) = 2·x1
POL(s(x1)) = 1 + x1
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
half(0) → 0
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
half(0) → 0
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
half(0) → 0
Used ordering:
Polynomial interpretation [25]:
POL(0) = 2
POL(half(x1)) = 2·x1
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RisEmptyProof
Q restricted rewrite system:
R is empty.
Q is empty.
The TRS R is empty. Hence, termination is trivially proven.