YES
Termination proof of ../tpdb/TRS/CSR/Ex25_Luc06.trs
Termination of the following Term Rewriting System could be proven:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(X)) → c(f(g(f(X))))
c(X) → d(X)
h(X) → c(d(X))
The replacement map contains the following entries:f: {1}
c: empty set
g: empty set
d: empty set
h: {1}
↳ CSR
↳ Lucas-Transformation
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(X)) → c(f(g(f(X))))
c(X) → d(X)
h(X) → c(d(X))
The replacement map contains the following entries:f: {1}
c: empty set
g: empty set
d: empty set
h: {1}
We applied the Lucas [26] to transform the context-sensitive TRS to an usual TRS.
↳ CSR
↳ Lucas-Transformation
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(X)) → c
c → d
h(X) → c
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(X)) → c
c → d
h(X) → c
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
f(f(X)) → c
c → d
h(X) → c
Used ordering:
Polynomial interpretation [25]:
POL(c) = 1
POL(d) = 0
POL(f(x1)) = 2 + 2·x1
POL(h(x1)) = 2 + x1
↳ CSR
↳ Lucas-Transformation
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RisEmptyProof
Q restricted rewrite system:
R is empty.
Q is empty.
The TRS R is empty. Hence, termination is trivially proven.