Term Rewriting System R:
[X, Y, Z]
f(X) -> cons(X, nf(g(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(X)
activate(X) -> X
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(X) -> G(X)
G(s(X)) -> G(X)
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nf(X)) -> F(X)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Size-Change Principle
→DP Problem 2
↳SCP
Dependency Pair:
G(s(X)) -> G(X)
Rules:
f(X) -> cons(X, nf(g(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(X)
activate(X) -> X
We number the DPs as follows:
- G(s(X)) -> G(X)
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
s(x1) -> s(x1)
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳Size-Change Principle
Dependency Pair:
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
Rules:
f(X) -> cons(X, nf(g(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(X)
activate(X) -> X
We number the DPs as follows:
- SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
cons(x1, x2) -> cons(x1, x2)
s(x1) -> s(x1)
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes