Term Rewriting System R:
[X, Y, Z]
f(X) -> cons(X, n_{f}(g(X)))
f(X) -> n_{f}(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(n_{f}(X)) -> f(X)
activate(X) -> X
Innermost 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(n_{f}(X)) -> F(X)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
→DP Problem 2
↳UsableRules
Dependency Pair:
G(s(X)) -> G(X)
Rules:
f(X) -> cons(X, n_{f}(g(X)))
f(X) -> n_{f}(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(n_{f}(X)) -> f(X)
activate(X) -> X
Strategy:
innermost
As we are in the innermost case, we can delete all 8 non-usable-rules.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 3
↳Size-Change Principle
→DP Problem 2
↳UsableRules
Dependency Pair:
G(s(X)) -> G(X)
Rule:
none
Strategy:
innermost
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):
D_{P}: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
s(x_{1}) -> s(x_{1})
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳Usable Rules (Innermost)
Dependency Pair:
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
Rules:
f(X) -> cons(X, n_{f}(g(X)))
f(X) -> n_{f}(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(n_{f}(X)) -> f(X)
activate(X) -> X
Strategy:
innermost
As we are in the innermost case, we can delete all 2 non-usable-rules.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳UsableRules
→DP Problem 4
↳Size-Change Principle
Dependency Pair:
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
Rules:
f(X) -> n_{f}(X)
f(X) -> cons(X, n_{f}(g(X)))
g(s(X)) -> s(s(g(X)))
g(0) -> s(0)
activate(X) -> X
activate(n_{f}(X)) -> f(X)
Strategy:
innermost
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):
D_{P}: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
cons(x_{1}, x_{2}) -> cons(x_{1}, x_{2})
s(x_{1}) -> s(x_{1})
We obtain no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes