Term Rewriting System R:
[X, Y, Z]
from(X) -> cons(X, n_{from}(s(X)))
from(X) -> n_{from}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{from}(X)) -> from(X)
activate(X) -> X
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{from}(X)) -> FROM(X)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pair:
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
Rules:
from(X) -> cons(X, n_{from}(s(X)))
from(X) -> n_{from}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{from}(X)) -> from(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
↳Size-Change Principle
Dependency Pair:
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
Rules:
from(X) -> cons(X, n_{from}(s(X)))
from(X) -> n_{from}(X)
activate(n_{from}(X)) -> from(X)
activate(X) -> 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