Term Rewriting System R:
[X, Y, Z]
from(X) -> cons(X, n_{from}(n_{s}(X)))
from(X) -> n_{from}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
s(X) -> n_{s}(X)
activate(n_{from}(X)) -> from(activate(X))
activate(n_{s}(X)) -> s(activate(X))
activate(X) -> X
Innermost Termination of R to be shown.
R
↳Removing Redundant Rules for Innermost Termination
Removing the following rules from R which left hand sides contain non normal subterms
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
R
↳RRRI
→TRS2
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
ACTIVATE(n_{from}(X)) -> FROM(activate(X))
ACTIVATE(n_{from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{s}(X)) -> S(activate(X))
ACTIVATE(n_{s}(X)) -> ACTIVATE(X)
Furthermore, R contains one SCC.
R
↳RRRI
→TRS2
↳DPs
→DP Problem 1
↳Size-Change Principle
Dependency Pairs:
ACTIVATE(n_{s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{from}(X)) -> ACTIVATE(X)
Rules:
from(X) -> cons(X, n_{from}(n_{s}(X)))
from(X) -> n_{from}(X)
sel(0, cons(X, Y)) -> X
s(X) -> n_{s}(X)
activate(n_{from}(X)) -> from(activate(X))
activate(n_{s}(X)) -> s(activate(X))
activate(X) -> X
We number the DPs as follows:
- ACTIVATE(n_{s}(X)) -> ACTIVATE(X)
- ACTIVATE(n_{from}(X)) -> ACTIVATE(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:
n_{from}(x_{1}) -> n_{from}(x_{1})
n_{s}(x_{1}) -> n_{s}(x_{1})
We obtain no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes