Term Rewriting System R:
[l, x, k, a, b, c]
f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(cons(x, k), l) -> G(k, l, cons(x, k))
G(a, b, c) -> F(a, cons(b, c))
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pairs:
G(a, b, c) -> F(a, cons(b, c))
F(cons(x, k), l) -> G(k, l, cons(x, k))
Rules:
f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))
Strategy:
innermost
As we are in the innermost case, we can delete all 3 non-usable-rules.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳Size-Change Principle
Dependency Pairs:
G(a, b, c) -> F(a, cons(b, c))
F(cons(x, k), l) -> G(k, l, cons(x, k))
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- G(a, b, c) -> F(a, cons(b, c))
- F(cons(x, k), l) -> G(k, l, cons(x, k))
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})
We obtain no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes