Term Rewriting System R:
[x, y]
f(c(s(x), y)) -> f(c(x, s(y)))
f(c(s(x), s(y))) -> g(c(x, y))
g(c(x, s(y))) -> g(c(s(x), y))
g(c(s(x), s(y))) -> f(c(x, y))
Innermost Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(c(s(x), s(y))) -> g(c(x, y))
g(c(s(x), s(y))) -> f(c(x, y))
where the Polynomial interpretation:
_{ }^{ }POL(c(x_{1}, x_{2})) | = x_{1} + x_{2}_{ }^{ } |
_{ }^{ }POL(g(x_{1})) | = x_{1}_{ }^{ } |
_{ }^{ }POL(s(x_{1})) | = 1 + x_{1}_{ }^{ } |
_{ }^{ }POL(f(x_{1})) | = x_{1}_{ }^{ } |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
G(c(x, s(y))) -> G(c(s(x), y))
F(c(s(x), y)) -> F(c(x, s(y)))
Furthermore, R contains two SCCs.
R
↳RRRPolo
→TRS2
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
→DP Problem 2
↳UsableRules
Dependency Pair:
G(c(x, s(y))) -> G(c(s(x), y))
Rules:
g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
Strategy:
innermost
As we are in the innermost case, we can delete all 2 non-usable-rules.
R
↳RRRPolo
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
...
→DP Problem 3
↳Size-Change Principle
→DP Problem 2
↳UsableRules
Dependency Pair:
G(c(x, s(y))) -> G(c(s(x), y))
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- G(c(x, s(y))) -> G(c(s(x), y))
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:
c(x_{1}, x_{2}) -> c(x_{2})
s(x_{1}) -> s(x_{1})
We obtain no new DP problems.
R
↳RRRPolo
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳Usable Rules (Innermost)
Dependency Pair:
F(c(s(x), y)) -> F(c(x, s(y)))
Rules:
g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
Strategy:
innermost
As we are in the innermost case, we can delete all 2 non-usable-rules.
R
↳RRRPolo
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳UsableRules
...
→DP Problem 4
↳Size-Change Principle
Dependency Pair:
F(c(s(x), y)) -> F(c(x, s(y)))
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- F(c(s(x), y)) -> F(c(x, s(y)))
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:
c(x_{1}, x_{2}) -> c(x_{1})
s(x_{1}) -> s(x_{1})
We obtain no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes