Term Rewriting System R:
[x, y]
f(x, c(y)) -> f(x, s(f(y, y)))
f(s(x), s(y)) -> f(x, s(c(s(y))))
Termination of R to be shown.
R
↳Overlay and local confluence Check
The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.
R
↳OC
→TRS2
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(x, c(y)) -> F(x, s(f(y, y)))
F(x, c(y)) -> F(y, y)
F(s(x), s(y)) -> F(x, s(c(s(y))))
Furthermore, R contains two SCCs.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
→DP Problem 2
↳UsableRules
Dependency Pair:
F(s(x), s(y)) -> F(x, s(c(s(y))))
Rules:
f(x, c(y)) -> f(x, s(f(y, y)))
f(s(x), s(y)) -> f(x, s(c(s(y))))
Strategy:
innermost
As we are in the innermost case, we can delete all 2 non-usable-rules.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
...
→DP Problem 3
↳Size-Change Principle
→DP Problem 2
↳UsableRules
Dependency Pair:
F(s(x), s(y)) -> F(x, s(c(s(y))))
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- F(s(x), s(y)) -> F(x, s(c(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}) -> c(x_{1})
s(x_{1}) -> s(x_{1})
We obtain no new DP problems.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳Usable Rules (Innermost)
Dependency Pair:
F(x, c(y)) -> F(y, y)
Rules:
f(x, c(y)) -> f(x, s(f(y, y)))
f(s(x), s(y)) -> f(x, s(c(s(y))))
Strategy:
innermost
As we are in the innermost case, we can delete all 2 non-usable-rules.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳UsableRules
...
→DP Problem 4
↳Size-Change Principle
Dependency Pair:
F(x, c(y)) -> F(y, y)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- F(x, c(y)) -> F(y, 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}) -> c(x_{1})
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes