Term Rewriting System R:
[x, y, z]
f(f(f(a, x), y), z) -> f(f(x, z), f(y, z))
f(f(b, x), y) -> x
f(c, y) -> 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(f(f(a, x), y), z) -> F(f(x, z), f(y, z))
F(f(f(a, x), y), z) -> F(x, z)
F(f(f(a, x), y), z) -> F(y, z)
Furthermore, R contains one SCC.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳Non Termination
Dependency Pairs:
F(f(f(a, x), y), z) -> F(y, z)
F(f(f(a, x), y), z) -> F(x, z)
F(f(f(a, x), y), z) -> F(f(x, z), f(y, z))
Rules:
f(f(f(a, x), y), z) -> f(f(x, z), f(y, z))
f(f(b, x), y) -> x
f(c, y) -> y
Strategy:
innermost
Found an infinite P-chain over R:
P =
F(f(f(a, x), y), z) -> F(y, z)
F(f(f(a, x), y), z) -> F(x, z)
F(f(f(a, x), y), z) -> F(f(x, z), f(y, z))
R =
f(f(f(a, x), y), z) -> f(f(x, z), f(y, z))
f(f(b, x), y) -> x
f(c, y) -> y
s = F(f(f(a, f(a, x'''')), c), f(c, f(f(a, x''''), f(f(a, f(a, x'''')), c))))
evaluates to t =F(f(f(a, f(a, x'''')), c), f(c, f(f(a, x''''), f(f(a, f(a, x'''')), c))))
Thus, s starts an infinite chain.
Non-Termination of R could be shown.
Duration:
0:00 minutes