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