Term Rewriting System R:
[x]
app(f, app(g, x)) -> app(g, app(f, app(f, x)))
app(f, app(h, x)) -> app(h, app(g, x))
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:
APP(f, app(g, x)) -> APP(g, app(f, app(f, x)))
APP(f, app(g, x)) -> APP(f, app(f, x))
APP(f, app(g, x)) -> APP(f, x)
APP(f, app(h, x)) -> APP(h, app(g, x))
APP(f, app(h, x)) -> APP(g, x)
Furthermore, R contains one SCC.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳A-Transformation
Dependency Pairs:
APP(f, app(g, x)) -> APP(f, x)
APP(f, app(g, x)) -> APP(f, app(f, x))
Rules:
app(f, app(g, x)) -> app(g, app(f, app(f, x)))
app(f, app(h, x)) -> app(h, app(g, x))
Strategy:
innermost
We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳ATrans
...
→DP Problem 2
↳Negative Polynomial Order
Dependency Pairs:
F(g(x)) -> F(x)
F(g(x)) -> F(f(x))
Rules:
f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
Strategy:
innermost
The following Dependency Pairs can be strictly oriented using the given order.
F(g(x)) -> F(x)
F(g(x)) -> F(f(x))
Moreover, the following usable rules (regarding the implicit AFS) are oriented.
f(h(x)) -> h(g(x))
f(g(x)) -> g(f(f(x)))
Used ordering:
Polynomial Order with Interpretation:
POL( F(x1) ) = x1
POL( g(x1) ) = x1 + 1
POL( f(x1) ) = x1
POL( h(x1) ) = 0
This results in one new DP problem.
R
↳OC
→TRS2
↳DPs
→DP Problem 1
↳ATrans
...
→DP Problem 3
↳Dependency Graph
Dependency Pair:
Rules:
f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes