Term Rewriting System R:
[x, y]
f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(g(x, y)) -> G(f(x), f(y))
F(g(x, y)) -> F(x)
F(g(x, y)) -> F(y)
F(h(x, y)) -> G(h(y, f(x)), h(x, f(y)))
F(h(x, y)) -> F(x)
F(h(x, y)) -> F(y)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Argument Filtering and Ordering

Dependency Pairs:

F(h(x, y)) -> F(y)
F(h(x, y)) -> F(x)
F(g(x, y)) -> F(y)
F(g(x, y)) -> F(x)

Rules:

f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(h(x, y)) -> F(y)
F(h(x, y)) -> F(x)

There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1, x2)) =  x1 + x2 POL(h(x1, x2)) =  1 + x1 + x2 POL(F(x1)) =  x1

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
h(x1, x2) -> h(x1, x2)
g(x1, x2) -> g(x1, x2)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
Argument Filtering and Ordering

Dependency Pairs:

F(g(x, y)) -> F(y)
F(g(x, y)) -> F(x)

Rules:

f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(g(x, y)) -> F(y)
F(g(x, y)) -> F(x)

There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1, x2)) =  1 + x1 + x2 POL(F(x1)) =  x1

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
g(x1, x2) -> g(x1, x2)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
...
→DP Problem 3
Dependency Graph

Dependency Pair:

Rules:

f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes