Term Rewriting System R:
[x, y]
f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(f(x)) -> F(g(f(x), x))
F(f(x)) -> G(f(x), x)
F(f(x)) -> F(h(f(x), f(x)))
F(f(x)) -> H(f(x), f(x))
H(x, x) -> G(x, 0)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Argument Filtering and Ordering

Dependency Pairs:

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

Rules:

f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

The following usable rules for innermost w.r.t. to the AFS can be oriented:

g(x, y) -> y
h(x, x) -> g(x, 0)
f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(F(x1)) =  x1 POL(f(x1)) =  1 + x1

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

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

Dependency Pair:

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

Rules:

f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

The following usable rules for innermost w.r.t. to the AFS can be oriented:

h(x, x) -> g(x, 0)
g(x, y) -> y

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(h) =  0 POL(F(x1)) =  x1 POL(f(x1)) =  1 + x1

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

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

Dependency Pair:

Rules:

f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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