Term Rewriting System R:
[x]
f(1) -> f(g(1))
f(f(x)) -> f(x)
g(0) -> g(f(0))
g(g(x)) -> g(x)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(1) -> F(g(1))
F(1) -> G(1)
G(0) -> G(f(0))
G(0) -> F(0)

Furthermore, R contains two SCCs.

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

Dependency Pair:

F(1) -> F(g(1))

Rules:

f(1) -> f(g(1))
f(f(x)) -> f(x)
g(0) -> g(f(0))
g(g(x)) -> g(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(1) -> F(g(1))

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

g(0) -> g(f(0))
g(g(x)) -> g(x)
f(1) -> f(g(1))
f(f(x)) -> f(x)

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

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 3`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pair:

Rules:

f(1) -> f(g(1))
f(f(x)) -> f(x)
g(0) -> g(f(0))
g(g(x)) -> g(x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

G(0) -> G(f(0))

Rules:

f(1) -> f(g(1))
f(f(x)) -> f(x)
g(0) -> g(f(0))
g(g(x)) -> g(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

G(0) -> G(f(0))

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

g(0) -> g(f(0))
g(g(x)) -> g(x)
f(1) -> f(g(1))
f(f(x)) -> f(x)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  1 POL(g(x1)) =  x1 POL(G(x1)) =  x1 POL(f) =  0

resulting in one new DP problem.
Used Argument Filtering System:
G(x1) -> G(x1)
f(x1) -> f
g(x1) -> g(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 4`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

f(1) -> f(g(1))
f(f(x)) -> f(x)
g(0) -> g(f(0))
g(g(x)) -> g(x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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