Term Rewriting System R:
[x]
a(c(d(x))) -> c(x)
u(b(d(d(x)))) -> b(x)
v(a(a(x))) -> u(v(x))
v(a(c(x))) -> u(b(d(x)))
v(c(x)) -> b(x)
w(a(a(x))) -> u(w(x))
w(a(c(x))) -> u(b(d(x)))
w(c(x)) -> b(x)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

V(a(a(x))) -> U(v(x))
V(a(a(x))) -> V(x)
V(a(c(x))) -> U(b(d(x)))
W(a(a(x))) -> U(w(x))
W(a(a(x))) -> W(x)
W(a(c(x))) -> U(b(d(x)))

Furthermore, R contains two SCCs.

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

Dependency Pair:

V(a(a(x))) -> V(x)

Rules:

a(c(d(x))) -> c(x)
u(b(d(d(x)))) -> b(x)
v(a(a(x))) -> u(v(x))
v(a(c(x))) -> u(b(d(x)))
v(c(x)) -> b(x)
w(a(a(x))) -> u(w(x))
w(a(c(x))) -> u(b(d(x)))
w(c(x)) -> b(x)

The following dependency pair can be strictly oriented:

V(a(a(x))) -> V(x)

The following rules can be oriented:

a(c(d(x))) -> c(x)
u(b(d(d(x)))) -> b(x)
v(a(a(x))) -> u(v(x))
v(a(c(x))) -> u(b(d(x)))
v(c(x)) -> b(x)
w(a(a(x))) -> u(w(x))
w(a(c(x))) -> u(b(d(x)))
w(c(x)) -> b(x)

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{d, a} > u
{d, a} > b
{d, a} > v
w > u
c > b

resulting in one new DP problem.
Used Argument Filtering System:
V(x1) -> V(x1)
a(x1) -> a(x1)
c(x1) -> c(x1)
d(x1) -> d(x1)
u(x1) -> u(x1)
b(x1) -> b(x1)
v(x1) -> v(x1)
w(x1) -> w(x1)

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

Dependency Pair:

Rules:

a(c(d(x))) -> c(x)
u(b(d(d(x)))) -> b(x)
v(a(a(x))) -> u(v(x))
v(a(c(x))) -> u(b(d(x)))
v(c(x)) -> b(x)
w(a(a(x))) -> u(w(x))
w(a(c(x))) -> u(b(d(x)))
w(c(x)) -> b(x)

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:

W(a(a(x))) -> W(x)

Rules:

a(c(d(x))) -> c(x)
u(b(d(d(x)))) -> b(x)
v(a(a(x))) -> u(v(x))
v(a(c(x))) -> u(b(d(x)))
v(c(x)) -> b(x)
w(a(a(x))) -> u(w(x))
w(a(c(x))) -> u(b(d(x)))
w(c(x)) -> b(x)

The following dependency pair can be strictly oriented:

W(a(a(x))) -> W(x)

The following rules can be oriented:

a(c(d(x))) -> c(x)
u(b(d(d(x)))) -> b(x)
v(a(a(x))) -> u(v(x))
v(a(c(x))) -> u(b(d(x)))
v(c(x)) -> b(x)
w(a(a(x))) -> u(w(x))
w(a(c(x))) -> u(b(d(x)))
w(c(x)) -> b(x)

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
a > d
a > u
a > v > {w, b}

resulting in one new DP problem.
Used Argument Filtering System:
W(x1) -> W(x1)
a(x1) -> a(x1)
c(x1) -> c(x1)
d(x1) -> d(x1)
u(x1) -> u(x1)
b(x1) -> b(x1)
v(x1) -> v(x1)
w(x1) -> w(x1)

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

Dependency Pair:

Rules:

a(c(d(x))) -> c(x)
u(b(d(d(x)))) -> b(x)
v(a(a(x))) -> u(v(x))
v(a(c(x))) -> u(b(d(x)))
v(c(x)) -> b(x)
w(a(a(x))) -> u(w(x))
w(a(c(x))) -> u(b(d(x)))
w(c(x)) -> b(x)

Using the Dependency Graph resulted in no new DP problems.

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