Term Rewriting System R:
[X, Y]
f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(X)) -> F(X)
G(cons(0, Y)) -> G(Y)
H(cons(X, Y)) -> H(g(cons(X, Y)))
H(cons(X, Y)) -> G(cons(X, Y))

Furthermore, R contains three SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

F(s(X)) -> F(X)

Rules:

f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))

The following dependency pair can be strictly oriented:

F(s(X)) -> F(X)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 4`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

Rules:

f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

G(cons(0, Y)) -> G(Y)

Rules:

f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))

The following dependency pair can be strictly oriented:

G(cons(0, Y)) -> G(Y)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 5`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

Rules:

f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳Argument Filtering and Ordering`

Dependency Pair:

H(cons(X, Y)) -> H(g(cons(X, Y)))

Rules:

f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))

The following dependency pair can be strictly oriented:

H(cons(X, Y)) -> H(g(cons(X, Y)))

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

g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{cons, g, s}

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`           →DP Problem 6`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

f(s(X)) -> f(X)
g(cons(0, Y)) -> g(Y)
g(cons(s(X), Y)) -> s(X)
h(cons(X, Y)) -> h(g(cons(X, Y)))

Using the Dependency Graph resulted in no new DP problems.

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