Term Rewriting System R:
[x, y]
f(0) -> 0
f(s(0)) -> s(0)
f(s(s(x))) -> p(h(g(x)))
f(s(s(x))) -> +(p(g(x)), q(g(x)))
g(0) -> pair(s(0), s(0))
g(s(x)) -> h(g(x))
g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
h(x) -> pair(+(p(x), q(x)), p(x))
p(pair(x, y)) -> x
q(pair(x, y)) -> y
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(s(x))) -> P(h(g(x)))
F(s(s(x))) -> H(g(x))
F(s(s(x))) -> G(x)
F(s(s(x))) -> +'(p(g(x)), q(g(x)))
F(s(s(x))) -> P(g(x))
F(s(s(x))) -> Q(g(x))
G(s(x)) -> H(g(x))
G(s(x)) -> G(x)
G(s(x)) -> +'(p(g(x)), q(g(x)))
G(s(x)) -> P(g(x))
G(s(x)) -> Q(g(x))
H(x) -> +'(p(x), q(x))
H(x) -> P(x)
H(x) -> Q(x)
+'(x, s(y)) -> +'(x, y)

Furthermore, R contains two SCCs.

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

Dependency Pair:

+'(x, s(y)) -> +'(x, y)

Rules:

f(0) -> 0
f(s(0)) -> s(0)
f(s(s(x))) -> p(h(g(x)))
f(s(s(x))) -> +(p(g(x)), q(g(x)))
g(0) -> pair(s(0), s(0))
g(s(x)) -> h(g(x))
g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
h(x) -> pair(+(p(x), q(x)), p(x))
p(pair(x, y)) -> x
q(pair(x, y)) -> y
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

The following dependency pair can be strictly oriented:

+'(x, s(y)) -> +'(x, y)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(s(x1)) =  1 + x1 POL(+'(x1, x2)) =  x1 + x2

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

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

Dependency Pair:

Rules:

f(0) -> 0
f(s(0)) -> s(0)
f(s(s(x))) -> p(h(g(x)))
f(s(s(x))) -> +(p(g(x)), q(g(x)))
g(0) -> pair(s(0), s(0))
g(s(x)) -> h(g(x))
g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
h(x) -> pair(+(p(x), q(x)), p(x))
p(pair(x, y)) -> x
q(pair(x, y)) -> y
+(x, 0) -> x
+(x, s(y)) -> s(+(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`

Dependency Pair:

G(s(x)) -> G(x)

Rules:

f(0) -> 0
f(s(0)) -> s(0)
f(s(s(x))) -> p(h(g(x)))
f(s(s(x))) -> +(p(g(x)), q(g(x)))
g(0) -> pair(s(0), s(0))
g(s(x)) -> h(g(x))
g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
h(x) -> pair(+(p(x), q(x)), p(x))
p(pair(x, y)) -> x
q(pair(x, y)) -> y
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

The following dependency pair can be strictly oriented:

G(s(x)) -> G(x)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(G(x1)) =  x1 POL(s(x1)) =  1 + x1

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

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

Dependency Pair:

Rules:

f(0) -> 0
f(s(0)) -> s(0)
f(s(s(x))) -> p(h(g(x)))
f(s(s(x))) -> +(p(g(x)), q(g(x)))
g(0) -> pair(s(0), s(0))
g(s(x)) -> h(g(x))
g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
h(x) -> pair(+(p(x), q(x)), p(x))
p(pair(x, y)) -> x
q(pair(x, y)) -> y
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

Using the Dependency Graph resulted in no new DP problems.

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