Term Rewriting System R:
[x, y]
f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(x)) -> F(x)
G(x, c(y)) -> G(x, y)
G(x, c(y)) -> G(x, if(f(x), c(g(s(x), y)), c(y)))
G(x, c(y)) -> IF(f(x), c(g(s(x), y)), c(y))
G(x, c(y)) -> F(x)
G(x, c(y)) -> G(s(x), y)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Remaining Obligation(s)`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pair:

F(s(x)) -> F(x)

Rules:

f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

• Dependency Pairs:

G(x, c(y)) -> G(s(x), y)
G(x, c(y)) -> G(x, if(f(x), c(g(s(x), y)), c(y)))
G(x, c(y)) -> G(x, y)

Rules:

f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Remaining Obligation(s)`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pair:

F(s(x)) -> F(x)

Rules:

f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

• Dependency Pairs:

G(x, c(y)) -> G(s(x), y)
G(x, c(y)) -> G(x, if(f(x), c(g(s(x), y)), c(y)))
G(x, c(y)) -> G(x, y)

Rules:

f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

Termination of R could not be shown.
Duration:
0:00 minutes