Term Rewriting System R:
[x, y, n, m]
eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

EQ(s(x), s(y)) -> EQ(x, y)
LE(s(x), s(y)) -> LE(x, y)
APP(add(n, x), y) -> APP(x, y)
MIN(add(n, add(m, x))) -> IFMIN(le(n, m), add(n, add(m, x)))
MIN(add(n, add(m, x))) -> LE(n, m)
IFMIN(true, add(n, add(m, x))) -> MIN(add(n, x))
IFMIN(false, add(n, add(m, x))) -> MIN(add(m, x))
RM(n, add(m, x)) -> IFRM(eq(n, m), n, add(m, x))
RM(n, add(m, x)) -> EQ(n, m)
IFRM(true, n, add(m, x)) -> RM(n, x)
IFRM(false, n, add(m, x)) -> RM(n, x)
MINSORT(add(n, x), y) -> IFMINSORT(eq(n, min(add(n, x))), add(n, x), y)
MINSORT(add(n, x), y) -> EQ(n, min(add(n, x)))
MINSORT(add(n, x), y) -> MIN(add(n, x))
IFMINSORT(true, add(n, x), y) -> MINSORT(app(rm(n, x), y), nil)
IFMINSORT(true, add(n, x), y) -> APP(rm(n, x), y)
IFMINSORT(true, add(n, x), y) -> RM(n, x)
IFMINSORT(false, add(n, x), y) -> MINSORT(x, add(n, y))

Furthermore, R contains six SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Remaining`
`       →DP Problem 5`
`         ↳Remaining`
`       →DP Problem 6`
`         ↳Remaining`

Dependency Pair:

EQ(s(x), s(y)) -> EQ(x, y)

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

The following dependency pair can be strictly oriented:

EQ(s(x), s(y)) -> EQ(x, y)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
EQ(x1, x2) -> EQ(x1, x2)
s(x1) -> s(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 7`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Remaining`
`       →DP Problem 5`
`         ↳Remaining`
`       →DP Problem 6`
`         ↳Remaining`

Dependency Pair:

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, 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`
`       →DP Problem 4`
`         ↳Remaining`
`       →DP Problem 5`
`         ↳Remaining`
`       →DP Problem 6`
`         ↳Remaining`

Dependency Pair:

LE(s(x), s(y)) -> LE(x, y)

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

The following dependency pair can be strictly oriented:

LE(s(x), s(y)) -> LE(x, y)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
LE(x1, x2) -> LE(x1, x2)
s(x1) -> s(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 8`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Remaining`
`       →DP Problem 5`
`         ↳Remaining`
`       →DP Problem 6`
`         ↳Remaining`

Dependency Pair:

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, 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`
`       →DP Problem 4`
`         ↳Remaining`
`       →DP Problem 5`
`         ↳Remaining`
`       →DP Problem 6`
`         ↳Remaining`

Dependency Pair:

APP(add(n, x), y) -> APP(x, y)

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

The following dependency pair can be strictly oriented:

APP(add(n, x), y) -> APP(x, y)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> APP(x1, x2)
add(x1, x2) -> add(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`           →DP Problem 9`
`             ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Remaining`
`       →DP Problem 5`
`         ↳Remaining`
`       →DP Problem 6`
`         ↳Remaining`

Dependency Pair:

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Remaining Obligation(s)`
`       →DP Problem 5`
`         ↳Remaining Obligation(s)`
`       →DP Problem 6`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pairs:

IFRM(false, n, add(m, x)) -> RM(n, x)
IFRM(true, n, add(m, x)) -> RM(n, x)
RM(n, add(m, x)) -> IFRM(eq(n, m), n, add(m, x))

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

• Dependency Pairs:

IFMIN(false, add(n, add(m, x))) -> MIN(add(m, x))
IFMIN(true, add(n, add(m, x))) -> MIN(add(n, x))
MIN(add(n, add(m, x))) -> IFMIN(le(n, m), add(n, add(m, x)))

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

• Dependency Pairs:

IFMINSORT(false, add(n, x), y) -> MINSORT(x, add(n, y))
IFMINSORT(true, add(n, x), y) -> MINSORT(app(rm(n, x), y), nil)
MINSORT(add(n, x), y) -> IFMINSORT(eq(n, min(add(n, x))), add(n, x), y)

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Remaining Obligation(s)`
`       →DP Problem 5`
`         ↳Remaining Obligation(s)`
`       →DP Problem 6`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pairs:

IFRM(false, n, add(m, x)) -> RM(n, x)
IFRM(true, n, add(m, x)) -> RM(n, x)
RM(n, add(m, x)) -> IFRM(eq(n, m), n, add(m, x))

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

• Dependency Pairs:

IFMIN(false, add(n, add(m, x))) -> MIN(add(m, x))
IFMIN(true, add(n, add(m, x))) -> MIN(add(n, x))
MIN(add(n, add(m, x))) -> IFMIN(le(n, m), add(n, add(m, x)))

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

• Dependency Pairs:

IFMINSORT(false, add(n, x), y) -> MINSORT(x, add(n, y))
IFMINSORT(true, add(n, x), y) -> MINSORT(app(rm(n, x), y), nil)
MINSORT(add(n, x), y) -> IFMINSORT(eq(n, min(add(n, x))), add(n, x), y)

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Remaining Obligation(s)`
`       →DP Problem 5`
`         ↳Remaining Obligation(s)`
`       →DP Problem 6`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pairs:

IFRM(false, n, add(m, x)) -> RM(n, x)
IFRM(true, n, add(m, x)) -> RM(n, x)
RM(n, add(m, x)) -> IFRM(eq(n, m), n, add(m, x))

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

• Dependency Pairs:

IFMIN(false, add(n, add(m, x))) -> MIN(add(m, x))
IFMIN(true, add(n, add(m, x))) -> MIN(add(n, x))
MIN(add(n, add(m, x))) -> IFMIN(le(n, m), add(n, add(m, x)))

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

• Dependency Pairs:

IFMINSORT(false, add(n, x), y) -> MINSORT(x, add(n, y))
IFMINSORT(true, add(n, x), y) -> MINSORT(app(rm(n, x), y), nil)
MINSORT(add(n, x), y) -> IFMINSORT(eq(n, min(add(n, x))), add(n, x), y)

Rules:

eq(0, 0) -> true
eq(0, s(x)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
min(add(n, nil)) -> n
min(add(n, add(m, x))) -> ifmin(le(n, m), add(n, add(m, x)))
ifmin(true, add(n, add(m, x))) -> min(add(n, x))
ifmin(false, add(n, add(m, x))) -> min(add(m, x))
rm(n, nil) -> nil
rm(n, add(m, x)) -> ifrm(eq(n, m), n, add(m, x))
ifrm(true, n, add(m, x)) -> rm(n, x)
ifrm(false, n, add(m, x)) -> add(m, rm(n, x))
minsort(nil, nil) -> nil
minsort(add(n, x), y) -> ifminsort(eq(n, min(add(n, x))), add(n, x), y)
ifminsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
ifminsort(false, add(n, x), y) -> minsort(x, add(n, y))

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