Term Rewriting System R:
[y, x, z]
app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(le, x)
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(eq, app(s, x)), app(s, y)) -> APP(eq, x)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(minsort, app(app(cons, x), y)) -> APP(cons, app(app(min, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(min, x)
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(del, app(app(min, x), y))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(if, app(app(le, x), y))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(le, x)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(min, y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(if, app(app(eq, x), y)), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(if, app(app(eq, x), y))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(eq, x)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(cons, y), app(app(del, x), z))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`

Dependency Pairs:

APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(cons, y), app(app(del, x), z))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(if, app(app(eq, x), y)), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(del, x), app(app(cons, y), z)) -> APP(app(if, app(app(eq, x), y)), z)
four new Dependency Pairs are created:

APP(app(del, 0), app(app(cons, 0), z)) -> APP(app(if, true), z)
APP(app(del, 0), app(app(cons, app(s, y'')), z)) -> APP(app(if, false), z)
APP(app(del, app(s, x'')), app(app(cons, 0), z)) -> APP(app(if, false), z)
APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Narrowing Transformation`

Dependency Pairs:

APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)
APP(app(del, app(s, x'')), app(app(cons, 0), z)) -> APP(app(if, false), z)
APP(app(del, 0), app(app(cons, app(s, y'')), z)) -> APP(app(if, false), z)
APP(app(del, 0), app(app(cons, 0), z)) -> APP(app(if, true), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(cons, y), app(app(del, x), z))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(del, x), app(app(cons, y), z)) -> APP(app(cons, y), app(app(del, x), z))
two new Dependency Pairs are created:

APP(app(del, x''), app(app(cons, y), nil)) -> APP(app(cons, y), nil)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 3`
`                 ↳Narrowing Transformation`

Dependency Pairs:

APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, app(s, x'')), app(app(cons, 0), z)) -> APP(app(if, false), z)
APP(app(del, 0), app(app(cons, app(s, y'')), z)) -> APP(app(if, false), z)
APP(app(del, 0), app(app(cons, 0), z)) -> APP(app(if, true), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(del, 0), app(app(cons, 0), z)) -> APP(app(if, true), z)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 4`
`                 ↳Narrowing Transformation`

Dependency Pairs:

APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)
APP(app(del, app(s, x'')), app(app(cons, 0), z)) -> APP(app(if, false), z)
APP(app(del, 0), app(app(cons, app(s, y'')), z)) -> APP(app(if, false), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(del, 0), app(app(cons, app(s, y'')), z)) -> APP(app(if, false), z)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 5`
`                 ↳Narrowing Transformation`

Dependency Pairs:

APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, app(s, x'')), app(app(cons, 0), z)) -> APP(app(if, false), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(del, app(s, x'')), app(app(cons, 0), z)) -> APP(app(if, false), z)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 6`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))

Termination of R could not be shown.
Duration:
0:02 minutes