Termination Proof using AProVE

Term Rewriting System R:
[x, y, w, z]
app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y)
APP(app(lt, app(s, x)), app(s, y)) -> APP(lt, x)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(if, app(app(lt, w), y))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(lt, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(lt, w)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(if, app(app(eq, w), y)), true)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(if, app(app(eq, w), y))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(eq, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(eq, w)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(eq, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(lt, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))
APP(app(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y)

Rules:

app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Strategy:

innermost

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

APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(eq, w), y)

no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Polynomial Ordering

Dependency Pairs:

APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(lt, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z)

Rules:

app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(lt, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(fork) = 0

POL(if) = 0

POL(0) = 0

POL(eq) = 0

POL(false) = 0

POL(null) = 0

POL(member) = 1

POL(lt) = 0

POL(true) = 0

POL(s) = 0

POL(app(x₁, x₂)) = x₁

POL(APP(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Polo

...

               →DP Problem 3

                 ↳Dependency Graph

Dependency Pairs:

APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x)
APP(app(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z)

Rules:

app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 2 DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Polo

...

               →DP Problem 4

                 ↳Polynomial Ordering

Dependency Pairs:

APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x)

Rules:

app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), z)
APP(app(member, w), app(app(app(fork, x), y), z)) -> APP(app(member, w), x)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(fork) = 1

POL(if) = 0

POL(0) = 0

POL(eq) = 0

POL(false) = 0

POL(null) = 0

POL(member) = 0

POL(lt) = 0

POL(true) = 0

POL(s) = 0

POL(app(x₁, x₂)) = x₁ + x₂

POL(APP(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Polo

...

               →DP Problem 6

                 ↳Dependency Graph

Dependency Pair:

Rules:

app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Polo

...

               →DP Problem 5

                 ↳Polynomial Ordering

Dependency Pair:

APP(app(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y)

Rules:

app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y)
app(app(lt, 0), app(s, y)) -> true
app(app(lt, y), 0) -> false
app(app(eq, x), x) -> true
app(app(eq, app(s, x)), 0) -> false
app(app(eq, 0), app(s, x)) -> false
app(app(member, w), null) -> false
app(app(member, w), app(app(app(fork, x), y), z)) -> app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

APP(app(lt, app(s, x)), app(s, y)) -> APP(app(lt, x), y)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(fork) = 0

POL(if) = 0

POL(0) = 0

POL(eq) = 0

POL(false) = 0

POL(null) = 0

POL(member) = 0

POL(lt) = 0

POL(true) = 0

POL(s) = 0

POL(app(x₁, x₂)) = 1 + x₂

POL(APP(x₁, x₂)) = x₂

resulting in one new DP problem.

Innermost Termination of R successfully shown.
Duration:
0:04 minutes

POL(fork)	= 0
POL(if)	= 0
POL(0)	= 0
POL(eq)	= 0
POL(false)	= 0
POL(null)	= 0
POL(member)	= 1
POL(lt)	= 0
POL(true)	= 0
POL(s)	= 0
POL(app(x₁, x₂))	= x₁
POL(APP(x₁, x₂))	= x₁