Termination Proof using AProVE

Term Rewriting System R:
[y, x]
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(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Innermost Termination of R to be shown.

     ↳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(minus, x), app(s, y)) -> APP(pred, app(app(minus, x), y))
APP(app(minus, x), app(s, y)) -> APP(app(minus, x), y)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(if_gcd, app(app(le, y), x)), app(s, x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(if_gcd, app(app(le, y), x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(le, y)
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, x), y)), app(s, y))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(gcd, app(app(minus, x), y))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(minus, x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(gcd, app(app(minus, y), x))
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(minus, y)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Nar

Dependency Pair:

APP(app(minus, x), app(s, y)) -> APP(app(minus, 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(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

The following dependency pair can be strictly oriented:

APP(app(minus, x), app(s, y)) -> APP(app(minus, 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(if_gcd) = 0

POL(0) = 0

POL(false) = 0

POL(pred) = 0

POL(minus) = 0

POL(true) = 0

POL(s) = 0

POL(le) = 0

POL(app(x₁, x₂)) = 1 + x₂

POL(APP(x₁, x₂)) = x₂

POL(gcd) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 3

             ↳Dependency Graph

       →DP Problem 2

         ↳Nar

Dependency Pair:

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(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, x), y)), app(s, y))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(if_gcd, app(app(le, y), x)), app(s, x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, 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(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(minus, x), y)

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, x), y)), app(s, y))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(if_gcd, app(app(le, y), x)), app(s, x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)

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(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 5

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, x), y)), app(s, y))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(if_gcd, app(app(le, y), x)), app(s, x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))

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(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

POL(if_gcd)	= 0
POL(0)	= 0
POL(false)	= 0
POL(pred)	= 0
POL(minus)	= 0
POL(true)	= 0
POL(s)	= 0
POL(le)	= 0
POL(app(x₁, x₂))	= 1 + x₂
POL(APP(x₁, x₂))	= x₂
POL(gcd)	= 0