Termination Proof using AProVE

Term Rewriting System R:
[X, ALPHA, BETA]
dx(X) -> one
dx(a) -> zero
dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) -> neg(dx(ALPHA))
dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

DX(plus(ALPHA, BETA)) -> DX(ALPHA)
DX(plus(ALPHA, BETA)) -> DX(BETA)
DX(times(ALPHA, BETA)) -> DX(ALPHA)
DX(times(ALPHA, BETA)) -> DX(BETA)
DX(minus(ALPHA, BETA)) -> DX(ALPHA)
DX(minus(ALPHA, BETA)) -> DX(BETA)
DX(neg(ALPHA)) -> DX(ALPHA)
DX(div(ALPHA, BETA)) -> DX(ALPHA)
DX(div(ALPHA, BETA)) -> DX(BETA)
DX(ln(ALPHA)) -> DX(ALPHA)
DX(exp(ALPHA, BETA)) -> DX(ALPHA)
DX(exp(ALPHA, BETA)) -> DX(BETA)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

Dependency Pairs:

DX(exp(ALPHA, BETA)) -> DX(BETA)
DX(exp(ALPHA, BETA)) -> DX(ALPHA)
DX(ln(ALPHA)) -> DX(ALPHA)
DX(div(ALPHA, BETA)) -> DX(BETA)
DX(div(ALPHA, BETA)) -> DX(ALPHA)
DX(neg(ALPHA)) -> DX(ALPHA)
DX(minus(ALPHA, BETA)) -> DX(BETA)
DX(minus(ALPHA, BETA)) -> DX(ALPHA)
DX(times(ALPHA, BETA)) -> DX(BETA)
DX(times(ALPHA, BETA)) -> DX(ALPHA)
DX(plus(ALPHA, BETA)) -> DX(BETA)
DX(plus(ALPHA, BETA)) -> DX(ALPHA)

Rules:

dx(X) -> one
dx(a) -> zero
dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) -> neg(dx(ALPHA))
dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

DX(neg(ALPHA)) -> DX(ALPHA)

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(exp(x₁, x₂)) = x₁ + x₂

POL(plus(x₁, x₂)) = x₁ + x₂

POL(DX(x₁)) = x₁

POL(neg(x₁)) = 1 + x₁

POL(times(x₁, x₂)) = x₁ + x₂

POL(minus(x₁, x₂)) = x₁ + x₂

POL(div(x₁, x₂)) = x₁ + x₂

POL(ln(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Polynomial Ordering

Dependency Pairs:

DX(exp(ALPHA, BETA)) -> DX(BETA)
DX(exp(ALPHA, BETA)) -> DX(ALPHA)
DX(ln(ALPHA)) -> DX(ALPHA)
DX(div(ALPHA, BETA)) -> DX(BETA)
DX(div(ALPHA, BETA)) -> DX(ALPHA)
DX(minus(ALPHA, BETA)) -> DX(BETA)
DX(minus(ALPHA, BETA)) -> DX(ALPHA)
DX(times(ALPHA, BETA)) -> DX(BETA)
DX(times(ALPHA, BETA)) -> DX(ALPHA)
DX(plus(ALPHA, BETA)) -> DX(BETA)
DX(plus(ALPHA, BETA)) -> DX(ALPHA)

Rules:

dx(X) -> one
dx(a) -> zero
dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) -> neg(dx(ALPHA))
dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

DX(minus(ALPHA, BETA)) -> DX(BETA)
DX(minus(ALPHA, BETA)) -> DX(ALPHA)

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(exp(x₁, x₂)) = x₁ + x₂

POL(plus(x₁, x₂)) = x₁ + x₂

POL(DX(x₁)) = x₁

POL(times(x₁, x₂)) = x₁ + x₂

POL(minus(x₁, x₂)) = 1 + x₁ + x₂

POL(div(x₁, x₂)) = x₁ + x₂

POL(ln(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Polo

...

               →DP Problem 3

                 ↳Polynomial Ordering

Dependency Pairs:

DX(exp(ALPHA, BETA)) -> DX(BETA)
DX(exp(ALPHA, BETA)) -> DX(ALPHA)
DX(ln(ALPHA)) -> DX(ALPHA)
DX(div(ALPHA, BETA)) -> DX(BETA)
DX(div(ALPHA, BETA)) -> DX(ALPHA)
DX(times(ALPHA, BETA)) -> DX(BETA)
DX(times(ALPHA, BETA)) -> DX(ALPHA)
DX(plus(ALPHA, BETA)) -> DX(BETA)
DX(plus(ALPHA, BETA)) -> DX(ALPHA)

Rules:

dx(X) -> one
dx(a) -> zero
dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) -> neg(dx(ALPHA))
dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

DX(times(ALPHA, BETA)) -> DX(BETA)
DX(times(ALPHA, BETA)) -> DX(ALPHA)

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(exp(x₁, x₂)) = x₁ + x₂

POL(plus(x₁, x₂)) = x₁ + x₂

POL(DX(x₁)) = x₁

POL(times(x₁, x₂)) = 1 + x₁ + x₂

POL(div(x₁, x₂)) = x₁ + x₂

POL(ln(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Polo

...

               →DP Problem 4

                 ↳Polynomial Ordering

Dependency Pairs:

Rules:

dx(X) -> one
dx(a) -> zero
dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) -> neg(dx(ALPHA))
dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

DX(plus(ALPHA, BETA)) -> DX(BETA)
DX(plus(ALPHA, BETA)) -> DX(ALPHA)

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(exp(x₁, x₂)) = x₁ + x₂

POL(plus(x₁, x₂)) = 1 + x₁ + x₂

POL(DX(x₁)) = x₁

POL(div(x₁, x₂)) = x₁ + x₂

POL(ln(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Polo

...

               →DP Problem 5

                 ↳Polynomial Ordering

Dependency Pairs:

Rules:

dx(X) -> one
dx(a) -> zero
dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) -> neg(dx(ALPHA))
dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

DX(exp(ALPHA, BETA)) -> DX(BETA)
DX(exp(ALPHA, BETA)) -> DX(ALPHA)

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(exp(x₁, x₂)) = 1 + x₁ + x₂

POL(DX(x₁)) = x₁

POL(div(x₁, x₂)) = x₁ + x₂

POL(ln(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Polo

...

               →DP Problem 6

                 ↳Polynomial Ordering

Dependency Pairs:

DX(ln(ALPHA)) -> DX(ALPHA)
DX(div(ALPHA, BETA)) -> DX(BETA)
DX(div(ALPHA, BETA)) -> DX(ALPHA)

Rules:

dx(X) -> one
dx(a) -> zero
dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) -> neg(dx(ALPHA))
dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

DX(ln(ALPHA)) -> DX(ALPHA)

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(DX(x₁)) = x₁

POL(div(x₁, x₂)) = x₁ + x₂

POL(ln(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Polo

...

               →DP Problem 7

                 ↳Polynomial Ordering

Dependency Pairs:

DX(div(ALPHA, BETA)) -> DX(BETA)
DX(div(ALPHA, BETA)) -> DX(ALPHA)

Rules:

dx(X) -> one
dx(a) -> zero
dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) -> neg(dx(ALPHA))
dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

DX(div(ALPHA, BETA)) -> DX(BETA)
DX(div(ALPHA, BETA)) -> DX(ALPHA)

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(DX(x₁)) = x₁

POL(div(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Polo

...

               →DP Problem 8

                 ↳Dependency Graph

Dependency Pair:

Rules:

dx(X) -> one
dx(a) -> zero
dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) -> neg(dx(ALPHA))
dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:06 minutes

POL(exp(x₁, x₂))	= x₁ + x₂
POL(plus(x₁, x₂))	= x₁ + x₂
POL(DX(x₁))	= x₁
POL(neg(x₁))	= 1 + x₁
POL(times(x₁, x₂))	= x₁ + x₂
POL(minus(x₁, x₂))	= x₁ + x₂
POL(div(x₁, x₂))	= x₁ + x₂
POL(ln(x₁))	= x₁

POL(exp(x₁, x₂))	= 1 + x₁ + x₂
POL(DX(x₁))	= x₁
POL(div(x₁, x₂))	= x₁ + x₂
POL(ln(x₁))	= x₁