Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z]
f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

G(s(X)) -> G(X)
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_f}(X)) -> F(activate(X))
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_g}(X)) -> G(activate(X))
ACTIVATE(n_{_g}(X)) -> ACTIVATE(X)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

Dependency Pair:

G(s(X)) -> G(X)

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

G(s(X)) -> G(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(G(x₁)) = x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 4

             ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

Dependency Pair:

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polynomial Ordering

       →DP Problem 3

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_g}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_g}(X)) -> ACTIVATE(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__f(x₁)) = x₁

POL(n__g(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 5

             ↳Polynomial Ordering

       →DP Problem 3

         ↳Nar

Dependency Pair:

ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__f(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 5

             ↳Polo

...

               →DP Problem 6

                 ↳Dependency Graph

       →DP Problem 3

         ↳Nar

Dependency Pair:

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Narrowing Transformation

Dependency Pair:

SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

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

SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))

three new Dependency Pairs are created:

SEL(s(X), cons(Y, n_{_f}(X''))) -> SEL(X, f(activate(X'')))
SEL(s(X), cons(Y, n_{_g}(X''))) -> SEL(X, g(activate(X'')))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Forward Instantiation Transformation

Dependency Pairs:

SEL(s(X), cons(Y, Z')) -> SEL(X, Z')
SEL(s(X), cons(Y, n_{_g}(X''))) -> SEL(X, g(activate(X'')))
SEL(s(X), cons(Y, n_{_f}(X''))) -> SEL(X, f(activate(X'')))

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

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

SEL(s(X), cons(Y, Z')) -> SEL(X, Z')

three new Dependency Pairs are created:

SEL(s(s(X'')), cons(Y, cons(Y'', n_{_f}(X'''')))) -> SEL(s(X''), cons(Y'', n_{_f}(X'''')))
SEL(s(s(X'')), cons(Y, cons(Y'', n_{_g}(X'''')))) -> SEL(s(X''), cons(Y'', n_{_g}(X'''')))
SEL(s(s(X'')), cons(Y, cons(Y'', Z'''))) -> SEL(s(X''), cons(Y'', Z'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳FwdInst

...

               →DP Problem 8

                 ↳Polynomial Ordering

Dependency Pairs:

SEL(s(s(X'')), cons(Y, cons(Y'', Z'''))) -> SEL(s(X''), cons(Y'', Z'''))
SEL(s(s(X'')), cons(Y, cons(Y'', n_{_g}(X'''')))) -> SEL(s(X''), cons(Y'', n_{_g}(X'''')))
SEL(s(s(X'')), cons(Y, cons(Y'', n_{_f}(X'''')))) -> SEL(s(X''), cons(Y'', n_{_f}(X'''')))
SEL(s(X), cons(Y, n_{_f}(X''))) -> SEL(X, f(activate(X'')))
SEL(s(X), cons(Y, n_{_g}(X''))) -> SEL(X, g(activate(X'')))

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

SEL(s(s(X'')), cons(Y, cons(Y'', Z'''))) -> SEL(s(X''), cons(Y'', Z'''))
SEL(s(s(X'')), cons(Y, cons(Y'', n_{_g}(X'''')))) -> SEL(s(X''), cons(Y'', n_{_g}(X'''')))
SEL(s(s(X'')), cons(Y, cons(Y'', n_{_f}(X'''')))) -> SEL(s(X''), cons(Y'', n_{_f}(X'''')))
SEL(s(X), cons(Y, n_{_g}(X''))) -> SEL(X, g(activate(X'')))

Additionally, the following usable rules for innermost can be oriented:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__f(x₁)) = 0

POL(activate(x₁)) = 1 + x₁

POL(0) = 0

POL(g(x₁)) = 0

POL(cons(x₁, x₂)) = 1 + x₂

POL(SEL(x₁, x₂)) = 1 + x₂

POL(n__g(x₁)) = 0

POL(s(x₁)) = 0

POL(f(x₁)) = 1

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳FwdInst

...

               →DP Problem 9

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pair:

SEL(s(X), cons(Y, n_{_f}(X''))) -> SEL(X, f(activate(X'')))

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:01 minutes

POL(n__f(x₁))	= x₁
POL(n__g(x₁))	= 1 + x₁
POL(ACTIVATE(x₁))	= x₁