Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z, X1, X2]
a_{_f}(X) -> cons(mark(X), f(g(X)))
a_{_f}(X) -> f(X)
a_{_g}(0) -> s(0)
a_{_g}(s(X)) -> s(s(a_{_g}(mark(X))))
a_{_g}(X) -> g(X)
a_{_sel}(0, cons(X, Y)) -> mark(X)
a_{_sel}(s(X), cons(Y, Z)) -> a_{_sel}(mark(X), mark(Z))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> a_{_g}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(s(X)) -> s(mark(X))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_F}(X) -> MARK(X)
A_{_G}(s(X)) -> A_{_G}(mark(X))
A_{_G}(s(X)) -> MARK(X)
A_{_SEL}(0, cons(X, Y)) -> MARK(X)
A_{_SEL}(s(X), cons(Y, Z)) -> A_{_SEL}(mark(X), mark(Z))
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(X)
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(Z)
MARK(f(X)) -> A_{_F}(mark(X))
MARK(f(X)) -> MARK(X)
MARK(g(X)) -> A_{_G}(mark(X))
MARK(g(X)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

A_{_SEL}(s(X), cons(Y, Z)) -> MARK(Z)
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(X)
A_{_SEL}(s(X), cons(Y, Z)) -> A_{_SEL}(mark(X), mark(Z))
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(0, cons(X, Y)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(g(X)) -> MARK(X)
A_{_G}(s(X)) -> MARK(X)
A_{_G}(s(X)) -> A_{_G}(mark(X))
MARK(g(X)) -> A_{_G}(mark(X))
MARK(f(X)) -> MARK(X)
MARK(f(X)) -> A_{_F}(mark(X))
A_{_F}(X) -> MARK(X)

Rules:

a_{_f}(X) -> cons(mark(X), f(g(X)))
a_{_f}(X) -> f(X)
a_{_g}(0) -> s(0)
a_{_g}(s(X)) -> s(s(a_{_g}(mark(X))))
a_{_g}(X) -> g(X)
a_{_sel}(0, cons(X, Y)) -> mark(X)
a_{_sel}(s(X), cons(Y, Z)) -> a_{_sel}(mark(X), mark(Z))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> a_{_g}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(s(X)) -> s(mark(X))

Strategy:

innermost

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

A_{_G}(s(X)) -> A_{_G}(mark(X))

six new Dependency Pairs are created:

A_{_G}(s(f(X''))) -> A_{_G}(a_{_f}(mark(X'')))
A_{_G}(s(g(X''))) -> A_{_G}(a_{_g}(mark(X'')))
A_{_G}(s(sel(X1', X2'))) -> A_{_G}(a_{_sel}(mark(X1'), mark(X2')))
A_{_G}(s(cons(X1', X2'))) -> A_{_G}(cons(mark(X1'), X2'))
A_{_G}(s(0)) -> A_{_G}(0)
A_{_G}(s(s(X''))) -> A_{_G}(s(mark(X'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

A_{_G}(s(s(X''))) -> A_{_G}(s(mark(X'')))
A_{_G}(s(sel(X1', X2'))) -> A_{_G}(a_{_sel}(mark(X1'), mark(X2')))
A_{_G}(s(g(X''))) -> A_{_G}(a_{_g}(mark(X'')))
A_{_G}(s(f(X''))) -> A_{_G}(a_{_f}(mark(X'')))
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(X)
A_{_SEL}(s(X), cons(Y, Z)) -> A_{_SEL}(mark(X), mark(Z))
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(0, cons(X, Y)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(g(X)) -> MARK(X)
A_{_G}(s(X)) -> MARK(X)
MARK(g(X)) -> A_{_G}(mark(X))
MARK(f(X)) -> MARK(X)
A_{_F}(X) -> MARK(X)
MARK(f(X)) -> A_{_F}(mark(X))
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(Z)

Rules:

a_{_f}(X) -> cons(mark(X), f(g(X)))
a_{_f}(X) -> f(X)
a_{_g}(0) -> s(0)
a_{_g}(s(X)) -> s(s(a_{_g}(mark(X))))
a_{_g}(X) -> g(X)
a_{_sel}(0, cons(X, Y)) -> mark(X)
a_{_sel}(s(X), cons(Y, Z)) -> a_{_sel}(mark(X), mark(Z))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> a_{_g}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(s(X)) -> s(mark(X))

Strategy:

innermost

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

A_{_SEL}(s(X), cons(Y, Z)) -> A_{_SEL}(mark(X), mark(Z))

12 new Dependency Pairs are created:

A_{_SEL}(s(f(X'')), cons(Y, Z)) -> A_{_SEL}(a_{_f}(mark(X'')), mark(Z))
A_{_SEL}(s(g(X'')), cons(Y, Z)) -> A_{_SEL}(a_{_g}(mark(X'')), mark(Z))
A_{_SEL}(s(sel(X1', X2')), cons(Y, Z)) -> A_{_SEL}(a_{_sel}(mark(X1'), mark(X2')), mark(Z))
A_{_SEL}(s(cons(X1', X2')), cons(Y, Z)) -> A_{_SEL}(cons(mark(X1'), X2'), mark(Z))
A_{_SEL}(s(0), cons(Y, Z)) -> A_{_SEL}(0, mark(Z))
A_{_SEL}(s(s(X'')), cons(Y, Z)) -> A_{_SEL}(s(mark(X'')), mark(Z))
A_{_SEL}(s(X), cons(Y, f(X''))) -> A_{_SEL}(mark(X), a_{_f}(mark(X'')))
A_{_SEL}(s(X), cons(Y, g(X''))) -> A_{_SEL}(mark(X), a_{_g}(mark(X'')))
A_{_SEL}(s(X), cons(Y, sel(X1', X2'))) -> A_{_SEL}(mark(X), a_{_sel}(mark(X1'), mark(X2')))
A_{_SEL}(s(X), cons(Y, cons(X1', X2'))) -> A_{_SEL}(mark(X), cons(mark(X1'), X2'))
A_{_SEL}(s(X), cons(Y, 0)) -> A_{_SEL}(mark(X), 0)
A_{_SEL}(s(X), cons(Y, s(X''))) -> A_{_SEL}(mark(X), s(mark(X'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Narrowing Transformation

Dependency Pairs:

A_{_SEL}(s(X), cons(Y, cons(X1', X2'))) -> A_{_SEL}(mark(X), cons(mark(X1'), X2'))
A_{_SEL}(s(X), cons(Y, sel(X1', X2'))) -> A_{_SEL}(mark(X), a_{_sel}(mark(X1'), mark(X2')))
A_{_SEL}(s(X), cons(Y, g(X''))) -> A_{_SEL}(mark(X), a_{_g}(mark(X'')))
A_{_SEL}(s(X), cons(Y, f(X''))) -> A_{_SEL}(mark(X), a_{_f}(mark(X'')))
A_{_SEL}(s(s(X'')), cons(Y, Z)) -> A_{_SEL}(s(mark(X'')), mark(Z))
A_{_SEL}(s(0), cons(Y, Z)) -> A_{_SEL}(0, mark(Z))
A_{_SEL}(s(sel(X1', X2')), cons(Y, Z)) -> A_{_SEL}(a_{_sel}(mark(X1'), mark(X2')), mark(Z))
A_{_SEL}(s(g(X'')), cons(Y, Z)) -> A_{_SEL}(a_{_g}(mark(X'')), mark(Z))
A_{_SEL}(s(f(X'')), cons(Y, Z)) -> A_{_SEL}(a_{_f}(mark(X'')), mark(Z))
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(Z)
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(X)
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(0, cons(X, Y)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(g(X)) -> MARK(X)
A_{_G}(s(sel(X1', X2'))) -> A_{_G}(a_{_sel}(mark(X1'), mark(X2')))
A_{_G}(s(g(X''))) -> A_{_G}(a_{_g}(mark(X'')))
A_{_G}(s(f(X''))) -> A_{_G}(a_{_f}(mark(X'')))
MARK(g(X)) -> A_{_G}(mark(X))
MARK(f(X)) -> MARK(X)
A_{_F}(X) -> MARK(X)
MARK(f(X)) -> A_{_F}(mark(X))
A_{_G}(s(X)) -> MARK(X)
A_{_G}(s(s(X''))) -> A_{_G}(s(mark(X'')))

Rules:

a_{_f}(X) -> cons(mark(X), f(g(X)))
a_{_f}(X) -> f(X)
a_{_g}(0) -> s(0)
a_{_g}(s(X)) -> s(s(a_{_g}(mark(X))))
a_{_g}(X) -> g(X)
a_{_sel}(0, cons(X, Y)) -> mark(X)
a_{_sel}(s(X), cons(Y, Z)) -> a_{_sel}(mark(X), mark(Z))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> a_{_g}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(s(X)) -> s(mark(X))

Strategy:

innermost

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

MARK(g(X)) -> A_{_G}(mark(X))

six new Dependency Pairs are created:

MARK(g(f(X''))) -> A_{_G}(a_{_f}(mark(X'')))
MARK(g(g(X''))) -> A_{_G}(a_{_g}(mark(X'')))
MARK(g(sel(X1', X2'))) -> A_{_G}(a_{_sel}(mark(X1'), mark(X2')))
MARK(g(cons(X1', X2'))) -> A_{_G}(cons(mark(X1'), X2'))
MARK(g(0)) -> A_{_G}(0)
MARK(g(s(X''))) -> A_{_G}(s(mark(X'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Narrowing Transformation

Dependency Pairs:

A_{_SEL}(s(X), cons(Y, sel(X1', X2'))) -> A_{_SEL}(mark(X), a_{_sel}(mark(X1'), mark(X2')))
A_{_SEL}(s(X), cons(Y, g(X''))) -> A_{_SEL}(mark(X), a_{_g}(mark(X'')))
A_{_SEL}(s(X), cons(Y, f(X''))) -> A_{_SEL}(mark(X), a_{_f}(mark(X'')))
A_{_SEL}(s(s(X'')), cons(Y, Z)) -> A_{_SEL}(s(mark(X'')), mark(Z))
A_{_SEL}(s(0), cons(Y, Z)) -> A_{_SEL}(0, mark(Z))
A_{_SEL}(s(sel(X1', X2')), cons(Y, Z)) -> A_{_SEL}(a_{_sel}(mark(X1'), mark(X2')), mark(Z))
A_{_SEL}(s(g(X'')), cons(Y, Z)) -> A_{_SEL}(a_{_g}(mark(X'')), mark(Z))
A_{_SEL}(s(f(X'')), cons(Y, Z)) -> A_{_SEL}(a_{_f}(mark(X'')), mark(Z))
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(Z)
MARK(g(s(X''))) -> A_{_G}(s(mark(X'')))
MARK(g(sel(X1', X2'))) -> A_{_G}(a_{_sel}(mark(X1'), mark(X2')))
A_{_G}(s(s(X''))) -> A_{_G}(s(mark(X'')))
A_{_G}(s(sel(X1', X2'))) -> A_{_G}(a_{_sel}(mark(X1'), mark(X2')))
A_{_G}(s(g(X''))) -> A_{_G}(a_{_g}(mark(X'')))
A_{_G}(s(f(X''))) -> A_{_G}(a_{_f}(mark(X'')))
MARK(g(g(X''))) -> A_{_G}(a_{_g}(mark(X'')))
A_{_G}(s(X)) -> MARK(X)
MARK(g(f(X''))) -> A_{_G}(a_{_f}(mark(X'')))
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(g(X)) -> MARK(X)
MARK(f(X)) -> MARK(X)
A_{_F}(X) -> MARK(X)
MARK(f(X)) -> A_{_F}(mark(X))
A_{_SEL}(0, cons(X, Y)) -> MARK(X)
A_{_SEL}(s(X), cons(Y, cons(X1', X2'))) -> A_{_SEL}(mark(X), cons(mark(X1'), X2'))

Rules:

a_{_f}(X) -> cons(mark(X), f(g(X)))
a_{_f}(X) -> f(X)
a_{_g}(0) -> s(0)
a_{_g}(s(X)) -> s(s(a_{_g}(mark(X))))
a_{_g}(X) -> g(X)
a_{_sel}(0, cons(X, Y)) -> mark(X)
a_{_sel}(s(X), cons(Y, Z)) -> a_{_sel}(mark(X), mark(Z))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> a_{_g}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(s(X)) -> s(mark(X))

Strategy:

innermost

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

MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))

12 new Dependency Pairs are created:

MARK(sel(f(X'), X2)) -> A_{_SEL}(a_{_f}(mark(X')), mark(X2))
MARK(sel(g(X'), X2)) -> A_{_SEL}(a_{_g}(mark(X')), mark(X2))
MARK(sel(sel(X1'', X2''), X2)) -> A_{_SEL}(a_{_sel}(mark(X1''), mark(X2'')), mark(X2))
MARK(sel(cons(X1'', X2''), X2)) -> A_{_SEL}(cons(mark(X1''), X2''), mark(X2))
MARK(sel(0, X2)) -> A_{_SEL}(0, mark(X2))
MARK(sel(s(X'), X2)) -> A_{_SEL}(s(mark(X')), mark(X2))
MARK(sel(X1, f(X'))) -> A_{_SEL}(mark(X1), a_{_f}(mark(X')))
MARK(sel(X1, g(X'))) -> A_{_SEL}(mark(X1), a_{_g}(mark(X')))
MARK(sel(X1, sel(X1'', X2''))) -> A_{_SEL}(mark(X1), a_{_sel}(mark(X1''), mark(X2'')))
MARK(sel(X1, cons(X1'', X2''))) -> A_{_SEL}(mark(X1), cons(mark(X1''), X2''))
MARK(sel(X1, 0)) -> A_{_SEL}(mark(X1), 0)
MARK(sel(X1, s(X'))) -> A_{_SEL}(mark(X1), s(mark(X')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

MARK(sel(X1, cons(X1'', X2''))) -> A_{_SEL}(mark(X1), cons(mark(X1''), X2''))
MARK(sel(X1, sel(X1'', X2''))) -> A_{_SEL}(mark(X1), a_{_sel}(mark(X1''), mark(X2'')))
MARK(sel(X1, g(X'))) -> A_{_SEL}(mark(X1), a_{_g}(mark(X')))
MARK(sel(X1, f(X'))) -> A_{_SEL}(mark(X1), a_{_f}(mark(X')))
MARK(sel(s(X'), X2)) -> A_{_SEL}(s(mark(X')), mark(X2))
MARK(sel(0, X2)) -> A_{_SEL}(0, mark(X2))
A_{_SEL}(s(X), cons(Y, cons(X1', X2'))) -> A_{_SEL}(mark(X), cons(mark(X1'), X2'))
A_{_SEL}(s(X), cons(Y, g(X''))) -> A_{_SEL}(mark(X), a_{_g}(mark(X'')))
A_{_SEL}(s(X), cons(Y, f(X''))) -> A_{_SEL}(mark(X), a_{_f}(mark(X'')))
A_{_SEL}(s(s(X'')), cons(Y, Z)) -> A_{_SEL}(s(mark(X'')), mark(Z))
A_{_SEL}(s(0), cons(Y, Z)) -> A_{_SEL}(0, mark(Z))
A_{_SEL}(s(sel(X1', X2')), cons(Y, Z)) -> A_{_SEL}(a_{_sel}(mark(X1'), mark(X2')), mark(Z))
A_{_SEL}(s(g(X'')), cons(Y, Z)) -> A_{_SEL}(a_{_g}(mark(X'')), mark(Z))
A_{_SEL}(s(f(X'')), cons(Y, Z)) -> A_{_SEL}(a_{_f}(mark(X'')), mark(Z))
MARK(sel(sel(X1'', X2''), X2)) -> A_{_SEL}(a_{_sel}(mark(X1''), mark(X2'')), mark(X2))
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(Z)
MARK(sel(g(X'), X2)) -> A_{_SEL}(a_{_g}(mark(X')), mark(X2))
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(X)
MARK(sel(f(X'), X2)) -> A_{_SEL}(a_{_f}(mark(X')), mark(X2))
MARK(g(s(X''))) -> A_{_G}(s(mark(X'')))
MARK(g(sel(X1', X2'))) -> A_{_G}(a_{_sel}(mark(X1'), mark(X2')))
A_{_G}(s(s(X''))) -> A_{_G}(s(mark(X'')))
A_{_G}(s(sel(X1', X2'))) -> A_{_G}(a_{_sel}(mark(X1'), mark(X2')))
A_{_G}(s(g(X''))) -> A_{_G}(a_{_g}(mark(X'')))
A_{_G}(s(f(X''))) -> A_{_G}(a_{_f}(mark(X'')))
MARK(g(g(X''))) -> A_{_G}(a_{_g}(mark(X'')))
A_{_G}(s(X)) -> MARK(X)
MARK(g(f(X''))) -> A_{_G}(a_{_f}(mark(X'')))
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
MARK(g(X)) -> MARK(X)
MARK(f(X)) -> MARK(X)
A_{_F}(X) -> MARK(X)
MARK(f(X)) -> A_{_F}(mark(X))
A_{_SEL}(0, cons(X, Y)) -> MARK(X)
A_{_SEL}(s(X), cons(Y, sel(X1', X2'))) -> A_{_SEL}(mark(X), a_{_sel}(mark(X1'), mark(X2')))

Rules:

a_{_f}(X) -> cons(mark(X), f(g(X)))
a_{_f}(X) -> f(X)
a_{_g}(0) -> s(0)
a_{_g}(s(X)) -> s(s(a_{_g}(mark(X))))
a_{_g}(X) -> g(X)
a_{_sel}(0, cons(X, Y)) -> mark(X)
a_{_sel}(s(X), cons(Y, Z)) -> a_{_sel}(mark(X), mark(Z))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> a_{_g}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(s(X)) -> s(mark(X))

Strategy:

innermost

The Proof could not be continued due to a Timeout.
Innermost Termination of R could not be shown.
Duration:
1:00 minutes