Termination Proof using AProVE

Term Rewriting System R:
[X]
a_{_f}(X) -> g(h(f(X)))
a_{_f}(X) -> f(X)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

MARK(f(X)) -> A_{_F}(mark(X))
MARK(f(X)) -> MARK(X)
MARK(h(X)) -> MARK(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

Dependency Pairs:

MARK(h(X)) -> MARK(X)
MARK(f(X)) -> MARK(X)

Rules:

a_{_f}(X) -> g(h(f(X)))
a_{_f}(X) -> f(X)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Strategy:

innermost

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

MARK(f(X)) -> MARK(X)

two new Dependency Pairs are created:

MARK(f(f(X''))) -> MARK(f(X''))
MARK(f(h(X''))) -> MARK(h(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳Forward Instantiation Transformation

Dependency Pairs:

MARK(f(h(X''))) -> MARK(h(X''))
MARK(f(f(X''))) -> MARK(f(X''))
MARK(h(X)) -> MARK(X)

Rules:

a_{_f}(X) -> g(h(f(X)))
a_{_f}(X) -> f(X)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Strategy:

innermost

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

MARK(h(X)) -> MARK(X)

three new Dependency Pairs are created:

MARK(h(h(X''))) -> MARK(h(X''))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(X''))) -> MARK(f(X''))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(h(X''))) -> MARK(h(X''))
MARK(f(h(X''))) -> MARK(h(X''))

Rules:

a_{_f}(X) -> g(h(f(X)))
a_{_f}(X) -> f(X)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Strategy:

innermost

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

MARK(f(f(X''))) -> MARK(f(X''))

two new Dependency Pairs are created:

MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(h(X''))) -> MARK(h(X''))
MARK(f(h(X''))) -> MARK(h(X''))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))

Rules:

a_{_f}(X) -> g(h(f(X)))
a_{_f}(X) -> f(X)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Strategy:

innermost

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

MARK(f(h(X''))) -> MARK(h(X''))

three new Dependency Pairs are created:

MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(h(X''))) -> MARK(h(X''))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))

Rules:

a_{_f}(X) -> g(h(f(X)))
a_{_f}(X) -> f(X)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Strategy:

innermost

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

MARK(h(h(X''))) -> MARK(h(X''))

three new Dependency Pairs are created:

MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

Dependency Pairs:

MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

Rules:

a_{_f}(X) -> g(h(f(X)))
a_{_f}(X) -> f(X)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Strategy:

innermost

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

MARK(h(f(f(X'''')))) -> MARK(f(f(X'''')))

two new Dependency Pairs are created:

MARK(h(f(f(f(X''''''))))) -> MARK(f(f(f(X''''''))))
MARK(h(f(f(h(X''''''))))) -> MARK(f(f(h(X''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

Dependency Pairs:

MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(h(f(f(h(X''''''))))) -> MARK(f(f(h(X''''''))))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(f(X''''''))))) -> MARK(f(f(f(X''''''))))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

Rules:

a_{_f}(X) -> g(h(f(X)))
a_{_f}(X) -> f(X)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Strategy:

innermost

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

MARK(h(f(h(X'''')))) -> MARK(f(h(X'''')))

three new Dependency Pairs are created:

MARK(h(f(h(h(X''''''))))) -> MARK(f(h(h(X''''''))))
MARK(h(f(h(f(f(X'''''''')))))) -> MARK(f(h(f(f(X'''''''')))))
MARK(h(f(h(f(h(X'''''''')))))) -> MARK(f(h(f(h(X'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 8

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(h(f(h(f(h(X'''''''')))))) -> MARK(f(h(f(h(X'''''''')))))
MARK(h(f(h(f(f(X'''''''')))))) -> MARK(f(h(f(f(X'''''''')))))
MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(h(f(f(h(X''''''))))) -> MARK(f(f(h(X''''''))))
MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(h(f(f(f(X''''''))))) -> MARK(f(f(f(X''''''))))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(f(h(h(X''''''))))) -> MARK(f(h(h(X''''''))))
MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

Rules:

a_{_f}(X) -> g(h(f(X)))
a_{_f}(X) -> f(X)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

MARK(h(f(h(f(h(X'''''''')))))) -> MARK(f(h(f(h(X'''''''')))))
MARK(h(f(h(f(f(X'''''''')))))) -> MARK(f(h(f(f(X'''''''')))))
MARK(h(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))
MARK(h(f(f(h(X''''''))))) -> MARK(f(f(h(X''''''))))
MARK(h(f(f(f(X''''''))))) -> MARK(f(f(f(X''''''))))
MARK(h(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(h(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(h(f(h(h(X''''''))))) -> MARK(f(h(h(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(MARK(x₁)) = 1 + x₁

POL(h(x₁)) = 1 + x₁

POL(f(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 9

                 ↳Dependency Graph

Dependency Pairs:

MARK(f(h(f(f(X''''''))))) -> MARK(h(f(f(X''''''))))
MARK(f(f(h(X'''')))) -> MARK(f(h(X'''')))
MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))
MARK(f(h(h(X'''')))) -> MARK(h(h(X'''')))
MARK(f(h(f(h(X''''''))))) -> MARK(h(f(h(X''''''))))

Rules:

a_{_f}(X) -> g(h(f(X)))
a_{_f}(X) -> f(X)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 10

                 ↳Polynomial Ordering

Dependency Pair:

MARK(f(f(f(X'''')))) -> MARK(f(f(X'''')))

Rules:

a_{_f}(X) -> g(h(f(X)))
a_{_f}(X) -> f(X)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

MARK(f(f(f(X'''')))) -> MARK(f(f(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(MARK(x₁)) = 1 + x₁

POL(f(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 11

                 ↳Dependency Graph

Dependency Pair:

Rules:

a_{_f}(X) -> g(h(f(X)))
a_{_f}(X) -> f(X)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes

POL(MARK(x₁))	= 1 + x₁
POL(h(x₁))	= 1 + x₁
POL(f(x₁))	= x₁