Termination Proof using AProVE

Term Rewriting System R:
[X]
active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

ACTIVE(g(X)) -> H(X)
ACTIVE(h(d)) -> G(c)
PROPER(g(X)) -> G(proper(X))
PROPER(g(X)) -> PROPER(X)
PROPER(h(X)) -> H(proper(X))
PROPER(h(X)) -> PROPER(X)
G(ok(X)) -> G(X)
H(ok(X)) -> H(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains four SCCs.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

Dependency Pair:

G(ok(X)) -> G(X)

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

G(ok(X)) -> G(X)

one new Dependency Pair is created:

G(ok(ok(X''))) -> G(ok(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 5

             ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

Dependency Pair:

G(ok(ok(X''))) -> G(ok(X''))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

G(ok(ok(X''))) -> G(ok(X''))

one new Dependency Pair is created:

G(ok(ok(ok(X'''')))) -> G(ok(ok(X'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 5

             ↳FwdInst

...

               →DP Problem 6

                 ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

Dependency Pair:

G(ok(ok(ok(X'''')))) -> G(ok(ok(X'''')))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

G(ok(ok(ok(X'''')))) -> G(ok(ok(X'''')))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.
Used Argument Filtering System:

G(x₁) -> G(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 5

             ↳FwdInst

...

               →DP Problem 7

                 ↳Dependency Graph

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

Dependency Pair:

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

Dependency Pair:

H(ok(X)) -> H(X)

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

H(ok(X)) -> H(X)

one new Dependency Pair is created:

H(ok(ok(X''))) -> H(ok(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 8

             ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

Dependency Pair:

H(ok(ok(X''))) -> H(ok(X''))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

H(ok(ok(X''))) -> H(ok(X''))

one new Dependency Pair is created:

H(ok(ok(ok(X'''')))) -> H(ok(ok(X'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 8

             ↳FwdInst

...

               →DP Problem 9

                 ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

Dependency Pair:

H(ok(ok(ok(X'''')))) -> H(ok(ok(X'''')))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

H(ok(ok(ok(X'''')))) -> H(ok(ok(X'''')))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.
Used Argument Filtering System:

H(x₁) -> H(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 8

             ↳FwdInst

...

               →DP Problem 10

                 ↳Dependency Graph

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

Dependency Pair:

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Forward Instantiation Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

PROPER(h(X)) -> PROPER(X)
PROPER(g(X)) -> PROPER(X)

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

PROPER(g(X)) -> PROPER(X)

two new Dependency Pairs are created:

PROPER(g(g(X''))) -> PROPER(g(X''))
PROPER(g(h(X''))) -> PROPER(h(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 11

             ↳Forward Instantiation Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

PROPER(g(h(X''))) -> PROPER(h(X''))
PROPER(g(g(X''))) -> PROPER(g(X''))
PROPER(h(X)) -> PROPER(X)

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

PROPER(h(X)) -> PROPER(X)

three new Dependency Pairs are created:

PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 11

             ↳FwdInst

...

               →DP Problem 12

                 ↳Forward Instantiation Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(X''))) -> PROPER(g(X''))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(g(h(X''))) -> PROPER(h(X''))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

PROPER(g(g(X''))) -> PROPER(g(X''))

two new Dependency Pairs are created:

PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 11

             ↳FwdInst

...

               →DP Problem 13

                 ↳Forward Instantiation Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(g(h(X''))) -> PROPER(h(X''))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

PROPER(g(h(X''))) -> PROPER(h(X''))

three new Dependency Pairs are created:

PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 11

             ↳FwdInst

...

               →DP Problem 14

                 ↳Forward Instantiation Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(h(X''))) -> PROPER(h(X''))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

PROPER(h(h(X''))) -> PROPER(h(X''))

three new Dependency Pairs are created:

PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 11

             ↳FwdInst

...

               →DP Problem 15

                 ↳Forward Instantiation Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

PROPER(h(g(g(X'''')))) -> PROPER(g(g(X'''')))

two new Dependency Pairs are created:

PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 11

             ↳FwdInst

...

               →DP Problem 16

                 ↳Forward Instantiation Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

PROPER(h(g(h(X'''')))) -> PROPER(g(h(X'''')))

three new Dependency Pairs are created:

PROPER(h(g(h(h(X''''''))))) -> PROPER(g(h(h(X''''''))))
PROPER(h(g(h(g(g(X'''''''')))))) -> PROPER(g(h(g(g(X'''''''')))))
PROPER(h(g(h(g(h(X'''''''')))))) -> PROPER(g(h(g(h(X'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 11

             ↳FwdInst

...

               →DP Problem 17

                 ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳Nar

Dependency Pairs:

PROPER(h(g(h(g(h(X'''''''')))))) -> PROPER(g(h(g(h(X'''''''')))))
PROPER(h(g(h(g(g(X'''''''')))))) -> PROPER(g(h(g(g(X'''''''')))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(h(g(h(h(X''''''))))) -> PROPER(g(h(h(X''''''))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(h(g(h(g(h(X'''''''')))))) -> PROPER(g(h(g(h(X'''''''')))))
PROPER(h(g(h(g(g(X'''''''')))))) -> PROPER(g(h(g(g(X'''''''')))))
PROPER(h(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))
PROPER(h(g(g(h(X''''''))))) -> PROPER(g(g(h(X''''''))))
PROPER(g(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(g(g(h(X'''')))) -> PROPER(g(h(X'''')))
PROPER(g(g(g(X'''')))) -> PROPER(g(g(X'''')))
PROPER(h(g(g(g(X''''''))))) -> PROPER(g(g(g(X''''''))))
PROPER(h(h(g(g(X''''''))))) -> PROPER(h(g(g(X''''''))))
PROPER(h(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(g(h(h(X'''')))) -> PROPER(h(h(X'''')))
PROPER(h(g(h(h(X''''''))))) -> PROPER(g(h(h(X''''''))))
PROPER(g(h(g(h(X''''''))))) -> PROPER(h(g(h(X''''''))))

The following usable rules for innermost w.r.t. to the AFS can be oriented:

g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

h > {ok, g}

resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
g(x₁) -> g(x₁)
h(x₁) -> h(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 11

             ↳FwdInst

...

               →DP Problem 18

                 ↳Dependency Graph

       →DP Problem 4

         ↳Nar

Dependency Pair:

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Narrowing Transformation

Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(mark(X)) -> TOP(proper(X))

four new Dependency Pairs are created:

TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))
TOP(mark(c)) -> TOP(ok(c))
TOP(mark(d)) -> TOP(ok(d))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Narrowing Transformation

Dependency Pairs:

TOP(mark(d)) -> TOP(ok(d))
TOP(mark(c)) -> TOP(ok(c))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(ok(X)) -> TOP(active(X))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(ok(X)) -> TOP(active(X))

three new Dependency Pairs are created:

TOP(ok(g(X''))) -> TOP(mark(h(X'')))
TOP(ok(c)) -> TOP(mark(d))
TOP(ok(h(d))) -> TOP(mark(g(c)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 20

                 ↳Narrowing Transformation

Dependency Pairs:

TOP(ok(h(d))) -> TOP(mark(g(c)))
TOP(ok(g(X''))) -> TOP(mark(h(X'')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(mark(g(X''))) -> TOP(g(proper(X'')))

four new Dependency Pairs are created:

TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(c))) -> TOP(g(ok(c)))
TOP(mark(g(d))) -> TOP(g(ok(d)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 21

                 ↳Rewriting Transformation

Dependency Pairs:

TOP(mark(g(d))) -> TOP(g(ok(d)))
TOP(mark(g(c))) -> TOP(g(ok(c)))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(ok(g(X''))) -> TOP(mark(h(X'')))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))
TOP(ok(h(d))) -> TOP(mark(g(c)))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(mark(g(c))) -> TOP(g(ok(c)))

one new Dependency Pair is created:

TOP(mark(g(c))) -> TOP(ok(g(c)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 22

                 ↳Rewriting Transformation

Dependency Pairs:

TOP(mark(g(c))) -> TOP(ok(g(c)))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(ok(h(d))) -> TOP(mark(g(c)))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(ok(g(X''))) -> TOP(mark(h(X'')))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))
TOP(mark(g(d))) -> TOP(g(ok(d)))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(mark(g(d))) -> TOP(g(ok(d)))

one new Dependency Pair is created:

TOP(mark(g(d))) -> TOP(ok(g(d)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 23

                 ↳Narrowing Transformation

Dependency Pairs:

TOP(mark(g(d))) -> TOP(ok(g(d)))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(ok(h(d))) -> TOP(mark(g(c)))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))
TOP(ok(g(X''))) -> TOP(mark(h(X'')))
TOP(mark(g(c))) -> TOP(ok(g(c)))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(mark(h(X''))) -> TOP(h(proper(X'')))

four new Dependency Pairs are created:

TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))
TOP(mark(h(c))) -> TOP(h(ok(c)))
TOP(mark(h(d))) -> TOP(h(ok(d)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 24

                 ↳Rewriting Transformation

Dependency Pairs:

TOP(mark(h(d))) -> TOP(h(ok(d)))
TOP(mark(h(c))) -> TOP(h(ok(c)))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(g(c))) -> TOP(ok(g(c)))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(ok(h(d))) -> TOP(mark(g(c)))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(ok(g(X''))) -> TOP(mark(h(X'')))
TOP(mark(g(d))) -> TOP(ok(g(d)))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(mark(h(c))) -> TOP(h(ok(c)))

one new Dependency Pair is created:

TOP(mark(h(c))) -> TOP(ok(h(c)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 25

                 ↳Rewriting Transformation

Dependency Pairs:

TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(g(d))) -> TOP(ok(g(d)))
TOP(mark(g(c))) -> TOP(ok(g(c)))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(ok(h(d))) -> TOP(mark(g(c)))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(ok(g(X''))) -> TOP(mark(h(X'')))
TOP(mark(h(d))) -> TOP(h(ok(d)))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(mark(h(d))) -> TOP(h(ok(d)))

one new Dependency Pair is created:

TOP(mark(h(d))) -> TOP(ok(h(d)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 26

                 ↳Forward Instantiation Transformation

Dependency Pairs:

TOP(mark(h(d))) -> TOP(ok(h(d)))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(g(d))) -> TOP(ok(g(d)))
TOP(mark(g(c))) -> TOP(ok(g(c)))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(ok(h(d))) -> TOP(mark(g(c)))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(ok(g(X''))) -> TOP(mark(h(X'')))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(ok(g(X''))) -> TOP(mark(h(X'')))

three new Dependency Pairs are created:

TOP(ok(g(g(X'''')))) -> TOP(mark(h(g(X''''))))
TOP(ok(g(h(X'''')))) -> TOP(mark(h(h(X''''))))
TOP(ok(g(d))) -> TOP(mark(h(d)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 27

                 ↳Narrowing Transformation

Dependency Pairs:

TOP(ok(g(h(X'''')))) -> TOP(mark(h(h(X''''))))
TOP(ok(g(g(X'''')))) -> TOP(mark(h(g(X''''))))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(ok(g(d))) -> TOP(mark(h(d)))
TOP(mark(g(d))) -> TOP(ok(g(d)))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(ok(h(d))) -> TOP(mark(g(c)))
TOP(mark(h(d))) -> TOP(ok(h(d)))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(ok(h(d))) -> TOP(mark(g(c)))

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 28

                 ↳Narrowing Transformation

Dependency Pairs:

TOP(ok(g(g(X'''')))) -> TOP(mark(h(g(X''''))))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(ok(g(d))) -> TOP(mark(h(d)))
TOP(mark(g(d))) -> TOP(ok(g(d)))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(ok(g(h(X'''')))) -> TOP(mark(h(h(X''''))))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(ok(g(d))) -> TOP(mark(h(d)))

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 29

                 ↳Argument Filtering and Ordering

Dependency Pairs:

TOP(ok(g(h(X'''')))) -> TOP(mark(h(h(X''''))))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(ok(g(g(X'''')))) -> TOP(mark(h(g(X''''))))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

TOP(ok(g(h(X'''')))) -> TOP(mark(h(h(X''''))))
TOP(mark(h(h(X')))) -> TOP(h(h(proper(X'))))
TOP(mark(h(g(X')))) -> TOP(h(g(proper(X'))))
TOP(mark(g(h(X')))) -> TOP(g(h(proper(X'))))
TOP(mark(g(g(X')))) -> TOP(g(g(proper(X'))))
TOP(ok(g(g(X'''')))) -> TOP(mark(h(g(X''''))))

The following usable rules for innermost w.r.t. to the AFS can be oriented:

proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

g > mark
g > h

resulting in one new DP problem.
Used Argument Filtering System:

TOP(x₁) -> TOP(x₁)
mark(x₁) -> mark(x₁)
g(x₁) -> g(x₁)
h(x₁) -> h(x₁)
proper(x₁) -> x₁
ok(x₁) -> x₁

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 30

                 ↳Dependency Graph

Dependency Pair:

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:03 minutes