Termination Proof using AProVE

Term Rewriting System R:
[x]
active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

ACTIVE(f(x)) -> F(f(x))
CHK(no(f(x))) -> F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
CHK(no(f(x))) -> CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))
CHK(no(f(x))) -> MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)
CHK(no(f(x))) -> F(f(f(f(f(f(f(f(f(f(X))))))))))
CHK(no(f(x))) -> F(f(f(f(f(f(f(f(f(X)))))))))
CHK(no(f(x))) -> F(f(f(f(f(f(f(f(X))))))))
CHK(no(f(x))) -> F(f(f(f(f(f(f(X)))))))
CHK(no(f(x))) -> F(f(f(f(f(f(X))))))
CHK(no(f(x))) -> F(f(f(f(f(X)))))
CHK(no(f(x))) -> F(f(f(f(X))))
CHK(no(f(x))) -> F(f(f(X)))
CHK(no(f(x))) -> F(f(X))
CHK(no(f(x))) -> F(X)
CHK(no(c)) -> ACTIVE(c)
MAT(f(x), f(y)) -> F(mat(x, y))
MAT(f(x), f(y)) -> MAT(x, y)
F(active(x)) -> ACTIVE(f(x))
F(active(x)) -> F(x)
F(no(x)) -> F(x)
F(mark(x)) -> F(x)
TP(mark(x)) -> TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
TP(mark(x)) -> CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))
TP(mark(x)) -> MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)
TP(mark(x)) -> F(f(f(f(f(f(f(f(f(f(X))))))))))
TP(mark(x)) -> F(f(f(f(f(f(f(f(f(X)))))))))
TP(mark(x)) -> F(f(f(f(f(f(f(f(X))))))))
TP(mark(x)) -> F(f(f(f(f(f(f(X)))))))
TP(mark(x)) -> F(f(f(f(f(f(X))))))
TP(mark(x)) -> F(f(f(f(f(X)))))
TP(mark(x)) -> F(f(f(f(X))))
TP(mark(x)) -> F(f(f(X)))
TP(mark(x)) -> F(f(X))
TP(mark(x)) -> F(X)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(mark(x)) -> F(x)
F(no(x)) -> F(x)
F(active(x)) -> F(x)

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

F(active(x)) -> F(x)

three new Dependency Pairs are created:

F(active(active(x''))) -> F(active(x''))
F(active(no(x''))) -> F(no(x''))
F(active(mark(x''))) -> F(mark(x''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(active(mark(x''))) -> F(mark(x''))
F(active(no(x''))) -> F(no(x''))
F(active(active(x''))) -> F(active(x''))
F(no(x)) -> F(x)
F(mark(x)) -> F(x)

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

F(no(x)) -> F(x)

five new Dependency Pairs are created:

F(no(no(x''))) -> F(no(x''))
F(no(mark(x''))) -> F(mark(x''))
F(no(active(active(x'''')))) -> F(active(active(x'''')))
F(no(active(no(x'''')))) -> F(active(no(x'''')))
F(no(active(mark(x'''')))) -> F(active(mark(x'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(no(active(mark(x'''')))) -> F(active(mark(x'''')))
F(no(active(no(x'''')))) -> F(active(no(x'''')))
F(no(active(active(x'''')))) -> F(active(active(x'''')))
F(no(mark(x''))) -> F(mark(x''))
F(no(no(x''))) -> F(no(x''))
F(active(no(x''))) -> F(no(x''))
F(active(active(x''))) -> F(active(x''))
F(mark(x)) -> F(x)
F(active(mark(x''))) -> F(mark(x''))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

F(mark(x)) -> F(x)

nine new Dependency Pairs are created:

F(mark(mark(x''))) -> F(mark(x''))
F(mark(active(active(x'''')))) -> F(active(active(x'''')))
F(mark(active(no(x'''')))) -> F(active(no(x'''')))
F(mark(active(mark(x'''')))) -> F(active(mark(x'''')))
F(mark(no(no(x'''')))) -> F(no(no(x'''')))
F(mark(no(mark(x'''')))) -> F(no(mark(x'''')))
F(mark(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(mark(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(mark(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(mark(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(no(active(no(x'''')))) -> F(active(no(x'''')))
F(mark(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(no(active(active(x'''')))) -> F(active(active(x'''')))
F(mark(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(mark(no(mark(x'''')))) -> F(no(mark(x'''')))
F(mark(no(no(x'''')))) -> F(no(no(x'''')))
F(mark(active(mark(x'''')))) -> F(active(mark(x'''')))
F(mark(active(no(x'''')))) -> F(active(no(x'''')))
F(no(mark(x''))) -> F(mark(x''))
F(no(no(x''))) -> F(no(x''))
F(active(no(x''))) -> F(no(x''))
F(active(active(x''))) -> F(active(x''))
F(mark(active(active(x'''')))) -> F(active(active(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(active(mark(x''))) -> F(mark(x''))
F(no(active(mark(x'''')))) -> F(active(mark(x'''')))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

F(active(no(x''))) -> F(no(x''))

five new Dependency Pairs are created:

F(active(no(no(x'''')))) -> F(no(no(x'''')))
F(active(no(mark(x'''')))) -> F(no(mark(x'''')))
F(active(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(active(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(active(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(mark(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(mark(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(mark(no(mark(x'''')))) -> F(no(mark(x'''')))
F(mark(no(no(x'''')))) -> F(no(no(x'''')))
F(mark(active(mark(x'''')))) -> F(active(mark(x'''')))
F(active(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(no(active(no(x'''')))) -> F(active(no(x'''')))
F(active(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(no(active(active(x'''')))) -> F(active(active(x'''')))
F(active(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(active(no(mark(x'''')))) -> F(no(mark(x'''')))
F(mark(active(no(x'''')))) -> F(active(no(x'''')))
F(no(mark(x''))) -> F(mark(x''))
F(no(no(x''))) -> F(no(x''))
F(active(no(no(x'''')))) -> F(no(no(x'''')))
F(active(active(x''))) -> F(active(x''))
F(mark(active(active(x'''')))) -> F(active(active(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(active(mark(x''))) -> F(mark(x''))
F(no(active(mark(x'''')))) -> F(active(mark(x'''')))
F(mark(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

F(active(mark(x''))) -> F(mark(x''))

nine new Dependency Pairs are created:

F(active(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(active(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(active(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(active(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(active(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(active(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(active(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(active(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(active(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 8

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(mark(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(active(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(active(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(mark(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(active(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(mark(no(mark(x'''')))) -> F(no(mark(x'''')))
F(active(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(mark(no(no(x'''')))) -> F(no(no(x'''')))
F(active(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(mark(active(mark(x'''')))) -> F(active(mark(x'''')))
F(active(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(mark(active(no(x'''')))) -> F(active(no(x'''')))
F(active(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(active(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(active(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(active(mark(x'''')))) -> F(active(mark(x'''')))
F(active(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(active(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(no(active(active(x'''')))) -> F(active(active(x'''')))
F(active(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(active(no(mark(x'''')))) -> F(no(mark(x'''')))
F(active(active(x''))) -> F(active(x''))
F(mark(active(active(x'''')))) -> F(active(active(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(no(mark(x''))) -> F(mark(x''))
F(no(no(x''))) -> F(no(x''))
F(active(no(no(x'''')))) -> F(no(no(x'''')))
F(no(active(no(x'''')))) -> F(active(no(x'''')))
F(mark(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

F(no(no(x''))) -> F(no(x''))

five new Dependency Pairs are created:

F(no(no(no(x'''')))) -> F(no(no(x'''')))
F(no(no(mark(x'''')))) -> F(no(mark(x'''')))
F(no(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(no(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(no(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 9

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(active(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(mark(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(active(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(mark(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(active(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(mark(no(mark(x'''')))) -> F(no(mark(x'''')))
F(active(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(no(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(no(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(no(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(mark(no(no(x'''')))) -> F(no(no(x'''')))
F(active(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(active(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(active(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(active(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(mark(active(mark(x'''')))) -> F(active(mark(x'''')))
F(active(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(no(active(no(x'''')))) -> F(active(no(x'''')))
F(active(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(no(active(active(x'''')))) -> F(active(active(x'''')))
F(active(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(active(no(mark(x'''')))) -> F(no(mark(x'''')))
F(mark(active(no(x'''')))) -> F(active(no(x'''')))
F(no(mark(x''))) -> F(mark(x''))
F(no(no(mark(x'''')))) -> F(no(mark(x'''')))
F(no(no(no(x'''')))) -> F(no(no(x'''')))
F(active(no(no(x'''')))) -> F(no(no(x'''')))
F(active(active(x''))) -> F(active(x''))
F(mark(active(active(x'''')))) -> F(active(active(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(active(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(active(mark(x'''')))) -> F(active(mark(x'''')))
F(mark(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

F(no(mark(x''))) -> F(mark(x''))

nine new Dependency Pairs are created:

F(no(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(no(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(no(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(no(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(no(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(no(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(no(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(no(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 10

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(active(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(active(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(no(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(mark(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(no(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(mark(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(no(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(no(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(no(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(mark(no(mark(x'''')))) -> F(no(mark(x'''')))
F(active(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(no(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(active(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(active(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(active(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(no(active(no(x'''')))) -> F(active(no(x'''')))
F(no(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(no(active(active(x'''')))) -> F(active(active(x'''')))
F(no(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(mark(no(no(x'''')))) -> F(no(no(x'''')))
F(active(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(active(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(active(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(active(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(mark(active(mark(x'''')))) -> F(active(mark(x'''')))
F(no(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(active(no(mark(x'''')))) -> F(no(mark(x'''')))
F(mark(active(no(x'''')))) -> F(active(no(x'''')))
F(no(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(no(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(no(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(no(mark(x'''')))) -> F(no(mark(x'''')))
F(no(no(no(x'''')))) -> F(no(no(x'''')))
F(active(no(no(x'''')))) -> F(no(no(x'''')))
F(active(active(x''))) -> F(active(x''))
F(mark(active(active(x'''')))) -> F(active(active(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(active(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(active(mark(x'''')))) -> F(active(mark(x'''')))
F(mark(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(active(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

F(mark(mark(x''))) -> F(mark(x''))

nine new Dependency Pairs are created:

F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(mark(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(mark(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(mark(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(mark(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(mark(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(mark(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(mark(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(mark(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 11

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(mark(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(mark(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(mark(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(mark(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(mark(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(active(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(active(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(mark(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(no(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(no(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(mark(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(no(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(no(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(no(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(no(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(mark(no(mark(x'''')))) -> F(no(mark(x'''')))
F(active(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(no(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(no(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(no(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(mark(no(no(x'''')))) -> F(no(no(x'''')))
F(active(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(active(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(active(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(active(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(mark(active(mark(x'''')))) -> F(active(mark(x'''')))
F(mark(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(mark(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(active(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(active(mark(x'''')))) -> F(active(mark(x'''')))
F(active(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(active(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(no(active(active(x'''')))) -> F(active(active(x'''')))
F(active(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(mark(active(no(x'''')))) -> F(active(no(x'''')))
F(no(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(no(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(active(no(mark(x'''')))) -> F(no(mark(x'''')))
F(active(active(x''))) -> F(active(x''))
F(mark(active(active(x'''')))) -> F(active(active(x'''')))
F(mark(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(no(mark(x'''')))) -> F(no(mark(x'''')))
F(no(no(no(x'''')))) -> F(no(no(x'''')))
F(active(no(no(x'''')))) -> F(no(no(x'''')))
F(no(active(no(x'''')))) -> F(active(no(x'''')))
F(mark(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(active(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

F(mark(no(no(x'''')))) -> F(no(no(x'''')))

five new Dependency Pairs are created:

F(mark(no(no(no(x''''''))))) -> F(no(no(no(x''''''))))
F(mark(no(no(mark(x''''''))))) -> F(no(no(mark(x''''''))))
F(mark(no(no(active(active(x'''''''')))))) -> F(no(no(active(active(x'''''''')))))
F(mark(no(no(active(no(x'''''''')))))) -> F(no(no(active(no(x'''''''')))))
F(mark(no(no(active(mark(x'''''''')))))) -> F(no(no(active(mark(x'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 12

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(mark(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(mark(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(mark(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(mark(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(mark(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(active(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(active(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(active(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(no(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(mark(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(no(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(mark(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(no(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(no(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(no(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(mark(no(mark(x'''')))) -> F(no(mark(x'''')))
F(active(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(no(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(mark(no(no(active(mark(x'''''''')))))) -> F(no(no(active(mark(x'''''''')))))
F(active(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(active(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(active(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(no(active(no(x'''')))) -> F(active(no(x'''')))
F(no(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(mark(no(no(active(no(x'''''''')))))) -> F(no(no(active(no(x'''''''')))))
F(no(active(active(x'''')))) -> F(active(active(x'''')))
F(no(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(mark(no(no(active(active(x'''''''')))))) -> F(no(no(active(active(x'''''''')))))
F(mark(no(no(mark(x''''''))))) -> F(no(no(mark(x''''''))))
F(mark(no(no(no(x''''''))))) -> F(no(no(no(x''''''))))
F(active(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(active(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(active(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(active(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(mark(active(mark(x'''')))) -> F(active(mark(x'''')))
F(no(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(no(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(no(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(active(no(mark(x'''')))) -> F(no(mark(x'''')))
F(mark(active(no(x'''')))) -> F(active(no(x'''')))
F(mark(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(no(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(no(mark(x'''')))) -> F(no(mark(x'''')))
F(no(no(no(x'''')))) -> F(no(no(x'''')))
F(active(no(no(x'''')))) -> F(no(no(x'''')))
F(active(active(x''))) -> F(active(x''))
F(mark(active(active(x'''')))) -> F(active(active(x'''')))
F(mark(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(active(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(active(mark(x'''')))) -> F(active(mark(x'''')))
F(mark(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(mark(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

F(mark(no(mark(x'''')))) -> F(no(mark(x'''')))

nine new Dependency Pairs are created:

F(mark(no(mark(mark(x''''''))))) -> F(no(mark(mark(x''''''))))
F(mark(no(mark(active(active(x'''''''')))))) -> F(no(mark(active(active(x'''''''')))))
F(mark(no(mark(active(no(x'''''''')))))) -> F(no(mark(active(no(x'''''''')))))
F(mark(no(mark(active(mark(x'''''''')))))) -> F(no(mark(active(mark(x'''''''')))))
F(mark(no(mark(no(no(x'''''''')))))) -> F(no(mark(no(no(x'''''''')))))
F(mark(no(mark(no(mark(x'''''''')))))) -> F(no(mark(no(mark(x'''''''')))))
F(mark(no(mark(no(active(active(x''''''''''))))))) -> F(no(mark(no(active(active(x''''''''''))))))
F(mark(no(mark(no(active(no(x''''''''''))))))) -> F(no(mark(no(active(no(x''''''''''))))))
F(mark(no(mark(no(active(mark(x''''''''''))))))) -> F(no(mark(no(active(mark(x''''''''''))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 13

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(mark(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(mark(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(mark(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(mark(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(active(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(active(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(active(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(mark(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(no(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(mark(no(mark(no(active(mark(x''''''''''))))))) -> F(no(mark(no(active(mark(x''''''''''))))))
F(no(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(mark(no(mark(no(active(no(x''''''''''))))))) -> F(no(mark(no(active(no(x''''''''''))))))
F(mark(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(no(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(mark(no(mark(no(active(active(x''''''''''))))))) -> F(no(mark(no(active(active(x''''''''''))))))
F(no(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(mark(no(mark(no(mark(x'''''''')))))) -> F(no(mark(no(mark(x'''''''')))))
F(no(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(mark(no(mark(no(no(x'''''''')))))) -> F(no(mark(no(no(x'''''''')))))
F(no(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(mark(no(mark(active(mark(x'''''''')))))) -> F(no(mark(active(mark(x'''''''')))))
F(mark(no(mark(active(no(x'''''''')))))) -> F(no(mark(active(no(x'''''''')))))
F(mark(no(mark(active(active(x'''''''')))))) -> F(no(mark(active(active(x'''''''')))))
F(mark(no(mark(mark(x''''''))))) -> F(no(mark(mark(x''''''))))
F(active(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(no(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(mark(no(no(active(mark(x'''''''')))))) -> F(no(no(active(mark(x'''''''')))))
F(no(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(mark(no(no(active(no(x'''''''')))))) -> F(no(no(active(no(x'''''''')))))
F(no(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(mark(no(no(active(active(x'''''''')))))) -> F(no(no(active(active(x'''''''')))))
F(mark(no(no(mark(x''''''))))) -> F(no(no(mark(x''''''))))
F(mark(no(no(no(x''''''))))) -> F(no(no(no(x''''''))))
F(active(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(active(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(active(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(active(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(mark(active(mark(x'''')))) -> F(active(mark(x'''')))
F(mark(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(mark(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(active(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(active(mark(x'''')))) -> F(active(mark(x'''')))
F(active(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(active(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(no(active(active(x'''')))) -> F(active(active(x'''')))
F(active(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(mark(active(no(x'''')))) -> F(active(no(x'''')))
F(no(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(no(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(active(no(mark(x'''')))) -> F(no(mark(x'''')))
F(active(active(x''))) -> F(active(x''))
F(mark(active(active(x'''')))) -> F(active(active(x'''')))
F(mark(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(no(mark(x'''')))) -> F(no(mark(x'''')))
F(no(no(no(x'''')))) -> F(no(no(x'''')))
F(active(no(no(x'''')))) -> F(no(no(x'''')))
F(no(active(no(x'''')))) -> F(active(no(x'''')))
F(mark(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(mark(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(active(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(active(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(active(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(active(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(active(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(active(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(active(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(active(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(active(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(active(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(active(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(active(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(active(no(mark(x'''')))) -> F(no(mark(x'''')))
F(active(active(x''))) -> F(active(x''))
F(active(no(no(x'''')))) -> F(no(no(x'''')))

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

active(f(x)) -> mark(f(f(x)))
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(active(x₁)) = 1 + x₁

POL(no(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(f(x₁)) = x₁

POL(F(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 14

                 ↳Dependency Graph

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(mark(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(mark(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(mark(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(mark(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(mark(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(no(mark(no(active(mark(x'''''''')))))) -> F(mark(no(active(mark(x'''''''')))))
F(mark(no(mark(no(active(mark(x''''''''''))))))) -> F(no(mark(no(active(mark(x''''''''''))))))
F(no(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))
F(mark(no(mark(no(active(no(x''''''''''))))))) -> F(no(mark(no(active(no(x''''''''''))))))
F(mark(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(no(mark(no(active(active(x'''''''')))))) -> F(mark(no(active(active(x'''''''')))))
F(mark(no(mark(no(active(active(x''''''''''))))))) -> F(no(mark(no(active(active(x''''''''''))))))
F(no(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(mark(no(mark(no(mark(x'''''''')))))) -> F(no(mark(no(mark(x'''''''')))))
F(no(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(mark(no(mark(no(no(x'''''''')))))) -> F(no(mark(no(no(x'''''''')))))
F(no(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(mark(no(mark(active(mark(x'''''''')))))) -> F(no(mark(active(mark(x'''''''')))))
F(mark(no(mark(active(no(x'''''''')))))) -> F(no(mark(active(no(x'''''''')))))
F(mark(no(mark(active(active(x'''''''')))))) -> F(no(mark(active(active(x'''''''')))))
F(mark(no(mark(mark(x''''''))))) -> F(no(mark(mark(x''''''))))
F(no(no(active(mark(x''''''))))) -> F(no(active(mark(x''''''))))
F(mark(no(no(active(mark(x'''''''')))))) -> F(no(no(active(mark(x'''''''')))))
F(no(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(mark(no(no(active(no(x'''''''')))))) -> F(no(no(active(no(x'''''''')))))
F(no(no(active(active(x''''''))))) -> F(no(active(active(x''''''))))
F(mark(no(no(active(active(x'''''''')))))) -> F(no(no(active(active(x'''''''')))))
F(mark(no(no(mark(x''''''))))) -> F(no(no(mark(x''''''))))
F(mark(no(no(no(x''''''))))) -> F(no(no(no(x''''''))))
F(mark(active(mark(x'''')))) -> F(active(mark(x'''')))
F(mark(mark(active(mark(x''''''))))) -> F(mark(active(mark(x''''''))))
F(mark(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(no(active(mark(x'''')))) -> F(active(mark(x'''')))
F(no(active(active(x'''')))) -> F(active(active(x'''')))
F(mark(active(no(x'''')))) -> F(active(no(x'''')))
F(no(mark(active(no(x''''''))))) -> F(mark(active(no(x''''''))))
F(no(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(mark(active(active(x'''')))) -> F(active(active(x'''')))
F(mark(mark(active(active(x''''''))))) -> F(mark(active(active(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(no(mark(x'''')))) -> F(no(mark(x'''')))
F(no(no(no(x'''')))) -> F(no(no(x'''')))
F(no(active(no(x'''')))) -> F(active(no(x'''')))
F(mark(no(active(no(x''''''))))) -> F(no(active(no(x''''''))))
F(mark(mark(no(active(no(x'''''''')))))) -> F(mark(no(active(no(x'''''''')))))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 15

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(mark(no(mark(no(mark(x'''''''')))))) -> F(no(mark(no(mark(x'''''''')))))
F(mark(no(mark(no(no(x'''''''')))))) -> F(no(mark(no(no(x'''''''')))))
F(no(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(mark(no(no(mark(x''''''))))) -> F(no(no(mark(x''''''))))
F(no(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(no(no(mark(x'''')))) -> F(no(mark(x'''')))
F(no(no(no(x'''')))) -> F(no(no(x'''')))
F(mark(no(no(no(x''''''))))) -> F(no(no(no(x''''''))))
F(mark(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(no(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(mark(no(mark(mark(x''''''))))) -> F(no(mark(mark(x''''''))))
F(mark(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(mark(no(mark(no(mark(x'''''''')))))) -> F(no(mark(no(mark(x'''''''')))))
F(mark(no(mark(no(no(x'''''''')))))) -> F(no(mark(no(no(x'''''''')))))
F(mark(no(no(mark(x''''''))))) -> F(no(no(mark(x''''''))))
F(mark(no(no(no(x''''''))))) -> F(no(no(no(x''''''))))
F(mark(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(mark(no(mark(mark(x''''''))))) -> F(no(mark(mark(x''''''))))
F(mark(mark(no(mark(x''''''))))) -> F(mark(no(mark(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(no(x₁)) = x₁

POL(mark(x₁)) = 1 + x₁

POL(F(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 16

                 ↳Dependency Graph

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pairs:

F(no(mark(no(mark(x''''''))))) -> F(mark(no(mark(x''''''))))
F(no(mark(no(no(x''''''))))) -> F(mark(no(no(x''''''))))
F(no(no(mark(x'''')))) -> F(no(mark(x'''')))
F(no(no(no(x'''')))) -> F(no(no(x'''')))
F(no(mark(mark(x'''')))) -> F(mark(mark(x'''')))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 17

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pair:

F(no(no(no(x'''')))) -> F(no(no(x'''')))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(no(no(no(x'''')))) -> F(no(no(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(no(x₁)) = 1 + x₁

POL(F(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 18

                 ↳Dependency Graph

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

Dependency Pair:

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Narrowing Transformation

       →DP Problem 3

         ↳Nar

Dependency Pair:

CHK(no(f(x))) -> CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

CHK(no(f(x))) -> CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))

two new Dependency Pairs are created:

CHK(no(f(f(y)))) -> CHK(f(mat(f(f(f(f(f(f(f(f(f(X))))))))), y)))
CHK(no(f(c))) -> CHK(no(c))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 19

             ↳Narrowing Transformation

       →DP Problem 3

         ↳Nar

Dependency Pair:

CHK(no(f(f(y)))) -> CHK(f(mat(f(f(f(f(f(f(f(f(f(X))))))))), y)))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

CHK(no(f(f(y)))) -> CHK(f(mat(f(f(f(f(f(f(f(f(f(X))))))))), y)))

no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Narrowing Transformation

Dependency Pair:

TP(mark(x)) -> TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

TP(mark(x)) -> TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

two new Dependency Pairs are created:

TP(mark(f(y))) -> TP(chk(f(mat(f(f(f(f(f(f(f(f(f(X))))))))), y))))
TP(mark(c)) -> TP(chk(no(c)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

           →DP Problem 20

             ↳Narrowing Transformation

Dependency Pairs:

TP(mark(c)) -> TP(chk(no(c)))
TP(mark(f(y))) -> TP(chk(f(mat(f(f(f(f(f(f(f(f(f(X))))))))), y))))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

TP(mark(f(y))) -> TP(chk(f(mat(f(f(f(f(f(f(f(f(f(X))))))))), y))))

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

           →DP Problem 20

             ↳Nar

...

               →DP Problem 21

                 ↳Narrowing Transformation

Dependency Pair:

TP(mark(c)) -> TP(chk(no(c)))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

TP(mark(c)) -> TP(chk(no(c)))

one new Dependency Pair is created:

TP(mark(c)) -> TP(active(c))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Nar

           →DP Problem 20

             ↳Nar

...

               →DP Problem 22

                 ↳Narrowing Transformation

Dependency Pair:

TP(mark(c)) -> TP(active(c))

Rules:

active(f(x)) -> mark(f(f(x)))
chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
chk(no(c)) -> active(c)
mat(f(x), f(y)) -> f(mat(x, y))
mat(f(x), c) -> no(c)
f(active(x)) -> active(f(x))
f(no(x)) -> no(f(x))
f(mark(x)) -> mark(f(x))
tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Strategy:

innermost

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

TP(mark(c)) -> TP(active(c))

no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

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

POL(active(x₁))	= 1 + x₁
POL(no(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(f(x₁))	= x₁
POL(F(x₁))	= 1 + x₁