Termination Proof using AProVE

Term Rewriting System R:
[X]
active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(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(c) -> F(g(c))
ACTIVE(c) -> G(c)
PROPER(f(X)) -> F(proper(X))
PROPER(f(X)) -> PROPER(X)
PROPER(g(X)) -> G(proper(X))
PROPER(g(X)) -> PROPER(X)
F(ok(X)) -> F(X)
G(ok(X)) -> G(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

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

Dependency Pair:

F(ok(X)) -> F(X)

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(ok(X)) -> F(X)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ok(x₁)) = 1 + x₁

POL(F(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 5

             ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

Dependency Pair:

Rules:

active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(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

         ↳Polo

       →DP Problem 2

         ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

Dependency Pair:

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

Rules:

active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(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(X)) -> G(X)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(G(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 6

             ↳Dependency Graph

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

Dependency Pair:

Rules:

active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(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

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polynomial Ordering

       →DP Problem 4

         ↳Nar

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(g(x₁)) = 1 + x₁

POL(PROPER(x₁)) = x₁

POL(f(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 7

             ↳Polynomial Ordering

       →DP Problem 4

         ↳Nar

Dependency Pair:

PROPER(f(X)) -> PROPER(X)

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(f(X)) -> PROPER(X)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(PROPER(x₁)) = x₁

POL(f(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 7

             ↳Polo

...

               →DP Problem 8

                 ↳Dependency Graph

       →DP Problem 4

         ↳Nar

Dependency Pair:

Rules:

active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(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

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(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))

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

           →DP Problem 9

             ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(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))

two new Dependency Pairs are created:

TOP(ok(c)) -> TOP(mark(f(g(c))))
TOP(ok(f(g(X'')))) -> TOP(mark(g(X'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

           →DP Problem 9

             ↳Nar

...

               →DP Problem 10

                 ↳Polynomial Ordering

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

g(ok(X)) -> ok(g(X))
f(ok(X)) -> ok(f(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(proper(x₁)) = x₁

POL(c) = 1

POL(g(x₁)) = 0

POL(mark(x₁)) = x₁

POL(ok(x₁)) = x₁

POL(TOP(x₁)) = 1 + x₁

POL(f(x₁)) = 1

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

           →DP Problem 9

             ↳Nar

...

               →DP Problem 11

                 ↳Polynomial Ordering

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

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

f(ok(X)) -> ok(f(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
g(ok(X)) -> ok(g(X))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(proper(x₁)) = x₁

POL(c) = 1

POL(g(x₁)) = 0

POL(mark(x₁)) = 1

POL(ok(x₁)) = x₁

POL(TOP(x₁)) = x₁

POL(f(x₁)) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

           →DP Problem 9

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

active(c) -> mark(f(g(c)))
active(f(g(X))) -> mark(g(X))
proper(c) -> ok(c)
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
f(ok(X)) -> ok(f(X))
g(ok(X)) -> ok(g(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(c)) -> TOP(mark(f(g(c))))

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

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

POL(g(x₁))	= 1 + x₁
POL(PROPER(x₁))	= x₁
POL(f(x₁))	= x₁

POL(proper(x₁))	= x₁
POL(c)	= 1
POL(g(x₁))	= 0
POL(mark(x₁))	= x₁
POL(ok(x₁))	= x₁
POL(TOP(x₁))	= 1 + x₁
POL(f(x₁))	= 1