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

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Remaining

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

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 5

             ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Remaining

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

         ↳Polo

       →DP Problem 2

         ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Remaining

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

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(H(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

         ↳Remaining

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

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polynomial Ordering

       →DP Problem 4

         ↳Remaining

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

The following dependency pair can be strictly oriented:

PROPER(h(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)) = x₁

POL(PROPER(x₁)) = x₁

POL(h(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

             ↳Polynomial Ordering

       →DP Problem 4

         ↳Remaining

Dependency Pair:

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

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

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

         ↳Remaining

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

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:
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

Innermost Termination of R could not be shown.
Duration:
0:00 minutes

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