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

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

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 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:

F(x₁) -> F(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 5

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

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

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

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 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

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 6

             ↳Dependency Graph

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳AFS

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 pairs can be strictly oriented:

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

PROPER(x₁) -> PROPER(x₁)
f(x₁) -> f(x₁)
g(x₁) -> g(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

           →DP Problem 7

             ↳Dependency Graph

       →DP Problem 4

         ↳AFS

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Argument Filtering and Ordering

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

The following dependency pair can be strictly oriented:

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

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

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

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

c > f > mark > proper
c > f > g

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

TOP(x₁) -> TOP(x₁)
mark(x₁) -> mark(x₁)
proper(x₁) -> proper(x₁)
ok(x₁) -> x₁
active(x₁) -> x₁
f(x₁) -> f
g(x₁) -> g

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

           →DP Problem 8

             ↳Narrowing Transformation

Dependency Pair:

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 no new DP problems.

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