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

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →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

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: 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 5

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →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

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

       →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

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: 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

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 6

             ↳Dependency Graph

       →DP Problem 3

         ↳AFS

       →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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and Ordering

       →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

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost 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)
h(x₁) -> h(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

         ↳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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Nar

           →DP Problem 8

             ↳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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 9

                 ↳Argument Filtering and Ordering

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

The following dependency pair can be strictly oriented:

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

The following usable rules for innermost can be oriented:

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

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

d > c
{mark, ok, h, proper, g}

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

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

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 10

                 ↳Argument Filtering and Ordering

Dependency Pairs:

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

The following dependency pairs can be strictly oriented:

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

The following usable rules for innermost can be oriented:

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

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

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 11

                 ↳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