Termination w.r.t. Q proof of /tmp/tmpADhZHg/Ex14_Luc06

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

ACTIVE(h(X)) → H(active(X))
H(ok(X)) → H(X)
ACTIVE(h(X)) → G(X, X)
G(mark(X1), X2) → G(X1, X2)
ACTIVE(g(X1, X2)) → G(active(X1), X2)
TOP(mark(X)) → PROPER(X)
ACTIVE(f(X1, X2)) → F(active(X1), X2)
F(ok(X1), ok(X2)) → F(X1, X2)
TOP(ok(X)) → ACTIVE(X)
PROPER(f(X1, X2)) → PROPER(X1)
PROPER(h(X)) → H(proper(X))
ACTIVE(f(X, X)) → H(a)
PROPER(g(X1, X2)) → PROPER(X2)
PROPER(f(X1, X2)) → PROPER(X2)
ACTIVE(g(a, X)) → F(b, X)
ACTIVE(h(X)) → ACTIVE(X)
F(mark(X1), X2) → F(X1, X2)
ACTIVE(f(X1, X2)) → ACTIVE(X1)
PROPER(g(X1, X2)) → PROPER(X1)
TOP(ok(X)) → TOP(active(X))
ACTIVE(g(X1, X2)) → ACTIVE(X1)
G(ok(X1), ok(X2)) → G(X1, X2)
PROPER(h(X)) → PROPER(X)
PROPER(g(X1, X2)) → G(proper(X1), proper(X2))
TOP(mark(X)) → TOP(proper(X))
H(mark(X)) → H(X)
PROPER(f(X1, X2)) → F(proper(X1), proper(X2))

The TRS R consists of the following rules:

active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(h(X)) → H(active(X))
H(ok(X)) → H(X)
ACTIVE(h(X)) → G(X, X)
G(mark(X1), X2) → G(X1, X2)
ACTIVE(g(X1, X2)) → G(active(X1), X2)
TOP(mark(X)) → PROPER(X)
ACTIVE(f(X1, X2)) → F(active(X1), X2)
F(ok(X1), ok(X2)) → F(X1, X2)
TOP(ok(X)) → ACTIVE(X)
PROPER(f(X1, X2)) → PROPER(X1)
PROPER(h(X)) → H(proper(X))
ACTIVE(f(X, X)) → H(a)
PROPER(g(X1, X2)) → PROPER(X2)
PROPER(f(X1, X2)) → PROPER(X2)
ACTIVE(g(a, X)) → F(b, X)
ACTIVE(h(X)) → ACTIVE(X)
F(mark(X1), X2) → F(X1, X2)
ACTIVE(f(X1, X2)) → ACTIVE(X1)
PROPER(g(X1, X2)) → PROPER(X1)
TOP(ok(X)) → TOP(active(X))
ACTIVE(g(X1, X2)) → ACTIVE(X1)
G(ok(X1), ok(X2)) → G(X1, X2)
PROPER(h(X)) → PROPER(X)
PROPER(g(X1, X2)) → G(proper(X1), proper(X2))
TOP(mark(X)) → TOP(proper(X))
H(mark(X)) → H(X)
PROPER(f(X1, X2)) → F(proper(X1), proper(X2))

The TRS R consists of the following rules:

active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 6 SCCs with 11 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

F(mark(X1), X2) → F(X1, X2)
F(ok(X1), ok(X2)) → F(X1, X2)

The TRS R consists of the following rules:

active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

F(mark(X1), X2) → F(X1, X2)
F(ok(X1), ok(X2)) → F(X1, X2)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

F(mark(X1), X2) → F(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
F(ok(X1), ok(X2)) → F(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

G(mark(X1), X2) → G(X1, X2)
G(ok(X1), ok(X2)) → G(X1, X2)

The TRS R consists of the following rules:

active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

G(mark(X1), X2) → G(X1, X2)
G(ok(X1), ok(X2)) → G(X1, X2)

From the DPs we obtained the following set of size-change graphs:

G(mark(X1), X2) → G(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
G(ok(X1), ok(X2)) → G(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

H(ok(X)) → H(X)
H(mark(X)) → H(X)

The TRS R consists of the following rules:

active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

H(ok(X)) → H(X)
H(mark(X)) → H(X)

From the DPs we obtained the following set of size-change graphs:

H(ok(X)) → H(X)
The graph contains the following edges 1 > 1
H(mark(X)) → H(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER(g(X1, X2)) → PROPER(X2)
PROPER(f(X1, X2)) → PROPER(X2)
PROPER(g(X1, X2)) → PROPER(X1)
PROPER(f(X1, X2)) → PROPER(X1)
PROPER(h(X)) → PROPER(X)

The TRS R consists of the following rules:

active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER(g(X1, X2)) → PROPER(X2)
PROPER(f(X1, X2)) → PROPER(X2)
PROPER(g(X1, X2)) → PROPER(X1)
PROPER(h(X)) → PROPER(X)
PROPER(f(X1, X2)) → PROPER(X1)

From the DPs we obtained the following set of size-change graphs:

PROPER(g(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(f(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(g(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(f(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(h(X)) → PROPER(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(g(X1, X2)) → ACTIVE(X1)
ACTIVE(h(X)) → ACTIVE(X)
ACTIVE(f(X1, X2)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(g(X1, X2)) → ACTIVE(X1)
ACTIVE(h(X)) → ACTIVE(X)
ACTIVE(f(X1, X2)) → ACTIVE(X1)

From the DPs we obtained the following set of size-change graphs:

ACTIVE(g(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(h(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(f(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(TOP(x₁)) = x₁
POL(a) = 0
POL(active(x₁)) = x₁
POL(b) = 0
POL(f(x₁, x₂)) = 2·x₁ + x₂
POL(g(x₁, x₂)) = x₁ + x₂
POL(h(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = 2·x₁
POL(proper(x₁)) = x₁

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
f(mark(X1), X2) → mark(f(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(mark(X1), X2) → mark(g(X1, X2))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
h(mark(X)) → mark(h(X))
h(ok(X)) → ok(h(X))
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(mark(X)) → TOP(proper(X)) at position [0] we obtained the following new rules:

TOP(mark(f(x0, x1))) → TOP(f(proper(x0), proper(x1)))
TOP(mark(h(x0))) → TOP(h(proper(x0)))
TOP(mark(g(x0, x1))) → TOP(g(proper(x0), proper(x1)))
TOP(mark(b)) → TOP(ok(b))
TOP(mark(a)) → TOP(ok(a))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(h(x0))) → TOP(h(proper(x0)))
TOP(mark(g(x0, x1))) → TOP(g(proper(x0), proper(x1)))
TOP(mark(f(x0, x1))) → TOP(f(proper(x0), proper(x1)))
TOP(mark(b)) → TOP(ok(b))
TOP(mark(a)) → TOP(ok(a))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
f(mark(X1), X2) → mark(f(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(mark(X1), X2) → mark(g(X1, X2))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
h(mark(X)) → mark(h(X))
h(ok(X)) → ok(h(X))
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(h(x0))) → TOP(h(proper(x0)))
TOP(mark(g(x0, x1))) → TOP(g(proper(x0), proper(x1)))
TOP(mark(f(x0, x1))) → TOP(f(proper(x0), proper(x1)))
TOP(mark(b)) → TOP(ok(b))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
f(mark(X1), X2) → mark(f(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(mark(X1), X2) → mark(g(X1, X2))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
h(mark(X)) → mark(h(X))
h(ok(X)) → ok(h(X))
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(ok(X)) → TOP(active(X)) at position [0] we obtained the following new rules:

TOP(ok(h(x0))) → TOP(h(active(x0)))
TOP(ok(g(a, x0))) → TOP(mark(f(b, x0)))
TOP(ok(f(x0, x0))) → TOP(mark(h(a)))
TOP(ok(f(x0, x1))) → TOP(f(active(x0), x1))
TOP(ok(h(x0))) → TOP(mark(g(x0, x0)))
TOP(ok(a)) → TOP(mark(b))
TOP(ok(g(x0, x1))) → TOP(g(active(x0), x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(h(x0))) → TOP(h(active(x0)))
TOP(ok(g(a, x0))) → TOP(mark(f(b, x0)))
TOP(mark(h(x0))) → TOP(h(proper(x0)))
TOP(mark(g(x0, x1))) → TOP(g(proper(x0), proper(x1)))
TOP(ok(f(x0, x0))) → TOP(mark(h(a)))
TOP(ok(f(x0, x1))) → TOP(f(active(x0), x1))
TOP(mark(f(x0, x1))) → TOP(f(proper(x0), proper(x1)))
TOP(ok(h(x0))) → TOP(mark(g(x0, x0)))
TOP(ok(a)) → TOP(mark(b))
TOP(ok(g(x0, x1))) → TOP(g(active(x0), x1))
TOP(mark(b)) → TOP(ok(b))

The TRS R consists of the following rules:

proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
f(mark(X1), X2) → mark(f(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(mark(X1), X2) → mark(g(X1, X2))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
h(mark(X)) → mark(h(X))
h(ok(X)) → ok(h(X))
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ SemLabProof

                                ↳ SemLabProof2

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(h(x0))) → TOP(h(active(x0)))
TOP(ok(g(a, x0))) → TOP(mark(f(b, x0)))
TOP(mark(h(x0))) → TOP(h(proper(x0)))
TOP(mark(g(x0, x1))) → TOP(g(proper(x0), proper(x1)))
TOP(ok(f(x0, x0))) → TOP(mark(h(a)))
TOP(ok(f(x0, x1))) → TOP(f(active(x0), x1))
TOP(mark(f(x0, x1))) → TOP(f(proper(x0), proper(x1)))
TOP(ok(h(x0))) → TOP(mark(g(x0, x0)))
TOP(ok(g(x0, x1))) → TOP(g(active(x0), x1))

The TRS R consists of the following rules:

proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
f(mark(X1), X2) → mark(f(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(mark(X1), X2) → mark(g(X1, X2))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
h(mark(X)) → mark(h(X))
h(ok(X)) → ok(h(X))
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We found the following model for the rules of the TRS R. Interpretation over the domain with elements from 0 to 1.active: 0
a: 0
g: 0
f: 0
h: 0
mark: 0
ok: x₀
b: 1
TOP: 0
proper: x₀
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(ok.0(g.0-1(a., x0))) → TOP.0(mark.0(f.1-1(b., x0)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.1-1(x0, x0))) → TOP.0(mark.0(h.0(a.)))
TOP.0(ok.0(g.0-0(a., x0))) → TOP.0(mark.0(f.1-0(b., x0)))
TOP.0(ok.0(h.0(x0))) → TOP.0(mark.0(g.0-0(x0, x0)))
TOP.0(ok.0(g.1-1(x0, x1))) → TOP.0(g.0-1(active.1(x0), x1))
TOP.0(ok.0(h.1(x0))) → TOP.0(mark.0(g.1-1(x0, x0)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(ok.0(f.0-0(x0, x0))) → TOP.0(mark.0(h.0(a.)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(ok.0(f.1-1(x0, x1))) → TOP.0(f.0-1(active.1(x0), x1))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(ok.0(h.1(x0))) → TOP.0(h.0(active.1(x0)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(ok.0(g.1-0(x0, x1))) → TOP.0(g.0-0(active.1(x0), x1))
TOP.0(ok.0(f.1-0(x0, x1))) → TOP.0(f.0-0(active.1(x0), x1))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))

The TRS R consists of the following rules:

proper.0(h.1(X)) → h.1(proper.1(X))
active.0(g.0-0(a., X)) → mark.0(f.1-0(b., X))
h.0(ok.0(X)) → ok.0(h.0(X))
active.0(f.0-0(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → mark.0(g.1-1(X, X))
proper.0(f.1-1(X1, X2)) → f.1-1(proper.1(X1), proper.1(X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
active.0(f.1-0(X1, X2)) → f.0-0(active.1(X1), X2)
proper.0(f.0-1(X1, X2)) → f.0-1(proper.0(X1), proper.1(X2))
proper.1(b.) → ok.1(b.)
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
active.0(h.0(X)) → h.0(active.0(X))
proper.0(g.1-1(X1, X2)) → g.1-1(proper.1(X1), proper.1(X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
active.0(f.0-0(X1, X2)) → f.0-0(active.0(X1), X2)
proper.0(f.1-0(X1, X2)) → f.1-0(proper.1(X1), proper.0(X2))
active.0(f.1-1(X1, X2)) → f.0-1(active.1(X1), X2)
active.0(g.0-1(a., X)) → mark.0(f.1-1(b., X))
proper.0(a.) → ok.0(a.)
active.0(g.1-0(X1, X2)) → g.0-0(active.1(X1), X2)
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
active.0(g.0-0(X1, X2)) → g.0-0(active.0(X1), X2)
proper.0(f.0-0(X1, X2)) → f.0-0(proper.0(X1), proper.0(X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
active.0(g.0-1(X1, X2)) → g.0-1(active.0(X1), X2)
proper.0(g.1-0(X1, X2)) → g.1-0(proper.1(X1), proper.0(X2))
active.0(g.1-1(X1, X2)) → g.0-1(active.1(X1), X2)
active.0(a.) → mark.1(b.)
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
active.0(f.0-1(X1, X2)) → f.0-1(active.0(X1), X2)
proper.0(g.0-0(X1, X2)) → g.0-0(proper.0(X1), proper.0(X2))
active.0(f.1-1(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → h.0(active.1(X))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
active.0(h.0(X)) → mark.0(g.0-0(X, X))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
proper.0(g.0-1(X1, X2)) → g.0-1(proper.0(X1), proper.1(X2))
proper.0(h.0(X)) → h.0(proper.0(X))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ SemLabProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                ↳ SemLabProof2

Q DP problem:
The TRS P consists of the following rules:

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(ok.0(g.0-1(a., x0))) → TOP.0(mark.0(f.1-1(b., x0)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.1-1(x0, x0))) → TOP.0(mark.0(h.0(a.)))
TOP.0(ok.0(g.0-0(a., x0))) → TOP.0(mark.0(f.1-0(b., x0)))
TOP.0(ok.0(h.0(x0))) → TOP.0(mark.0(g.0-0(x0, x0)))
TOP.0(ok.0(g.1-1(x0, x1))) → TOP.0(g.0-1(active.1(x0), x1))
TOP.0(ok.0(h.1(x0))) → TOP.0(mark.0(g.1-1(x0, x0)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(ok.0(f.0-0(x0, x0))) → TOP.0(mark.0(h.0(a.)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(ok.0(f.1-1(x0, x1))) → TOP.0(f.0-1(active.1(x0), x1))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(ok.0(h.1(x0))) → TOP.0(h.0(active.1(x0)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(ok.0(g.1-0(x0, x1))) → TOP.0(g.0-0(active.1(x0), x1))
TOP.0(ok.0(f.1-0(x0, x1))) → TOP.0(f.0-0(active.1(x0), x1))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))

The TRS R consists of the following rules:

proper.0(h.1(X)) → h.1(proper.1(X))
active.0(g.0-0(a., X)) → mark.0(f.1-0(b., X))
h.0(ok.0(X)) → ok.0(h.0(X))
active.0(f.0-0(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → mark.0(g.1-1(X, X))
proper.0(f.1-1(X1, X2)) → f.1-1(proper.1(X1), proper.1(X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
active.0(f.1-0(X1, X2)) → f.0-0(active.1(X1), X2)
proper.0(f.0-1(X1, X2)) → f.0-1(proper.0(X1), proper.1(X2))
proper.1(b.) → ok.1(b.)
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
active.0(h.0(X)) → h.0(active.0(X))
proper.0(g.1-1(X1, X2)) → g.1-1(proper.1(X1), proper.1(X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
active.0(f.0-0(X1, X2)) → f.0-0(active.0(X1), X2)
proper.0(f.1-0(X1, X2)) → f.1-0(proper.1(X1), proper.0(X2))
active.0(f.1-1(X1, X2)) → f.0-1(active.1(X1), X2)
active.0(g.0-1(a., X)) → mark.0(f.1-1(b., X))
proper.0(a.) → ok.0(a.)
active.0(g.1-0(X1, X2)) → g.0-0(active.1(X1), X2)
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
active.0(g.0-0(X1, X2)) → g.0-0(active.0(X1), X2)
proper.0(f.0-0(X1, X2)) → f.0-0(proper.0(X1), proper.0(X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
active.0(g.0-1(X1, X2)) → g.0-1(active.0(X1), X2)
proper.0(g.1-0(X1, X2)) → g.1-0(proper.1(X1), proper.0(X2))
active.0(g.1-1(X1, X2)) → g.0-1(active.1(X1), X2)
active.0(a.) → mark.1(b.)
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
active.0(f.0-1(X1, X2)) → f.0-1(active.0(X1), X2)
proper.0(g.0-0(X1, X2)) → g.0-0(proper.0(X1), proper.0(X2))
active.0(f.1-1(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → h.0(active.1(X))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
active.0(h.0(X)) → mark.0(g.0-0(X, X))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
proper.0(g.0-1(X1, X2)) → g.0-1(proper.0(X1), proper.1(X2))
proper.0(h.0(X)) → h.0(proper.0(X))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ SemLabProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                ↳ SemLabProof2

Q DP problem:
The TRS P consists of the following rules:

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(g.0-0(a., x0))) → TOP.0(mark.0(f.1-0(b., x0)))
TOP.0(ok.0(h.0(x0))) → TOP.0(mark.0(g.0-0(x0, x0)))
TOP.0(ok.0(h.1(x0))) → TOP.0(mark.0(g.1-1(x0, x0)))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(g.0-1(a., x0))) → TOP.0(mark.0(f.1-1(b., x0)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(ok.0(f.0-0(x0, x0))) → TOP.0(mark.0(h.0(a.)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(f.1-1(x0, x0))) → TOP.0(mark.0(h.0(a.)))

The TRS R consists of the following rules:

proper.0(h.1(X)) → h.1(proper.1(X))
active.0(g.0-0(a., X)) → mark.0(f.1-0(b., X))
h.0(ok.0(X)) → ok.0(h.0(X))
active.0(f.0-0(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → mark.0(g.1-1(X, X))
proper.0(f.1-1(X1, X2)) → f.1-1(proper.1(X1), proper.1(X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
active.0(f.1-0(X1, X2)) → f.0-0(active.1(X1), X2)
proper.0(f.0-1(X1, X2)) → f.0-1(proper.0(X1), proper.1(X2))
proper.1(b.) → ok.1(b.)
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
active.0(h.0(X)) → h.0(active.0(X))
proper.0(g.1-1(X1, X2)) → g.1-1(proper.1(X1), proper.1(X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
active.0(f.0-0(X1, X2)) → f.0-0(active.0(X1), X2)
proper.0(f.1-0(X1, X2)) → f.1-0(proper.1(X1), proper.0(X2))
active.0(f.1-1(X1, X2)) → f.0-1(active.1(X1), X2)
active.0(g.0-1(a., X)) → mark.0(f.1-1(b., X))
proper.0(a.) → ok.0(a.)
active.0(g.1-0(X1, X2)) → g.0-0(active.1(X1), X2)
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
active.0(g.0-0(X1, X2)) → g.0-0(active.0(X1), X2)
proper.0(f.0-0(X1, X2)) → f.0-0(proper.0(X1), proper.0(X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
active.0(g.0-1(X1, X2)) → g.0-1(active.0(X1), X2)
proper.0(g.1-0(X1, X2)) → g.1-0(proper.1(X1), proper.0(X2))
active.0(g.1-1(X1, X2)) → g.0-1(active.1(X1), X2)
active.0(a.) → mark.1(b.)
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
active.0(f.0-1(X1, X2)) → f.0-1(active.0(X1), X2)
proper.0(g.0-0(X1, X2)) → g.0-0(proper.0(X1), proper.0(X2))
active.0(f.1-1(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → h.0(active.1(X))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
active.0(h.0(X)) → mark.0(g.0-0(X, X))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
proper.0(g.0-1(X1, X2)) → g.0-1(proper.0(X1), proper.1(X2))
proper.0(h.0(X)) → h.0(proper.0(X))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP.0(ok.0(h.0(x0))) → TOP.0(mark.0(g.0-0(x0, x0)))
TOP.0(ok.0(h.1(x0))) → TOP.0(mark.0(g.1-1(x0, x0)))

The remaining pairs can at least be oriented weakly.

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(g.0-0(a., x0))) → TOP.0(mark.0(f.1-0(b., x0)))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(g.0-1(a., x0))) → TOP.0(mark.0(f.1-1(b., x0)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(ok.0(f.0-0(x0, x0))) → TOP.0(mark.0(h.0(a.)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(f.1-1(x0, x0))) → TOP.0(mark.0(h.0(a.)))

Used ordering: Polynomial interpretation [25]:

POL(TOP.0(x₁)) = x₁
POL(a.) = 0
POL(active.0(x₁)) = 0
POL(active.1(x₁)) = 0
POL(b.) = 0
POL(f.0-0(x₁, x₂)) = 1
POL(f.0-1(x₁, x₂)) = 1
POL(f.1-0(x₁, x₂)) = 0
POL(f.1-1(x₁, x₂)) = 1
POL(g.0-0(x₁, x₂)) = 0
POL(g.0-1(x₁, x₂)) = 1 + x₂
POL(g.1-0(x₁, x₂)) = 0
POL(g.1-1(x₁, x₂)) = 0
POL(h.0(x₁)) = 1
POL(h.1(x₁)) = 1
POL(mark.0(x₁)) = x₁
POL(mark.1(x₁)) = 0
POL(ok.0(x₁)) = x₁
POL(ok.1(x₁)) = x₁
POL(proper.0(x₁)) = 0
POL(proper.1(x₁)) = x₁

The following usable rules [17] were oriented:

f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
proper.1(b.) → ok.1(b.)
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
h.0(ok.0(X)) → ok.0(h.0(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ SemLabProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                ↳ SemLabProof2

Q DP problem:
The TRS P consists of the following rules:

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(g.0-0(a., x0))) → TOP.0(mark.0(f.1-0(b., x0)))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(g.0-1(a., x0))) → TOP.0(mark.0(f.1-1(b., x0)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(ok.0(f.0-0(x0, x0))) → TOP.0(mark.0(h.0(a.)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(f.1-1(x0, x0))) → TOP.0(mark.0(h.0(a.)))

The TRS R consists of the following rules:

proper.0(h.1(X)) → h.1(proper.1(X))
active.0(g.0-0(a., X)) → mark.0(f.1-0(b., X))
h.0(ok.0(X)) → ok.0(h.0(X))
active.0(f.0-0(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → mark.0(g.1-1(X, X))
proper.0(f.1-1(X1, X2)) → f.1-1(proper.1(X1), proper.1(X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
active.0(f.1-0(X1, X2)) → f.0-0(active.1(X1), X2)
proper.0(f.0-1(X1, X2)) → f.0-1(proper.0(X1), proper.1(X2))
proper.1(b.) → ok.1(b.)
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
active.0(h.0(X)) → h.0(active.0(X))
proper.0(g.1-1(X1, X2)) → g.1-1(proper.1(X1), proper.1(X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
active.0(f.0-0(X1, X2)) → f.0-0(active.0(X1), X2)
proper.0(f.1-0(X1, X2)) → f.1-0(proper.1(X1), proper.0(X2))
active.0(f.1-1(X1, X2)) → f.0-1(active.1(X1), X2)
active.0(g.0-1(a., X)) → mark.0(f.1-1(b., X))
proper.0(a.) → ok.0(a.)
active.0(g.1-0(X1, X2)) → g.0-0(active.1(X1), X2)
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
active.0(g.0-0(X1, X2)) → g.0-0(active.0(X1), X2)
proper.0(f.0-0(X1, X2)) → f.0-0(proper.0(X1), proper.0(X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
active.0(g.0-1(X1, X2)) → g.0-1(active.0(X1), X2)
proper.0(g.1-0(X1, X2)) → g.1-0(proper.1(X1), proper.0(X2))
active.0(g.1-1(X1, X2)) → g.0-1(active.1(X1), X2)
active.0(a.) → mark.1(b.)
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
active.0(f.0-1(X1, X2)) → f.0-1(active.0(X1), X2)
proper.0(g.0-0(X1, X2)) → g.0-0(proper.0(X1), proper.0(X2))
active.0(f.1-1(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → h.0(active.1(X))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
active.0(h.0(X)) → mark.0(g.0-0(X, X))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
proper.0(g.0-1(X1, X2)) → g.0-1(proper.0(X1), proper.1(X2))
proper.0(h.0(X)) → h.0(proper.0(X))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP.0(ok.0(f.0-0(x0, x0))) → TOP.0(mark.0(h.0(a.)))

The remaining pairs can at least be oriented weakly.

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(g.0-0(a., x0))) → TOP.0(mark.0(f.1-0(b., x0)))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(g.0-1(a., x0))) → TOP.0(mark.0(f.1-1(b., x0)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(f.1-1(x0, x0))) → TOP.0(mark.0(h.0(a.)))

Used ordering: Polynomial interpretation [25]:

POL(TOP.0(x₁)) = x₁
POL(a.) = 0
POL(active.0(x₁)) = 0
POL(active.1(x₁)) = 0
POL(b.) = 0
POL(f.0-0(x₁, x₂)) = 1
POL(f.0-1(x₁, x₂)) = 0
POL(f.1-0(x₁, x₂)) = 0
POL(f.1-1(x₁, x₂)) = 0
POL(g.0-0(x₁, x₂)) = 0
POL(g.0-1(x₁, x₂)) = 0
POL(g.1-0(x₁, x₂)) = 0
POL(g.1-1(x₁, x₂)) = 0
POL(h.0(x₁)) = 0
POL(h.1(x₁)) = 0
POL(mark.0(x₁)) = x₁
POL(mark.1(x₁)) = 0
POL(ok.0(x₁)) = x₁
POL(ok.1(x₁)) = 0
POL(proper.0(x₁)) = 0
POL(proper.1(x₁)) = 0

The following usable rules [17] were oriented:

f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
h.0(ok.0(X)) → ok.0(h.0(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ SemLabProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                ↳ SemLabProof2

Q DP problem:
The TRS P consists of the following rules:

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(g.0-0(a., x0))) → TOP.0(mark.0(f.1-0(b., x0)))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(g.0-1(a., x0))) → TOP.0(mark.0(f.1-1(b., x0)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(f.1-1(x0, x0))) → TOP.0(mark.0(h.0(a.)))

The TRS R consists of the following rules:

proper.0(h.1(X)) → h.1(proper.1(X))
active.0(g.0-0(a., X)) → mark.0(f.1-0(b., X))
h.0(ok.0(X)) → ok.0(h.0(X))
active.0(f.0-0(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → mark.0(g.1-1(X, X))
proper.0(f.1-1(X1, X2)) → f.1-1(proper.1(X1), proper.1(X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
active.0(f.1-0(X1, X2)) → f.0-0(active.1(X1), X2)
proper.0(f.0-1(X1, X2)) → f.0-1(proper.0(X1), proper.1(X2))
proper.1(b.) → ok.1(b.)
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
active.0(h.0(X)) → h.0(active.0(X))
proper.0(g.1-1(X1, X2)) → g.1-1(proper.1(X1), proper.1(X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
active.0(f.0-0(X1, X2)) → f.0-0(active.0(X1), X2)
proper.0(f.1-0(X1, X2)) → f.1-0(proper.1(X1), proper.0(X2))
active.0(f.1-1(X1, X2)) → f.0-1(active.1(X1), X2)
active.0(g.0-1(a., X)) → mark.0(f.1-1(b., X))
proper.0(a.) → ok.0(a.)
active.0(g.1-0(X1, X2)) → g.0-0(active.1(X1), X2)
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
active.0(g.0-0(X1, X2)) → g.0-0(active.0(X1), X2)
proper.0(f.0-0(X1, X2)) → f.0-0(proper.0(X1), proper.0(X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
active.0(g.0-1(X1, X2)) → g.0-1(active.0(X1), X2)
proper.0(g.1-0(X1, X2)) → g.1-0(proper.1(X1), proper.0(X2))
active.0(g.1-1(X1, X2)) → g.0-1(active.1(X1), X2)
active.0(a.) → mark.1(b.)
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
active.0(f.0-1(X1, X2)) → f.0-1(active.0(X1), X2)
proper.0(g.0-0(X1, X2)) → g.0-0(proper.0(X1), proper.0(X2))
active.0(f.1-1(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → h.0(active.1(X))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
active.0(h.0(X)) → mark.0(g.0-0(X, X))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
proper.0(g.0-1(X1, X2)) → g.0-1(proper.0(X1), proper.1(X2))
proper.0(h.0(X)) → h.0(proper.0(X))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP.0(ok.0(g.0-1(a., x0))) → TOP.0(mark.0(f.1-1(b., x0)))

The remaining pairs can at least be oriented weakly.

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(g.0-0(a., x0))) → TOP.0(mark.0(f.1-0(b., x0)))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(f.1-1(x0, x0))) → TOP.0(mark.0(h.0(a.)))

Used ordering: Polynomial interpretation [25]:

POL(TOP.0(x₁)) = x₁
POL(a.) = 0
POL(active.0(x₁)) = 0
POL(active.1(x₁)) = 0
POL(b.) = 1
POL(f.0-0(x₁, x₂)) = 0
POL(f.0-1(x₁, x₂)) = 0
POL(f.1-0(x₁, x₂)) = 0
POL(f.1-1(x₁, x₂)) = 0
POL(g.0-0(x₁, x₂)) = 0
POL(g.0-1(x₁, x₂)) = 1
POL(g.1-0(x₁, x₂)) = 0
POL(g.1-1(x₁, x₂)) = 0
POL(h.0(x₁)) = 0
POL(h.1(x₁)) = 0
POL(mark.0(x₁)) = x₁
POL(mark.1(x₁)) = 0
POL(ok.0(x₁)) = x₁
POL(ok.1(x₁)) = 1 + x₁
POL(proper.0(x₁)) = 0
POL(proper.1(x₁)) = 1 + x₁

The following usable rules [17] were oriented:

f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
h.0(ok.0(X)) → ok.0(h.0(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ SemLabProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                ↳ SemLabProof2

Q DP problem:
The TRS P consists of the following rules:

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(g.0-0(a., x0))) → TOP.0(mark.0(f.1-0(b., x0)))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(f.1-1(x0, x0))) → TOP.0(mark.0(h.0(a.)))

The TRS R consists of the following rules:

proper.0(h.1(X)) → h.1(proper.1(X))
active.0(g.0-0(a., X)) → mark.0(f.1-0(b., X))
h.0(ok.0(X)) → ok.0(h.0(X))
active.0(f.0-0(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → mark.0(g.1-1(X, X))
proper.0(f.1-1(X1, X2)) → f.1-1(proper.1(X1), proper.1(X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
active.0(f.1-0(X1, X2)) → f.0-0(active.1(X1), X2)
proper.0(f.0-1(X1, X2)) → f.0-1(proper.0(X1), proper.1(X2))
proper.1(b.) → ok.1(b.)
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
active.0(h.0(X)) → h.0(active.0(X))
proper.0(g.1-1(X1, X2)) → g.1-1(proper.1(X1), proper.1(X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
active.0(f.0-0(X1, X2)) → f.0-0(active.0(X1), X2)
proper.0(f.1-0(X1, X2)) → f.1-0(proper.1(X1), proper.0(X2))
active.0(f.1-1(X1, X2)) → f.0-1(active.1(X1), X2)
active.0(g.0-1(a., X)) → mark.0(f.1-1(b., X))
proper.0(a.) → ok.0(a.)
active.0(g.1-0(X1, X2)) → g.0-0(active.1(X1), X2)
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
active.0(g.0-0(X1, X2)) → g.0-0(active.0(X1), X2)
proper.0(f.0-0(X1, X2)) → f.0-0(proper.0(X1), proper.0(X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
active.0(g.0-1(X1, X2)) → g.0-1(active.0(X1), X2)
proper.0(g.1-0(X1, X2)) → g.1-0(proper.1(X1), proper.0(X2))
active.0(g.1-1(X1, X2)) → g.0-1(active.1(X1), X2)
active.0(a.) → mark.1(b.)
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
active.0(f.0-1(X1, X2)) → f.0-1(active.0(X1), X2)
proper.0(g.0-0(X1, X2)) → g.0-0(proper.0(X1), proper.0(X2))
active.0(f.1-1(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → h.0(active.1(X))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
active.0(h.0(X)) → mark.0(g.0-0(X, X))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
proper.0(g.0-1(X1, X2)) → g.0-1(proper.0(X1), proper.1(X2))
proper.0(h.0(X)) → h.0(proper.0(X))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP.0(ok.0(f.1-1(x0, x0))) → TOP.0(mark.0(h.0(a.)))

The remaining pairs can at least be oriented weakly.

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(g.0-0(a., x0))) → TOP.0(mark.0(f.1-0(b., x0)))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))

Used ordering: Polynomial interpretation [25]:

POL(TOP.0(x₁)) = x₁
POL(a.) = 0
POL(active.0(x₁)) = 0
POL(active.1(x₁)) = 0
POL(b.) = 0
POL(f.0-0(x₁, x₂)) = 0
POL(f.0-1(x₁, x₂)) = 1
POL(f.1-0(x₁, x₂)) = 0
POL(f.1-1(x₁, x₂)) = 1
POL(g.0-0(x₁, x₂)) = 0
POL(g.0-1(x₁, x₂)) = 0
POL(g.1-0(x₁, x₂)) = 0
POL(g.1-1(x₁, x₂)) = 0
POL(h.0(x₁)) = 0
POL(h.1(x₁)) = 0
POL(mark.0(x₁)) = x₁
POL(mark.1(x₁)) = 0
POL(ok.0(x₁)) = x₁
POL(ok.1(x₁)) = 0
POL(proper.0(x₁)) = 0
POL(proper.1(x₁)) = 0

The following usable rules [17] were oriented:

f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
h.0(ok.0(X)) → ok.0(h.0(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ SemLabProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                ↳ SemLabProof2

Q DP problem:
The TRS P consists of the following rules:

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(g.0-0(a., x0))) → TOP.0(mark.0(f.1-0(b., x0)))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))

The TRS R consists of the following rules:

proper.0(h.1(X)) → h.1(proper.1(X))
active.0(g.0-0(a., X)) → mark.0(f.1-0(b., X))
h.0(ok.0(X)) → ok.0(h.0(X))
active.0(f.0-0(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → mark.0(g.1-1(X, X))
proper.0(f.1-1(X1, X2)) → f.1-1(proper.1(X1), proper.1(X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
active.0(f.1-0(X1, X2)) → f.0-0(active.1(X1), X2)
proper.0(f.0-1(X1, X2)) → f.0-1(proper.0(X1), proper.1(X2))
proper.1(b.) → ok.1(b.)
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
active.0(h.0(X)) → h.0(active.0(X))
proper.0(g.1-1(X1, X2)) → g.1-1(proper.1(X1), proper.1(X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
active.0(f.0-0(X1, X2)) → f.0-0(active.0(X1), X2)
proper.0(f.1-0(X1, X2)) → f.1-0(proper.1(X1), proper.0(X2))
active.0(f.1-1(X1, X2)) → f.0-1(active.1(X1), X2)
active.0(g.0-1(a., X)) → mark.0(f.1-1(b., X))
proper.0(a.) → ok.0(a.)
active.0(g.1-0(X1, X2)) → g.0-0(active.1(X1), X2)
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
active.0(g.0-0(X1, X2)) → g.0-0(active.0(X1), X2)
proper.0(f.0-0(X1, X2)) → f.0-0(proper.0(X1), proper.0(X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
active.0(g.0-1(X1, X2)) → g.0-1(active.0(X1), X2)
proper.0(g.1-0(X1, X2)) → g.1-0(proper.1(X1), proper.0(X2))
active.0(g.1-1(X1, X2)) → g.0-1(active.1(X1), X2)
active.0(a.) → mark.1(b.)
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
active.0(f.0-1(X1, X2)) → f.0-1(active.0(X1), X2)
proper.0(g.0-0(X1, X2)) → g.0-0(proper.0(X1), proper.0(X2))
active.0(f.1-1(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → h.0(active.1(X))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
active.0(h.0(X)) → mark.0(g.0-0(X, X))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
proper.0(g.0-1(X1, X2)) → g.0-1(proper.0(X1), proper.1(X2))
proper.0(h.0(X)) → h.0(proper.0(X))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP.0(ok.0(g.0-0(a., x0))) → TOP.0(mark.0(f.1-0(b., x0)))

The remaining pairs can at least be oriented weakly.

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))

Used ordering: Polynomial interpretation [25]:

POL(TOP.0(x₁)) = x₁
POL(a.) = 0
POL(active.0(x₁)) = 0
POL(active.1(x₁)) = 0
POL(b.) = 0
POL(f.0-0(x₁, x₂)) = 0
POL(f.0-1(x₁, x₂)) = 0
POL(f.1-0(x₁, x₂)) = 0
POL(f.1-1(x₁, x₂)) = 0
POL(g.0-0(x₁, x₂)) = 1
POL(g.0-1(x₁, x₂)) = x₂
POL(g.1-0(x₁, x₂)) = 0
POL(g.1-1(x₁, x₂)) = 0
POL(h.0(x₁)) = 0
POL(h.1(x₁)) = 0
POL(mark.0(x₁)) = x₁
POL(mark.1(x₁)) = 0
POL(ok.0(x₁)) = x₁
POL(ok.1(x₁)) = x₁
POL(proper.0(x₁)) = 0
POL(proper.1(x₁)) = 0

The following usable rules [17] were oriented:

f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
proper.1(b.) → ok.1(b.)
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
h.0(ok.0(X)) → ok.0(h.0(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ SemLabProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                ↳ SemLabProof2

Q DP problem:
The TRS P consists of the following rules:

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))

The TRS R consists of the following rules:

proper.0(h.1(X)) → h.1(proper.1(X))
active.0(g.0-0(a., X)) → mark.0(f.1-0(b., X))
h.0(ok.0(X)) → ok.0(h.0(X))
active.0(f.0-0(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → mark.0(g.1-1(X, X))
proper.0(f.1-1(X1, X2)) → f.1-1(proper.1(X1), proper.1(X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
active.0(f.1-0(X1, X2)) → f.0-0(active.1(X1), X2)
proper.0(f.0-1(X1, X2)) → f.0-1(proper.0(X1), proper.1(X2))
proper.1(b.) → ok.1(b.)
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
active.0(h.0(X)) → h.0(active.0(X))
proper.0(g.1-1(X1, X2)) → g.1-1(proper.1(X1), proper.1(X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
active.0(f.0-0(X1, X2)) → f.0-0(active.0(X1), X2)
proper.0(f.1-0(X1, X2)) → f.1-0(proper.1(X1), proper.0(X2))
active.0(f.1-1(X1, X2)) → f.0-1(active.1(X1), X2)
active.0(g.0-1(a., X)) → mark.0(f.1-1(b., X))
proper.0(a.) → ok.0(a.)
active.0(g.1-0(X1, X2)) → g.0-0(active.1(X1), X2)
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
active.0(g.0-0(X1, X2)) → g.0-0(active.0(X1), X2)
proper.0(f.0-0(X1, X2)) → f.0-0(proper.0(X1), proper.0(X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
active.0(g.0-1(X1, X2)) → g.0-1(active.0(X1), X2)
proper.0(g.1-0(X1, X2)) → g.1-0(proper.1(X1), proper.0(X2))
active.0(g.1-1(X1, X2)) → g.0-1(active.1(X1), X2)
active.0(a.) → mark.1(b.)
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
active.0(f.0-1(X1, X2)) → f.0-1(active.0(X1), X2)
proper.0(g.0-0(X1, X2)) → g.0-0(proper.0(X1), proper.0(X2))
active.0(f.1-1(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → h.0(active.1(X))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
active.0(h.0(X)) → mark.0(g.0-0(X, X))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
proper.0(g.0-1(X1, X2)) → g.0-1(proper.0(X1), proper.1(X2))
proper.0(h.0(X)) → h.0(proper.0(X))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP.0(mark.0(g.1-0(x0, x1))) → TOP.0(g.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(g.1-1(x0, x1))) → TOP.0(g.1-1(proper.1(x0), proper.1(x1)))

The remaining pairs can at least be oriented weakly.

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))

Used ordering: Polynomial interpretation [25]:

POL(TOP.0(x₁)) = x₁
POL(a.) = 1
POL(active.0(x₁)) = x₁
POL(active.1(x₁)) = 0
POL(b.) = 0
POL(f.0-0(x₁, x₂)) = 1
POL(f.0-1(x₁, x₂)) = x₁
POL(f.1-0(x₁, x₂)) = 1
POL(f.1-1(x₁, x₂)) = 1
POL(g.0-0(x₁, x₂)) = x₁
POL(g.0-1(x₁, x₂)) = x₁
POL(g.1-0(x₁, x₂)) = 0
POL(g.1-1(x₁, x₂)) = 0
POL(h.0(x₁)) = 1
POL(h.1(x₁)) = 1
POL(mark.0(x₁)) = 1
POL(mark.1(x₁)) = 1
POL(ok.0(x₁)) = x₁
POL(ok.1(x₁)) = 0
POL(proper.0(x₁)) = 1
POL(proper.1(x₁)) = 0

The following usable rules [17] were oriented:

f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
active.0(h.0(X)) → mark.0(g.0-0(X, X))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
active.0(h.1(X)) → h.0(active.1(X))
active.0(f.1-1(X, X)) → mark.0(h.0(a.))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))
proper.0(g.0-1(X1, X2)) → g.0-1(proper.0(X1), proper.1(X2))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
proper.0(h.0(X)) → h.0(proper.0(X))
proper.0(f.0-0(X1, X2)) → f.0-0(proper.0(X1), proper.0(X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
active.0(g.0-0(X1, X2)) → g.0-0(active.0(X1), X2)
proper.0(a.) → ok.0(a.)
active.0(g.1-0(X1, X2)) → g.0-0(active.1(X1), X2)
active.0(f.1-1(X1, X2)) → f.0-1(active.1(X1), X2)
active.0(g.0-1(a., X)) → mark.0(f.1-1(b., X))
active.0(f.0-1(X1, X2)) → f.0-1(active.0(X1), X2)
proper.0(g.0-0(X1, X2)) → g.0-0(proper.0(X1), proper.0(X2))
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
active.0(g.1-1(X1, X2)) → g.0-1(active.1(X1), X2)
active.0(a.) → mark.1(b.)
active.0(g.0-1(X1, X2)) → g.0-1(active.0(X1), X2)
proper.0(g.1-0(X1, X2)) → g.1-0(proper.1(X1), proper.0(X2))
proper.0(f.0-1(X1, X2)) → f.0-1(proper.0(X1), proper.1(X2))
active.0(f.1-0(X1, X2)) → f.0-0(active.1(X1), X2)
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
proper.0(f.1-1(X1, X2)) → f.1-1(proper.1(X1), proper.1(X2))
active.0(h.1(X)) → mark.0(g.1-1(X, X))
proper.0(f.1-0(X1, X2)) → f.1-0(proper.1(X1), proper.0(X2))
active.0(f.0-0(X1, X2)) → f.0-0(active.0(X1), X2)
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
proper.0(g.1-1(X1, X2)) → g.1-1(proper.1(X1), proper.1(X2))
active.0(h.0(X)) → h.0(active.0(X))
h.0(mark.1(X)) → mark.0(h.1(X))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
proper.0(h.1(X)) → h.1(proper.1(X))
active.0(g.0-0(a., X)) → mark.0(f.1-0(b., X))
h.0(ok.0(X)) → ok.0(h.0(X))
active.0(f.0-0(X, X)) → mark.0(h.0(a.))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ SemLabProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                ↳ SemLabProof2

Q DP problem:
The TRS P consists of the following rules:

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))

The TRS R consists of the following rules:

proper.0(h.1(X)) → h.1(proper.1(X))
active.0(g.0-0(a., X)) → mark.0(f.1-0(b., X))
h.0(ok.0(X)) → ok.0(h.0(X))
active.0(f.0-0(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → mark.0(g.1-1(X, X))
proper.0(f.1-1(X1, X2)) → f.1-1(proper.1(X1), proper.1(X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
active.0(f.1-0(X1, X2)) → f.0-0(active.1(X1), X2)
proper.0(f.0-1(X1, X2)) → f.0-1(proper.0(X1), proper.1(X2))
proper.1(b.) → ok.1(b.)
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
active.0(h.0(X)) → h.0(active.0(X))
proper.0(g.1-1(X1, X2)) → g.1-1(proper.1(X1), proper.1(X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
active.0(f.0-0(X1, X2)) → f.0-0(active.0(X1), X2)
proper.0(f.1-0(X1, X2)) → f.1-0(proper.1(X1), proper.0(X2))
active.0(f.1-1(X1, X2)) → f.0-1(active.1(X1), X2)
active.0(g.0-1(a., X)) → mark.0(f.1-1(b., X))
proper.0(a.) → ok.0(a.)
active.0(g.1-0(X1, X2)) → g.0-0(active.1(X1), X2)
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
active.0(g.0-0(X1, X2)) → g.0-0(active.0(X1), X2)
proper.0(f.0-0(X1, X2)) → f.0-0(proper.0(X1), proper.0(X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
active.0(g.0-1(X1, X2)) → g.0-1(active.0(X1), X2)
proper.0(g.1-0(X1, X2)) → g.1-0(proper.1(X1), proper.0(X2))
active.0(g.1-1(X1, X2)) → g.0-1(active.1(X1), X2)
active.0(a.) → mark.1(b.)
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
active.0(f.0-1(X1, X2)) → f.0-1(active.0(X1), X2)
proper.0(g.0-0(X1, X2)) → g.0-0(proper.0(X1), proper.0(X2))
active.0(f.1-1(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → h.0(active.1(X))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
active.0(h.0(X)) → mark.0(g.0-0(X, X))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
proper.0(g.0-1(X1, X2)) → g.0-1(proper.0(X1), proper.1(X2))
proper.0(h.0(X)) → h.0(proper.0(X))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP.0(mark.0(f.1-0(x0, x1))) → TOP.0(f.1-0(proper.1(x0), proper.0(x1)))
TOP.0(mark.0(h.1(x0))) → TOP.0(h.1(proper.1(x0)))
TOP.0(mark.0(f.1-1(x0, x1))) → TOP.0(f.1-1(proper.1(x0), proper.1(x1)))

The remaining pairs can at least be oriented weakly.

TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))

Used ordering: Polynomial interpretation [25]:

POL(TOP.0(x₁)) = x₁
POL(a.) = 0
POL(active.0(x₁)) = 0
POL(active.1(x₁)) = 0
POL(b.) = 1
POL(f.0-0(x₁, x₂)) = 1
POL(f.0-1(x₁, x₂)) = 1
POL(f.1-0(x₁, x₂)) = 0
POL(f.1-1(x₁, x₂)) = 0
POL(g.0-0(x₁, x₂)) = 1
POL(g.0-1(x₁, x₂)) = 1
POL(g.1-0(x₁, x₂)) = 0
POL(g.1-1(x₁, x₂)) = 0
POL(h.0(x₁)) = 1
POL(h.1(x₁)) = 0
POL(mark.0(x₁)) = 1
POL(mark.1(x₁)) = 0
POL(ok.0(x₁)) = x₁
POL(ok.1(x₁)) = 0
POL(proper.0(x₁)) = 0
POL(proper.1(x₁)) = x₁

The following usable rules [17] were oriented:

f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
h.0(ok.0(X)) → ok.0(h.0(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ SemLabProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                ↳ SemLabProof2

Q DP problem:
The TRS P consists of the following rules:

TOP.0(mark.0(g.0-1(x0, x1))) → TOP.0(g.0-1(proper.0(x0), proper.1(x1)))
TOP.0(ok.0(g.0-1(x0, x1))) → TOP.0(g.0-1(active.0(x0), x1))
TOP.0(mark.0(f.0-1(x0, x1))) → TOP.0(f.0-1(proper.0(x0), proper.1(x1)))
TOP.0(mark.0(g.0-0(x0, x1))) → TOP.0(g.0-0(proper.0(x0), proper.0(x1)))
TOP.0(ok.0(f.0-1(x0, x1))) → TOP.0(f.0-1(active.0(x0), x1))
TOP.0(ok.0(g.0-0(x0, x1))) → TOP.0(g.0-0(active.0(x0), x1))
TOP.0(mark.0(h.0(x0))) → TOP.0(h.0(proper.0(x0)))
TOP.0(ok.0(h.0(x0))) → TOP.0(h.0(active.0(x0)))
TOP.0(ok.0(f.0-0(x0, x1))) → TOP.0(f.0-0(active.0(x0), x1))
TOP.0(mark.0(f.0-0(x0, x1))) → TOP.0(f.0-0(proper.0(x0), proper.0(x1)))

The TRS R consists of the following rules:

proper.0(h.1(X)) → h.1(proper.1(X))
active.0(g.0-0(a., X)) → mark.0(f.1-0(b., X))
h.0(ok.0(X)) → ok.0(h.0(X))
active.0(f.0-0(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → mark.0(g.1-1(X, X))
proper.0(f.1-1(X1, X2)) → f.1-1(proper.1(X1), proper.1(X2))
g.0-1(mark.1(X1), X2) → mark.0(g.1-1(X1, X2))
g.0-0(ok.0(X1), ok.0(X2)) → ok.0(g.0-0(X1, X2))
g.1-0(ok.1(X1), ok.0(X2)) → ok.0(g.1-0(X1, X2))
active.0(f.1-0(X1, X2)) → f.0-0(active.1(X1), X2)
proper.0(f.0-1(X1, X2)) → f.0-1(proper.0(X1), proper.1(X2))
proper.1(b.) → ok.1(b.)
f.0-0(mark.1(X1), X2) → mark.0(f.1-0(X1, X2))
f.0-1(ok.0(X1), ok.1(X2)) → ok.0(f.0-1(X1, X2))
h.0(mark.1(X)) → mark.0(h.1(X))
active.0(h.0(X)) → h.0(active.0(X))
proper.0(g.1-1(X1, X2)) → g.1-1(proper.1(X1), proper.1(X2))
f.0-0(ok.0(X1), ok.0(X2)) → ok.0(f.0-0(X1, X2))
active.0(f.0-0(X1, X2)) → f.0-0(active.0(X1), X2)
proper.0(f.1-0(X1, X2)) → f.1-0(proper.1(X1), proper.0(X2))
active.0(f.1-1(X1, X2)) → f.0-1(active.1(X1), X2)
active.0(g.0-1(a., X)) → mark.0(f.1-1(b., X))
proper.0(a.) → ok.0(a.)
active.0(g.1-0(X1, X2)) → g.0-0(active.1(X1), X2)
g.1-1(ok.1(X1), ok.1(X2)) → ok.0(g.1-1(X1, X2))
active.0(g.0-0(X1, X2)) → g.0-0(active.0(X1), X2)
proper.0(f.0-0(X1, X2)) → f.0-0(proper.0(X1), proper.0(X2))
g.0-0(mark.1(X1), X2) → mark.0(g.1-0(X1, X2))
active.0(g.0-1(X1, X2)) → g.0-1(active.0(X1), X2)
proper.0(g.1-0(X1, X2)) → g.1-0(proper.1(X1), proper.0(X2))
active.0(g.1-1(X1, X2)) → g.0-1(active.1(X1), X2)
active.0(a.) → mark.1(b.)
h.0(mark.0(X)) → mark.0(h.0(X))
h.1(ok.1(X)) → ok.0(h.1(X))
active.0(f.0-1(X1, X2)) → f.0-1(active.0(X1), X2)
proper.0(g.0-0(X1, X2)) → g.0-0(proper.0(X1), proper.0(X2))
active.0(f.1-1(X, X)) → mark.0(h.0(a.))
active.0(h.1(X)) → h.0(active.1(X))
f.1-0(ok.1(X1), ok.0(X2)) → ok.0(f.1-0(X1, X2))
f.1-1(ok.1(X1), ok.1(X2)) → ok.0(f.1-1(X1, X2))
f.0-0(mark.0(X1), X2) → mark.0(f.0-0(X1, X2))
f.0-1(mark.1(X1), X2) → mark.0(f.1-1(X1, X2))
f.0-1(mark.0(X1), X2) → mark.0(f.0-1(X1, X2))
active.0(h.0(X)) → mark.0(g.0-0(X, X))
g.0-1(mark.0(X1), X2) → mark.0(g.0-1(X1, X2))
proper.0(g.0-1(X1, X2)) → g.0-1(proper.0(X1), proper.1(X2))
proper.0(h.0(X)) → h.0(proper.0(X))
g.0-0(mark.0(X1), X2) → mark.0(g.0-0(X1, X2))
g.0-1(ok.0(X1), ok.1(X2)) → ok.0(g.0-1(X1, X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
As can be seen after transforming the QDP problem by semantic labelling [33] and then some rule deleting processors, only certain labelled rules and pairs can be used. Hence, we only have to consider all unlabelled pairs and rules (without the decreasing rules for quasi-models).

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ SemLabProof

                                ↳ SemLabProof2

                                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(h(x0))) → TOP(h(active(x0)))
TOP(mark(h(x0))) → TOP(h(proper(x0)))
TOP(mark(g(x0, x1))) → TOP(g(proper(x0), proper(x1)))
TOP(ok(f(x0, x1))) → TOP(f(active(x0), x1))
TOP(mark(f(x0, x1))) → TOP(f(proper(x0), proper(x1)))
TOP(ok(g(x0, x1))) → TOP(g(active(x0), x1))

The TRS R consists of the following rules:

proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
f(mark(X1), X2) → mark(f(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(mark(X1), X2) → mark(g(X1, X2))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
h(mark(X)) → mark(h(X))
h(ok(X)) → ok(h(X))
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.