Termination Proof using AProVE

Term Rewriting System R:
[X, Y, X1, X2]
active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

ACTIVE(f(g(X), Y)) -> F(X, f(g(X), Y))
ACTIVE(f(X1, X2)) -> F(active(X1), X2)
ACTIVE(f(X1, X2)) -> ACTIVE(X1)
ACTIVE(g(X)) -> G(active(X))
ACTIVE(g(X)) -> ACTIVE(X)
F(mark(X1), X2) -> F(X1, X2)
F(ok(X1), ok(X2)) -> F(X1, X2)
G(mark(X)) -> G(X)
G(ok(X)) -> G(X)
PROPER(f(X1, X2)) -> F(proper(X1), proper(X2))
PROPER(f(X1, X2)) -> PROPER(X1)
PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(g(X)) -> G(proper(X))
PROPER(g(X)) -> PROPER(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 five SCCs.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳Remaining

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

Dependency Pairs:

F(ok(X1), ok(X2)) -> F(X1, X2)
F(mark(X1), X2) -> F(X1, X2)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pair can be strictly oriented:

F(ok(X1), ok(X2)) -> F(X1, X2)

The following rules can be oriented:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

F(x₁, x₂) -> F(x₁, x₂)
ok(x₁) -> ok(x₁)
mark(x₁) -> x₁
active(x₁) -> active(x₁)
f(x₁, x₂) -> x₁
g(x₁) -> x₁
proper(x₁) -> proper(x₁)
top(x₁) -> top

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
F(mark(X1), X2) -> F(X1, X2)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
G(ok(X)) -> G(X)
G(mark(X)) -> G(X)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(f(X1, X2)) -> ACTIVE(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(g(X)) -> PROPER(X)
PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(f(X1, X2)) -> PROPER(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
F(mark(X1), X2) -> F(X1, X2)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
G(ok(X)) -> G(X)
G(mark(X)) -> G(X)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(f(X1, X2)) -> ACTIVE(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(g(X)) -> PROPER(X)
PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(f(X1, X2)) -> PROPER(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
F(mark(X1), X2) -> F(X1, X2)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
G(ok(X)) -> G(X)
G(mark(X)) -> G(X)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(f(X1, X2)) -> ACTIVE(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(g(X)) -> PROPER(X)
PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(f(X1, X2)) -> PROPER(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
F(mark(X1), X2) -> F(X1, X2)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
G(ok(X)) -> G(X)
G(mark(X)) -> G(X)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(f(X1, X2)) -> ACTIVE(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(g(X)) -> PROPER(X)
PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(f(X1, X2)) -> PROPER(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
F(mark(X1), X2) -> F(X1, X2)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
G(ok(X)) -> G(X)
G(mark(X)) -> G(X)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(f(X1, X2)) -> ACTIVE(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(g(X)) -> PROPER(X)
PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(f(X1, X2)) -> PROPER(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

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