(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__H(X) → A__G(mark(X), X)
A__H(X) → MARK(X)
A__G(a, X) → A__F(b, X)
A__F(X, X) → A__H(a__a)
A__F(X, X) → A__A
MARK(h(X)) → A__H(mark(X))
MARK(h(X)) → MARK(X)
MARK(g(X1, X2)) → A__G(mark(X1), X2)
MARK(g(X1, X2)) → MARK(X1)
MARK(a) → A__A
MARK(f(X1, X2)) → A__F(mark(X1), X2)
MARK(f(X1, X2)) → MARK(X1)
The TRS R consists of the following rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__G(a, X) → A__F(b, X)
A__F(X, X) → A__H(a__a)
A__H(X) → A__G(mark(X), X)
A__H(X) → MARK(X)
MARK(h(X)) → A__H(mark(X))
MARK(h(X)) → MARK(X)
MARK(g(X1, X2)) → A__G(mark(X1), X2)
MARK(g(X1, X2)) → MARK(X1)
MARK(f(X1, X2)) → A__F(mark(X1), X2)
MARK(f(X1, X2)) → MARK(X1)
The TRS R consists of the following rules:
a__h(X) → a__g(mark(X), X)
a__g(a, X) → a__f(b, X)
a__f(X, X) → a__h(a__a)
a__a → b
mark(h(X)) → a__h(mark(X))
mark(g(X1, X2)) → a__g(mark(X1), X2)
mark(a) → a__a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → b
a__h(X) → h(X)
a__g(X1, X2) → g(X1, X2)
a__a → a
a__f(X1, X2) → f(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.