(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(d(x)) → d(b(x))
a(x) → b(b(b(x)))
b(d(b(x))) → a(c(x))
c(x) → d(x)
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(d(x)) → B(x)
A(x) → B(b(b(x)))
A(x) → B(b(x))
A(x) → B(x)
B(d(b(x))) → A(c(x))
B(d(b(x))) → C(x)
The TRS R consists of the following rules:
a(d(x)) → d(b(x))
a(x) → b(b(b(x)))
b(d(b(x))) → a(c(x))
c(x) → d(x)
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 1 less node.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(d(b(x))) → A(c(x))
A(d(x)) → B(x)
A(x) → B(b(b(x)))
A(x) → B(b(x))
A(x) → B(x)
The TRS R consists of the following rules:
a(d(x)) → d(b(x))
a(x) → b(b(b(x)))
b(d(b(x))) → a(c(x))
c(x) → d(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
A(d(x)) → B(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
c(x) → d(x)
a(x) → b(b(b(x)))
b(d(b(x))) → a(c(x))
a(d(x)) → d(b(x))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(d(b(x))) → A(c(x))
A(x) → B(b(b(x)))
A(x) → B(b(x))
A(x) → B(x)
The TRS R consists of the following rules:
a(d(x)) → d(b(x))
a(x) → b(b(b(x)))
b(d(b(x))) → a(c(x))
c(x) → d(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
A(x) → B(b(b(x)))
A(x) → B(b(x))
A(x) → B(x)
Strictly oriented rules of the TRS R:
b(d(b(x))) → a(c(x))
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = 3 + x1
POL(B(x1)) = x1
POL(a(x1)) = 3 + x1
POL(b(x1)) = 1 + x1
POL(c(x1)) = 3·x1
POL(d(x1)) = 3·x1
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(d(b(x))) → A(c(x))
The TRS R consists of the following rules:
a(d(x)) → d(b(x))
a(x) → b(b(b(x)))
c(x) → d(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(10) TRUE