(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
p(a(x0), p(b(x1), p(a(x2), x3))) → p(x2, p(a(a(x0)), p(b(x1), x3)))
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:
P(a(x0), p(b(x1), p(a(x2), x3))) → P(x2, p(a(a(x0)), p(b(x1), x3)))
P(a(x0), p(b(x1), p(a(x2), x3))) → P(a(a(x0)), p(b(x1), x3))
P(a(x0), p(b(x1), p(a(x2), x3))) → P(b(x1), x3)
The TRS R consists of the following rules:
p(a(x0), p(b(x1), p(a(x2), x3))) → p(x2, p(a(a(x0)), p(b(x1), x3)))
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:
P(a(x0), p(b(x1), p(a(x2), x3))) → P(a(a(x0)), p(b(x1), x3))
P(a(x0), p(b(x1), p(a(x2), x3))) → P(x2, p(a(a(x0)), p(b(x1), x3)))
The TRS R consists of the following rules:
p(a(x0), p(b(x1), p(a(x2), x3))) → p(x2, p(a(a(x0)), p(b(x1), x3)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
P(a(x0), p(b(x1), p(a(x2), x3))) → P(a(a(x0)), p(b(x1), x3))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
P(
x1,
x2) =
P(
x2)
a(
x1) =
x1
p(
x1,
x2) =
p(
x2)
b(
x1) =
b(
x1)
Lexicographic Path Order [LPO].
Precedence:
P1 > p1 > b1
The following usable rules [FROCOS05] were oriented:
p(a(x0), p(b(x1), p(a(x2), x3))) → p(x2, p(a(a(x0)), p(b(x1), x3)))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
P(a(x0), p(b(x1), p(a(x2), x3))) → P(x2, p(a(a(x0)), p(b(x1), x3)))
The TRS R consists of the following rules:
p(a(x0), p(b(x1), p(a(x2), x3))) → p(x2, p(a(a(x0)), p(b(x1), x3)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.