(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), y) → f(x, s(c(y)))
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:
F(x, c(y)) → F(x, s(f(y, y)))
F(x, c(y)) → F(y, y)
F(s(x), y) → F(x, s(c(y)))
The TRS R consists of the following rules:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), y) → f(x, s(c(y)))
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 2 SCCs with 1 less node.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), y) → F(x, s(c(y)))
The TRS R consists of the following rules:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), y) → f(x, s(c(y)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(s(x), y) → F(x, s(c(y)))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
F(
x1)
s(
x1) =
s(
x1)
c(
x1) =
c(
x1)
f(
x1,
x2) =
f
Recursive path order with status [RPO].
Quasi-Precedence:
[s1, c1] > F1 > f
Status:
F1: multiset
s1: multiset
c1: multiset
f: multiset
The following usable rules [FROCOS05] were oriented:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), y) → f(x, s(c(y)))
(7) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), y) → f(x, s(c(y)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(9) TRUE
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, c(y)) → F(y, y)
The TRS R consists of the following rules:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), y) → f(x, s(c(y)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(x, c(y)) → F(y, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
F(
x1,
x2)
c(
x1) =
c(
x1)
f(
x1,
x2) =
x2
s(
x1) =
s
Recursive path order with status [RPO].
Quasi-Precedence:
[F2, c1] > s
Status:
F2: multiset
c1: multiset
s: multiset
The following usable rules [FROCOS05] were oriented:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), y) → f(x, s(c(y)))
(12) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), y) → f(x, s(c(y)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(14) TRUE