(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), s(y)) → f(x, s(c(s(y))))
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) 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), s(y)) → f(x, s(c(s(y))))
The set Q consists of the following terms:
f(x0, c(x1))
f(s(x0), s(x1))
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) 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), s(y)) → F(x, s(c(s(y))))
The TRS R consists of the following rules:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), s(y)) → f(x, s(c(s(y))))
The set Q consists of the following terms:
f(x0, c(x1))
f(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), s(y)) → F(x, s(c(s(y))))
The TRS R consists of the following rules:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), s(y)) → f(x, s(c(s(y))))
The set Q consists of the following terms:
f(x0, c(x1))
f(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(8) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(s(x), s(y)) → F(x, s(c(s(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)
s(
x1) =
s(
x1)
c(
x1) =
c
f(
x1,
x2) =
f
Lexicographic path order with status [LPO].
Quasi-Precedence:
F2 > [s1, c, f]
Status:
c: []
f: []
s1: [1]
F2: [2,1]
The following usable rules [FROCOS05] were oriented:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), s(y)) → f(x, s(c(s(y))))
(9) 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), s(y)) → f(x, s(c(s(y))))
The set Q consists of the following terms:
f(x0, c(x1))
f(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(10) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(11) TRUE
(12) 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), s(y)) → f(x, s(c(s(y))))
The set Q consists of the following terms:
f(x0, c(x1))
f(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(13) 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) =
x1
s(
x1) =
s(
x1)
Lexicographic path order with status [LPO].
Quasi-Precedence:
s1 > [F2, c1]
Status:
c1: [1]
s1: [1]
F2: [2,1]
The following usable rules [FROCOS05] were oriented:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), s(y)) → f(x, s(c(s(y))))
(14) 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), s(y)) → f(x, s(c(s(y))))
The set Q consists of the following terms:
f(x0, c(x1))
f(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(15) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(16) TRUE