(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
cond(true, x) → cond(odd(x), p(x))
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
p(0) → 0
p(s(x)) → x
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
p(0) → 0
p(s(x)) → x
The TRS R 2 is
cond(true, x) → cond(odd(x), p(x))
The signature Sigma is {
cond}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
cond(true, x) → cond(odd(x), p(x))
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
p(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
cond(true, x0)
odd(0)
odd(s(0))
odd(s(s(x0)))
p(0)
p(s(x0))
(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:
COND(true, x) → COND(odd(x), p(x))
COND(true, x) → ODD(x)
COND(true, x) → P(x)
ODD(s(s(x))) → ODD(x)
The TRS R consists of the following rules:
cond(true, x) → cond(odd(x), p(x))
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
p(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
cond(true, x0)
odd(0)
odd(s(0))
odd(s(s(x0)))
p(0)
p(s(x0))
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 2 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ODD(s(s(x))) → ODD(x)
The TRS R consists of the following rules:
cond(true, x) → cond(odd(x), p(x))
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
p(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
cond(true, x0)
odd(0)
odd(s(0))
odd(s(s(x0)))
p(0)
p(s(x0))
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.
ODD(s(s(x))) → ODD(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ODD(
x1) =
x1
s(
x1) =
s(
x1)
Lexicographic Path Order [LPO].
Precedence:
trivial
The following usable rules [FROCOS05] were oriented:
none
(9) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
cond(true, x) → cond(odd(x), p(x))
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
p(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
cond(true, x0)
odd(0)
odd(s(0))
odd(s(s(x0)))
p(0)
p(s(x0))
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:
COND(true, x) → COND(odd(x), p(x))
The TRS R consists of the following rules:
cond(true, x) → cond(odd(x), p(x))
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
p(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
cond(true, x0)
odd(0)
odd(s(0))
odd(s(s(x0)))
p(0)
p(s(x0))
We have to consider all minimal (P,Q,R)-chains.