(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
bin(x, 0) → s(0)
bin(0, s(y)) → 0
bin(s(x), s(y)) → +(bin(x, s(y)), bin(x, 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:
bin(x, 0) → s(0)
bin(0, s(y)) → 0
bin(s(x), s(y)) → +(bin(x, s(y)), bin(x, y))
The set Q consists of the following terms:
bin(x0, 0)
bin(0, s(x0))
bin(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:
BIN(s(x), s(y)) → BIN(x, s(y))
BIN(s(x), s(y)) → BIN(x, y)
The TRS R consists of the following rules:
bin(x, 0) → s(0)
bin(0, s(y)) → 0
bin(s(x), s(y)) → +(bin(x, s(y)), bin(x, y))
The set Q consists of the following terms:
bin(x0, 0)
bin(0, s(x0))
bin(s(x0), s(x1))
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.
BIN(s(x), s(y)) → BIN(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
BIN(
x1,
x2) =
x2
s(
x1) =
s(
x1)
bin(
x1,
x2) =
bin(
x2)
0 =
0
+(
x1,
x2) =
+(
x2)
Recursive path order with status [RPO].
Quasi-Precedence:
[s1, bin1, 0, +1]
Status:
s1: [1]
bin1: [1]
0: multiset
+1: [1]
The following usable rules [FROCOS05] were oriented:
bin(x, 0) → s(0)
bin(0, s(y)) → 0
bin(s(x), s(y)) → +(bin(x, s(y)), bin(x, y))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
BIN(s(x), s(y)) → BIN(x, s(y))
The TRS R consists of the following rules:
bin(x, 0) → s(0)
bin(0, s(y)) → 0
bin(s(x), s(y)) → +(bin(x, s(y)), bin(x, y))
The set Q consists of the following terms:
bin(x0, 0)
bin(0, s(x0))
bin(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(7) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
BIN(s(x), s(y)) → BIN(x, s(y))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
BIN(
x1,
x2) =
x1
s(
x1) =
s(
x1)
bin(
x1,
x2) =
bin
0 =
0
+(
x1,
x2) =
x2
Recursive path order with status [RPO].
Quasi-Precedence:
bin > s1 > 0
Status:
s1: multiset
bin: []
0: multiset
The following usable rules [FROCOS05] were oriented:
bin(x, 0) → s(0)
bin(0, s(y)) → 0
bin(s(x), s(y)) → +(bin(x, s(y)), bin(x, y))
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
bin(x, 0) → s(0)
bin(0, s(y)) → 0
bin(s(x), s(y)) → +(bin(x, s(y)), bin(x, y))
The set Q consists of the following terms:
bin(x0, 0)
bin(0, s(x0))
bin(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) TRUE