(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
quot(0, s(y), s(z)) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(quot(x, s(z), s(z)))
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:
quot(0, s(y), s(z)) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(quot(x, s(z), s(z)))
The set Q consists of the following terms:
quot(0, s(x0), s(x1))
quot(s(x0), s(x1), x2)
quot(x0, 0, 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:
QUOT(s(x), s(y), z) → QUOT(x, y, z)
QUOT(x, 0, s(z)) → QUOT(x, s(z), s(z))
The TRS R consists of the following rules:
quot(0, s(y), s(z)) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(quot(x, s(z), s(z)))
The set Q consists of the following terms:
quot(0, s(x0), s(x1))
quot(s(x0), s(x1), x2)
quot(x0, 0, 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.
QUOT(s(x), s(y), z) → QUOT(x, y, z)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
QUOT(
x1,
x2,
x3) =
QUOT(
x1,
x3)
s(
x1) =
s(
x1)
0 =
0
Recursive path order with status [RPO].
Quasi-Precedence:
[QUOT2, s1, 0]
Status:
QUOT2: multiset
s1: multiset
0: multiset
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
QUOT(x, 0, s(z)) → QUOT(x, s(z), s(z))
The TRS R consists of the following rules:
quot(0, s(y), s(z)) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(quot(x, s(z), s(z)))
The set Q consists of the following terms:
quot(0, s(x0), s(x1))
quot(s(x0), s(x1), x2)
quot(x0, 0, s(x1))
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(8) TRUE