(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))
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:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))
The set Q consists of the following terms:
minus(x0, x1)
cond(x0, x1, x0)
min(0, x0)
min(x0, 0)
min(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:
MINUS(x, y) → COND(min(x, y), x, y)
MINUS(x, y) → MIN(x, y)
COND(y, x, y) → MINUS(x, s(y))
MIN(s(u), s(v)) → MIN(u, v)
The TRS R consists of the following rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))
The set Q consists of the following terms:
minus(x0, x1)
cond(x0, x1, x0)
min(0, x0)
min(x0, 0)
min(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:
MIN(s(u), s(v)) → MIN(u, v)
The TRS R consists of the following rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))
The set Q consists of the following terms:
minus(x0, x1)
cond(x0, x1, x0)
min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
COND(y, x, y) → MINUS(x, s(y))
MINUS(x, y) → COND(min(x, y), x, y)
The TRS R consists of the following rules:
minus(x, y) → cond(min(x, y), x, y)
cond(y, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))
The set Q consists of the following terms:
minus(x0, x1)
cond(x0, x1, x0)
min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.