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