(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
p(m, n, s(r)) → p(m, r, n)
p(m, s(n), 0) → p(0, n, m)
p(m, 0, 0) → m
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:
p(m, n, s(r)) → p(m, r, n)
p(m, s(n), 0) → p(0, n, m)
p(m, 0, 0) → m
The set Q consists of the following terms:
p(x0, x1, s(x2))
p(x0, s(x1), 0)
p(x0, 0, 0)
(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:
P(m, n, s(r)) → P(m, r, n)
P(m, s(n), 0) → P(0, n, m)
The TRS R consists of the following rules:
p(m, n, s(r)) → p(m, r, n)
p(m, s(n), 0) → p(0, n, m)
p(m, 0, 0) → m
The set Q consists of the following terms:
p(x0, x1, s(x2))
p(x0, s(x1), 0)
p(x0, 0, 0)
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.
P(m, n, s(r)) → P(m, r, n)
P(m, s(n), 0) → P(0, n, m)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
0 > [P3, s1]
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
p(m, n, s(r)) → p(m, r, n)
p(m, s(n), 0) → p(0, n, m)
p(m, 0, 0) → m
The set Q consists of the following terms:
p(x0, x1, s(x2))
p(x0, s(x1), 0)
p(x0, 0, 0)
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE