(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(nil) → nil
f(.(nil, y)) → .(nil, f(y))
f(.(.(x, y), z)) → f(.(x, .(y, z)))
g(nil) → nil
g(.(x, nil)) → .(g(x), nil)
g(.(x, .(y, z))) → g(.(.(x, y), 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:
f(nil) → nil
f(.(nil, y)) → .(nil, f(y))
f(.(.(x, y), z)) → f(.(x, .(y, z)))
g(nil) → nil
g(.(x, nil)) → .(g(x), nil)
g(.(x, .(y, z))) → g(.(.(x, y), z))
The set Q consists of the following terms:
f(nil)
f(.(nil, x0))
f(.(.(x0, x1), x2))
g(nil)
g(.(x0, nil))
g(.(x0, .(x1, x2)))
(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:
F(.(nil, y)) → F(y)
F(.(.(x, y), z)) → F(.(x, .(y, z)))
G(.(x, nil)) → G(x)
G(.(x, .(y, z))) → G(.(.(x, y), z))
The TRS R consists of the following rules:
f(nil) → nil
f(.(nil, y)) → .(nil, f(y))
f(.(.(x, y), z)) → f(.(x, .(y, z)))
g(nil) → nil
g(.(x, nil)) → .(g(x), nil)
g(.(x, .(y, z))) → g(.(.(x, y), z))
The set Q consists of the following terms:
f(nil)
f(.(nil, x0))
f(.(.(x0, x1), x2))
g(nil)
g(.(x0, nil))
g(.(x0, .(x1, x2)))
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.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(.(x, .(y, z))) → G(.(.(x, y), z))
G(.(x, nil)) → G(x)
The TRS R consists of the following rules:
f(nil) → nil
f(.(nil, y)) → .(nil, f(y))
f(.(.(x, y), z)) → f(.(x, .(y, z)))
g(nil) → nil
g(.(x, nil)) → .(g(x), nil)
g(.(x, .(y, z))) → g(.(.(x, y), z))
The set Q consists of the following terms:
f(nil)
f(.(nil, x0))
f(.(.(x0, x1), x2))
g(nil)
g(.(x0, nil))
g(.(x0, .(x1, x2)))
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(.(.(x, y), z)) → F(.(x, .(y, z)))
F(.(nil, y)) → F(y)
The TRS R consists of the following rules:
f(nil) → nil
f(.(nil, y)) → .(nil, f(y))
f(.(.(x, y), z)) → f(.(x, .(y, z)))
g(nil) → nil
g(.(x, nil)) → .(g(x), nil)
g(.(x, .(y, z))) → g(.(.(x, y), z))
The set Q consists of the following terms:
f(nil)
f(.(nil, x0))
f(.(.(x0, x1), x2))
g(nil)
g(.(x0, nil))
g(.(x0, .(x1, x2)))
We have to consider all minimal (P,Q,R)-chains.