(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, y, f(z, u, v)) → f(f(x, y, z), u, f(x, y, v))
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, y, f(z, u, v)) → F(f(x, y, z), u, f(x, y, v))
F(x, y, f(z, u, v)) → F(x, y, z)
F(x, y, f(z, u, v)) → F(x, y, v)
The TRS R consists of the following rules:
f(x, y, f(z, u, v)) → f(f(x, y, z), u, f(x, y, v))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) NonLoopProof (EQUIVALENT transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0,
σ' = [ ], and μ' = [x4 / f(x4, x3, x0), x3 / x1, x0 / x4, x1 / x3] on the rule
F(f(x4, x3, x0), x1, f(x4, x3, x2))[ ]n[ ] → F(f(x4, x3, x0), x1, f(x4, x3, x2))[ ]n[x4 / f(x4, x3, x0), x3 / x1, x0 / x4, x1 / x3]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Simplify mu) - Instantiate mu - Instantiation
F(x, y, f(z, u, v))[ ]n[ ] → F(f(x, y, z), u, f(x, y, v))[ ]n[ ]
by OriginalRule from TRS P
(4) NO