(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, y) → f(x, x)
f(s(x), y) → f(y, x)
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(x, x)
F(s(x), y) → F(y, x)
The TRS R consists of the following rules:
f(x, y) → f(x, x)
f(s(x), y) → f(y, x)
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' = [x0 / x1] on the rule
F(x1, x1)[ ]n[ ] → F(x1, x1)[ ]n[x0 / x1]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Simplify mu) - Instantiate mu - Instantiation
F(x, y)[ ]n[ ] → F(x, x)[ ]n[ ]
by OriginalRule from TRS P
(4) NO