(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(X, g(X)) → f(1, g(X))
g(1) → g(0)
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, g(X)) → F(1, g(X))
G(1) → G(0)
The TRS R consists of the following rules:
f(X, g(X)) → f(1, g(X))
g(1) → g(0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(X, g(X)) → F(1, g(X))
The TRS R consists of the following rules:
f(X, g(X)) → f(1, g(X))
g(1) → g(0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) 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' = [ ] on the rule
F(1, g(1))[ ]n[x0 / 1] → F(1, g(1))[ ]n[x0 / 1]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Simplify mu) - Instantiate mu - Instantiation
F(X, g(X))[ ]n[ ] → F(1, g(X))[ ]n[ ]
by OriginalRule from TRS P
(6) NO