(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(X), Y) → f(X, n__f(g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
activate(n__f(X1, X2)) → f(X1, X2)
activate(X) → 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(g(X), Y) → F(X, n__f(g(X), activate(Y)))
F(g(X), Y) → ACTIVATE(Y)
ACTIVATE(n__f(X1, X2)) → F(X1, X2)
The TRS R consists of the following rules:
f(g(X), Y) → f(X, n__f(g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
activate(n__f(X1, X2)) → f(X1, X2)
activate(X) → 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 / activate(x0)] on the rule
ACTIVATE(n__f(g(g(x1)), activate(x0)))[ ]n[ ] → ACTIVATE(n__f(g(g(x1)), activate(x0)))[ ]n[x0 / activate(x0)]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Simplify mu) - Instantiate mu - Instantiation
ACTIVATE(n__f(g(g(y1)), x0))[ ]n[ ] → ACTIVATE(n__f(g(g(y1)), activate(x0)))[ ]n[ ]
by Narrowing at position: []
intermediate steps: Instantiation - Instantiation
ACTIVATE(n__f(X1, X2))[ ]n[ ] → F(X1, X2)[ ]n[ ]
by OriginalRule from TRS P
intermediate steps: Instantiation - Instantiation
F(g(g(y1)), x0)[ ]n[ ] → ACTIVATE(n__f(g(g(y1)), activate(x0)))[ ]n[ ]
by Narrowing at position: []
intermediate steps: Instantiation - Instantiation
F(g(X), Y)[ ]n[ ] → F(X, n__f(g(X), activate(Y)))[ ]n[ ]
by OriginalRule from TRS P
intermediate steps: Instantiation - Instantiation
F(g(X), Y)[ ]n[ ] → ACTIVATE(Y)[ ]n[ ]
by OriginalRule from TRS P
(4) NO