(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(n__a, X, X) → f(activate(X), b, n__b)
b → a
a → n__a
b → n__b
activate(n__a) → a
activate(n__b) → b
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(n__a, X, X) → F(activate(X), b, n__b)
F(n__a, X, X) → ACTIVATE(X)
F(n__a, X, X) → B
B → A
ACTIVATE(n__a) → A
ACTIVATE(n__b) → B
The TRS R consists of the following rules:
f(n__a, X, X) → f(activate(X), b, n__b)
b → a
a → n__a
b → n__b
activate(n__a) → a
activate(n__b) → b
activate(X) → X
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 5 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(n__a, X, X) → F(activate(X), b, n__b)
The TRS R consists of the following rules:
f(n__a, X, X) → f(activate(X), b, n__b)
b → a
a → n__a
b → n__b
activate(n__a) → a
activate(n__b) → b
activate(X) → X
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(n__a, n__b, n__b)[ ]n[ ] → F(n__a, n__b, n__b)[ ]n[ ]
This rule is correct for the QDP as the following derivation shows:
F(n__a, n__b, n__b)[ ]n[ ] → F(n__a, n__b, n__b)[ ]n[ ]
by Rewrite t
F(n__a, n__b, n__b)[ ]n[ ] → F(b, n__b, n__b)[ ]n[ ]
by Narrowing at position: [0]
intermediate steps: Instantiation
F(n__a, x0, x0)[ ]n[ ] → F(activate(x0), n__b, n__b)[ ]n[ ]
by Narrowing at position: [1]
intermediate steps: Instantiation
F(n__a, X, X)[ ]n[ ] → F(activate(X), b, n__b)[ ]n[ ]
by OriginalRule from TRS P
b[ ]n[ ] → n__b[ ]n[ ]
by OriginalRule from TRS R
activate(n__b)[ ]n[ ] → b[ ]n[ ]
by OriginalRule from TRS R
(6) NO