(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(X, n__g(X), Y) → f(activate(Y), activate(Y), activate(Y))
g(b) → c
b → c
g(X) → n__g(X)
activate(n__g(X)) → g(X)
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(X, n__g(X), Y) → F(activate(Y), activate(Y), activate(Y))
F(X, n__g(X), Y) → ACTIVATE(Y)
ACTIVATE(n__g(X)) → G(X)
The TRS R consists of the following rules:
f(X, n__g(X), Y) → f(activate(Y), activate(Y), activate(Y))
g(b) → c
b → c
g(X) → n__g(X)
activate(n__g(X)) → g(X)
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 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(X, n__g(X), Y) → F(activate(Y), activate(Y), activate(Y))
The TRS R consists of the following rules:
f(X, n__g(X), Y) → f(activate(Y), activate(Y), activate(Y))
g(b) → c
b → c
g(X) → n__g(X)
activate(n__g(X)) → g(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) NonLoopProof (COMPLETE transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0,
σ' = [ ], and μ' = [x0 / c] on the rule
F(c, n__g(c), n__g(b))[ ]n[ ] → F(c, n__g(c), n__g(b))[ ]n[x0 / c]
This rule is correct for the QDP as the following derivation shows:
F(c, n__g(c), n__g(b))[ ]n[ ] → F(c, n__g(c), n__g(b))[ ]n[x0 / c]
by Equivalency by Simplifying Mu with µ1: [x0 / c] µ2: [ ]
intermediate steps: Instantiate mu
F(x0, n__g(x0), n__g(b))[ ]n[ ] → F(c, n__g(c), n__g(b))[ ]n[ ]
by Narrowing at position: [2]
F(x0, n__g(x0), n__g(b))[ ]n[ ] → F(c, n__g(c), activate(n__g(b)))[ ]n[ ]
by Narrowing at position: [1]
F(x0, n__g(x0), n__g(b))[ ]n[ ] → F(c, activate(n__g(c)), activate(n__g(b)))[ ]n[ ]
by Narrowing at position: [1,0,0]
F(x0, n__g(x0), n__g(b))[ ]n[ ] → F(c, activate(n__g(b)), activate(n__g(b)))[ ]n[ ]
by Narrowing at position: [0]
intermediate steps: Instantiation - Instantiation
F(x0, n__g(x0), n__g(y0))[ ]n[ ] → F(g(y0), activate(n__g(y0)), activate(n__g(y0)))[ ]n[ ]
by Narrowing at position: [0]
intermediate steps: Instantiation - Instantiation
F(X, n__g(X), Y)[ ]n[ ] → F(activate(Y), activate(Y), activate(Y))[ ]n[ ]
by Rule from TRS P
intermediate steps: Instantiation
activate(n__g(X))[ ]n[ ] → g(X)[ ]n[ ]
by Rule from TRS R
g(b)[ ]n[ ] → c[ ]n[ ]
by Rule from TRS R
b[ ]n[ ] → c[ ]n[ ]
by Rule from TRS R
intermediate steps: Instantiation - Instantiation
activate(X)[ ]n[ ] → X[ ]n[ ]
by Rule from TRS R
intermediate steps: Instantiation - Instantiation
activate(X)[ ]n[ ] → X[ ]n[ ]
by Rule from TRS R
(6) NO