(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
h(X) → g(X, X)
g(a, X) → f(b, activate(X))
f(X, X) → h(a)
a → 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:
H(X) → G(X, X)
G(a, X) → F(b, activate(X))
G(a, X) → ACTIVATE(X)
F(X, X) → H(a)
F(X, X) → A
The TRS R consists of the following rules:
h(X) → g(X, X)
g(a, X) → f(b, activate(X))
f(X, X) → h(a)
a → 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 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(a, X) → F(b, activate(X))
F(X, X) → H(a)
H(X) → G(X, X)
The TRS R consists of the following rules:
h(X) → g(X, X)
g(a, X) → f(b, activate(X))
f(X, X) → h(a)
a → 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' = [x0 / b] on the rule
F(b, b)[ ]n[ ] → F(b, b)[ ]n[x0 / b]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Simplify mu) - Instantiate mu
F(x0, x0)[ ]n[ ] → F(b, b)[ ]n[ ]
by Rewrite t
F(x0, x0)[ ]n[ ] → F(b, activate(a))[ ]n[ ]
by Narrowing at position: []
intermediate steps: Instantiation
F(X, X)[ ]n[ ] → H(a)[ ]n[ ]
by OriginalRule from TRS P
H(a)[ ]n[ ] → F(b, activate(a))[ ]n[ ]
by Narrowing at position: []
intermediate steps: Instantiation - Instantiation
H(X)[ ]n[ ] → G(X, X)[ ]n[ ]
by OriginalRule from TRS P
intermediate steps: Instantiation - Instantiation
G(a, X)[ ]n[ ] → F(b, activate(X))[ ]n[ ]
by OriginalRule from TRS P
(6) NO