(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, x) → f(i(x), g(g(x)))
f(x, y) → x
g(x) → i(x)
f(x, i(x)) → f(x, x)
f(i(x), i(g(x))) → a
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 1
POL(f(x1, x2)) = 2 + 2·x1 + 2·x2
POL(g(x1)) = x1
POL(i(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(x, y) → x
f(i(x), i(g(x))) → a
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, x) → f(i(x), g(g(x)))
g(x) → i(x)
f(x, i(x)) → f(x, x)
Q is empty.
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, x) → F(i(x), g(g(x)))
F(x, x) → G(g(x))
F(x, x) → G(x)
F(x, i(x)) → F(x, x)
The TRS R consists of the following rules:
f(x, x) → f(i(x), g(g(x)))
g(x) → i(x)
f(x, i(x)) → f(x, x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, i(x)) → F(x, x)
F(x, x) → F(i(x), g(g(x)))
The TRS R consists of the following rules:
f(x, x) → f(i(x), g(g(x)))
g(x) → i(x)
f(x, i(x)) → f(x, x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) 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 / i(x0)] on the rule
F(i(x0), i(i(x0)))[ ]n[ ] → F(i(x0), i(i(x0)))[ ]n[x0 / i(x0)]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Simplify mu) - Instantiate mu
F(x0, i(x0))[ ]n[ ] → F(i(x0), i(i(x0)))[ ]n[ ]
by Rewrite t
F(x0, i(x0))[ ]n[ ] → F(i(x0), g(g(x0)))[ ]n[ ]
by Narrowing at position: []
intermediate steps: Instantiation
F(x, i(x))[ ]n[ ] → F(x, x)[ ]n[ ]
by OriginalRule from TRS P
intermediate steps: Instantiation - Instantiation
F(x, x)[ ]n[ ] → F(i(x), g(g(x)))[ ]n[ ]
by OriginalRule from TRS P
(8) NO