(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a) → f(b)
g(b) → g(a)
f(x) → g(x)
Q is empty.
(1) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a'(f(x)) → b'(f(x))
b'(g(x)) → a'(g(x))
f(x) → g(x)
Q is empty.
(3) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a'(f(x)) → b'(f(x))
b'(g(x)) → a'(g(x))
f(x) → g(x)
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 0
POL(b) = 0
POL(f(x1)) = 1 + x1
POL(g(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(x) → g(x)
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a) → f(b)
g(b) → g(a)
Q is empty.
(7) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none
The TRS R 2 is
f(a) → f(b)
g(b) → g(a)
The signature Sigma is {
f,
g}
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a) → f(b)
g(b) → g(a)
The set Q consists of the following terms:
f(a)
g(b)
(9) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a) → F(b)
G(b) → G(a)
The TRS R consists of the following rules:
f(a) → f(b)
g(b) → g(a)
The set Q consists of the following terms:
f(a)
g(b)
We have to consider all minimal (P,Q,R)-chains.
(11) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
(12) TRUE