(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a) → g(h(a))
h(g(x)) → g(h(f(x)))
k(x, h(x), a) → h(x)
k(f(x), y, x) → f(x)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 0
POL(f(x1)) = x1
POL(g(x1)) = 2·x1
POL(h(x1)) = x1
POL(k(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + x3
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
k(x, h(x), a) → h(x)
k(f(x), y, x) → f(x)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a) → g(h(a))
h(g(x)) → g(h(f(x)))
Q is empty.
(3) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a) → g(h(a))
h(g(x)) → g(h(f(x)))
The set Q consists of the following terms:
f(a)
h(g(x0))
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a) → H(a)
H(g(x)) → H(f(x))
H(g(x)) → F(x)
The TRS R consists of the following rules:
f(a) → g(h(a))
h(g(x)) → g(h(f(x)))
The set Q consists of the following terms:
f(a)
h(g(x0))
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
H(g(x)) → H(f(x))
The TRS R consists of the following rules:
f(a) → g(h(a))
h(g(x)) → g(h(f(x)))
The set Q consists of the following terms:
f(a)
h(g(x0))
We have to consider all minimal (P,Q,R)-chains.
(9) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
H(g(x)) → H(f(x))
The TRS R consists of the following rules:
f(a) → g(h(a))
The set Q consists of the following terms:
f(a)
h(g(x0))
We have to consider all minimal (P,Q,R)-chains.
(11) RFCMatchBoundsDPProof (EQUIVALENT transformation)
Finiteness of the DP problem can be shown by a matchbound of 1.
As the DP problem is minimal we only have to initialize the certificate graph by the rules of P:
H(g(x)) → H(f(x))
To find matches we regarded all rules of R and P:
f(a) → g(h(a))
H(g(x)) → H(f(x))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
31, 32, 33, 34, 35, 36
Node 31 is start node and node 32 is final node.
Those nodes are connected through the following edges:
- 31 to 33 labelled H_1(0)
- 31 to 36 labelled H_1(1)
- 32 to 32 labelled #_1(0)
- 33 to 32 labelled f_1(0)
- 33 to 34 labelled g_1(1)
- 34 to 35 labelled h_1(1)
- 35 to 32 labelled a(1)
- 36 to 34 labelled f_1(1)
(12) YES