(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, a(a(b(b(y))))) → f(a(a(a(b(b(b(x)))))), y)
f(a(x), y) → f(x, a(y))
f(b(x), y) → f(x, b(y))
Q is empty.
(1) 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(x, a(a(b(b(y))))) → f(a(a(a(b(b(b(x)))))), y)
f(a(x), y) → f(x, a(y))
f(b(x), y) → f(x, b(y))
The signature Sigma is {
f}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, a(a(b(b(y))))) → f(a(a(a(b(b(b(x)))))), y)
f(a(x), y) → f(x, a(y))
f(b(x), y) → f(x, b(y))
The set Q consists of the following terms:
f(x0, a(a(b(b(x1)))))
f(a(x0), x1)
f(b(x0), x1)
(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, a(a(b(b(y))))) → F(a(a(a(b(b(b(x)))))), y)
F(a(x), y) → F(x, a(y))
F(b(x), y) → F(x, b(y))
The TRS R consists of the following rules:
f(x, a(a(b(b(y))))) → f(a(a(a(b(b(b(x)))))), y)
f(a(x), y) → f(x, a(y))
f(b(x), y) → f(x, b(y))
The set Q consists of the following terms:
f(x0, a(a(b(b(x1)))))
f(a(x0), x1)
f(b(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(5) 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.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, a(a(b(b(y))))) → F(a(a(a(b(b(b(x)))))), y)
F(a(x), y) → F(x, a(y))
F(b(x), y) → F(x, b(y))
R is empty.
The set Q consists of the following terms:
f(x0, a(a(b(b(x1)))))
f(a(x0), x1)
f(b(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(7) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(x0, a(a(b(b(x1)))))
f(a(x0), x1)
f(b(x0), x1)
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, a(a(b(b(y))))) → F(a(a(a(b(b(b(x)))))), y)
F(a(x), y) → F(x, a(y))
F(b(x), y) → F(x, b(y))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) QDPToSRSProof (SOUND transformation)
The finiteness of this DP problem is implied by strong termination of a SRS due to [UNKNOWN].
(10) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(b(b(x)))) → b(b(b(a(a(a(x))))))
Q is empty.
(11) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(a(b(b(x)))) → A(a(a(x)))
A(a(b(b(x)))) → A(a(x))
A(a(b(b(x)))) → A(x)
The TRS R consists of the following rules:
a(a(b(b(x)))) → b(b(b(a(a(a(x))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) MNOCProof (EQUIVALENT transformation)
We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(a(b(b(x)))) → A(a(a(x)))
A(a(b(b(x)))) → A(a(x))
A(a(b(b(x)))) → A(x)
The TRS R consists of the following rules:
a(a(b(b(x)))) → b(b(b(a(a(a(x))))))
The set Q consists of the following terms:
a(a(b(b(x0))))
We have to consider all minimal (P,Q,R)-chains.
(15) RFCMatchBoundsDPProof (EQUIVALENT transformation)
Finiteness of the DP problem can be shown by a matchbound of 4.
As the DP problem is minimal we only have to initialize the certificate graph by the rules of P:
A(a(b(b(x)))) → A(a(a(x)))
A(a(b(b(x)))) → A(a(x))
A(a(b(b(x)))) → A(x)
To find matches we regarded all rules of R and P:
a(a(b(b(x)))) → b(b(b(a(a(a(x))))))
A(a(b(b(x)))) → A(a(a(x)))
A(a(b(b(x)))) → A(a(x))
A(a(b(b(x)))) → A(x)
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85
Node 44 is start node and node 45 is final node.
Those nodes are connected through the following edges:
- 44 to 46 labelled A_1(0)
- 44 to 47 labelled A_1(0)
- 44 to 45 labelled A_1(0), A_1(1)
- 44 to 48 labelled A_1(1)
- 44 to 49 labelled A_1(1)
- 44 to 51 labelled A_1(1), A_1(2)
- 44 to 55 labelled A_1(2)
- 44 to 56 labelled A_1(2)
- 44 to 57 labelled A_1(2)
- 44 to 67 labelled A_1(3)
- 44 to 68 labelled A_1(3)
- 44 to 59 labelled A_1(3)
- 44 to 63 labelled A_1(3)
- 44 to 79 labelled A_1(4)
- 44 to 80 labelled A_1(4)
- 44 to 75 labelled A_1(4)
- 45 to 45 labelled #_1(0)
- 46 to 47 labelled a_1(0)
- 46 to 50 labelled b_1(1)
- 47 to 45 labelled a_1(0)
- 47 to 50 labelled b_1(1)
- 48 to 49 labelled a_1(1)
- 48 to 50 labelled b_1(1)
- 48 to 62 labelled b_1(2)
- 49 to 45 labelled a_1(1)
- 49 to 50 labelled b_1(1)
- 49 to 51 labelled a_1(1)
- 50 to 51 labelled b_1(1)
- 51 to 52 labelled b_1(1)
- 52 to 53 labelled a_1(1)
- 52 to 57 labelled b_1(2)
- 53 to 54 labelled a_1(1)
- 53 to 50 labelled b_1(1)
- 54 to 45 labelled a_1(1)
- 54 to 50 labelled b_1(1)
- 55 to 56 labelled a_1(2)
- 55 to 62 labelled b_1(2)
- 55 to 69 labelled b_1(3)
- 56 to 51 labelled a_1(2)
- 56 to 57 labelled a_1(2)
- 57 to 58 labelled b_1(2)
- 58 to 59 labelled b_1(2)
- 59 to 60 labelled a_1(2)
- 60 to 61 labelled a_1(2)
- 60 to 62 labelled b_1(2)
- 61 to 51 labelled a_1(2)
- 62 to 63 labelled b_1(2)
- 63 to 64 labelled b_1(2)
- 64 to 65 labelled a_1(2)
- 65 to 66 labelled a_1(2)
- 65 to 69 labelled b_1(3)
- 66 to 57 labelled a_1(2)
- 67 to 68 labelled a_1(3)
- 68 to 59 labelled a_1(3)
- 68 to 63 labelled a_1(3)
- 68 to 74 labelled b_1(3)
- 69 to 70 labelled b_1(3)
- 70 to 71 labelled b_1(3)
- 71 to 72 labelled a_1(3)
- 71 to 81 labelled b_1(4)
- 72 to 73 labelled a_1(3)
- 73 to 59 labelled a_1(3)
- 73 to 74 labelled b_1(3)
- 74 to 75 labelled b_1(3)
- 75 to 76 labelled b_1(3)
- 76 to 77 labelled a_1(3)
- 77 to 78 labelled a_1(3)
- 78 to 63 labelled a_1(3)
- 79 to 80 labelled a_1(4)
- 80 to 75 labelled a_1(4)
- 81 to 82 labelled b_1(4)
- 82 to 83 labelled b_1(4)
- 83 to 84 labelled a_1(4)
- 84 to 85 labelled a_1(4)
- 85 to 75 labelled a_1(4)
(16) YES