(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.

(5) UsableRulesReductionPairsProof (EQUIVALENT transformation)

First, we A-transformed [FROCOS05] the QDP-Problem. Then we obtain the following A-transformed DP problem.
The pairs P are:

a1(a(b(a(a(b(a(x))))))) → a1(a(a(b(x))))
a1(a(b(a(a(b(a(x))))))) → a1(a(b(a(a(a(b(x)))))))
a1(a(b(a(a(b(a(x))))))) → a1(a(b(x)))

and the Q and R are:
Q restricted rewrite system:
The TRS R consists of the following rules:

a(a(b(a(a(b(a(x))))))) → a(b(a(a(b(a(a(a(b(x)))))))))

Q is empty.

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

The following rules are removed from R:

f(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) → f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x)))))))))

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(a(x₁)) = x₁   
POL(a1(x₁)) = 2·x₁   
POL(b(x₁)) = x₁   

(7) RFCMatchBoundsDPProof (EQUIVALENT transformation)

Finiteness of the DP problem can be shown by a matchbound of 2.
As the DP problem is minimal we only have to initialize the certificate graph by the rules of P:

a1(a(b(a(a(b(a(x))))))) → a1(a(a(b(x))))
a1(a(b(a(a(b(a(x))))))) → a1(a(b(a(a(a(b(x)))))))
a1(a(b(a(a(b(a(x))))))) → a1(a(b(x)))

To find matches we regarded all rules of R and P:

a(a(b(a(a(b(a(x))))))) → a(b(a(a(b(a(a(a(b(x)))))))))
a1(a(b(a(a(b(a(x))))))) → a1(a(a(b(x))))
a1(a(b(a(a(b(a(x))))))) → a1(a(b(a(a(a(b(x)))))))
a1(a(b(a(a(b(a(x))))))) → a1(a(b(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, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104

Node 44 is start node and node 45 is final node.

Those nodes are connected through the following edges:

• 44 to 46 labelled a1_1(0)
• 44 to 49 labelled a1_1(0)
• 44 to 47 labelled a1_1(0)
• 44 to 55 labelled a1_1(1)
• 44 to 58 labelled a1_1(1)
• 44 to 56 labelled a1_1(1)
• 44 to 80 labelled a1_1(2)
• 44 to 83 labelled a1_1(2)
• 44 to 81 labelled a1_1(2)
• 45 to 45 labelled #_1(0)
• 46 to 47 labelled a_1(0)
• 46 to 64 labelled a_1(1)
• 47 to 48 labelled a_1(0)
• 48 to 45 labelled b_1(0)
• 49 to 50 labelled a_1(0)
• 50 to 51 labelled b_1(0)
• 51 to 52 labelled a_1(0)
• 51 to 72 labelled a_1(1)
• 52 to 53 labelled a_1(0)
• 52 to 64 labelled a_1(1)
• 53 to 54 labelled a_1(0)
• 54 to 45 labelled b_1(0)
• 55 to 56 labelled a_1(1)
• 55 to 64 labelled a_1(1)
• 56 to 57 labelled a_1(1)
• 57 to 45 labelled b_1(1)
• 57 to 69 labelled b_1(1)
• 57 to 66 labelled b_1(1)
• 57 to 89 labelled b_1(1)
• 57 to 97 labelled b_1(1)
• 58 to 59 labelled a_1(1)
• 59 to 60 labelled b_1(1)
• 60 to 61 labelled a_1(1)
• 60 to 89 labelled a_1(2)
• 61 to 62 labelled a_1(1)
• 61 to 64 labelled a_1(1)
• 62 to 63 labelled a_1(1)
• 63 to 45 labelled b_1(1)
• 63 to 69 labelled b_1(1)
• 63 to 66 labelled b_1(1)
• 63 to 89 labelled b_1(1)
• 63 to 97 labelled b_1(1)
• 64 to 65 labelled b_1(1)
• 65 to 66 labelled a_1(1)
• 65 to 97 labelled a_1(2)
• 66 to 67 labelled a_1(1)
• 67 to 68 labelled b_1(1)
• 68 to 69 labelled a_1(1)
• 68 to 89 labelled a_1(2)
• 69 to 70 labelled a_1(1)
• 69 to 64 labelled a_1(1)
• 70 to 71 labelled a_1(1)
• 71 to 45 labelled b_1(1)
• 72 to 73 labelled b_1(1)
• 73 to 74 labelled a_1(1)
• 73 to 97 labelled a_1(2)
• 74 to 75 labelled a_1(1)
• 75 to 76 labelled b_1(1)
• 76 to 77 labelled a_1(1)
• 76 to 89 labelled a_1(2)
• 77 to 78 labelled a_1(1)
• 77 to 64 labelled a_1(1)
• 78 to 79 labelled a_1(1)
• 79 to 69 labelled b_1(1)
• 79 to 89 labelled b_1(1)
• 80 to 81 labelled a_1(2)
• 80 to 64 labelled a_1(1)
• 81 to 82 labelled a_1(2)
• 82 to 69 labelled b_1(2)
• 82 to 66 labelled b_1(2)
• 82 to 89 labelled b_1(2)
• 82 to 97 labelled b_1(2)
• 83 to 84 labelled a_1(2)
• 84 to 85 labelled b_1(2)
• 85 to 86 labelled a_1(2)
• 85 to 89 labelled a_1(2)
• 86 to 87 labelled a_1(2)
• 86 to 64 labelled a_1(1)
• 87 to 88 labelled a_1(2)
• 88 to 69 labelled b_1(2)
• 88 to 66 labelled b_1(2)
• 88 to 89 labelled b_1(2)
• 88 to 97 labelled b_1(2)
• 89 to 90 labelled b_1(2)
• 90 to 91 labelled a_1(2)
• 90 to 97 labelled a_1(2)
• 91 to 92 labelled a_1(2)
• 92 to 93 labelled b_1(2)
• 93 to 94 labelled a_1(2)
• 93 to 89 labelled a_1(2)
• 94 to 95 labelled a_1(2)
• 94 to 64 labelled a_1(1)
• 95 to 96 labelled a_1(2)
• 96 to 69 labelled b_1(2)
• 96 to 89 labelled b_1(2)
• 97 to 98 labelled b_1(2)
• 98 to 99 labelled a_1(2)
• 99 to 100 labelled a_1(2)
• 100 to 101 labelled b_1(2)
• 101 to 102 labelled a_1(2)
• 102 to 103 labelled a_1(2)
• 103 to 104 labelled a_1(2)
• 104 to 66 labelled b_1(2)
• 104 to 97 labelled b_1(2)