Certification Problem

Input (TPDB SRS_Standard/Trafo_06/un10)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  b#(b(a(b(b(a(b(b(b(b(x1))))))))))	→	b#(b(a(b(b(a(b(b(a(b(x1))))))))))	(7)
  b#(b(a(b(b(a(b(b(b(b(x1))))))))))	→	b#(b(a(b(b(a(b(x1)))))))	(5)
  b#(b(a(b(b(a(b(b(b(b(x1))))))))))	→	b#(b(a(b(x1))))	(3)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals

  [a(x1)]	=	0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 · x1 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  [b(x1)]	=	0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 · x1 + 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
  [b#(x1)]	=	0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 · x1 + 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

  together with the usable rule

  b(b(a(b(b(a(b(b(b(b(x1))))))))))

  →

  b(b(b(b(b(a(b(b(a(b(b(a(b(x1)))))))))))))

  (1)

  (w.r.t. the implicit argument filter of the reduction pair), the pairs

  b#(b(a(b(b(a(b(b(b(b(x1))))))))))	→	b#(b(a(b(b(a(b(b(a(b(x1))))))))))	(7)
  b#(b(a(b(b(a(b(b(b(b(x1))))))))))	→	b#(b(a(b(b(a(b(x1)))))))	(5)
  b#(b(a(b(b(a(b(b(b(b(x1))))))))))	→	b#(b(a(b(x1))))	(3)

  could be deleted.

  1.1.1.1 P is empty

  There are no pairs anymore.