Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z004)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by matchbox @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  a#(c(b(x1)))

  →

  a#(x1)

  (3)

  1.1.1 Monotonic Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

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

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

  a#(c(b(x1)))

  →

  a#(x1)

  (3)

  and no rules could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

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

  →

  a#(b(a(x1)))

  (6)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

  [c(x1)]	=	0 0 1 1 · x1 + 0 0
  [b(x1)]	=	0 0 1 0 · x1 + 2 0
  [a(x1)]	=	0 1 0 1 · x1 + 0 1
  [a#(x1)]	=	0 2 0 0 · x1 + 1 0

  together with the usable rules

  a(b(a(a(x1))))	→	c(b(a(b(a(x1)))))	(1)
  a(c(b(x1)))	→	a(a(b(c(b(a(x1))))))	(2)

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

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

  →

  a#(b(a(x1)))

  (6)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.