Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z018)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by NaTT @ 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#(b(x1))

  →

  a#(x1)

  (5)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

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

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

  a#(b(x1))

  →

  a#(x1)

  (5)

  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

  b#(a(x1))	→	b#(x1)	(9)
  b#(c(x1))	→	b#(b(x1))	(8)
  b#(c(x1))	→	b#(x1)	(4)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [a(x1)]	=	x1 + 7720
  [b(x1)]	=	x1 + 0
  [c(x1)]	=	x1 + 0
  [a#(x1)]	=	0
  [b#(x1)]	=	x1 + 0

  together with the usable rules

  a(b(x1))	→	b(c(a(x1)))	(1)
  b(a(x1))	→	a(c(b(x1)))	(3)
  b(c(x1))	→	c(b(b(x1)))	(2)

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

  b#(a(x1))

  →

  b#(x1)

  (9)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    b#(c(x1))	→	b#(x1)	(4)
    b#(c(x1))	→	b#(b(x1))	(8)

    1.1.2.1.1 Reduction Pair Processor with Usable Rules

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

    [a(x1)]	=	1 0 1 0 · x1 + 32529 32528
    [b(x1)]	=	1 1 0 1 · x1 + 1 0
    [c(x1)]	=	0 0 0 1 · x1 + 0 1
    [a#(x1)]	=	0 0
    [b#(x1)]	=	0 1 0 0 · x1 + 0 0

    together with the usable rules

    a(b(x1))	→	b(c(a(x1)))	(1)
    b(a(x1))	→	a(c(b(x1)))	(3)
    b(c(x1))	→	c(b(b(x1)))	(2)

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

    b#(c(x1))	→	b#(x1)	(4)
    b#(c(x1))	→	b#(b(x1))	(8)

    could be deleted.

    1.1.2.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.