Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z009)

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

  d#(c(x1))

  →

  d#(a(x1))

  (5)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

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

  together with the usable rules

  a(c(x1))	→	c(a(x1))	(4)
  a(x1)	→	b(c(x1))	(1)
  a(b(x1))	→	b(a(x1))	(2)

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

  d#(c(x1))

  →

  d#(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

  a#(c(x1))	→	a#(x1)	(8)
  a#(b(x1))	→	a#(x1)	(7)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

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

  together with the usable rules

  a(c(x1))	→	c(a(x1))	(4)
  a(x1)	→	b(c(x1))	(1)
  a(b(x1))	→	b(a(x1))	(2)

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

  a#(b(x1))

  →

  a#(x1)

  (7)

  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

    a#(c(x1))

    →

    a#(x1)

    (8)

    1.1.2.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [a(x1)]	=	x1 + 1
    [d(x1)]	=	0
    [b(x1)]	=	1
    [c(x1)]	=	x1 + 7720
    [d#(x1)]	=	x1 + 0
    [a#(x1)]	=	x1 + 0

    together with the usable rules

    a(c(x1))	→	c(a(x1))	(4)
    a(x1)	→	b(c(x1))	(1)
    a(b(x1))	→	b(a(x1))	(2)

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

    a#(c(x1))

    →

    a#(x1)

    (8)

    could be deleted.

    1.1.2.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.