Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z015)

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 1 component.

• The 1^st component contains the pair

  a#(b(x1))	→	b#(a(a(a(x1))))	(10)
  b#(a(x1))	→	a#(a(x1))	(9)
  a#(a(x1))	→	b#(x1)	(8)
  a#(b(x1))	→	a#(a(x1))	(5)
  a#(b(x1))	→	a#(a(a(x1)))	(7)
  a#(b(x1))	→	a#(x1)	(6)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

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

  together with the usable rules

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

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

  a#(b(x1))	→	b#(a(a(a(x1))))	(10)
  a#(b(x1))	→	a#(a(x1))	(5)
  a#(b(x1))	→	a#(a(a(x1)))	(7)
  a#(b(x1))	→	a#(x1)	(6)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

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

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the argument filter

    π(a#)	=	1
    π(b#)	=	1

    in combination with the following Weighted Path Order with the following precedence and status

    prec(a)	=	0		status(a)	=	[1]		list-extension(a)	=	Lex
    prec(b)	=	1		status(b)	=	[]		list-extension(b)	=	Lex
    prec(c)	=	0		status(c)	=	[]		list-extension(c)	=	Lex

    and the following Max-polynomial interpretation

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

    together with the usable rules

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

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

    a#(a(x1))

    →

    b#(x1)

    (8)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.