Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z019)

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

  c#(a(x1))	→	a#(c(x1))	(11)
  a#(b(x1))	→	a#(x1)	(10)
  b#(c(x1))	→	c#(b(b(x1)))	(9)
  b#(c(x1))	→	b#(x1)	(8)
  a#(b(x1))	→	b#(c(a(x1)))	(5)
  c#(a(x1))	→	c#(x1)	(7)
  a#(b(x1))	→	c#(a(x1))	(6)
  b#(c(x1))	→	b#(b(x1))	(4)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [a(x1)]	=	x1 + 0
  [b(x1)]	=	x1 + 3
  [c(x1)]	=	0
  [c#(x1)]	=	1
  [a#(x1)]	=	x1 + 0
  [b#(x1)]	=	2

  together with the usable rules

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

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

  c#(a(x1))	→	a#(c(x1))	(11)
  a#(b(x1))	→	a#(x1)	(10)
  b#(c(x1))	→	c#(b(b(x1)))	(9)
  a#(b(x1))	→	b#(c(a(x1)))	(5)
  a#(b(x1))	→	c#(a(x1))	(6)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 2 components.

  • The 1^st component contains the pair

    c#(a(x1))

    →

    c#(x1)

    (7)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [a(x1)]	=	x1 + 1
    [b(x1)]	=	40651
    [c(x1)]	=	x1 + 0
    [c#(x1)]	=	x1 + 1
    [a#(x1)]	=	x1 + 0
    [b#(x1)]	=	2

    together with the usable rules

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

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

    c#(a(x1))

    →

    c#(x1)

    (7)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.

  • The 2^nd component contains the pair

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

    1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

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

    together with the usable rules

    a(b(x1))	→	b(c(a(x1)))	(1)
    c(a(x1))	→	a(c(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)	(8)
    b#(c(x1))	→	b#(b(x1))	(4)

    could be deleted.

    1.1.1.1.2.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.