Certification Problem

Input (TPDB TRS_Standard/Endrullis_06/pair2simple1)

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

  p#(a(x0),p(a(b(x1)),x2))	→	p#(a(b(a(x2))),p(a(a(x1)),x2))	(3)
  p#(a(x0),p(a(b(x1)),x2))	→	p#(a(a(x1)),x2)	(2)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [a(x1)]	=	1
  [b(x1)]	=	x1 + 1
  [p#(x1, x2)]	=	x2 + 0
  [p(x1, x2)]	=	x1 + x2 + 38

  together with the usable rule

  p(a(x0),p(a(b(x1)),x2))

  →

  p(a(b(a(x2))),p(a(a(x1)),x2))

  (1)

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

  p#(a(x0),p(a(b(x1)),x2))

  →

  p#(a(a(x1)),x2)

  (2)

  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

    p#(a(x0),p(a(b(x1)),x2))

    →

    p#(a(b(a(x2))),p(a(a(x1)),x2))

    (3)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the argument filter

    π(a)

    =

    1

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

    prec(b)	=	2		status(b)	=	[1]		list-extension(b)	=	Lex
    prec(p#)	=	1		status(p#)	=	[1, 2]		list-extension(p#)	=	Lex
    prec(p)	=	2		status(p)	=	[2, 1]		list-extension(p)	=	Lex

    and the following Max-polynomial interpretation

    [b(x1)]	=	x1 + 0
    [p#(x1, x2)]	=	max(x1 + 0, x2 + 0, 0)
    [p(x1, x2)]	=	max(x1 + 0, x2 + 0, 0)

    together with the usable rule

    p(a(x0),p(a(b(x1)),x2))

    →

    p(a(b(a(x2))),p(a(a(x1)),x2))

    (1)

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

    p#(a(x0),p(a(b(x1)),x2))

    →

    p#(a(b(a(x2))),p(a(a(x1)),x2))

    (3)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.