Certification Problem

Input (TPDB TRS_Standard/AG01/#3.49)

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

  g#(c(s(x),s(y)))	→	f#(c(x,y))	(8)
  f#(c(s(x),s(y)))	→	g#(c(x,y))	(7)
  f#(c(s(x),y))	→	f#(c(x,s(y)))	(6)
  g#(c(x,s(y)))	→	g#(c(s(x),y))	(5)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [c(x1, x2)]	=	x1 + x2 + 36466
  [f(x1)]	=	0
  [f#(x1)]	=	x1 + 1
  [g#(x1)]	=	x1 + 0
  [g(x1)]	=	0

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

  g#(c(s(x),s(y)))	→	f#(c(x,y))	(8)
  f#(c(s(x),s(y)))	→	g#(c(x,y))	(7)

  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

    f#(c(s(x),y))

    →

    f#(c(x,s(y)))

    (6)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	x1 + 1
    [c(x1, x2)]	=	x1 + 1424
    [f(x1)]	=	0
    [f#(x1)]	=	x1 + 1
    [g#(x1)]	=	0
    [g(x1)]	=	0

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

    f#(c(s(x),y))

    →

    f#(c(x,s(y)))

    (6)

    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

    g#(c(x,s(y)))

    →

    g#(c(s(x),y))

    (5)

    1.1.1.1.2 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	x1 + 1
    [c(x1, x2)]	=	x2 + 1424
    [f(x1)]	=	0
    [f#(x1)]	=	1
    [g#(x1)]	=	x1 + 0
    [g(x1)]	=	0

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

    g#(c(x,s(y)))

    →

    g#(c(s(x),y))

    (5)

    could be deleted.

    1.1.1.1.2.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.