Certification Problem

Input (TPDB TRS_Standard/Secret_06_TRS/gen-18)

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#(y,x,f(z))	→	b#(f(b(z,x)),z)	(7)
  c#(y,x,f(z))	→	b#(z,x)	(6)
  b#(b(y,z),c(a,a,a))	→	c#(z,y,z)	(5)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [a]	=	0
  [b(x1, x2)]	=	max(x1 + 1, x2 + 1, 0)
  [c(x1, x2, x3)]	=	max(x1 + 1, x2 + 2, x3 + 2, 0)
  [f(x1)]	=	x1 + 0
  [f#(x1)]	=	0
  [c#(x1, x2, x3)]	=	max(x2 + 28102, x3 + 28102, 0)
  [b#(x1, x2)]	=	max(x1 + 28101, 0)

  together with the usable rules

  b(b(y,z),c(a,a,a))	→	f(c(z,y,z))	(1)
  c(y,x,f(z))	→	b(f(b(z,x)),z)	(3)
  f(b(b(a,z),c(a,x,y)))	→	z	(2)

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

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

  →

  b#(z,x)

  (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

    b#(b(y,z),c(a,a,a))	→	c#(z,y,z)	(5)
    c#(y,x,f(z))	→	b#(f(b(z,x)),z)	(7)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the argument filter

    π(f#)	=	1
    π(b#)	=	1

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

    prec(a)	=	0		status(a)	=	[]		list-extension(a)	=	Lex
    prec(b)	=	2		status(b)	=	[]		list-extension(b)	=	Lex
    prec(c)	=	3		status(c)	=	[]		list-extension(c)	=	Lex
    prec(f)	=	1		status(f)	=	[]		list-extension(f)	=	Lex
    prec(c#)	=	1		status(c#)	=	[]		list-extension(c#)	=	Lex

    and the following Max-polynomial interpretation

    [a]	=	2
    [b(x1, x2)]	=	max(x1 + 7723, x2 + 7723, 0)
    [c(x1, x2, x3)]	=	max(x1 + 1, x2 + 15446, x3 + 15446, 0)
    [f(x1)]	=	x1 + 0
    [c#(x1, x2, x3)]	=	max(x1 + 7723, x2 + 7723, x3 + 7723, 0)

    together with the usable rules

    b(b(y,z),c(a,a,a))	→	f(c(z,y,z))	(1)
    c(y,x,f(z))	→	b(f(b(z,x)),z)	(3)
    f(b(b(a,z),c(a,x,y)))	→	z	(2)

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

    b#(b(y,z),c(a,a,a))

    →

    c#(z,y,z)

    (5)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.