Certification Problem

Input (TPDB TRS_Standard/Secret_06_TRS/gen-9)

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 2 components.

• The 1^st component contains the pair

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

  →

  f#(b(y,z))

  (4)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the argument filter
  in combination with the following Weighted Path Order with the following precedence and status

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

  and the following Max-polynomial interpretation

  [a]	=	32287
  [b(x1, x2)]	=	max(x1 + 161442, x2 + 32288, 0)
  [c(x1, x2, x3)]	=	max(x1 + 129153, x2 + 161442, x3 + 32288, 0)
  [f(x1)]	=	x1 + 64576
  [f#(x1)]	=	x1 + 1
  [c#(x1, x2, x3)]	=	max(x1 + 1, 0)
  [b#(x1, x2)]	=	max(x1 + 1, x2 + 1, 0)

  together with the usable rules

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

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

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

  →

  f#(b(y,z))

  (4)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  b#(z,b(c(a,y,a),f(f(x))))	→	c#(c(y,a,z),z,x)	(9)
  c#(c(z,y,a),a,a)	→	b#(z,y)	(5)
  b#(z,b(c(a,y,a),f(f(x))))	→	c#(y,a,z)	(6)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

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

  together with the usable rules

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

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

  b#(z,b(c(a,y,a),f(f(x))))	→	c#(c(y,a,z),z,x)	(9)
  c#(c(z,y,a),a,a)	→	b#(z,y)	(5)
  b#(z,b(c(a,y,a),f(f(x))))	→	c#(y,a,z)	(6)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.