Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex5_Zan97_FR)

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

  if#(false,X,Y)	→	activate#(Y)	(13)
  activate#(n__f(X))	→	f#(activate(X))	(12)
  f#(X)	→	if#(X,c,n__f(n__true))	(10)
  activate#(n__f(X))	→	activate#(X)	(9)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [activate(x1)]	=	1
  [n__true]	=	0
  [activate#(x1)]	=	x1 + 0
  [false]	=	4
  [c]	=	0
  [true]	=	1
  [f(x1)]	=	1
  [if(x1, x2, x3)]	=	x1 + x3 + 0
  [n__f(x1)]	=	x1 + 1
  [f#(x1)]	=	1
  [if#(x1, x2, x3)]	=	x3 + 0
  [true#]	=	0

  together with the usable rules

  true	→	n__true	(5)
  activate(n__true)	→	true	(7)

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

  activate#(n__f(X))

  →

  activate#(X)

  (9)

  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

    f#(X)	→	if#(X,c,n__f(n__true))	(10)
    if#(false,X,Y)	→	activate#(Y)	(13)
    activate#(n__f(X))	→	f#(activate(X))	(12)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [activate(x1)]	=	x1 + 1
    [n__true]	=	0
    [activate#(x1)]	=	x1 + 3
    [false]	=	4
    [c]	=	0
    [true]	=	1
    [f(x1)]	=	x1 + 1
    [if(x1, x2, x3)]	=	x1 + x2 + x3 + 0
    [n__f(x1)]	=	x1 + 1
    [f#(x1)]	=	x1 + 2
    [if#(x1, x2, x3)]	=	x1 + x3 + 0
    [true#]	=	0

    together with the usable rules

    f(X)	→	n__f(X)	(4)
    activate(X)	→	X	(8)
    f(X)	→	if(X,c,n__f(n__true))	(1)
    if(false,X,Y)	→	activate(Y)	(3)
    true	→	n__true	(5)
    activate(n__true)	→	true	(7)
    activate(n__f(X))	→	f(activate(X))	(6)
    if(true,X,Y)	→	X	(2)

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

    f#(X)	→	if#(X,c,n__f(n__true))	(10)
    if#(false,X,Y)	→	activate#(Y)	(13)
    activate#(n__f(X))	→	f#(activate(X))	(12)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.