Certification Problem

Input (TPDB TRS_Standard/AotoYamada_05/002)

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

  app#(app(filter,f),app(app(cons,y),ys))	→	app#(f,y)	(13)
  app#(app(app(filtersub,false),f),app(app(cons,y),ys))	→	app#(app(filter,f),ys)	(12)
  app#(app(filter,f),app(app(cons,y),ys))	→	app#(app(app(filtersub,app(f,y)),f),app(app(cons,y),ys))	(11)
  app#(app(app(filtersub,true),f),app(app(cons,y),ys))	→	app#(app(filter,f),ys)	(6)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [false]	=	1
  [true]	=	34723
  [filtersub]	=	1
  [nil]	=	17221
  [app#(x1, x2)]	=	x1 + 0
  [cons]	=	1
  [filter]	=	1
  [app(x1, x2)]	=	x2 + 22248

  together with the usable rules

  app(app(app(filtersub,false),f),app(app(cons,y),ys))	→	app(app(filter,f),ys)	(4)
  app(app(filter,f),nil)	→	nil	(1)
  app(app(app(filtersub,true),f),app(app(cons,y),ys))	→	app(app(cons,y),app(app(filter,f),ys))	(3)
  app(app(filter,f),app(app(cons,y),ys))	→	app(app(app(filtersub,app(f,y)),f),app(app(cons,y),ys))	(2)

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

  app#(app(filter,f),app(app(cons,y),ys))

  →

  app#(f,y)

  (13)

  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

    app#(app(app(filtersub,false),f),app(app(cons,y),ys))	→	app#(app(filter,f),ys)	(12)
    app#(app(app(filtersub,true),f),app(app(cons,y),ys))	→	app#(app(filter,f),ys)	(6)
    app#(app(filter,f),app(app(cons,y),ys))	→	app#(app(app(filtersub,app(f,y)),f),app(app(cons,y),ys))	(11)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [false]	=	1
    [true]	=	58713
    [filtersub]	=	1
    [nil]	=	17221
    [app#(x1, x2)]	=	x1 + x2 + 0
    [cons]	=	1
    [filter]	=	1
    [app(x1, x2)]	=	x2 + 6996

    together with the usable rules

    app(app(app(filtersub,false),f),app(app(cons,y),ys))	→	app(app(filter,f),ys)	(4)
    app(app(filter,f),nil)	→	nil	(1)
    app(app(app(filtersub,true),f),app(app(cons,y),ys))	→	app(app(cons,y),app(app(filter,f),ys))	(3)
    app(app(filter,f),app(app(cons,y),ys))	→	app(app(app(filtersub,app(f,y)),f),app(app(cons,y),ys))	(2)

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

    app#(app(app(filtersub,false),f),app(app(cons,y),ys))	→	app#(app(filter,f),ys)	(12)
    app#(app(app(filtersub,true),f),app(app(cons,y),ys))	→	app#(app(filter,f),ys)	(6)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.