Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/motivation)

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

  app#(app(map,fun),app(app(cons,x),xs))	→	app#(fun,x)	(15)
  app#(app(app(app(filter2,false),fun),x),xs)	→	app#(app(filter,fun),xs)	(24)
  app#(app(filter,fun),app(app(cons,x),xs))	→	app#(fun,x)	(13)
  app#(app(map,fun),app(app(cons,x),xs))	→	app#(app(map,fun),xs)	(12)
  app#(app(app(app(filter2,true),fun),x),xs)	→	app#(app(filter,fun),xs)	(22)
  app#(app(filter,fun),app(app(cons,x),xs))	→	app#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(20)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [h]	=	1
  [false]	=	35919
  [true]	=	282
  [f]	=	35141
  [filter2]	=	32335
  [nil]	=	36466
  [app#(x1, x2)]	=	x2 + 0
  [map]	=	33428
  [cons]	=	26372
  [filter]	=	85957
  [g]	=	39558
  [app(x1, x2)]	=	x1 + x2 + 8851

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

  app#(app(map,fun),app(app(cons,x),xs))	→	app#(fun,x)	(15)
  app#(app(filter,fun),app(app(cons,x),xs))	→	app#(fun,x)	(13)
  app#(app(map,fun),app(app(cons,x),xs))	→	app#(app(map,fun),xs)	(12)
  app#(app(filter,fun),app(app(cons,x),xs))	→	app#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(20)

  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

  app#(g,app(g,x))

  →

  app#(g,app(h,app(g,x)))

  (25)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [h]	=	591
  [false]	=	0
  [true]	=	0
  [f]	=	0
  [filter2]	=	28183
  [nil]	=	0
  [app#(x1, x2)]	=	x2 + 0
  [map]	=	3564
  [cons]	=	203
  [filter]	=	29865
  [g]	=	3038
  [app(x1, x2)]	=	x1 + 841

  together with the usable rules

  app(app(map,fun),nil)	→	nil	(4)
  app(app(app(app(filter2,true),fun),x),xs)	→	app(app(cons,x),app(app(filter,fun),xs))	(8)
  app(g,app(h,app(g,x)))	→	app(g,x)	(1)
  app(h,app(h,x))	→	app(h,app(app(f,app(h,x)),x))	(3)
  app(app(map,fun),app(app(cons,x),xs))	→	app(app(cons,app(fun,x)),app(app(map,fun),xs))	(5)
  app(app(filter,fun),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(fun,x)),fun),x),xs)	(7)
  app(app(app(app(filter2,false),fun),x),xs)	→	app(app(filter,fun),xs)	(9)
  app(app(filter,fun),nil)	→	nil	(6)
  app(g,app(g,x))	→	app(g,app(h,app(g,x)))	(2)

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

  app#(g,app(g,x))

  →

  app#(g,app(h,app(g,x)))

  (25)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.